Finite-Deformation Modeling of Elastodynamics and Smart Materials with Nonlinear Electro-Magneto-Elastic Coupling

Home , Finite strain theory, Linear elasticity, Materials with memory

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Robert Lindsey Lowe, B.S., M.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2015

Dissertation Committee:

Prof. Sheng-Tao John Yu, Advisor Prof. Marcelo J. Dapino Prof. Daniel A. Mendelsohn Prof. Amos Gilat Dr. Kelly S. Carney c Copyright by

Robert Lindsey Lowe

2015 Abstract

Eulerian formulations of the equations of ﬁnite-deformation solid dynamics are ideal for numerical implementation in modern high-resolution shock-capturing schemes.

These powerful numerical techniques – traditionally employed in unsteady compressible ﬂow applications – are becoming increasingly popular in the computational solid mechanics community. Their primary appeal is an exceptional ability to capture the evolution and interaction of nonlinear traveling waves. Currently, however, Eulerian models for the nonlinear dynamics of rods, beams, plates, membranes, and other elastic structures are currently unavailable in the literature.

The need for these reduced-order (1-D and 2-D) Eulerian structural models motivates the first part of this dissertation, where a comprehensive perturbation theory is used to develop a 1-D Eulerian model for nonlinear waves in elastic rods. The leading- order equations in the perturbation formalism are (i) verified using a control-volume analysis, (ii) linearized to recover a classical model for longitudinal waves in ultrasonic horns, and (iii) solved numerically using the novel space-time Conservation Element and Solution Element (CESE) method for first-order hyperbolic systems. Numeri- cal simulations of several benchmark problems demonstrate that the CESE method effectively captures shocks, rarefactions, and contact discontinuities.

ii The second part of this dissertation focuses on another emerging area of ﬁnite- deformation mechanics: magnetoelectric polymer composites (MEPCs). A distinguishing feature of MEPCs is the tantalizing ability to electrically control their magnetization, or, conversely, magnetically control their polarization. Leveraging this magnetoelectric coupling could potentially impact numerous technologies, including information storage, spintronics, sensing, actuation, and energy harvesting. Most of the research on MEPCs to date, however, has focused on optimizing the magnitude of the magnetoelectric coupling through iterative design. Substantially less activity has occurred in the way of mathematical modeling and experimental characterization at ﬁnite strains, which are needed to advance fundamental understanding of MEPCs and encourage their technological implementation.

The aforementioned need motivates the second part of this dissertation, where a finite-strain theoretical framework is developed for modeling soft magnetoelectric composites. Finite deformations, electro-magneto-elastic coupling, and material nonlinearities are incorporated into the model. A particular emphasis is placed on the development of tractable constitutive equations to facilitate material characterization in the laboratory. Accordingly, a catalogue of free energies and constitutive equations is presented, each employing a different set of independent variables. The ramifications of invariance, angular momentum, incompressibility, and material symmetry are explored, and a representative (neo-Hookean-type) free energy with full electro-magneto-elastic coupling is posed.

iii Dedicatedinlovingmemoryofmyfather

iv Acknowledgments

This dissertation represents the culmination of a rich and rewarding journey. I have reached this point by standing on the shoulders of others. I will never be able to repay my debt of gratitude in kind, but I will do my best to pay it forward.

I would like to begin by expressing my sincere appreciation to my advisor, Pro- fessor Sheng-Tao John Yu, for his guidance, advice, support, and friendship through the years. I thoroughly enjoyed our conversations – both technical and otherwise – and the wisdom and perspective I gained through our interactions. Prof. Yu’s words of encouragement always gave me hope and inspiration when times were challenging.

I proﬁted greatly from his expertise in numerical methods and programming.

I would also like to thank my former advisor, Professor Stephen Bechtel, who was forced to retire far too young due to the devastating eﬀects of Parkinson’s dis- ease. Prof. Bechtel taught me the importance of rigour, clarity, and perfectionism in scientiﬁc work as well as the art of technical writing. These lessons have had an indelible impression on me. I feel privileged to have been selected by him to convert his prized set of continuum mechanics course notes into a published textbook. Prof.

Bechtel’s unique perspective on the topic and legacy as a continuum mechanician live on through our book.

I would like to express my appreciation to Professor Marcelo Dapino, Professor

Dan Mendelsohn, Professor Amos Gilat, and Dr. Kelly Carney for serving on my

v doctoral committee. Each of them has willingly oﬀered support, guidance, and advice when I needed it. Prof. Dapino’s input on the smart materials portion of this dissertation brought a much-needed shift toward developing theories that would be useful in practical applications. I am thankful for the opportunity to have taken Prof.

Mendelsohn’s course in elastic wave propagation and Prof. Gilat’s course in plasticity, both of which were foundational in my research. Their help and friendship over the years are also acknowledged. Dr. Carney’s comments on the computational aspects of my work are also much appreciated; I look forward to working together more in the coming years.

I would like to acknowledge several faculty members at OSU who have provided mentoring, guidance, support, and friendship, including Professors Carlos Castro,

Brian Harper, Gary Kinzel, Cheena Srinivasan, and Denny Guenther. A hearty thank you is also due to Ms. Janeen Sands and Mr. Nick Breckenridge in the Graduate

Advising Oﬃce. Brieﬂy stated, both are tremendous assets to our Department. I am particularly thankful for Janeen’s compassion, willingness to listen, and helpful advice, which helped me weed my way though several challenging situations.

I would like to express my gratitude to several friends with whom I’ve had the pleasure of collaborating: First, I’d like to thank Dr. Po-Hsien Lin for performing the numerical computations in Chapter 2. I’ve enjoyed and proﬁted greatly from our conversations about hyperbolic systems and numerical methods. Secondly, I’d like to thank Professors Monon Mahboob and Md. Zahabul Islam for a fruitful international collaboration that resulted in three journal publications on the mechanics of nanoscale structures. Thirdly, I’d like to thank Professor Sushma Santapuri for

vi numerous discussions on the nuances of continuum electrodynamics and an enjoy-

able collaboration in this ﬁeld. I would also like to thank Dr. Lixiang Yang for a

productive collaboration during his time at OSU and for many helpful conversations.

Next, I would like to thank Professor Prasad Mokashi for his close friendship and support throughout the years. I would also like to thank Dr. Hafez Tari, who has become a dear friend to my wife and I in but a few short years in Columbus. I extend many thanks to my amigos Professor Chris Cooley, Mr. Tom Walters, Dr. Jeremy

Seidt, Dr. David Bilyeu, and Dr. Krista Kecskemety for sharing the joys (mostly) and frustrations (some) of graduate life over hot and cold beverages alike. A special thanks as well to my friends outside of graduate school – Shawn, Kyle, Nate, Brad, and many others – for road trips, rounds of golf, and chatting about sports and other topics over the occasional frosty beverage.

I acknowledge the generous ﬁnancial support I received throughout my doctoral studies, most notably a Presidential Fellowship from the OSU Graduate School, a

Graduate Teaching Fellowship from ASME, and a research assistantship from NSF. I am grateful to Department Chairman Ahmet Selamet and Associate Chairman Dan

Mendelsohn for their ﬁnancial support during several times of need. I would also

like to thank former Department Chairman Cheena Srinivasan and former Associate

Chairman Gary Kinzel for the opportunity to serve as an independent course in-

structor. I also gratefully acknowledge travel support from The Honor Society of Phi

Kappa Phi and the OSU Council of Graduate Students.

I would like to extend a special thanks to my family, particularly my wife Karen

and my mother Kristine. Karen – words cannot convey how grateful I am for your

enduring love, support, patience, understanding, and encouragement during this long

vii journey. We did it! Mom – I will never be able to repay you for your unwavering

love, devotion, and selﬂessness, but know that it is felt and cherished. I truly hope

I can be half the parent you are. A loving thank you is also due to my Aunt Doey,

whose weekly phone calls always provided me with a few good laughs, some lively chat about sports, and all the latest news in Bryan, Ohio. A heartfelt thank you to my Uncle John & Aunt Diane and cousins Kirk & Amy, who lovingly looked out for me during my years in Columbus. And last, but certainly not least, much love and

many thanks to my in-laws Ed, Barbara, Thomas, and Kim for their love, support,

and unwavering belief in me.

I am saddened that my father Bob, who passed away in 1997, is not here to

celebrate this milestone with me; I dedicate this dissertation to his memory.

Lastly, I thank God for His presence, for always being by my side, and for bringing

all of these outstanding people into my life. I am truly blessed beyond words.

viii Vita

October 16, 1980 ...... Born- Toledo,OH

June 1999 ...... Diploma, Bryan High School, Bryan, OH

September 1999 - May 2003 ...... Presidential Scholar, T.J. Smull Col- lege of Engineering, Ohio Northern University, Ada, OH

May 2003 ...... B.S. Mechanical Engineering, with High Distinction, Ohio Northern Uni- versity, Ada, OH

August 2005 ...... M.S. Mechanical Engineering, The Ohio State University, Columbus, OH

September 2006 - present ...... Ph.D. Student, ASME Graduate Teaching Fellow, and Presidential Fellow, The Ohio State University, Columbus, OH

Publications

Books

S.E. Bechtel and R.L. Lowe. “Fundamentals of Continuum Mechanics: With Appli- cations to Mechanical, Thermomechanical, and Smart Materials.” Elsevier Academic Press, San Diego, Nov. 2014.

ix Refereed Journal Articles

R.L. Lowe, S.-T.J. Yu, L. Yang, and S.E. Bechtel. “Modal and Characteristics- Based Approaches for Modeling Elastic Waves Induced by Time-Dependent Boundary Conditions.” Journal of Sound and Vibration 333(3), pp. 873-886, Feb. 2014.

S. Santapuri, R.L. Lowe, S.E. Bechtel, and M.J. Dapino. “Thermodynamic Mod- eling of Fully Coupled Finite-Deformation Thermo-Electro-Magneto-Mechanical Be- havior for Multifunctional Applications.” International Journal of Engineering Sci- ence 72, pp. 117-139, Nov. 2013.

M.Z. Islam, M. Mahboob, R.L. Lowe, and S.E. Bechtel. “Characterization of the Thermal Expansion Properties of Graphene Using Molecular Dynamics Simulations.” Journal of Physics D: Applied Physics 46(43), 435302, Oct. 2013.

M. Mahboob, M.Z. Islam, R.L. Lowe, and S.E. Bechtel. “Molecular Dynamics and Atomistic Finite Element Simulation Studies of the Eﬀect of Stone-Wales Defects on the Mechanical Properties of Carbon Nanotubes.” Nanoscience and Nanotechnology Letters 5(9), pp. 941-951, Sept. 2013.

S.-T.J. Yu, L. Yang, R.L. Lowe, and S.E. Bechtel. “Numerical Simulation of Linear and Nonlinear Waves in Hypoelastic Solids by the CESE Method.” Wave Motion 47(3), pp. 168-182, Apr. 2010.

L. Yang, R.L. Lowe, S.-T.J. Yu, and S.E. Bechtel. “Numerical Solution by the CESE Method of a First-Order Hyperbolic Form of the Equations of Dynamic Nonlinear Elasticity.” ASME Journal of Vibration and Acoustics 132(5), 051003, Oct. 2010.

Fields of Study

Major Field: Mechanical Engineering

Studies in: Theoretical & Applied Mechanics

x Table of Contents

Page

Abstract...... ii

Dedication...... iv

Acknowledgments...... v

Vita...... ix

ListofTables...... xv

ListofFigures...... xvi

1. AnIntroductiontoEulerianApproachestoElastodynamics...... 1

1.1Overviewandresearchopportunity...... 1 1.2Backgroundandpreliminaries...... 3 1.2.1 Eulerian forms of the fundamental laws of continuum mechanics 3 1.2.2 ConstitutivemodelinginEulerianﬁniteelasticity...... 6 1.2.3 A brief review of Eulerian and Lagrangian approaches to com- putationalmechanics...... 10 1.2.4 AbriefoverviewoftheCESEmethod...... 13 1.2.5 Existing 1-D models for the nonlinear elastodynamics of rods 16 1.3Objectives,structure,andnovelcontributions...... 17

2. An Eulerian Model for Nonlinear Waves in Elastic Rods, Solved Numeri- callybytheCESEMethod...... 20

2.1Introduction...... 21 2.2The2-Dmathematicalmodel:Eulerianformulation...... 27 2.2.1 Descriptionoftheproblem...... 27 2.2.2 3-Dgoverningequations...... 27

xi 2.2.3 3-D boundary conditions at the lateral surface ...... 29 2.2.4 Specializationofthemodelto2-D...... 30 2.3Derivationoftheleading-order1-Dmodel...... 33 2.3.1 Thedimensionless2-Dmodel...... 33 2.3.2 Perturbationformalism...... 36 2.3.3 Theelementary1-Dtheory(leading-ordermodel)...... 39 2.4Analternativedevelopmentoftheelementary1-Dtheory...... 43 2.4.1 Conservationofmass...... 46 2.4.2 Balanceoflinearmomentum...... 47 2.5Linearizationoftheelementary1-Dtheory...... 49 2.6Numericalimplementationoftheelementary1-Dtheory...... 52 2.6.1 Mathematicalstructureofthemodel...... 52 2.6.2 Conservativeform...... 55 2.6.3 TheCESEmethod...... 57 2.6.4 Benchmarkproblems...... 58

3. Modal and Characteristics-Based Approaches for Modeling Elastic Waves Induced by Time-Dependent Boundary Conditions ...... 69

3.1Introduction...... 71 3.2Illustrativeelastodynamicproblem...... 74 3.3Solutionoftheproblembymodalanalysis...... 76 3.3.1 Theconcentratedbodyforcemethod(CBFM)...... 77 3.3.2 The homogeneous eigenfunction expansion method (HEEM) 81 3.3.3 TheMindlin-Goodmanmethod...... 82 3.4Solutionoftheproblembythemethodofcharacteristics...... 85 3.4.1 Mathematical structure of the ﬁrst-order linear system . . . 86 3.4.2 Implementationofthemethodofcharacteristics...... 88 3.4.3 Resultsanddiscussion...... 91

4. ConclusionsandFutureWorkinEulerianElastodynamics...... 93

4.1Conclusions...... 93 4.2Futurework...... 98 4.2.1 Finite-deformationelasticity...... 98 4.2.2 Fluid-structureinteraction...... 100 4.2.3 Finite-deformationplasticity...... 100

5. AnIntroductiontoSmartPolymers...... 106

5.1Overviewandresearchopportunity...... 106 5.2Abriefreviewofseveralclassesofsmartpolymers...... 108

xii 5.2.1 Magnetorheologicalelastomers...... 108 5.2.2 Magnetoelectricpolymers...... 111 5.3Objectives,structure,andnovelcontributions...... 114

6. Revisiting the Fundamental Laws of Continuum Electrodynamics .... 116

6.1Introduction...... 116 6.2 Primitive, material, and integral versions of the fundamental laws . 119 6.2.1 Notationandnomenclature...... 120 6.2.2 Conservationofmass...... 121 6.2.3 Balanceoflinearmomentum...... 123 6.2.4 Balanceofangularmomentum...... 125 6.2.5 Firstlawofthermodynamics...... 127 6.2.6 Secondlawofthermodynamics...... 129 6.2.7 Conservationofelectriccharge...... 131 6.2.8 Faraday’slaw...... 134 6.2.9 Gauss’slawformagnetism...... 135 6.2.10Gauss’slawforelectricity...... 136 6.2.11 Ampère-Maxwelllaw...... 138 6.3Thelocalizationtheorem...... 140 6.4 Pointwise versions of the fundamental laws ...... 140 6.4.1 Eulerianforms...... 141 6.4.2 Lagrangianforms...... 145 6.5 Transformations between Eulerian and Lagrangian quantities . . . 149 6.6Someobservations...... 150 6.7 Two-way coupling between thermomechanics and electromagnetism 158 6.7.1 Effectiveelectromagneticfields...... 158 6.7.2 Electromagneticallyinducedcouplingterms...... 161 6.8MaxwellstressandtotalCauchystress...... 162

7. A Finite-Deformation Framework for Nonlinear Electro-Magneto-Elastic Materials, with Application to Magnetoelectric Polymer Composites . . . 164

7.1Introduction...... 164 7.2 Kinematics, fundamental laws, and boundary conditions ...... 170 7.2.1 Kinematics...... 171 7.2.2 Fundamental laws ...... 172 7.3Constitutiveequations...... 178 7.3.1 EnergyformulationswithEulerianIVs...... 179 7.3.2 EnergyformulationswithLagrangianIVs...... 189 7.3.3 Augmentedfreeenergyformulations...... 199 7.3.4 Invariancerequirements...... 204

xiii 7.3.5 Incompressibility ...... 205 7.3.6 Materialsymmetry...... 206 7.3.7 Arepresentativefreeenergyfunction...... 211

8. ConclusionsandFutureWorkinSmartPolymers...... 213

8.1Conclusions...... 213 8.2Futurework...... 214

Appendices 217

A. Supplement to Chapter 2 ...... 217

A.1 Extended mathematical structure of the perturbation formalism . . 217 A.2Thelinearizedtheory...... 221 A.2.1Mathematicalstructure...... 222 A.2.2Theequationofmotionforaninitiallystraightrod..... 224 A.2.3Thed’Alembertsolutionoftheinitial-valueproblem.... 224 A.2.4Velocity,stress,strain,density,andarea...... 227 A.2.5Impact...... 229 A.2.6Separation...... 232

B. Supplement to Chapter 3 ...... 234

B.1ReformulationoftheproblembytheCBFM...... 234 B.2ThespecializedCBFM/HEEMsolution...... 236 B.3ThespecializedMindlin-Goodmansolution...... 237 B.4Convergenceandterm-by-termdiﬀerentiation...... 238

C. Supplement to Chapter 6 ...... 241

D. Supplement to Chapter 7 ...... 260

Bibliography...... 283

xiv List of Tables

Table Page

3.1 Material properties, geometric parameters, and boundary conditions . 76

6.1 Units for various electrical, magnetic, mechanical, and thermal quantities.154

7.1FreeenergiescorrespondingtodiﬀerentsetsofEulerianIVs..... 180

7.2 Constitutive equations for energy formulations with Eulerian IVs . . . 181

7.3 Constitutive equations for energy formulations with Lagrangian IVs . 190

7.4 Constitutive equations for ‘augmented’ energy formulations with La- grangianIVs...... 201

7.5Invariantconstitutiveequations...... 205

xv List of Figures

Figure Page

2.1 A control volume (dashed lines) occupied at time t by a differential element of a tapered rod (solid lines). The axial control surfaces are fixed in space but undergo cross-sectional area variation. The top and bottom lateral control surfaces co-deform with the lateral boundaries ofthedifferentialelement...... 44

2.2 Low-speed (20 m/s) impact problem. Snapshot t =3μs after impact illustrating the analytical (solid) and numerical (circles) wave proﬁles for (a) density, (b) velocity, (c) area, and (d) stress. Both the striker bar (initially moving at 20 m/s) and the incident bar (initially stationary) are composed of 7075-T651 aluminum alloy. The impact interface is located at z = 5 cm. Two weak shocks travel away from the impact interface, both propagating compression and lateral expansion in their wake...... 60

2.3 High-speed (6000 m/s) impact problem. Snapshot t =4.8 μsafter impact illustrating the analytical (solid) and numerical (circles) wave profiles for (a) density, (b) velocity, (c) area, and (d) stress. Both the striker bar (initially moving at 6000 m/s) and the incident bar (initially stationary) are composed of 7075-T651 aluminum alloy. The impact interface is located at z = 5 cm. Two strong shocks travel away from the impact interface, both propagating compression and lateral expansion in their wake. A contact wave (or contact discontinuity) is also clearly discernible in the density, area, and stress profiles. Substantial discrepancies in wave amplitudes and wave speeds are observed, owing totheeffectsofnonlinearity...... 61

xvi 2.4 A comparison of the numerical predictions of three diﬀerent objective rates – Jaumann-Zaremba (circles), Oldroyd (squares), and Cotter- Rivlin (diamonds) – with an analytical solution (solid line) of a low- speed (20 m/s) impact. As expected, good agreement is observed at lowspeeds(linearregime)...... 64

2.5 A comparison of the numerical predictions of three different objective rates – Jaumann-Zaremba (circles), Oldroyd (squares), and Cotter- Rivlin (diamonds) – with an analytical solution (solid line) of a high- speed (6000 m/s) impact. As expected, different rates yield substantially different physical predictions, and all deviate appreciably from theanalyticalsolution,owingtotheeffectsofnonlinearity...... 64

2.6 Low-speed separation problem (Vo = 10 m/s). Snapshot at t =4μs illustrating the analytical (solid) and numerical (circles) wave proﬁles for (a) density, (b) velocity, (c) area, and (d) stress. The rod is composed of UNS C15720 copper alloy. Two weak rarefactions travel away from the separation interface at z = 5 cm, both propagating decompression, tension,andlateralcontractionintheirwake...... 66

2.7 High-speed separation problem (Vo = 1000 m/s). Snapshot at t = 6 μs illustrating the analytical (solid) and numerical (circles) wave proﬁles for (a) density, (b) velocity, (c) area, and (d) stress. The rod is composed of UNS C15720 copper alloy. Two strong rarefactions travel away from the separation interface at z = 5 cm, both propagating decompression,tension,andlateralcontractionintheirwake...... 67

3.1 (a) A snapshot at t = 240 μs illustrating a propagating stress wave generated at the forced boundary traveling leftward prior to reﬂection. (b)-(e) Zoomed regions near the forced boundary, indicated by the box in (a). Solutions were obtained via the concentrated body force method and the homogeneous eigenfunction expansion method (solid lines), the Mindlin-Goodman method (dashed lines), and the method of characteristics (squares). The mode superposition solutions were truncated at (b) 500 modes, (c) 1000 modes, (d) 2500 modes, and (e) 5000 modes...... 79

xvii 3.2 (a) A snapshot at t =4.1 ms illustrating the stress proﬁle resulting from the interference of leftward-propagating incident waves from the forced end with reﬂected waves traveling inward from the boundaries. (b)-(e) Zoomed regions near the forced boundary, indicated by the box in (a). Solutions were obtained via the concentrated body force method and the homogeneous eigenfunction expansion method (solid lines), the Mindlin-Goodman method (dashed lines), and the method of characteristics (squares). The mode superposition solutions were truncated at (b) 500 modes, (c) 1000 modes, (d) 2500 modes, and (e) 5000 modes. Note that the stresses are still well within the elastic regimeforthisparticularmaterial...... 80

3.3 The characteristic mesh for two time steps, illustrated for the special case N = 4. Along the line t = 0 spanning 0 ≤ ξ ≤ L, we introduce N nodes uniformly spaced Δξ apart. A right-running characteristic emerges from the left boundary node (i =1), a left-running characteristic emerges from the right boundary node (i = N), and both left-running and right-running characteristics emerge from the interior nodes. As these characteristics advance in time, pairs of them intersect, ﬁrst at time t =Δt,thenatt =2Δt,andsoon,withthetime step Δt dictated by the nodal spacing Δξ and the constant slope ±c of the characteristics. Arrows denote the direction that information propagates along the characteristic lines, and ﬁlled circles designate nodes...... 91

6.1 Subsets S1 and S2 as seen in the present conﬁguration of body B. Subset S1 is an open volume P bounded by a closed surface ∂P, while subset S2 is an open surface Q bounded by a closed curve ∂Q..... 121

xviii Chapter 1: An Introduction to Eulerian Approaches to Elastodynamics

This chapter begins the ﬁrst part of the dissertation, where the focus is on Eulerian

approaches to mathematically modeling and numerically simulating inertia-driven

problems in solid mechanics.

1.1 Overview and research opportunity

Finite-element codes that employ a co-deforming mesh (Lagrangian approach) currently dominate the landscape in computational solid mechanics. However, numerical algorithms that employ a fixed mesh (Eulerian approach) are beginning to attract significant attention – particularly for solving inertia-driven (dynamic) problems such as impacts and blasts in finite-deformation elasticity [1–16] and plasticity [17–32].

Generally speaking, Eulerian approaches employ a deformed-configuration (Eulerian) formulation of the governing equations, which are then discretized on a fixed mesh and solved numerically by a finite-difference or finite-volume scheme.

Eulerian approaches are appealing for several reasons. First, the mathematical structure of the governing equations enables the use of modern numerical methods for ﬁrst-order hyperbolic systems, which are explicitly designed for resolving highly

1 nonlinear, inertia-driven physics. Consequently, Eulerian approaches are adept at tracking the evolution and interaction of nonlinear waves such as shocks and rarefactions. Secondly, fixed grids are generally regarded as the most natural setting for shock capturing [30]. Thirdly, since the continuum is deforming through a stationary grid, arbitrarily large strains and rotations can be accommodated without excessive grid distortion or remeshing. Finally, Eulerian approaches to solid dynamics have opened the door for solving multiphase problems on a single fixed mesh with a single solver, greatly simplifying the numerical treatment of solid-fluid interfaces in

ﬂuid-structure interaction problems [33–40].

Rod-like structures are used in dynamic applications such as ultrasonics, acoustic wave sensing, ballistic impact, structural analysis, nondestructive evaluation, ﬂaw detection, high-strain-rate material testing, and percussion drilling. To model the nonlinear elastic waves that often arise in these applications, approximate 1-D rod theories are developed from the full 3-D dynamic ﬁnite elasticity theory [41–66]. The primary appeal of specialized 1-D theories over the general 3-D theory is the reduced order of the governing equations, which depend on just one spatial coordinate and time. Practically, this translates to more straightforward computational implementation and reduced code execution times.

To the best of our knowledge, elastodynamic rod theories formulated in the present conﬁguration (i.e., the Eulerian representation) are presently unavailable in the literature, with the exception of some related ideas set forth by Yang and coworkers [67, 68]. An Eulerian rod theory is attractive for several reasons: First, it could be implemented in a computer code employing a ﬁxed grid and a modern numerical

2 method for capturing hyperbolic wave physics. The resulting framework could poten-

tially achieve unprecedented shock capturing and wave tracking capabilities in elastic

rod-like structures. Secondly, it could be useful for validating new ﬁnite-diﬀerence

and ﬁnite-volume codes in a simple solid-dynamics setting. Lastly, it could be in-

corporated within a unified Eulerian computational architecture for fluid-structure interaction, wherein the dynamics of the solid and fluid phases are simultaneously

tracked on a single ﬁxed mesh.

The remainder of this chapter is structured as follows: In Section 1.2, we pro-

vide some relevant background from continuum mechanics, ﬁnite elasticity, numerical

methods, and structural vibrations, with a particular focus on Eulerian aspects of

modeling and computation. Then, in Section 1.3, we highlight the objectives and

novel contributions of this part of the dissertation.

1.2 Background and preliminaries

1.2.1 Eulerian forms of the fundamental laws of continuum mechanics

In continuum mechanics, the motion

x = χ(X,t) (1.1)

maps each particle in the body B from its reference position X to its present position

x. Consequently, the body itself is mapped from its reference (or undeformed) conﬁg-

uration to its present (or deformed) conﬁguration. In the deformed conﬁguration, the

body occupies open volume R bounded by closed surface ∂R, and subset S occupies

open volume P bounded by closed surface ∂P.

3 Conservation of mass

Conservation of mass postulates that the mass of every subset of the body is

constant throughout its motion, or, equivalently, the time rate of change of the mass

of every subset is zero. This can be expressed in Eulerian integral form as [69] d ρdv =0, (1.2) dt P where ρ is the mass density and dv is a volume element, both in the present conﬁg- uration. Note that to perform the integration in (1.2), ρ is regarded in its Eulerian description, i.e., as a function of present position x and time t.

Provided the density is suﬃciently smooth, the Eulerian diﬀerential form of conservation of mass follows from (1.2) as [69]

∂ρ + div(ρv)=0, (1.3a) ∂t or, alternatively,

∂ρ +(v · grad)ρ + ρ divv =0. (1.3b) ∂t

In (1.3a) and (1.3b), v is the velocity, “grad” denotes the Eulerian gradient operator

(i.e., the gradient calculated with respect to the present conﬁguration), “div” denotes the Eulerian divergence operator, and ( ) · ( ) denotes an inner product. Equation

(1.3a) is referred to as the conservative form of conservation of mass, while (1.3b) is called the non-conservative form [70].

4 Balance of linear momentum

Balance of linear momentum postulates that the time rate of change of the linear

momentum of any subset of the body is equal to the resultant external force acting

on that subset. This can be expressed in Eulerian integral form as [69] d vρdv = f m ρdv + t da, (1.4) dt P P ∂P where f m is the mechanically induced body force (per unit mass), t is the traction

(per unit deformed area), and da is a surface element in the present conﬁguration.

Note that f m accounts for long-range forces such as gravity, while t accounts for

short-range contact forces acting on the surface of the subset.

Provided the fields are sufficiently smooth, the Eulerian differential form of balance

of linear momentum follows from (1.4) as [69]

∂(ρv) +divρv ⊗ v = ρf m +divT, (1.5a) ∂t or, alternatively, ∂v ρ +(v · grad)v = ρf m +divT, (1.5b) ∂t where T is the Cauchy stress and ( ) ⊗ ( ) denotes the dyadic (tensor) product of two vectors. Equations (1.5a) and (1.5b) are referred to the conservative and non- conservative forms, respectively, of balance of linear momentum.

Balance of angular momentum

Balance of angular momentum postulatesthatthetimerateofchangeofthe angular momentum of any subset of the body about the origin is equal to the resultant

5 external moment acting on that subset about the origin. This can be expressed in

Eulerian integral form as [69] d x × vρdv = x × f m ρdv + x × t da, (1.6) dt P P ∂P

where x is the present position of a continuum particle and ( ) × ( ) denotes the cross

product of two vectors. The local form of (1.6) is [69]

T = TT. (1.7)

Hence, the Cauchy stress tensor is symmetric.

1.2.2 Constitutive modeling in Eulerian ﬁnite elasticity

In ﬁnite elasticity, hyperelastic (or Green-elastic) materials are those for which the

Cauchy stress T is derivable from a strain energy potential W . In the isotropic case, the strain energy W (per unit reference volume) is a function of the deformation only through the principal invariants I1, I2,andI3 of the left Cauchy-Green deformation tensor B, i.e.,

1 2 2 I1 =trB,I2 = tr B − tr B ,I3 =detB, 2 where

∂χ B = FFT, F =Gradχ ≡ . ∂X

(The tensor F, called the deformation gradient, plays a pivotal role in continuum kinematics.) It can then be shown that the Cauchy stress T for a compressible,

6 isotropic, nonlinearly elastic material is given by [69]

−1 T = β0 I + β1 B + β−1 B , (1.8) where − 1 − 1 1 2 ∂W ∂W 2 ∂W 2 ∂W β0 =2I3 I2 + I3 ,β1 =2I3 ,β−1 = −2I3 . ∂I2 ∂I3 ∂I1 ∂I2

Hence, the constitutive equation (1.8) is of the customary ‘stress-strain’ variety, relating the Cauchy stress T to a large-deformation strain measure B (and, implicitly, its invariants).

Many strain energy functions for compressible, isotropic, nonlinearly elastic materials have been proposed in the literature, e.g, Ogden, Blatz-Ko, neo-Hookean,

Mooney-Rivlin, and Levinson-Burgess [69,71]. As an example, the compressible neo-

Hookean strain energy function for rubbery materials is [71, p. 247] μ 1 −γ W = I1 − 3 + I − 1 , 2 γ 3 where γ = ν/(1 − 2ν), and μ and ν are the shear modulus and Poisson’s ratio evalu-

ated at small strains.

A key observation is that the ﬁnite-strain constitutive equation (1.8) implicitly

introduces a new unknown (the deformation gradient F), so that (1.8) together with

the Eulerian conservation laws (1.3) and (1.5) do not form a closed system of equa-

tions. This incompatibility is rooted in the fact that the stress in an elastic solid

depends on the strain relative to the undeformed reference conﬁguration.‘Memory’

of this undeformed reference conﬁguration is embedded in F. However, the Eule-

rian ﬁrst principles (1.3) and (1.5) – formulated in terms of present velocity – do

7 not contain any primitive kinematic quantities that allow us to directly connect with

F =Gradx, the undeformed reference conﬁguration, and, most importantly, the stress-strain constitutive equation (1.8).

As discussed in [16], several methods have been proposed to either bridge or circumvent the inherent incompatibility between the the Eulerian fundamental laws

(1.3) & (1.5) and the hyperelastic ‘stress-strain’ constitutive equation (1.8):

(i) Transport of the deformation gradient. Equations (1.3), (1.5), and (1.8)

are augmented by an evolution (advection) equation for the deformation gradi-

ent, e.g., [9,15,72]

∂F +(v · grad)F = (grad v)F, (1.9) ∂t

or, alternatively, its inverse, e.g., [17,20,25]

∂F−1 +grad(F−1 v)=0. (1.10) ∂t

Loosely speaking, (1.9) and (1.10) close the system of equations by connecting

the deformation gradient F to the velocity v. Note that both (1.9) and (1.10)

are subject to compatibility conditions – the former a divergence constraint and

the latter a curl constraint [28] – that ensure the deformation gradient F can be

integrated to produce a unique displacement ﬁeld.

We remark that (1.9) and (1.10) are only representative examples of advec-

tion equations for F and its inverse; refer to [1,2,7,10] for alternative versions.

8 Also see the discussions in [28, 73]. In any event, the basic idea of advect-

ing F and its inverse is now commonly employed in Eulerian ﬁnite elasticity

(see [3,4,8,11,13,14]) and plasticity (see [18,26,28,30]).

(ii) Transport of the reference map. Equations (1.3), (1.5), and (1.8) are aug-

mented by an evolution (advection) equation for the reference map [16,74]

∂ξ +(v · grad)ξ = 0. (1.11) ∂t

Essentially, the reference map is the inverse of the motion (1.1), i.e., X = ξ(x,t);

it accepts the present position x and time t as inputs, and gives the reference

position X as output. Initially, at t =0,

ξ(x,t=0)= x.

Thus, ξ(x,t) is an Eulerian ﬁeld that provides “memory” of a particle’s reference

position. As such, it enables a straightforward calculation of the deformation

gradient F (i.e., F−1 = ∂ξ/∂x is calculated, then inverted). Thus, Eqs. (1.3),

(1.5), (1.8), and (1.11) together form a closed system.

We mention that the basic idea of evolving the reference map was indepen-

dently set forth by Cottet et al. [75] and Kamrin [76] around the same time.

Recent applications of this approach can be found in [39, 40, 77].

(iii) Hypoelasticity. The hyperelastic stress-strain constitutive equation (1.8) is

discarded altogether in favor of a constitutive equation where, loosely, the rate

of stress is related to the rate of strain. Rate-type models for elastic behavior,

originally proposed in the seminal work of Truesdell [78,79], are typically referred

9 to as hypoelastic. A popular example in the literature is [5]

DT = λ(tr D)I +2μD, (1.12) Dt

where λ and μ are the Lam´e constants, and D/Dt is a suitably invariant (i.e.,

objective) rate of the Cauchy stress tensor. (The particular choice of objective

rate is crucial to the validity and robustness of (1.12); refer, for instance, to [5,

80,81].) By directly relating the stress to the velocity, the constitutive equation

(1.12), together with the Eulerian conservation laws (1.3) and (1.5), form a

closed system of equations. Rate-type constitutive models for elastic response

are widely used, for instance, in industrial and commercial ﬁnite-element codes

[25, 80]; some recent work and associated applications can be found in [5, 6, 10,

19,21–24,27,29,31,32,80–83].

1.2.3 A brief review of Eulerian and Lagrangian approaches to computational mechanics

There is a lack of agreement in the computational mechanics community regarding the optimal approach for solving inertia-driven problems with ﬁnite deformations and constitutive nonlinearities. In these situations (e.g., blasts and impacts), tracking the interaction of both linear and nonlinear waves is of central importance, as is minimizing numerical dissipation and dispersion near steep gradients and solution discontinuities.

The players in this debate are divided along multiple lines, with the Eulerian,

Lagrangian, arbitrary Lagrangian-Eulerian (ALE), and meshless factions being the most notable. Not surprisingly, each of these aforementioned approaches has its own

10 merits and drawbacks. In what follows, we give a brief overview of the Eulerian and

Lagrangian approaches; more extensive discussions on these and other approaches can be found in [15, 16, 24, 84], the review articles [85, 86], and the extensive list of references therein.

Lagrangian approaches

Lagrangian approaches solve the Lagrangian form of the fundamental laws, which correspond to the reference (undeformed) configuration of the continuum. The nodes in a Lagrangian mesh coincide with the material points in the continuum; thus, the mesh is affixed to and co-deforms with the continuum. Consequently, Lagrangian approaches naturally track the evolution of boundaries and interfaces during the deformation. They also allow for more straightforward implementation of boundary conditions, as they are imposed on the undeformed reference configuration.

However, during large deformations, a Lagrangian mesh can become excessively distorted or even entangled. To remedy this, some algorithms check mesh quality after each time step and, if needed, call a remeshing algorithm. Remeshing, however, comes at a cost, as the interpolation of data to the new mesh is computationally expensive and introduces numerical error.

Eulerian approaches

Eulerian approaches use a mesh that is ﬁxed in space, so that the continuum deforms through the grid. Consequently, mesh distortion is no longer a concern – no matter how large the deformation – and thus the need for expensive remeshing and interpolation is circumvented. Eulerian methods solve an Eulerian formulation

11 of the fundamental laws and constitutive equations,1 whichcorrespondtothepresent

(deformed) conﬁguration of the continuum; refer to Sections 1.2.1 and 1.2.2.

In dynamic finite elasticity, Eulerian formulations of the governing equations can be expressed as a fully conservative first-order hyperbolic system (FOHS) [25]. Sys- tems of equations with this type of mathematical structure (e.g., the Euler equations) have been studied and solved numerically for many years in the high-speed gas dynamics community, where flows are typically inertia-dominated, compressible, unsteady, and highly nonlinear. A suite of modern numerical methods for FOHSs have been developed to accurately resolve this complex physics, faithfully track the evolution and interaction of nonlinear traveling waves like rarefactions and shocks, and capture discontinuities and sharp gradients in the solution variables. Thus, the equations of dynamic finite elasticity, when cast in their Eulerian form (refer again to Sections

1.2.1 and 1.2.2), are amenable to the rich mathematical and numerical formalism available for FOHSs and shock capturing.2

Among the numerical methods commonly used for FOHSs, modern high-resolution shock-capturing schemes are of particular interest. A typical high-resolution scheme consists of an underlying finite-volume method coupled with a Riemann solver for computing the upwind flux at cell boundaries. For instance, Barton & Drikakis [14] solve an Eulerian formulation of nonlinear elasticity using a finite-volume method

1Kamrin et al. [16] refer to this as a “fully Eulerian” approach, in contrast to the “partially Eulerian” remap procedures commonly employed in commercial hydrocodes; see [86,87] for additional details on the latter. 2Eulerian formulations of the governing equations have a ﬁrst-order structure (with velocity and stress as the primitive unknowns), whereas Lagrangian formulations have a second-order structure (with displacement and stress as the primitive unknowns). Hence, Lagrangian formulations are not amenable to the powerful numerical schemes designed for ﬁrst-order hyperbolic systems.

12 coupled with a ﬁfth-order weighted essentially non-oscillatory (WENO) reconstruc-

tion procedure, a linearized Riemann solver for computing intercell ﬂuxes, and a

third-order Runge-Kutta method for time integration. Other contributions – both

theoretical and computational – to Eulerian ﬁnite elasticity and plasticity can be

found in [1–16] and [17–32], respectively.

1.2.4 A brief overview of the CESE method

In this dissertation, we pursue an Eulerian approach to elastodynamics, with the novel space-time Conservation Element and Solution Element (CESE) method as its numerical backbone.3 The CESE method is designed from the ground up to solve conservative, first-order, hyperbolic systems of PDEs. (Refer to [88–91] for the pioneering CESE schemes, [92, 93] for Courant-number-insensitive CESE schemes, and [94–96] for higher-order CESE schemes. The interested reader is also encouraged to consult [94,97] and the references therein for extensive reviews, comparisons, and assessments of the aforementioned CESE schemes.) The CESE method differs markedly from customary finite-difference, finite-volume, and high-resolution shock- capturing schemes in concept, structure, and methodology. For instance, standard high-resolution schemes are based on a finite-volume method coupled with (i) a high- order reconstruction procedure that approximates the values of discrete variables at cell boundaries and (ii) a Riemann solver that uses these values to compute the numerical flux at cell boundaries. On the other hand, the CESE method, whose tenet is a unified treatment of space and time,

(a) discretizes space and time using a ﬁxed, staggered, space-time mesh;

3Perhaps ‘semi-Eulerian’ is a more ﬁtting description; as will soon be apparent, although we solve the Eulerian form of the governing equations, our mesh is not ﬁxed in space, but rather space-time.

13 (b) enforces ﬂux conservation in both space and time;

(d) does not require a reconstruction procedure to interpolate or extrapolate the

numerical values of conserved variables at cell boundaries;

(e) does not require a physics-speciﬁc Riemann solver to compute the spatial gradi-

ents (numerical ﬂuxes) of conserved variables at cell boundaries.

Instead, conserved variables and their spatial gradients are calculated primitively as

part of the core CESE scheme. The absence of a Riemann solver and a reconstruction

step translates to more straightforward implementation, simpler logic, and lower op-

eration counts. Furthermore, unlike its counterparts that employ a Riemann solver,

the CESE method is physics independent. That is, the CESE method is able to solve

generic systems of ﬁrst-order hyperbolic PDEs without ad hoc customization.

For illustrative purposes, we now consider the 1-D scalar analog of the conservative

forms of the 3-D balance laws (1.3a) and (1.5a), given by

∂u ∂f + =0. (1.13) ∂t ∂z

In general, the ﬂux f depends on the conserved variable u so that f(u). The CESE

method regards x1 ≡ z and x2 ≡ t as the coordinates of a two-dimensional Euclidean space E 2, where customary mathematical operations such as gradient, divergence, and curl hold. It then follows that

∇·h =0, (1.14)

14 where ∂ ∂ ∇≡ , , h ≡ (f, u) , ∂z ∂t

and h is referred to as the space-time ﬂux vector. Note that space and time are treated

on the same footing. Integrating (1.14) over a ﬁxed space-time region V gives ∇·h dv =0, V and subsequent use of the divergence theorem leads to h · n da =0, (1.15) ∂V

where ∂V is the boundary of V . Equation (1.15) is a statement of space-time ﬂux

conservation. The CESE method enforces (1.15) over each Conservation Element

(CE) and approximates h with discrete values within each Solution Element (SE);

refer to [88, 97] for additional details.4 Numerous explicit time-marching schemes follow from the core concept of space-time ﬂux conservation (1.15), e.g., the a scheme, the a- scheme, the c scheme, and the c-τ scheme; refer to [94,97] for additional details.

Ultimately, these schemes diﬀer from one another primarily in the manner in which

n they approximate (uz)j , the discrete counterpart of ∂u/∂z at mesh node (j, n).

Additional details regarding the speciﬁc scheme employed in this dissertation, its accuracy and stability, the use of weighting functions (for oscillation suppression at

4In contrast, conventional high-resolution shock-capturing schemes typically discretize the spatial domain into non-overlapping volume elements and integrate Eq. (1.13) over a representative element, à la the finite-volume method. Flux is thus conserved in space, not space-time. Also, spatial and temporal integration are handled separately: The first (temporal) term in Eq. (1.13) is typically discretized using a Runge-Kutta method. On the other hand, the numerical fluxes in the second (spatial) term are computed using a Riemann solver coupled with a reconstruction procedure that approximates the variation of u within a representative volume element.

15 solution discontinuities), source term integration, and boundary condition treatment

can be found in Section 2.6.3 of Chapter 2.

1.2.5 Existing 1-D models for the nonlinear elastodynamics of rods

Rod-like structures are used in applications such as ultrasonics, acoustic wave

sensing, ballistic impact, structural analysis, DNA mechanics, nondestructive eval-

uation, ﬂaw detection, high-strain-rate material testing, and percussion drilling. Of

particular interest in this dissertation are nonlinear waves that propagate longitudinal

or extensional motion (i.e., axisymmetric axial-radial motion) in slender elastic rods.

To model these types of waves in rod-like structures, approximate 1-D theories are

developed from the full 3-D dynamic ﬁnite elasticity theory.5 The primary appeal of specialized 1-D theories over the general 3-D theory is the reduced order of the governing equations, which depend on just one spatial coordinate and time. Practically, this translates to more straightforward computational implementation and reduced code execution times.

Numerous approaches have been used to develop 1-D theories for nonlinear longitudinal waves in elastic rods. For instance, perturbation techniques that exploit the slenderness of the rod have been employed by Antman & Warner [41], Nariboli [42],

Howard [43], Benveniste & Lubliner [44], Benveniste [45], Samsonov et al. [46], Dai et al. [47, 48], and de Lima & Hamilton [49]. Therein, asymptotic solutions to the

3-D theory are constructed by solving a series of 1-D problems, each corresponding to a successively higher-order correction in the asymptotics. Perturbation schemes

5By ‘1-D theory,’ we mean a theory whose primitive variables depend on only one spatial coordinate and time.

16 have also been used by Hay [98] for the special case of static deformations and Nari-

boli [99], Rogge [100], Bostr¨om [101], and Stephen et al. [102] for the special case of

small deformations and linear constitutive equations.

An altogether diﬀerent approach is adopted by Ostrovskii & Sutin [50], Soerensen

et al. [51], Clarkson et al. [52], Coleman & Newman [53], Cohen & Dai [54], and

Dai [55], wherein 1-D kinematics are posited at the outset and energy principles

are subsequently used to deduce the 1-D equations of motion. Wright [56, 57] also

employs energy methods, although he posits 2-D kinematics at the outset and reduces

the equations of motion to 1-D through a cross-sectional averaging technique. 1-D rod

theories have also been developed using the concepts of a Cosserat curve, convected

coordinates, and deformable directors; refer, for instance, to [58–66].

The nonlinear rod theories discussed in the preceding paragraphs are formulated in

the Lagrangian representation, with respect to the reference conﬁguration of the rod.

To the best of our knowledge, rod theories formulated in the present conﬁguration,or

Eulerian representation, are presently unavailable in the literature, with the exception

of some fundamental ideas set forth by Yang and coworkers in [67, 68]. (The same

holds true for other elastic structural elements, such as beams, plates, membranes,

and shells.)

1.3 Objectives, structure, and novel contributions

Chapters 2 and 3 are intended to be self-contained studies of two diﬀerent yet related aspects of structural elastodynamics. In Chapter 2, a comprehensive perturbation theory is used to develop an Eulerian model for nonlinear longitudinal waves in tapered elastic rods – ideal for ﬁxed-grid computational techniques such as

17 finite-difference methods, finite-volume methods, and modern high-resolution shock-

capturing schemes. The leading-order equations in our perturbation formalism are veriﬁed using a control-volume analysis, then linearized to recover a classical model for longitudinal waves in ultrasonic horns. We assess the mathematical structure of our leading-order model, deduce the eigenvalues and eigenvectors of the Jacobian matrix, and verify its hyperbolicity. Numerical solutions are obtained using the novel space-time Conservation Element and Solution Element (CESE) method. In our simulations, the CESE method is shown to eﬀectively capture shocks, rarefactions, and contact discontinuities.

In Chapter 3, we further investigate the linearized versions of the leading-order equations derived in Chapter 2. This set of linear hyperbolic PDEs is sufficiently general to model a large class of forced vibration and elastic wave propagation problems. In the context of an initial-value/boundary-value problem (IVBVP) with time- dependent boundary conditions, solutions are obtained via three different modal approaches and the method of characteristics. The modal approaches reformulate a second-order (displacement-stress) formulation of the IVBVP to remove the time- dependent inhomogeneity from the boundary and enable separation of variables, but only do so in an approximate sense. The resulting reformulated problems are thus mathematically inexact, leading to solutions with spurious oscillations and significant overshoot at the forced boundary, similar to the Gibbs phenomenon. We demonstrate that these oscillations and overshoot are physical manifestations of a series solution for stress, obtained from term-by-term differentiation, that is not uniformly convergent, as required by the formalism of mathematical analysis. In contrast, the method of characteristics solution of the first-order (velocity-stress) formulation of the problem

18 is exact to within machine precision, yielding no artiﬁcial discontinuities, spurious oscillations, or unphysical overshoot.

19 Chapter 2: An Eulerian Model for Nonlinear Waves in Elastic Rods, Solved Numerically by the CESE Method

Much of the work in this chapter is presented in R.L. Lowe, P.-H. Lin, S.-T.J. Yu, and S.E. Bechtel, “An Eulerian Model for Nonlinear Waves in Elastic Rods, Solved

Numerically by the CESE Method,” to be submitted to the International Journal of

Solids and Structures.

In this chapter, we use a comprehensive perturbation theory to develop an Eule- rian model for nonlinear longitudinal waves in tapered elastic rods. The equations of motion, when cast in their Eulerian form, are ideal for fixed-grid computational techniques such as finite-difference methods, finite-volume methods, and modern high- resolution shock-capturing schemes. The leading-order equations in our perturbation formalism are verified using a control-volume analysis, then linearized to recover a classical model for longitudinal waves in ultrasonic horns. We assess the mathematical structure of our leading-order model, deduce the eigenvalues and eigenvectors of the

Jacobian matrix, and verify its hyperbolicity. Solutions are obtained using the space- time Conservation Element and Solution Element (CESE) method, a novel numerical

20 technique for ﬁrst-order hyperbolic systems. In our simulations, the CESE method is

shown to eﬀectively capture shocks, rarefactions, and contact discontinuities.

2.1 Introduction

In recent years, there has been a growing interest in the use of modern numerical methods for first-order hyperbolic systems – traditionally employed in unsteady compressible flow applications – to simulate the dynamic behavior of elastic solids, espe- cially in the finite-deformation regime [1–4,6–8,10,11,13,14,16–18,20,21,23–28,30,32].

In particular, a class of techniques referred to as high-resolution shock-capturing schemes, or high-resolution schemes for short, have received considerable attention.

A typical high-resolution scheme consists of an underlying ﬁnite-volume method coupled with an advanced algorithm for computing the upwind ﬂux at cell boundaries, a notoriously challenging aspect in the computational treatment of hyperbolic (inertia- dominated) problems [103,104]. The main appeal of these techniques is their exceptional ability to capture the evolution and interaction of nonlinear traveling waves.

For instance, shock waves are crisply resolved without excessive numerical dissipation

(e.g., smearing and clipping) or numerical dispersion (e.g., artiﬁcial oscillations) near the wavefront.

From a computational perspective, shock-capturing schemes are most naturally implemented on a fixed (stationary) mesh [30].6 A fixed mesh, in turn, most naturally mates with an Eulerian formulation of the mathematical model, corresponding to the present configuration of the deforming continuum. Eulerian formulations of dynamic

6In other words, the computational mesh is ﬁxed in space, and the continuum deforms through the mesh.

21 ﬁnite elasticity have been developed or employed in [3–6, 8–15], many of which were

inspired by the seminal work of Truesdell [78], Godunov & Romenskii (ﬁrst in [1] and

later in [7]), Plohr & Sharp [2], and, more recently, Kamrin et al. [16]. Refer to [17–32]

for analogous examples of Eulerian formulations of dynamic ﬁnite plasticity.

Many of the aforementioned Eulerian finite-elasticity models have been numerically implemented using high-resolution schemes [3,4,8,10,11,13,14,16]. As a representative example, Barton & Drikakis [14] numerically solve Godunov & Romenskii’s elasticity model [7] using a finite-volume method coupled with a fifth-order weighted essentially non-oscillatory (WENO) reconstruction procedure, a linearized Riemann solver for computing intercell fluxes, and a third-order Runge-Kutta method for time integration.

We remark that Eulerian formulations of dynamic nonlinear elasticity – and

their associated implementation in finite-difference and finite-volume codes – are, at

present, less prevalent than their Lagrangian counterparts. This is due to the nearly

ubiquitous use of Lagrangian-type formulations in ﬁnite-element codes (with [9,15,22]

being several notable exceptions), which currently dominate the landscape in compu-

tational solid mechanics. We also remark that Lagrangian formulations do not have

a ﬁrst-order mathematical structure, and are thus not readily amenable to numerical

treatment via high-resolution schemes for ﬁrst-order hyperbolic systems.

Rod-like structures are used in dynamic applications such as ultrasonics, acoustic

wave sensing, ballistic impact, structural analysis, DNA mechanics, nondestructive

evaluation, ﬂaw detection, high-strain-rate material testing, and percussion drilling.

Of particular interest in this chapter are nonlinear waves that propagate longitudinal

or extensional motion (i.e., axisymmetric axial-radial motion) in slender elastic rods.

22 To model these types of waves in rod-like structures, approximate 1-D theories are developed from the full 3-D dynamic ﬁnite elasticity theory. The primary appeal of specialized 1-D theories over the general 3-D theory is the reduced order of the governing equations, which depend on just one spatial coordinate and time. Practically, this translates to more straightforward computational implementation and reduced code execution times.

Howard [43], Benveniste & Lubliner [44], Benveniste [45], Samsonov et al. [46], Dai et al. [47, 48], and de Lima & Hamilton [49]. Therein, asymptotic solutions to the

3-D theory are constructed by solving a series of 1-D problems, each corresponding to a successively higher-order correction in the asymptotics. Perturbation schemes have also been used by Hay [98] for the special case of static deformations and Nari- boli [99], Rogge [100], Bostr¨om [101], and Stephen et al. [102] for the special case of small deformations and linear constitutive equations.

An altogether diﬀerent approach is adopted by Ostrovskii & Sutin [50], Soerensen et al. [51], Clarkson et al. [52], Coleman & Newman [53], Cohen & Dai [54], and

Dai [55], wherein 1-D kinematics are posited at the outset and energy principles are subsequently used to deduce the 1-D equations of motion. Wright [56, 57] also employs energy methods, although he posits 2-D kinematics at the outset and reduces the equations of motion to 1-D through a cross-sectional averaging technique. 1-D rod theories have also been developed using the concepts of a Cosserat curve, convected coordinates, and deformable directors; refer, for instance, to [58–66].

23 The nonlinear rod theories discussed in the preceding paragraphs are formulated in the Lagrangian representation, with respect to the reference (undeformed) conﬁgu-

ration of the rod. To the best of our knowledge, rod theories formulated in the present

(deformed) conﬁguration,orEulerian representation, are presently unavailable in the

literature, with the exception of some related ideas set forth by the present authors

in [67, 68]. Hence, the aim of this chapter, as well as its main contribution, is the

development of an Eulerian rod theory for the ﬁrst time.

Starting with a fully 3-D Eulerian formulation of dynamic nonlinear elasticity, we

employ a perturbation scheme developed by Bechtel and coworkers [105, 106] that

exploits axisymmetry, slenderness of the structure, and low-frequency waves.7,8 The so-called elementary 1-D theory for a tapered, or variable-cross-section, rod is ex- tracted at leading order in the asymptotics. To leading order, (a) plane sections remain planar and (b) axial motion is accompanied by expansion and contraction of the circular cross section, consistent with the Poisson eﬀect. However, the lateral inertia and transverse shear accompanying this radial expansion and contraction are absent at leading order, instead emerging as higher-order, or weak, eﬀects.9

A key feature of our Eulerian rod theory is its amenability to numerical solution

via a high-resolution shock-capturing scheme on a ﬁxed computational mesh.This

7We refer to this as the long-wave/slender-structure assumption. 8In the development of theories for low-frequency waves in slender rods, it is customary – based on physical insight – to assume at the outset that (i) lateral inertia and transverse shear are negligible and (ii) plane cross sections remain planar and normal to the longitudinal axis. Perturbation schemes allow engineers and scientists to assess the validity and limitations of these (and other) simplifying assumptions by explicitly quantifying at what order in the asymptotics different physical effects emerge. 9As the long-wave/slender-structure assumption is relaxed, i.e., as the rod thickens and the frequency of the waves increases, lateral inertia becomes a leading-order, or dominant, effect; see, for instance, [107] and [108, pp. 77 and 120-121].

24 setting is ideal for capturing the evolution and interaction of nonlinear waves that

arise in important physical applications [30]. Another key feature of our rod theory

is its ability to be incorporated within a unified Eulerian computational architecture for fluid-structure interaction [33–40], wherein the motions of the solid and fluid phases are both tracked on a single fixed mesh. We further remark that mathematical models for simple geometries like rods provide excellent testbeds for benchmarking, validation, and verification. For instance, our Eulerian rod theory provides a simple solid-dynamics setting in which the performance of new finite-difference and finite- volume codes could be tested, representing a useful departure from customary fluid- dynamics-based benchmarking.

In this chapter, we numerically solve our Eulerian rod theory using the novel space-time Conservation Element and Solution Element (CESE) method. The CESE method is designed from the ground up to solve conservative, ﬁrst-order, hyperbolic systems of PDEs. (Refer to [88–91] for the pioneering CESE schemes, [92, 93] for Courant-number-insensitive CESE schemes, and [94–96] for higher-order CESE schemes. The interested reader is also encouraged to consult [94, 97] and the references therein for extensive reviews, comparisons, and assessments of the aforementioned CESE schemes.)

The CESE method diﬀers markedly from customary high-resolution schemes in

concept, structure, and methodology. For instance, standard high-resolution schemes

are based on a ﬁnite-volume method coupled with (i) a high-order reconstruction

procedure that approximates the values of discrete variables at cell boundaries and

(ii) a Riemann solver that uses these values to compute the numerical ﬂux at cell

25 boundaries. On the other hand, the CESE method, whose tenet is a uniﬁed treatment

of space and time,

(a) discretizes space and time using a ﬁxed, staggered, space-time mesh;

(b) enforces ﬂux conservation in both space and time;

(d) does not require a reconstruction procedure to interpolate or extrapolate the

numerical values of conserved variables at cell boundaries;

(e) does not require a physics-speciﬁc Riemann solver to compute the spatial gradi-

ents (numerical ﬂuxes) of conserved variables at cell boundaries.

Instead, conserved variables and their spatial gradients are calculated primitively as

part of the core CESE scheme. The absence of a Riemann solver and a reconstruction

step translates to more straightforward implementation, simpler logic, and lower op-

eration counts. Furthermore, unlike its counterparts that employ a Riemann solver,

the CESE method is physics independent. That is, the CESE method is able to solve

generic systems of ﬁrst-order hyperbolic PDEs, regardless of the physical setting and

without ad hoc customizations.

The remainder of this chapter is structured as follows: In Section 2.2, we present

the governing equations and boundary conditions in their fully 3-D form, then spe-

cialize to 2-D axisymmetric axial-radial motion. In Section 2.3, we use a slender-rod

perturbation scheme to formally obtain the 1-D elementary theory at leading order

in the asymptotics. In Section 2.4, a control-volume approach – inspired by ﬂuid

dynamics – is used as an alternative means to obtain the 1-D elementary theory. In

26 Section 2.5, the 1-D elementary theory is linearized to recover a classical model for

longitudinal waves in an ultrasonic horn (i.e., a tapered linearly elastic rod). In Sec-

tion 2.6, we assess the mathematical structure of the 1-D elementary theory and solve

it numerically using the CESE method. Several prototypical elastic wave propagation

problems are simulated to demonstrate the performance of our model and our code.

2.2 The 2-D mathematical model: Eulerian formulation

2.2.1 Description of the problem

A slender elastic rod undergoes longitudinal (axisymmetric axial-radial) motion.

It has a traction-free lateral surface, is composed of a homogeneous isotropic elastic material, and, for the sake of generality, has a tapered (axially varying) circular cross section. The rod is assumed to remain isothermal as it deforms so that thermal eﬀects can be neglected.

2.2.2 3-D governing equations

The position of a representative continuum particle in the rod’s present (deformed) conﬁguration is given by the cylindrical polar coordinates (r, θ, z), with r the radial coordinate, θ the circumferential coordinate, and z the axial coordinate. The z-axis is coincident with the longitudinal axis of symmetry of the rod. As our equations of motion, we adopt the 3-D Eulerian (present conﬁguration) statements of the fundamental laws of continuum mechanics, i.e.,

27 conservation of mass

∂ρ +(v · grad)ρ + ρ divv =0, (2.1a) ∂t balance of linear and angular momentum ∂v ρ +(v · grad) v =divT + ρg, T = TT, (2.1b) ∂t where ρ is the mass density in the present conﬁguration, v is the velocity, T is the

Cauchy stress, and g is the acceleration due to gravity. All ﬁelds are Eulerian, i.e.,

functions of present position (r, θ, z)andtimet. In Eqs. (2.1a) and (2.1b), “grad”

denotes the Eulerian gradient operator (i.e., the gradient calculated with respect

to the present conﬁguration), “div” denotes the Eulerian divergence operator, ∂/∂t

denotes partial diﬀerentiation with respect to time, ( ) · ( ) denotes an inner product,

and TT denotes the transpose of the Cauchy stress tensor T.

We characterize the isotropic elastic response of the material with a rate-type

constitutive equation for the evolution of the Cauchy stress [5,6,10,78–82]:

DT = λ (tr D) I +2μD, (2.2) Dt where I is the identity tensor, λ and μ are the dilatational and shear moduli, “tr” 1 denotes the trace of a tensor, D = L + LT is the rate of deformation tensor, 2 i.e., the symmetric part of the Eulerian velocity gradient L =gradv,andD/Dt is a suitably invariant (i.e., objective) rate of the Cauchy stress tensor. Of the numerous objective rates that have appeared in the literature, for the sake of simplicity, we

28 select the Gordon-Schowalter derivative [109]

D ∂ (·)= (·)+(v · grad) (·)+(·) W − W (·) − a (·) D + D (·) , (2.3) Dt ∂t

1 where W = L − LT is the vorticity tensor, i.e., the skew part of the Eulerian 2 velocity gradient, and the rate parameter a ∈ [−1, 1]. By setting a to the integer values 1, 0, and -1, Eq. (2.3) becomes the Oldroyd (upper convected) rate, Jaumann-

Zaremba (corotational) rate, and Cotter-Rivlin (lower convected) rate, respectively.10

Use of Eq. (2.3) in Eq. (2.2) yields our constitutive model:

∂T +(v · grad) T + TW − WT − a (TD + DT)=λ (tr D) I +2μD. (2.4) ∂t

The constitutive model (2.4) together with the fundamental laws (2.1a)-(2.1b) con-

stitute the 3-D equations of motion governing the nonlinear elastodynamics of the

rod. In Section 2.2.3 that follows, we complete our formulation of the 3-D model by

specifying the boundary conditions at the lateral surface; we defer speciﬁcation of the

the initial conditions, and the boundary conditions at the ends, until Section 2.6.4.

2.2.3 3-D boundary conditions at the lateral surface

The lateral surface of the rod is described by the equation [105,106,110]

f(r, z, t)=φ(z, t) − r =0, (2.5)

10The Oldroyd rate can be interpreted as the material time derivative of the Cauchy stress with respect to the body-ﬁxed basis gi ⊗ gj,wherei and j are free indices, () ⊗ () denotes the tensor product of two vectors, and g1, g2, g3 are the covariant basis vectors spanning Euclidean 3-space. Similarly, the Cotter-Rivlin rate can be interpreted as the material time derivative of the Cauchy stress with respect to the body-ﬁxed basis gi ⊗ gj ,whereg1, g2, g3 are the contravariant basis vectors spanning Euclidean 3-space.

29 where φ(z, t) is the radius of the rod, which varies with both axial position z and

time t. The boundary conditions at the lateral surface are [105,106,110] ∂f +(v · grad)f = 0 (2.6a) ∂t ∂ and

t = 0, (2.6b) ∂ where t = Tn is the traction, n is the outward unit normal, and ∂ denotes that a function is evaluated at the lateral surface. Elaborating, the kinematic boundary condition (2.6a) ensures that as the rod deforms, its lateral surface (as deﬁned in (2.5)) always consists of the same continuum particles. The kinetic boundary condition

(2.6b) speciﬁes a traction-free lateral surface.

2.2.4 Specialization of the model to 2-D

We hereafter specialize to axial-radial motion that is axisymmetric and torsionless

(also referred to herein as longitudinal or extensional motion). It follows, then, that the dependent variables ρ, v,andT can be expressed in cylindrical polar components as

ρ = ρ(r, z, t), (2.7a)

v = vr(r, z, t) er + vz(r, z, t) ez, (2.7b)

30 and

T = Trr(r, z, t) er ⊗ er + Trz(r, z, t) er ⊗ ez + ez ⊗ er

+ Tθθ(r, z, t) eθ ⊗ eθ + Tzz(r, z, t) ez ⊗ ez, (2.7c) wherewehaveusedsymmetryofT (as required for balance of angular momentum; see (2.1b)2)andpositedapriorithat

(i) vθ, Trθ,andTθz vanish identically, and

(ii) the remaining dependent variables (ρ, vr, vz, Trr, Trz, Tθθ,andTzz) are inde-

pendent of the circumferential coordinate θ.

Note that postulates (i) and (ii) do not violate the governing equations or boundary conditions. In Eqs. (2.7a)-(2.7c) and postulates (i) and (ii), () ⊗() denotes the tensor product of two vectors; er, eθ,andez are the unit vectors in the radial, circumferential, and axial directions; vr, vθ,andvz are the radial, circumferential, and axial components of velocity; and Trr, Tθθ, Tzz, Trz, Trθ,andTθz are the radial, circumferential, axial, axial-radial shear, radial-circumferential shear, and axial-circumferential shear components of the Cauchy stress.

The governing equations (2.1a), (2.1b), and (2.4) are written in cylindrical polar components and – by exploiting (2.7a)-(2.7c) – subsequently specialized to 2-D:

vr ρ,t + vr ρ,r + vz ρ,z + ρ vr,r + + vz,z =0, (2.8a) r

1 ρ (vr,t + vr vr,r + vz vr,z )=Trr,r + Trz,z + (Trr − Tθθ) , (2.8b) r

31 Trz ρ (vz,t + vr vz,r + vz vz,z)=Trz,r + Tzz,z + + ρg, (2.8c) r

Trr,t + vr Trr,r + vz Trr,z + Trz (vz,r − vr,z) − a 2Trr vr,r + Trz (vr,z + vz,r) vr = λ vr,r + + vz,z +2μvr,r , (2.8d) r

1 Trz,t + vr Trz,r + vz Trz,z + (Trr − Tzz)(vr,z − vz,r) − a Trz (vr,r + vz,z) 2 1 + (Trr + Tzz)(vr,z + vz,r) = μ (vr,z + vz,r), (2.8e) 2

2a vr vr Tθθ,t + vr Tθθ,r + vz Tθθ,z − vr Tθθ = λ vr,r + + vz,z +2μ , (2.8f) r r r

Tzz,t + vr Tzz,r + vz Tzz,z + Trz (vr,z − vz,r) − a 2Tzz vz,z + Trz (vr,z + vz,r) vr = λ vr,r + + vz,z +2μvz,z, (2.8g) r noting that the gravitational acceleration g is aligned with the positive ez axis, i.e., g = gez. Note that the circumferential component of balance of linear momentum is trivially satisﬁed, and thus does not appear in Eqs. (2.8a)-(2.8g). The compact notation ( ),t,(),r,(),z denotes partial diﬀerentiation with respect to time, the radial coordinate, and the axial coordinate, respectively. For instance,

def ∂vr(r, z, t) vr,t = . ∂t

32 The cylindrical polar components of the kinematic and kinetic boundary conditions (2.6a) and (2.6b), upon using (2.7a)-(2.7c) to specialize to 2-D, are [105]

− φ,t + φ,z vz r=φ vr r=φ = 0 (2.9a) and

− − Trr r=φ φ,z Trz r=φ =0,Trz r=φ φ,z Tzz r=φ =0, (2.9b)

respectively, where r=φ denotes that a function is evaluated at the lateral surface.

Note that the second terms in (2.9b)1 and (2.9b)2 are a consequence of the tapered geometry of the rod (and vanish for the special case of a straight rod).

2.3 Derivation of the leading-order 1-D model

2.3.1 The dimensionless 2-D model

To obtain the dimensionless version of the 2-D model developed in Section 2.2.4, we scale the independent radial, axial, and time variables as

r = r0 r,˜ z = z 0 z,˜ t = t0 t,˜ (2.10) wherer ˜,˜z, t˜ are the dimensionless independent variables. The characteristic radial dimension r0 is generally the radius or diameter of the rod; the characteristic axial dimension z 0 is generally a representative wavelength, or, in some instances, the length

of the rod; and the characteristic time t0 is generally a representative wave period.

33 It is convenient to deﬁne a positive nondimensional parameter

r0 = , (2.11) z 0

which can be interpreted as a slenderness ratio (if z 0 is taken to be the length of the rod) or a dimensionless wavenumber (if z 0 is taken to be a representative wavelength).

Slenderness of the rod and low-frequency waves ensure that is suﬃciently small [110];

werefertothisasthelong-wave/slender-structure approximation.

The dependent variables are then scaled as [105,106,110]

r0 z 0 z 0 ρ = ρ 0 ρ,˜ φ = r0 φ,˜ vr = v˜r = v˜r,vz = v˜z, t0 t0 t0

f0 ˜ f0 ˜ f0 ˜ f0 ˜ Trr = 2 Trr,Trz = 2 Trz,Tθθ = 2 Tθθ,Tzz = 2 Tzz, (2.12) r0 r0 r0 r0

where ρ 0 is the characteristic density, f0 is the characteristic force, and the dimensionless dependent variables are denoted by tildes.

Use of the scalings (2.10)-(2.12) in the 2-D ﬁeld equations (2.8a)-(2.8g) and 2-D lateral-surface boundary conditions (2.9a)-(2.9b) leads to v˜r ρ˜,t˜ +˜vr ρ˜,r˜ +˜vz ρ˜,z˜ +˜ρ v˜r,r˜ + +˜vz,z˜ =0, (2.13a) r˜

1 2 1 ρ˜ v˜r,t˜ +˜vr v˜r,r˜ +˜vz v˜r,z˜ = Lo T˜rr,r˜ + T˜rz,z˜ + T˜rr − T˜θθ , (2.13b) 2 r˜

T˜rz 1 ρ˜ v˜ ˜ +˜vr v˜z,r˜ +˜vz v˜z,z˜ =Lo T˜rz,r˜ + T˜zz,z˜ + + ρ,˜ (2.13c) z,t r˜ Fr

34 2 T˜rr,t˜ +˜vr T˜rr,r˜ +˜vz T˜rr,z˜ + T˜rz v˜z,r˜ − v˜r,z˜ − a 2T˜rr v˜r,r˜ 2 v˜r + T˜rz v˜r,z˜ +˜vz,r˜ =Λ v˜r,r˜ + +˜vz,z˜ +2M˜vr,r˜ , (2.13d) r˜

1 1 T˜ ˜ +˜vr T˜rz,r˜ +˜vz T˜rz,z˜ + (T˜rr − T˜zz) v˜r,z˜ − v˜z,r˜ − a T˜rz (˜vr,r˜ +˜vz,z˜) rz,t 2 2 1 1 1 + (T˜rr + T˜zz) v˜r,z˜ + v˜z,r˜ =M v˜r,z˜ + v˜z,r˜ , (2.13e) 2 2 2

2a v˜r v˜r T˜θθ,t˜ +˜vr T˜θθ,r˜ +˜vz T˜θθ,z˜ − v˜r T˜θθ =Λv˜r,r˜ + +˜vz,z˜ +2M , (2.13f) r˜ r˜ r˜

2 T˜zz,t˜ +˜vr T˜zz,r˜ +˜vz T˜zz,z˜ + T˜rz v˜r,z˜ − v˜z,r˜ − a 2T˜zz v˜z,z˜ 2 v˜r + T˜rz v˜r,z˜ +˜vz,r˜ =Λ v˜r,r˜ + +˜vz,z˜ +2M˜vz,z˜ , (2.13g) r˜

˜ ˜ − φ,t˜ + φ,z˜ v˜z r˜=φ˜ v˜r r˜=φ˜ =0, (2.13h)

˜ − 2 ˜ ˜ ˜ − ˜ ˜ Trr r˜=φ˜ φ,z˜ Trz r˜=φ˜ =0, Trz r˜=φ˜ φ,z˜ Tzz r˜=φ˜ =0. (2.13i)

Equations (2.13a)-(2.13i) constitute the dimensionless 2-D model. The nondimensional groups

2 2 2 f0 t0 z 0 λr0 μr0 Lo = 2 2 , Fr = 2 , Λ= , M= (2.14) ρ 0 r0 z 0 gt0 f0 f0 appearing in (2.13a)-(2.13i) are dimensionless combinations of material properties (λ,

μ); ambient conditions (g); and characteristic length (r0, z 0), time (t0), density (ρ 0), and force (f0) scales. In particular, we note that the Froude number Fr quantiﬁes the

35 relative importance of inertial eﬀects to gravitational eﬀects, the product of Lo and Λ

quantifies the relative importance of elastic dilatational effects to inertial effects, and

the product of Lo and M quantiﬁes the relative importance of elastic shear eﬀects to

inertial eﬀects.

2.3.2 Perturbation formalism

Following Bechtel and coworkers [105,106] (see also Benveniste [45]), our perturbation formalism employs the following power series expansions of the dimensionless primitive ﬁeld variables:

2n+m 2n+1 n,m v˜r r,˜ z,˜ t˜ = r˜ v˜r z,˜ t˜ n≥0 m≥0 0,0 0,1 =˜r v˜r z,˜ t˜ + r˜v˜r z,˜ t˜ 2 0,2 3 1,0 3 + r˜v˜r z,˜ t˜ +˜r v˜r z,˜ t˜ + O( ), (2.15a)

2n+m 2n n,m v˜z r,˜ z,˜ t˜ = r˜ v˜z z,˜ t˜ n≥0 m≥0 0,0 0,1 =˜vz z,˜ t˜ + v˜z z,˜ t˜ 2 0,2 2 1,0 3 + v˜z z,˜ t˜ +˜r v˜z z,˜ t˜ + O( ), (2.15b)

2n+m 2n n,m T˜rr r,˜ z,˜ t˜ = r˜ T˜rr z,˜ t˜ , (2.15c) n≥0 m≥0

2n+m 2n+1 n,m T˜rz r,˜ z,˜ t˜ = r˜ T˜rz z,˜ t˜ , (2.15d) n≥0 m≥0

36 2n+m 2n n,m T˜θθ r,˜ z,˜ t˜ = r˜ T˜θθ z,˜ t˜ , (2.15e) n≥0 m≥0

2n+m 2n n,m T˜zz r,˜ z,˜ t˜ = r˜ T˜zz z,˜ t˜ , (2.15f) n≥0 m≥0

φ˜ z,˜ t˜ = m φ˜(m) z,˜ t˜ , (2.15g) m≥0

ρ˜ r,˜ z,˜ t˜ = 2n+m r˜2n ρ˜ n,m z,˜ t˜ . (2.15h) n≥0 m≥0

The asymptotic expansions (2.15a)-(2.15h) are formal approximations to the exact

solutions of the dimensionless 2-D problem presented in Section 2.3.1. In (2.15a)-

(2.15h), is the small parameter deﬁned in (2.11), m and n are nonnegative integers,

n,m n,m n,m andv ˜r , T˜zz ,˜ρ , etc. are the dimensionless modal coeﬃcients. Details regarding the development and subtleties of these expansions can be found in [105, 106, 110].

However, before proceeding, we brieﬂy elaborate on several notable features of their construction:

(i) Each primitive dependent variable is expanded in both andr ˜, with the dimen-

n,m n,m n,m sionless modal coeﬃcientsv ˜r , T˜zz ,˜ρ , etc. functions ofz ˜ and t˜ alone. It

is thus tacit that the primitive dependent variables admit a “separation” of the

radial coordinate from the axial coordinate and time.

(ii) The expansions reﬂect two fundamental physical features of the problem: slen-

derness of the circular-cylindrical geometry and axisymmetry of the motion. For

instance, to ensure axisymmetry, some expansions (e.g., those forv ˜z and T˜rr)

37 are even inr ˜ at each order of , while others (e.g., those forv ˜r and T˜rz)areodd

inr ˜ at each order of .

In the physical applications that we model in this chapter (refer ahead to Section

2.6.4), the dimensionless groups Lo, Λ, and M are O(1), i.e., dominant, while 1/Fr

is typically O( 5) or higher, i.e., weak.11 Said differently, the geometry and material properties of the rods employed in our simulations dictate a regime where inertia and elasticity are dominant (leading-order effects) and therefore couple in our model, and gravity is weak (a higher-order effect). With these scalings of the dimensionless groups, and use of the power series expansions (2.15a)-(2.15h) in Eqs. (2.13a)-(2.13i), the 2-D problem decouples into a series of 1-D problems; refer to Appendix A.1. These

1-D problems are generated by collecting the coeﬃcients of the O(1), O( ), O( 2),

O( 2 r˜), O( 2 r˜2), etc. terms in Eqs. (A.1)-(A.10) in Appendix A.1.

The coeﬃcients of the O(1), or zeroth-order, terms constitute what we call the

leading-order problem. Our perturbation method ensures that, for a given regime

(i.e., balance of physical eﬀects), if the leading-order problem is closed (i.e., same

number of equations and unknowns), then closure also occurs at all higher orders in

the perturbation [106]. The solutions of each closed 1-D problem in the perturbation

hierarchy can be used to construct asymptotic solutions of the fully 2-D problem to

any desired order of accuracy.12

11The magnitudes of the dimensionless groups Lo, Fr, Λ, and M (refer to Eq. (2.14)) are expressed in powers of in order to specify at what order in the perturbation inertia, elasticity, and gravity ﬁrst alter the kinematics and kinetics of the rod. The relative ordering of Lo, Fr, Λ, and M constitutes a regime [105,106,110]. 12Explicitly, the 1-D solutions are used in the power series expansions (2.15a)-(2.15h) to build approximate solutions to the fully 2-D dimensionlesss ﬁelds, which are then converted back to their dimensional counterparts using (2.12).

38 In this chapter, our primary interest is in the leading-order behavior of the rod, which we investigate in the upcoming section. See [106, 110] for solutions of higher- order problems in a perturbation theory and the role of weak physical eﬀects.

2.3.3 The elementary 1-D theory (leading-order model)

The elementary 1-D theory, or leading-order model, is obtained by collecting the coeﬃcients of the O(1), or zeroth-order, terms in Eqs. (A.1)-(A.10) in Appendix A.1, which are13

0,0 0,0 0,0 0,0 0,0 ρ˜,t˜ +2˜ρ v˜r + ρ˜ v˜z ,z˜ =0, (2.16a)

0,0 0,0 T˜rr = T˜θθ , (2.16b)

0,0 0,0 0,0 0,0 ˜ 0,0 ˜ 0,0 ρ˜ v˜z,t˜ +˜vz v˜z,z˜ =LoTzz,z˜ +2Trz , (2.16c)

˜ 0,0 0,0 ˜ 0,0 − 0,0 ˜ 0,0 0,0 0,0 0,0 Trr,t˜ +˜vz Trr,z˜ 2a v˜r Trr =Λ2˜vr +˜vz,z˜ +2M˜vr , (2.16d)

˜ 0,0 0,0 ˜ 0,0 − ˜ 0,0 0,0 0,0 0,0 0,0 Tzz,t˜ +˜vz Tzz,z˜ 2a Tzz v˜z,z˜ =Λ2˜vr +˜vz,z˜ +2M˜vz,z˜ , (2.16e)

˜(0) 0,0 ˜(0) − 0,0 ˜(0) φ,t˜ +˜vz φ,z˜ v˜r φ =0, (2.16f)

0,0 T˜rr =0, (2.16g)

13Note that (A.5) and (A.6) do not contribute to the leading-order problem: (A.5) only enters at O( 2) and higher orders, while the leading-order term in (A.6) is formally identical to (2.16d) and, as a consequence of (2.16b), redundant.

39 (0) 0,0 0,0 (0) φ˜ T˜rz − T˜zz φ˜,z˜ =0. (2.16h)

Equations (2.16a)-(2.16h) are a closed system of eight independent equations in eight 0,0 ˜(0) 0,0 0,0 0,0 0,0 0,0 0,0 primitive unknowns (˜ρ , φ ,˜vr ,˜vz , T˜rr , T˜rz , T˜θθ , T˜zz ), all functions of dimensionless axial coordinatez ˜ and dimensionless time t˜.14 Thus, for the balance of physical eﬀects considered in this chapter (i.e., inertia and elasticity are dominant, and gravity is weak), we obtain closure at leading order. Upon examining the leading- order equations (2.16a)-(2.16h), the power series expansions (2.15a)-(2.15h), and the scalings (2.10)-(2.12), we infer that, to leading order:

(i) plane cross sections remain planar and normal to the longitudinal axis (the axial

velocity vz is independent of the radial coordinate r to leading order);

(ii) axial motion is accompanied by expansion and contraction of the circular cross

section, consistent with the Poisson eﬀect;

(iii) expansion and contraction of the circular cross section are not accompanied by

lateral inertia or transverse shear, which instead emerge as higher-order correc-

tions;

(iv) radial and circumferential stresses vanish, rendering the state of stress purely

uniaxial to leading order.

Equations (2.16b) and (2.16f)-(2.16h) are algebraic in the sense that they can be

0,0 0,0 0,0 0,0 solved for T˜θθ ,˜vr , T˜rr , T˜rz , respectively, in terms of the remaining unknowns,

14The leading-order problem (2.16a)-(2.16h) only contains modal variables with the superscripts (0,0) or (0), a consequence of the bookkeeping aﬀorded by the superscript (n, m) numbering scheme (recall (2.15a)-(2.15h)).

40 i.e.,

˜(0) 0,0 ˜ (0) ˜ 0,0 ˜(0) 0,0 φ,t˜ +˜vz φ,z˜ 0,0 0,0 0,0 Tzz φ,z˜ v˜r = , T˜rr = T˜θθ =0, T˜rz = . (2.17) φ˜ (0) φ˜(0)

Use of (2.17) in the leading-order 1-D statements of conservation of mass (2.16a), balance of axial momentum (2.16c), and the radial (2.16d) and axial (2.16e) components of the constitutive model, after some manipulation, leads to

0,0 (0) 0,0 0,0 (0) ρ˜ A˜ + ρ˜ v˜z A˜ =0, (2.18a) ,t˜ ,z˜

0,0 0,0 (0) 0,0 0,0 2 (0) 0,0 (0) ρ˜ v˜z A˜ + ρ˜ v˜z A˜ =LoT˜zz A˜ , (2.18b) ,t˜ ,z˜ ,z˜

(0) 0,0 (0) (0) 0,0 A˜ ˜ + v˜z A˜ =(1− 2ν) A˜ v˜z,z˜ , (2.18c) ,t ,z˜

0,0 0,0 0,0 0,0 0,0 T˜ ˜ + v˜z T˜zz = E +(1+2a) T˜zz v˜z,z˜ , (2.18d) zz,t ,z˜ wherewehaveused

2 2 A˜ z,˜ t˜ = π φ˜ z,˜ t˜ = π φ˜(0) + 2π φ˜ (0) φ˜ (1) + O( 2), A˜ (0) z,˜ t˜ A˜ (1) z,˜ t˜ a power series expansion for the dimensionless cross-sectional area. Note that

Λ λ M(2M + 3Λ) r2 ν = = , E = = 0 E (2.19) 2(Λ + M) 2(λ + μ) M+Λ f0

41 are Poisson’s ratio and dimensionless Young’s modulus. Using the scalings (2.10)-

(2.12) and (2.19), Eqs. (2.18a)-(2.18d) can be rewritten in dimensional form:

∂(ρA) ∂(ρvzA) + =0, (2.20a) ∂t ∂z

2 ∂(ρvzA) ∂(ρv A − TzzA) + z =0, (2.20b) ∂t ∂z

∂A ∂(vzA) ∂vz + =(1− 2ν)A , (2.20c) ∂t ∂z ∂z

∂Tzz ∂ (vzTzz) ∂vz + = E +(1+2a) Tzz . (2.20d) ∂t ∂z ∂z

Note that we have reverted to the conventional notation for partial derivatives and

omitted the superscripts denoting a zeroth-order modal variable (for the sake of clarity

and notational brevity).

Equations (2.20a)-(2.20d) – a closed system of four partial diﬀerential equations

in four primitive unknowns: density ρ(z, t), axial velocity vz(z, t), cross-sectional area A(z, t), and axial (Cauchy) stress Tzz(z, t) – constitute our leading-order 1-D model. Recall that z is the axial coordinate, t is time, ν is Poisson’s ratio, E is

Young’s modulus, and a is the rate parameter. Equations (2.20a) and (2.20b) are

1-D statements of the ﬁrst principles of conservation of mass and balance of axial

momentum, while Eqs. (2.20c) and (2.20d) that follow from the constitutive model

are 1-D evolution equations for the cross-sectional area and stress.15 To validate the elementary 1-D theory (2.20a)-(2.20d):

15Compare (2.20a)-(2.20b) with the quasi-1D Euler equations for inviscid ﬂuid ﬂow in a duct with area variation; see, e.g., [70, pp. 27-28].

42 (i) In Section 2.4, as an alternative to the approach presented in this section, we em-

ploy a control-volume analysis to obtain the 1-D ﬁrst principles (2.20a)-(2.20b).

(ii) In Section 2.5, we demonstrate that our leading-order 1-D model (2.20a)-(2.20d)

linearizes to a classical model for linear elastic waves in a tapered waveguide.

2.4 An alternative development of the elementary 1-D theory

The leading-order 1-D statements of conservation of mass (2.20a) and balance of axial momentum (2.20b) were derived from the fully 2-D problem (2.8a)-(2.9b) using a perturbation scheme. In this section, we present an alternative approach for developing (2.20a) and (2.20b), wherein mass and force balances are applied to a control volume occupied by a diﬀerential element of the rod. Motivated by slenderness, a free lateral boundary, and torsionless axisymmetry, we aprioriposit the following

1-D ﬁelds for the density, velocity, cross-sectional area, and Cauchy stress:

ρ = ρ(z, t), v = vz(z, t) ez,A= A(z, t), T = Tzz(z, t) ez ⊗ ez, where ( ) ⊗ ( ) denotes the tensor product of two vectors. These postulates are consistent with our scalings and perturbation formalism (refer to Section 2.3) in that the primitive variables ρ, vz, A, Tzz are independent of r to leading order, and vr, Trr,

Trz, Tθθ vanish at leading order.

We proceed by considering a control volume (CV) occupied at time t by a diﬀer- ential element (DE) of the tapered rod, as shown in Figure 2.1. The left and right axial control surfaces (CS) at z − dz/2andz + dz/2 (hereafter referred to as CS 1 and

CS 2, respectively), although axially stationary, vary in area with time t; refer ahead

43 Figure 2.1: A control volume (dashed lines) occupied at time t by a differential element of a tapered rod (solid lines). The axial control surfaces are fixed in space but undergo cross-sectional area variation. The top and bottom lateral control surfaces co-deform with the lateral boundaries of the differential element.

44 to Eqs. (2.21)3 and (2.22)3. The top and bottom lateral control surfaces (hereafter

referred to as CS 3 and CS 4) conform to this area variation and thus co-deform with the lateral boundaries of the DE; hence, there is no mass ﬂow through CS 3 and CS 4.

Using ﬁrst-order Taylor series expansions, we approximate the values of the primitive variables at the left axial control surface CS 1:

∂ρ dz ∂vz dz ρ = ρ − , v = vz − ez, z− dz z− dz 2 ∂z 2 2 ∂z 2 ∂A dz ∂Tzz dz A = A − , T = Tzz − ez ⊗ ez, z− dz z− dz 2 ∂z 2 2 ∂z 2 n = − ez, (2.21) z− dz 2 and the right axial control surface CS 2:

∂ρ dz ∂vz dz ρ = ρ + , v = vz + ez, z+ dz z+ dz 2 ∂z 2 2 ∂z 2 ∂A dz ∂Tzz dz A = A + , T = Tzz + ez ⊗ ez, z+ dz z+ dz 2 ∂z 2 2 ∂z 2 n = ez. (2.22) z+ dz 2

Recall that n is the outward unit normal. Equations (2.21) and (2.22) imply that

∂vz dz ∂vz dz (v · n) = −vz + , (v · n) = vz + , (2.23) z− dz z+ dz 2 ∂z 2 2 ∂z 2 while use of Cauchy’s stress theorem t = Tn and the relationship (a ⊗ b)c =(b · c)a gives the traction on the axial control surfaces:

∂Tzz dz ∂Tzz dz t = −Tzz + ez, t = Tzz + ez. (2.24) z− dz z+ dz 2 ∂z 2 2 ∂z 2

45 2.4.1 Conservation of mass

At the outset, we employ ∂ ρdv + ρ (w · n) da =0, (2.25) ∂t CV CS an integral statement of conservation of mass for a deforming control volume [111, pp.

210-212], where w is the velocity of the rod relative to the control surface. In what follows, we apply Eq. (2.25) term by term to the deforming control volume shown in

Figure 2.1.

For the ﬁrst term in Eq. (2.25), the volume integral is approximated using the midpoint rule, i.e.,

∂ ∂ ∂(ρA) ρdv = ρA dz = dz. (2.26) ∂t ∂t z ∂t CV

For the second term in Eq. (2.25), we exploit stationary axial control surfaces, no mass ﬂux through the lateral control surfaces, and no dependence of the density or velocity on the cross-sectional coordinates (r, θ) to deduce

∂A dz ρ (w · n) da = ρ(v · n) A + z+ dz 2 ∂z 2 CS ∂A dz + ρ(v · n) A − . (2.27) z− dz 2 ∂z 2

Subsequent use of Eqs. (2.21)-(2.23) in Eq. (2.27) leads to ∂(ρvzA) 1 ∂ρ ∂vz ∂A ρ (w · n) da = dz + dz3 . (2.28) ∂z 4 ∂z ∂z ∂z CS

46 Substituting results (2.26) and (2.28) into Eq. (2.25), dividing by dz, and jettisoning the higher-order term (i.e., letting dz → 0) gives

∂(ρA) ∂(ρvzA) + = 0 (2.29) ∂t ∂z as our 1-D statement of conservation of mass. As expected, (2.29) is formally identical to (2.20a).

2.4.2 Balance of linear momentum

At the outset, we employ ∂ ρv dv + ρv (w · n) da = t da, ∂t CV CS CS an integral statement of balance of linear momentum for a deforming control volume

in the absence of body forces [111, p. 225], of which only the axial component ∂ ρvz dv + ρvz (w · n) da = tz da (2.30) ∂t CV CS CS is nontrivial. In what follows, each term in Eq. (2.30) is applied individually to the deforming control volume in Figure 2.1.

For the ﬁrst term in Eq. (2.30), the volume integral is approximated using the midpoint rule, i.e.,

∂ ∂ ∂(ρvzA) ρvz dv = ρvzAdz = dz. (2.31) ∂t ∂t z ∂t CV

For the second term in Eq. (2.30), we exploit stationary axial control surfaces, no mass ﬂux through the lateral control surfaces, and no dependence of the density or

47 velocity on the cross-sectional coordinates to deduce

∂A dz ρvz (w · n) da = ρvz(v · n) A + z+ dz 2 ∂z 2 CS ∂A dz + ρvz(v · n) A − . z− dz 2 ∂z 2

Use of Eqs. (2.21)-(2.23) in the preceding expression leads to 2 2 ∂(ρvz A) 1 ∂ρ ∂vz ρvz (w · n) da = dz + A ∂z 4 ∂z ∂z CS 2 ∂vz ∂A ∂ρ ∂vz ∂A 3 + ρ +2vz dz . (2.32) ∂z ∂z ∂z ∂z ∂z

Finally, for the third term in Eq. (2.30), we exploit traction-free lateral control surfaces and no dependence of the traction on the cross-sectional coordinates to arrive at

∂A dz ∂A dz tz da = tz A + + tz A − , z+ dz z− dz 2 ∂z 2 2 ∂z 2 CS which, upon use of Eq. (2.24), becomes ∂(TzzA) tz da = dz. (2.33) ∂z CS

Combining results (2.31)-(2.33) in Eq. (2.30), dividing by dz, and jettisoning the higher-order terms (i.e., letting dz → 0) gives

2 ∂(ρvzA) ∂(ρv A) ∂(TzzA) + z = (2.34) ∂t ∂z ∂z

48 as our 1-D statement of balance of axial momentum. As expected, (2.34) is formally

identical to (2.20b).

2.5 Linearization of the elementary 1-D theory

In this section, we establish contact with a classical theory for tapered elastic

waveguides by linearizing our leading-order 1-D model (2.20a)-(2.20d) presented in

Section 2.3.3. We proceed by expanding each of the primitive variables ρ(z, t), vz(z, t),

A(z, t), and Tzz(z, t) into a power series in terms of the small parameter = r0/z 0

(0) (0) (0) (0) (refer to Eq. (2.11)) around the solutions ρ (z, t), vz (z, t), A (z, t), and Tzz (z, t):

ρ(z, t)=ρ (0)(z, t)+ ρ(1)(z, t)+ 2ρ (2)(z, t)+O( 3),

(0) (1) 2 (2) 3 vz(z, t)=vz (z, t)+ vz (z, t)+ vz (z, t)+O( ),

A(z, t)=A(0)(z, t)+ A(1)(z, t)+ 2A(2)(z, t)+O( 3),

(0) (1) 2 (2) 3 Tzz(z, t)=Tzz (z, t)+ Tzz (z, t)+ Tzz (z, t)+O( ). (2.35)

(1) (1) (1) (1) In (2.35), ρ (z, t), vz (z, t), A (z, t), and Tzz (z, t) are the ﬁrst-order corrections of the density, axial velocity, cross-sectional area, and axial stress, respectively, and

(2) (2) (2) (2) ρ (z, t), vz (z, t), A (z, t), and Tzz (z, t) are the second-order corrections.

The solutions we choose to perturb about are those corresponding to the undeformed, stress-free reference conﬁguration of the rod, i.e.,

(0) (0) (0) (0) ρ = ρR,vz =0,A= AR (z),Tzz =0,

49 where AR(z) is the axially varying (tapered) cross-sectional area of the rod in its undeformed reference conﬁguration, and ρR is the uniform reference density. Note that AR(z)andρR are speciﬁed quantities. It follows that

(1) 2 (2) 3 ρ(z, t)=ρR + ρ (z, t)+ ρ (z, t)+O( ),

(1) 2 (2) 3 vz(z, t)= vz (z, t)+ vz (z, t)+O( ),

(1) 2 (2) 3 A(z, t)=AR (z)+ A (z, t)+ A (z, t)+O( ),

(1) 2 (2) 3 Tzz(z, t)= Tzz (z, t)+ Tzz (z, t)+O( ). (2.36)

Substituting the power series expansions (2.36) into the nonlinear 1-D model

(2.20a)-(2.20d), then collecting the coeﬃcients of the leading-order (i.e., O( )) terms, leads to the linearized 1-D problem:

∂ρ ∂vz +(1− 2ν)ρR =0, (2.37a) ∂t ∂z

∂vz 1 ∂Tzz Tzz dAR − = , (2.37b) ∂t ρR ∂z ρR AR dz

∂A ∂vz dAR +2νAR = − vz , (2.37c) ∂t ∂z dz

∂Tzz ∂vz − E =0, (2.37d) ∂t ∂z where d/dz denotes the derivative of a function of a single variable z, and the superscripts denoting ﬁrst-order corrections have been omitted for notational brevity.

Equations (2.37a)-(2.37d) are linear 1-D statements of conservation of mass, balance

50 of axial momentum, and the radial and axial components of the constitutive model.

Equations (2.37a) and (2.37c) are novel, as density and cross-sectional area are cus-

tomarily assumed to remain constant in linear rod theories [108].

Integrating Eq. (2.37d) with respect to time, discarding the arbitrary function of

z that arises during integration,16 and combining the resulting expression with Eq.

(2.37b) leads to the following displacement equation of motion:

2 2 1 ∂ uz 1 dAR ∂uz ∂ uz 2 2 = + 2 , (2.38) c ∂t AR dz ∂z ∂z where uz(z, t) is the axial displacement, c = E/ρR is the bar velocity, i.e., the propagation speed of longitudinal waves in a slender elastic rod with a uniform cross section, and

∂uz vz = . ∂t

Though developed from entirely diﬀerent perspectives, our linearized equation of motion (2.38) is formally identical to the classical linear model for longitudinal waves in a horn (i.e., tapered elastic waveguide); refer, for instance, to Webster [112], Donnell

[113], Eisner [114], Rogge [100], and Graﬀ [108, pp. 108-111]. Tapered waveguides are often employed in ultrasonics applications to amplify kinematic quantities such as displacement and velocity.

Before proceeding, we comment on several special cases of (2.37a)-(2.37d) and verify their consistency with physical observations:

16In order for the rod to be stress free when it is undeformed, the arbitrary function of z that arises during integration must vanish.

51 (i) If the rod is straight in its undeformed reference conﬁguration, then dAR/dz =0,

the source terms on the right-hand side of (2.37a)-(2.37d) vanish, and the equa-

tion of motion (2.38) reverts to the classical second-order linear wave equation,

as expected.

(ii) If, in addition to dAR/dz = 0, we enforce incompressibility (i.e., ν =0.5), then

(2.37a) demands that the density remains constant, as expected.

(iii) If, in addition to dAR/dz = 0, we enforce no lateral expansion or contraction of

the rod (i.e., ν = 0), then (2.37c) demands that the cross-sectional area remains

constant, as expected.

Additional analysis on the linear system (2.37a)-(2.37d), including its mathematical structure and typical analytical methods, can be found in Appendix A.2.

2.6 Numerical implementation of the elementary 1-D theory

2.6.1 Mathematical structure of the model

Returning to the nonlinear theory, the elementary 1-D model (2.20a)-(2.20d) is rewritten as follows:

∂ρ ∂ρ ∂vz + vz +(1− 2ν)ρ =0, (2.39a) ∂t ∂z ∂z

∂vz ∂vz Tzz ∂A 1 ∂Tzz + vz − − =0, (2.39b) ∂t ∂z ρA ∂z ρ ∂z

∂A ∂A ∂vz + vz +2νA =0, (2.39c) ∂t ∂z ∂z

52 ∂Tzz ∂Tzz ∂vz + vz − (E +2aTzz) =0. (2.39d) ∂t ∂z ∂z

Equations (2.39a)-(2.39d) are then expressed compactly as a homogeneous, ﬁrst-order, quasi-linear system:

∂Uˆ ∂Uˆ + Aˆ = 0, (2.40a) ∂t ∂z

where the matrix of primitive variables Uˆ and the Jacobian matrix Aˆ are ⎡ ⎤ ⎛ ⎞ vz f1 00 ⎢ ⎥ ρ ⎢ ⎥ ⎜ ⎟ ⎢ Tzz 1 ⎥ ⎜ ⎟ ⎢ − − ⎥ ⎜ vz ⎟ ⎢ 0 vz ⎥ Uˆ = ⎜ ⎟ , Aˆ = ⎢ ρA ρ ⎥ , (2.40b) ⎝ A ⎠ ⎢ ⎥ ⎢ 02νA vz 0 ⎥ Tzz ⎣ ⎦ 0 −f2 0 vz with

f1 =(1− 2ν)ρ, f2 = E +2aTzz.

The quasi-linear form (2.40a)-(2.40b) is the form that is used to determine the underlying mathematical structure of Eqs. (2.20a)-(2.20d) [115]. The eigenvalues of

Aˆ are

ξˆ1,2 = vz, ξˆ3,4 = vz ± Δ , (2.41a)

53 and the corresponding eigenvectors are ⎛ ⎞ ⎛ ⎞ 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ qˆ1 = ⎜ 1 ⎟ , qˆ2 = ⎜ 1 ⎟ , ⎝ ⎠ ⎝ ⎠ −Tzz −Tzz A A ⎛ ⎞ ⎛ ⎞ 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ Δ ⎟ ⎜ Δ ⎟ ⎜ ⎟ ⎜ − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ f1 ⎟ ⎜ f1 ⎟ ⎜ ⎟ ⎜ ⎟ qˆ3 = ⎜ 2νA ⎟ , qˆ4 = ⎜ 2νA ⎟ , (2.41b) ⎜ ⎟ ⎜ ⎟ ⎜ f1 ⎟ ⎜ f1 ⎟ ⎝ ⎠ ⎝ ⎠ −f2 −f2 f1 f1 where ( E +2(a − ν)Tzz Δ= . ρ

Note that the ﬁrst term in Δ is formally identical to the bar velocity c in the linear theory (refer to Eq. (2.38)), while the second term includes contributions from the

Poisson eﬀect, stress convection, and compressibility.

In the elastic regime (i.e., prior to yield), the argument of Δ is greater than zero for the materials of interest in this chapter. It follows that the eigenvalues are real. Real eigenvalues and linearly independent eigenvectors imply that our model is hyperbolic [70]. In the context of a ﬁrst-order hyperbolic system, the eigenvalues of

Aˆ represent the characteristic speeds, while the eigenvectors are used to transform between the primitive variables and the characteristic variables. The interested reader is referred to [6, 70, 115–118] for additional details on ﬁrst-order hyperbolic systems and the mathematical structure of quasi-linear PDEs.

54 2.6.2 Conservative form

Equations (2.20a)-(2.20d) are rewritten and expressed compactly as

∂U ∂F + = S, (2.42a) ∂t ∂z where the matrices of unknowns U, ﬂuxes F, and sources S are ⎛ ⎞ ⎛ ⎞ ρA ρvzA ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 − ⎟ ⎜ ρvzA ⎟ ⎜ ρvz A TzzA ⎟ [ U ]=⎜ ⎟ , [ F ]=⎜ ⎟ , ⎝ A ⎠ ⎝ vzA ⎠

Tzz vz(Tzz − E)

⎛ ⎞ 0 ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ [ S ]=⎜ ∂vz ⎟ . (2.42b) ⎜ − ⎟ ⎜ (1 2ν)A ⎟ ⎝ ∂z ⎠ ∂vz (1 + 2a) Tzz ∂z

Equations (2.42a)-(2.42b) are referred to as the conservative form with a source.

This is the form of the governing equations that will be solved numerically by the

Conservation Element and Solution Element method in Section 2.6.4, provided Eqs.

(2.42a)-(2.42b) have a suitable eigenstructure.

Proceeding, Eq. (2.42a) is rewritten as

∂U ∂U + A = S, (2.43a) ∂t ∂z

55 where ⎡ ⎤ 0100 ⎢ ⎥ ⎢ 2 ⎥ ⎢ −v 2vz −Tzz −A ⎥ ⎢ z ⎥ ∂F ⎢ ⎥ ⎢ ⎥ [ A ]= = ⎢ vz 1 ⎥ (2.43b) ∂U ⎢ − vz 0 ⎥ ⎢ ρ ρ ⎥ ⎣ ⎦ (Tzz − E) vz Tzz − E − 0 vz ρA ρA is the Jacobian of the ﬂux F.17 The eigenvalues of A are

ξ1,2 = vz,ξ3,4 = vz ± δ, where ( E − 2Tzz δ = . ρ

In the elastic regime (i.e., prior to yield), the argument of δ is greater than zero for the materials of interest in this chapter. Hence, the eigenvalues are real. The

17We note that (2.43a)-(2.43b) is not quasi-linear since the source S contains partial derivatives of the unknowns [115]; cf. (2.40a)-(2.40b), which is quasi-linear.

56 corresponding eigenvectors are ⎛ ⎞ ⎛ ⎞ 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ vz ⎟ ⎜ vz ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ [q1 ]=⎜ ⎟ , [q2 ]=⎜ 1 ⎟ , ⎝ 0 ⎠ ⎝ ⎠ 0 −Tzz A ⎛ ⎞ ⎛ ⎞ 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ vz + δ ⎟ ⎜ vz − δ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ [q3 ]=⎜ ⎟ , [q4 ]=⎜ ⎟ . ⎜ ρ ⎟ ⎜ ρ ⎟ ⎝ ⎠ ⎝ ⎠ Tzz − E Tzz − E ρA ρA

Since the eigenvectors of A are linearly independent and the eigenvalues are real, the conservative form (2.42a)-(2.42b) can be solved numerically by the CESE method.

2.6.3 The CESE method

The conservative form (2.42a)-(2.42b) is solved numerically using the space-time

Conservation Element and Solution Element (CESE) method; refer to Section 1.2.4 of Chapter 1 for a general overview. Recall that a distinguishing feature of the CESE method is space-time ﬂux conservation. Numerous explicit time-marching schemes follow from this core concept of space-time ﬂux conservation, e.g., the a scheme, the a- scheme, the c scheme, and the c-τ scheme; refer to [94,97] for additional details.

In this chapter, we employ the second-order-accurate a- scheme [88]. Its numerical stability is ensured provided the Courant-Friedrichs-Lewy (CFL) number is less than 1. The W-2 weighting function [93] is incorporated into the numerical framework to suppress oscillations near solution discontinuities. The source terms in (2.42a)-(2.42b) are integrated using the approach discussed in [119, 120], while

57 the non-reﬂecting boundary conditions are treated using similar techniques to those

presented in [90,121].

2.6.4 Benchmark problems

In this section, we numerically simulate a series of elastic wave propagation problems. Our goal is to elicit the major capabilities of our model and test the performance of our numerical method. The problems we have selected incorporate evolutionary wave physics, unsteadiness, advection, compressibility, strong and weak rarefactions, and strong and weak shocks. Some problems are designed to replicate physical systems, while others are more conceptual. Numerical solutions of the nonlinear model are compared with analytical solutions of the linearized model (refer to Appendix

A.2).

Low-speed and high-speed impact

At time t = 0, a semi-inﬁnite rod rigidly translating longitudinally rightward

with speed Vo (the striker bar) collinearly impacts a stationary semi-inﬁnite rod (the

incident bar). Hence, the initial velocity proﬁle is ) Vo,zL where z = L is the impact interface. We take L = 5 cm. Both rods are composed of

7075-T651 aluminum alloy (Young’s modulus E =71.7 GPa, Poisson’s ratio ν =0.33,

3 and initial density ρR = 2810 kg/m ), have identical initial diameters dR =0.5in, and are considered semi-inﬁnite so that we may disregard wave reﬂection at the ends.

58 After impact, for t>0, the rods remain attached; we enforce the customary ‘stick’ contact conditions at the interface, i.e., continuity of axial velocity and axial stress.

Figure 2.2 illustrates the density, velocity, cross-sectional area, and stress proﬁles t =3μs after a low-speed impact of Vo =20m/s.Twoweak shocks (or shock- like waves) are clearly discernible, one traveling leftward into the striker bar and the other traveling rightward into the incident bar. Both shocks propagate com- paction (density increase), lateral expansion (area increase), and compression (negative stresses); the leftward-traveling shock propagates deceleration in the striker bar, while the rightward-traveling shock propagates acceleration in the incident bar. The stresses remain well within the elastic regime, comfortably below the yield strength of the material.

Good agreement is observed between the analytical and numerical solutions in

Figure 2.2, including accurate capture of the wave speeds and wave amplitudes, as expected at low speeds (i.e., in the linear regime). Sharp wavefronts are crisply resolved, with minimal numerical dissipation (smearing) and numerical dispersion

(spurious oscillations). One notable disparity, however, is what appears to be ‘ar- tiﬁcial’ undershoot and overshoot near the impact interface in Figures 2.2(a) and

2.2(c), respectively. Similar behavior has been observed in previous impact simulations [10, 11, 14], and are explained by Barton [14] as being analogous to ‘heating errors’ in gas dynamics. However, our subsequent simulation of a high-speed impact provides an alternative perspective.

Figure 2.3 illustrates the density, velocity, cross-sectional area, and stress proﬁles t =4.8 μs after a high-speed impact of Vo = 6000 m/s. (Although more conceptual in

59 2812 20

) 15 3 (m/s) z

(kg/m 2811 ρ 10

Density 5 Axial velocity v

2810 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (a) (b)

−4 x 10 1.269 0 ) 2 1.2685

−50 (MPa)

1.268 zz

1.2675 −100

1.267 Axial stress T Cross−sectional area A (m

1.2665 −150 0 0.05 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (c) (d)

Figure 2.2: Low-speed (20 m/s) impact problem. Snapshot t =3μs after impact illustrating the analytical (solid) and numerical (circles) wave proﬁles for (a) density, (b) velocity, (c) area, and (d) stress. Both the striker bar (initially moving at 20 m/s) and the incident bar (initially stationary) are composed of 7075-T651 aluminum alloy. The impact interface is located at z = 5 cm. Two weak shocks travel away from the impact interface, both propagating compression and lateral expansion in their wake.

60 3400 6000

3300 5000 ) 3

3200 (m/s) 4000 z (kg/m

ρ 3100 3000

3000 2000 Density Axial velocity v 2900 1000

2800 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (a) (b)

−4 4 x 10 x 10 2 0 ) 2

1.8 −1 (MPa) zz −2 1.6

−3 1.4 Axial stress T

Cross−sectional area A (m −4 1.2 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (c) (d)

Figure 2.3: High-speed (6000 m/s) impact problem. Snapshot t =4.8 μs after impact illustrating the analytical (solid) and numerical (circles) wave profiles for (a) density, (b) velocity, (c) area, and (d) stress. Both the striker bar (initially moving at 6000 m/s) and the incident bar (initially stationary) are composed of 7075-T651 aluminum alloy. The impact interface is located at z = 5 cm. Two strong shocks travel away from the impact interface, both propagating compression and lateral expansion in their wake. A contact wave (or contact discontinuity) is also clearly discernible in the density, area, and stress profiles. Substantial discrepancies in wave amplitudes and wave speeds are observed, owing to the effects of nonlinearity.

61 nature than the low-speed impact,18 this simulation is quite useful for (i) revealing a clear picture of the underlying wave structure and (ii) eliciting interesting features of nonlinearity.) A contact wave (or contact discontinuity ) is clearly discernible in addition to two strong shocks. This wave structure is consistent with the four eigenvalues

(wave speeds) ( E − 2Tzz ξ1,2 = vz,ξ3,4 = vz ± δ, δ = (2.44) ρ of the conservative system of PDEs solved by the CESE method; refer to Section

2.6.2. One shock (corresponding to the eigenvalue vz − δ) travels leftward into the striker bar, while the other shock (corresponding to the eigenvalue vz + δ)andthe contact wave (corresponding to the repeated eigenvalue vz) both travel rightward into the incident bar. Based on this wave structure, which is more discernible in the high-speed simulations (cf. Figures 2.2 and 2.3), we feel that the apparent ‘artiﬁcial’ undershoot and overshoot in the low-speed simulations (Figures 2.2(a) and 2.2(c)) are actually trivial (nearly stationary) contact waves near the impact interface.

Substantial discrepancies in wave amplitudes and wave speeds are observed in

Figure 2.3, owing to the effects of nonlinearity. The differences in the latter are clarified by comparing the eigenvalues (wave speeds) of the linearized system (refer to Appendix A.2.1) ( E ξ1,2 =0,ξ3,4 = ± (2.45) ρR

18An impact of this magnitude would cause the material to yield and thus, strictly speaking, exceed the capabilities of our purely elastic model. That said, the practice of neglecting a material’s yield strength to elicit interesting nonlinear physics is common in computational ﬁnite elasticity [3,10,11,13,14].

62 to the eigenvalues (2.44) of the conservative nonlinear system in both the low-speed and high-speed scenarios.

In the previous simulations, we employed the Jaumann-Zaremba rate by setting a = 0 in the 1-D constitutive equation (2.20d); refer also to Eq. (2.3). Figures

2.4 and 2.5, on the other hand, compare the physical predictions of the Jaumann rate to two other representative rates, namely the Oldroyd rate (a =1)andthe

Cotter-Rivlin rate (a = −1); refer again to Eq. (2.3). It is well known that different objective rates yield different physical predictions [81,122,123], and this trend holds true in the high-speed simulation (Figure 2.5), where the effects of nonlinearity are significant. The variation in wave amplitude from one rate to another in Figure 2.5 is expected since the rate parameter a appears as a source term in (2.42a)-(2.42b). This amplitude variation, in turn, impacts the magnitude of the eigenvalues (2.44), i.e., the propagation speed of the shocks and contact wave. Conversely, in our low-speed simulation – where the effects of nonlinearity are insignificant – excellent agreement is observed in wave amplitudes and waves speeds among all of the rates; refer to Figure

2.4.

In all of the preceding simulations, a uniform grid is employed over the ﬁnite computational domain 0

63 20 0 Analytical Jaumann 15 Oldroyd Cotter−Rivlin

(m/s) −50 (MPa) z zz 10

−100 5 Axial velocity v Axial stress T

0 −150 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (a) (b)

Figure 2.4: A comparison of the numerical predictions of three diﬀerent objective rates – Jaumann-Zaremba (circles), Oldroyd (squares), and Cotter-Rivlin (diamonds) – with an analytical solution (solid line) of a low-speed (20 m/s) impact. As expected, good agreement is observed at low speeds (linear regime).

4 x 10 6000 0

5000 −1

(m/s) 4000 −2 (MPa) z zz 3000 −3

2000 Analytical −4 Jaumann Axial velocity v 1000 Oldroyd Axial stress T −5 Cotter−Rivlin

0 −6 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (a) (b)

Figure 2.5: A comparison of the numerical predictions of three different objective rates – Jaumann-Zaremba (circles), Oldroyd (squares), and Cotter-Rivlin (diamonds) – with an analytical solution (solid line) of a high-speed (6000 m/s) impact. As expected, different rates yield substantially different physical predictions, and all deviate appreciably from the analytical solution, owing to the effects of nonlinearity.

64 Low-speed and high-speed separation

At time t = 0, an event occurs at location z = L along a previously stress-free

inﬁnite rod, causing the regions on both sides of z = L to separate. Hence, the initial

velocity proﬁle is ) −Vo,zL

We take L = 5 cm. The rod is composed of UNS C15720 copper alloy (Young’s modulus E = 113 GPa, Poisson’s ratio ν =0.35, and initial density ρR = 8810

3 kg/m ), has an initial diameter dR =0.75 in, and is regarded as inﬁnite so that we may disregard wave reﬂection at its ends. For t>0, we enforce continuity of axial velocity and axial stress at z = L. We note that symmetric separation is a challenging test problem that often causes advanced numerical methods to fail [11].

Figure 2.6 illustrates the density, velocity, cross-sectional area, and stress pro-

ﬁles t =4μs after a low-speed separation (Vo =10m/s).Twoweak rarefactions

(or rarefaction-like waves), one traveling leftward and the other traveling rightward, propagate a density decrease in their wake. Both rarefactions also propagate lateral contraction and tensile stresses. The leftward-traveling rarefaction accelerates the region left of z = 5 cm, while the rightward-traveling rarefaction decelerates the region right of z = 5 cm. Good agreement is observed between the analytical and numerical solutions. As with the impact problem, overshoot and undershoot in the density and area proﬁles accompany the (trivial) contact wave near the impact interface. The stresses remain well within the elastic regime, comfortably below the yield strength of the material.

65 8812 10

8810

) 5 3 (m/s) 8808 z (kg/m

ρ 0 8806

Density −5 8804 Axial velocity v

8802 −10 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (a) (b)

−4 x 10 2.851 350 ) 2 2.85 300 2.849 250

2.848 (MPa) zz 200 2.847 150 2.846 100 2.845 Axial stress T 50

Cross−sectional area A (m 2.844 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (c) (d)

Figure 2.6: Low-speed separation problem (Vo = 10 m/s). Snapshot at t =4μs illustrating the analytical (solid) and numerical (circles) wave proﬁles for (a) density, (b) velocity, (c) area, and (d) stress. The rod is composed of UNS C15720 copper alloy. Two weak rarefactions travel away from the separation interface at z =5cm, both propagating decompression, tension, and lateral contraction in their wake.

66 9000 1000

8800

) 500 3 (m/s) 8600 z (kg/m

ρ 0 8400

Density −500 8200 Axial velocity v

8000 −1000 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (a) (b)

−4 4 x 10 x 10 3 4 ) 2

2.8 3 (MPa) zz 2.6 2

2.4 1 Axial stress T Cross−sectional area A (m

2.2 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Axial position z (m) Axial position z (m) (c) (d)

Figure 2.7: High-speed separation problem (Vo = 1000 m/s). Snapshot at t =6μs illustrating the analytical (solid) and numerical (circles) wave proﬁles for (a) density, (b) velocity, (c) area, and (d) stress. The rod is composed of UNS C15720 copper alloy. Two strong rarefactions travel away from the separation interface at z =5cm, both propagating decompression, tension, and lateral contraction in their wake.

67 Figure 2.7 illustrates the density, velocity, cross-sectional area, and stress proﬁles t =6μs after a high-speed separation (Vo = 1000 m/s). Much of the same behavior observed in the low-impact situation is mirrored here, with the notable exception that the expansion waves are now strong rarefactions. This is evidenced by a softening of the steep ‘jumps’ in Figure 2.6 to more gradual variations in Figure 2.7.

In all of the preceding simulations, a uniform grid is employed over the ﬁnite computational domain 0

Zaremba (corotational) rate is chosen by setting a = 0 in the constitutive equation

(2.20d); refer also to Eq. (2.3). In the low-speed simulations, the time step is Δt =

0.02 μs, and the CFL number is about 0.72. In the high-speed simulations, the time step is Δt =0.02 μs, and the CFL number is about 0.92.

68 Chapter 3: Modal and Characteristics-Based Approaches for Modeling Elastic Waves Induced by Time-Dependent Boundary Conditions

Much of the work in this chapter was presented in [124]: R.L. Lowe, S.-T.J. Yu,

L. Yang, and S.E. Bechtel, “Modal and characteristics-based approaches for modeling

elastic waves induced by time-dependent boundary conditions,” Journal of Sound and

Vibration 333(3), 873-886, 2014.

In this chapter, we systematically develop a characteristics-based approach for solving elastic wave problems with time-dependent traction boundary conditions. A generalized mathematical model for this important class of problems is expressed as a set of first-order, linear, hyperbolic partial differential equations. We analyze the mathematical structure of this first-order linear system, verify its hyperbolicity, derive its characteristic form, and deduce its eigenvalues, eigenvectors, and Riemann invariants. The eigenvalues correspond to the wave speeds, while the Riemann invariants are used to construct a solution by the method of characteristics.

We benchmark the method of characteristics against several popular modal approaches. Two of these, which we refer to as the concentrated body force method

69 (CBFM) and the homogeneous eigenfunction expansion method (HEEM), were developed to simplify the well-established but tedious Mindlin-Goodman method. To homogenize the boundary conditions and enable modal analysis, the CBFM and HEEM

forgo the usual formalism of linear transformations (`a la Mindlin-Goodman) in favor

of intuitive modeling assumptions and postulated solution structures. We ﬁnd, how-

ever, that these approaches introduce an artiﬁcial stress discontinuity at the forced

boundary in their reformulated problems. When these reformulated problems are

solved by modal analysis, spurious oscillations and signiﬁcant overshoot, similar to

the Gibbs phenomenon, emerge in the stress proﬁle at the artiﬁcial discontinuity.

We demonstrate that these oscillations and overshoot are physical manifestations of

a series solution for stress, obtained from term-by-term diﬀerentiation, that is not

uniformly convergent, as required by the formalism of mathematical analysis.

The method of characteristics solution, on the other hand, is exact to within

machine precision, yielding no artiﬁcial discontinuities, spurious oscillations, or un-

physical overshoot. Unlike the modal approaches, the method of characteristics solves the ﬁrst-order problem with time-dependent boundary conditions ‘as is’ without any reformulation, restructuring, or postulated solution structures. Further, its solutions require no post-processing: no a posteriori solution treatment like l’Hˆopital’s rule

to accommodate resonance (resonant behavior is inherently captured without the

emergence of singularities), no term-by-term diﬀerentiation to deduce stress (stress

is primitive in a ﬁrst-order velocity-stress formulation of elastodynamics), and no convergence tests.

70 3.1 Introduction

The study of elastic waves in structural elements such as rods, beams, and plates remains an important and active area of research in engineering, physics, and applied mathematics. Modern applications include structural dynamics, high-power ultrasonics, acoustic metamaterials, damage and defect detection, seismology, and high-strain- rate material characterization, to name a few. Of particular importance in many of these applications are linear propagating waves induced by time-dependent traction boundary conditions. The mathematical models that characterize these systems are hyperbolic and admit traveling wave solutions.

Analytical methods are often used to obtain closed-form solutions of these models

to enable design, optimization, and other predictive capabilities. For instance, inte-

gral transforms have been employed [125–127], although in some cases the transforms

do not exist or their inverses are unknown [128,129]. Modal analysis (or mode super-

position) based on eigenfunction expansion has enjoyed more widespread use in the

vibrations community, in part because of its ease of implementation. Additionally, its

solution structure inherently reveals rich physics (e.g., wavenumbers, natural frequen-

cies, and mode shapes) important to the design, analysis, control, and optimization

of dynamic systems. However, modal analysis breaks down when applied directly to

problems with time-dependent boundary conditions [128, 130–132]. To circumvent

this diﬃculty and enable separation of variables, the original problem with time-

dependent boundary conditions is reformulated to render the boundary conditions

stationary.

71 Arguably the most widely used technique for reformulating this important class of vibration problems is the so-called Mindlin-Goodman method [128]. The Mindlin-

Goodman method uses a linear transformation of the primitive dependent variable to shift the time-dependent inhomogeneity from the boundary to the governing equation and initial conditions, yielding a reformulated problem with homogeneous boundary conditions in a new dependent variable. (This basic idea was presented earlier, e.g.,

[133], but saw application to vibration problems for the ﬁrst time in [128].) This approach has been applied to Euler-Bernoulli beams [128, 134], Timoshenko beams

[135, 136], rods [137], plates [138, 139], and general elastic bodies [130], to name a few, and is routinely covered in textbooks on vibrations [131] and partial diﬀerential equations [140]. The shifting function in the linear transformation, originally proposed to be a polynomial19 in [128], may assume other forms, provided it satisﬁes the time-

dependent boundary conditions and adheres to certain continuity requirements. This

ﬂexibility has led to the development of numerous successors, e.g., [141–145].

A simpler, more intuitive approach for shifting the time-dependent inhomogene-

ity from the boundary is presented in [146, pp. 443-446], which we hereafter call

the concentrated body force method (CBFM). In lieu of a linear transformation, the

CBFM homogenizes the boundary conditions by modeling the time-dependent ex-

citation as a concentrated body force in the governing equation. Accordingly, the

reformulated problem is amenable to separation of variables. Another approach for

solving initial-boundary value problems with time-dependent boundary conditions is

presented in [140, pp. 140-143], which we hereafter call the homogeneous eigenfunction

19In particular, a polynomial one order lower than the number of boundary conditions (or, equivalently, one order lower than the order of the highest spatial derivative) so that it can be determined by the boundary conditions alone.

72 expansion method (HEEM). In this approach, the primitive dependent variable and

its partial derivatives are expanded in the eigenfunctions of the associated homoge-

neous problem. Demanding that these eigenfunction expansions satisfy the governing

equation leads to an ordinary diﬀerential equation in the modal coordinates of the

primitive dependent variable, from which the general mode superposition solution

follows.

In this chapter, we compare the Mindlin-Goodman, concentrated body force, and

homogeneous eigenfunction expansion methods. In the context of a generalized elas-

todynamic problem with time-dependent boundary conditions, we investigate how

these techniques reformulate the original problem to permit separation of variables,

assess their performance, and identify their limitations. Our primary contributions

are assessing the mathematical equivalency of the reformulated problems, identifying

limitations on the term-by-term diﬀerentiability of their series solutions, examining

their convergence characteristics, and explicitly showing, through a numerical ex-

ample, the practical physical implications of convergence issues. We also assess the ability of these series solutions to satisfy the governing equation (in the classical sense) and the boundary conditions.

Another key contribution of this chapter is our presentation of the method of characteristics as a compelling alternative to modal analysis for solving elastodynamic problems with time-dependent boundary conditions. Despite several successful applications in vibrations and wave propagation (see, for example, [147–151]), the method of characteristics continues to receive limited attention in the solid mechanics community. In this chapter, we demonstrate its proﬁciency at solving elastic wave problems with time-dependent boundary conditions by explicitly illustrating its advantages and

73 limitations relative to more traditional approaches. We also show that formulating

the mathematical model as ﬁrst-order hyperbolic system (i.e., a hyperbolic system of

ﬁrst-order partial diﬀerential equations) reveals compelling wave physics not readily

ascertained from associated higher-order formulations.

The remainder of this chapter is structured as follows: In Section 3.2, we present

a generalized model for elastic waves induced by time-dependent traction boundary

conditions. This model is ﬂexible enough to capture a large class of propagating waves.

In Section 3.3, we present mode superposition solutions of the generalized model using the Mindlin-Goodman, concentrated body force, and homogeneous eigenfunction expansion methods. We then show, through a numerical example, that several of these methods exhibit spurious oscillations near the forced boundary. We identify the origins of this behavior and discuss strategies for mitigating it. Lastly, in Section

3.4, we present a ﬁrst-order hyperbolic formulation of the generalized problem, deduce its Riemann invariants, and construct a solution using the method of characteristics.

Through a numerical example, we explicitly show the appeal of this approach for this particular class of problems, then discuss its limitations. Appendix B provides supplementary details for many of the derivations presented in this chapter.

3.2 Illustrative elastodynamic problem

We proceed by considering the following generalized 1-D model for small-amplitude, non-dispersive, linear elastic waves generated by time-dependent traction boundary conditions:

1 ∂2η ∂2η = , (3.1a) c2 ∂t2 ∂ξ2

74 ∂η ∂η η(0,t)=0,a = f(t),η(ξ,0) = 0, =0. (3.1b) ∂ξ ξ=L ∂t t=0

In (3.1a)-(3.1b), the generalized displacement η(ξ,t) is a function of the generalized spatial coordinate ξ and time t; the generalized phase velocity c and the generalized stiﬀness a are both constants. The homogeneous, isotropic, linearly elastic propagation medium is ﬁxed at ξ = 0, subjected to a time-dependent driving force f(t)at

ξ = L, and, for the sake of simplicity, taken to be initially undeformed and static.

The second-order linear wave equation (3.1a), a prototypical hyperbolic partial differential equation (PDE), arises frequently in science and engineering, and models a large class of dispersionless forced vibration and elastic wave propagation problems. Different choices of the generalized quantities in problem (3.1a)-(3.1b) lead to various special cases, e.g., transverse motion in taut strings, longitudinal waves in slender rods, torsional waves in circular shafts, longitudinal waves in the through- thickness direction of a thin semi-infinite plate, and axisymmetric torsional waves in thin cylindrical shells [108,152]. Refer also to Section 2.5. For instance, by setting

ξ ≡ x, η ≡ u, c ≡ E/ρ, a ≡ EA, f(t) ≡ F sin(Ωt) (3.2)

in (3.1a)-(3.1b), we recover a model for harmonically induced longitudinal vibrations

in a slender elastic rod, where u(x, t) is the axial displacement, x is the axial coor-

dinate, E is Young’s modulus, ρ is the density, A is the cross-sectional area, L is the length, F is the amplitude of the periodic driving force, and Ω is the driving frequency.

For the sake of generality, we ﬁrst present our solutions to (3.1a)-(3.1b) in the upcoming sections in terms of the generalized quantities η, ξ, a, etc. In order to obtain

75 Table 3.1: Material properties, geometric parameters, and boundary conditions Parameter Symbol Magnitude Units Young’s modulusa E 200 GPa Densitya ρ 7870 kg· m−3 Length L 2m Cross-sectional area A 25 cm2 Amplitude of driving force F 15 kN Cyclic frequency of driving forceb Ω9.45 kHz

aAISI 1020 low-carbon steel b The angular frequency of the driving force is taken to be Ω = ω8,the eighth natural frequency of the rod. This value is then expressed as a cyclic frequency.

numerical data for plots, we then specialize our results to the particular physical

example (3.2) with the parameters shown in Table 3.1. Note that the rod is driven

at a resonant frequency, a physical situation of interest in engineering practice, e.g.,

ultrasonic horns [108].

3.3 Solution of the problem by modal analysis

Separation of variables fails when applied directly to problems like (3.1a)-(3.1b) with time-dependent boundary conditions. To obviate this incompatibility and enable mode superposition, the original problem is reformulated to render the boundary conditions homogeneous. In what follows, we present three diﬀerent approaches for reformulating the problem. For each approach, we examine the mathematical equivalency of the reformulated problem, the convergence characteristics and term-by-term diﬀerentiability of its series solutions, the ability of these series solutions to satisfy

76 the governing equation (in the classical sense) and the boundary conditions, and, in

the context of a numerical example, the physical implications of convergence issues.

3.3.1 The concentrated body force method (CBFM)

Following the CBFM as described in [146, pp. 443-446], we reformulate the original problem (3.1a)-(3.1b) by modeling the time-dependent traction boundary condition as a concentrated body force in the governing equation (refer to Appendix B.1 for details):

1 ∂2η ∂2η − = g(ξ,t), (3.3a) c2 ∂t2 ∂ξ2

∂η ∂η η(0,t)=0, =0,η(ξ,0) = 0, =0, (3.3b) ∂ξ ξ=L ∂t t=0 where

1 g(ξ,t)= f(t) δ(ξ − L) (3.3c) a and δ(ξ − L) is the spatial Dirac delta function. The reformulated problem (3.3a)-

(3.3c), obtained via a modeling assumption rather than a linear transformation, is approximate, i.e., not mathematically equivalent to the original problem. The solution of (3.3a)-(3.3c) follows from standard mode superposition methods for inhomogeneous

PDEs:

∞ η(ξ,t)= ηm(t) Xm(ξ), (3.4a) m=1

77 where

t 2 2c m+1 ηm(t)= (−1) sin [ωm(t − τ)]f(τ)dτ (3.4b) aLωm 0

are the modal coordinates,

(2m − 1)π αm = ,Xm(ξ)=sin(αm ξ),ωm = cαm (3.4c) 2L are the eigenvalues (wavenumbers), eigenfunctions (mode shapes), and natural frequencies, respectively, and the positive integer m corresponds to the m-th harmonic

(vibrational mode).

As we show in Appendix B.4, although the series solution (3.4a)-(3.4c) for the generalized displacement is uniformly convergent, its term-by-term differentiated proge- nies are not (they converge in a weaker sense). From the perspective of mathematical analysis (see, for example, [153]), term-by-term differentiation of a series is formally valid if and only if the resulting term-by-term differentiated series is uniformly convergent. Further, (3.4a)-(3.4c) does not satisfy the governing PDE (3.1a) in the classical sense (the twice term-by-term differentiated series does not converge uniformly; see

Appendix B.4) and is unable to capture the time-dependent traction boundary condition (3.1b)2. To illustrate the physical implications of these convergence issues, we

(i) specialize result (3.4a)-(3.4c) to the particular physical example (3.2) with the parameters provided in Table 3.1, (ii) differentiate the displacement term by term with respect to the axial coordinate to deduce the axial stress (see Appendix B.2 for details), and (iii) exhibit axial stress profiles in the rod at two different snapshots of time (see Figs. 3.1 and 3.2).

78 7 (b)7 (c) 10 6.5 6.5 (a) 6 6 5 5.5 5.5 0 1.96 1.98 2 1.96 1.98 2

-5 7 (d) 7 (e) Axial stress [MPa] Axial stress -10 6.5 6.5 0 0.5 1 1.5 2 Axial position [m] 6 6 5.5 5.5

1.96 1.98 2 1.96 1.98 2

Figure 3.1: (a) A snapshot at t = 240 μs illustrating a propagating stress wave generated at the forced boundary traveling leftward prior to reﬂection. (b)-(e) Zoomed regions near the forced boundary, indicated by the box in (a). Solutions were obtained via the concentrated body force method and the homogeneous eigenfunction expansion method (solid lines), the Mindlin-Goodman method (dashed lines), and the method of characteristics (squares). The mode superposition solutions were truncated at (b) 500 modes, (c) 1000 modes, (d) 2500 modes, and (e) 5000 modes.

As illustrated in Figs. 3.1(b)-(e) and 3.2(b)-(e), the CBFM renders unphysical predictions in the region neighboring the forced boundary. The origins of this behavior are rooted in the CBFM’s reformulation of the problem. By modeling the time- dependent force f(t) as a concentrated body force (delta function) in the governing equation and homogenizing the traction boundary condition (cf. (3.1a)-(3.1b) and

(3.3a)-(3.3b)), an artificial stress discontinuity is introduced at the forced end in the reformulated problem. The mode superposition solution of the reformulated problem, in turn, attempts to capture this artificial jump, resulting in spurious oscillations and significant overshoot in the stress profile near the discontinuity, analogous to the

79 −5 −5 −5.5 −5.5 10 −6 −6 5 −6.5 −6.5 (b) −7 −7 (c) 0 1.96 1.98 2 1.96 1.98 2

-5 −5 −5 (a) Axial stress [MPa] Axial stress −5.5 −5.5 -10 0 0.5 1 1.5 2 −6 −6 Axial position [m] −6.5 −6.5 (d) (e) −7 −7 1.96 1.98 2 1.96 1.98 2

Figure 3.2: (a) A snapshot at t =4.1 ms illustrating the stress proﬁle resulting from the interference of leftward-propagating incident waves from the forced end with reﬂected waves traveling inward from the boundaries. (b)-(e) Zoomed regions near the forced boundary, indicated by the box in (a). Solutions were obtained via the concentrated body force method and the homogeneous eigenfunction expansion method (solid lines), the Mindlin-Goodman method (dashed lines), and the method of characteristics (squares). The mode superposition solutions were truncated at (b) 500 modes, (c) 1000 modes, (d) 2500 modes, and (e) 5000 modes. Note that the stresses are still well within the elastic regime for this particular material.

Gibbs phenomenon. The Gibbs phenomenon is a prototypical example of non-uniform convergence.

As shown in Figs. 3.1(b)-(e) and 3.2(b)-(e), the number of modes retained in the series solution for stress has a strong eﬀect on the predicted behavior near the forced boundary. In particular, truncating at a higher number of modes generally results in

(i) diminished oscillation amplitudes, (ii) increased oscillation frequencies, and (iii) propagation of the maximum overshoot outward toward the boundary.20 This is not

20No change in the magnitude of the maximum overshoot (about 9 percent) is observed with a change in the number of modes.

80 surprising: higher modes have higher frequencies, which are useful for approximating

sharp gradients like the artiﬁcial discontinuity being captured here. As the number

of modes tends to inﬁnity, the oscillations continue to diminish, and the maximum

overshoot approaches the artiﬁcial jump at the boundary. Thus, if the concentrated

body force method is employed judiciously (e.g., if a suﬃcient number of modes

are retained before truncation), the oscillations and overshoot can be conﬁned to a

suﬃciently small region near the boundary, at least for engineering purposes.

3.3.2 The homogeneous eigenfunction expansion method (HEEM)

Following the HEEM as described in [140, pp. 140-143], we expand the primitive dependent variable and its partial derivatives as follows:

∞ ∞ ∂2η(ξ,t) η(ξ,t)= ηm(t) Xm(ξ), 2 = ym(t) Xm(ξ), m=1 ∂t m=1

∞ ∂2η(ξ,t) 2 = zm(t) Xm(ξ), (3.5) ∂ξ m=1 where ηm(t), ym(t), and zm(t) are the modal coordinates, and

(2m − 1)π αm = ,Xm(ξ)=sin(αm ξ) 2L are the eigenvalues and eigenfunctions of the homogeneous problem associated with

(3.1a)-(3.1b). It follows from orthogonality of the eigenfunctions that

L L 2 2 ∂2η(ξ,t) ηm(t)= η(ξ,t) sin(αm ξ)dξ, ym(t)= sin(αm ξ)dξ, L L ∂t2 0 0

81 L 2 ∂2η(ξ,t) zm(t)= sin(αm ξ)dξ. (3.6) L ∂ξ2 0

Demanding that expansions (3.5)2 and (3.5)3 satisfy the governing equation (3.1a) for

2 all integers m gives an expression relating the modal coordinates, i.e., ym(t)=c zm(t).

Successive use of (3.6), integration by parts, and the boundary conditions (3.1b)1,2

leads to an ordinary diﬀerential equation in the modal coordinate ηm(t), that is, 2 2 d ηm 2 2c − m+1 dηm 2 + ωm ηm = ( 1) f(t),ηm(0) = 0, =0, (3.7) dt aL dt t=0 where ωm = cαm.

Remarkably, the solution of (3.7) is (3.4b), so that its use in the eigenfunction expansion (3.5)1 leads to (3.4a)-(3.4c), the same general solution as the CBFM. This conﬂuence of the CBFM and HEEM solutions stems from the following: The HEEM’s solution construction (3.5)1 incorrectly predicts a stress-free boundary at the forced end and is thus inherently incompatible with the time-dependent boundary condition

(3.1b)2. As an artifact of this inconsistent solution construction, an artiﬁcial discon-

tinuity arises at the forced boundary. Recall that the same artiﬁcial discontinuity

arose using the CBFM, but was instead due to the modeling approximation used to

reformulate the problem.

3.3.3 The Mindlin-Goodman method

The methods presented in Sections 3.3.1 and 3.3.2 were developed to simplify the more established, but more laborious, Mindlin-Goodman method [128]. The discussion and analysis that follow are inspired by [143,144].

82 The Mindlin-Goodman method uses the linear transformation η(ξ,t)=v(ξ,t)+ w(ξ,t) to transform (3.1a)-(3.1b) into a mathematically equivalent problem with homogeneous boundary conditions in a new dependent variable w. Apart from diﬀer- entiability requirements and satisfying the boundary conditions ∂v v(0,t)=0,a = f(t), ∂ξ ξ=L the shifting function v is arbitrary. The most popular choice in the literature is

ξ v(ξ,t)= f(t), a which is the solution of the quasi-static boundary value problem 2 ∂ v ∂v 2 =0,v(0,t)=0,a = f(t) ∂ξ ∂ξ ξ=L associated with (3.1a)-(3.1b). (The solution of the quasi-static problem represents the steady-state spatial distribution of η after transient eﬀects have subsided.) As discussed in [154], this special case of the Mindlin-Goodman method makes contact with the Williams or mode-acceleration method [155,156].21 For this choice of v,the transformed problem in w becomes

1 ∂2w ∂2w − = h(ξ,t), (3.8a) c2 ∂t2 ∂ξ2

∂w ∂w w(0,t)=0, =0,w(ξ,0) = φ(ξ), = ψ(ξ), (3.8b) ∂ξ ξ=L ∂t t=0

21The Williams method gives the solution of the dynamic problem as an eigenfunction expansion about the quasi-static or steady-state solution. It has been shown to decrease the number of modes to convergence in many situations. Successful applications include those illustrated in [129, 157].

83 where 2 − ξ d f − ξ − ξ df h(ξ,t)= 2 2 ,φ(ξ)= f(0),ψ(ξ)= (3.8c) c a dt a a dt t=0

Note that f(t) must be twice diﬀerentiable. A comparison of (3.1a)-(3.1b) and (3.8a)-

(3.8c) reveals that the Mindlin-Goodman transformation shifts the inhomogeneity from the boundary to both the governing equation (via the ‘forcing function’ h(ξ,t)) and the initial conditions (via the ‘initial displacement’ φ(ξ) and the ‘initial velocity’

ψ(ξ)).

The separation of variables solution of the reformulated problem (3.8a)-(3.8c) is

∞ ξ η(ξ,t)= f(t)+ wm(t) Xm(ξ), (3.9a) a m=1 where

t 2 ψm c wm(t)= φm cos(ωmt)+ sin(ωmt)+ sin [ωm(t − τ)]hm(τ)dτ (3.9b) ωm ωm 0

are the modal coordinates,

(2m − 1)π αm = ,Xm(ξ)=sin(αm ξ),ωm = cαm (3.9c) 2L are the eigenvalues (wavenumbers), eigenfunctions (mode shapes), and natural frequencies, respectively, and the positive integer m corresponds to the m-th harmonic

(vibrational mode). In (3.9b), φm, ψm,andhm(t) are the Fourier coeﬃcients of φ(ξ),

ψ(ξ), and h(ξ,t) (refer to (3.8c)) over 0 ≤ ξ ≤ L with respect to the basis functions * + sin(αm ξ),m=1, 2, 3, ... .

84 In Appendix B.4, we show that the series solution (3.9a)-(3.9c) for the generalized displacement is uniformly convergent. Upon specializing to the physical example

(3.2), we ﬁnd that the axial stress, obtained from term-by-term diﬀerentiation of the axial displacement, is also uniformly convergent (see Appendices B.3 and B.4,

Figs. 3.1 and 3.2). However, in the more general case given by (3.9a)-(3.9c), term- by-term diﬀerentiation of the generalized displacement η(ξ,t) does not always result in a uniformly convergent series. (From the perspective of mathematical analysis

[153], uniform convergence is required for term-by-term diﬀerentiation to be a valid operation.) To resolve this issue, and, moreover, to ensure that the series solution for displacement satisﬁes the governing PDE in the classical sense, we direct the reader to several recent papers by Wu [144, 145].

3.4 Solution of the problem by the method of characteristics

In this section, we depart from modal analysis and systematically develop a characteristics-based approach for solving elastic wave problems with time-dependent boundary conditions. Accordingly, the second-order hyperbolic PDE (3.1a) is rewritten as a pair of ﬁrst-order PDEs, i.e.,

∂α ∂β ∂β ∂α − c2 =0, − =0, (3.10a) ∂t ∂ξ ∂t ∂ξ and the initial and boundary conditions (3.1b) are re-expressed as

α(0,t)=0,aβ(L, t)=f(t),α(ξ,0) = 0,β(ξ,0) = 0, (3.10b)

85 where

∂η ∂η α = ,β= . (3.11) ∂t ∂ξ

In this ﬁrst-order formulation of the problem, α(ξ,t)andβ(ξ,t) are the primitive dependent variables. In what follows, we use linear algebra to explore the mathematical structure of the ﬁrst-order linear system (3.10a) and verify its hyperbolicity. We then derive its characteristic form and deduce its Riemann invariants, which are required to construct a solution via the method of characteristics.

3.4.1 Mathematical structure of the ﬁrst-order linear system

Before we proceed, it is convenient to express the ﬁrst-order linear system (3.10a)

in matrix form, i.e.,

∂U ∂U + A = 0, (3.12) ∂t ∂ξ where 2 α 0 −c U = , A = . (3.13) β −10

The eigenvalues and corresponding eigenvectors of A are −c c λ1 = c, λ2 = −c, m1 = , m2 = . 1 1

Since the eigenvalues are real and distinct, and the eigenvectors are linearly independent, the ﬁrst-order linear system (3.12) is strictly hyperbolic. In a hyperbolic

86 system, the eigenvalues correspond to the wave speeds. We deﬁne c 0 −cc Λ = , M = , 0 −c 11 where the eigenvalues of A populate the diagonal elements of Λ, and the corresponding eigenvectors populate the columns of M. For later use, we compute ⎛ ⎞ 1 ⎜ − 1 ⎟ −1 1 ⎜ c ⎟ M = ⎝ ⎠ . (3.14) 2 1 1 c

The ﬁrst-order linear system (3.12) is then algebraically manipulated, i.e.,

∂U ∂U M−1 + M−1AM M−1 = 0, ∂t ∂ξ to diagonalize A and recover the characteristic form

∂U˜ ∂U˜ + Λ = 0, (3.15) ∂t ∂ξ

where

U˜ = M−1 U, Λ = M−1AM. (3.16)

−1 Note that (3.16)1 is a consequence of M being independent of ξ and t. It follows from (3.13)1, (3.14), and (3.16)1 that ⎛ ⎞ α(ξ,t) ⎜ − + β(ξ,t) ⎟ u˜1(ξ,t) 1 ⎜ c ⎟ U˜ = = ⎝ ⎠ , (3.17) u˜2(ξ,t) 2 α(ξ,t) + β(ξ,t) c

87 whereu ˜1 andu ˜2 are the characteristic variables. Whenu ˜1 andu ˜2 are written explicitly in terms of the primitive dependent variables α and β (see the right-hand side of

(3.17)), they are referred to as the Riemann invariants.

The matrix expression (3.15) represents a decoupled pair of linear advection equations, i.e.,

∂u˜1 ∂u˜1 ∂u˜2 ∂u˜2 + c =0, − c =0. (3.18) ∂t ∂ξ ∂t ∂ξ

From (3.18), it follows thatu ˜1 andu ˜2 are constant along families of characteristics with slope dξ/dt = c and dξ/dt = −c, respectively, where ±c are the characteristic

speeds, i.e., the rate at which information advects along a particular characteristic.22

3.4.2 Implementation of the method of characteristics

In what follows, we illustrate how to construct a solution of the initial-boundary value problem (3.10a)-(3.10b) using the method of characteristics. The ξ − t space- time domain is discretized using the characteristic mesh shown in Fig. 3.3. As the characteristics emanating from t = 0 advance in time, pairs of them intersect, ﬁrst at time t =Δt. As shown in Fig. 3.3, at t =Δt,theN − 1 points of intersection

all lie on the interior of the ξ − t domain. To calculate the primitive variables α(ξ,t)

and β(ξ,t) at any of these interior nodes, we solve the algebraic equations (3.17)1 and

(3.17)2 for α and β. For example, at node (Δξ/2, Δt), we have

α(Δξ/2, Δt)=c [˜u2(Δξ,0) − u˜1(0, 0)] ,

β(Δξ/2, Δt)=˜u1(0, 0) +u ˜2(Δξ,0). (3.19)

22Note that the slopes ±c of the characteristics are constant so that they are lines, not curves. This is true in general for linear systems of PDEs.

88 Note that the Riemann invariantsu ˜1 (constant as it advects along the right-running characteristic with slope dξ/dt = c emanating from (0, 0); see Fig. 3.3) andu ˜2

(constant as it advects along the left-running characteristic with slope dξ/dt = −c

emanating from (Δξ,0); see Fig. 3.3) are evaluated at the previous time step. The numeric values ofu ˜1(0, 0) andu ˜2(Δξ,0) are obtained by using the initial conditions

(3.10b)3 and (3.10b)4 in (3.17).

Advancing in time, left-running and right-running characteristics emerge from the N − 1 interior nodes at time t =Δt. As shown in Fig. 3.3, at time t =2Δt,a left-running characteristic intersects the left boundary, a right-running characteristic intersects the right boundary, and pairs of characteristics intersect at the N−2 interior nodes. At the left boundary node (0, 2Δt), we use boundary condition (3.10b)1 in

Eq. (3.17)2 to deduce

α(0, 2Δt)=0,β(0, 2Δt)=2˜u2(Δξ/2, Δt)=2˜u2(Δξ,0).

Similarly, at the right boundary node (L, 2Δt), we use boundary condition (3.10b)2

in Eq. (3.17)1 to deduce

c α(L, 2Δt)= f (2Δt) − 2c u˜1((2N − 3)Δξ/2, Δt) a c = f (2Δt) − 2c u˜1((N − 2)Δξ,0), a

1 β(L, 2Δt)= f (2Δt). a

At the N − 2 interior nodes, we calculate α and β using a procedure similar to that used to obtain (3.19). As we continue to march forward in time beyond t =2Δt,the

89 methodology described above is repeated. Hence, for a large number of time steps, a

computational algorithm is the most eﬃcient means of implementation.

Unlike the finite-difference method, the finite-volume method, and the finite- element method, the method of characteristics is not a numerical method, per se, but rather a semi-analytical approach. To be more precise, the method of characteristics solves a pair of algebraic equations (refer to Eq. (3.17)) at each of the space-time nodes in the characteristic mesh (refer to Fig. 3.3). These algebraic equations are exact, as they are obtained from characteristic analysis rather than approximations such as interpolation or finite differences (via Taylor series expansions). Their solutions are exact to within machine precision (i.e., computer round-off error incurred during floating point arithmetic); however, as these solutions are only available at each of the space-time nodes in the characteristic mesh (again refer to Fig. 3.3), they are not closed form. The method of characteristics is inherently free of truncation error, numerical diffusion, and numerical dispersion, and numerical stability is not a concern. The magnitude of the nodal spacing Δξ and the time step Δt have no bearing on the accuracy of the solution at a particular space-time node; in fact, Δξ and Δt only affect the solution density (i.e., the number of locations in space-time where solutions are available).23

23We note that this is true for linear systems of PDEs only, where the characteristics are straight lines in x-t space, and their points of intersection (where solutions are obtained) can be deduced explicitly apriori. However, for nonlinear systems, where the characteristics are curves in x-t space (the slope of a characteristic at a given point in x-t space depends on the values of the solution variables at that particular point), approximations enter the picture, and Δξ and Δt ultimately play an important role in the accuracy of the solution.

90 2ǻt

t ǻt

Time ǻt

i= 1 i= N 0 L Generalized coordinate ȟ

Figure 3.3: The characteristic mesh for two time steps, illustrated for the special case N = 4. Along the line t = 0 spanning 0 ≤ ξ ≤ L, we introduce N nodes uniformly spaced Δξ apart. A right-running characteristic emerges from the left boundary node (i =1), a left-running characteristic emerges from the right boundary node (i = N), and both left-running and right-running characteristics emerge from the interior nodes. As these characteristics advance in time, pairs of them intersect, ﬁrst at time t =Δt,thenatt =2Δt, and so on, with the time step Δt dictated by the nodal spacing Δξ and the constant slope ±c of the characteristics. Arrows denote the direction that information propagates along the characteristic lines, and ﬁlled circles designate nodes.

3.4.3 Results and discussion

We now specialize our results to the particular physical example (3.2) with the parameters listed in Table 3.1; for our nodal spacing, we select Δx = 5 mm, which

in turn gives a time step of Δt =1μs (recall that Δx/Δt = c). In this context,

α becomes ∂u/∂t (the axial velocity v)andβ becomes ∂u/∂x (the axial strain ).

The axial stress σ in the rod is then deduced algebraically from the axial strain

91 using Hooke’s law, σ = E [108]. Hence, velocity v and stress σ are the primitive dependent variables in the ﬁrst-order formulation of the governing equations solved by the method of characteristics. (For more on ﬁrst-order velocity-stress formulations of elastodynamics, see [118].) Conversely, in the second-order formulation of the governing equations solved by modal analysis, displacement u is the primitive dependent variable, with stress σ and velocity v relegated to deduced response variables, i.e., partial derivatives of displacement.

As expected, the method of characteristics captures the time-dependent traction boundary condition without suﬀering from spurious behavior (see Figs. 3.1 and

3.2). Unlike the modal analysis methods studied in this chapter (CBFM, HEEM,

Mindlin-Goodman), the method of characteristics solves the original problem with time-dependent boundary conditions ‘as is’ without any reformulation, restructuring, or postulated solution structure. Furthermore, its solutions require no post- processing: no a posteriori solution treatment like l’Hôpital’s rule to accommodate singularities due to resonance, no term-by-term differentiation to deduce stress (stress is primitive in a first-order formulation of elastodynamics), and no convergence tests.

It is also eﬃcient and straightforward to implement.

On the other hand, the method of characteristics is conﬁned to physical systems that are modeled by ﬁrst-order hyperbolic systems of PDEs. Wavenumbers, natural frequencies, and mode shapes are not readily available. Its solutions are in terms of velocity and stress, so numerical integration is required to obtain displacement.

Finally, unlike modal analysis, its solutions are discrete, not closed form.

92 Chapter 4: Conclusions and Future Work in Eulerian Elastodynamics

4.1 Conclusions

In Chapter 2, we presented an elementary 1-D nonlinear theory for longitudinal waves in elastic rods. This elementary 1-D theory was derived from a fully 3-D formulation of dynamic nonlinear elasticity, and emerged as the leading-order equations in a slender-structure perturbation scheme. A distinguishing feature of our elementary theory is its development in the Eulerian representation, i.e., with respect to the present (or deformed) configuration of the rod. Eulerian formulations of the equations of motion are ideal for fixed-grid numerical techniques such as finite-difference methods, finite-volume methods, and high-resolution shock-capturing schemes from

CFD, which have recently attracted considerable attention in the computational solid dynamics community. Several other notable features of our leading-order model are the following:

(a) The boundary conditions at the lateral surface were incorporated directly into our

mathematical framework. (Conversely, in many previous rod theories, the lateral

boundary conditions are not satisﬁed, as discussed in [47,48].) As a consequence,

the cross-sectional area of the rod appears explicitly as a physical unknown in our

93 governing equations, and is hence deduced primitively as part of the numerical

solution of our model. Thus, we can inherently capture the evolving lateral surface

of the rod without post-processing data or using an interface tracking algorithm

(e.g., a level set method).

(b) Our leading-order model linearized to the classical equation of motion for a ta-

pered rod, often called the Webster horn equation [114]. Hence, our leading-order

model can be interpreted as a generalization of the Webster horn equation to ﬁnite

deformations. (We note that our linearized model includes two novel evolution

equations – one for density, the other for cross-sectional area – that are unavail-

able in existing linear rod theories, which customarily assume that density and

cross-sectional area do not vary with time.)

Analysis of the mathematical structure of this system revealed (i) its inherent

hyperbolicity and (ii) contributions to the characteristic speeds (wave speeds)

from the Poisson eﬀect, stress convection, and compressibility.

(d) The physical eﬀects of radial expansion and contraction (i.e., the Poisson eﬀect),

axial inertia, and axial stress emerge at leading order, whereas lateral inertia,

transverse shear stress, radial stress, and circumferential stress are relegated to

higher-order corrections.

A foreseeable criticism that could be levied against our model is the use of a hy-

poelastic (rate-type) constitutive equation; refer to Sections 1.2.2 and 2.2.2. Most of

the popular objective rates (e.g., Jaumann-Zaremba) produce hypoelastic constitu-

tive models that violate the second law of thermodynamics. That is, they cannot be

94 integrated to give a stress-strain relationship that is derivable from a thermodynamic

energy potential [16,158], and they may predict unphysical energy dissipation and/or

residual stresses in a closed elastic loading-unloading cycle [159–161]. Some rates

have even been shown to produce spurious oscillations [162,163]. Also, a hypoelastic

constitutive equation with the Jaumann rate is restricted to ﬁnite deformations with

small strains [161] – perfectly appropriate for modeling the deformation of metals and

other hard materials, but not soft solids such as rubbers and elastomers.24 Other limitations of our model include a passive ambient (no interaction with the surrounding medium), torsionless axisymmetry, isothermal deformation, homogeneity, isotropy, a slender structure, and long waves.

Despite these limitations, our model is compact, structurally straightforward, and readily amenable to numerical implementation and use in practical applications. To- ward this end, the numerical simulation of several prototypical nonlinear wave propagation problems were demonstrated using the space-time Conservation Element and

Solution Element (CESE) method, a novel technique for ﬁrst-order hyperbolic systems. In our simulations, the CESE method eﬀectively resolved both strong and weak shocks, rarefactions, and contact discontinuities.

As an outgrowth of the work in Chapter 2, Chapter 3 investigated propagating waves generated by time-dependent traction boundary conditions in elastic structural elements. We reported solutions of a generalized model – developed using the linearized theory of Chapter 2 – for this important class of problems by means of the well-established Mindlin-Goodman method and two of its successors, which we referred to as the concentrated body force method (CBFM) and the homogeneous

24A new objective rate called the logarithmic rate has been shown to remedy the aforementioned issues; see [5,19,29,80,82,164] and other related work by these authors.

95 eigenfunction expansion method (HEEM). Each of these methods employed a different strategy for homogenizing the time-dependent boundary conditions in order to enable modal analysis. We found that these diﬀerent approaches impacted (i) the mathematical equivalency of the reformulated problem, (ii) the convergence and term-by-term diﬀerentiability of the mode superposition (series) solutions, and (iii) the ability of these series solutions to satisfy the governing equation (in a classical sense) and the time-dependent boundary conditions.

The Mindlin-Goodman method used a linear transformation to obtain a mathematically exact reformulated problem with homogeneous boundary conditions. The

CBFM, on the other hand, jettisoned this linear transformation in favor of a modeling assumption that embedded the surface traction as a body force (forcing function) in the governing equation. We showed that this modeling assumption on the kinetics, although straightforward and intuitive, introduced an artiﬁcial stress discontinuity at the forced boundary, resulting in a mathematically inexact reformulated problem.

Remarkably, the mode superposition solution of the reformulated problem was uniformly convergent. However, when it was differentiated term by term to deduce the stress, spurious oscillations and overshoot resembling the Gibbs phenomenon emerged near the discontinuity. We found that these oscillations and overshoot were physical manifestations of a term-by-term-differentiated series that was not uniformly convergent and hence not a rigorous representation of the first derivative, per the formalism of mathematical analysis.

The HEEM employed a diﬀerent strategy, expanding the displacement and its partial derivatives in the eigenfunctions of the associated homogeneous problem. Re- markably, we found that this method rendered identical solutions to the CBFM,

96 bearing the same artiﬁcial stress discontinuity accompanied by the same unphysi-

cal oscillations and overshoot. This time, however, the discontinuity, oscillations, and

overshoot were artifacts of the solution construction, which inherently and incorrectly

predicted a stress-free forced boundary. In spite of these drawbacks, we found that judicious application of the CBFM and HEEM yielded good solutions for engineering purposes. In particular, when enough high-frequency modes were retained in the series solutions for stress, the oscillations predicted by the CBFM and HEEM were confined to a sufficiently small region near the artificial discontinuity.

Another important contribution of this chapter was the systematic development of a characteristics-based approach for solving a first-order hyperbolic formulation of the problem. We explored the mathematical structure of the first-order system to determine its eigenvalues and eigenvectors, verify its hyperbolicity, and, from its characteristic form, deduce the Riemann invariants. The eigenvalues revealed the wave speeds, while the Riemann invariants were used to construct a solution via the method of characteristics. We found that the method of characteristics solution was exact to within machine precision and captured the time-dependent traction boundary condition without artificial discontinuities, spurious oscillations, or unphysical overshoot. Unlike its modal analysis counterparts, the method of characteristics solved the first-order problem with time-dependent boundary conditions ‘as is’ without any reformulation, restructuring, or postulated solution structure. Further, its solutions required no post-processing: no a posteriori solution treatment like l’Hôpital’s rule to accommodate resonance (resonant behavior was inherently captured without the emergence of singularities), no term-by-term differentiation to deduce stress (stress

97 is primitive in a ﬁrst-order velocity-stress formulation of elastodynamics), and no convergence tests.

4.2 Future work

4.2.1 Finite-deformation elasticity

Regarding outgrowth of the current work, one exciting avenue involves developing a novel Eulerian framework for modeling and simulating the dynamics of soft elastic materials and structures, where geometric and constitutive nonlinearities play a prominent role. (Somewhat surprisingly, nearly all of the work in Eulerian ﬁnite elasticity has been focused on metals and other hard materials [3,8–11,13–15], which are typically limited to small strains. A notable exception is the recent work of Kamrin et al. [16].) Accordingly, the main thrust of this work will be capturing nonlinear waves such as shocks and rarefactions in rubber, elastomers, biological tissues, polymer gels, and other compliant elastic solids. Envisioned applications include automotive and aerospace components, defense-related technologies, and biomechanics.

Following Kamrin et al. [16] (refer also to Sections 1.2.1 and 1.2.2 of this dissertation), a 3-D Eulerian model for the dynamics of compressible, isotropic, nonlinearly elastic materials will be adopted: conservation of mass and linear momentum

∂ρ ∂(ρv) + div(ρv)=0, +div ρv ⊗ v =divT, ∂t ∂t

98 the evolution of the reference map

∂(ρξ) + div(ρξ ⊗ v)=0, ∂t

the ﬁnite-strain constitutive equation

−1 T = β0 I + β1 B + β−1 B , − 1 − 1 1 2 ∂W ∂W 2 ∂W 2 ∂W β0 =2I3 I2 + I3 ,β1 =2I3 ,β−1 = −2I3 , ∂I2 ∂I3 ∂I1 ∂I2 and the algebraic relations

∂ξ = F−1, B = FFT. ∂x

Note that this is a fully conservative ﬁrst-order system, suitable for numerical solution via the Conservation Element and Solution Element (CESE) method. Recall from

Sections 1.2.1 and 1.2.2 that ρ is the density in the present conﬁguration, v is the velocity, T is the Cauchy stress, ξ is the reference map, F is the deformation gradient,

B is the left Cauchy-Green deformation tensor, I1, I2,andI3 are the invariants of B,

W (I1,I2,I3) is the strain energy (e.g., neo-Hookean or Mooney-Rivlin), and x and X

are the present and reference positions of a representative continuum particle.

Some gentle extensions of this work include incorporating dissipative (viscoelas-

tic) eﬀects through a modiﬁed non-equilibrium framework, comparing the perfor-

mance of the CESE method to other high-resolution shock-capturing schemes, and

developing reduced-order Eulerian models for soft elastic structures (e.g., beams and

membranes).

99 4.2.2 Fluid-structure interaction

The compliant structural model developed in Section 4.2.1 could be coupled to a Navier-Stokes solver to model fluid-structure interaction problems. Of particular interest are soft structures interacting dynamically with a viscous ambient. A key feature of the resulting code will be the solution of the coupled fluid-solid problem on a single fixed mesh using a single solver, which promises to simplify and improve existing multi-mesh/multi-solver approaches. Targeted applications include traumatic brain injury (e.g., concussions), biomedical applications of ultrasound (e.g., ultrasound- aided drug delivery), natural phenomena (e.g., flapping of insect wings), automotive and aerospace systems (e.g., airbag deployment and fluid flow in flexible hoses), and hemodynamics.

4.2.3 Finite-deformation plasticity

Another exciting future research direction involves extending the existing mathematical framework and computer code to dynamic ﬁnite-deformation plasticity. This would lay the groundwork for simulating metal forming, ultrasound-aided manufac- turing processes (e.g., welding, wire drawing, and forging), and high-rate loading events (e.g., impacts, blasts, detonations, and penetrations) for defense-related applications.

A uniﬁed approach to constitutive modeling in ﬁnite plasticity remains elusive.

Essentially, there are four standard approaches, each diﬀering on the form of the elastic-plastic decomposition of the deformation employed at the outset:

100 (1) One approach is based on an additive decomposition of the rate of deformation

tensor D into an elastic part and a plastic part [165]. This approach is common

in commercial and industrial ﬁnite-element codes [25, 80].

(2) Another approach is based on a multiplicative decomposition of the deformation

gradient F into an elastic part and a plastic part [166]. This approach is arguably

the most compatible with the physical basis of crystal plasticity [167].

(3) Yet another approach is based on an additive decomposition of the Lagrangian

strain tensor E into an elastic part and a plastic part [168].

(4) A fourth approach is based on Prandtl-Reuss (i.e., inﬁnitesimal, rate-independent)

plasticity, which begins with an additive decomposition of the inﬁnitesimal strain

tensor . The development proceeds with J2-ﬂow theory and isotropic hardening;

the equations are then inverted from strain rate to stress rate and generalized to

ﬁnite deformations [6].

Assessing these four diﬀerent approaches – and selecting what we feel to be the optimal choice for our long-term purposes – will constitute much of our preliminary research in this area. Along these lines, in what follows, we further examine Approach (4).

The corresponding model appears in [6], but a detailed illustration of its development is neither provided in [6] nor elsewhere in the literature. We attempt to clarify the details of its development here.

Based on experimental observation, inﬁnitesimal plasticity admits the additive decomposition

e p dεij = dεij + dεij,

101 e p where dεij and dεij are the inﬁnitesimal elastic and plastic strain increments, respectively. The elastic strain increment can be obtained from isotropic linear elasticity:

e 1+ν ν dε = dσij − dσkk δij, ij E E where dσij is the stress increment, δij is the Kronecker delta, E is Young’s modulus, and ν is Poisson’s ratio.

p To determine the plastic strain increment dεij, we consider the equation of the yield surface of a material that undergoes strain-rate-independent isotropic strain- hardening:

p F (Sij, ε¯ )=0, (4.1)

p where F (Sij, ε¯ ) is a scalar-valued yield function whose form is made explicit by the yield criterion, e.g., von Mises or Tresca. Sij is the deviatoric part of stress, i.e.,

1 Sij = σij − σkk δij. 3

That the yield function F depends on only the deviatoric part of the stress reﬂects the customary assumption of yield insensitivity to hydrostatic pressure. Note that the yield surface is the union of all points in deviatoric stress space that satisfy Eq. (4.1).

The effective plastic strainε ¯p, defined here as , def 2 p p ε¯p = dε¯p,dε¯p = dε dε , 3 ij ij quantifies the plastic strain accumulated during the deformation history. It represents the scalar hardening parameter in (4.1).

102 The associated ﬂow rule

p ∂F dεij = dλ (4.2) ∂Sij implies normality of the plastic strain increment to the yield surface deﬁned in deviatoric stress space. In Eq. (4.2), dλ is a scalar function that, loosely speaking, represents the magnitude of the plastic strain increment. The loading criteria are

F<0 elastic deformation ∂F F =0, dSij > 0 plastic loading ∂Sij ∂F F =0, dSij = 0 neutral loading ∂Sij ∂F F =0, dSij < 0 elastic unloading ∂Sij

Plastic strain only accrues during plastic loading; otherwise, the plastic strain in-

p crement dεij vanishes. Hence, our adoption of the plastic loading criteria is tacit throughout the remainder of this section.

As strain-rate-insensitive materials harden during plastic deformation, points on

the original yield surface remain on all subsequent yield surfaces. This observation,

p together with a ﬁrst-order Taylor series expansion of F (Sij, ε¯ ), imply the consistency condition

∂F ∂F p dF = dSij + p dε¯ =0. (4.3) ∂Sij ∂ε¯

Use of Eq. (4.2) in (4.3) leads to

∂F ij ∂Sij dS dλ = − 1 . (4.4) ∂F 2 ∂F ∂F 2 p ∂ε¯ 3 ∂Sij ∂Sij

103 We employ the von Mises yield criterion, i.e., J2-ﬂow theory, which makes the yield

p function F (Sij, ε¯ ) in Eq. (4.1) explicit:

p 1 1 y p 2 F (Sij, ε¯ )= SijSij − σ (¯ε ) =0, (4.5) 2 3 where J2 = SijSij/2 is the second invariant of the deviatoric stress (related to the energy of distortion), and σy(¯ε p) is the yield stress in uniaxial tension, which evolves with eﬀective plastic strain as the material hardens during plastic deformation. For choice (4.5), it follows that

y ∂F 2 y y ∂F −2 y dσ = Sij,Sij dSij = σ dσ , p = σ p , ∂Sij 3 ∂ε¯ 3 dε¯ and use of these results in Eq. (4.4) leads to

y p 3 dσ dεij = y dσy Sij. 2 σ dε¯p

For a linear strain-hardening material, the tensile yield stress increases linearly with the eﬀective plastic strain, i.e.,

y p y p σ (¯ε )=σo + BSH ε¯ ,

y where the initial tensile yield stress σo and the strength coeﬃcient BSH are material- dependent constants. It follows that the plastic strain increment is

p 3 dσ¯ dεij = Sij, 2 BSH σ¯ where we have introduced the eﬀective stress , def 3 σ¯ = Sij Sij . 2

104 Thus, the total strain increment (elastic + plastic) is

1+ν ν 3 dσ¯ dεij = dσij − dσkk δij + Sij. (4.6) E E 2 BSH σ¯

Equation (4.6) can be inverted to give the deviatoric stress increment, or, equivalently, its rate

2 Skl ε˙kl S˙ij =2με˙ij − με˙kkδij − 3μ Sij, (4.7) 3 BSH 3 2μ + 2 Smn Smn where μ is the shear modulus. The inﬁnitesimal elastic-plastic constitutive equation

(4.7) is generalized to the ﬁnite-deformation regime by (i) replacing the stress tensor

σij with its ﬁnite Eulerian analog Tij, the Cauchy stress, (ii) replacing the inﬁnitesimal strain increment dεij with the rate of deformation Dij, which is the work conjugate of the Cauchy stress, and (iii) employing an objective rate D/Dt of the stress. The objective rate ensures that the constitutive equation is invariant under an arbitrary superposed rigid body motion. The resulting constitutive equation is

D 2 Sij =2μDij − μDkkδij − β(s)SklDklSij, Dt 3 where s = SmnSmn and ⎧ ⎪ ⎪ 0ifF<0 ⎪ ∂F ⎨ ij 0ifF =0 and ∂Sij dS < 0 β(s)= ∂F (4.8) ⎪ 0ifF =0 and dσij =0 ⎪ ∂σij ⎩⎪ 6μ2 ∂F ij (3μ+BSH)s if F =0 and ∂σij dσ > 0

Note that the four rows of Eq. (4.8) correspond to elastic deformation, elastic unloading, neutral loading, and plastic loading, respectively.

105 Chapter 5: An Introduction to Smart Polymers

This chapter begins the second part of the dissertation, where the focus shifts to modeling smart polymers in general and magnetoelectric polymers in particular. Our study of elasticity from the ﬁrst part of this dissertation thus continues, but with an important new wrinkle: the deforming elastic continuum now responds to and interacts with electromagnetic ﬁelds. The study of these interactions forms the basis of Chapters 6 and 7.

5.1 Overview and research opportunity

Smart polymers (i.e., soft active materials) have received considerable attention due to their potential to enable powerful new technologies that are currently inacces- sible with existing smart materials. For instance, compared to piezoelectric ceramics such as lead zirconate titanate (PZT) and magnetostrictive alloys such as Terfenol-D, smart polymers are generally lighter, less expensive, less brittle, and more compliant [169–172]. Consequently, they are more formable and easier to manufacture [173].

Additionally, soft smart materials are better suited for applications requiring (i) large elastic deformations and (ii) mechanical properties (e.g., elastic stiffness) that can be modified rapidly and reversibly in response to electromagnetic fields.

106 As an example, magnetorheological elastomers (MREs) – typically fabricated by dispersing micron-sized magnetizable particles in a rubbery matrix – undergo large recoverable strains and mechanically stiﬀen in response to an external magnetic ﬁeld.

In fact, their actuation strains and magneto-mechanical stiﬀening are often several orders of magnitude larger than those of conventional magnetostrictive materials [174].

Adaptive properties such as controllable stiﬀness make MREs ideal candidates for unprecedented vibration-mitigating, noise-suppressing, and energy-harvesting technologies [175–179].

Magnetoelectric (ME) polymers, on the other hand, are a comparatively novel class of smart composites that are beginning to attract signiﬁcant attention. A distinguishing feature of ME polymers is the tantalizing ability to electrically control their magnetization, or, conversely, magnetically control their polarization. Lever- aging this magnetoelectric coupling could potentially impact numerous technologies, including information storage, spintronics, sensing, actuation, and energy harvesting [169,171,172,180–185].

Most of the research on soft ME polymer composites to date has occurred in the applied physics and materials science communities. The primary emphasis of this work has been on tailoring the design of the composites to optimize the magnitude of the ME coupling. Substantially less activity has occurred in the way of mathematical modeling and experimental characterization at ﬁnite strains. A lack of experimental data on the mechanical behavior of ME polymers has impeded the development of robust ﬁnite-strain constitutive equations, which, in turn, has hindered technological implementation. Progress in these areas is vital to advance our fundamental understanding of ME polymers, facilitate their design and optimization, and encourage

107 their implementation in the next generation of intelligent systems, structures, and

devices.

The remainder of this chapter is structured as follows: In Section 5.2, we pro-

vide some relevant background on ME polymers and their more well-known MRE

counterparts. In particular, we provide an overview of how MREs and ME polymers

are fabricated, brieﬂy review previous theoretical and experimental work, and discuss

some potential applications. Then, in Section 5.3, we highlight the objectives and

novel contributions of this part of the dissertation.

5.2 A brief review of several classes of smart polymers

5.2.1 Magnetorheological elastomers

Magnetorheological elastomers (MREs), often referred to in the literature as magneto- active polymers, are composite materials that typically consist of ferromagnetic (magnetizable) particles dispersed in a rubbery (elastomeric) matrix. A key feature of

MREs is their strong magnetoelastic coupling, driven by interactions between the magnetized particles [173, 176, 179, 186]. This magnetoelastic coupling manifests itself macroscopically in two important ways: Firstly, MREs are ‘magnetostrictive’ in that they undergo large elastic deformations (e.g., length changes) in response to the application of a magnetic field. Secondly, MREs exhibit field-dependent mechanical properties (e.g., elastic stiffness) that can be modified rapidly and reversibly by a magnetic field.

108 Applications and fabrication

The prospect of developing adaptive devices with controllable stiﬀness makes

MREs ideal candidates for automotive applications related to vibration control and noise suppression. Along these lines, variable-stiﬀness shock absorbers, mounts, bush- ings, suspensions, and clutches have been envisioned or developed [175–177]; sensors, haptic devices, and variable-stiﬀness actuators have also been suggested as promising applications of MREs [178,179].

The most common matrix materials are elastomers (e.g., silicone and polyurethane) and natural rubber, which are highly elastic, poor conductors, and non-magnetizable

[175,177,187]. Carbonyl iron, which has a high magnetic saturation and a high magnetic permeability, is typically used for the ferromagnetic ﬁller particles [187, 188], although pure iron, cobalt, iron alloys, nickel alloys, and even rare-earth alloys such as

Terfenol-D have also been used [177,187,188]. Particle size is generally on the order of

1to5μm [179,187,189], with volume fractions typically less than 30% [179,187]. De- pending on the nature of the curing process, the particles may be distributed randomly

(isotropically) or in chain-like arrangements (transversely isotropically) [179,190–192].

For the latter, substantial enhancement of the ﬁeld-induced stiﬀening can be realized if the MRE is loaded along the preferred direction [179,190–192].

Modeling and experiments

Experiments on MREs have been undertaken by Jolly et al. [193], Ginder et al. [194], Bellan & Bossis [195], Gong et al. [196], Varga et al. [197], Boczkowska

& Awietjan [198], and Danas et al. [199]. (Additional references to experimental work can be found in [187, 190, 191, 199, 200].) The general consensus is that the

109 physical behavior of MRE composites depends the choice of material for the matrix and particles, as well as the size, shape, orientation, volume fraction, distribution, and alignment of the particles. For instance, an increase in particle volume fraction enhances magnetostriction but reduces compliance [173,187].

Broad theoretical foundations for the mechanics of a deforming continuum in the presence of an electromagnetic ﬁeld were laid in the pioneering monograph of Trues- dell & Toupin [201]. For the special case of ﬁnite-deformation magnetoelasticity, much of the seminal theoretical work was set forth by Tiersten [202,203], Brown [204], and

Maugin & Eringen [205, 206]. These (and other) early works have been collected, summarized, critiqued, and extended in the monographs of Pao [207], Maugin [208],

Eringen & Maugin [209], Hutter & van de Ven [210], and Kovetz [211]. Further developments in modeling ﬁnite-deformation magnetoelasticity are showcased in the recent papers of Dorfmann & Ogden [188,212–214] (summarized nicely in their monograph [174]), Kankanala & Triantafyllidis [176], Steigmann [215], Ericksen [216], Bus- tamante et al. [217], and Ott´enio et al. [189]. Numerical solutions of boundary-value problems have been pursued, for instance, in [218–220].

Current trends in finite-deformation magnetoelasticity include incorporating viscoelasticity [192, 221] and anisotropy [179, 190, 192, 199] into the modeling framework. The former accounts for magneto-mechanical energy dissipation, while the latter accounts for the preferred direction associated with the chain-like microstructure ‘frozen in’ when the MRE is cured in the presence of a magnetic field. Another important area of research involves finite-deformation homogenization theories that account for the effect of the microstructure on the macroscopic response of the composite. Along these lines, we mention the work of deBotton et al. [222], Galipeau and

110 Ponte Casta˜neda [173, 177, 223, 224], Javili et al. [225], Chatzigeorgiou et al. [226],

and Galipeau et al. [186]. Instabilities have been investigated recently by Rudykh &

Bertoldi [191] and Danas & Triantafyllidis [178], while Bustamante & Shariﬀ [200]

have developed a novel principal-axis formulation to facilitate the experimental char-

acterization of MREs, which remains an ongoing challenge in the ﬁeld.

5.2.2 Magnetoelectric polymers

In this section, we discuss magnetoelectric (ME) polymer composites, a novel and compelling class of engineered materials. Generally, ME composites have both piezoelectric and magnetostrictive constituents and exploit the so-called ‘product property’ concept [227] to generate ME coupling as a secondary, or indirect, effect. For instance, when an electric field is applied to a ME composite, strain is induced in the piezoelectric phase. When this strain is transmitted to the magnetostrictive phase, a magnetization results. Thus, the electric field indirectly induces a magnetization in the composite, mediated by the exchange of elastic deformation between the two constituents, neither of which is magnetoelectric in and of itself.

Applications and fabrication

Polymer-based ME composites are generally lighter, less expensive, easier to manufacture, and more compliant than ceramic/alloy-based ME composites, making them well suited for applications requiring large deformations, e.g., stretchable electronics, soft machines, non-planar conﬁgurations, memory devices, and soft robots [169–172].

111 Another promising application of ME polymers lies in the realm of information stor-

age, where the magnetoelectric eﬀect could be harnessed to allow data to be writ-

ten electrically and read magnetically [171, 172, 182]. Other envisioned applications

include high-sensitivity magnetic ﬁeld detectors, magneto-electric transducers, spintronic devices, solid-state transformers, microwave devices, wireless energy transfer, sensors, attenuators, ﬁlters, and vibration energy harvesting [169,180,181,183–185].

ME polymer composites are most often constructed by (i) dispersing magnetostrictive particles in an electro-active polymer matrix (particulate or granular composites) or (ii) stacking alternating layers of electro-active polymer and magnetostrictive alloy into a sandwich-like structure (laminated composites or bi-layers) [170, 183]. Com- mon materials include PVDF and P(VDF-TrFE) for the electro-active polymer; and iron oxide, cobalt ferrite, Terfenol-D, and Metlgas for the magnetostrictive alloy. In general, laminated composites attain the largest ME couplings at the lowest applied magnetic ﬁelds [170].

Modeling and experiments

Polymer-based ME composites require a ﬁnite-strain mathematical framework that incorporates both geometric and constitutive nonlinearities. In a rather general sense, the theoretical foundations needed to facilitate this type of modeling have been collected, summarized, reviewed, and extended in the monographs of Pao [207],

Maugin [208], Eringen & Maugin [209], Hutter & van de Ven [210], and Kovetz [211], which also provide extensive sets of references to the seminal work in continuum electrodynamics (e.g., Toupin [228], Truesdell & Toupin [201], and Brown [204]). There

112 is a gap, however, between the degree of generality and rigour aﬀorded by these theoretically oriented monographs and the level of specialization and simplicity required for convenient implementation in practical applications.

More recently, Liu [229] employed an energy-based (variational) approach to derive a ﬁnite-deformation theory for nonlinear magnetoelectric elastomers. Importantly, he demonstrates how Maxwell stresses and geometric nonlinearities combine to generate

‘apparent’ ME coupling, an idea he and coworkers cleverly leverage in [171, 172].

Liu’s free energy function, however, does not include coupling between electricity, magnetism, and elasticity, rendering these eﬀects uncoupled in his constitutive model.

Polarization and magnetization are selected as the independent variables (IVs).

It has been argued that constitutive models that use the electric ﬁeld or electric displacement as an IV (rather than polarization), and the magnetic ﬁeld or magnetic induction as an IV (rather than magnetization) – and the converse – can be advan- tageous in certain situations [176, 199, 213, 215, 230, 231]. To address this, Santapuri

& Lowe et al. [232] developed a comprehensive catalogue of free energies and constitutive equations for fully coupled thermo-electro-magneto-mechanical materials in the ﬁnite-deformation regime. However, invariance, incompressibility, material symmetry, explicit forms of the free energy, and other important considerations needed for technological implementation are not pursued in this work (although the issue of invariance was later addressed by Santapuri in [233]).

On the computational front, a nonlinear ﬁnite-element framework was recently set forth by Guo & Ghosh [234] for the coupled electro-magneto-mechanical problem.

An important contribution of this work is the inclusion of fully dynamic deformations

113 and electromagnetic ﬁelds, a notable departure from the customary quasi-static as-

sumption. The model employed by the authors includes a one-way coupling between

electromagnetism and mechanical deformations via the deformation-dependent ‘rest-

frame’ electromagnetic ﬁelds, excluding coupling in the momentum equations (via the

Maxwell stress) and the ﬁnite-strain constitutive model.

5.3 Objectives, structure, and novel contributions

Chapters 6 and 7 are intended to be self-contained studies of two different aspects of the modeling of soft smart materials at finite strains. In Chapter 6, we revisit the fundamental laws of continuum electrodynamics, which govern the behavior of smart polymers. To researchers trained in continuum mechanics, it is customary to progress from the primitive (verbal) statement of a law, to its material or global form (the law applied to a representative subset of the body, independent of its configuration), to its integral form (the law applied to the subset in its reference or present configuration), and finally to its differential form (the law applied pointwise throughout the entire body). Somewhat surprisingly, this systematic progression – which is particularly useful from a pedagogical perspective – is absent in the customary development of the fundamental equations of continuum electrodynamics. We thus revisit and re- examine these basic laws. Novel perspectives emerge during our progression from primitive statements to pointwise equations.

In Chapter 7, we develop a ﬁnite-strain theoretical framework for modeling soft

magnetoelectric composites, with practical applications in mind. Finite deformations,

electro-magneto-elastic coupling, and material nonlinearities will be incorporated into

the model. In the spirit of Dorfmann & Ogden [174,213,235], a particular emphasis

114 will be placed on the ground-up development of tractable constitutive equations to facilitate material characterization in the laboratory. Accordingly, a catalogue of free energies and constitutive equations will be presented, each employing a different set of independent variables. As the optimal choice of IV varies from one scenario to another [176,199,213,215,230,231], our extensive catalogue will provide theoreticians and experimentalists with the needed flexibility. The ramifications of invariance, angular momentum, incompressibility, and material symmetry will be explored, and a representative (neo-Hookean-type) free energy with full electro-magneto-elastic coupling will be posed.

115 Chapter 6: Revisiting the Fundamental Laws of Continuum Electrodynamics

Much of the work in this chapter was presented in [69]: S.E. Bechtel and R.L.

Lowe, Fundamentals of Continuum Mechanics: With Applications to Mechanical,

Thermomechanical, and Smart Materials, Academic Press, San Diego, 2014.

The emergence of smart polymers and their concomitant technological implementation has led to renewed interest in the basic equations that govern their behavior. In this chapter, we follow suit and revisit the fundamental laws of continuum electrodynamics. Novel perspectives emerge during our transparent and systematic progression from primitive statements to pointwise equations.

6.1 Introduction

The well-known and well-established linear theories of piezoelectricity and piezo-

magnetism are insuﬃcient for modeling nonlinear electro-mechanical and magneto-

mechanical interactions in soft smart materials undergoing ﬁnite deformations, much

like linear elasticity is incapable of capturing the response of rubbery materials at

116 large strains. Continuum electrodynamics, on the other hand, provides an appropri-

ate mathematical framework for modeling the ﬁnite deformations, nonlinear electro-

magneto-elastic coupling, and ﬁeld-dependent material properties exhibited by smart

polymers.25

Broadly speaking, continuum electrodynamics is concerned with the interaction of mechanical deformations and electromagnetic fields in continuous media at finite strains. The theory is quite general and accounts for a diverse array of physics including dynamic electromagnetic fields and mechanical deformations, polarization, magnetization, current flow, and heat flux. The fundamental laws of continuum electrodynamics consist of the familiar conservation laws of mechanics and thermodynamics together with Maxwell’s equations of electromagnetism. A two-way coupling between these sets of equations drives the interaction between mechanics, thermodynamics, and electromagnetism.

Some of the pioneering work in continuum electrodynamics can be attributed to

Truesdell & Toupin [201] and Penﬁeld & Haus [236]. Along these lines, we mention the seminal work of Toupin [228,237,238] on elastic dielectrics and Brown [204] on magnetoelastic materials. Other important contributions to continuum electrodynamics, including thermodynamic aspects of the theory, can be found in [239–248]. Much of the pioneering research in the ﬁeld was later collected, summarized, reviewed, and extended in the monographs of Pao [207], Maugin [208], Eringen & Maugin [209],

Tiersten [249], and Hutter & van de Ven [210].

25Smart polymers exhibit geometric and constitutive nonlinearities at ﬁnite strains. Hence, nonlinear descriptions of the kinematics (strain measures), kinetics (stress measures), and constitutive response (ﬁeld-dependent material properties) are required.

117 More recently, the emergence of smart polymers (as well as electrorheological and magnetorheological ﬂuids) has spurred renewed interest in the theoretical foundations of continuum electrodynamics; see, for instance, the modern monographs of Kovetz

[211], Yang [250], Ogden & Steigmann [251], and Dorfmann & Ogden [174], as well as the recent work of Rinaldi & Brenner [252], Ericksen [253–255], Fosdick & Tang [256],

Steigmann [257], Maugin [258], and Liu [229, 259]. In this chapter, we follow suit, revisiting and re-examining the fundamental laws of continuum electrodynamics for a deformable, polarizable, magnetizable continuum.

For the first time, we present the fundamental laws at four different levels: primitive, material, integral, and differential. To be explicit, primitive refers to the law stated in words, in its most basic form; material (or, alternatively, global or system-

wise) refers to the law applied to the body, independent of its conﬁguration; integral

refers to the law applied to the body as seen in a particular conﬁguration, either

reference (Lagrangian form) or present (Eulerian form); and differential refers to the localized Lagrangian and Eulerian forms of the law, which are valid pointwise throughout the body. Both the Lagrangian and Eulerian forms have their own merit, with the optimal form depending on the material (e.g., solid or fluid), physical application (e.g., geometry of the device or structure), and solution methodology (e.g., semi-inverse or numerical; finite element or finite difference).

Novel perspectives emerge during our transparent and systematic progression from primitive statements to pointwise equations. For instance, we illustrate how the material form, a crucial but oft-overlooked intermediate form,

(i) highlights the role of both free and bound charge – as well as free, bound, and

polarization current – when modeling polarizable and magnetizable materials;

118 (ii) clariﬁes the connections between Eulerian and Lagrangian integral representa-

tions of global electromagnetic quantities such as electric charge, electric ﬂux,

electromotive force, magnetic ﬂux, and magnetic ﬁeld;

(iii) illustrates and explains the absence of form invariance between Eulerian and La-

grangian integral representations of the electric ﬂux and magnetic ﬁeld (whereas

other thermo-electro-magneto-mechanical quantities such as mass, linear mo-

mentum, angular momentum, kinetic energy, internal energy, entropy, electric

charge, electromotive force, and magnetic ﬂux are form invariant);

(iv) allows for the systematic derivation of transformations between local Eulerian

and Lagrangian quantities;

(v) shows how the lack of form invariance in (iii) ensures dimensional homogeneity

at the global level as well as proper and self-consistent transformations (refer to

(iv)) between Eulerian and Lagrangian electromagnetic ﬁelds at the local level;

(vi) explains the unexpected absence of Lagrangian-Eulerian form invariance in the

customary algebraic relationship between the local electric displacement, electric

ﬁeld, and polarization (and, similarly, the algebraic relationship between the

local magnetic induction, magnetic ﬁeld, and magnetization).

6.2 Primitive, material, and integral versions of the fundamental laws

In this section, we present the primitive, material, and integral versions of the fundamental laws of continuum electrodynamics. Recall that primitive refers to the law

119 stated in words, in its most basic form; material refers to the law applied to the body, independent of its configuration; and integral refers to the law applied to the body as seen in a particular configuration, either reference (Lagrangian form) or present (Eu- lerian form). The fundamental laws are valid for all deformable continua: conductor or insulator, electrically charged or electrically neutral, solid or fluid, conservative or dissipative, polarizable or non-polarizable, magnetizable or non-magnetizable.

Collectively, the fundamental laws consist of the conservation laws of mechanics and thermodynamics together with Maxwell’s equations of electromagnetism. A two-way coupling between these sets of equations drives the interaction between mechanics, thermodynamics, and electromagnetism; this coupling is examined in detail in Section 6.7.

6.2.1 Notation and nomenclature

We now present the notation necessary to describe the geometry of the deformable, polarizable, magnetizable body. We label the body B and two arbitrary subsets S1

and S2. Subset S1 is bounded by a closed surface, while subset S2 is bounded by a closed curve; see Figure 6.1. As will soon be evident, both closed surfaces and closed curves are required to formulate integral statements of the fundamental laws of continuum electrodynamics.

In the reference conﬁguration, body B occupies open volume RR of Euclidean 3-

3 space E , bounded by closed surface ∂RR. Subset S1 occupies open volume PR ⊂RR, bounded by closed surface ∂ PR, and subset S2 occupies open surface QR ⊂RR, bounded by closed curve ∂QR.

120 In the present conﬁguration at time t,bodyB occupies open volume R, bounded

by closed surface ∂ R. Subset S1 occupies open volume P⊂R, bounded by closed surface ∂ P, and subset S2 occupies open surface Q⊂R, bounded by closed curve

∂Q.

Figure 6.1: Subsets S1 and S2 as seen in the present conﬁguration of body B. Subset S1 is an open volume P bounded by a closed surface ∂P, while subset S2 is an open surface Q bounded by a closed curve ∂Q.

6.2.2 Conservation of mass

Primitively, conservation of mass postulates that the mass M of every subset of the body B is constant throughout its motion, or, equivalently,thetimerateofchange of the mass of every subset is zero. Applying this primitive statement to arbitrary

121 subset S1 allows us to express conservation of mass mathematically in material form:

d M(S1,t)=0 or M(S1) ≡ independent of t. (6.1) dt

Specializing to a continuum allows the mass M of subset S1 to be expressed in

Eulerian and Lagrangian integral forms, i.e., ⎧ ⎪ ⎪ ⎪ ρdv ⎨⎪ P M(S1)= dm = (6.2) ⎪ S ⎪ 1 ⎪ ρR dV ⎩⎪ PR

Thus, specializing to a continuum is tantamount to a smoothness assumption on the mass M. The Eulerian integral representation (top of (6.2)) corresponds to subset

S1 as seen in its present conﬁguration, while the Lagrangian integral representation

26 (bottom of (6.2)) corresponds to S1 as seen in its reference configuration. In (6.2), dv and dV are volume elements in the present and reference configurations, and ρ and ρR are the mass densities in the present and reference configurations. Note that ρ has units of mass per present volume, while ρR has units of mass per reference volume

(see Table 6.1). Both ρ and ρR are bounded, continuous functions of space and time.

To perform the integrations in (6.2), it is natural to consider ρ in its spatial description, i.e., as a function of x and t, and to consider ρR in its referential description, i.e., as a function of X and t, although both ρ and ρR can be expressed using either a spatial or a referential description. Note that X and x are the reference and present positions of a continuum particle, related through the motion x = χ(X,t), and t is time.

26 We are free to label the volume occupied by subset S1 by its present volume P or its reference volume PR.

122 Use of (6.2) in (6.1) leads to Eulerian and Lagrangian integral representations of conservation of mass: d ρdv =0, (6.3a) dt P

ρdv = ρR dV. (6.3b)

P PR

These integral statements are valid for any open volume P in the present conﬁguration

and corresponding open volume PR in the reference conﬁguration. Note that (6.3a) and (6.3b) are identical to their counterparts in the thermomechanical theory, which can be found, for instance, in [69].

6.2.3 Balance of linear momentum

Balance of linear momentum postulates that the time rate of change of the linear momentum L of any subset of the body is equal to the resultant external force f acting on that subset. Applying this primitive statement to subset S1 gives the material form:

d L(S1,t)=f (S1,t). (6.4) dt

123 Assuming smoothness of L, we write its Eulerian and Lagrangian integral represen-

tations: ⎧ ⎪ ⎪ ⎪ vρdv ⎨⎪ L S P ( 1,t)=⎪ (6.5) ⎪ ⎪ vρR dV ⎩⎪ PR where v is the velocity of a continuum particle at the present time t,andtheinte- grands are continuous, bounded functions of space and time.

It is assumed that the resultant external force f can be additively decomposed

into a body force and a contact force. Following [207–210,247], the eﬀects of electro-

magnetism are modeled through an electromagnetic contribution to the body force

so that ⎧ ⎪ ⎪ ⎪ (f m + f em)ρdv + t da ⎨⎪ P ∂P f (S1,t)= (6.6) ⎪ ⎪ m em ⎪ (f + f )ρR dV + tR dA ⎩⎪ PR ∂PR where f m is the mechanically induced body force per unit mass, f em is the electromagnetically induced body force per unit mass (more on this in Section 6.7.2), t and tR are the Eulerian (true) and Lagrangian (nominal) surface tractions, and da and

dA are area elements in the present and reference conﬁgurations. Elaborating, t is

the traction acting on surface ∂P in the present conﬁguration measured per unit area

of ∂P,whereastR is the traction acting on surface ∂P in the present conﬁguration

124 but measured per unit area of the corresponding surface ∂PR in the reference conﬁg- uration. Thus, t has units of force per present area, while tR has units of force per reference area (see Table 6.1).

Use of (6.5) and (6.6) in (6.4) gives Eulerian and Lagrangian integral representations of balance of linear momentum: d vρdv = (f m + f em)ρdv + t da, (6.7a) dt P P ∂P

d m em vρR dV = (f + f )ρR dV + tR dA. (6.7b) dt PR PR ∂PR

Comparing (6.7a) and (6.7b) to their counterparts in the thermomechanical theory

(which can be found, for instance, in [69]), we see that (6.7a) and (6.7b) contain an additional body force f em.

6.2.4 Balance of angular momentum

Balance of angular momentum postulatesthatthetimerateofchangeofthe

angular momentum H0 of any subset of the body about the origin 0 is equal to the

resultant external moment M0 acting on that subset about the origin 0. In material form:

d H0 (S1,t,0)=M0 (S1,t,0). (6.8) dt

125 Assuming that H0 is smooth leads to Eulerian and Lagrangian integral representations of the angular momentum about 0, i.e., ⎧ ⎪ ⎪ ⎪ x × vρdv ⎨⎪ P H0(S1,t,0)= (6.9) ⎪ ⎪ ⎪ x × vρR dV ⎩⎪ PR where ( ) × ( ) denotes the cross product of two vectors. The integrands in (6.9)

are continuous, bounded functions of space and time. Similarly, smoothness of M0

implies that ⎧ ⎪ ⎪ ⎪ x × (f m + f em)ρdv + x × t da + cem ρdv ⎨⎪ P ∂P P M0 (S1,t,0)= ⎪ ⎪ m em em ⎪ x × (f + f )ρR dV + x × tR dA + c ρR dV ⎩⎪ PR ∂PR PR (6.10)

Following [207–210, 247], an electromagnetically induced body couple (torque) per unit mass cem is included in (6.10) to model the eﬀects of electromagnetism; more on

this in Section 6.7.2. Note that the ﬁrst two terms in (6.10)1 and (6.10)2 represent the moment about 0 due to the resultant external force f, the ﬁrst term being a contribution from the body force and the second term being a contribution from the contact force.

Use of (6.9) and (6.10) in (6.8) gives Eulerian and Lagrangian integral representations of balance of angular momentum: d x × vρdv = x × (f m + f em)ρdv + x × t da + cem ρdv, (6.11a) dt P P ∂P P

126 d m em em x×vρR dV = x×(f + f )ρR dV + x×tR dA + c ρR dV. (6.11b) dt PR PR ∂PR PR

Comparing (6.11a) and (6.11b) to their counterparts in the thermomechanical theory

(which can be found, for instance, in [69]), we see that (6.11a) and (6.11b) contain additional moments due to (i) the electromagnetic body force f em and (ii) the electromagnetic body couple cem.

6.2.5 First law of thermodynamics

The ﬁrst law of thermodynamics (or conservation of energy) postulates that the time rate of change of the total energy (i.e., kinetic energy K plus internal energy E) of any subset of the body is equal to the rate of work W generated by the resultant external force acting on that subset plus the rate of all auxiliary energies A (e.g., heat, electromagnetic, chemical) entering or exiting that subset. In material form:

d K(S1,t)+E(S1,t) = W (S1,t)+A(S1,t). (6.12) dt

Assuming that the kinetic and internal energies of part S1 are smooth implies that ⎧ ⎧ ⎪ ⎪ ⎪ 1 ⎪ ⎪ v · vρdv ⎪ ερdv ⎨⎪ 2 ⎨⎪ S P S P K( 1,t)=⎪ E( 1,t)=⎪ (6.13) ⎪ 1 ⎪ ⎪ v · vρR dV ⎪ ερR dV ⎩⎪ 2 ⎩⎪ PR PR where ( ) · ( ) denotes the inner product of two vectors, ε is the speciﬁc internal energy (or internal energy per unit mass), and the integrands are bounded, continuous functions of space and time. The rate of work W generated by the resultant external force f can be additively decomposed into contributions from the body force and the

127 contact force, i.e., ⎧ ⎪ ⎪ ⎪ (f m + f em) · v ρdv + t · v da ⎨⎪ P ∂P W (S1,t)= (6.14) ⎪ ⎪ m em ⎪ (f + f ) · v ρR dV + tR · v dA ⎩⎪ PR ∂PR

Following [207–210], the auxiliary energy rate A is additively decomposed into three contributions, two from radiation and one from conduction: (i) the rate of heat absorption throughout the volume, (ii) the rate of electromagnetic energy absorption throughout the volume, and (iii) the rate of heat entering through the boundary, i.e., ⎧ ⎪ ⎪ ⎪ rt ρdv + rem ρdv − hda ⎨⎪ P P ∂P A(S1,t)= (6.15) ⎪ ⎪ t em ⎪ r ρR dV + r ρR dV − hR dA ⎩⎪ PR PR ∂PR where rt is the speciﬁc heat supply rate, rem is the speciﬁc electromagnetic energy supply rate (more on this in Section 6.7.2), and h and hR are the Eulerian and

Lagrangian heat flux rates. Elaborating, h is the rate of heat flow out of the present boundary ∂P measured per unit area of the present boundary ∂P; hR is the rate of heat flow out of the present boundary ∂P, but instead measured per unit area of the corresponding boundary ∂PR in the reference configuration. Thus, h has units of energy per time per present area, while hR has units of energy per time per reference area (see Table 6.1).

128 Use of (6.13)-(6.15) in (6.12) yields Eulerian and Lagrangian integral representations of the ﬁrst law of thermodynamics: d 1 d v · vρdv + ερdv = (f m + f em) · v ρdv + t · v da dt 2 dt P P P ∂P + (rt + rem)ρdv − hda, (6.16a) P ∂P

d 1 d m em v · vρR dV + ερR dV = (f + f ) · v ρR dV + tR · v dA dt 2 dt PR PR PR ∂PR t em + (r + r )ρR dV − hR dA.

PR ∂PR (6.16b)

Comparing (6.16a) and (6.16b) to their counterparts in the thermomechanical theory

(which can be found, for instance, in [69]), we see that (6.16a) and (6.16b) contain additional energy contributions from (i) the work due to the electromagnetic body force f em and (ii) the electromagnetic energy supply rem.

6.2.6 Second law of thermodynamics

We adopt the Clausius-Duhem inequality as our particular statement of the second law of thermodynamics. The Clausius-Duhem inequality postulates that the rate of change of the entropy N of any subset of the body exceeds (or, in the absence of electro-magneto-mechanical eﬀects, equals) the rate of entropy generation R due to the radiative heat supply minus the rate of entropy loss H due to the outward heat

ﬂux. Applying this primitive statement of the second law to subset S1 yields the

129 material form:

d N (S1,t) ≥R(S1,t) −H(S1,t). (6.17) dt

Specializing to a continuum and assuming smoothness of N (S1,t), R(S1,t), and

H(S1,t) allows us to write ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ rt ⎪ ηρdv ⎪ ρdv ⎨⎪ ⎨⎪ Θ P P N (S1,t)= R(S1,t)= (6.18) ⎪ ⎪ t ⎪ ⎪ r ⎪ ηρR dV ⎪ ρR dV ⎩⎪ ⎩⎪ Θ PR PR and ⎧ ⎪ ⎪ h ⎪ da ⎨⎪ Θ H S ∂P ( 1,t)=⎪ (6.19) ⎪ hR ⎪ dA ⎩⎪ Θ ∂PR where η is the speciﬁc entropy (or entropy per unit mass) and Θ is the absolute temperature. The Eulerian integral representations of N (S1,t), R(S1,t), and H(S1,t)

(top of (6.18) and (6.19)) correspond to subset S1 as seen in its present conﬁguration, while the Lagrangian integral representations of these quantities (bottom of (6.18) and (6.19)) correspond to subset S1 as seen in its reference conﬁguration.

Use of (6.18) and (6.19) in (6.17) leads to Eulerian and Lagrangian integral representations of the Clausius-Duhem inequality: d rt h ηρ dv ≥ ρdv − da, (6.20a) dt Θ Θ P P ∂P

130 t d r hR ηρR dV ≥ ρR dV − dA. (6.20b) dt Θ Θ PR PR ∂PR

Note that these integral statements are valid for any open volume P bounded by closed surface ∂P in the present conﬁguration, or corresponding open volume PR

and closed surface ∂PR in the reference conﬁguration. Also note that (6.20a) and

(6.20b) are identical to their counterparts in the thermomechanical theory, which can be found, for instance, in [69].

6.2.7 Conservation of electric charge

Conservation of charge postulates that the time rate of change of the total electric

charge (i.e., free charge Σ plus bound charge Σb) within any closed surface is equal to the sum of the free (or conductive) current J , the bound current Jb,andthe polarization current Jp entering that surface [260]. Applying this primitive statement of the law to subset S1 (an open volume bounded by a closed surface) allows us to express conservation of charge mathematically in material form:

d Σ(S1,t)+Σb (S1,t) = J (S1,t)+Jb (S1,t)+Jp (S1,t). (6.21) dt

Loosely, free charges are unpaired and ‘free’ to move; this motion gives rise to the conductive current. Conversely, bound charges are paired, and are thus ‘bound’ to a particular atom; when a material is magnetized or experiences a time varying polarization, the bound charges realign, giving rise to the bound current and the polarization current, respectively [174].

Physically, the conductive current J (S1,t)alwaysentersandexitssubsetS1 at time t through its present surface ∂ P, but we are free to label this surface by its

131 reference location ∂ PR instead. Similarly, the free charge Σ(S1,t) always resides within present volume P, but we are free to label this volume by its reference location

PR instead. Exploiting this freedom in how the geometry of S1 is labeled allows us to write Eulerian and Lagrangian integral representations of the free charge, bound charge, free current, polarization current, and bound current, i.e., [174,260] ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ σ∗ dv ⎪ − div p∗ dv ⎨⎪ ⎨⎪ P P Σ(S1,t)= Σb (S1,t)= (6.22) ⎪ ⎪ ⎪ ⎪ ⎪ σR dV ⎪ − Div pR dV ⎩⎪ ⎩⎪ PR PR

⎧ ⎧ ⎪ ⎪ ⎪ ⎪ d ⎪ − j∗ · n da ⎪ − p∗ · n da ⎨⎪ ⎨⎪ dt ∂P ∂P J(S1,t)= Jp(S1,t)= (6.23) ⎪ ⎪ ⎪ ⎪ d ⎪ − jR · N dA ⎪ − pR · N dA ⎩⎪ ⎩⎪ dt ∂PR ∂PR

⎧ ⎪ ⎪ ⎪ − (curl m∗) · n da ⎨⎪ J S ∂P b ( 2,t)=⎪ (6.24) ⎪ ⎪ − (Curl mR) · N dA ⎩⎪ ∂PR where n and N are outward unit normals in the present and reference conﬁgurations,

“div” and “curl” denote the Eulerian divergence and curl (i.e., the divergence and curl calculated with respect to the present conﬁguration), and “Div” and “Curl” denote the Lagrangian divergence and curl (i.e., the divergence and curl calculated with respect to the reference conﬁguration). Note that the minus signs in (6.23) and

(6.24) are required to maintain consistency in our sign convention: positive J denotes

132 current ﬂowing into the boundary (see (6.21)), whereas positive j∗ · n implies that current is ﬂowing out of the boundary (n is an outward unit normal).

∗ In (6.22)-(6.24), σ and σR are denoted the Eulerian and Lagrangian free charge

∗ ∗ density, j and jR the Eulerian and Lagrangian conductive current density, p and pR

∗ the Eulerian and Lagrangian electric polarization, and m and mR the Eulerian and

Lagrangian magnetization (or magnetic polarization). All are bounded, continuous functions of space and time. Recall that the Eulerian and Lagrangian representations of a particular quantity are fundamentally diﬀerent since they are associated with diﬀerent labels for the geometry of the subset: σ∗ has units of charge per present

∗ volume, while σR has units of charge per reference volume; j has units of current per

∗ present area, while jR has units of current per reference area; p has units of charge

∗ per present area, while pR has units of charge per reference area; and m has units of current per present length, while mR has units of current per reference length (see

Table 6.1).

σ∗, j∗,andp∗ are often called eﬀective electromagnetic ﬁelds in the literature

to signify that they are measured with respect to a co-moving or rest frame, i.e., one affixed to but not deforming with the continuum [207,209,236]. In this chapter, an effective electromagnetic field is denoted by a superscript asterisk. In Section

6.7.1, we present transformations that relate the effective electromagnetic fields to the standard electromagnetic fields, the latter being measured with respect to a stationary or laboratory frame rather than a co-moving frame.

133 Use of (6.22)-(6.24) in (6.21) leads to Eulerian and Lagrangian integral representations of conservation of charge: d σ∗ dv = − j∗ · n da − (curl m∗) · n da, (6.25a) dt P ∂P ∂P

d σR dV = − jR · N dA − (Curl mR) · N dA. (6.25b) dt PR ∂PR ∂PR

These integral statements are valid for any open volume P bounded by closed surface

∂P in the present conﬁguration, or corresponding open volume PR and closed surface

∂PR in the reference conﬁguration. Note that to perform the integrations in (6.25a)

and (6.25b), it is more natural to consider σ∗ and j∗ in their spatial descriptions, i.e., as functions of x and t,andσR and jR in their referential descriptions, i.e., as functions of X and t. Recall that X and x are the reference and present positions of

a continuum particle, related through the motion x = χ(X,t).

6.2.8 Faraday’s law

Faraday’s law postulates that the time rate of change of the magnetic ﬂux B through any open surface is equal and opposite the electromotive force E induced in the closed curve bounding that surface. Mathematically, in material form for subset

S2 (an open surface bounded by a closed curve), this amounts to

d B(S2,t)=−E(S2,t). (6.26) dt

134 Smoothness of B and E imply that ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ b∗ · n da ⎪ e∗ · l dl ⎨⎪ ⎨⎪ Q ∂Q B(S2,t)= E (S2,t)= (6.27) ⎪ ⎪ ⎪ ⎪ ⎪ bR · N dA ⎪ eR · lR dL ⎩⎪ ⎩⎪ QR ∂QR where dl and dL are line elements in the present and reference conﬁgurations, l and lR are unit tangents in the present and reference conﬁgurations (related to n and

∗ N in a right-handed sense), b and bR are the Eulerian and Lagrangian magnetic

∗ ﬂux density (or magnetic induction), and e and eR are the Eulerian and Lagrangian electric ﬁeld. Refer to Table 6.1 for their respective units. Use of (6.27) in (6.26) leads to Eulerian and Lagrangian representations of Faraday’s law: d b∗ · n da = − e∗ · l dl, (6.28a) dt Q ∂Q

d bR · N dA = − eR · lR dL. (6.28b) dt QR ∂QR

Note that these integral statements are valid for any open surface Q bounded by a closed curve ∂Q in the present conﬁguration, or corresponding open surface QR and closed curve ∂QR in the reference conﬁguration.

6.2.9 Gauss’s law for magnetism

Gauss’s law for magnetism (a statement of conservation of magnetic ﬂux, or, alternatively, the absence of magnetic monopoles) postulates that the magnetic ﬂux

135 B through any closed surface is zero, i.e.,

B(S1,t)=0. (6.29)

Assuming that the magnetic ﬂux B(S1,t) through the closed surface of S1 at time t is smooth allows us to write: ⎧ ⎪ ⎪ ⎪ b∗ · n da ⎨⎪ ∂P B(S1,t)= (6.30) ⎪ ⎪ ⎪ bR · N dA ⎩⎪ ∂PR

Use of (6.30) in (6.29) gives Eulerian and Lagrangian representations of Gauss’s law for magnetism: b∗ · n da =0, (6.31a) ∂P

bR · N dA =0. (6.31b)

∂PR

6.2.10 Gauss’s law for electricity

Gauss’s law for electricity postulates that the electric ﬂux F through any closed

surface is proportional to the total electric charge (i.e., free charge Σ plus bound

charge Σb) enclosed within that surface, i.e.,

Σ(S1,t)+Σb (S1,t) F (S1,t)= , (6.32) o

136 where o is the electric permittivity in vacuo. Assuming that the electric ﬂux F is

smooth implies that ⎧ ⎪ ⎪ ⎪ e∗ · n da ⎨⎪ ∂P F (S1,t)= (6.33) ⎪ ⎪ −1 ⎪ JC eR · N dA ⎩⎪ ∂PR where J is the determinant of the deformation gradient F =Gradx,andC−1 is the inverse of the right Cauchy-Green deformation tensor C = FTF. Note that unlike all of the global quantities presented heretofore (e.g., the magnetic ﬂux B and the electromotive force E in (6.27)), the electric ﬂux (6.33) is not form invariant, i.e., ∗ F (S1,t)= e · n da = eR · N dA.

∂P ∂PR

We elaborate on this important observation in Section 6.6.

Recall that the free charge Σ and the bound charge Σb associated with subset S1

at time t are given in (6.22). Subsequent use of (6.22) and (6.33) in (6.32) leads to

Eulerian and Lagrangian representations of Gauss’s law for electricity: d∗ · n da = σ∗ dv, (6.34a) ∂P P

dR · N dA = σR dV, (6.34b)

∂PR PR

137 ∗ where the Eulerian electric displacement d and Lagrangian electric displacement dR

are introduced through the algebraic relationships (see, for instance, [230])

∗ ∗ ∗ −1 d = p + o e , dR = pR + o J C eR. (6.35)

Note that the algebraic relationships (6.35)1 and (6.35)2 are not form invariant, which we again discuss in Section 6.6.

6.2.11 Amp`ere-Maxwell law

The Amp`ere-Maxwell law postulates that the time rate of change of the electric

ﬂux F through any open surface plus the free current J , bound current Jb,and polarization current Jp passing through that surface is proportional to the magnetic

ﬁeld T around the closed curve bounding that surface. Applying this primitive statement of the law to subset S2 (an open surface bounded by a closed curve) allows us to express the Amp`ere-Maxwell law mathematically in material form: d μo o F (S2,t)+J (S2,t)+Jb (S2,t)+Jp (S2,t) = T (S2,t), (6.36) dt

138 where μo is the magnetic permeability in vacuo. Smoothness allows us to write

[174,260] ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ e∗ · n da ⎪ j∗ · n da ⎨⎪ ⎨⎪ Q Q F (S2,t)= J (S2,t)= (6.37) ⎪ ⎪ ⎪ −1 ⎪ ⎪ JC eR · N dA ⎪ jR · N dA ⎩⎪ ⎩⎪ QR QR ⎧ ⎧ ⎪ ⎪ ⎪ d ⎪ ⎪ p∗ · n da ⎪ (curl m∗) · n da ⎨⎪ dt ⎨⎪ Q Q Jp(S2,t)= Jb (S2,t)= (6.38) ⎪ ⎪ ⎪ d ⎪ ⎪ pR · N dA ⎪ (Curl mR) · N dA ⎩⎪ dt ⎩⎪ QR QR

⎧ ⎪ ⎪ ⎪ b∗ · l dl ⎨⎪ ∂Q T (S2,t)= (6.39) ⎪ ⎪ 1 ⎪ CbR · lR dL ⎩⎪ J ∂QR

Note that the magnetic ﬁeld T , like the electric ﬂux F, is not form invariant; refer

ahead to Section 6.6. Use of (6.37)-(6.39) in (6.36) leads to d d∗ · n da + j∗ · n da = h∗ · l dl, (6.40a) dt Q Q ∂Q

d dR · N dA + jR · N dA = hR · lR dL, (6.40b) dt QR QR ∂QR

139 ∗ where we have used (6.35) to introduce d and dR, and the algebraic relationships

(see, for instance, [214])

∗ 1 ∗ ∗ 1 h = b − m , hR = CbR − mR (6.41) μo μoJ

∗ to introduce the Eulerian magnetic ﬁeld h and the Lagrangian magnetic ﬁeld hR.

6.3 The localization theorem

Throughout the remainder of this chapter, we frequently make use of the localization theorem: If φ is a continuous scalar-, vector-, or tensor-valued ﬁeld in the open region R occupied by the body B and φdv = 0 (6.42) P for any subset P of R, then it is necessary and suﬃcient that

φ = 0 (6.43) in R. Said diﬀerently, if (6.42) holds for any arbitrary subset of the body, then the integrand φ vanishes everywhere throughout the body, and vice versa. Refer to [69] for a detailed proof.

6.4 Pointwise versions of the fundamental laws

In this section, we derive pointwise versions of the integral balance laws developed in Section 6.2.

140 6.4.1 Eulerian forms

We begin by recalling the Eulerian integral forms of the ﬁrst principles: conservation of mass d ρdv =0, (6.44a) dt P balance of linear momentum d vρdv = (f m + f em)ρdv + t da, (6.44b) dt P P ∂P

balance of angular momentum d x × vρdv = x × (f m + f em)ρdv + x × t da + cem ρdv, (6.44c) dt P P ∂P P

ﬁrst law of thermodynamics d 1 d v · vρdv + ερdv = (f m + f em) · v ρdv + t · v da dt 2 dt P P P ∂P + (rt + rem)ρdv − hda, (6.44d) P ∂P second law of thermodynamics d rt h ηρ dv ≥ ρdv − da, (6.44e) dt Θ Θ P P ∂P conservation of electric charge d σ∗ dv = − j∗ · n da − (curl m∗) · n da, (6.44f) dt P ∂P ∂P

141 Gauss’s law for magnetism b∗ · n da =0, (6.44g) ∂P

Faraday’s law d b∗ · n da = − e∗ · l dl, (6.44h) dt Q ∂Q

Gauss’s law for electricity d∗ · n da = σ∗ dv, (6.44i) ∂P P

Amp`ere-Maxwell law d d∗ · n da + j∗ · n da = h∗ · l dl. (6.44j) dt Q Q ∂Q

To obtain pointwise versions of the integral equations (6.44a)-(6.44j), we make use of the transport theorem for surface integrals (refer to Proof C.1 in Appendix C),

d a · n da = a +curl(a × v)+v(div a) · n da, (6.45) dt Q Q

the transport theorem for volume integrals [69], d φdv = φ˙ + φ div v dv, (6.46) dt P P

Stokes’s theorem [261,262], a · l dl = (curl a) · n da, (6.47) ∂Q Q

142 the divergence theorem [261,262], a · n da = div a dv, (6.48) ∂P P

and the localization theorem (refer to Section 6.3). In (6.45)-(6.48), φ(x,t)and

a(x,t) are arbitrary scalar-valued and vector-valued functions, respectively, of present

position x and time t, v is the velocity, “div” denotes the Eulerian divergence (i.e.,

the divergence calculated with respect to the present conﬁguration), “curl” denotes

the Eulerian curl,

∂ a = a(x,t) ∂t denotes the Eulerian time derivative, that is, the partial derivative of the spatial description of a with respect to time t,and

a˙ = a +(v · grad) a denotes the material time derivative of a(x,t). Also useful are the relations

t = Tn,h= q · n, (6.49) where T is the ‘Cauchy stress’ and q is the Eulerian heat ﬂux vector. Note that the proofs of (6.49)1 and (6.49)2 in a thermo-electro-magneto-mechanical setting are essentially identical to the corresponding proofs in a thermomechanical setting; refer to [69] for details. With these tools in hand, it can be shown (refer, for instance, to Proofs C.2–C.4) that the pointwise variants of the Eulerian integral equations

143 (6.44a)-(6.44j) are

ρ˙ + ρ div v =0, (6.50a)

ρv˙ = ρ (f m + f em)+divT, (6.50b)

ρΓem + T − TT = 0, (6.50c)

ρε˙ = T · L + ρ rt + rem − div q, (6.50d)

rt q ρη˙ ≥ ρ − div , (6.50e) Θ Θ

σ˙ ∗ + σ∗ div v +divj∗ =0, (6.50f)

div b∗ =0, (6.50g)

curl e∗ = − (b∗) − curl (b∗ × v), (6.50h)

div d∗ = σ∗, (6.50i)

curl h∗ =(d∗) +curl(d∗ × v)+σ∗v + j∗. (6.50j)

Equations (6.50g)-(6.50j) are often referred to as Maxwell’s equations for a deformable, polarizable, magnetizable continuum. It can be shown that conservation of

144 charge (6.50f) is implicitly contained in Maxwell’s equations (6.50g)-(6.50j); refer to

Proof C.5. Note that

L =gradv is the Eulerian velocity gradient, and Γem is a skew tensor whose corresponding axial vector is cem, i.e.,

Γem a = cem × a for any vector a. Also note that in deriving the pointwise version of the ﬁrst law of thermodynamics (6.50d) from its integral counterpart (6.44d), we have made use of the Eulerian form of the energy theorem for continuum electrodynamics [69]: d 1 (f m + f em) · vρdv + t · v da − v · vρdv = T · L dv. (6.51) dt 2 P ∂P P P

6.4.2 Lagrangian forms

Recall from Section 6.2 the Lagrangian integral forms of the ﬁrst principles: conservation of mass

ρdv = ρR dV, (6.52a)

P PR balance of linear momentum d m em vρR dV = (f + f )ρR dV + tR dA, (6.52b) dt PR PR ∂PR

145 balance of angular momentum d m em em x×vρR dV = x×(f + f )ρR dV + x×tR dA + c ρR dV, (6.52c) dt PR PR ∂PR PR

ﬁrst law of thermodynamics d 1 d m em v · vρR dV + ερR dV = (f + f ) · v ρR dV + tR · v dA dt 2 dt PR PR PR ∂PR t em + (r + r )ρR dV − hR dA,

PR ∂PR (6.52d) second law of thermodynamics

t d r hR ηρR dV ≥ ρR dV − dA, (6.52e) dt Θ Θ PR PR ∂PR conservation of electric charge d σR dV = − jR · N dA − (Curl mR) · N dA, (6.52f) dt PR ∂PR ∂PR

Gauss’s law for magnetism

bR · N dA =0, (6.52g)

∂PR

Faraday’s law d bR · N dA = − eR · lR dL, (6.52h) dt QR ∂QR

146 Gauss’s law for electricity

dR · N dA = σR dV, (6.52i)

∂PR PR

Amp`ere-Maxwell law d dR · N dA + jR · N dA = hR · lR dL. (6.52j) dt QR QR ∂QR

An important feature of the Lagrangian integral conservation laws (6.52a)-(6.52j) is that the regions PR and QR occupied by subsets S1 and S2 in the reference conﬁgu- ration are ﬁxed. Hence, the regions of integration PR and QR do not change with time, so that time derivatives of Lagrangian surface and volume integrals can be passed directly inside of the integrals. Conversely, with the Eulerian integral conservation laws

(6.44a)-(6.44j), the regions of integration P and Q change with time. Thus, to take time derivatives of Eulerian integrals whose limits of integration vary with time, the transport theorems (6.45) and (6.46) were employed. The transport theorems can be interpreted as generalizations of Leibniz’s rule in multivariable calculus to surface and volume integrations.

The continuous, bounded nature of the integrands in (6.52a)-(6.52j) enables the divergence and Stokes’s theorems, and the requirement that (6.52a)-(6.52j) be continuous and global, i.e., true for the entire body and all subsets, enables the localization theorem. With the traction tR and heat ﬂux hR dependent on surface geometry only through the outward unit normal N, so that [69]

tR = PN,hR = qR · N,

147 it can be shown that application of the divergence, Stokes, and localization theorems to the Lagrangian integral equations (6.52a)-(6.52j) leads to the Lagrangian pointwise equations

ρJ = ρR, (6.53a)

m em ρR v˙ = ρR (f + f )+DivP, (6.53b)

em T T ρR Γ + PF − FP = 0, (6.53c)

t em ρR ε˙ = P · Grad v + ρR r + r − Div qR, (6.53d)

t r qR ρR η˙ ≥ ρR − Div , (6.53e) Θ Θ

σ˙ R +DivjR =0, (6.53f)

Div bR =0, (6.53g)

Curl eR = − b˙ R, (6.53h)

Div dR = σR, (6.53i)

Curl hR = d˙ R + jR. (6.53j)

148 Note that it can be shown that conservation of charge (6.53f) is not independent of Maxwell’s equations (6.53g)-(6.53j), but rather implicitly contained in them. In

(6.53a)-(6.53j), P is the ‘first Piola-Kirchhoff stress,’ qR is the Lagrangian heat flux vector, “Div” is the Lagrangian divergence (i.e., the divergence calculated with respect to the reference configuration), “Curl” is the Lagrangian curl, and

∂ a˙ = a(X,t) ∂t

is the material derivative of an arbitrary vector a = a(X,t), i.e., the partial derivative

of the referential description of a with respect to time t.Also,

Γem a = cem × a for any vector a,thatis,Γem is a skew tensor whose corresponding axial vector is cem. Note that in deriving the pointwise version of the ﬁrst law of thermodynamics

(6.53d) from its integral counterpart (6.52d), we have made use of the Lagrangian form of the energy theorem for continuum electrodynamics [69]: m em d 1 (f + f )·v ρR dV + tR ·vdA − v·vρR dV = P·Grad vdV. (6.54) dt 2 PR ∂PR PR PR

Refer to Proof C.6.

6.5 Transformations between Eulerian and Lagrangian quantities

The Eulerian and Lagrangian representations of various thermal, electrical, magnetic, and mechanical quantities appearing throughout this chapter, along with their corresponding units, are listed in Table 6.1. (Note that in Table 6.1, we employ the

149 following notation for the fundamental units: M is mass, LP is present length, LR is reference length, T is time, θ is temperature, and C is charge.)27 It can be shown that the Eulerian and Lagrangian quantities in Table 6.1 are related through the following linear algebraic transformations:

T ∗ −1 ∗ −1 ∗ ∗ eR = F e , pR = J F p , dR = J F d ,σR = Jσ ,

T ∗ T ∗ −1 ∗ hR = F h , mR = F m , bR = J F b ,ρR = Jρ, (6.55)

−1 ∗ −T −1 jR = J F j , P = J TF , qR = J F q.

Note that J, the determinant of the deformation gradient F, is a measure of dilatation or volume change.

We remark that the transformations in (6.55) follow by equating the Eulerian and Lagrangian integral representations of global quantities such as mass M,force f, bound charge Σb, bound current Jb, electric flux F, and magnetic flux B that are defined for the subset or system; refer to Proofs C.7–C.9 in Appendix C for additional details.

6.6 Some observations

Novel perspectives on the fundamental laws have emerged during our systematic progression from primitive statements to pointwise equations. First, it is observed that the integral representations of most global quantities, including mass M, linear

27When validating the dimensional homogeneity of an equation in ﬁnite-deformation theories,

it is useful in our view to diﬀerentiate the fundamental units of length from one another, with LP

corresponding to the present conﬁguration and LR corresponding to the reference conﬁguration.

150 momentum L, angular momentum H0, kinetic energy K,internalenergyE, electromotive force E, and magnetic flux B,areform invariant. That is, the Eulerian and Lagrangian integral representations are formally identical, differing only in the configuration they are referred to (present or reference). For instance, ∗ E (S2,t)= e · l dl = eR · lR dL (6.56)

∂Q ∂QR and ∗ B(S1,t)= b · n da = bR · N dA (6.57)

∂P ∂PR are both form invariant. In contrast, the Eulerian and Lagrangian integral representations of the electric ﬂux F and the magnetic ﬁeld T are not form invariant, i.e., ∗ −1 F (S1,t)= e · n da = JC eR · N dA, (6.58)

∂P ∂PR

∗ 1 T (S2,t)= b · l dl = CbR · lR dL. (6.59) J ∂Q ∂QR

The lack of form invariance in (6.58) and (6.59) is necessary to ensure that (a) the transformations (6.55) between the local Eulerian and Lagrangian electromagnetic

ﬁelds are self-consistent and (b) dimensional homogeneity is preserved throughout the theory. We illustrate (a) and (b) using a speciﬁc example in the following paragraph.

Equation (6.56) for the electromotive force is taken to be fundamental. Using

Table 6.1, it can be veriﬁed that (6.56) is dimensionally homogeneous, with each term

151 having the same fundamental units. Recall from Section 6.5 that the transformation

T ∗ eR = F e (6.60) follows from (6.56); refer to Proof C.7 in Appendix C. We use this transformation to investigate the lack of form invariance in (6.58). The electric ﬂux is deﬁned as def ∗ F (S1,t) = e · n da. (6.61) ∂P

The Lagrangian counterpart of (6.61) is obtained by using transformation (6.60) and performing a change of independent variable from x to X to convert the Eulerian integration to a Lagrangian integration, i.e., def ∗ F (S1,t) = e · n da (deﬁnition of electric ﬂux) ∂P −T = F eR · n da (transformation (6.60)) ∂P −T −T = F eR · J F N dA (Nanson’s formula)

∂PR −1 −T = J F F eR · N dA (deﬁnition of tensor transpose)

∂PR −1 = J C eR · N dA (deﬁnition of C)

∂PR

We have thus veriﬁed the lack of form invariance in (6.58). Note that the presence of

J C−1 in the Lagrangian integral ensures that (6.58) is dimensionally homogeneous, with each term having the same fundamental units. Thus, to summarize, we have

152 demonstrated that ∗ F (S1,t)= e · n da = eR · N dA,

∂P ∂PR i.e., the electric ﬂux is not form invariant, contrary to what one might have expected at the outset. A similar procedure can be used to verify that the magnetic ﬁeld (6.59) is also not form invariant.

On another front, recall the algebraic relationship between the Eulerian electric displacement d, Eulerian electric ﬁeld e, and Eulerian polarization p, i.e.,

∗ ∗ ∗ d = p + o e , (6.62) which is dimensionally homogeneous and regarded as fundamental. The Lagrangian counterpart of (6.62) is obtained using the transformations developed in Section 6.5:

−1 dR = pR + o J C eR. (6.63)

See also [230]. The obvious lack of form invariance between (6.62) and (6.63) ensures that (6.63) is dimensionally homogeneous. It also ensures that Gauss’s law for electricity is form invariant and dimensionally homogeneous; refer to the derivations of (6.34a) and (6.34b) in Section 6.2.10. Similar arguments hold for the algebraic relationships (6.41)1 and (6.41)2.

153 Table 6.1: Units for various electrical, magnetic, mechanical, and thermal quantities.

Type Quantity Form Symbol Fund. Units Derived Units SI Units

C2T2 capacitance farad Vacuum permittivity o 3 MLP pres. len. meter ML2 force pres. len. newton volt P × ≡ Lag. eR 2 Electric ﬁeld LRT C charge ref. len. coulomb meter

MLP force newton volt Eul. e∗ ≡ CT2 charge coulomb meter

154 C charge coulomb Lag. pR 2 2 Electric polarization LR ref. area meter Electrical ∗ C charge coulomb Eul. p 2 2 LP pres. area meter C charge coulomb Lag. dR 2 2 Electric displacement LR ref. area meter

∗ C charge coulomb Eul. d 2 2 LP pres. area meter C charge coulomb Lag. σR 3 3 Free charge density LR ref. vol. meter

∗ C charge coulomb Eul. σ 3 3 LP pres. vol. meter

Continued on the next page . . . Table6.1–Continued...

Type Quantity Form Symbol Fund. Units Derived Units SI Units

C current ampere Lag. jR 2 2 Electrical Free current density LRT ref. area meter

∗ C current ampere Eul. j 2 2 LPT pres. area meter

MLP inductance henry Vacuum permeability μo C2 pres. length meter C current ampere Lag. hR LRT ref. len. meter 155 Magnetic field C current ampere Eul. h∗ LPT pres. len. meter Magnetic C current ampere Lag. mR Magnetization LRT ref. len. meter C current ampere Eul. m∗ LPT pres. len. meter ML2 magnetic flux weber P ≡ Lag. bR 2 2 tesla Magnetic flux density LRTC ref. area meter M magnetic flux weber Eul. b∗ ≡ tesla TC pres. area meter2

Continued on the next page . . . Table6.1–Continued...

Type Quantity Form Symbol Fund. Units Derived Units SI Units

M mass kilogram Lag. ρR 3 3 Mass density LR ref. vol. meter M mass kilogram Eul. ρ 3 3 LP pres. vol. meter

MLP force newton ≡ Lag. tR 2 2 2 pascal Traction LRT ref. area meter M force newton ≡ Eul. t 2 2 pascal LPT pres. area meter 156 Mechanical MLP force newton ≡ Lag. P 2 2 2 pascal Stress LRT ref. area meter M force newton ≡ Eul. T 2 2 pascal LPT pres. area meter

LP meter Velocity v T second

LP Deformation gradient F LR 2 LP Green’s deformation C 2 LR

Continued on the next page . . . Table6.1–Continued...

Type Quantity Form Symbol Fund. Units Derived Units SI Units

3 Mechanical LP Volumetric deformation J 3 LR 2 MLP energy watt Lag. hR 2 3 · 2 Heat ﬂux rate LRT ref. area time meter M energy watt Eul. h T3 pres. area · time meter2 2 MLP energy watt qR Lag. 2 3 · 2 LRT ref. area time meter

157 Heat ﬂux vector Thermal M energy watt Eul. q T3 pres. area · time meter2

Temperature Θ θ kelvin

L2 energy joule Speciﬁc internal energy ε P T2 mass kilogram L2 energy joule Speciﬁc entropy η P T2 θ mass · temp. kilogram · kelvin 6.7 Two-way coupling between thermomechanics and electromagnetism

Eulerian formulations of continuum electrodynamics incorporate a two-way coupling between the thermomechanical conservation laws (6.50a)-(6.50d) and Maxwell’s equations (6.50g)-(6.50j). This coupling drives the interaction between mechanics, thermodynamics, and electromagnetism. On one hand, the coupling terms f em, Γem, and rem in the conservation laws model the force, torque, and energy exerted by elec-

tromagnetic ﬁelds on the deforming continuum [209, 210]. On the other hand, the

rest-frame electromagnetic fields e∗, d∗, p∗, h∗, b∗, m∗, σ∗,andj∗ model the effect of motion and mechanical deformation on the laboratory-frame electromagnetic fields e, d, p, h, b, m, σ,andj [207, 209, 236]. (Conversely, in Lagrangian formulations, there is no notion of rest-frame or laboratory-frame electromagnetic fields. Thus, the coupling is just one-way.)

6.7.1 Eﬀective electromagnetic ﬁelds

Eulerian modeling of deformable thermo-electro-magneto-mechanical materials involves two sets of Eulerian electromagnetic fields: the effective fields e∗, d∗, p∗, h∗, b∗, m∗, σ∗,andj∗, distinguished in our notation by superscript asterisks, and the standard fields e, d, p, h, b, m, σ,andj. The effective fields are the electromagnetic fields acting on the deforming continuum as seen in its present configuration, measured with respect to a co-moving or rest frame, i.e., a frame affixed to but not deforming with the continuum [207, 209, 236]. The standard fields also act on the deforming continuum as seen in its present configuration, but are instead measured with respect to a fixed or laboratory frame [207,209,236]. In the absence of motion,

158 the effective fields reduce to the standard fields; that is, when v = 0,thene∗ collapses to e, d∗ collapses to d,andsoon.

Various interaction models have been presented in the literature that relate the effective fields to the standard fields. In this section, we present four representative theories, each based on a different set of physical principles or empirical postulates.

Minkowski interaction model

The Minkowski model is motivated by Einstein’s special theory of relativity and appeals to classical rigid body electrodynamics [207,209,210]. In this approximation, the effective fields are related to the standard fields through semi-relativistic inverse

Lorentz transformations:

e∗ = e + v × b, h∗ = h − v × d, d∗ = d, b∗ = b,

p∗ = p, m∗ = m + v × p, j∗ = j − σv,σ∗ = σ. (6.64)

Lorentz interaction model

The Lorentz model is motivated by Lorentz’s theory of electrons [263], which postulates that a body consists of an infinitely large number of rapidly moving charged particles. The motion of these charged particles, in turn, generates rapidly fluctuating microscopic electromagnetic fields. The microscopic fields averaged over a small time interval and an infinitesimal volume are defined as the corresponding macroscopic

159 ﬁelds. This leads to the following relationships:

∗ ∗ ∗ ∗ e = e + v × b, h = h − o v × e, d = d, b = b,

p∗ = p, m∗ = m, j∗ = j − σv,σ∗ = σ. (6.65)

Statistical interaction model

The Statistical model [264] is a modiﬁcation of Lorentz’s theory, wherein the charged particles are grouped into stable structures (e.g., atoms, molecules, or ions).

The field effects of the charged particles within each stable structure are represented by microscopic electric and magnetic multipole moments (e.g., dipole, quadrupole, and octupole moments), and the macroscopic polarization and magnetization fields are defined as statistical averages of these multipole moments over a large number of stable structures. The transformations corresponding to this model are identical to

Minkowski’s, i.e.,

e∗ = e + v × b, h∗ = h − v × d, d∗ = d, b∗ = b,

p∗ = p, m∗ = m + v × p, j∗ = j − σv,σ∗ = σ. (6.66)

Chu interaction model

The Chu model [265] is based on the postulate that deforming bodies contribute to electromagnetic phenomena by acting, in a macroscopic sense, as electric and magnetic dipole sources for the electromagnetic ﬁelds. The transformations for the

160 Chu model are

∗ ∗ ∗ ∗ e = e + μo v × h, h = h − o v × e, d = d, b = b,

p∗ = p, m∗ = m, j∗ = j − σv,σ∗ = σ. (6.67)

6.7.2 Electromagnetically induced coupling terms

The electromagnetic body force f em, body couple Γem, and energy supply rate rem

in the thermomechanical conservation laws (6.50a)-(6.50d) model the force, torque,

and energy exerted by electromagnetic ﬁelds on the deforming continuum [209,210].

The mathematical forms of the coupling terms are motivated by either atomic physics

or empirical postulates [207, 209, 210], as was the case with the eﬀective ﬁelds (refer

to the interaction models in Section 6.7.1). As an example, the coupling terms in the

Minkowski interaction model for a polarizable, magnetizable, deformable continuum

are, in Eulerian form [210],

em ∗ ∗ ∗ ∗ ∗ T ∗ ∗ T ∗ ∗ ∗ ∗ ∗ ρf = σ e + j ×b + (grad e ) p + μo (grad h ) m + ˚d ×b + d ×˚b , (6.68a)

em ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ρΓ = e ⊗ p − p ⊗ e + μo h ⊗ m − m ⊗ h , (6.68b)

˙∗ ˙ ∗ em ∗ ∗ ∗ p ∗ m ρr = j · e + ρe · + ρμo h · , (6.68c) ρ ρ where ( ) ⊗ ( ) denotes the dyadic product of two vectors, “grad” is the Eulerian

gradient, and

˚a = a +curl(a × v)+v (div a) (6.69)

161 is a convected rate of an arbitrary vector-valued function a(x,t) of present position x and time t (cf. (6.69) with the integrand in the transport theorem (6.45)).

6.8 Maxwell stress and total Cauchy stress

To simplify the presentation of the theory and facilitate its use in applications, we now deﬁne a total Cauchy (true) stress tensor [174]

T = T + Tem (6.70) that consists of two contributions: one from the ‘Cauchy stress’ T associated with the surface traction t, and the other from the ‘Maxwell stress’ Tem.TheMaxwell stress is deﬁned so that its divergence is ρf em and twice its skew part is ρΓem, i.e.,

div Tem = ρf em, Tem − (Tem)T = ρΓem. (6.71)

Using (6.71), it can be shown that the Maxwell stress corresponding to the Minkowski interaction model (6.68a)-(6.68b) is

em ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ T = e ⊗ d + h ⊗ b − o e · e + μo h · h I; (6.72) 2 refer to Proof C.10 in Appendix C. As a consequence of (6.70) and (6.71), the Eulerian forms of conservation of linear and angular momentum (6.50b)-(6.50c) simplify to

ρv˙ = ρf m +divT , T = T T, (6.73)

162 regardless of the choice of interaction model (i.e., form of the Maxwell stress). Al-

though the total Cauchy stress T is symmetric (as a consequence of balance of angular

em momentum (6.73)2), its constituents T and T are generally not symmetric.

The superiority of the ‘total stress’ T has been demonstrated (and championed)

by Dorfmann et al. [174, 213, 214, 217, 230, 235, 266] and Triantafyllidis [176], and its

use is now customary in the modern electroelasticity and magnetoelasticity litera-

ture. Besides symmetry, a key advantage of working with T instead of T (and thus

(6.73) instead of (6.50b)-(6.50c)) is that the electromagnetic body force, body couple, and Maxwell stress need not be speciﬁed explicitly via an interaction model (as was done here in (6.68a)-(6.68c) and (6.72) for completeness). Consequently, one need not quantify how the total stress T is additively decomposed into a short-range

‘mechanical’ part T and a long-range ‘electromagnetic’ part Tem, a long-standing source of debate in the literature. (This issue remains unresolved due to the inability to separate and measure the ‘mechanical’ and ‘electromagnetic’ contributions in a laboratory28 [174,200,207,210,237,252,256,266].)

28Along these lines, Toupin [237] remarked “Any division of ... stress ... into electromagnetic and mechanical components is bound to be somewhat arbitrary ...”

163 Chapter 7: A Finite-Deformation Framework for Nonlinear Electro-Magneto-Elastic Materials, with Application to Magnetoelectric Polymer Composites

Soft magnetoelectric (ME) materials that couple ferroelectricity and ferromagnetism have recently attracted significant attention due to their unique properties and technological promise. In this chapter, we develop a theoretical framework suitable for modeling this intriguing class of materials. Finite deformations, electro- magneto-elastic coupling, and material nonlinearities are explicitly incorporated into our model. A particular emphasis is placed on the ground-up development of constitutive equations to facilitate material characterization in the laboratory. Accordingly, a catalogue of free energies and constitutive equations is presented, each employing a different set of independent variables. The ramifications of invariance, angular momentum, incompressibility, and material symmetry are explored, and a representative

(neo-Hookean-type) free energy with full electro-magneto-elastic coupling is posed.

7.1 Introduction

Materials that exhibit magnetoelectric (ME) coupling, wherein the material po- larizes in response to a magnetic ﬁeld (direct ME eﬀect) or magnetizes in response

164 to an electric field (converse ME effect), have attracted significant attention in recent

years [170, 180–184, 267–275]. Much of this interest stems from the tantalizing abil-

ity to electrically control a material’s ferromagnetism, or, conversely, magnetically control a material’s ferroelectricity [275]. If leveraged, this capability could lead to the development of unprecedented new technologies. One promising example lies in the realm of information storage, where the magnetoelectric effect could be harnessed to allow data to be written electrically and read magnetically [171, 172, 182]. Other envisioned applications include high-sensitivity magnetic field detectors, magnetoelectric transducers, spintronic devices, solid-state transformers, microwave devices, wireless energy transfer, sensors, attenuators, filters, and vibration energy harvesting [169,180,181,183–185].

A curious and distinguishing feature of magnetoelectric materials is the coexistence and coupling of ferroelectricity and ferromagnetism. Accordingly, ME materials are often dubbed ‘multiferroic’ to signify their simultaneous exhibition (and coupling) of two ferroic orders. Although naturally occurring single-phase ME materials were ﬁrst discovered as early as the 1950s, their technological implementation was hindered by

ME coupling that was either too weak, or occurred at temperatures that were too low, to be useful in practical devices. (Typical ME coeﬃcients for single-phase materials range from about 1 to 20 mV/cm Oe [170].) High electrical conductivities proved to be another technological impediment, rendering naturally occurring ME materials poor candidates for information storage applications [169].

More recently, researchers have turned toward composites as a means to engineer materials with strong magnetoelectric coupling. In fact, ME coeﬃcients several orders of magnitude higher than those of single-phase materials have been attained [170].

165 Generally, ME composites have both piezoelectric and magnetostrictive constituents and exploit the so-called ‘product property’ concept [227] to generate ME coupling as a secondary, or indirect, eﬀect. For instance, when an electric ﬁeld is applied to a ME composite, strain is induced in the piezoelectric phase. When this strain is transmitted to the magnetostrictive phase, a magnetization results. Thus, the electric

ﬁeld indirectly induces a magnetization in the composite, mediated by the exchange of elastic deformation between the two constituents, neither of which is magnetoelectric in and of itself.

Much of the research on ME composites to date has occurred in the applied physics and materials science communities. The primary emphasis of this work has been on tailoring the design of the composites to optimize the magnitude of the ME coupling.

Numerous design-related aspects have been explored, including different structures, configurations, constituent materials, dimensions, interfacial bonding mechanisms, and fabrication techniques [181]. Most often, ME composites are constructed by (i) dispersing particles in a matrix (particulate or granular composites) or (ii) layering different materials into a sandwich-like structure (laminated composites or multi- layers). Particulate composites generally exhibit weaker ME coupling than their laminated counterparts, with the highest ME coefficients measuring less than 100 mV/cm Oe [181]. This limitation is due to factors such as chemical reactions between constituents, percolation, leakage currents, and mechanical defects [181].

Laminated composites, on the other hand, do not suﬀer from the majority of these drawbacks. Furthermore, their ME coupling is enhanced by large interfacial surface areas, which enhance the transfer of strain from one constituent to another [180].

Brittle ceramics such as lead zirconate titanate (PZT) for the piezoelectric phase and

166 magnetic alloys such as Terfenol-D for the magnetostrictive phase are the most popular choices [169, 170]. Other common materials include PMN-PT (lead magnesium niobate–lead titanate), PZN-PT (lead zirconate niobate–lead titanate), and barium titanate for the piezoelectric phase; and Galfenol (iron-gallium alloy), Permendur

(iron-cobalt alloy), Metlgas (metallic glass), nickel ferrite, and cobalt ferrite for the magnetostrictive phase [169,183,276]. Representative ME coeﬃcients for several layered ceramic/alloy ME composites are about 1.01 V/cm Oe for a PMN-PT/Galfenol laminate [276] and about 3.1 V/cm Oe for an aluminum nitride/ferromagnetic glass laminate [277].

Ceramic/alloy-based magnetoelectric composites such as those discussed in the preceding paragraph often suﬀer from undesirable dielectric and eddy current losses at material interfaces [170, 183]. Also, ceramic/alloy-based composites are typically expensive, dense, and brittle [170–172]. Consequently, they are challenging to fabri- cate and limited to small-deformation applications. Polymer-based ME composites, on the other hand, are generally lighter, less expensive, easier to manufacture, and highly compliant, making them well suited for applications requiring large deformations, e.g., stretchable electronics, soft machines, non-planar conﬁgurations, memory devices, and soft robots [169–172].

ME polymer composites are most often constructed by (i) dispersing magnetostrictive particles in an electro-active polymer matrix (particulate composites) or (ii) stacking alternating layers of electro-active polymer and magnetostrictive alloy into a sandwich-like structure (laminated composites) [170,183]. Common materials include

PVDF and P(VDF-TrFE) for the electro-active polymer; and iron oxide, cobalt ferrite, Terfenol-D, and Metlgas for the magnetostrictive alloy. In general, the laminated

167 composites attain the largest ME couplings at the lowest applied magnetic ﬁelds [170].

For instance, a coupling coeﬃcient of 21.46 V/cm Oe was recently obtained in a

PVDF/Metglas laminated composite at a magnetic ﬁeld of only 8 Oe [170,278]; values of less than 50 mV/cm Oe are standard for particulate composites [170], although somewhat higher values, e.g., 300 mV/cm Oe, have been reported [279].

Numerous mathematical models have been developed for (hard) ceramic/alloy- based ME composites, e.g., [280–292], most of which focus on aspects of constitutive modeling, such as determining ‘effective’ material properties or applying homogenization techniques. These models are generally confined to small deformations and employ linear constitutive equations, standard simplifications for hard materials.

Less activity has occurred in the way of modeling and characterizing (soft) polymer- based ME composites, which require a ﬁnite-deformation framework that incorporates both geometric and constitutive nonlinearities. In a rather general sense, the theoretical foundations needed to facilitate this type of modeling have been collected, summarized, reviewed, and extended in the monographs of Pao [207], Maugin [208],

Eringen & Maugin [209], Hutter & van de Ven [210], and Kovetz [211], which also provide extensive sets of references to the seminal work in continuum electrodynamics (e.g., Toupin [228], Truesdell & Toupin [201], and Brown [204]). There is a gap, however, between the degree of generality and rigour aﬀorded by these theoretically oriented monographs and the level of specialization and simplicity required for convenient implementation in practical applications.

168 ‘apparent’ ME coupling, an idea he and coworkers cleverly leverage in [171, 172].29

Liu’s free energy function, however, does not include coupling between electricity,

magnetism, and elasticity, rendering these eﬀects uncoupled in his constitutive model.

Polarization and magnetization are selected as the independent variables (IVs).

It has been argued that constitutive models that use the electric ﬁeld or electric

displacement as an IV (rather than polarization), and the magnetic ﬁeld or magnetic

induction as an IV (rather than magnetization) – and the converse – can be advan- tageous in certain situations [176, 199, 213, 215, 230, 231]. To address this, Santapuri

On the computational front, a nonlinear ﬁnite-element framework was recently set forth by Guo & Ghosh [234] for the coupled electro-magneto-mechanical problem.

An important contribution of this work is the inclusion of fully dynamic deformations

and electromagnetic ﬁelds, a notable departure from the customary quasi-static as-

sumption. The model employed by the authors includes a one-way coupling between

29Liu & Sharma et al. [171,172] recently proposed a novel mechanism for generating strong magnetoelectric coupling in soft materials. The authors show that by embedding charge in a nonlinearly elastic dielectric and applying a magnetic ﬁeld, Maxwell stresses work in tandem with geometric nonlinearities to produce a strong ‘apparent’ ME coupling without the need for piezoelectric or magnetostrictive constituents.

169 electromagnetism and mechanical deformations via the deformation-dependent ‘rest-

frame’ electromagnetic ﬁelds, excluding coupling in the momentum equations (via the

Maxwell stress) and the ﬁnite-strain constitutive model.

It is thus the intent of this chapter to build upon the aforementioned work in

ﬁnite-strain electro-magneto-elasticity and develop a convenient theoretical frame-

work for modeling soft ME composites, with practical applications in mind. Finite

deformations, electro-magneto-elastic coupling, and material nonlinearities will be incorporated into the model. In the spirit of Dorfmann & Ogden [174, 213, 235], a particular emphasis will be placed on the ground-up development of (simple forms of the) constitutive equations to facilitate material characterization in the laboratory.

Accordingly, a catalogue of free energies and constitutive equations will be presented, each employing a different set of independent variables. As the optimal choice of IV varies from one scenario to another [176, 199, 213, 215, 230, 231], our extensive catalogue will provide theoreticians and experimentalists with the needed flexibility. The ramifications of invariance, angular momentum, incompressibility, and material symmetry will be explored, and a representative (neo-Hookean-type) free energy with full electro-magneto-elastic coupling will be posed.

7.2 Kinematics, fundamental laws, and boundary conditions

In this chapter, we consider a soft magnetoelectric (ME) composite consisting of micron-sized magnetostrictive (e.g., Terfenol-D or Galfenol) and piezoelectric (e.g.,

PZT or barium titanate) particles dispersed in a natural rubber matrix. Analogous to rubber elasticity, a ﬁnite-deformation description of the kinematics, kinetics, fundamental laws, and constitutive response is needed to accommodate the geometric

170 (e.g., strain-displacement) and material (e.g., stress-strain) nonlinearities that accom-

pany large elastic strains. Our key assumptions include quasi-static electromagnetic

ﬁelds (electrostatics), isothermal static deformations, no rate eﬀects, no hysteresis,

no gradient eﬀects, and a random distribution of particles.

In this section, we specialize the mathematical framework set forth in Chapter

6 to obtain appropriate descriptions of the kinetics, kinematics, and fundamental

laws for soft ME materials. Finite-strain constitutive equations are then developed in

Section 7.3 by appealing to thermodynamics, invariance requirements, and material symmetry.

7.2.1 Kinematics

In ﬁnite-deformation theory, the motion

x = χ(X) (7.1)

maps each particle in the body from its reference position X to its present position

x. Consequently, the body itself is mapped from its reference conﬁguration RR to its present conﬁguration R. As is customary, the motion (7.1) is assumed to be smooth and invertible. Note that we have restricted our attention to the static case by omitting time dependence in (7.1).

The deformation, or change in local geometry, is given by the deformation gradient

def ∂χ F =Gradχ = , ∂X

171 where Grad denotes the gradient with respect to the reference conﬁguration. The

Jacobian of the deformation gradient, i.e.,

J =detF > 0, is a measure of local dilatation or volume change of the material. The (symmetric) left and right Cauchy-Green deformation tensors

def def B = FFT, C = FTF (7.2) are alternative measures of deformation that will appear frequently during our development of the constitutive equations in Section 7.3.

7.2.2 Fundamental laws

Maxwell’s equations

The electromagnetic balance laws (or Maxwell’s equations) for a polarizable, magnetizable, nonlinearly elastic material consist of pointwise statements of Faraday’s law, the Amp`ere-Maxwell law, Gauss’s law for magnetism, and Gauss’s law for electricity, whose most general forms are given in (6.50g)-(6.50j) in Chapter 6. Specializ- ing to an electrically neutral insulator (so that the free charge σ and the free current j both vanish) and electrostatics30 (so that the electromagnetic ﬁelds are quasi-static) reduces (6.50g)-(6.50j) to

curl e = 0, curl h = 0, div d =0, div b =0. (7.3)

30By specializing to static deformations, the rest-frame electromagnetic fields in the theory of Chapter 6 collapse to the laboratory-frame fields. Further, specializing to electrostatics implies that the electromagnetic fields are time-independent. Together, these assumptions lead to dramatic simplifications of the fully dynamic theory of Chapter 6.

172 Conservation of charge is trivially satisfied. Equations (7.3)1–(7.3)4 are the Eulerian forms of Maxwell’s equations. The Eulerian (or true) electric field e, magnetic field h, electric displacement d, and magnetic induction b are functions of present position x and correspond to the present configuration of the deforming continuum. Note that lowercase letters (e.g., curl and div) denote a mathematical operation with respect to x.

Recall from Chapter 6 that the Eulerian polarization p and magnetization m (both per unit present volume) are deﬁned through the customary algebraic relationships

d = p + o e, b = μo m + h , (7.4)

where o and μo are the electric permittivity and magnetic permeability, respectively, in vacuo.

The corresponding Lagrangian forms of the electromagnetic balance laws, specialized from the more general forms (6.53g)-(6.53j) in Chapter 6 to quasi-static ﬁelds and no free charges or currents, are

Curl eR = 0, Curl hR = 0, Div dR =0, Div bR =0. (7.5)

In (7.5)1–(7.5)4, the subscript R denotes a Lagrangian (or pull-back) ﬁeld, i.e., a

ﬁeld corresponding to the reference conﬁguration of the deforming continuum. All

Lagrangian ﬁelds are functions of reference position X. Note that uppercase letters

(e.g., Curl and Div) denote a mathematical operation with respect to X.

173 Recall from Chapter 6 that the transformations between the Eulerian and La- grangian electromagnetic ﬁelds are given by

T −1 −1 eR = F e, dR = J F d, pR = J F p,

T −1 T hR = F h, bR = J F b, mR = F m, (7.6)

where FT and F−1 denote the transpose and inverse, respectively, of the deformation

gradient tensor F. It then follows that the Lagrangian form of (7.4) is

−1 −1 dR = pR + o J C eR, bR = μo J C hR + mR . (7.7)

Mechanical balance laws and electromagnetic coupling terms

The mechanical balance laws consist of pointwise statements of conservation of mass, linear momentum, and angular momentum, specialized to static equilibrium and no mechanical body forces, i.e.,

em em T ρJ = ρR, div T + ρf = 0,ρΓ + T − T = 0. (7.8)

Refer to Chapter 6. In (7.8)1–(7.8)3, ρR and ρ are the mass densities in the reference and present conﬁgurations, T is the ‘Cauchy stress’ tensor, f em is the electromagnetic body force (per unit mass), and Γem is the electromagnetic body couple (per unit mass). The mechanical balance laws (7.8)1–(7.8)3 are complemented by the energy balance equation

em ρR ε˙ = J T · L + ρR r , (7.9)

174 where ( ) · ( ) denotes the inner product of two vectors, and a superposed dot now

denotes a virtual increment (or virtual change) rather than a rate as in previous

chapters. Equation (7.9) is the Lagrangian form of the ﬁrst law of thermodynamics

(6.50d) from Chapter 6 specialized to no thermal eﬀects and quasi-static ﬁelds and

deformations. In this quasi-static context, Eq. (7.9) can be interpreted as a ‘virtual’

energy balance [174], with ε the internal energy (per unit mass), L =gradu the Eule-

rian gradient of the ‘virtual’ displacement increment u,andrem the electromagnetic energy increment.

The electromagnetic coupling terms f em, Γem,andrem appearing in the mechanical balance laws (7.8)1–(7.8)3 and energy balance (7.9) arise due to interactions between the electromagnetic ﬁelds and the deforming continuum. The mathematical forms of f em, Γem,andrem are made explicit by selecting one of the many interaction models in the literature, e.g., Minkowski, Lorentz, or Chu; refer to [207, 209, 210, 236, 245, 258] for additional details. Here, we select the popular

Minkowski interaction model (refer to (6.68a)-(6.68c) in Chapter 6), which – when specialized to quasi-static ﬁelds and static deformations, and in the absence of free current and free charge – becomes

em T T ρf = (grad e) p + μo (grad h) m, (7.10a)

em ρΓ =(e ⊗ p − p ⊗ e)+μo (h ⊗ m − m ⊗ h) , (7.10b)

em ρR r = e · p˙ r + μo h · m˙ r, (7.10c)

175 where ( )⊗( ) denotes the dyadic product of two vectors and grad denotes the gradient

with respect to x. Note that in (7.10c), following Dorfmann & Ogden [174], we

have introduced the Eulerian polarization per unit reference volume pr and Eulerian magnetization per unit reference volume mr, which are related to their ‘per unit present volume’ counterparts by

def def pr = J p, mr = Jm. (7.11)

Note the difference between the fields defined in (7.11) and the Lagrangian (or pull-

back) ﬁelds pR and mR deﬁned in (7.6).

As is now customary, following Kovetz [211], we introduce the ‘total Cauchy stress’

T , which consists of contributions from the ‘Cauchy stress’ T and the ‘Maxwell stress’

Tem, i.e.,

def T = T + Tem. (7.12)

The Maxwell stress Tem is deﬁned so that its divergence is ρf em (the electromagnetic body force per unit present volume) and twice its skew part is ρΓem (the electromagnetic body couple per unit present volume), i.e.,

def def div Tem = ρf em, Tem − (Tem)T = ρΓem. (7.13)

Using (7.10a), (7.10b), and (7.13), it can be shown that the Maxwell stress corresponding to the Minkowski interaction model is31

em 1 T = e ⊗ d + h ⊗ b − o e · e + μo h · h I. (7.14) 2

31Note that, in general, the ‘Maxwell stress’ varies from one interaction model to another.

176 As a consequence of (7.12) and (7.13), the mechanical balance laws (7.8)1–(7.8)3 can be rewritten as

T ρJ = ρR, div T = 0, T = T , (7.15a) regardless of the choice of interaction model (Maxwell stress). Although the total

Cauchy stress T is symmetric (as a consequence of balance of angular momentum

em (7.15a)3), its constituents T and T are generally not symmetric.

The Eulerian forms of linear and angular momentum (7.15a)2 and (7.15a)3 can be written in Lagrangian form in terms of the ‘total ﬁrst Piola-Kirchhoﬀ stress’ P,

Div P = 0, P FT = FP T, (7.15b) or the ‘total second Piola-Kirchhoﬀ stress’ S,

Div FS = 0, S = S T. (7.15c)

Note that the true and nominal forms of the total stress are related through the transformations

P = J T F−T, S = J F−1T F−T. (7.16)

The superiority of the ‘total stress’ T has been demonstrated (and championed) by Dorfmann et al. [174,213,214,217,230,235,266] and Triantafyllidis [176], and its use is now customary in the modern electroelasticity and magnetoelasticity literature.32

Besides symmetry, a key advantage of working with T instead of T (and thus (7.15a)

instead of (7.8)) is that the electromagnetic body force, body couple, and Maxwell

32Refer to Rinaldi & Brenner [252] for criticisms on the use of the ‘total stress’ tensor.

177 stress need not be speciﬁed explicitly via an interaction model (as was done here in

(7.10a)-(7.10c) and (7.14) for completeness). Consequently, one need not quantify

how the total stress T is additively decomposed into a ‘mechanical’ part T and an

‘electromagnetic’ part Tem, a long-standing source of debate in the literature. (This

issue remains unresolved due to the inability to separate and measure the ‘mechanical’

and ‘electromagnetic’ contributions in a laboratory [174, 200, 207, 210, 237, 252, 256,

266].) The utility of working with the total stress will also emerge when developing

the constitutive equations in Section 7.3 and implementing boundary conditions.

To complete our model, we substitute the electromagnetic energy (7.10c) into the

virtual energy balance (7.9) to obtain

−T ρR ε˙ = J TF · F˙ + e · p˙ r + μo h · m˙ r, (7.17) where we have used the deﬁnition of the trace, i.e., A · B =tr(ABT), and the relationship F˙ = LF. Note that in (7.17), conjugate pairs of mechanical, electrical, and magnetic quantities contribute to virtual changes in the internal energy. The

‘Cauchy stress’ T (or, more precisely, the ‘ﬁrst Piola-Kirchhoﬀ stress’ P ≡ J TF−T) and deformation gradient F are the mechanical work conjugates, the Eulerian electric

ﬁeld e and Eulerian polarization per unit reference volume pr are the electrical work conjugates, and the Eulerian magnetic ﬁeld h and Eulerian magnetization per unit reference volume mr are the magnetic work conjugates.

7.3 Constitutive equations

To complete the model, Maxwell’s equations ((7.3)1–(7.3)4 in Eulerian form or

(7.5)1–(7.5)4 in Lagrangian form) and the mechanical balance laws (7.15a)1–(7.15a)3

178 must be complemented by constitutive equations that describe the response of the

magnetoelectric (ME) polymer. Recall from Section 7.2 that, for the sake of simplicity,

we restrict our attention to nonlinear electro-magneto-elastic response that is free of

hysteresis, energy dissipation, rate dependence, and gradient eﬀects. Hence, only

fully reversible processes are possible.

7.3.1 Energy formulations with Eulerian IVs

In this section, we develop a catalogue of constitutive equations, with each formulation employing a diﬀerent free energy and set of Eulerian independent variables

(IVs); refer to Tables 7.1 and 7.2. As the setting is quasi-static and isothermal, the virtual energy balance (7.17) plays the customary role of the second law of thermodynamics, placing restrictions on the forms of the constitutive equations (`alaColeman and Noll [240,241,293]). In what follows, these restrictions are highlighted for several representative formulations.

179 Table 7.1: Free energies corresponding to diﬀerent sets of Eulerian IVs

IVs Energy Legendre transformation

F, pr, mr ε N/A

F, p, m εˆ N/A

Fph Fph F, pr, h ψ ρR ψ = ρR ε − μo h · mr

F, p, h ψˆFph N/A

Fem Fem F, e, mr ψ ρR ψ = ρR ε − e · pr

F, e, m ψˆFem N/A

Feh Feh F, e, h ψ ρR ψ = ρR ε − e · pr − μo h · mr

Feb Feb Feh 1 F, e, b ψ ρR ψ = ρR ψ − μo J m · m 2

Fdh Fdh Feh 1 F, d, h ψ ρR ψ = ρR ψ − J p · p 2 o

Fdb Fdb Feh 1 1 F, d, b ψ ρR ψ = ρR ψ − J p · p − μo J m · m 2 o 2

180 Table 7.2: Constitutive equations for energy formulations with Eulerian IVs

IVs ‘Cauchy stress’ ‘Maxwell stress’ Electric DV Magnetic DV

∂ε T em ∂ε ρR ∂ε F, pr, mr ρ F T e = ρR h = ∂F ∂pr μo ∂mr ∂εˆ T em ∂εˆ ρ ∂εˆ F, p, m ρ F T − e · p + μo h · m I e = ρ h = ∂F ∂p μo ∂m

Fph Fph Fph ∂ψ T em ∂ψ ρ ∂ψ F, pr, h ρ F T e = ρR m = − ∂F ∂pr μo ∂h 181 ∂ψˆFph ∂ψˆFph ρ ∂ψˆFph F, p, h ρ FT Tem − (e · p)I e = ρ m = − ∂F ∂p μo ∂h

Fem Fem Fem ∂ψ T em ∂ψ ρR ∂ψ F, e, mr ρ F T p = − ρ h = ∂F ∂e μo ∂mr ˆFem ˆFem ˆFem ∂ψ T em ∂ψ ρ ∂ψ F, e, m ρ F T − μo (h · m)I p = − ρ h = ∂F ∂e μo ∂m ∂ψFeh ∂ψFeh ρ ∂ψFeh F, e, h ρ FT Tem p = − ρ m = − ∂F ∂e μo ∂h

Feb Feb Feb ∂ψ T em 1 ∂ψ ∂ψ F, e, b ρ F T + μo (m · m)I p = − ρ m = − ρ ∂F 2 ∂e ∂b

Continued on the next page . . . Table7.2–Continued...

IVs ‘Cauchy stress’ ‘Maxwell stress’ Electric DV Magnetic DV

Fdh Fdh Fdh ∂ψ T em 1 ∂ψ ρ ∂ψ F, d, h ρ F T + (p · p)I p = − o ρ m = − ∂F 2 o ∂d μo ∂h Fdb Fdb Fdb ∂ψ T em 1 1 ∂ψ ∂ψ F, d, b ρ F T + p · p + μo m · m I p = − o ρ m = − ρ ∂F 2 o 2 ∂d ∂b em 1 Note that T = e ⊗ d + h ⊗ b − o e · e + μo h · h I 2 182 Internal energy formulation

Analogous to classical thermodynamics, the internal energy (per unit mass) ε in

(7.17) is regarded as the fundamental energy potential. The natural independent

variables are those appearing as increments in (7.17). Hence,

ε = ε F, pr, mr , i.e., the internal energy is regarded as a function of the deformation gradient F,

Eulerian polarization (per unit reference volume) pr, and Eulerian magnetization

(per unit reference volume) mr. Use of the chain rule on ε(F, pr, mr)gives

∂ε ∂ε ∂ε ε˙ = · F˙ + · p˙ r + · m˙ r, ∂F ∂pr ∂mr and substitution of this result into the virtual energy balance (7.17) gives −T ∂ε ∂ε ∂ε JTF − ρR · F˙ + e − ρR · p˙ r + μo h − ρR · m˙ r =0. ∂F ∂pr ∂mr (7.18)

Analogous to Coleman and Noll’s requirement that the second law of thermodynamics be satisﬁed for all thermodynamic processes [240, 241, 293], we demand that

(7.18) hold for all virtual electro-magneto-elastic processes. Since the coeﬃcients of the virtual increments are independent of the increments themselves, and the increments may be varied independently and are arbitrary, it follows that the coeﬃcients vanish, i.e.,

∂ε T ∂ε ρR ∂ε T = ρ F , e = ρR , h = . (7.19) ∂F ∂pr μo ∂mr

183 The constitutive equations (7.19) are consistent with those presented in [69,174,232].

The Maxwell stress for this formulation is given by (7.14), and thus the total Cauchy

(true) stress may be written as

em ∂ε T 1 T = T + T = ρ F + e ⊗ d + h ⊗ b − o e · e + μo h · h I. ∂F 2

We collectively coin the set of independent variables {F, pr, mr}, the energy potential

ε(F, pr, mr), and the constitutive equations (7.19) the internal energy formulation.

Refer to Table 7.2.

Electric ﬁeld–magnetic ﬁeld formulation

In this formulation, we promote the Eulerian electric ﬁeld e and Eulerian magnetic

ﬁeld h to independent variables, and concomitantly relegate the Eulerian polarization

(per unit reference volume) pr and Eulerian magnetization (per unit reference volume)

Feh mr to dependent variables. The free energy (per unit mass) ψ (F, e, h) is deﬁned as the Legendre transformation of the internal energy (per unit mass) ε(F, pr, mr) with respect to the electromagnetic variables, from pr to e and mr to h,

Feh ρR ψ = ρR ε − e · pr − μo h · mr. (7.20)

Refer to Table 7.1. The incremental form of (7.20) is

Feh ρR ψ˙ = ρR ε˙ − e · p˙ r − pr · e˙ − μo h · m˙ r − μo mr · h˙ ,

184 which, upon substitution into (7.17), gives the virtual energy balance for this formu-

lation:

Feh −T ρR ψ˙ = J TF · F˙ − pr · e˙ − μo mr · h˙ . (7.21)

Compare (7.21) with (7.17), and note that (i) the work conjugate of F˙ is unchanged and (ii) e and h now appear in incremental form, i.e., as natural independent variables.

Use of the chain rule on the free energy ψFeh gives

∂ψFeh ∂ψFeh ∂ψFeh ψ˙ Feh = · F˙ + · e˙ + · h˙ , ∂F ∂e ∂h and subsequent substitution of this result into (7.21) leads to Feh Feh Feh −T ∂ψ ∂ψ ∂ψ JTF − ρR · F˙ − pr + ρR · e˙ − μo mr + ρR · h˙ =0. ∂F ∂e ∂h

It follows from standard arguments that

Feh Feh Feh ∂ψ T ∂ψ ρR ∂ψ T = ρ F , pr = − ρR , mr = − . (7.22a) ∂F ∂e μo ∂h

Upon use of (7.11), we may alternatively write (7.22a)2 and (7.22a)3 as

∂ψFeh ρ ∂ψFeh p = − ρ , m = − . (7.22b) ∂e μo ∂h

The constitutive equations (7.22a) and (7.22b) are consistent with those presented

in [69,174,232]. The Maxwell stress for this formulation is given by (7.14), and thus

the total Cauchy stress may be written as

Feh em ∂ψ T 1 T = T + T = ρ F + e ⊗ d + h ⊗ b − o e · e + μo h · h I. ∂F 2

185 We collectively coin the set of independent variables {F, e, h}, the energy potential

ψFeh(F, e, h), and the constitutive equations (7.22a)-(7.22b) the deformation–electric

ﬁeld–magnetic ﬁeld formulation. Refer to Table 7.2.

Electric displacement–magnetic induction formulation

Since the Eulerian electric displacement d and magnetic induction b are not part of a conjugate pair in (7.17), a conventional Legendre transformation of the internal energy ε cannot be used to introduce them as independent variables. To circumvent this, a ‘Legendre-type’ transformation is used to deﬁne a new free energy ψFdb that employs d and b as independent variables, i.e., [174]

Fdb Feh 1 1 ρR ψ = ρR ψ − J p · p − μo J m · m. (7.23) 2 o 2

Refer to Table 7.1. Note that a similar strategy was proposed by Santapuri & Lowe

et al. in [69, 232]. The Legendre-type transformation (7.23) can be written in incre-

mental form as Fdb Feh 1 1 −T 1 ρR ψ˙ = ρR ψ˙ − J p · p + μo m · m F · F˙ − pr · p˙ − μo mr · m˙ , 2 o 2 o wherewehaveused

J˙ = J F−T · F˙ . (7.24)

Refer to Proof D.1 in Appendix D. Use of the incremental form of (7.23) in (7.21) leads to the virtual energy balance for this formulation:

Fdb −T 1 ρR ψ˙ = J TF¯ · F˙ − pr · d˙ − mr · b˙ , (7.25) o

186 where d and b have been introduced using the algebraic relationships in (7.4). Exami- nation of (7.25) reveals that the conjugate stress to the virtual deformation increment

F˙ is now def 1 1 T¯ = T − p · p + μo m · m I. (7.26) 2 o 2

Also note that d and b appear as increments in (7.25), i.e., as natural independent variables, with pr and mr, respectively, as their work conjugates.

Use of the chain rule on the free energy ψFdb gives

∂ψFdb ∂ψFdb ∂ψFdb ψ˙ Fdb = · F˙ + · d˙ + · b˙ , ∂F ∂d ∂b and subsequent substitution of this result into (7.25) leads to Fdb Fdb Fdb −T ∂ψ 1 ∂ψ ∂ψ JTF¯ − ρR · F˙ − pr + ρR · d˙ − mr + ρR · b˙ =0. ∂F o ∂d ∂b

It follows from standard arguments that

Fdb Fdb Fdb ∂ψ T ∂ψ ∂ψ T¯ = ρ F , p = − o ρ , m = − ρ , (7.27) ∂F ∂d ∂b where we have used (7.11). The constitutive equations (7.27) are consistent with those presented in [174] for the special cases of electroelasticity and magnetoelasticity. We collectively coin the set of independent variables {F, d, b}, the energy potential ψFdb(F, d, b), and the constitutive equations (7.27) the deformation–electric displacement–magnetic induction formulation. Refer to Table 7.2.

It is important to note that the ‘Cauchy stress’ in this formulation (denoted by T¯ , the work conjugate of F˙ in (7.25)) diﬀers from the ‘Cauchy stress’ in the previous two formulations (denoted by T, the work conjugate of F˙ in (7.17) and (7.21)). Hence, it

187 follows that the ‘Maxwell stress’ T¯ em in this formulation will diﬀer from the ‘Maxwell stress’ Tem in the previous formulations, since the total true stress must remain the

same, i.e.,

T = T + Tem = T¯ + T¯ em.

Refer to [174] for additional details. Deﬁnition (7.26) then implies that em em 1 1 T¯ = T + p · p + μo m · m I, 2 o 2 where Tem is given in (7.14).

Polarization–magnetization formulation

Recall that the internal energy formulation used pr and mr, the Eulerian polarization and magnetization per unit reference volume, as the electromagnetic independent variables. Suppose that we instead wish to use p and m, the Eulerian polarization and magnetization per unit present volume, as the independent variables. Follow- ing [69,232], we use deﬁnition (7.11), the product rule, and result (7.24) to obtain

˙ −T p˙ r = J p = J˙p + J p˙ = J p F · F˙ + J p˙ (7.28a)

and

˙ −T m˙ r = J m = J˙m + J m˙ = J m F · F˙ + J m˙ . (7.28b)

Results (7.28a) and (7.28b) allow the virtual energy balance (7.17) to be rewritten as

−T ρR εˆ˙ = J TFˆ · F˙ + J e · p˙ + μo J h · m˙ , (7.29)

188 wherewehavedeﬁned

def Tˆ = T + e · p + μo h · m I.

Note that p and m appear as increments in (7.29), i.e., as natural independent vari-

ables, and that the ‘modiﬁed’ internal energyε ˆ is a function of F, p,andm. Following

the standard procedure, it can be shown that

∂εˆ ∂εˆ ρ ∂εˆ Tˆ = ρ FT, e = ρ , h = . (7.30) ∂F ∂p μo ∂m

Since

T = T + Tem = Tˆ + Tˆ em, the Maxwell stress corresponding to this formulation is

em em Tˆ = T − e · p + μo h · m I, where Tem is given in (7.14).

7.3.2 Energy formulations with Lagrangian IVs

In this section, anticipating the restrictions that will soon be imposed by invariance requirements (refer ahead to Section 7.3.4), we modify the energy potentials and constitutive equations presented in Section 7.3.1 (and catalogued in Tables 7.1 and

7.2) to accommodate Lagrangian (rather than Eulerian) quantities as independent variables. In what follows, we illustrate this process for several representative formulations; the free energies and constitutive equations for the remaining formulations are presented in Table 7.3.

189 Table 7.3: Constitutive equations for energy formulations with Lagrangian IVs

IVs Total true stress T (‘Cauchy’ plus ‘Maxwell’) Electric DV Magnetic DV Fpm Fpm Fpm ∂E T 1 1 1 −T ∂E ∂E F, pR, mR ρ F + o e ⊗ e − (e · e)I + b ⊗ b − (b · b)I e = ρR F b = ρF ∂F 2 μo 2 ∂pR ∂mR Fph Fph Fph ∂Ψ T 1 1 −T ∂Ψ ρ ∂Ψ F, pR, hR ρ F + o e ⊗ e − (e · e)I + μo h ⊗ h − (h · h)I e = ρR F m = − F ∂F 2 2 ∂pR μo ∂hR

190 Fem Fem Fem ∂E T 1 1 1 ∂E ∂E F, eR, mR ρ F + o e ⊗ e − (e · e)I + b ⊗ b − (b · b)I p = − ρF b = ρF ∂F 2 μo 2 ∂eR ∂mR Feh Feh Feh ∂Ψ T 1 1 ∂Ψ ρ ∂Ψ F, eR, hR ρ F + o e ⊗ e − (e · e)I + μo h ⊗ h − (h · h)I p = − ρF m = − F ∂F 2 2 ∂eR μo ∂hR Feb Feb Feb ∂Ψ T 1 1 1 ∂Ψ −T ∂Ψ F, eR, bR ρ F + o e ⊗ e − (e · e)I + b ⊗ b − (b · b)I p = − ρF m = − ρR F ∂F 2 μo 2 ∂eR ∂bR Fdh Fdh Fdh ∂Ψ T 1 1 1 −T ∂Ψ ρ ∂Ψ F, dR, hR ρ F + d ⊗ d − (d · d)I + μo h ⊗ h − (h · h)I p = − o ρR F m = − F ∂F o 2 2 ∂dR μo ∂hR Fdb Fdb Fdb ∂Ψ T 1 1 1 1 −T ∂Ψ −T ∂Ψ F, dR, bR ρ F + d ⊗ d − (d · d)I + b ⊗ b − (b · b)I p = − o ρR F m = − ρR F ∂F o 2 μo 2 ∂dR ∂bR Electric ﬁeld–magnetic ﬁeld formulation

Recall from Section 7.3.1 that the constitutive equations for the deformation– electric ﬁeld–magnetic ﬁeld formulation are

∂ψFeh ∂ψFeh ρ ∂ψFeh T = ρ FT, p = − ρ , m = − , (7.31) ∂F ∂e μo ∂h where the free energy (per unit mass) ψFeh is a function of the deformation gradient

F, Eulerian electric ﬁeld e, and Eulerian magnetic ﬁeld h. With the aid of (7.6), we

Feh deﬁne a new free energy (per unit mass) Ψ that uses the Lagrangian electric ﬁeld eR

and magnetic ﬁeld hR as independent variables in lieu of their Eulerian counterparts e and h:

Feh Feh −T −T def Feh ψ (F, e, h)=ψ (F, F eR, F hR) =Ψ (F, eR, hR).

The change of independent variables from e to eR and h to hR implies that the partial derivatives in (7.31) may be rewritten as

Feh Feh ∂ψ ∂Ψ 1 μo = − (e ⊗ p)F−T − (h ⊗ m)F−T (7.32a) ∂F ∂F ρ ρ and

∂ψFeh ∂ΨFeh ∂ψFeh ∂ΨFeh = F , = F . (7.32b) ∂e ∂eR ∂h ∂hR

191 Refer to Proofs D.2 and D.3 in Appendix D for additional details. As a consequence

of (7.32a) and (7.32b), the constitutive equations (7.31)1–(7.31)3 become

Feh ∂Ψ T T = ρ F − e ⊗ p − μo h ⊗ m, ∂F

∂ΨFeh ρ ∂ΨFeh p = − ρF , m = − F . (7.33) ∂eR μo ∂hR

Recall from Table 7.2 that the ‘Maxwell stress’ for this formulation is Tem,which is given by (7.14). It then follows that the total true stress (‘Cauchy’ T plus ‘Maxwell’

Tem)is Feh ∂Ψ T 1 1 T = ρ F + o e ⊗ e − (e · e)I + μo h ⊗ h − (h · h)I , (7.34) ∂F 2 2 where we have used the algebraic relations (7.4) to simplify the result.

The last two terms on the right-hand side of (7.34) are formally identical to the

Maxwell stress in vacuo. As discussed in [215], the Maxwell stress in vacuo is associated with the so-called ‘applied’ electric and magnetic ﬁelds, which exist independently of the body and are unaﬀected by its presence. In the absence of material, the free energy ΨFeh vanishes (as do the polarization p and magnetization m)sothat

the total stress (7.34) reduces to the Maxwell stress in vacuo [176,235].

Electric displacement–magnetic induction formulation

Recall from Section 7.3.1 that the constitutive equations for the deformation– electric displacement–magnetic induction formulation are

Fdb Fdb Fdb ∂ψ T ∂ψ ∂ψ T¯ = ρ F , p = − o ρ , m = − ρ , (7.35) ∂F ∂d ∂b

192 where the free energy (per unit mass) ψFdb is a function of the deformation gradient

F, Eulerian electric displacement d, and Eulerian magnetic induction b. With the aid of (7.6), we deﬁne a new free energy (per unit mass) ΨFdb that uses the Lagrangian electric displacement dR and magnetic induction bR as independent variables in lieu of their Eulerian counterparts d and b:

Fdb Fdb −1 −1 def Fdb ψ (F, d, b)=ψ (F,J FdR,J FbR) =Ψ (F, dR, bR).

The change of independent variables from d to dR and b to bR implies that the partial derivatives in (7.35) may be rewritten as Fdb Fdb ∂ψ ∂Ψ 1 1 −T = + (p ⊗ dR) − (p · d)F ∂F ∂F o ρ J 1 1 −T + (m ⊗ bR) − (m · b)F (7.36a) ρ J and

∂ψFdb ∂ΨFdb ∂ψFdb ∂ΨFdb = J F−T , = J F−T . (7.36b) ∂d ∂dR ∂b ∂bR

Refer to Proofs D.4 and D.5 in Appendix D for additional details. As a consequence of (7.36a) and (7.36b), the constitutive equations (7.35)1–(7.35)3 become

∂ΨFdb 1 T¯ = ρ FT + p ⊗ d − (p · d)I + m ⊗ b − (m · b)I, ∂F o

Fdb Fdb −T ∂Ψ −T ∂Ψ p = − o ρR F , m = − ρR F , (7.37) ∂dR ∂bR where we have used the relationships

S(a ⊗ b)=(Sa) ⊗ b, (a ⊗ b)S = a ⊗ (ST b), (7.38)

193 the transformations (7.6), and conservation of mass (7.15a)1.

Recall from Table 7.2 that the ‘Maxwell stress’ for this formulation is

em 1 1 −1 T¯ = e ⊗ d + h ⊗ b − o e · e + μo h · h I + p · p + μo m · m I. 2 2 o

It then follows that the total true stress (‘Cauchy’ T¯ plus ‘Maxwell’ T¯ em)is ∂ΨFdb 1 1 1 1 T = ρ FT + d ⊗ d − (d · d)I + b ⊗ b − (b · b)I , (7.39) ∂F o 2 μo 2 where we have used the algebraic relations (7.4) to simplify the result. As before, the last two terms in the total stress (7.39) are formally identical to the Maxwell stress in vacuo.33

Polarization–magnetization formulation

Recall from Section 7.3.1 that the constitutive equations for the deformation– polarization–magnetization formulation are

∂εˆ ∂εˆ ρ ∂εˆ Tˆ = ρ FT, e = ρ , h = , (7.40) ∂F ∂p μo ∂m where the internal energy (per unit mass)ε ˆ is a function of the Eulerian polarization p and magnetization m (per unit present volume) and the deformation gradient F.

Using (7.6), we deﬁne a new internal energy (per unit mass) E that employs the

Lagrangian polarization pR and magnetization mR as independent variables instead of their Eulerian counterparts p and m:

−1 −T def εˆ(F, p, m)=ˆε(F,J FpR, F mR) = E (F, pR, mR).

33Note that in vacuo, where the polarization p and magnetization m both vanish, the algebraic relations (7.4) collapse to the well-known aether relations d = o e and b = μo h.

194 It can be shown (refer to Proofs D.6 and D.7 in Appendix D) that as a consequence

of this change of independent variables, the partial derivatives ofε ˆ in (7.40) can be

rewritten as

∂εˆ ∂E 1 1 −T μo −T = − e ⊗ pR + (e · p)F + (m ⊗ h)F ∂F ∂F ρJ ρ ρ

and

∂εˆ ∂E ∂εˆ ∂E = J F−T , = F . ∂p ∂pR ∂m ∂mR

It then follows that the constitutive equations (7.40) become

∂E T Tˆ = ρ F − e ⊗ p + μo m ⊗ h +(e · p)I, ∂F

−T ∂E ρ ∂E e = ρR F , h = F . (7.41) ∂pR μo ∂mR

Recall from Table 7.2 that the ‘Maxwell stress’ for this formulation is

em 1 Tˆ = e ⊗ d + h ⊗ b − o e · e + μo h · h I − e · p + μo h · m I. 2

Hence, the total true stress (‘Cauchy’ Tˆ plus ‘Maxwell’ Tˆ em)is ∂E T 1 T = ρ F + o e ⊗ e − (e · e)I + μo m ⊗ h − (m · h)I ∂F 2

1 + h ⊗ b − μo (h · h)I. 2

195 Subsequent use of the algebraic relations (7.4) leads to a more convenient representation of the total stress, i.e., ∂E T 1 1 T = ρ F − μo m ⊗ m − (m · m)I + o e ⊗ e − (e · e)I ∂F 2 2 1 1 + b ⊗ b − (b · b)I . (7.42) μo 2

Note, however, that unlike (7.34) and (7.39), (7.42) is not in the standard form

∂ T = ρ (energy potential) FT + Maxwell stress in vacuo, ∂F which arose naturally in the previous formulations. To remedy this, the second term in (7.42) is absorbed into the energy potential as follows.

We begin by using (7.16)1 to rewrite the ﬁrst two terms in (7.42) in nominal form as

∂E −T 1 −T P = ρR − μo J (m ⊗ m)F + μo J (m · m)F + ... (7.43) ∂F 2

Subsequent use of the results

∂ (J m · m)=J (m · m)F−T − 2J (m ⊗ m)F−T ∂F

and

−1 J m · m = J mR · C mR

from Appendix D (refer to Proofs D.8 and D.9) allows us to rewrite (7.43) as ∂ μo J −1 P = ρR E + mR · C mR + ... (7.44) ∂F 2ρR

196 Then, by deﬁning the ‘augmented’ energy

Fpm def μo J −1 E (F, pR, mR) = E (F, pR, mR)+ mR · C mR (7.45) 2ρR and converting (7.44) back to its Eulerian (true) form using (7.16)1, the total Cauchy stress may be expressed as Fpm ∂E T 1 1 1 T = ρ F + o e ⊗ e − (e · e)I + b ⊗ b − (b · b)I . (7.46) ∂F 2 μo 2

The last two terms in the total stress (7.46) are formally identical to the Maxwell stress in vacuo, as desired.

It follows from partial diﬀerentiation of the augmented energy (7.45) that

Fpm Fpm −T ∂E ∂E e = ρR F , b = ρF , (7.47) ∂pR ∂mR where we have used the algebraic relations (7.4), the constitutive equations (7.41), and the result

∂ −1 −1 mR · C mR =2C mR. ∂mR

Refer to Proof D.10 in Appendix D.

An alternative development of the polarization–magnetization formulation

Contrast (7.40), our starting point in the previous section, with the internal energy formulation of Section 7.3.1, whose constitutive equations are

∂ε T ∂ε ρR ∂ε T = ρ F , e = ρR , h = . (7.48) ∂F ∂pr μo ∂mr

197 For the latter, the internal energy (per unit mass) ε is a function of the Eulerian

polarization pr and magnetization mr, both per unit reference volume. In what follows, we show that changing the independent variables in ε from pr to pR and mr

to mR leads to the same constitutive equations (7.46)-(7.47) developed in the previous section.

Equations (7.6) and (7.11) are used to deﬁne a new internal energy (per unit mass) E that employs the Lagrangian polarization pR and magnetization mR (instead of their Eulerian counterparts pr and mr) as independent variables:

−T def ε(F, pr, mr)=ε(F, FpR,JF mR) = E (F, pR, mR).

Using the chain rule, it can be shown (refer to Proof D.11 in Appendix D) that

∂ε ∂E 1 μo −T = − e ⊗ pR + m ⊗ h − m · h F . ∂F ∂F ρJ ρ

Consequently, the constitutive equation (7.48)1 for the ‘Cauchy stress’ can be rewritten as

∂E T T = ρ F − e ⊗ p + μo m ⊗ h − (m · h)I . ∂F

Recall from Table 7.2 that the ‘Maxwell stress’ for this formulation is

em 1 T = e ⊗ d + h ⊗ b − o e · e + μo h · h I. 2

198 It then follows that the total true stress (‘Cauchy’ T plus ‘Maxwell’ Tem), after some manipulation, is ∂E T 1 1 T = ρ F − μo m ⊗ m − (m · m)I + o e ⊗ e − (e · e)I ∂F 2 2 1 1 + b ⊗ b − (b · b)I , μo 2 which is identical to (7.42), as expected. Equations (7.46) and (7.47) subsequently follow using the procedure described in the previous section.

7.3.3 Augmented free energy formulations

In the previous section, we showed that regardless of the choice of independent variables, the total stress can always be written in the form

∂ T = ρ (free energy potential) FT + Maxwell stress in vacuo. ∂F

Refer to Table 7.3. In what follows, using similar ideas to those presented in [174,213,

235], we illustrate how – in certain instances – the Maxwell stress in vacuo can be absorbed into the free energy so that the total stress can be expressed in the compact form

∂ P = ‘augmented’ free energy potential . ∂F

199 Note that of the seven formulations presented in Table 7.3, only four are amenable to

this treatment.34 Refer to Table 7.4 for these four formulations and their correspond-

ing constitutive equations, which are now derived in detail for two representative

cases.

Electric ﬁeld–magnetic ﬁeld formulation

Recall from Table 7.3 that for this formulation, the total Cauchy stress is Feh ∂Ψ T 1 1 T = ρ F + o e ⊗ e − (e · e)I + μo h ⊗ h − (h · h)I . (7.49) ∂F 2 2

Also recall that the free energy ΨFeh employs the deformation gradient F, Lagrangian electric ﬁeld eR, and Lagrangian magnetic ﬁeld hR as independent variables. Using

(7.16)1, the total true stress (7.49) can be expressed in nominal form as Feh ∂Ψ 1 −T 1 −T P = ρR + o J e ⊗ e − (e · e)I F + μo J h ⊗ h − (h · h)I F . ∂F 2 2 (7.50)

Use of the results (refer to Proofs D.12 and D.13 in Appendix D)

∂ −T −T −1 (J e · e)=J (e · e)F − 2J (e ⊗ e)F ,Je · e = J eR · C eR ∂F

and

∂ −T −T −1 (J h · h)=J (h · h)F − 2J (h ⊗ h)F ,Jh · h = J hR · C hR ∂F

34 For instance, in the {F, eR, mR} formulation, the magnetic part of the Maxwell stress (which Fem depends on b) cannot be absorbed into the free energy E (which instead depends on mR).

200 Table 7.4: Constitutive equations for ‘augmented’ energy formulations with Lagrangian IVs

IVs ‘Augmented’ free energy (per unit reference volume) Total stress Electric DV Magnetic DV

Feh Feh Feh Feh def Feh 1 −1 1 −1 ∂E ∂E ∂E F, eR, hR E = ρR Ψ − o J eR · C eR − μo J hR · C hR P = dR = − bR = −

201 2 2 ∂F ∂eR ∂hR

Feb Feb Feb Feb def Feb 1 −1 1 ∂E ∂E ∂E F, eR, bR E = ρR Ψ − o J eR · C eR + bR · CbR P = dR = − hR = 2 2μoJ ∂F ∂eR ∂bR

Fdh Fdh Fdh Fdh def Fdh 1 1 −1 ∂E ∂E ∂E F, dR, hR E = ρR Ψ + dR · CdR − μo J hR · C hR P = eR = bR = − 2 oJ 2 ∂F ∂dR ∂hR

Fdb Fdb Fdb Fdb def Fdb 1 1 ∂E ∂E ∂E F, dR, bR E = ρR Ψ + dR · CdR + bR · CbR P = eR = hR = 2 oJ 2μoJ ∂F ∂dR ∂bR allows us to deﬁne an ‘augmented’ free energy (per unit reference volume)

Feh def Feh 1 −1 1 −1 E = ρR Ψ − o J eR · C eR − μo J hR · C hR (7.51) 2 2 and rewrite (7.50) compactly as

∂EFeh 1 ∂EFeh P = or T = FT. (7.52) ∂F J ∂F

Partial diﬀerentiation of the augmented free energy (7.51) with respect to the electric

field eR and magnetic field hR (holding F fixed) leads to

∂EFeh ∂EFeh dR = − , bR = − , (7.53) ∂eR ∂hR where we have used the transformations (7.6), the algebraic relations (7.7), the constitutive equations (7.33), and the results

∂ −1 −1 ∂ −1 −1 eR · C eR =2C eR, hR · C hR =2C hR. ∂eR ∂hR

RefertoProofsD.14andD.15inAppendix D for additional details.

Electric displacement–magnetic induction formulation

Recall from Table 7.3 that the total stress for this formulation can be expressed in Eulerian (true) form as ∂ΨFdb 1 1 1 1 T = ρ FT + d ⊗ d − (d · d)I + b ⊗ b − (b · b)I ∂F o 2 μo 2 or – upon using (7.6), (7.16)1, and (7.38)2 – in Lagrangian (nominal) form as Fdb ∂Ψ 1 J −T 1 J −T P = ρR + d ⊗ dR − (d · d)F + b ⊗ bR − (b · b)F . ∂F o 2 μo 2

202 Similar to what was done in the previous section, we deﬁne an ‘augmented’ free energy

(per unit reference volume)

Fdb def Fdb 1 1 E = ρR Ψ + dR · CdR + bR · CbR (7.54) 2 oJ 2μoJ that allows the total stress to be written compactly as

∂EFdb 1 ∂EFdb P = or T = FT, (7.55) ∂F J ∂F wherewehaveusedtheresults

∂ −T −1 (J d · d)=2d ⊗ dR − J (d · d)F ,Jd · d = J dR · CdR ∂F

and

∂ −T −1 (J b · b)=2b ⊗ bR − J (b · b)F ,Jb · b = J bR · CbR. ∂F

Refer to Proofs D.16 and D.17 in Appendix D. Partial diﬀerentiation of the ‘augmented’ free energy (7.54) with respect to the electric displacement dR and magnetic induction bR (holding F ﬁxed) leads to

∂EFdb ∂EFdb eR = , hR = , (7.56) ∂dR ∂bR where we have used the transformations (7.6), the algebraic relations (7.7), the constitutive equations (7.37), and the results

∂ ∂ dR · CdR =2CdR, bR · CbR =2CbR. ∂dR ∂bR

RefertoProofsD.18andD.19inAppendix D for additional details.

203 7.3.4 Invariance requirements

As is standard in ﬁnite-strain theory, we require the constitutive equations in Table

7.4 to be invariant under all possible superposed rigid body motions. To satisfy this requirement, the augmented free energies E Feh, E Feb, E Fdh,andE Fdb are regarded as functions of the right Cauchy-Green deformation tensor C = FTF instead of the deformation gradient F [69,174]. (The electric and magnetic fields remain unaltered as they are already in Lagrangian form.) A notational change to E Ceh, E Ceb, E Cdh, and E Cdb reflects this modified dependence.

For the electric ﬁeld–magnetic ﬁeld formulation, a change of independent variable

from F to C implies that the constitutive equation for the total Cauchy stress becomes

2 ∂ECeh T = F FT. (7.57a) J ∂C

Note that (7.57a) ensures balance of angular momentum (7.15a)3 is satisﬁed, i.e., the total Cauchy stress is symmetric. Use of (7.16) gives the total ﬁrst and second

Piola-Kirchhoﬀ stresses:

∂ECeh ∂ECeh P =2F , S =2 . (7.57b) ∂C ∂C

The total stresses for the other formulations in Table 7.4 are formally identical to (7.57a) and (7.57b); refer to Table 7.5. Also, the constitutive equations for the electric and magnetic dependent variables in Table 7.4 are unaltered by the change of independent variable, apart from the free energies depending on C instead of F; refer again to Table 7.5.

204 Table 7.5: Invariant constitutive equations

IVs Total stress Electric DV Magnetic DV

∂ECeh ∂ECeh ∂ECeh C, eR, hR S =2 dR = − bR = − ∂C ∂eR ∂hR ∂ECeb ∂ECeb ∂ECeb C, eR, bR S =2 dR = − hR = ∂C ∂eR ∂bR ∂ECdh ∂ECdh ∂ECdh C, dR, hR S =2 eR = bR = − ∂C ∂dR ∂hR ∂ECdb ∂ECdb ∂ECdb C, dR, bR S =2 eR = hR = ∂C ∂dR ∂bR

7.3.5 Incompressibility

Based on experimental observations, it is customary to assume that natural rubber and rubber-like (elastomeric) materials are incompressible, unless the loading is extreme [174, 294–297]. Hence, only isochoric (volume-preserving) deformations are possible. Tacit, then, is the restriction that the density remain constant, i.e.,

ρ = ρR ≡ constant. (7.58)

Restriction (7.58), in tandem with conservation of mass (7.15a)1, gives the incompressibility constraint

J =1. (7.59)

205 Using arguments similar to [69, pp. 237-240], it can be shown that the total

Cauchy stress for an incompressible magnetoelectric material (for the electric ﬁeld– magnetic ﬁeld formulation) is

∂ECeh T =2F FT − pI, (7.60a) ∂C where the constraint pressure p is a Lagrange multiplier. In other words, p is a primitive unknown and takes on whatever value is necessary to maintain the incompressibility constraint (7.59). Hence, the ﬁrst term in (7.60a) corresponds to the response determined by the constitutive equation, and the second term corresponds to the response associated with maintaining the incompressibility constraint (7.59).

It follows from (7.16) and (7.60a) that the total Piola-Kirchhoﬀ stresses are

∂ECeh ∂ECeh P =2F − pF−T, S =2 − p C−1. (7.60b) ∂C ∂C

The stresses for the remaining formulations in Table 7.5 are formally identical to

(7.60a) and (7.60b). Also, the constitutive equations for the electric and magnetic dependent variables in Table 7.5 are structurally unaltered by the incompressibility constraint (7.59).

7.3.6 Material symmetry

Recall that in this chapter, we are modeling a soft magnetoelectric (ME) composite consisting of magnetostrictive (e.g., Terfenol-D or Galfenol) and piezoelectric (e.g.,

PZT or barium titanate) particles dispersed in a natural rubber matrix. For simplicity, we consider a random distribution of particles so that the composite may be regarded as isotropic. (This mirrors the case of a magnetorheological elastomer cured in the

206 absence of a magnetic field, which results in a random distribution of ferromagnetic particles and an effectively isotropic composite [179, 190–192].) Additionally, the piezoelectric and magnetostrictive particles are sufficiently small and distributed in such a fashion that, in a ‘macroscopic’ sense, we may regard the ME composite as effectively homogeneous [174,227]. (We recognize and acknowledge the limitations of this assumption,35 and regard it simply as a starting point.) With these simplifications in hand, we now explore the ramifications of isotropy on a representative formulation from Table 7.5.

Electric ﬁeld–magnetic ﬁeld formulation

Recall that the augmented free energy E Ceh is a function of the right Cauchy-

Green deformation C, the Lagrangian electric ﬁeld eR, and the Lagrangian magnetic

Ceh ﬁeld hR. However, isotropy imposes the restriction that E depends on C, eR,and hR only through the following eleven invariants:

1 2 2 I1 =trC,I2 = tr C − tr C ,I3 =detC, 2

2 I4 = eR · eR,I5 = eR · CeR,I6 = eR · C eR,

2 I7 = hR · hR,I8 = hR · ChR,I9 = hR · C hR,

2 I10 =(eR · hR) ,I11 =(eR · hR)(eR · ChR).

Note that C2 ≡ CC. In developing this list, we found the work of Spencer [299] and Zheng [300] on representation theory (the theory of invariants) to be particularly

35Experimental evidence indicates that the microstructural (heterogeneous) features of a smart composite (e.g., volume fraction, size, shape, and orientation of the particles) signiﬁcantly impact its macroscopic performance [177,191,199,226,298].

207 helpful, as was the work of Bustamante et al. [190, 301, 302], Danas et al. [199],

and Saxena et al. [192] on transversely isotropic electro-active and magneto-active

elastomers. Note that our treatment of I10 and I11 is analogous to that employed in [199] and discussed in [190]. Also note that our omission of

2 I12 =(eR · hR)(eR · C hR), which should be included in the list of invariants according to Zheng [300], is a consequence of Bustamante’s proof of its redundancy [187].

To accommodate the change of independent variables from C, eR,andhR to their eleven invariants, the constitutive equations for this formulation (refer to Table 7.5) are rewritten using the chain rule:

11 Ceh 11 Ceh ∂Ii ∂E ∂Ii ∂E S =2 , dR = − , i=1 ∂C ∂Ii i=1 ∂eR ∂Ii

11 Ceh ∂Ii ∂E bR = − . (7.61) i=1 ∂hR ∂Ii

The notation for the free energy E Ceh remains unaltered, with the new dependence on I1,I2,...,I11 understood. The (non-vanishing) partial derivatives of the invariants

208 with respect to C (holding eR and hR ﬁxed) are

∂I1 ∂I2 ∂I3 −1 = I, = I1 I − C, = I3 C , ∂C ∂C ∂C

∂I5 ∂I6 = eR ⊗ eR, = C eR ⊗ eR + eR ⊗ eR C, ∂C ∂C

∂I8 ∂I9 = hR ⊗ hR, = C hR ⊗ hR + hR ⊗ hR C, ∂C ∂C

∂I11 1 = eR · hR eR ⊗ hR + hR ⊗ eR . (7.62) ∂C 2

It then follows from (7.61)1 and (7.62) that the total Cauchy stress is Ceh Ceh Ceh 1 ∂E ∂E 2 ∂E T = 2 B +2 I1B − B +2 I3 I J ∂I1 ∂I2 ∂I3

∂ECeh ∂ECeh +2 Be ⊗ Be +2 B2 e ⊗ Be + Be ⊗ B2 e ∂I5 ∂I6

∂ECeh ∂ECeh +2 Bh ⊗ Bh +2 B2 h ⊗ Bh + Bh ⊗ B2 h ∂I8 ∂I9 ∂ECeh + e · Bh Be ⊗ Bh + Bh ⊗ Be , (7.63) ∂I11

where we have used deﬁnition (7.2) of the (symmetric) left and right Cauchy-Green

deformation tensors B and C, transformations (7.6) and (7.16), and relationships

(7.38).

209 The (non-vanishing) partial derivatives of the invariants with respect to eR (holding C and hR ﬁxed) are

∂I4 ∂I5 ∂I6 2 =2eR, =2CeR, =2C eR, ∂eR ∂eR ∂eR

∂I10 ∂I11 =2(eR · hR)hR, =(eR · hR)ChR +(eR · ChR)hR, (7.64) ∂eR ∂eR

while the (non-vanishing) partial derivatives of the invariants with respect to hR

(holding C and eR ﬁxed) are

∂I7 ∂I8 ∂I9 2 =2hR, =2ChR, =2C hR, ∂hR ∂hR ∂hR

∂I10 ∂I11 =2(eR · hR)eR, =(eR · hR)CeR +(eR · ChR)eR. (7.65) ∂hR ∂hR

It follows from (7.61)2 and (7.64) that the Eulerian electric displacement is ) 1 ∂ECeh ∂ECeh ∂ECeh d = − 2 Be +2 B2 e +2 B3 e J ∂I4 ∂I5 ∂I6 ∂ECeh ∂ECeh +2 (e · Bh)Bh + (e · Bh)B2h +(e · B2 h)Bh , (7.66) ∂I10 ∂I11 and, similarly, from (7.61)3 and (7.65) that the Eulerian magnetic induction is ) 1 ∂ECeh ∂ECeh ∂ECeh b = − 2 Bh +2 B2 h +2 B3 h J ∂I7 ∂I8 ∂I9 ∂ECeh ∂ECeh +2 (e · Bh)Be + (e · Bh)B2 e +(e · B2 h)Be . (7.67) ∂I10 ∂I11

The constitutive equations (7.63), (7.66), and (7.67) are for the compressible theory.Intheincompressible theory (refer to Section 7.3.5), the constraint J =1is

210 enforced. Also, the constraint response −p I is added onto (7.63), so that the total

Cauchy stress becomes

Ceh Ceh ∂E ∂E 2 T =2 B +2 I1B − B − pI ∂I1 ∂I2

∂ECeh ∂ECeh +2 Be ⊗ Be +2 B2 e ⊗ Be + Be ⊗ B2 e ∂I5 ∂I6

∂ECeh ∂ECeh +2 Bh ⊗ Bh +2 B2 h ⊗ Bh + Bh ⊗ B2 h ∂I8 ∂I9

∂ECeh + e · Bh Be ⊗ Bh + Bh ⊗ Be . (7.68) ∂I11

Note that the contribution from the third term in (7.63) was absorbed into the La-

grange multiplier p. Also, since J =1,wehave ) ∂ECeh ∂ECeh ∂ECeh d = − 2 Be +2 B2 e +2 B3 e ∂I4 ∂I5 ∂I6 ∂ECeh ∂ECeh +2 (e · Bh)Bh + (e · Bh)B2 h +(e · B2 h)Bh (7.69) ∂I10 ∂I11 and ) ∂ECeh ∂ECeh ∂ECeh b = − 2 Bh +2 B2 h +2 B3 h ∂I7 ∂I8 ∂I9 ∂ECeh ∂ECeh +2 (e · Bh)Be + (e · Bh)B2 e +(e · B2 h)Be . (7.70) ∂I10 ∂I11

7.3.7 A representative free energy function

In the spirit of [174], as a representative free energy function, we present a generalized neo-Hookean model with purely elastic, electric, and magnetic contributions,

211 together with coupled electro-elastic, magneto-elastic, and magneto-electric contribu-

tions:

√ Ceh μ o μo o μo E = (I1 − 3) + (αI4 + βI5)+ (γI7 + δI8)+ χ I10. (7.71) 2 2 2 2

In (7.71), μ is the shear modulus at small strains; o and μo are the electric permittivity and magnetic permeability in vacuo; and α, β, γ, δ,andχ are dimensionless

parameters that can be adjusted to ﬁt experimental data. Use of (7.71) in (7.63)

gives the total Cauchy stress (for the compressible case): 1 T = μB + o β Be ⊗ Be + μo δ Bh ⊗ Bh . (7.72) J

The second and third terms in (7.72) represent electrical and magnetic contributions to the total stress, which tend to increase the eﬀective mechanical stiﬀness of the composite.

212 Chapter 8: Conclusions and Future Work in Smart Polymers

8.1 Conclusions

In Chapter 6, we revisited the fundamental laws of continuum electrodynamics,

which govern the behavior of smart polymers. For the ﬁrst time, we presented the

fundamental laws at four diﬀerent levels: primitive, material, integral, and diﬀeren-

tial. Primitive refers to the law stated in words, in its most basic form; material (or

global) refers to the law applied to the body, independent of its conﬁguration; integral

refers to the law applied to the body as seen in a particular conﬁguration, either

reference (Lagrangian form) or present (Eulerian form); and diﬀerential refers to the

localized Lagrangian and Eulerian forms of the law, valid pointwise throughout the

body. Novel perspectives emerged during our transparent and systematic progression

from primitive statements to pointwise equations. For instance, we illustrated and ex-

plained the unexpected absence of form invariance between Eulerian and Lagrangian

integral representations of the electric ﬂux and magnetic ﬁeld.

In Chapter 7, a ﬁnite-strain theoretical framework for modeling soft magnetoelec-

tric composites was developed, with practical applications in mind. Finite deforma-

tions, electro-magneto-elastic coupling, and material nonlinearities were incorporated

into the model. A particular emphasis was placed on developing tractable constitutive

213 equations to facilitate material characterization in the laboratory. Toward this end, an

extensive catalogue of free energies and constitutive equations was presented, each em-

ploying a diﬀerent set of independent variables. As the optimal choice of independent

variable changes from one scenario to another, this catalogue will provide theoreti-

cians and experimentalists with the needed ﬂexibility. The ramiﬁcations of invariance,

angular momentum, incompressibility, and material symmetry were explored, and a

representative (neo-Hookean-type) free energy with full electro-magneto-elastic cou-

pling was posed. It is hoped that the research presented in Chapter 7 will serve as a

starting point for (i) experimentally characterizing magnetoelectric polymers at ﬁnite

strain and (ii) developing a fundamental understanding of their response through the

solution of simple boundary-value problems.

8.2 Future work

The ﬁeld of magnetoelectric polymers (MEPs) is replete with open problems and technological promise. In what follows, some potential future directions are highlighted.

(1) Solution of boundary-value problems for MEPs. The lack of experimental data

on the ﬁnite-strain mechanics of MEPs remains a key obstacle in the ﬁeld. As

suggested by Dorfmann & Ogden [213,214], solutions of boundary-value problems

can be used to guide the design of experiments and encourage the generation of

experimental data. Currently, to the best of our knowledge, no solutions of ﬁnite-

deformation boundary-value problems for MEPs are available in the literature.

214 Toward this end, it would be immensely beneﬁcial to develop analytical (semi-

inverse) and numerical solutions for some simple MEP boundary-value problems,

starting perhaps with uniaxial extension and simple shear.

(2) MEP fabrication and experimental characterization. A lack of experimental data

on the mechanical behavior of MEPs has impeded the development of robust

ﬁnite-strain constitutive equations,36 which, in turn, has hindered technological

implementation. Progress in these areas is vital to advance fundamental un-

derstanding of MEPs, enable the development of computational tools, facilitate

design and optimization, and encourage the implementation of MEPs in the next

generation of intelligent systems, structures, and devices. It is hoped that the

research presented in Chapter 7 will serve as a starting point for these endeavors.

(3) Theoretical modeling of MEPs. Several avenues could be pursued to enhance the

mathematical model for MEPs developed in Chapter 7:

(a) Viscoelasticity. Polymers and elastomers often depart from the elastic ideal,

exhibiting strain-rate dependence and mechanical dissipation [221]. Hence, a

3-D ﬁnite-deformation continuum framework for the viscoelastic response of

MEPs could prove useful.

(b) Anisotropy. As discussed in Chapters 5 and 7, when a particle-ﬁlled smart

composite (such as a magnetorheological elastomer) is cured in the presence of

an electromagnetic ﬁeld, the particles can form chain-like arrangements that

are ‘frozen in’ [179,190–192]. The resulting ‘preferred direction’ is associated

with enhanced electro-magneto-mechanical response. Hence, a transversely

36Experimental data is needed to determine suitable forms of the free energy function.

215 isotropic constitutive model for MEPs that relaxes Chapter 7’s assumption

of isotropy could be useful in practical applications.

(heterogeneous) features of a smart composite (e.g., volume fraction, size,

shape, and orientation of the particles) signiﬁcantly impact its macroscopic

performance [177, 191, 199, 226, 298]. Hence, a ‘homogenization theory’ that

accounts for the eﬀect of the microstructure on the macroscopic response of

the composite could be useful.

(d) Dynamic eﬀects. The customary assumption of quasi-static electromagnetic

ﬁelds and static deformations employed in Chapter 7 is based on the typical

disparity between the electromagnetic and mechanical time scales. How-

ever, in a situation where, say, a structure is vibrating at a suﬃciently high

frequency, this standard assumption could break down. Hence, in such a sit-

uation, it is necessary to relax the quasi-static restriction and adopt the fully

dynamic theory of Chapter 6. Of course, a fully dynamic theory introduces

numerous complexities, one being the deformation-dependent ‘eﬀective’ Eu-

lerian electromagnetic ﬁelds.

216 Appendix A: Supplement to Chapter 2

A.1 Extended mathematical structure of the perturbation formalism

Here we present the extended mathematical structure of the perturbation for-

malism set forth in Section 2.3, obtained by inserting the power series expansions

(2.15a)-(2.15h) into the dimensionless 2-D model (2.13a)-(2.13i). Note that the 2-D

problem decouples into a series of 1-D problems, each corresponding to a successively

higher order in the asymptotics. These 1-D problems are generated by collecting the

coeﬃcients of the O(1), O( ), O( 2), O( 2 r˜2), etc. terms in Eqs. (A.1)-(A.10) that follow: conservation of mass 0,0 0,0 0,0 0,0 0,0 0,1 0,0 0,1 0,1 0,0 ρ˜,t˜ +2˜ρ v˜r + ρ˜ v˜z ,z˜ + ρ˜,t˜ +2˜ρ v˜r +2˜ρ v˜r 0,0 0,1 0,1 0,0 2 0,2 0,0 0,2 0,1 0,1 + ρ˜ v˜z ,z˜ + ρ˜ v˜z ,z˜ + ρ˜,t˜ +2˜ρ v˜r +2˜ρ v˜r 0,2 0,0 0,0 0,2 0,1 0,1 0,2 0,0 +2˜ρ v˜r + ρ˜ v˜z ,z˜ + ρ˜ v˜z ,z˜ + ρ˜ v˜z ,z˜ 2 1,0 0,0 1,0 1,0 0,0 0,0 1,0 1,0 0,0 +˜r ρ˜,t˜ +4˜ρ v˜r +4˜ρ v˜r + ρ˜ v˜z ,z˜ + ρ˜ v˜z ,z˜

+ O( 3)=0 (A.1)

217 radial component of balance of linear momentum 0,0 2 0,0 1 0,0 2 r˜ ρ˜0,0 v˜ + v˜ 0,0 +˜v 0,0 v˜ =Lo T˜ 0,0 − T˜ r,t˜ r z r,z˜ r˜ rr θθ 1 0,1 1,0 0,0 + T˜ 0,1 − T˜ + 2 r˜ 3T˜ 1,0 − T˜ + T˜ r˜ rr θθ rr θθ rz,z˜ 1 0,2 + T˜ 0,2 − T˜ + O( 3)(A.2) r˜ rr θθ axial component of balance of linear momentum 0,0 0,0 0,0 0,0 0,0 0,1 0,0 0,1 0,1 0,0 ρ˜ v˜z,t˜ +˜vz v˜z,z˜ + ρ˜ v˜z,t˜ +˜vz v˜z,z˜ +˜vz v˜z,z˜ 0,1 0,0 0,0 0,0 2 0,0 0,2 0,0 0,2 0,1 0,1 +˜ρ v˜z,t˜ +˜vz v˜z,z˜ + ρ˜ v˜z,t˜ +˜vz v˜z,z˜ +˜vz v˜z,z˜ 0,2 0,0 2 1,0 0,0 1,0 1,0 0,0 0,0 +˜vz v˜z,z˜ +˜r v˜z,t˜ +˜vz v˜z,z˜ +˜vz v˜z,z˜ +2˜vr 0,1 0,1 0,0 0,1 0,1 0,0 0,2 0,0 0,0 0,0 +˜ρ v˜z,t˜ +˜vz v˜z,z˜ +˜vz v˜z,z˜ +˜ρ v˜z,t˜ +˜vz v˜z,z˜ 2 1,0 0,0 0,0 0,0 +˜r ρ˜ v˜z,t˜ +˜vz v˜z,z˜ 0,0 0,0 0,1 0,1 =Lo T˜zz,z˜ +2T˜rz + T˜zz,z˜ +2T˜rz 2 0,2 0,2 2 1,0 1,0 3 + T˜zz,z˜ +2T˜rz +˜r T˜zz,z˜ +4T˜rz + O( )(A.3) radial component of the constitutive equation

218 ˜ 0,0 0,0 ˜ 0,0 − 0,0 ˜ 0,0 ˜ 0,1 0,0 ˜ 0,1 0,1 ˜ 0,0 Trr,t˜ +˜vz Trr,z˜ 2a v˜r Trr + Trr,t˜ +˜vz Trr,z˜ +˜vz Trr,z˜ − 0,0 ˜ 0,1 0,1 ˜ 0,0 2 ˜ 0,2 0,0 ˜ 0,2 0,1 ˜ 0,1 2a v˜r Trr +˜vr Trr + Trr,t˜ +˜vz Trr,z˜ +˜vz Trr,z˜ 0,2 ˜ 0,0 − 0,0 ˜ 0,2 0,1 ˜ 0,1 0,2 ˜ 0,0 2 ˜ 1,0 +˜vz Trr,z˜ 2a v˜r Trr +˜vr Trr +˜vr Trr +˜r Trr,t˜

0,0 1,0 1,0 0,0 0,0 1,0 1,0 0,0 +2(1− a)˜vr T˜rr − 6a v˜r T˜rr +˜vz T˜rr,z˜ +˜vz T˜rr,z˜ 1,0 0,0 0,0 0,0 +2(1− a)˜vz T˜rz − (1 + a) T˜rz v˜r,z˜

0,0 0,0 0,0 0,1 0,1 =Λ2˜vr +˜vz,z˜ +2M˜vr + Λ 2˜vr +˜vz,z˜

* 0,1 2 0,2 0,2 0,2 +2M˜vr + Λ 2˜vr +˜vz,z˜ +2M˜vr

+ 2 1,0 1,0 1,0 3 +˜r Λ 4˜vr +˜vz,z˜ +6M˜vr + O( )(A.4) transverse shear component of the constitutive equation 0,0 0,0 1 0,0 2 r˜ T˜ +(1− a)˜v 0,0 T˜ 0,0 +˜v 0,0 T˜ + (1 − a) T˜ 0,0 v˜ rz,t˜ r rz z rz,z˜ 2 rr r,z˜ 1 0,0 0,0 − (1 + a) T˜ 0,0 v˜ +(1− a)˜v 1,0 T˜ 0,0 − (1 + a)˜v 1,0 T˜ 0,0 − a T˜ 0,0 v˜ 2 zz r,z˜ z zz z rr rz z,z˜ 2 1,0 0,0 3 = r˜ M 2˜vz +˜vr,z˜ + O( )(A.5) circumferential component of the constitutive equation

219 ˜ 0,0 0,0 ˜ 0,0 − 0,0 ˜ 0,0 ˜ 0,1 0,0 ˜ 0,1 0,1 ˜ 0,0 Tθθ,t˜ +˜vz Tθθ,z˜ 2a v˜r Tθθ + Tθθ,t˜ +˜vz Tθθ,z˜ +˜vz Tθθ,z˜ − 0,0 ˜ 0,1 0,1 ˜ 0,0 2 ˜ 0,2 0,0 ˜ 0,2 0,1 ˜ 0,1 2a v˜r Tθθ +˜vr Tθθ + Tθθ,t˜ +˜vz Tθθ,z˜ +˜vz Tθθ,z˜ 0,2 ˜ 0,0 − 0,0 ˜ 0,2 0,1 ˜ 0,1 0,2 ˜ 0,0 2 ˜ 1,0 +˜vz Tθθ,z˜ 2a v˜r Tθθ +˜vr Tθθ +˜vr Tθθ +˜r Tθθ,t˜ 0,0 1,0 1,0 0,0 0,0 1,0 1,0 0,0 +˜vz T˜θθ,z˜ +˜vz T˜θθ,z˜ +2(1− a)˜vr T˜θθ − 2a v˜r T˜θθ 0,0 0,0 0,0 0,1 0,1 =Λ2˜vr +˜vz,z˜ +2M˜vr + Λ 2˜vr +˜vz,z˜

* 0,1 2 0,2 0,2 0,2 +2M˜vr + Λ 2˜vr +˜vz,z˜ +2M˜vr

+ 2 1,0 1,0 1,0 3 +˜r Λ 4˜vr +˜vz,z˜ +2M˜vr + O( )(A.6) axial component of the constitutive equation ˜ 0,0 0,0 ˜ 0,0 − ˜ 0,0 0,0 ˜ 0,1 0,0 ˜ 0,1 0,1 ˜ 0,0 Tzz,t˜ +˜vz Tzz,z˜ 2a Tzz v˜z,z˜ + Tzz,t˜ +˜vz Tzz,z˜ +˜vz Tzz,z˜ − ˜ 0,0 0,1 ˜ 0,1 0,0 2 ˜ 0,2 0,0 ˜ 0,2 0,1 ˜ 0,1 2a Tzz v˜z,z˜ + Tzz v˜z,z˜ + Tzz,t˜ +˜vz Tzz,z˜ +˜vz Tzz,z˜ 0,2 ˜ 0,0 − ˜ 0,0 0,2 ˜ 0,1 0,1 ˜ 0,2 0,0 2 ˜ 1,0 +˜vz Tzz,z˜ 2a Tzz v˜z,z˜ + Tzz v˜z,z˜ + Tzz v˜z,z˜ +˜r Tzz,t˜

0,0 1,0 0,0 1,0 1,0 0,0 0,0 0,0 +2˜vr T˜zz +˜vz T˜zz,z˜ +˜vz T˜zz,z˜ +(1− a) T˜rz v˜r,z˜ 1,0 0,0 0,0 1,0 1,0 0,0 − 2(1+a)˜vz T˜rz − 2a T˜zz v˜z,z˜ + T˜zz v˜z,z˜

0,0 0,0 0,0 0,1 0,1 =Λ2˜vr +˜vz,z˜ +2M˜vz,z˜ + Λ 2˜vr +˜vz,z˜

* 0,1 2 0,2 0,2 0,2 +2M˜vz,z˜ + Λ 2˜vr +˜vz,z˜ +2M˜vz,z˜

+ 2 1,0 1,0 1,0 3 +˜r Λ 4˜vr +˜vz,z˜ +2M˜vz,z˜ + O( )(A.7)

220 kinematic boundary condition ˜ (0) 0,0 ˜ (0) − 0,0 ˜ (0) ˜ (1) 0,0 ˜(1) 0,1 ˜(0) φ,t˜ +˜vz φ,z˜ v˜r φ + φ,t˜ +˜vz φ,z˜ +˜vz φ,z˜ − 0,0 ˜(1) − 0,1 ˜ (0) 2 ˜ (2) 0,0 ˜ (2) 0,1 ˜(1) v˜r φ v˜r φ + φ,t˜ +˜vz φ,z˜ +˜vz φ,z˜

0,2 (0) 0,0 (2) 0,1 (1) 0,2 (0) +˜vz φ˜,z˜ − v˜r φ˜ − v˜r φ˜ − v˜r φ˜ 2 (0) 1,0 (0) 1,0 (0) 3 + φ˜ v˜z φ˜,z˜ − v˜r φ˜ + O( )=0 (A.8) radial component of the kinetic boundary condition 2 0,0 0,1 2 0,2 (0) 1,0 (0) (0) 0,0 T˜rr + T˜rr + T˜rr + φ˜ T˜rr − φ˜ φ˜,z˜ T˜rz

+ O( 3)=0 (A.9) axial component of the kinetic boundary condition (0) 0,0 0,0 (0) (0) 0,1 (1) 0,0 0,0 (1) 0,1 (0) φ˜ T˜rz − T˜zz φ˜,z˜ + φ˜ T˜rz + φ˜ T˜rz − T˜zz φ˜,z˜ − T˜zz φ˜,z˜ 3 2 (0) 0,2 (1) 0,1 (2) 0,0 (0) 1,0 + φ˜ T˜rz + φ˜ T˜rz + φ˜ T˜rz + φ˜ T˜rz 2 0,0 (2) 0,1 (1) 0,2 (0) (0) 1,0 (0) − T˜zz φ˜,z˜ − T˜zz φ˜,z˜ − T˜zz φ˜,z˜ − φ˜ T˜zz φ˜,z˜

+ O( 3) = 0 (A.10)

A.2 The linearized theory

In this section, we elaborate on some aspects of the linearized theory of Section

2.5. In particular, we discuss its mathematical structure, specialize the displacement equation of motion for a tapered horn to an initially straight geometry, discuss typical

221 analytical methods, and present detailed analytical solutions of several problems from

Section 2.6.4.

A.2.1 Mathematical structure

Equations (2.37a)-(2.37d) can be written compactly as an inhomogeneous linear

system:

∂U ∂U + A = S, ∂t ∂z where the matrix of primitive variables U, the Jacobian matrix A, and the source terms S are given by ⎛ ⎞ ⎡ ⎤

⎜ ⎟ ⎢ 0(1− 2ν)ρR 00⎥ ⎜ ρ ⎟ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎢ − 1 ⎥ ⎜ vz ⎟ ⎢ 000⎥ ⎢ ρR ⎥ ⎜ ⎟ ⎢ ⎥ [ U ]=⎜ ⎟ , [ A ]=⎢ ⎥ , ⎜ ⎟ ⎢ ⎥ ⎜ A ⎟ ⎢ ⎥ ⎜ ⎟ ⎢ 02νAR 00⎥ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ Tzz 0 −E 00 and ⎛ ⎞ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Tzz dAR ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ρR AR dz ⎟ [ S ]=⎜ ⎟ , ⎜ ⎟ ⎜ ⎟ ⎜ − dAR ⎟ ⎜ vz ⎟ ⎜ dz ⎟ ⎜ ⎟ ⎝ ⎠ 0

222 respectively. Note that neither the Jacobian A nor the source S contains partial

derivatives of the primitive variables (recall that AR is a speciﬁed function of z,and thus not a primitive variable). Also note that the Jacobian A is singular, a consequence of the primitive variables being independent of the density and area ﬂuxes

(the ﬁrst and third columns of A are empty). The eigenvalues of A are ( E ξ1,2 =0,ξ3,4 = ± = ± c, ρR where c is the customary bar velocity, and the corresponding eigenvectors are ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ [q1 ]=⎜ ⎟ , [q2 ]=⎜ ⎟ , ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 0 0

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ c ⎟ ⎜ c ⎟ ⎜ ⎟ ⎜ − ⎟ ⎜ (1 − 2ν)ρR ⎟ ⎜ (1 − 2ν)ρR ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ [q3 ]=⎜ ⎟ , [q4 ]=⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎜ 2ν AR ⎟ ⎜ 2ν AR ⎟ ⎜ − ⎟ ⎜ − ⎟ ⎜ 1 2ν ρR ⎟ ⎜ 1 2ν ρR ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ − c − c 1 − 2ν 1 − 2ν

The eigenvalues are real, and the eigenvectors are linearly independent, which together imply that the linear system (2.37a)-(2.37d) is hyperbolic. Note that the two

223 vanishing eigenvalues are a consequence of the Jacobian A being singular, with two

columns of zeros.

A.2.2 The equation of motion for an initially straight rod

If, rather than tapered, the rod is straight in its undeformed reference conﬁgu- ration, then the initial cross-sectional area AR is constant and dAR/dz =0.Inthis case, the displacement equation of motion (2.38) in the linearized theory reduces to

2 2 1 ∂ uz ∂ uz = , (A.11) c 2 ∂t2 ∂z2 which we identify as the second-order linear wave equation governing the propagation of small-amplitude, non-dispersive, longitudinal waves in a linearly elastic rod. Note that this equation can be obtained using a ‘mechanics-of-materials’ approach [108, pp. 75-77] or as a special case of Navier’s equation of linear elasticity [152, pp.

55-57]. Using the standard ‘sign of the discriminant’ technique for classifying second- order PDEs [303, pp. 174-175], it is straightforward to show that both forms of the displacement equation of motion, i.e., (2.38) for horns and (A.11) for straight rods, are hyperbolic, as is well known.

A.2.3 The d’Alembert solution of the initial-value problem

The equation of motion for an initially straight rod

2 2 1 ∂ uz ∂ uz = , (A.12a) c 2 ∂t2 ∂z2

224 together with the initial conditions ∂uz uz(z, 0) = φ(z),vz(z, 0) = = ψ(z), (A.12b) ∂t t=0 constitute an initial-value problem.Notethatφ(z)andψ(z) are the prescribed initial displacement and initial velocity, respectively.

The classical d’Alembert solution of the initial-value problem (A.12a)-(A.12b) is obtained as follows (also refer to [108, pp. 13-17]). Upon a change of independent variables,

η = z − ct, ξ = z + ct, the equation of motion (A.12a) can be rewritten as

2 ∂ uz =0. (A.13) ∂ξ∂η

Integrating (A.13) twice leads to

uz(ξ,η)=f(ξ)+g(η), or, upon changing back to the original independent variables,

uz(z, t)=f(z + ct)+g(z − ct), (A.14) where f and g are arbitrary functions of integration. The particular forms of f and

g are dictated by the initial displacement φ and the initial velocity ψ in (A.12b), as

illustrated in the following paragraph.

225 Diﬀerentiating (A.14) with respect to time, with the aid of the chain rule, leads to

∂uz = cf (z + ct) − cg(z − ct), (A.15) ∂t where a superscript primes denotes diﬀerentiation with respect to the argument of the function. Evaluating (A.15) at the initial condition (A.12b)2 gives

cf (z) − cg(z)=ψ(z). (A.16)

Integrating (A.16) between a ﬁxed lower limit of integration b and a variable upper limit of integration z, and subsequently using the ﬁrst fundamental theorem of calculus, leads to

z 1 f(z) − g(z)= ψ(s)ds, (A.17) c b where s is the integration variable. Next, we switch gears and evaluate (A.14) at the initial condition (A.12b)1,whichgives

f(z)+g(z)=φ(z). (A.18)

Equations (A.17) and (A.18) can be solved for f and g:

z+ct 1 1 f(z + ct)= φ(z + ct)+ ψ(s)ds, (A.19a) 2 2c b

z−ct 1 1 g(z − ct)= φ(z − ct) − ψ(s)ds, (A.19b) 2 2c b

226 wherewehavechangedtheargumentoff from z to z + ct, and the argument of g

from z to z − ct. Finally, substitution of (A.19a) and (A.19b) into (A.14) leads to the

d’Alembert solution of the initial-value problem:

z+ct 1 1 uz(z, t)= φ(z + ct)+φ(z − ct) + ψ(s)ds. (A.20) 2 2c z−ct

Note that (A.20) can also be obtained using integral transforms, as shown by Graﬀ

[108, pp. 17-19]. Also note that the domain and boundary conditions have not yet been speciﬁed. As will soon become apparent, the d’Alembert solution (A.20) is quite useful for inﬁnite domains, where separation of variables fails (the Sturm-Liouville eigenvalue problem cannot be posed without boundary conditions). However, for

ﬁnite domains, where wave reﬂection and transmission at the boundaries leads to complex interactions, separation of variables is generally preferable.

A.2.4 Velocity, stress, strain, density, and area

Now that a solution for the displacement ﬁeld uz(z, t) that satisﬁes the initial-value problem (A.12a)-(A.12b) has been determined in Section A.2.3, the axial velocity and axial strain can be calculated using

∂uz ∂uz vz(z, t)= ,ezz(z, t)= (A.21) ∂t ∂z from linear elasticity theory. The velocity (A.21)1 and strain (A.21)2 are then subsequently used to deduce the stress, density, and cross-sectional area, as illustrated in the following paragraphs. Note that the ability to calculate the evolution of density and cross-sectional area is novel for a linear setting, as both quantities are customarily regarded as constant in linear rod theories.

227 The density is deduced by integrating Eq. (2.37a) with respect to time, i.e.,

ρ(z, t)=−(1 − 2ν) ρR ezz + h(z), (A.22) where h(z) is an arbitrary function of integration. We expect (A.22) to hold for all linearly elastic materials, including those that are incompressible (ν =0.5), in which case the density remains constant (ρ = ρR). This condition demands that h(z)=ρR, and thus (A.22) becomes

ρ(z, t)=ρR 1 − (1 − 2ν)ezz . (A.23)

Note that in the absence of strain (i.e., if ezz = 0) or if the material is incompressible

(i.e., if ν =0.5), the density ρ reverts to its reference value ρR, as expected. Also, it can be seen that Eq. (A.23) predicts a density increase under a compressive strain, as expected, noting that Poisson’s ratio ν ranges between 0.5to−1toensureelastic stability (and thus 1 − 2ν>0). Equation (A.23) holds for both straight and tapered rods.

The cross-sectional area is obtained by specializing Eq. (2.37c) to an initially straight rod (i.e., dAR/dz = 0) and then integrating with respect to time, i.e.,

A(z, t)=−2νAR ezz + k(z), (A.24) where k(z) is an arbitrary function of integration. Equation (A.24) must hold for all linearly elastic materials, including those that are incapable of lateral expansion or contraction (ν = 0), in which case the area must remain unchanged (A = AR). This

228 condition demands that k(z)=AR, so that (A.24) becomes

A(z, t)=AR 1 − 2νezz . (A.25)

Note that in the absence of strain (i.e., if ezz =0)orifthematerialcannotlaterally expand or contract (i.e., if ν = 0), the cross-sectional area A reverts to its reference value AR, as expected. Equation (A.25) only holds for straight rods (where AR is constant), although a similar procedure could be employed to deduce A(z, t) for an initially tapered proﬁle (where AR is a function of z).

Lastly, to obtain the stress, Eq. (2.37d) is integrated with respect to time, which yields

Tzz(z, t)=Eezz + l(z), where l(z) is an arbitrary function of integration. In the absence of strain, the stress must vanish. This implies that l(z) = 0 and thus

Tzz(z, t)=Eezz. (A.26)

We identify (A.26) as Hooke’s law, i.e., a specialization of the 3-D constitutive equation for an isotropic linear elastic material to a state of ‘longitudinal stress,’ where all stress components vanish except Tzz [152, p. 57]. Of course, Eq. (A.26) holds regardless of the rod’s cross-sectional proﬁle.

A.2.5 Impact

At time t = 0, a rod rigidly translating longitudinally rightward with incident speed Vo collinearly impacts a rod rigidly translating longitudinally leftward with

229 incident speed Vo. Both rods are composed of the same material (with properties E,

ν, ρR), have identical (initially uniform) cross-sectional areas AR, and are considered semi-inﬁnite so that we may disregard wave reﬂection at the ends. After impact, for t>0, we enforce the customary ‘stick’ contact conditions at the interface, i.e., continuity of axial velocity and axial stress.

The initial-value problem consists of the displacement equation of motion

2 2 1 ∂ uz ∂ uz = (A.27a) c 2 ∂t2 ∂z2 together with the initial conditions ⎧ ⎪ ⎪ ⎨ Vo,zL

Note that z = L is the location of the interface. It is useful to express the initial velocity (A.27b)2 in terms of the Heaviside step function, i.e.,

vz(z, 0) = Vo 1 − 2H(z − L) , (A.27c)

where ⎧ ⎪ ⎪ ⎨ 1,s>0 H(s)=⎪ ⎪ ⎩ 0,s<0

The Heaviside step function vanishes if its argument is negative and has a value of unity if its argument is positive. The general solution of the initial-value problem

230 (A.27a)-(A.27c) is given by the d’Alembert solution (A.20). Upon making the iden-

tiﬁcations

φ =0,ψ(z)=Vo 1 − 2H(z − L) ,

the d’Alembert solution (A.20) specializes to

z+ct Vo uz(z, t)= 1 − 2H(s − L) ds. (A.28) 2c z−ct

The velocity and strain follow from substituting the displacement (A.28) into Eqs.

(A.21)1 and (A.21)2 and using the Leibniz integral rule:

∂uz vz(z, t)= = Vo 1 − H(z − L − ct) − H(z − L + ct) , (A.29) ∂t

∂uz Vo ezz(z, t)= = H(z − L − ct) − H(z − L + ct) . (A.30) ∂z c

Note that the terms H(z − L + ct)andH(z − L − ct) correspond to waves traveling leftward and rightward, respectively. The density, cross-sectional area, and stress then follow from substituting the strain (A.30) into Eqs. (A.23), (A.25), and (A.26), respectively: ) (1 − 2ν)Vo ρ(z, t)=ρR 1 − H(z − L − ct) − H(z − L + ct) , (A.31) c

) 2νVo A(z, t)=AR 1 − H(z − L − ct) − H(z − L + ct) , (A.32) c

EVo Tzz(z, t)= H(z − L − ct) − H(z − L + ct) . (A.33) c

231 We emphasize that since both rods are semi-inﬁnite, there is no need to worry about

wave reﬂection at the ends. Also, since both rods are composed of the same material

and have identical cross-sectional areas, there is no impedance mismatch and thus complete transmission at the interface (refer to [108, pp. 83-84] and [152, pp. 26-29]).

Hence, there is no need to modify the solutions (A.28)-(A.33).

A.2.6 Separation

At time t = 0, an event occurs at location z = L along the length of an inﬁnite rod

that causes the regions on either side of z = L to separate. The rod has homogeneous

elastic properties (E, ν, ρR), a uniform initial cross-sectional area AR, and is regarded as inﬁnite so that we may disregard wave reﬂection at its ends. For t>0, we enforce continuity of axial velocity and axial stress at z = L.

The initial-value problem consists of the displacement equation of motion

2 2 1 ∂ uz ∂ uz = (A.34a) c 2 ∂t2 ∂z2 together with the initial conditions ⎧ ⎪ ⎪ ⎨ −Vo,zL

For mathematical convenience, we express the initial velocity (A.34b)2 in terms of the Heaviside step function:

vz(z, 0) = Vo 2H(z − L) − 1 . (A.34c)

232 Then, by identifying

φ =0,ψ(z)=Vo 2H(z − L) − 1 , the d’Alembert solution (A.20) of the initial-value problem (A.34a)-(A.34c) can be written as

z+ct Vo uz(z, t)= 2H(s − L) − 1 ds. (A.35) 2c z−ct

Substitution of the displacement (A.35) into Eqs. (A.21)1 and (A.21)2, and subsequent use of the Leibniz integral rule, leads to the velocity and strain:

∂uz vz(z, t)= = Vo H(z − L + ct)+H(z − L − ct) − 1 , (A.36) ∂t

∂uz Vo ezz(z, t)= = H(z − L + ct) − H(z − L − ct) . (A.37) ∂z c

The density, cross-sectional area, and stress follow from substitution of the strain

(A.37) into Eqs. (A.23), (A.25), and (A.26), respectively: ) (1 − 2ν)Vo ρ(z, t)=ρR 1 − H(z − L + ct) − H(z − L − ct) , (A.38) c

) 2νVo A(z, t)=AR 1 − H(z − L + ct) − H(z − L − ct) , (A.39) c

EVo Tzz(z, t)= H(z − L + ct) − H(z − L − ct) . (A.40) c

233 Appendix B: Supplement to Chapter 3

B.1 Reformulation of the problem by the CBFM

The reformulation of the generalized problem (3.1a)-(3.1b) by the concentrated body force method (CBFM) is most straightforwardly illustrated by ﬁrst specializing

(3.1a)-(3.1b) to a particular physical example, implementing the CBFM modeling assumption, then generalizing the result. Hence, we ﬁrst specialize (3.1a)-(3.1b) to

(3.2), longitudinal waves in a slender rod: 2 2 1 ∂ u ∂ u ∂u 2 2 = 2 ,u(0,t)=0,EA = f(t), c ∂t ∂x ∂x x=L ∂u u(x, 0) = 0, =0. (B.1) ∂t t=0 Recall that u is the axial displacement, x is the axial coordinate, t is time, c = E/ρ is the phase velocity, E is Young’s modulus, ρ is the density, A is the cross-sectional area, and L is the length. For this particular physical example, the governing equation

(B.1)1 is obtained by combining the following statements of balance of linear momentum, strain-displacement, and stress-strain from isotropic linear elasticity [108]:

∂2u ∂σ ∂u ρ = , = ,σ= E . (B.2) ∂t2 ∂x ∂x

234 Loosely, the objective of the CBFM is to homogenize the time-dependent boundary condition (B.1)3 by accounting for the force f(t) in a modiﬁed form of the governing equation (B.1)1. This modiﬁed form is obtained by modeling f(t) as a concen-

trated body force and subsequently performing an axial force balance on a diﬀerential

element of the rod: ∂σ ∂2u −σA + σ + dx A + f(t) δ(x − L) dx = ρAdx , (B.3) ∂x ∂t2 where σ is the axial stress, dx is the length of the diﬀerential element, and δ(x − L)

is the spatial Dirac delta function. After some algebra and use of (B.2)2 and (B.2)3,

(B.3) becomes

1 ∂2u ∂2u 1 − = f(t) δ(x − L), (B.4a) c2 ∂t2 ∂x2 EA with ∂u ∂u u(0,t)=0, =0,u(x, 0) = 0, =0. (B.4b) ∂x x=L ∂t t=0

Using (3.2), (B.4a)-(B.4b) can be straightforwardly generalized to obtain (3.3a)-

(3.3c).

Note that this reformulation procedure could have been demonstrated in the context of other physical problems modeled by (3.1a)-(3.1b), e.g., transverse motion in taut strings, torsional waves in circular shafts, longitudinal waves in the through- thickness direction of a thin semi-inﬁnite plate, and axisymmetric torsional waves in thin cylindrical shells.

235 B.2 The specialized CBFM/HEEM solution

Specializing the generalized CBFM/HEEM solution (3.4a)-(3.4c) to the particular physical example (3.2) allows us to write

∞ u(x, t)= um(t) sin(αm x)(B.5) m=1 and evaluate the convolution integral for the modal coordinates:

t 2 2Fc m+1 um(t)= (−1) sin [ωm(t − τ)] sin(Ωτ)dτ EALωm 0 − Ω 2F sin(Ωt) ω sin(ωmt) − m+1 m = ( 1) 2 2 . (B.6) ρAL ωm − Ω

For a slender rod, the axial stress σ follows from term-by-term diﬀerentiation of the axial displacement u (refer to (B.2)2 and (B.2)3):

∞ ∂u(x, t) σ(x, t)=E = E αm um(t)cos(αm x). (B.7) ∂x m=1

If the rod is driven at its r-th natural frequency, i.e., Ω = ωr,wherer is a positive integer, then a singularity arises in the r-th modal coordinate ur(t). To accommodate this singularity, we apply l’Hˆopital’s rule to (B.6), which leads to − F − r+1 sin(ωrt) ωrt cos(ωrt) ur(t)= ( 1) 2 . (B.8) ρAL ωr

Hence, when performing the summations in (B.5) and (B.7), the modal coordinate

(B.8) is used for the resonant mode (m = r), while the modal coordinates (B.6) are

used for the non-resonant modes (m = r).

236 B.3 The specialized Mindlin-Goodman solution

Specializing the generalized Mindlin-Goodman solution (3.9a)-(3.9c) to the particular physical example (3.2) allows us to write

∞ F u(x, t)= x sin(Ωt)+ wm(t) sin(αm x). (B.9) EA m=1

It follows from (3.8c) that

F Ω F Ω2 φ(x)=0,ψ(x)=− x, h(x, t)= sin(Ωt) x, EA c2EA whose Fourier coeﬃcients are

L 2 φm = φ(x) sin(αm x)dx =0, L 0 L 2 8FLΩ (−1)m+1 ψm = ψ(x) sin(αm x)dx = − , L π2EA (2m − 1)2 0 L 2 8FLΩ2 (−1)m+1 hm(t)= h(x, t) sin(αm x)dx = sin(Ωt), L π2c2EA (2m − 1)2 0 allowing us to evaluate the convolution integral (3.9b) for the modal coordinates:

− m+1 − 8FL ( 1) Ω wm(t)= 2 2 sin(ωm t) π EA (2m − 1) ωm 2 m+1 Ω − − m 8FLΩ ( 1) sin(Ωt) ωm sin(ω t) + 2 2 2 2 . (B.10) π EA (2m − 1) ωm − Ω

237 For a slender rod, the axial stress σ follows from term-by-term diﬀerentiation of the

axial displacement u (refer to (B.2)2 and (B.2)3):

∞ ∂u(x, t) F σ(x, t)=E = sin(Ωt)+E αm wm(t)cos(αm x). (B.11) ∂x A m=1

If the rod is driven at its r-th natural frequency, i.e., Ω = ωr,wherer is a positive integer, then a singularity arises in the r-th modal coordinate wr(t). To accommodate this singularity, we apply l’Hˆopital’s rule to (B.10), which leads to

4FL (−1)r+1 wr(t)=− sin(ωrt)+ωr t cos(ωrt) . (B.12) π2EA (2r − 1)2

Hence, when performing the summations in (B.9) and (B.11), the modal coordinate

(B.12) is used for the resonant mode (m = r), while the modal coordinates (B.10) are used for the non-resonant modes (m = r).

B.4 Convergence and term-by-term diﬀerentiation

The generalized CBFM/HEEM solution

It follows from the Weierstrass M-test that the Fourier sine series (3.4a)-(3.4c) for

η(ξ,t) is uniformly convergent if the series t ∞ ∞ 2 2c m+1 |ηm(t)| = (−1) sin [ωm(t − τ)]f(τ)dτ (B.13) aLωm m=1 m=1 0 converges. We proceed by arguing that the integral in (B.13) is O(m−1) for loadings f(t) of physical interest (e.g., periodic, constant, polynomial, impulse) so that the

−2 modal coordinates ηm(t)areO(m ). We then argue that the behavior of (B.13) is

238 dictated by the series

∞ 1 (B.14) − 2 m=1 (2m 1)

Using customary techniques from mathematical analysis (e.g., the integral test), we

ﬁnd that this series converges. Hence, η(ξ,t) is uniformly convergent.

If η(ξ,t) is diﬀerentiated once term by term with respect to ξ, then it follows from the Weierstrass M-test that the resulting Fourier cosine series is uniformly convergent if t ∞ ∞ 2c m+1 |αm ηm(t)| = (−1) sin [ωm(t − τ)]f(τ)dτ . (B.15) aL m=1 m=1 0

Using similar arguments, we deduce that the behavior of (B.15) is dictated by the series

∞ 1 − (B.16) m=1 2m 1 which diverges per the integral test. This result also holds for term-by-term differentiation of η(ξ,t) with respect to t. Hence, the once-term-by-term differentiated series are not uniformly convergent and not rigorous representations of the first derivative of η(ξ,t) with respect to ξ or t. Non-uniformly convergent behavior is expected from higher-order derivatives of η(ξ,t) as well.

The generalized Mindlin-Goodman solution

239 It follows from the Weierstrass M-test that the Fourier sine series (3.9a)-(3.9c) for

η(ξ,t) is uniformly convergent if the series t ∞ ∞ 2 ψm c |wm(t)| = φm cos(ωmt)+ sin(ωm t)+ sin [ωm(t − τ)]hm(τ)dτ ωm ωm m=1 m=1 0 (B.17)

−2 converges. We argue that the φm term (O(m )) dominates the convergence behavior of wm(t). It then follows that the convergence of (B.17) is dictated by a series of the type (B.14) so that η(ξ,t) is uniformly convergent.

Term-by-term diﬀerentiation of η(ξ,t) is more subtle. Again, the φm term, now

O(m−1), dominates so that the term-by-term diﬀerentiated series is not uniformly convergent. However, in a scenario where φm vanishes (e.g., the physical example

−2 (3.2); see Section B.3), the ψm term (O(m )) dominates so that the term-by-term diﬀerentiated series is uniformly convergent.

240 Appendix C: Supplement to Chapter 6

In this section of the appendix, we provide detailed proofs of selected results from

Chapter 6.

Proof C.1

Prove the transport theorem for surface integrals. That is, show that

d a · n da = a +curl(a × v)+v(div a) · n da. dt Q Q

Solution

We begin by using the relationship

J N dA = FT n da between n da and N dA, together with a change of independent variable from x to

X, to convert the Eulerian integration to a Lagrangian integration, i.e., d d a · n da = a · J F−T N dA. (C.1) dt dt Q QR

241 We emphasize that the integrand on the left-hand side of (C.1) is a function of x and t, while the integrand on the right-hand side of (C.1) is a function of X and t. Then,

it follows that d d a · n da = J F−1 a · N dA dt dt Q QR = J F˙−1 a · N dA

QR = J˙F−1 a + J F˙−1 a + J F−1 a˙ · N dA. (C.2)

We now individually examine the three terms on the right-hand side of (C.2). For the ﬁrst term, we have J˙ F−1 a · N dA = J (div v) F−1 a · N dA

QR QR = a (div v) · J F−T N dA.

For the second term, we have J F˙−1 a · N dA = − J F−1 La · N dA

QR QR = − La · J F−T N dA

QR = − (grad v)a · J F−T N dA.

242 For the third term, we have J F−1 a˙ · N dA = a˙ · J F−T N dA.

QR QR

Assembling the preceding results yields

d a · n da = a˙ + a (div v) − (grad v)a · J F−T N dA. dt Q QR

We convert the Lagrangian integration back to an Eulerian integration with a change of independent variable from X to x and use of the relationship between N dA and n da:

d a · n da = a˙ + a (div v) − (grad v)a · n da. (C.3) dt Q Q

Use of

curl (a × v) = (grad a)v − (grad v)a + a(div v) − v(div a) and

a˙ = a + (grad a) v in Eq. (C.3) leads to

d a · n da = a +curl(a × v)+v(div a) · n da. dt Q Q

Proof C.2

243 Starting with the Eulerian statement of balance of angular momentum in integral form, d x × vρdv = x × (f m + f em)ρdv + x × t da + cem ρdv, dt P P ∂P P derive the corresponding pointwise form

ρΓem + T − TT = 0.

Solution

We begin with the Eulerian integral form of balance of angular momentum, i.e., d x × vρdv = x × (f m + f em)ρdv + x × t da + cem ρdv. (C.4) dt P P ∂P P

We ﬁrst consider the left-hand side of (C.4), which, after use of the transport theorem

(6.46), the product rule, and the local form of conservation of mass (6.50a), becomes

d x × ρv dv = x ×˙ ρv +(x × ρv)divv dv dt P P = x × ρv˙ + ρ(v × v)+x × ρ˙ + ρ div v v dv P = x × ρv˙ dv. (C.5) P

Next, we consider the second term on the right-hand side of (C.4), which, after use of t = Tn and a fundamental result from tensor calculus [261], becomes x × t da = x × (Tn) da = x × div T + τ dv. (C.6) ∂P ∂P P

244 Note that

T − TT a = τ × a for any vector a in E 3. In other words, τ is the axial vector corresponding to the skew tensor T − TT.

Using results (C.5) and (C.6) in the original integral equation (C.4), we arrive at x × ρ f m + f em +divT − ρv˙ + τ + ρcem dv = 0. (C.7) P

By virtue of (6.50b), the local form of balance of linear momentum, (C.7) reduces to τ + ρcem dv = 0. P

Since the integrand is continuous and P is arbitrary, the localization theorem implies that

τ + ρcem = 0.

It then follows that

τ × a + ρcem × a = 0 (C.8) for any vector a in E 3.Sinceτ is the axial vector corresponding to the skew tensor

T − TT and cem is the axial vector corresponding to the skew tensor Γem,(C.8) becomes

T − TT + ρΓem a = 0.

245 Since a is arbitrary, and the coeﬃcient of a is independent of a itself, we conclude that

T − TT + ρΓem = 0.

Hence, the Cauchy stress is not symmetric.

Proof C.3

Starting with the Eulerian statement of conservation of charge in integral form, d σ∗ dv = − j∗ · n da − (curl m∗) · n da, dt P ∂P ∂P derive the corresponding pointwise form

σ˙ ∗ + σ∗ div v +divj∗ =0.

Solution

We begin with the Eulerian integral form of conservation of charge, i.e., d σ∗ dv = − j∗ · n da − (curl m∗) · n da. (C.9) dt P ∂P ∂P

Upon use of the transport theorem (6.46), the left-hand side of (C.9) becomes d σ∗ dv = σ˙ ∗ + σ∗ div v dv. (C.10) dt P P

246 Use of the divergence theorem (6.48) on the right-hand side of (C.9) leads to − j∗ · n da − (curl m∗) · n da = − div j∗ dv, (C.11) ∂P ∂P P noting that the divergence of the curl of a vector vanishes. Substituting results (C.10) and (C.11) into (C.9), and subsequently using the localization theorem of Section 6.3, gives the pointwise form of conservation of charge:

σ˙ ∗ + σ∗ div v +divj∗ =0.

Proof C.4

Starting with the Eulerian statement of the Amp`ere-Maxwell law in integral form, d d∗ · n da + j∗ · n da = h∗ · l dl, dt Q Q ∂Q derive the corresponding pointwise form

curl h∗ =(d∗) +curl(d∗ × v)+σ∗v + j∗.

Solution

We begin with the Eulerian integral form of the Amp`ere-Maxwell law, i.e., d d∗ · n da + j∗ · n da = h∗ · l dl. (C.12) dt Q Q ∂Q

We consider the ﬁrst term on the left-hand side of (C.12), which, after use of the transport theorem (6.45) and the pointwise form of Gauss’s law for electricity (6.50i),

247 becomes

d d∗ · n da = (d∗) +curl(d∗ × v)+v(div d∗) · n da dt Q Q = (d∗) +curl(d∗ × v)+σ∗ v · n da. Q

Next, we consider the right-hand side of (C.12), which, after use of Stokes’s theorem

(6.47), becomes h∗ · l dl = (curl h∗) · n da. ∂Q Q

Substitution of the preceding results into (C.12) leads to

(d∗) +curl(d∗ × v)+σ∗ v + j∗ − curl h∗ · n da =0. Q

Since the integrand is continuous and Q is arbitrary, the localization theorem implies

that

(d∗) +curl(d∗ × v)+σ∗ v + j∗ − curl h∗ · n =0.

Furthermore, since the coeﬃcient of n is independent of n,andn is arbitrary, it follows that the coeﬃcient must vanish, i.e.,

curl h∗ =(d∗) +curl(d∗ × v)+σ∗ v + j∗.

Proof C.5

248 Show that conservation of charge (6.50f) is a consequence of Maxwell’s equations

(6.50g)-(6.50j).

Solution

Taking the divergence of the Amp`ere-Maxwell law (6.50j), and noting that the divergence of the curl of a vector vanishes, leads to

div (d∗) + div (σv)+divj∗ =0.

Continuity of d enables the order of partial diﬀerentiation to be exchanged, so that

div (d∗) = div d∗ .

This, together with the result

div (σ∗ v)=σ∗ div v + v · grad σ∗

from tensor calculus, implies that

(div d∗) + v · grad σ∗ + σ∗ div v +divj∗ =0.

We then invoke Gauss’s law for electricity (6.50i) and the relationship

σ˙ ∗ =(σ∗) + v · grad σ∗

to recover

σ˙ ∗ + σ∗ div v +divj∗ =0, the pointwise Eulerian statement of conservation of charge.

249 Proof C.6

Starting with the Lagrangian statement of the ﬁrst law of thermodynamics (or con-

servation of energy) in integral form, d 1 d m em v · vρR dV + ερR dV = (f + f ) · v ρR dV + tR · v dA dt 2 dt PR PR PR ∂PR t em + (r + r )ρR dV − hR dA,

PR ∂PR derive the corresponding pointwise form

t em ρR ε˙ = P · Grad v + ρR r + r − Div qR.

Solution

We begin with the Lagrangian integral form of the ﬁrst law of thermodynamics, i.e., d 1 d m em v · vρR dV + ερR dV = (f + f ) · v ρR dV + tR · v dA dt 2 dt PR PR PR ∂PR t em + (r + r )ρR dV − hR dA.

PR ∂PR

By consolidating the mechanical energy contributions into a single term, the energy theorem (6.54) dramatically simpliﬁes the above statement of the ﬁrst law: d t em ερR dV = P · Grad v dV + (r + r )ρR dV − hR dA. (C.13) dt PR PR PR ∂PR

250 Working with the left-hand side of (C.13): d ερR dV = ερ˙ R dV (PR is ﬁxed) dt PR PR = ερ˙R + ρR ε˙ dV (product rule)

= ρR εdV˙ (ρR is independent of time).

Now working with the third term on the right-hand side of (C.13):

hR dA = qR · N dA (hR = qR · N)

∂PR ∂PR

= Div qR dV (divergence theorem (6.48)).

Assembling the above results, we have

t em ρR ε˙ − P · Grad v − ρR (r + r )+DivqR dV =0.

Since the integrand is continuous and the region of integration PR is arbitrary, it follows from the localization theorem that

t em ρR ε˙ − P · Grad v − ρR (r + r )+DivqR =0, or

t em ρR ε˙ = P · Grad v + ρR (r + r ) − Div qR.

251 Proof C.7

T ∗ Verify that eR = F e .

Solution

Recall from (6.27)2 that depending on whether we label the closed curve enclosing subset S2 by its present location ∂Q or its reference location ∂QR, the electromotive force E induced in the boundary of S2 has the following Eulerian and Lagrangian integral representations: ⎧ ⎪ ⎪ ⎪ e∗ · l dl ⎨⎪ ∂Q E (S2,t)= ⎪ ⎪ ⎪ eR · lR dL ⎩⎪ ∂QR

It follows that ∗ e · l dl = eR · lR dL. (C.14)

∂Q ∂QR

Upon a change of independent variable from x to X, the left-hand side of (C.14)

becomes ∗ ∗ e · l dl = e · FlR dL, (C.15)

∂Q ∂QR wherewehaveused

dl = FdlR,

i.e., the deformation gradient F linearly maps each line element dlR = lR dL in the reference conﬁguration into a line element dl = l dl in the present conﬁguration.

252 (Recall that lR and l are unit vectors in the reference and present configurations, and dL and dl are the infinitesimal lengths of the line elements.) Substitution of (C.15) into (C.14), and subsequent use of the definition of the transpose of a tensor, leads to

T ∗ F e − eR · lR dL =0.

∂QR

Since the integrand is continuous and ∂QR is arbitrary, the localization theorem of

Section 6.3 implies that

T ∗ F e − eR · lR =0.

Since the coeﬃcient of lR is independent of lR,andlR is arbitrary, it follows that

T ∗ eR = F e . (C.16)

Note that relationship (C.16) can also be obtained by starting with the Eulerian and

Lagrangian integral representations of the electric ﬂux F in (6.33) and using similar arguments to those employed here.

Proof C.8

∗ Verify that σR = Jσ .

Solution

Recall from (6.22)1 that depending on whether we label the volume occupied by subset

S1 by its present location P or its reference location PR, the free charge Σ within S1

253 has the following Eulerian and Lagrangian integral representations: ⎧ ⎪ ⎪ ⎪ σ∗ dv ⎨⎪ P Σ(S1,t)= ⎪ ⎪ ⎪ σR dV ⎩⎪ PR

It follows that ∗ σ dv = σR dV. (C.17)

P PR

The left-hand side of (C.17), after a change of independent variable from x to X and use of the relationship dv = JdV, becomes σ∗ dv = σ∗JdV. (C.18)

P PR

Substitution of (C.18) into (C.17) and use of the localization theorem leads to

∗ σR = Jσ .

Proof C.9

−1 ∗ Verify that bR = J F b .

Solution

Recall from (6.30) that depending on whether we label the closed surface bounding subset S1 by its present location ∂P or its reference location ∂PR, the magnetic ﬂux

B through the boundary of S1 has the following Eulerian and Lagrangian integral

254 representations: ⎧ ⎪ ⎪ ⎪ b∗ · n da ⎨⎪ ∂P B(S1,t)= ⎪ ⎪ ⎪ bR · N dA ⎩⎪ ∂PR

It follows that ∗ b · n da = bR · N dA. (C.19)

∂P ∂PR

Upon a change of independent variable from X to x, the right-hand side of (C.19) becomes 1 T bR · N dA = bR · F n da, (C.20) J ∂PR ∂P wherewehaveusedNanson’sformula:

J N dA = FT n da.

Substitution of (C.20) into (C.19), and subsequent use of the deﬁnition of the transpose of a tensor, leads to

∗ 1 b − FbR · n da =0. J ∂P

The localization theorem then implies that

∗ 1 b − FbR · n =0. J

255 Since the coeﬃcient of n is independent of n,andn is arbitrary, it follows that

−1 ∗ bR = J F b .

Note that this relationship can also be obtained starting from the Eulerian and La- grangian integral representations of the magnetic ﬁeld T in (6.39).

Proof C.10

Prove that the divergence of the Maxwell stress tensor Tem in (6.72) is equal to the electromagnetic body force vector ρf em in (6.68a). Then show that twice the skew part of the Maxwell stress is equal to the electromagnetic body couple ρΓem in (6.68b).

Solution

For clarify of presentation, we omit the superscript asterisks that denote an eﬀective electromagnetic ﬁeld. Taking the divergence of the Maxwell stress tensor (6.72) gives em 1 div T =div e ⊗ d + h ⊗ b − o e · e + μo h · h I . 2

The divergence is distributive over tensor addition, which implies that

em 1 div T = div (e ⊗ d)+div(h ⊗ b) − o div (e·e)I + μo div (h·h)I . (C.21) 2

The ﬁrst term in (C.21) simpliﬁes to

: σ div (e ⊗ d)=(divd) e + (grad e) d = σe + (grad e) d. (C.22)

256 In a similar fashion, the second term reduces to

: 0 div (h ⊗ b)=(divb) h + (grad h) b = (grad h) b. (C.23)

The third term in (C.21) simpliﬁes to

1 1 T o div (e · e)I = o grad (e · e)= o (grad e) e. (C.24) 2 2

In a similar fashion, the fourth term reduces to

1 T μo div (h · h)I = μo (grad h) h. (C.25) 2

Combining results (C.22)-(C.25) in (C.21), we obtain

em T T div T = σe + (grad e) d + (grad h) b − o (grad e) e + μo (grad h) h . (C.26)

The relationships

d = p + o e, b = μo (h + m)

allow us to rewrite (C.26) as

T T div Tem = σe + grad e − (grad e) d + grad h − (grad h) b

T T + (grad e) p + μo (grad h) m. (C.27)

Invoking the deﬁnition of the curl, i.e.,

T (curl f) × g = grad f − (grad f) g

257 for any vectors f and g, allows (C.27) to be rewritten as

em T T div T = σe +(curle)×d +(curlh)×b + (grad e) p + μo (grad h) m. (C.28)

Faraday’s law (6.50h) and the Amp`ere-Maxwell law (6.50j), i.e.,

curl e = −b − curl (b × v), curl h = d +curl(d × v)+σv + j, are then used in (C.28), which, noting anticommutativity of the vector product, leads to

em T T div T = σe + j × b + (grad e) p + μo (grad h) m + d +curl(d × v)+σv × b + d × b +curl(b × v) .

It then follows that

em T T em div T = σe + j × b + (grad e) p + μo (grad h) m + ˚d × b + d ×˚b = ρf , where ˚b and ˚d represent convected rates as deﬁned in (6.69). Hence, the divergence of the Maxwell stress is equal to the electromagnetic body force.

For the second part of the proof, it follows from (6.72) that

Tem − (Tem)T = e ⊗ d − (e ⊗ d)T + h ⊗ b − (h ⊗ b)T, or, upon use of the relation (f ⊗ g)T = g ⊗ f,

Tem − (Tem)T = e ⊗ d − d ⊗ e + h ⊗ b − b ⊗ h. (C.29)

258 Substituting the algebraic relations

d = p + o e, b = μo (h + m)

into (C.29) leads to

em em T em T − (T ) =(e ⊗ p − p ⊗ e)+μo(h ⊗ m − m ⊗ h)=ρΓ .

Hence, twice the skew part of the Maxwell stress is equal to the electromagnetic body couple.

259 Appendix D: Supplement to Chapter 7

In this part of the appendix, we provide detailed proofs of selected results from

Chapter 7. A useful relation that will be used in many of these proofs is [174]

−1 ∂FAi −1 −1 = −FAj FBi , (D.1) ∂FjB where FiA is the Cartesian component form of the deformation gradient F.Notethat

F can be written in indicial notation as

F = FiA ei ⊗ eA,

where ei and eA are the Cartesian basis vectors associated with the present and ref-

erence conﬁgurations, respectively. Also note that lowercase subscripts pertain to the

present conﬁguration and uppercase subscripts pertain to the reference conﬁguration.

Another result that will be frequently called upon is [69]

dJ = J F−T, dF where J =detF. Alternatively, in indicial notation,

dJ −1 = JFAi . (D.2) dFiA

260 Proof D.1

Verify that J˙ = J F−T · F˙ .

Solution

As is customary, this proof is performed in a dynamic setting (where v = 0):

J˙ = J div v = J tr(grad v)=J tr FF˙ −1 = J tr F−TF˙ T = J F−T · F˙ .

In the context of Chapter 7, where only static deformations are considered, we inter- pret this result as a relation between virtual changes in J and F (arising from the imposition of a virtual displacement).

Proof D.2

Verify result (7.32a), i.e.,

Feh Feh ∂ψ ∂Ψ 1 μo = − (e ⊗ p)F−T − (h ⊗ m)F−T. ∂F ∂F ρ ρ

Solution

In this proof, the deformation gradient F, Lagrangian electric ﬁeld eR, and Lagrangian magnetic ﬁeld hR are regarded as the independent variables. Hence, it follows from the chain rule that

T T ∂ΨFeh ∂ψFeh ∂e ∂ψFeh ∂h ∂ψFeh = + + . (D.3) ∂F ∂F ∂F ∂e ∂F ∂h

261 We proceed by individually examining the second and third terms on the right-hand-

side of (D.3). For the second term, we have

Feh −1 Feh ∂ei ∂ψ ∂ (FAi eA) ∂ψ = (transformation (7.6)1) ∂FjB ∂ei ∂FjB ∂ei

−1 1 ∂FAi = − eA pi (result (7.22b)1) ρ ∂FjB

1 −1 −1 = F eA pi F (result (D.1)) ρ Aj Bi

1 −1 = ej pi F (transformation (7.6)1) ρ Bi

Converting this result back to direct notation gives

T ∂e ∂ψFeh 1 = (e ⊗ p)F−T. (D.4) ∂F ∂e ρ

For the third term, we write

Feh −1 Feh ∂hi ∂ψ ∂ (FAi hA) ∂ψ = (transformation (7.6)4) ∂FjB ∂hi ∂FjB ∂hi

−1 μo ∂FAi = − hA mi (result (7.22b)2) ρ ∂FjB

μo −1 −1 = F hA mi F (result (D.1)) ρ Aj Bi

μo −1 = hj mi F (transformation (7.6)4) ρ Bi

Converting this result back to direct notation gives T Feh ∂h ∂ψ μo = (h ⊗ m)F−T. (D.5) ∂F ∂h ρ

262 Use of (D.4) and (D.5) in (D.3) gives the anticipated result, i.e.,

Feh Feh ∂ψ ∂Ψ 1 μo = − (e ⊗ p)F−T − (h ⊗ m)F−T. ∂F ∂F ρ ρ

Proof D.3

Verify results (7.32b)1 and (7.32b)2, i.e.,

∂ψFeh ∂ΨFeh ∂ψFeh ∂ΨFeh = F , = F . ∂e ∂eR ∂h ∂hR

Solution

We first prove result (7.32b)1. In what follows, the deformation gradient F,Eule- rian electric field e, and Eulerian magnetic field h are regarded as the independent variables. It follows from the chain rule that Feh T Feh ∂ψ ∂eR ∂Ψ = . (D.6) ∂e ∂e ∂eR

To proceed, we write (D.6) in indicial notation and use transformation (7.6)1 to obtain

Feh Feh Feh Feh Feh ∂ψ ∂eA ∂Ψ ∂ (FjAej) ∂Ψ ∂Ψ ∂Ψ = = = δij FjA = FiA , ∂ei ∂ei ∂eA ∂ei ∂eA ∂eA ∂eA where δij is the Kronecker delta. Re-expressing this result in direct notation gives

∂ψFeh ∂ΨFeh = F . ∂e ∂eR

Since hR transforms to h in the same manner that eR transforms to e (refer to

(7.6)), it follows that the proof of (7.32b)2 will be formally identical to the proof of

263 (7.32b)1.Consequently,

∂ψFeh ∂ΨFeh = F . ∂h ∂hR

Proof D.4

Verify result (7.36a), i.e., Fdb Fdb ∂ψ ∂Ψ 1 1 −T 1 1 −T = + (p ⊗ dR) − (p · d)F + (m ⊗ bR) − (m · b)F . ∂F ∂F o ρ J ρ J

Solution

In this proof, the deformation gradient F, Lagrangian electric displacement dR,and

Lagrangian magnetic induction bR are regarded as the independent variables. Hence, it follows from the chain rule that

T T ∂ΨFdb ∂ψFdb ∂d ∂ψFdb ∂b ∂ψFdb = + + . (D.7) ∂F ∂F ∂F ∂d ∂F ∂b

We proceed by individually examining the second and third terms on the right-hand-

side of (D.7). For the second term, we have

Fdb −1 Fdb ∂di ∂ψ ∂ (J FiA dA) ∂ψ = (transformation (7.6)2) ∂FjB ∂di ∂FjB ∂di −1 1 dJ 1 ∂FiA = − pi dA FiA + (result (7.27)2) o ρ dFjB J ∂FjB 1 −1 −1 −1 = − pi dA − J FBj FiA + J δij δAB (result (D.2)) o ρ 1 −1 −1 = − − pi diFBj + J pj dB (transformation (7.6)2) o ρ

264 Converting this result back to direct notation gives T Fdb ∂d ∂ψ 1 −T 1 = − − (p · d)F + (p ⊗ dR) . (D.8) ∂F ∂d o ρ J

For the third term, we write

Fdb −1 Fdb ∂bi ∂ψ ∂ (J FiA bA) ∂ψ = (transformation (7.6)5) ∂FjB ∂bi ∂FjB ∂bi −1 1 dJ 1 ∂FiA = − mi bA FiA + (result (7.27)3) ρ dFjB J ∂FjB 1 −1 −1 −1 = − mi bA − J F FiA + J δij δAB (result (D.2)) ρ Bj 1 −1 −1 = − − mi biF + J mj bB (transformation (7.6)5) ρ Bj

Converting this result back to direct notation gives T Fdb ∂b ∂ψ 1 −T 1 = − − (m · b)F + (m ⊗ bR) . (D.9) ∂F ∂b ρ J

Use of (D.8) and (D.9) in (D.7) gives the anticipated result, i.e., Fdb Fdb ∂ψ ∂Ψ 1 1 −T 1 1 −T = + (p ⊗ dR) − (p · d)F + (m ⊗ bR) − (m · b)F . ∂F ∂F o ρ J ρ J

Proof D.5

Verify results (7.36b)1 and (7.36b)2, i.e.,

∂ψFdb ∂ΨFdb ∂ψFdb ∂ΨFdb = J F−T , = J F−T . ∂d ∂dR ∂b ∂bR

Solution

265 We ﬁrst prove result (7.36b)1. In what follows, the deformation gradient F,Eule- rian electric displacement d, and Eulerian magnetic induction b are regarded as the independent variables. It follows from the chain rule that Fdb T Fdb ∂ψ ∂dR ∂Ψ = . (D.10) ∂d ∂d ∂dR

To proceed, we write (D.10) in indicial notation and use transformation (7.6)2 to obtain

Fdb Fdb −1 Fdb Fdb Fdb ∂ψ ∂dA ∂Ψ ∂ (JFAj dj) ∂Ψ −1 ∂Ψ −1 ∂Ψ = = = JFAj δji = JFAi . ∂di ∂di ∂dA ∂di ∂dA ∂dA ∂dA

Re-expressing this result in direct notation gives

∂ψFdb ∂ΨFdb = J F−T . ∂d ∂dR

Since bR transforms to b in the same manner that dR transforms to d (refer to

(7.6)), it follows that the proof of (7.36b)2 will be formally identical to the proof of

(7.36b)1.Consequently,

∂ψFdb ∂ΨFdb = J F−T . ∂b ∂bR

Proof D.6

Verify that

∂εˆ ∂E 1 1 −T μo −T = − e ⊗ pR + (e · p)F + (m ⊗ h)F . ∂F ∂F ρJ ρ ρ

266 Solution

In this proof, the deformation gradient F, Lagrangian polarization pR, and Lagrangian magnetization mR are regarded as the independent variables. Hence, use of the chain rule on the internal energy gives

T T ∂E ∂εˆ ∂p ∂εˆ ∂m ∂εˆ = + + , (D.11) ∂F ∂F ∂F ∂p ∂F ∂m noting that

−1 −T εˆ(F, p, m)=ˆε(F,J FpR, F mR), which follows from (7.6). We proceed by individually examining the second and third terms on the right-hand-side of (D.11). For the second term, we have

−1 ∂pi ∂εˆ ∂ (J FiA pA) ∂εˆ = (transformation (7.6)3) ∂FjB ∂pi ∂FjB ∂pi −1 1 dJ 1 ∂FiA = ei pA FiA + (result (7.30)2) ρ dFjB J ∂FjB 1 −1 −1 −1 = ei pA − J F FiA + J δij δAB (result (D.2)) ρ Bj 1 −1 −1 = − ei pi F + J ej pB (transformation (7.6)3) ρ Bj

Converting this result back to direct notation gives

T ∂p ∂εˆ 1 1 −T = e ⊗ pR − (e · p)F . (D.12) ∂F ∂p ρJ ρ

267 For the third term, we write

−1 ∂mi ∂εˆ ∂ (FAi mA) ∂εˆ = (transformation (7.6)6) ∂FjB ∂mi ∂FjB ∂mi

−1 μo ∂FAi = hi mA (result (7.30)3) ρ ∂FjB

μo −1 −1 = − F mA hi F (result (D.1)) ρ Aj Bi

μo −1 = − mj hi F (transformation (7.6)6) ρ Bi

Converting this result back to direct notation gives

T ∂m ∂εˆ μo = − (m ⊗ h)F−T. (D.13) ∂F ∂m ρ

Use of (D.12) and (D.13) in (D.11) gives the anticipated result, i.e.,

∂εˆ ∂E 1 1 −T μo −T = − e ⊗ pR + (e · p)F + (m ⊗ h)F . ∂F ∂F ρJ ρ ρ

Proof D.7

Verify that

∂εˆ ∂E ∂εˆ ∂E = J F−T , = F . ∂p ∂pR ∂m ∂mR

Solution

We ﬁrst prove the ﬁrst result. In what follows, the deformation gradient F,Eule- rian polarization p, and Eulerian magnetization m are regarded as the independent

268 variables. It follows from the chain rule that

T ∂εˆ ∂pR ∂E = . (D.14) ∂p ∂p ∂pR

To proceed, we write (D.14) in indicial notation and use transformation (7.6)3 to obtain

−1 ∂εˆ ∂pA ∂E ∂ (JFAj pj) ∂E −1 ∂E −1 ∂E = = = JFAj δji = JFAi . ∂pi ∂pi ∂pA ∂pi ∂pA ∂pA ∂pA

Re-expressing this in direct notation gives

∂εˆ ∂E = J F−T . ∂p ∂pR

For the second result, we have

∂εˆ ∂mA ∂E ∂ (FjAmj) ∂E ∂E ∂E = = = δij FjA = FiA , ∂mi ∂mi ∂mA ∂mi ∂mA ∂mA ∂mA which can be re-expressed in direct notation as

∂εˆ ∂E = F . ∂m ∂mR

Proof D.8

Show that

∂ (J m · m)=J (m · m)F−T − 2J (m ⊗ m)F−T. ∂F

Solution

269 This proof is performed in indicial notation. Using transformation (7.6)6 and the product rule, we obtain

∂ ∂ −1 −1 Jmi mi = JFAi mA FDi mD ∂FjB ∂FjB

−1 −1 dJ −1 −1 ∂FAi −1 −1 ∂FDi = mA mD FAi FDi + J FDi + JFAi dFjB ∂FjB ∂FjB

As a consequence of results (D.1) and (D.2), the right-hand side of the above expression becomes

−1 −1 −1 −1 −1 −1 −1 −1 −1 JFAi mA FDi mD FBj − JFAj mA FDi mD FBi − JFAi mA FDj mD FBi .

Subsequent simpliﬁcations using transformation (7.6)6 allow us to conclude that

∂ −1 −1 Jmi mi = Jmi mi FBj − 2Jmj mi FBi , ∂FjB which, when re-expressed in direct notation, is the desired result:

∂ (J m · m)=J (m · m)F−T − 2J (m ⊗ m)F−T. ∂F

Proof D.9

−1 Conﬁrm that m · m = mR · C mR.

Solution

270 We have

−T −T m · m = F mR · F mR (transformation (7.6)6)

−1 −T = mR · F F mR (deﬁnition of the transpose of a tensor)

−1 = mR · C mR (deﬁnition of C)

noting that

−1 C−1 = FTF = F−1F−T.

Proof D.10

Verify that

∂ −1 −1 mR · C mR =2C mR. ∂mR

Solution

This proof is performed in indicial notation. We have:

∂ −1 −1 mA CAB mB = CAB mA δBD + mB δAD ∂mD

−1 −1 = CAD mA + CDB mB

−1 −1 = CDA mA + CDB mB

−1 =2CDA mA,

271 where we have exploited symmetry of the right Cauchy-Green deformation tensor C and arbitrariness of the dummy (repeated) indices.

Proof D.11

Show that

∂ε ∂E 1 μo −T = − e ⊗ pR + m ⊗ h − m · h F . ∂F ∂F ρJ ρ

Solution

T T ∂E ∂ε ∂pr ∂ε ∂mr ∂ε = + + , (D.15) ∂F ∂F ∂F ∂pr ∂F ∂mr noting that

−T ε(F, pr, mr)=ε(F, FpR,JF mR), which follows from (7.6) and (7.11).

272 We proceed by individually examining the second and third terms on the right- hand-side of (D.15). For the second term, we have

∂(pr)i ∂ε ∂ (FiA pA) ∂ε = (transformations (7.6)3 and (7.11)1) ∂FjB ∂(pr)i ∂FjB ∂(pr)i

1 ∂FiA = ei pA (result (7.19)2) ρJ ∂FjB

1 = δij ei δAB pA ρJ

1 = ej pB ρJ

Converting this result back to direct notation gives

T ∂pr ∂ε 1 = e ⊗ pR. (D.16) ∂F ∂pr ρJ

For the third term, we write

−1 ∂(mr)i ∂ε ∂ (JF mA) ∂ε = Ai ∂FjB ∂(mr)i ∂FjB ∂(mr)i

−1 μo ∂FAi dJ −1 = mA hi J + FAi ρJ ∂FjB dFjB μo −1 −1 −1 −1 = − F mA hiF + F mA hi F ρ Aj Bi Ai Bj μo −1 −1 = − mj hi F + mi hi F ρ Bi Bj

Converting this result back to direct notation gives

T ∂mr ∂ε μo = − m ⊗ h − m · h F−T. (D.17) ∂F ∂mr ρ

273 Use of (D.16) and (D.17) in (D.15) gives the anticipated result, i.e.,

∂ε ∂E 1 μo −T = − e ⊗ pR + m ⊗ h − m · h F . ∂F ∂F ρJ ρ

Proof D.12

Show that

∂ (J e · e)=J (e · e)F−T − 2J (e ⊗ e)F−T ∂F

and

∂ (J h · h)=J (h · h)F−T − 2J (h ⊗ h)F−T. ∂F

Solution

This proof is performed in indicial notation. Using transformation (7.6)1 and the product rule, we obtain

∂ ∂ −1 −1 Jei ei = JFAi eA FDi eD ∂FjB ∂FjB

−1 −1 dJ −1 −1 ∂FAi −1 −1 ∂FDi = eA eD FAi FDi + J FDi + JFAi dFjB ∂FjB ∂FjB

As a consequence of results (D.1) and (D.2), the right-hand side of the above expression becomes

−1 −1 −1 −1 −1 −1 −1 −1 −1 JFAi eA FDi eD FBj − JFAj eA FDi eD FBi − JFAi eA FDj eD FBi .

274 Subsequent simpliﬁcations using transformation (7.6)6 allow us to conclude that

∂ −1 −1 Jei ei = Jei ei FBj − 2Jej ei FBi , ∂FjB which, when re-expressed in direct notation, is the desired result:

∂ (J e · e)=J (e · e)F−T − 2J (e ⊗ e)F−T. ∂F

Since hR transforms to h in the same manner that eR transforms to e (refer to

(7.6)), the two proofs will be formally identical. By this same logic, we expect Proof

D.8 to be formally identical as well, which is indeed the case.

Proof D.13

−1 −1 Conﬁrm that e · e = eR · C eR and h · h = hR · C hR.

Solution

We have

−T −T e · e = F eR · F eR (transformation (7.6)1)

−1 −T = eR · F F eR (deﬁnition of the transpose of a tensor)

−1 = eR · C eR (deﬁnition of C) noting that

−1 C−1 = FTF = F−1F−T.

275 Since hR transforms to h in the same manner that eR transforms to e (refer to (7.6)), the two proofs will be formally identical.

Proof D.14

Verify that

∂ −1 −1 eR · C eR =2C eR ∂eR

and

∂ −1 −1 hR · C hR =2C hR. ∂hR

Solution

Working in indicial notation, we have:

∂ −1 −1 eA CAB eB = CAB eA δBD + eB δAD ∂eD

−1 −1 = CAD eA + CDB eB

−1 −1 = CDA eA + CDB eB

−1 =2CDA eA, where we have exploited symmetry of the right Cauchy-Green deformation tensor

C and arbitrariness of the dummy (repeated) indices. The second proof is formally identical to the ﬁrst.

276 Proof D.15

Show that partial diﬀerentiation of the ‘augmented’ free energy

Feh def Feh 1 −1 1 −1 E = ρR Ψ − o J eR · C eR − μo J hR · C hR 2 2

with respect to the electric ﬁeld eR and magnetic ﬁeld hR leads to

∂EFeh ∂EFeh dR = − , bR = − . ∂eR ∂hR

Solution

Feh Taking a partial derivative of E with respect to eR (holding the other independent variables F and hR ﬁxed) gives

Feh Feh ∂E ∂Ψ 1 ∂ −1 = ρR − o J eR · C eR . ∂eR ∂eR 2 ∂eR

Subsequent use of constitutive equation (7.33)2, transformation (7.6)3, and the result

∂ −1 −1 eR · C eR =2C eR ∂eR

from Proof D.14 leads to

Feh ∂E −1 = − pR − o J C eR. ∂eR

It then follows from the algebraic relation (7.7)1 that

∂EFeh dR = − . ∂eR

277 Feh Now, taking a partial derivative of E with respect to hR (holding the other independent variables F and eR ﬁxed) yields

Feh Feh ∂E ∂Ψ 1 ∂ −1 = ρR − μo J hR · C hR . ∂hR ∂hR 2 ∂hR

Subsequent use of constitutive equation (7.33)3, transformation (7.6)6, and the result

∂ −1 −1 hR · C hR =2C hR ∂hR

from Proof D.14 leads to

Feh ∂E −1 = − μo J C mR + hR . ∂hR

It then follows from algebraic relation (7.7)2 that

∂EFeh bR = − . ∂hR

Proof D.16

Verify that

∂ −T (J d · d)=2d ⊗ dR − J (d · d)F ∂F

and

∂ −T (J b · b)=2b ⊗ bR − J (b · b)F . ∂F

Solution

278 This proof is performed in indicial notation. Using transformation (7.6)2 and the product rule, we obtain

∂ ∂ −1 Jdi di = J FiA dA FiB dB ∂FjD ∂FjD −1 dJ 1 ∂FiA 1 ∂FiB = dA dB FiA FiB + FiB + FiA dFjD J ∂FjD J ∂FjD

As a consequence of result (D.2), the right-hand side of the above expression becomes

−1 −1 −1 −1 − J FiA dA FiB dB FDj + J δijFiB δAD dA dB + J δijFiA δBD dB dA, or, after some simpliﬁcation,

−1 −1 −1 −1 − J FiA dA FiB dB FDj + J FjB dB dD + J FjAdA dD.

Subsequent use of transformation (7.6)2 allow us to conclude that

∂ −1 Jdi di = − Jdi di FDj +2dj dD, ∂FjD which, when re-expressed in direct notation, is the desired result:

∂ −T (Jd · d)=2d ⊗ dR − J (d · d)F . ∂F

Since bR transforms to b in the same manner that dR transforms to d (refer to (7.6)), the two proofs will be formally identical.

Proof D.17

−1 −1 Conﬁrm that J d · d = J dR · CdR and J b · b = J bR · CbR.

279 Solution

We have

−1 J d · d = J FdR · FdR (transformation (7.6)2)

−1 T = J dR · F FdR (deﬁnition of the transpose of a tensor)

−1 = J dR · CdR (deﬁnition of C)

Since bR transforms to b in the same manner that dR transforms to d (refer to (7.6)), the two proofs will be formally identical.

Proof D.18

Verify that

∂ dR · CdR =2CdR ∂dR

and

∂ bR · CbR =2CbR. ∂bR

Solution

280 Working in indicial notation, we have:

∂ dA CAB dB = CAB dA δBD + dB δAD ∂dD

= CAD dA + CDB dB

= CDA dA + CDB dB

=2CDA dA, where we have exploited symmetry of the right Cauchy-Green deformation tensor

C and arbitrariness of the dummy (repeated) indices. The second proof is formally identical to the ﬁrst.

Proof D.19

Show that partial diﬀerentiation of the ‘augmented’ free energy

Fdb def Fdb 1 1 E = ρR Ψ + dR · CdR + bR · CbR 2 oJ 2μoJ

with respect to the electric displacement dR and magnetic induction bR leads to

∂EFdb ∂EFdb eR = , hR = . ∂dR ∂bR

Solution

Fdb Taking a partial derivative of E with respect to dR (holding the other independent variables F and bR ﬁxed) gives

∂EFdb ∂ΨFdb 1 ∂ = ρR + dR · CdR . ∂dR ∂dR 2 oJ ∂dR

281 Subsequent use of constitutive equation (7.37)2, transformation (7.6)3, and the result

∂ dR · CdR =2CdR ∂dR

from Proof D.18 leads to

∂EFdb 1 = C dR − pR . ∂dR oJ

It then follows from algebraic relation (7.7)1 that

∂EFdb eR = . ∂dR

Fdb Now, taking a partial derivative of E with respect to bR (holding the other independent variables F and dR ﬁxed) yields

∂EFdb ∂ΨFdb 1 ∂ = ρR + bR · CbR . ∂bR ∂bR 2μoJ ∂bR

Subsequent use of constitutive equation (7.37)3, transformation (7.6)6, and the result

∂ bR · CbR =2CbR ∂bR

from Proof D.18 leads to

∂EFdb 1 = CbR − mR. ∂bR μoJ

It then follows from algebraic relation (7.7)2 that

∂EFdb hR = . ∂bR

282 Bibliography

[1] S.K. Godunov and E.I. Romenskii. Nonstationary equations of nonlinear elasticity theory in Eulerian coordinates. Journal of Applied Mechanics and Technical Physics, 13(6):868–884, 1972.

[2] B.J. Plohr and D.H. Sharp. A conservative Eulerian formulation of the equations for elastic ﬂow. Advances in Applied Mathematics, 9(4):481–499, 1988.

[3] P. Le Floch and F. Olsson. A second-order Godunov method for the conservation laws of nonlinear elastodynamics. IMPACT of Computing in Science and Engineering, 2(4):318–354, 1990.

[4] X. Garaizar. Solution of a Riemann problem for elasticity. Journal of Elasticity, 26(1):43–63, 1991.

[5] H. Xiao, O.T. Bruhns, and A. Meyers. Existence and uniqueness of the integrable-exactly hypoelastic equation ˚τ ∗ = λ(trD)I +2μD and its signiﬁ- cance to ﬁnite inelasticity. Acta Mechanica, 138(1-2):31–50, 1999.

[6] A.G. Kulikovskii, N.V. Pogorelov, and A.Y. Semenov. Mathematical Aspects of Numerical Solution of Hyperbolic Systems. CRC Press, Boca Raton, 2001.

[7] S.K. Godunov and E.I. Romenskii. Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/Plenum Publishers, New York, 2003.

[8] G.H. Miller. An iterative Riemann solver for systems of hyperbolic conservation laws, with application to hyperelastic solid mechanics. Journal of Computa- tional Physics, 193(1):198–225, 2004.

[9] D. Demarco and E.N. Dvorkin. An Eulerian ﬁnite element formulation for modelling stationary ﬁnite strain elastic deformation processes. International Journal for Numerical Methods in Engineering, 62(8):1038–1063, 2005.

[10] S.L. Gavrilyuk, N. Favrie, and R. Saurel. Modelling wave dynamics of compressible elastic materials. Journal of Computational Physics, 227(5):2941–2969, 2008.

283 [11] V.A. Titarev, E. Romenski, and E.F. Toro. MUSTA-type upwind ﬂuxes for nonlinear elasticity. International Journal for Numerical Methods in Engineering, 73(7):897–926, 2008.

[12] K.R. Rajagopal and A.R. Srinivasa. On a class of non-dissipative materials that are not hyperelastic. Proceedings of the Royal Society of London A, 465(2102):493–500, 2009.

[13] P.T. Barton, D. Drikakis, E. Romenski, and V.A. Titarev. Exact and approximate solutions of Riemann problems in non-linear elasticity. Journal of Computational Physics, 228(18):7046–7068, 2009.

[14] P.T. Barton and D. Drikakis. An Eulerian method for multi-component problems in non-linear elasticity with sliding interfaces. Journal of Computational Physics, 229(15):5518–5540, 2010.

[15] R. Duddu, L.L. Lavier, T.J.R. Hughes, and V.M. Calo. A ﬁnite strain Eu- lerian formulation for compressible and nearly incompressible hyperelasticity using high-order B-spline ﬁnite elements. International Journal for Numerical Methods in Engineering, 89(6):762–785, 2012.

[16]K.Kamrin,C.H.Rycroft,andJ.-C.Nave. Reference map technique for ﬁnite- strain elasticity and ﬂuid-solid interaction. Journal of the Mechanics and Physics of Solids, 60(11):1952–1969, 2012.

[17] J.A. Trangenstein and P. Colella. A higher-order Godunov method for modeling ﬁnite deformation in elastic-plastic solids. Communications on Pure and Applied Mathematics, 44(1):41–100, 1991.

[18] B.J. Plohr and D.H. Sharp. A conservative formulation for plasticity. Advances in Applied Mathematics, 13(4):462–493, 1992.

[19] O.T. Bruhns, H. Xiao, and A. Meyers. Self-consistent Eulerian rate type elastoplasticity models based upon the logarithmic stress rate. International Journal of Plasticity, 15(5):479–520, 1999.

[20] G.H. Miller and P. Colella. A high-order Eulerian Godunov method for elastic- plastic ﬂow in solids. Journal of Computational Physics, 167(1):131–176, 2001.

[21] H.S. Udaykumar, L. Tran, D.M. Belk, and K.J. Vanden. An Eulerian method for computation of multimaterial impact with ENO shock-capturing and sharp interfaces. Journal of Computational Physics, 186(1):136–177, 2003.

[22] S. Gomaa, T.-L. Sham, and E. Krempl. Finite element formulation for ﬁnite deformation, isotropic viscoplasticity theory based on overstress (FVBO). In- ternational Journal of Solids and Structures, 41(13):3607–3624, 2004.

284 [23] B. Mehmandoust and A.-R. Pishevar. An Eulerian particle level set method for compressible deforming solids with arbitrary EOS. International Journal for Numerical Methods in Engineering, 79(10):1175–1202, 2009.

[24] Q. Chen, J. Wang, and K. Liu. Improved CE/SE scheme with particle level set method for numerical simulation of spall fracture due to high-velocity impact. Journal of Computational Physics, 229(19):7503–7519, 2010.

[25] D.J. Hill, D. Pullin, M. Ortiz, and D. Meiron. An Eulerian hybrid WENO centered-diﬀerence solver for elastic-plastic solids. Journal of Computational Physics, 229(24):9053–9072, 2010.

[26] P.T. Barton, D. Drikakis, and E.I. Romenski. An Eulerian ﬁnite-volume scheme for large elastoplastic deformations in solids. International Journal for Numer- ical Methods in Engineering, 81(4):453–484, 2010.

[27] C.H. Rycroft and F. Gibou. Simulations of a stretching bar using a plasticity model from the shear transformation zone theory. Journal of Computational Physics, 231(5):2155–2179, 2012.

[28] P.T. Barton, R. Deiterding, D. Meiron, and D. Pullin. Eulerian adaptive ﬁnite- diﬀerence method for high-velocity impact and penetration problems. Journal of Computational Physics, 240:76–99, 2013.

[29] A. Eshraghi, K.D. Papoulia, and H. Jahed. Eulerian framework for inelasticity based on the Jaumann rate and a hyperelastic constitutive relation – Part I: Rate-form hyperelasticity. ASME Journal of Applied Mechanics, 80(2):021027, 2013.

[30] A. L´opez Ortega, M. Lombardini, D.I. Pullin, and D.I. Meiron. Numerical simulation of elastic-plastic solid mechanics using an Eulerian stretch tensor approach and HLLD Riemann solver. Journal of Computational Physics, 257, Part A:414–441, 2014.

[31] C.H. Rycroft, Y. Sui, and E. Bouchbinder. An Eulerian projec- tion method for quasi-static elastoplasticity. arXiv e-prints, 2014. http://arxiv.org/abs/1409.2173v1.

[32] N.K. Rai, A. Kapahi, and H.S. Udaykumar. Treatment of contact separation in Eulerian high-speed multimaterial dynamic simulations. International Journal for Numerical Methods in Engineering, 100(11):793–813, 2014.

[33] G.H. Miller and P. Colella. A conservative three-dimensional Eulerian method for coupled solid-ﬂuid shock capturing. Journal of Computational Physics, 183(1):26–82, 2002.

285 [34] H. Zhao, J.B. Freund, and R.D. Moser. A fixed-mesh method for incompressible flow-structure systems with finite solid deformations. Journal of Computational Physics, 227(6):3114–3140, 2008.

[35] N. Favrie, S.L. Gavrilyuk, and R. Saurel. Solid-ﬂuid diﬀuse interface model in cases of extreme deformations. Journal of Computational Physics, 228(16):6037–6077, 2009.

[36] P.T. Barton, B. Obadia, and D. Drikakis. A conservative level-set based method for compressible solid/ﬂuid problems on ﬁxed grids. Journal of Computational Physics, 230(21):7867–7890, 2011.

[37] K.-H. Kim and J.J. Yoh. A particle level-set based Eulerian method for multimaterial detonation simulation of high explosive and metal conﬁnements. Pro- ceedings of the Combustion Institute, 34(2):2025–2033, 2013.

[38] S. Schoch, K. Nordin-Bates, and N. Nikiforakis. An Eulerian algorithm for coupled simulations of elastoplastic-solids and condensed-phase explosives. Journal of Computational Physics, 252:163–194, 2013.

[39] Y. Gorsse, A. Iollo, T. Milcent, and H. Telib. A simple Cartesian scheme for compressible multimaterials. Journal of Computational Physics, 272:772–798, 2014.

[40] B. Valkov, C.H. Rycroft, and K. Kamrin. Eulerian method for multiphase interactions of soft solid bodies in ﬂuids. ASME Journal of Applied Mechanics, 82(4):041011, 2015.

[41] S.S. Antman and W.H. Warner. Dynamical theory of hyperelastic rods. Archive for Rational Mechanics and Analysis, 23(2):135–162, 1966.

[42] G.A. Nariboli. Nonlinear longitudinal dispersive waves in elastic rods. Journal of Mathematical and Physical Sciences, 4:64–73, 1970.

[43] I.C. Howard. An approximation for slowly varying waves in elastic rods. IMA Journal of Applied Mathematics, 7(1):47–60, 1971.

[44] Y. Benveniste and J. Lubliner. Perturbation solutions for wave propagation in nonlinearly elastic rods. International Journal of Solids and Structures, 8(9):1115–1138, 1972.

[45] Y. Benveniste. Wave propagation in a nonlinearly elastic compressible rod with variable cross section. Acta Mechanica, 22(3-4):197–208, 1975.

286 [46] A.M. Samsonov, G.V. Dreiden, A.V. Porubov, and I.V. Semenova. Longitudinal-strain soliton focusing in a narrowing nonlinearly elastic rod. Physical Review B, 57(10):5778–5787, 1998.

[47] H.-H. Dai and Y. Huo. Asymptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods. Acta Mechanica, 157(1- 4):97–112, 2002.

[48] H.-H. Dai and X. Fan. Asymptotically approximate model equations for weakly nonlinear long waves in compressible elastic rods and their comparisons with other simpliﬁed model equations. Mathematics and Mechanics of Solids, 9(1):61–79, 2004.

[49] W.J.N. de Lima and M.F. Hamilton. Finite amplitude waves in isotropic elastic waveguides with arbitrary constant cross-sectional area. Wave Motion, 41(1):1– 11, 2005.

[50] L.A. Ostrovskii and A.M. Sutin. Nonlinear elastic waves in rods. PMM Journal of Applied Mathematics and Mechanics, 41(3):543–549, 1977.

[51] M.P. Soerensen, P.L. Christiansen, and P.S. Lomdahl. Solitary waves on nonlinear elastic rods. I. The Journal of the Acoustical Society of America, 76(3):871– 879, 1984.

[52] P.A. Clarkson, R.J. LeVeque, and R. Saxton. Solitary-wave interactions in elastic rods. Studies in Applied Mathematics, 75(2):95–121, 1986.

[53] B.D. Coleman and D.C. Newman. On waves in slender elastic rods. Archive for Rational Mechanics and Analysis, 109(1):39–61, 1990.

[54] H. Cohen and H.-H. Dai. Nonlinear axisymmetric waves in compressible hyperelastic rods: long ﬁnite amplitude waves. Acta Mechanica, 100(3-4):223–239, 1993.

[55] H.-H. Dai. Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod. Acta Mechanica, 127(1-4):193–207, 1998.

[56] T.W. Wright. Nonlinear waves in rods. In D.E. Carlson and R.T. Shield, editors, Proceedings of the IUTAM Symposium on Finite Elasticity, pages 423–443, The Hague, 1982. Martinus Nijhoﬀ.

[57] T.W. Wright. Nonlinear waves in a rod: Results for incompressible elastic materials. Studies in Applied Mathematics, 72:149–160, 1985.

[58] J.L. Ericksen and C. Truesdell. Exact theory of stress and strain in rods and shells. Archive for Rational Mechanics and Analysis, 1(1):295–323, 1958.

287 [59] H. Cohen. A non-linear theory of elastic directed curves. International Journal of Engineering Science, 4(5):511–524, 1966. [60] A.E. Green and N. Laws. A general theory of rods. Proceedings of the Royal Society of London A, 293(1433):145–155, 1966. [61] S.S. Antman. Kirchhoff’s problem for nonlinearly elastic rods. Quarterly of Applied Mathematics, 32(3):221–240, 1974. [62] A.E. Green, P.M. Naghdi, and M.L. Wenner. On the theory of rods. I. Deriva- tions from the three-dimensional equations. Proceedings of the Royal Society of London A, 337(1611):451–483, 1974. [63] A.E. Green, P.M. Naghdi, and M.L. Wenner. On the theory of rods. II. De- velopments by direct approach. Proceedings of the Royal Society of London A, 337(1611):485–507, 1974. [64] P.M. Naghdi and M.B. Rubin. Constrained theories of rods. Journal of Elas- ticity, 14(4):343–361, 1984. [65] O.M. O’Reilly. On constitutive relations for elastic rods. International Journal of Solids and Structures, 35(11):1009–1024, 1998. [66] T.J. Healey. Material symmetry and chirality in nonlinearly elastic rods. Math- ematics and Mechanics of Solids, 7(4):405–420, 2002. [67] S.-T.J. Yu, L. Yang, R.L. Lowe, and S.E. Bechtel. Numerical simulation of linear and nonlinear waves in hypoelastic solids by the CESE method. Wave Motion, 47(3):168–182, 2010. [68] L. Yang, R.L. Lowe, S.-T.J. Yu, and S.E. Bechtel. Numerical solution by the CESE method of a first-order hyperbolic form of the equations of dynamic nonlinear elasticity. ASME Journal of Vibration and Acoustics, 132(5):051003, 2010. [69] S.E. Bechtel and R.L. Lowe. Fundamentals of Continuum Mechanics.Academic Press, San Diego, 2014. [70] E.F Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin, 2nd edition, 1999. [71] G.A. Holzapfel. Nonlinear Solid Mechanics: A Continuum Approach for Engi- neering. Wiley, Chichester, 2000. [72] C. Liu and N.J. Walkington. An Eulerian description of fluids containing viscoelastic particles. Archive for Rational Mechanics and Analysis, 159(3):229–252, 2001.

288 [73] G. Kluth and B. Despr´es. Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme. Journal of Computational Physics, 229(24):9092–9118, 2010.

[74] K. Kamrin and J.-C. Nave. An Eulerian approach to the simulation of deformable solids: Application to ﬁnite-strain elasticity. arXiv e-prints, 2009. http://arxiv.org/abs/0901.3799v2.

[75] G.-H. Cottet, E. Maitre, and T. Milcent. Eulerian formulation and level set models for incompressible ﬂuid-structure interaction. ESAIM: Mathematical Modelling and Numerical Analysis, 42(3):471–492, 2008.

[76] K.N. Kamrin. Stochastic and Deterministic Models for Dense Granular Flow. PhD thesis, Massachusetts Institute of Technology, 2008.

[77] D.I.W. Levin, J. Litven, G.L. Jones, S. Sueda, and D.K. Pai. Eulerian solid simulation with contact. ACM Transactions on Graphics, 30(4):36:1–36:10, 2011.

[78] C. Truesdell. The simplest rate theory of pure elasticity. Communications on Pure and Applied Mathematics, 8(1):123–132, 1955.

[79] C. Truesdell. Hypo-elasticity. Journal of Rational Mechanics and Analysis, 4(1):83–133, 1955.

[80] R. Lin. Numerical study of consistency of rate constitutive equations with elasticity at ﬁnite deformation. International Journal for Numerical Methods in Engineering, 55(9):1053–1077, 2002.

[81] X. Zhou and K.K. Tamma. On the applicability and stress update formulations for corotational stress rate hypoelasticity constitutive models. Finite Elements in Analysis and Design, 39(8):783–816, 2003.

[82] H. Xiao, O.T. Bruhns, and A. Meyers. Hypo-elasticity model based upon the logarithmic stress rate. Journal of Elasticity, 47(1):51–68, 1997.

[83] B. Bernstein and K. Rajagopal. Thermodynamics of hypoelasticity. Zeitschrift f¨ur angewandte Mathematik und Physik (ZAMP), 59(3):537–553, 2008.

[84] S. Ghosh and N. Kikuchi. An arbitrary Lagrangian-Eulerian ﬁnite element method for large deformation analysis of elastic-viscoplastic solids. Computer Methods in Applied Mechanics and Engineering, 86(2):127–188, 1991.

[85] C.E. Anderson, Jr. An overview of the theory of hydrocodes. International Journal of Impact Engineering, 5(1-4):33–59, 1987.

289 [86] D.J. Benson. Computational methods in Lagrangian and Eulerian hydrocodes. Computer Methods in Applied Mechanics and Engineering, 99(2-3):235–394, 1992. [87] D.J. Benson. A multi-material Eulerian formulation for the eﬃcient solution of impact and penetration problems. Computational Mechanics, 15(6):558–571, 1995. [88] S.-C. Chang. The method of space-time conservation element and solution element – A new approach for solving the Navier-Stokes and Euler equations. Journal of Computational Physics, 119(2):295–324, 1995. [89] S.-C. Chang, X.-Y. Wang, and C.-Y. Chow. The space-time conservation element and solution element method: A new high-resolution and genuinely multi- dimensional paradigm for solving conservation laws. Journal of Computational Physics, 156(1):89–136, 1999. [90] X.-Y. Wang and S.-C. Chang. A 2D non-splitting unstructured triangular mesh Euler solver based on the space-time conservation element and solution element method. Computational Fluid Dynamics Journal, 8(2):309–325, 1999. [91] Z.-C. Zhang, S.T.J. Yu, and S.-C. Chang. A space-time conservation element and solution element method for solving the two- and three-dimensional unsteady Euler equations using quadrilateral and hexahedral meshes. Journal of Computational Physics, 175(1):168–199, 2002. [92] S.-C. Chang. Courant number insensitive CE/SE schemes. In Proceedings of the 38th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, pages AIAA Paper 2002–3890, 2002. [93] S.-C. Chang and X.Y. Wang. Multi-dimensional Courant number insensitive CE/SE Euler solvers for applications involving highly nonuniform meshes. In Proceedings of the 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, pages AIAA Paper 2003–5285, 2003. [94] S.-C. Chang. A new approach for constructing highly stable high order CESE schemes. In Proceedings of the 48th AIAA Aerospace Sciences Meeting, pages AIAA Paper 2010–543, 2010. [95] D. Bilyeu. A Higher-Order Conservation Element Solution Element Method for Solving Hyperbolic Diﬀerential Equations on Unstructured Meshes. PhD thesis, The Ohio State University, 2014. [96] D.L. Bilyeu, S.-T.J. Yu, Y.-Y. Chen, and J.-L. Cambier. A two-dimensional fourth-order unstructured-meshed Euler solver based on the CESE method. Journal of Computational Physics, 257, Part A:981–999, 2014.

290 [97] Y.-Y. Chen. A Multi-Physics Software Framework on Hybrid Parallel Comput- ing for High-Fidelity Solutions of Conservation Laws. PhD thesis, The Ohio State University, 2011.

[98] G.E. Hay. The ﬁnite displacement of thin rods. Transactions of the American Mathematical Society, 51(1):65–102, 1942.

[99] G.A. Nariboli. Asymptotic theory of wave-motion in rods. Zeitschrift f¨ur angewandte Mathematik und Mechanik (ZAMM), 49(9):525–531, 1969.

[100] T. Rogge. Longitudinal wave propagation in bars with variable cross section. Zeitschrift f¨ur angewandte Mathematik und Physik (ZAMP), 22(2):299–307, 1971.

[101] A. Bostr¨om. On wave equations for elastic rods. Zeitschrift f¨ur angewandte Mathematik und Mechanik (ZAMM), 80(4):245–251, 2000.

[102] N.G. Stephen, K.F. Lai, K. Young, and K.T. Chan. Longitudinal vibrations in circular rods: A systematic approach. Journal of Sound and Vibration, 331(1):107–116, 2012.

[103] B.P. Leonard. The ULTIMATE conservative diﬀerence scheme applied to unsteady one-dimensional advection. Computer Methods in Applied Mechanics and Engineering, 88(1):17–74, 1991.

[104] M.A. Alves, P.J. Oliveira, and F.T. Pinho. A convergent and universally bounded interpolation scheme for the treatment of advection. International Journal for Numerical Methods in Fluids, 41(1):47–75, 2003.

[105] K.D. Bolinger. Pointwise Closure Models for Slender, Non-Newtonian Free Jets. PhD thesis, The Ohio State University, 1989.

[106] S.E. Bechtel, J.Z. Cao, and M.G. Forest. Practical application of a higher order perturbation theory for slender viscoelastic jets and ﬁbers. Journal of Non-Newtonian Fluid Mechanics, 41(3):201–273, 1992.

[107] G. Carta. Correction to Bishop’s approximate method for the propagation of longitudinal waves in bars of generic cross-section. European Journal of Mechanics A/Solids, 36:156–162, 2012.

[108] K.F. Graﬀ. Wave Motion in Elastic Solids. Dover Publications, Mineola, 1991.

[109] R.J. Gordon and W.R. Schowalter. Anisotropic ﬂuid theory: A diﬀerent approach to the dumbbell theory of dilute polymer solutions. Transactions of The Society of Rheology, 16(1):79–97, 1972.

291 [110] S.E. Bechtel, K.D. Bolinger, J.Z. Cao, and M.G. Forest. Torsional eﬀects in high-order viscoelastic thin-ﬁlament models. SIAM Journal on Applied Mathe- matics, 55(1):58–99, 1995.

[111] B.R. Munson, T.H. Okiishi, W.W. Huebsch, and A.P. Rothmayer. Fundamen- tals of Fluid Mechanics. Wiley, Hoboken, 7th edition, 2013.

[112] A.G. Webster. Acoustical impedance and the theory of horns and of the phono- graph. Proceedings of the National Academy of Sciences of the United States of America, 5(7):275–282, 1919.

[113] L.H. Donnell. Longitudinal wave transmission and impact. Transactions of the American Society of Mechanical Engineers, 52:153–167, 1930.

[114] E. Eisner. Complete solutions of the “Webster” horn equation. The Journal of the Acoustical Society of America, 41(4B):1126–1146, 1967.

[115] D.D. Joseph, M. Renardy, and J.-C. Saut. Hyperbolicity and change of type in the ﬂow of viscoelastic ﬂuids. Archive for Rational Mechanics and Analysis, 87(3):213–251, 1985.

[116] R.F. Warming, R.M. Beam, and B.J. Hyett. Diagonalization and simultaneous symmetrization of the gas-dynamic matrices. Mathematics of Computation, 29(132):1037–1045, 1975.

[117] R.J. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, 2002.

[118] Y.-Y. Chen, L. Yang, and S.-T. Yu. Hyperbolicity of velocity-stress equations for waves in anisotropic elastic solids. Journal of Elasticity, 106(2):149–164, 2012.

[119] S.-T. Yu and S.-C. Chang. Treatments of stiﬀ source terms in conservation laws by the method of space-time conservation element/solution element. In Proceedings of the 35th AIAA Aerospace Sciences Meeting, pages AIAA Paper 97–0435, 1997.

[120] S.-T. Yu, S.-C. Chang, P.C.E. Jorgenson, S.-J. Park, and M.-C. Lai. Treating stiﬀ source terms in conservation laws by the space-time conservation element and solution element method. In C.-H. Bruneau, editor, Proceedings of the Sixteenth International Conference on Numerical Methods in Fluid Dynamics, volume 515 of Lecture Notes in Physics, pages 433–438. Springer, Berlin, 1998.

[121] S.-C. Chang, A. Himansu, C.Y. Loh, X.-Y. Wang, S.-T. Yu, and P.C.E. Jorgen- son. Robust and simple non-reﬂecting boundary conditions for the space-time

292 conservation element and solution element method. In A Collection of Technical Papers, 13th AIAA CFD Conference, pages AIAA Paper 97–2077, 1997.

[122] S.E. Bechtel, K.J. Lin, and M.G. Forest. Eﬀective stress rates of viscoelastic free jets. Journal of Non-Newtonian Fluid Mechanics, 26(1):1–41, 1987.

[123] A. Meyers, H. Xiao, and O.T. Bruhns. Choice of objective rate in single parameter hypoelastic deformation cycles. Computers & Structures, 84(17-18):1134– 1140, 2006.

[124] R.L. Lowe, S.-T.J. Yu, L. Yang, and S.E. Bechtel. Modal and characteristics- based approaches for modeling elastic waves induced by time-dependent boundary conditions. Journal of Sound and Vibration, 333(3):873–886, 2014.

[125] G.A. Nothmann. Vibration of a cantilever beam with prescribed end motion. ASME Journal of Applied Mechanics, 15:327–334, 1948.

[126] T.C. Yen and S. Kao. Vibration of beam-mass systems with time-dependent boundary conditions. ASME Journal of Applied Mechanics, 26:353–356, 1959.

[127] K. Yang. A uniﬁed solution for longitudinal wave propagation in an elastic rod. Journal of Sound and Vibration, 314(1-2):307–329, 2008.

[128] R.D. Mindlin and L.E. Goodman. Beam vibrations with time-dependent boundary conditions. ASME Journal of Applied Mechanics, 17:377–380, 1950.

[129] H. Reismann. On the forced motion of elastic solids. Applied Scientiﬁc Research, 18:156–165, 1968.

[130] J.G. Berry and P.M. Naghdi. On the vibration of elastic bodies having time- dependent boundary conditions. Quarterly of Applied Mathematics, 14:43–50, 1956.

[131] L. Meirovitch. Principles and Techniques of Vibrations. Prentice Hall, Upper Saddle River, 1997.

[132] H. Benaroya. Mechanical Vibration: Analysis, Uncertainties, and Control. Prentice Hall, Upper Saddle River, 1998.

[133] R. Courant and D. Hilbert. Methods of Mathematical Physics, Vol. I. Wiley, New York, 1953.

[134] S.Y. Lee and S.M. Lin. Dynamic analysis of nonuniform beams with time- dependent elastic boundary conditions. ASME Journal of Applied Mechanics, 63(2):474–478, 1996.

293 [135] G. Herrmann. Forced motions of Timoshenko beams. ASME Journal of Applied Mechanics, 22:53–56, 1955.

[136] S.Y. Lee and S.M. Lin. Non-uniform Timoshenko beams with time-dependent elastic boundary conditions. Journal of Sound and Vibration, 217(2):223–238, 1998.

[137] G. Herrmann. Forced motions of elastic rods. ASME Journal of Applied Me- chanics, 21:221–224, 1954.

[138] J. Ramachandran. Dynamic response of a plate to time-dependent boundary conditions. Nuclear Engineering and Design, 21(3):339–349, 1972.

[139] S.M. Lin. The closed-form solution for the forced vibration of non-uniform plates with distributed time-dependent boundary conditions. Journal of Sound and Vibration, 232(3):493–509, 2000.

[140] W.A. Strauss. Partial Diﬀerential Equations: An Introduction. Wiley, New York, 1992.

[141] C.R. Edstrom. The vibrating beam with nonhomogeneous boundary conditions. ASME Journal of Applied Mechanics, 48(3):669–670, 1981.

[142] D.A. Grant. Beam vibrations with time-dependent boundary conditions. Jour- nal of Sound and Vibration, 89(4):519–522, 1983.

[143] G.L. Anderson and C.R. Thomas. A forced vibration problem involving time derivatives in the boundary conditions. Journal of Sound and Vibration, 14(2):193–214, 1971.

[144] S.R. Wu. Classical solutions of forced vibration of rod and beam driven by displacement boundary conditions. Journal of Sound and Vibration, 279(1- 2):481–486, 2005.

[145] S.R. Wu. Classical solutions of forced vibration of rectangular plate driven by displacement boundary conditions. Journal of Sound and Vibration, 291(3- 5):1104–1121, 2006.

[146] L. Meirovitch. Fundamentals of Vibrations. McGraw-Hill, New York, 2001.

[147] R.D. Swope and W.F. Ames. Vibrations of a moving threadline. Journal of the Franklin Institute, 275(1):36–55, 1963.

[148] P.C. Chou and R.W. Mortimer. Solution of one-dimensional elastic wave problems by the method of characteristics. ASME Journal of Applied Mechanics, 34(3):745–750, 1967.

294 [149] K.H. Parker and C.J.H. Jones. Forward and backward running waves in the arteries: Analysis using the method of characteristics. ASME Journal of Biome- chanical Engineering, 112(3):322–326, 1990.

[150] W.D. Zhu and B.Z. Guo. Free and forced vibration of an axially moving string with an arbitrary velocity proﬁle. ASME Journal of Applied Mechan- ics, 65(4):901–907, 1998.

[151] W.D. Zhu and N.A. Zheng. Exact response of a translating string with arbitrarily varying length under general excitation. ASME Journal of Applied Mechanics, 75(3):031003, 2008.

[152] J.D. Achenbach. Wave Propagation in Elastic Solids. Elsevier, Amsterdam, 1975.

[153] W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, New York, ﬁrst edition, 1953.

[154] S. Chandra. Some comments on the Mindlin-Goodman method of solving vibrations problems with time dependent boundary conditions. Journal of Sound and Vibration, 67(2):275–277, 1979.

[155] D. Williams. Displacements of a linear elastic system under a given transient load. The Aeronautical Quarterly, 1:123–136, 1949.

[156] R.E. Cornwell, R.R. Craig, and C.P. Johnson. On the application of the mode- acceleration method to structural engineering problems. Earthquake Engineer- ing & Structural Dynamics, 11(5):679–688, 1983.

[157] H. Reismann. Forced motion of elastic plates. ASME Journal of Applied Me- chanics, 35(3):510–515, 1968.

[158] J.C. Simo and K.S. Pister. Remarks on rate constitutive equations for ﬁnite deformation problems: computational implications. Computer Methods in Applied Mechanics and Engineering, 46(2):201–215, 1984.

[159] J.C. Simo and M. Ortiz. A uniﬁed approach to ﬁnite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Computer Methods in Applied Mechanics and Engineering, 49(2):221–245, 1985.

[160] M. Koji´c and K.-J. Bathe. Studies of ﬁnite element procedures – Stress solution of a closed elastic strain path with stretching and shearing using the updated Lagrangian Jaumann formulation. Computers & Structures, 26(1-2):175–179, 1987.

295 [161] G. Weber and L. Anand. Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Computer Methods in Applied Mechanics and Engineering, 79(2):173–202, 1990.

[162] J.K. Dienes. On the analysis of rotation and stress rate in deforming bodies. Acta Mechanica, 32(4):217–232, 1979.

[163] G.C. Johnson and D.J. Bammann. A discussion of stress rates in ﬁnite deformation problems. International Journal of Solids and Structures, 20(8):725–737, 1984.

[164] H. Xiao, O.T. Bruhns, and A. Meyers. Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica, 124(1-4):89–105, 1997.

[165] R.M. McMeeking and J.R. Rice. Finite-element formulations for problems of large elastic-plastic deformation. International Journal of Solids and Structures, 11(5):601–616, 1975.

[166] E.H. Lee. Elastic-plastic deformation at ﬁnite strains. ASME Journal of Applied Mechanics, 36(1):1–6, 1969.

[167] B. Moran, M. Ortiz, and C.F. Shih. Formulation of implicit ﬁnite element methods for multiplicative ﬁnite deformation plasticity. International Journal for Numerical Methods in Engineering, 29(3):483–514, 1990.

[168] A.E. Green and P.M. Naghdi. A general theory of an elastic-plastic continuum. Archive for Rational Mechanics and Analysis, 18(4):251–281, 1965.

[169] J.F. Scott. Applications of magnetoelectrics. Journal of Materials Chemistry, 22(11):4567–4574, 2012.

[170] P. Martins and S. Lanceros-M´endez. Polymer-based magnetoelectric materials. Advanced Functional Materials, 23(27):3371–3385, 2013.

[171] L. Liu and P. Sharma. Giant and universal magnetoelectric coupling in soft materials and concomitant ramiﬁcations for materials science and biology. Physical Review E, 88(4):040601, 2013.

[172] Z. Alameh, Q. Deng, L. Liu, and P. Sharma. Using electrets to design concurrent magnetoelectricity and piezoelectricity in soft materials. Journal of Materials Research, 30(1):93–100, 2015.

[173] E. Galipeau and P. Ponte Casta˜neda. Giant ﬁeld-induced strains in magnetoactive elastomer composites. Proceedings of the Royal Society of London A, 469(2158):20130385, 2013.

296 [174] L. Dorfmann and R.W. Ogden. Nonlinear Theory of Electroelastic and Magne- toelastic Interactions. Springer, New York, 2014.

[175] I.A. Brigadnov and A. Dorfmann. Mathematical modeling of magneto-sensitive elastomers. International Journal of Solids and Structures, 40(18):4659–4674, 2003.

[176] S.V. Kankanala and N. Triantafyllidis. On ﬁnitely strained magnetorheological elastomers. Journal of the Mechanics and Physics of Solids, 52(12):2869–2908, 2004.

[177] P. Ponte Casta˜neda and E. Galipeau. Homogenization-based constitutive models for magnetorheological elastomers at ﬁnite strain. Journal of the Mechanics and Physics of Solids, 59(2):194–215, 2011.

[178] K. Danas and N. Triantafyllidis. Instability of a magnetoelastic layer resting on a non-magnetic substrate. Journal of the Mechanics and Physics of Solids, 69:67–83, 2014.

[179] P. Saxena, J.-P. Pelteret, and P. Steinmann. Modelling of iron-ﬁlled magneto- active polymers with a dispersed chain-like microstructure. European Journal of Mechanics A/Solids, 50:132–151, 2015.

[180] N.A. Spaldin and M. Fiebig. The renaissance of magnetoelectric multiferroics. Science, 309(5733):391–392, 2005.

[181] M. Fiebig. Revival of the magnetoelectric eﬀect. Journal of Physics D: Applied Physics, 38(8):R123–R152, 2005.

[182] W. Eerenstein, N.D. Mathur, and J.F. Scott. Multiferroic and magnetoelectric materials. Nature, 442(7104):759–765, 2006.

[183] C.-W. Nan, M.I. Bichurin, S. Dong, D. Viehland, and G. Srinivasan. Multi- ferroic magnetoelectric composites: Historical perspective, status, and future directions. Journal of Applied Physics, 103(3):031101, 2008.

[184] C.A.F. Vaz, J. Hoﬀman, C.H. Ahn, and R. Ramesh. Magnetoelectric coupling eﬀects in multiferroic complex oxide composite structures. Advanced Materials, 22(26-27):2900–2918, 2010.

[185] S.D. Moss, J.E. McLeod, I.G. Powlesland, and S.C. Galea. A bi-axial magnetoelectric vibration energy harvester. Sensors and Actuators A: Physical, 175:165–168, 2012.

297 [186] E. Galipeau, S. Rudykh, G. deBotton, and P. Ponte Casta˜neda. Magnetoactive elastomers with periodic and random microstructures. International Journal of Solids and Structures, 51(18):3012–3024, 2014.

[187] R. Bustamante. Mathematical Modelling of Non-linear Magneto- and Electro- active Rubber-like Materials. PhD thesis, University of Glasgow, 2007.

[188] A. Dorfmann and R.W. Ogden. Magnetoelastic modelling of elastomers. Euro- pean Journal of Mechanics A/Solids, 22(4):497–507, 2003.

[189] M. Ott´enio, M. Destrade, and R.W. Ogden. Incremental magnetoelastic deformations, with application to surface instability. Journal of Elasticity, 90(1):19– 42, 2008.

[190] R. Bustamante. Transversely isotropic nonlinear magneto-active elastomers. Acta Mechanica, 210(3-4):183–214, 2010.

[191] S. Rudykh and K. Bertoldi. Stability of anisotropic magnetorheological elastomers in ﬁnite deformations: A micromechanical approach. Journal of the Mechanics and Physics of Solids, 61(4):949–967, 2013.

[192] P. Saxena, M. Hossain, and P. Steinmann. Nonlinear magneto-viscoelasticity of transversally isotropic magneto-active polymers. Proceedings of the Royal Society of London A, 470(2166):20140082, 2014.

[193] M.R. Jolly, J.D. Carlson, and B.C. Mu˜noz. A model of the behaviour of magnetorheological materials. Smart Materials and Structures, 5(5):607–614, 1996.

[194] J.M. Ginder, M.E. Nichols, L.D. Elie, and J.L. Tardiﬀ. Magnetorheological elastomers: properties and applications. In Proceedings of SPIE, volume 3675, pages 131–138, 1999.

[195] C. Bellan and G. Bossis. Field dependence of viscoelastic properties of MR elastomers. International Journal of Modern Physics B, 16(17-18):2447–2453, 2002.

[196] X.L. Gong, X.Z. Zhang, and P.Q. Zhang. Fabrication and characterization of isotropic magnetorheological elastomers. Polymer Testing, 24(5):669–676, 2005.

[197] Z. Varga, G. Filipcsei, and M. Zr´ınyi. Magnetic ﬁeld sensitive functional elastomers with tuneable elastic modulus. Polymer, 47(1):227–233, 2006.

[198] A. Boczkowska and S.F. Awietjan. Smart composites of urethane elastomers with carbonyl iron. Journal of Materials Science, 44(15):4104–4111, 2009.

298 [199] K. Danas, S.V. Kankanala, and N. Triantafyllidis. Experiments and modeling of iron-particle-ﬁlled magnetorheological elastomers. Journal of the Mechanics and Physics of Solids, 60(1):120–138, 2012.

[200] R. Bustamante and M.H.B.M. Shariﬀ. A principal axis formulation for nonlinear magnetoelastic deformations: Isotropic bodies. European Journal of Mechanics A/Solids, 50:17–27, 2015.

[201] C. Truesdell and R.A. Toupin. The classical ﬁeld theories. In S. Fl¨ugge, editor, Handbuch der Physik, volume III/1, pages 226–858. Springer, Berlin, 1960.

[202] H.F. Tiersten. Coupled magnetomechanical equations for magnetically saturated insulators. Journal of Mathematical Physics, 5(9):1298–1318, 1964.

[203] H.F. Tiersten. Variational principle for saturated magnetoelastic insulators. Journal of Mathematical Physics, 6(5):779–787, 1965.

[204] W.F. Brown. Magnetoelastic Interactions. Springer, New York, 1966.

[205] G.A. Maugin and A.C. Eringen. Deformable magnetically saturated media. I. Field equations. Journal of Mathematical Physics, 13(2):143–155, 1972.

[206] G.A. Maugin and A.C. Eringen. Deformable magnetically saturated media. II. Constitutive theory. Journal of Mathematical Physics, 13(9):1334–1347, 1972.

[207] Y.-H. Pao. Electromagnetic forces in deformable continua. In S. Nemat-Nasser, editor, Mechanics Today, volume 4, pages 209–305. Pergamon Press, New York, 1978.

[208] G.A. Maugin. Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam, 1988.

[209] A.C. Eringen and G.A. Maugin. Electrodynamics of Continua I: Foundations and Solid Media. Springer, New York, 1990.

[210] K. Hutter, A.A.F. van de Ven, and A. Ursescu. Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids. Springer, Berlin and Heidelberg, 2006.

[211] A. Kovetz. Electromagnetic Theory. Oxford University Press, New York, 2000.

[212] A. Dorfmann and R.W. Ogden. Nonlinear magnetoelastic deformations of elastomers. Acta Mechanica, 167(1-2):13–28, 2004.

[213] A. Dorfmann and R.W. Ogden. Nonlinear magnetoelastic deformations. The Quarterly Journal of Mechanics and Applied Mathematics, 57(4):599–622, 2004.

299 [214] A. Dorfmann and R.W. Ogden. Some problems in nonlinear magnetoelasticity. Zeitschrift f¨ur angewandte Mathematik und Physik (ZAMP), 56(4):718–745, 2005.

[215] D.J. Steigmann. Equilibrium theory for magnetic elastomers and magnetoelastic membranes. International Journal of Non-Linear Mechanics, 39(7):1193–1216, 2004.

[216] J.L. Ericksen. A modiﬁed theory of magnetic eﬀects in elastic materials. Math- ematics and Mechanics of Solids, 11(1):23–47, 2006.

[217] R. Bustamante, A. Dorfmann, and R.W. Ogden. On variational formulations in nonlinear magnetoelastostatics. Mathematics and Mechanics of Solids, 13(8):725–745, 2008.

[218] M.I. Barham, D.A. White, and D.J. Steigmann. Finite element modeling of the deformation of magnetoelastic ﬁlm. Journal of Computational Physics, 229(18):6193–6207, 2010.

[219] R. Bustamante, A. Dorfmann, and R.W. Ogden. Numerical solution of ﬁnite geometry boundary-value problems in nonlinear magnetoelasticity. International Journal of Solids and Structures, 48(6):874–883, 2011.

[220] E. Salas and R. Bustamante. Numerical solution of some boundary value problems in nonlinear magneto-elasticity. Journal of Intelligent Material Systems and Structures, 26(2):156–171, 2015.

[221] P. Saxena, M. Hossain, and P. Steinmann. A theory of ﬁnite deformation magneto-viscoelasticity. International Journal of Solids and Structures, 50(24):3886–3897, 2013.

[222] G. deBotton, L. Tevet-Deree, and E.A. Socolsky. Electroactive heterogeneous polymers: analysis and applications to laminated composites. Mechanics of Advanced Materials and Structures, 14(1):13–22, 2007.

[223] E. Galipeau and P. Ponte Casta˜neda. The eﬀect of particle shape and distribution on the macroscopic behavior of magnetoelastic composites. International Journal of Solids and Structures, 49(1):1–17, 2012.

[224] E. Galipeau and P. Ponte Castañeda. A finite-strain constitutive model for magnetorheological elastomers: Magnetic torques and fiber rotations. Journal of the Mechanics and Physics of Solids, 61(4):1065–1090, 2013.

[225] A. Javili, G. Chatzigeorgiou, and P. Steinmann. Computational homogenization in magneto-mechanics. International Journal of Solids and Structures, 50(25- 26):4197–4216, 2013.

300 [226] G. Chatzigeorgiou, A. Javili, and P. Steinmann. Uniﬁed magnetomechanical homogenization framework with application to magnetorheological elastomers. Mathematics and Mechanics of Solids, 19(2):193–211, 2014.

[227] J. van Suchtelen. Product properties: a new application of composite materials. Philips Research Reports, 27:28–37, 1972.

[228] R.A. Toupin. The elastic dielectric. Journal of Rational Mechanics and Analy- sis, 5(6):849–915, 1956.

[229] L. Liu. An energy formulation of continuum magneto-electro-elasticity with applications. Journal of the Mechanics and Physics of Solids, 63:451–480, 2014.

[230] A. Dorfmann and R.W. Ogden. Nonlinear electroelastic deformations. Journal of Elasticity, 82(2):99–127, 2006.

[231] S. Rudykh, K. Bhattacharya, and G. deBotton. Multiscale instabilities in soft heterogeneous dielectric elastomers. Proceedings of the Royal Society of London A, 470(2162):20130618, 2014.

[232] S. Santapuri, R.L. Lowe, S.E. Bechtel, and M.J. Dapino. Thermodynamic modeling of fully coupled ﬁnite-deformation thermo-electro-magneto-mechanical behavior for multifunctional applications. International Journal of Engineering Science, 72:117–139, 2013.

[233] S. Santapuri. Uniﬁed thermo-electro-magneto-mechanical framework for characterization of reversible and irreversible multiphysical processes. arXiv e-prints, 2015. http://arxiv.org/abs/1504.00646v1.

[234] S. Guo and S. Ghosh. A ﬁnite element model for coupled 3D transient electromagnetic and structural dynamics problems. Computational Mechanics, 54(2):407–424, 2014.

[235] A. Dorfmann and R.W. Ogden. Nonlinear electroelasticity. Acta Mechanica, 174(3-4):167–183, 2005.

[236] P. Penﬁeld, Jr. and H.A. Haus. Electrodynamics of Moving Media.TheMIT Press, Cambridge, 1967.

[237] R.A. Toupin. Stress tensors in elastic dielectrics. Archive for Rational Mechanics and Analysis, 5(1):440–452, 1960.

[238] R.A. Toupin. A dynamical theory of elastic dielectrics. International Journal of Engineering Science, 1(1):101–126, 1963.

301 [239] C.T. Tai. A study of electrodynamics of moving media. Proceedings of the IEEE, 52(6):685–689, 1964.

[240] B.D. Coleman and E.H. Dill. On the thermodynamics of electromagnetic ﬁelds in materials with memory. Archive for Rational Mechanics and Anal- ysis, 41(2):132–162, 1971.

[241] B.D. Coleman and E.H. Dill. Thermodynamic restrictions on the constitutive equations of electromagnetic theory. Zeitschrift f¨ur angewandte Mathematik und Physik (ZAMP), 22(4):691–702, 1971.

[242] I.-S. Liu and I. Müller. On the thermodynamics and thermostatics of fluids in electromagnetic fields. Archive for Rational Mechanics and Analysis, 46(2):149– 176, 1972.

[243] H.F. Tiersten and C.F. Tsai. On the interaction of the electromagnetic ﬁeld with heat conducting deformable insulators. Journal of Mathematical Physics, 13(3):361–378, 1972.

[244] K. Hutter and Y.-H. Pao. A dynamic theory for magnetizable elastic solids with thermal and electrical conduction. Journal of Elasticity, 4(2):89–114, 1974.

[245] Y.-H. Pao and K. Hutter. Electrodynamics for moving elastic solids and viscous ﬂuids. Proceedings of the IEEE, 63(7):1011–1021, 1975.

[246] G.A. Maugin and A.C. Eringen. On the equations of the electrodynamics of deformable bodies of ﬁnite extent. Journal de M´ecanique, 16(1):101–147, 1977.

[247] A.E. Green and P.M. Naghdi. Aspects of the second law of thermodynamics in the presence of electromagnetic eﬀects. The Quarterly Journal of Mechanics and Applied Mathematics, 37(2):179–193, 1984.

[248] A.E. Green and P.M. Naghdi. A uniﬁed procedure for construction of theories of deformable media. I. Classical continuum physics. Proceedings of the Royal Society of London A, 448(1934):335–356, 1995.

[249] H.F. Tiersten. A Development of the Equations of Electromagnetism in Material Continua. Springer, New York, 1990.

[250] J. Yang. An Introduction to the Theory of Piezoelectricity. Springer, New York, 2005.

[251] R.W. Ogden and D.J. Steigmann, editors. Mechanics and Electrodynamics of Magneto- and Electro-Elastic Materials, volume 527 of CISM Courses and Lec- tures. Springer, Wien, 2011.

302 [252] C. Rinaldi and H. Brenner. Body versus surface forces in continuum mechanics: Is the Maxwell stress tensor a physically objective Cauchy stress? Physical Review E, 65(3):036615, 2002.

[253] J.L. Ericksen. Electromagnetic eﬀects in thermoelastic materials. Mathematics and Mechanics of Solids, 7(2):165–189, 2002.

[254] J.L. Ericksen. On formulating and assessing continuum theories of electromagnetic ﬁelds in elastic materials. Journal of Elasticity, 87(2-3):95–108, 2007.

[255] J.L. Ericksen. Magnetizable and polarizable elastic materials. Mathematics and Mechanics of Solids, 13(1):38–54, 2008.

[256] R. Fosdick and H. Tang. Electrodynamics and thermomechanics of material bodies. Journal of Elasticity, 88(3):255–297, 2007.

[257] D.J. Steigmann. On the formulation of balance laws for electromagnetic continua. Mathematics and Mechanics of Solids, 14(4):390–402, 2009.

[258] G.A. Maugin. On modelling electromagnetomechanical interactions in deformable solids. International Journal of Advances in Engineering Sciences and Applied Mathematics, 1(1):25–32, 2009.

[259] L. Liu. On energy formulations of electrostatics for continuum media. Journal of the Mechanics and Physics of Solids, 61(4):968–990, 2013.

[260] K. Hutter. On thermodynamics and thermostatics of viscous thermoelastic solids in the electromagnetic ﬁelds. A Lagrangian formulation. Archive for Rational Mechanics and Analysis, 58(4):339–368, 1975.

[261] P. Chadwick. Continuum Mechanics. Wiley, New York, 1976.

[262] M.E. Gurtin. An Introduction to Continuum Mechanics. Academic Press, San Diego, 2003.

[263] H.A. Lorentz. The Theory of Electrons. Dover, New York, 2nd edition, 2004.

[264] S.R. de Groot and L.G. Suttorp. Foundations of Electrodynamics. North- Holland Publishing Company, Amsterdam, 1972.

[265] R.M. Fano, L.J. Chu, and R.B. Adler. Electromagnetic ﬁelds, energy, and forces. The MIT Press, Cambridge, 1968.

[266] R. Bustamante, A. Dorfmann, and R.W. Ogden. On electric body forces and Maxwell stresses in nonlinearly electroelastic solids. International Journal of Engineering Science, 47(11-12):1131–1141, 2009.

303 [267] S.-W. Cheong and M. Mostovoy. Multiferroics: a magnetic twist for ferroelectricity. Nature Materials, 6(1):13–20, 2007.

[268] R. Ramesh and N.A. Spaldin. Multiferroics: progress and prospects in thin ﬁlms. Nature Materials, 6(1):21–29, 2007.

[269] J. Zhai, Z. Xing, S. Dong, J. Li, and D. Viehland. Magnetoelectric laminate composites: an overview. Journal of the American Ceramic Society, 91(2):351– 358, 2008.

[270] J.-P. Rivera. A short review of the magnetoelectric eﬀect and related experimental techniques on single phase (multi-) ferroics. The European Physical Journal B, 71(3):299–313, 2009.

[271] G. Srinivasan. Magnetoelectric composites. Annual Review of Materials Re- search, 40:153–178, 2010.

[272] G. Lawes and G. Srinivasan. Introduction to magnetoelectric coupling and multiferroic ﬁlms. Journal of Physics D: Applied Physics, 44(24):243001, 2011.

[273] J. Ma, J. Hu, Z. Li, and C.-W. Nan. Recent progress in multiferroic magnetoelectric composites: from bulk to thin ﬁlms. Advanced Materials, 23(9):1062– 1087, 2011.

[274] J.P. Velev, S.S. Jaswal, and E.Y. Tsymbal. Multi-ferroic and magnetoelectric materials and interfaces. Philosophical Transactions of the Royal Society A, 369(1948):3069–3097, 2011.

[275] C.A.F. Vaz. Electric ﬁeld control of magnetism in multiferroic heterostructures. Journal of Physics: Condensed Matter, 24(33):333201, 2012.

[276] S. Dong, J. Zhai, N. Wang, F. Bai, J. Li, D. Viehland, and T.A. Lograsso. Fe– Ga/Pb(Mg1/3Nb2/3)O3–PbTiO3 magnetoelectric laminate composites. Applied Physics Letters, 87(22):222504, 2005.

[277] H. Greve, E. Woltermann, H.-J. Quenzer, B. Wagner, and E. Quandt. Giant magnetoelectric coeﬃcients in (Fe90Co10)78Si12B10–AlN thin ﬁlm composites. Applied Physics Letters, 96(18):182501, 2010.

[278] Z. Fang, S.G. Lu, F. Li, S. Datta, Q.M. Zhang, and M. El Tahchi. Enhancing the magnetoelectric response of Metglas/polyvinylidene fluoride laminates by exploiting the flux concentration effect. Applied Physics Letters, 95(11):112903, 2009.

304 [279] C.-W. Nan, N. Cai, Z. Shi, J. Zhai, G. Liu, and Y. Lin. Large magnetoelectric response in multiferroic polymer-based composites. Physical Review B, 71(1):014102, 2005.

[280] G. Harsh´e, J.P. Dougherty, and R.E. Newnham. Theoretical modelling of multilayer magnetoelectric composites. International Journal of Applied Electro- magnetics in Materials, 4(2):145–159, 1993.

[281] G. Harsh´e, J.P. Dougherty, and R.E. Newnham. Theoretical modelling of 3-0/0- 3 magnetoelectric composites. International Journal of Applied Electromagnet- ics in Materials, 4(2):161–171, 1993.

[282] M. Avellaneda and G. Harsh´e. Magnetoelectric eﬀect in piezoelectric/magnetostrictive multilayer (2-2) composites. Journal of Intelligent Ma- terial Systems and Structures, 5(4):501–513, 1994.

[283] C.-W. Nan. Magnetoelectric eﬀect in composites of piezoelectric and piezomagnetic phases. Physical Review B, 50(9):6082–6088, 1994.

[284] Y. Benveniste. Magnetoelectric eﬀect in ﬁbrous composites with piezoelectric and piezomagnetic phases. Physical Review B, 51(22):16424–16427, 1995.

[285] J.H. Huang and W.-S. Kuo. The analysis of piezoelectric/piezomagnetic composite materials containing ellipsoidal inclusions. Journal of Applied Physics, 81(3):1378–1386, 1997.

[286] J.Y. Li. Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. International Journal of Engineering Science, 38(18):1993–2011, 2000.

[287] J. Aboudi. Micromechanical analysis of fully coupled electro-magneto-thermoelastic multiphase composites. Smart Materials and Structures, 10(5):867–877, 2001.

[288] E. Pan. Exact solution for simply supported and multilayered magneto-electroelastic plates. ASME Journal of Applied Mechanics, 68(4):608–618, 2001.

[289] G.R. Buchanan. Layered versus multiphase magneto-electro-elastic composites. Composites Part B: Engineering, 35(5):413–420, 2004.

[290] J. Lee, J.G. Boyd IV, and D.C. Lagoudas. Eﬀective properties of three-phase electro-magneto-elastic composites. International Journal of Engineering Sci- ence, 43(10):790–825, 2005.

305 [291] A.K. Soh and J.X. Liu. On the constitutive equations of magnetoelectroelastic solids. Journal of Intelligent Material Systems and Structures, 16(7-8):597–602, 2005.

[292] L.D. P´erez-Fern´andez, J. Bravo-Castillero, R. Rodr´ıguez-Ramos, and F.J. Sabina. On the constitutive relations and energy potentials of linear thermo- magneto-electro-elasticity. Mechanics Research Communications, 36(3):343– 350, 2009.

[293] B.D. Coleman and W. Noll. The thermodynamics of elastic materials with heat conduction and viscosity. Archive for Rational Mechanics and Analysis, 13(1):167–178, 1963.

[294] L. Anand. A constitutive model for compressible elastomeric solids. Computa- tional Mechanics, 18(5):339–355, 1996.

[295] M.C. Boyce and E.M. Arruda. Constitutive models of rubber elasticity: a review. Rubber Chemistry and Technology, 73(3):504–523, 2000.

[296] J.E. Bischoﬀ, E.M. Arruda, and K. Grosh. A new constitutive model for the compressibility of elastomers at ﬁnite deformations. Rubber Chemistry and Technology, 74(4):541–559, 2001.

[297] R.W. Ogden. Mechanics of rubberlike solids. In W. Gutkowski and T.A. Kowalewski, editors, Mechanics of the 21st Century, pages 263–274. Springer Netherlands, 2005.

[298] O. Lopez-Pamies. Elastic dielectric composites: Theory and application to particle-ﬁlled ideal dielectrics. Journal of the Mechanics and Physics of Solids, 64:61–82, 2014.

[299] A.J.M. Spencer. Theory of invariants. In A.C. Eringen, editor, Continuum Physics, volume 1, pages 239–353. Academic Press, New York, 1971.

[300] Q.-S. Zheng. Theory of representations for tensor functions – A uniﬁed invariant approach to constitutive equations. ASME Applied Mechanics Reviews, 47(11):545–587, 1994.

[301] R. Bustamante. Transversely isotropic non-linear electro-active elastomers. Acta Mechanica, 206(3-4):237–259, 2009.

[302] R. Bustamante and J. Merodio. Constitutive structure in coupled non-linear electro-elasticity: Invariant descriptions and constitutive restrictions. Interna- tional Journal of Non-Linear Mechanics, 46(10):1315–1323, 2011.

306 [303] S.J. Farlow. Partial Diﬀerential Equations for Scientists and Engineers.Dover, New York, 1993.

307