1 Hooke's Law, Stiffness, and Compliance
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Arxiv:1910.01953V1 [Cond-Mat.Soft] 4 Oct 2019 Tion of the Load
Geometric charges and nonlinear elasticity of soft metamaterials Yohai Bar-Sinai,1 Gabriele Librandi,1 Katia Bertoldi,1 and Michael Moshe2, ∗ 1School of Engineering and Applied Sciences, Harvard University, Cambridge MA 02138 2Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel 91904 Problems of flexible mechanical metamaterials, and highly deformable porous solids in general, are rich and complex due to nonlinear mechanics and nontrivial geometrical effects. While numeric approaches are successful, analytic tools and conceptual frameworks are largely lacking. Using an analogy with electrostatics, and building on recent developments in a nonlinear geometric formu- lation of elasticity, we develop a formalism that maps the elastic problem into that of nonlinear interaction of elastic charges. This approach offers an intuitive conceptual framework, qualitatively explaining the linear response, the onset of mechanical instability and aspects of the post-instability state. Apart from intuition, the formalism also quantitatively reproduces full numeric simulations of several prototypical structures. Possible applications of the tools developed in this work for the study of ordered and disordered porous mechanical metamaterials are discussed. I. INTRODUCTION tic metamaterials out of an underlying nonlinear theory of elasticity. The hallmark of condensed matter physics, as de- A theoretical analysis of the elastic problem requires scribed by P.W. Anderson in his paper \More is Dif- solving the nonlinear equations of elasticity while satis- ferent" [1], is the emergence of collective phenomena out fying the multiple free boundary conditions on the holes of well understood simple interactions between material edges - a seemingly hopeless task from an analytic per- elements. Within the ever increasing list of such systems, spective. -
Phys 140A Lecture 8–Elastic Strain, Compliance, and Stiffness
Lecture 8 Elastic strains, compliance, and stiffness Review for exam Stress: force applied to a unit area Strain: deformation resulting from stress Purpose of today’s derivations: Generalize Hooke’s law to a 3D body that may be subjected to any arbitrary force. We imagine 3 orthogonal vectors 풙̂, 풚̂, 풛̂ embedded in a solid before we have deformed it. After we have deformed the solid, these vectors might be of different length and they might be pointing in different directions. We describe these deformed vectors 풙′, 풚′, 풛′ in the following way: ′ 풙 = (1 + 휖푥푥)풙̂ + 휖푥푦풚̂ + 휖푥푧풛̂ ′ 풚 = 휖푦푥풙̂ + (1 + 휖푦푦)풚̂ + 휖푦푧풛̂ ′ 풛 = 휖푧푥풙̂ + 휖푧푦풚̂ + (1 + 휖푧푧)풛̂ The components 휖훼훽 define the deformation. They are dimensionless and have values much smaller than 1 in most instances in solid state physics. How one ‘reads’ these vectors is (for example) 휖푥푥force along x, deformation along x; 휖푥푦shear xy-plane along x or y direction (depending which unit vector it is next to) The new axes have new lengths given by (for example): ′ ′ 2 2 2 풙 ∙ 풙 = 1 + 2휖푥푥 + 휖푥푥 + 휖푥푦 + 휖푥푧 Usually we are dealing with tiny deformations so 2nd order terms are often dropped. Deformation also changes the volume of a solid. Considering our unit cube, originally it had a volume of 1. After distortion, it has volume 1 + 휖푥푥 휖푥푦 휖푥푧 ′ ′ ′ ′ 푉 = 풙 ∙ 풚 × 풛 = | 휖푦푥 1 + 휖푦푦 휖푦푧 | ≈ 1 + 휖푥푥 + 휖푦푦 + 휖푧푧 휖푧푥 휖푧푦 1 + 휖푧푧 Dilation (훿) is given by: 푉 − 푉′ 훿 ≡ ≅ 휖 + 휖 + 휖 푉 푥푥 푦푦 푧푧 (2nd order terms have been dropped because we are working in a regime of small distortion which is almost always the appropriate one for solid state physics. -
19.1 Attitude Determination and Control Systems Scott R. Starin
19.1 Attitude Determination and Control Systems Scott R. Starin, NASA Goddard Space Flight Center John Eterno, Southwest Research Institute In the year 1900, Galveston, Texas, was a bustling direct hit as Ike came ashore. Almost 200 people in the community of approximately 40,000 people. The Caribbean and the United States lost their lives; a former capital of the Republic of Texas remained a tragedy to be sure, but far less deadly than the 1900 trade center for the state and was one of the largest storm. This time, people were prepared, having cotton ports in the United States. On September 8 of received excellent warning from the GOES satellite that year, however, a powerful hurricane struck network. The Geostationary Operational Environmental Galveston island, tearing the Weather Bureau wind Satellites have been a continuous monitor of the gauge away as the winds exceeded 100 mph and world’s weather since 1975, and they have since been bringing a storm surge that flooded the entire city. The joined by other Earth-observing satellites. This weather worst natural disaster in United States’ history—even surveillance to which so many now owe their lives is today—the hurricane caused the deaths of between possible in part because of the ability to point 6000 and 8000 people. Critical in the events that led to accurately and steadily at the Earth below. The such a terrible loss of life was the lack of precise importance of accurately pointing spacecraft to our knowledge of the strength of the storm before it hit. daily lives is pervasive, yet somehow escapes the notice of most people. -
Crack Tip Elements and the J Integral
EN234: Computational methods in Structural and Solid Mechanics Homework 3: Crack tip elements and the J-integral Due Wed Oct 7, 2015 School of Engineering Brown University The purpose of this homework is to help understand how to handle element interpolation functions and integration schemes in more detail, as well as to explore some applications of FEA to fracture mechanics. In this homework you will solve a simple linear elastic fracture mechanics problem. You might find it helpful to review some of the basic ideas and terminology associated with linear elastic fracture mechanics here (in particular, recall the definitions of stress intensity factor and the nature of crack-tip fields in elastic solids). Also check the relations between energy release rate and stress intensities, and the background on the J integral here. 1. One of the challenges in using finite elements to solve a problem with cracks is that the stress field at a crack tip is singular. Standard finite element interpolation functions are designed so that stresses remain finite a everywhere in the element. Various types of special b c ‘crack tip’ elements have been designed that 3L/4 incorporate the singularity. One way to produce a L/4 singularity (the method used in ABAQUS) is to mesh L the region just near the crack tip with 8 noded elements, with a special arrangement of nodal points: (i) Three of the nodes (nodes 1,4 and 8 in the figure) are connected together, and (ii) the mid-side nodes 2 and 7 are moved to the quarter-point location on the element side. -
Beam Element Stiffness Matrices
Beam Element Stiffness Matrices CEE 421L. Matrix Structural Analysis Department of Civil and Environmental Engineering Duke University Henri P. Gavin Fall, 2014 Truss elements carry only axial forces. Beam elements carry shear forces and bending moments. Frame elements carry shear forces, bending moments, and axial forces. This document presents the development of beam element stiffness matrices in local coordinates. 1 A simply supported beam carrying end-moments Consider a simply supported beam resisting moments M1 and M2 applied at its ends. The flexibility relates the end rotations {θ1, θ2} to the end moments {M1,M2}: θ1 F11 F12 M1 = . θ2 F21 F22 M2 The flexibility coefficients, Fij, may be obtained from Castigliano’s 2nd Theo- ∗ rem, θi = ∂U (Mi)/∂Mi. First Column Second Column 2 CEE 421L. Matrix Structural Analysis – Duke University – Fall 2014 – H.P. Gavin The applied moments M1 and M2 are in equilibrium with the reactions forces V1 and V2; V1 = (M1 + M2)/L and V2 = −(M1 + M2)/L M + M x ! x V (x) = 1 2 M(x) = M − 1 + M L 1 L 2 L The total potential energy of a beam with these forces and moments is: 1Z L M 2 1Z L V 2 U = dx + dx 2 0 EI 2 0 G(A/α) By Castigliano’s Theorem, ∂U θ1 = ∂M1 ∂M(x) ∂V (x) Z L M(x) Z L V (x) = ∂M1 dx + ∂M1 dx 0 EI 0 G(A/α) x 2 x x Z L Z L Z L Z L L − 1 dx αdx L L − 1 dx αdx = + M1 + + M2 0 EI 0 GAL2 0 EI 0 GAL2 and ∂U θ2 = ∂M2 ∂M(x) ∂V (x) Z L M(x) Z L V (x) = ∂M2 dx + ∂M2 dx 0 EI 0 G(A/α) x x x 2 Z L Z L Z L Z L L L − 1 dx αdx L dx αdx = + M1 + + M2 0 EI 0 GAL2 0 EI 0 GAL2 -
Linear Elasticity
10 Linear elasticity When you bend a stick the reaction grows noticeably stronger the further you go — until it perhaps breaks with a snap. If you release the bending force before it breaks, the stick straightens out again and you can bend it again and again without it changing its reaction or its shape. That is elasticity. Robert Hooke (1635–1703). In elementary mechanics the elasticity of a spring is expressed by Hooke’s law English physicist. Worked on elasticity, built tele- which says that the force necessary to stretch or compress a spring is propor- scopes, and the discovered tional to how much it is stretched or compressed. In continuous elastic materials diffraction of light. The Hooke’s law implies that stress is proportional to strain. Some materials that we famous law which bears his name is from 1660. usually think of as highly elastic, for example rubber, do not obey Hooke’s law He stated already in 1678 except under very small deformation. When stresses grow large, most materials the inverse square law for gravity, over which he deform more than predicted by Hooke’s law. The proper treatment of non-linear got involved in a bitter elasticity goes far beyond the simple linear elasticity which we shall discuss in controversy with Newton. this book. The elastic properties of continuous materials are determined by the under- lying molecular structure, but the relation between material properties and the molecular structure and arrangement in solids is complicated, to say the least. Luckily, there are broad classes of materials that may be described by a few Thomas Young (1773–1829). -
Linear Elastostatics
6.161.8 LINEAR ELASTOSTATICS J.R.Barber Department of Mechanical Engineering, University of Michigan, USA Keywords Linear elasticity, Hooke’s law, stress functions, uniqueness, existence, variational methods, boundary- value problems, singularities, dislocations, asymptotic fields, anisotropic materials. Contents 1. Introduction 1.1. Notation for position, displacement and strain 1.2. Rigid-body displacement 1.3. Strain, rotation and dilatation 1.4. Compatibility of strain 2. Traction and stress 2.1. Equilibrium of stresses 3. Transformation of coordinates 4. Hooke’s law 4.1. Equilibrium equations in terms of displacements 5. Loading and boundary conditions 5.1. Saint-Venant’s principle 5.1.1. Weak boundary conditions 5.2. Body force 5.3. Thermal expansion, transformation strains and initial stress 6. Strain energy and variational methods 6.1. Potential energy of the external forces 6.2. Theorem of minimum total potential energy 6.2.1. Rayleigh-Ritz approximations and the finite element method 6.3. Castigliano’s second theorem 6.4. Betti’s reciprocal theorem 6.4.1. Applications of Betti’s theorem 6.5. Uniqueness and existence of solution 6.5.1. Singularities 7. Two-dimensional problems 7.1. Plane stress 7.2. Airy stress function 7.2.1. Airy function in polar coordinates 7.3. Complex variable formulation 7.3.1. Boundary tractions 7.3.2. Laurant series and conformal mapping 7.4. Antiplane problems 8. Solution of boundary-value problems 1 8.1. The corrective problem 8.2. The Saint-Venant problem 9. The prismatic bar under shear and torsion 9.1. Torsion 9.1.1. Multiply-connected bodies 9.2. -
Introduction to Stiffness Analysis Stiffness Analysis Procedure
Introduction to Stiffness Analysis displacements. The number of unknowns in the stiffness method of The stiffness method of analysis is analysis is known as the degree of the basis of all commercial kinematic indeterminacy, which structural analysis programs. refers to the number of node/joint Focus of this chapter will be displacements that are unknown development of stiffness equations and are needed to describe the that only take into account bending displaced shape of the structure. deformations, i.e., ignore axial One major advantage of the member, a.k.a. slope-deflection stiffness method of analysis is that method. the kinematic degrees of freedom In the stiffness method of analysis, are well-defined. we write equilibrium equations in 1 2 terms of unknown joint (node) Definitions and Terminology Stiffness Analysis Procedure Positive Sign Convention: Counterclockwise moments and The steps to be followed in rotations along with transverse forces and displacements in the performing a stiffness analysis can positive y-axis direction. be summarized as: 1. Determine the needed displace- Fixed-End Forces: Forces at the “fixed” supports of the kinema- ment unknowns at the nodes/ tically determinate structure. joints and label them d1, d2, …, d in sequence where n = the Member-End Forces: Calculated n forces at the end of each element/ number of displacement member resulting from the unknowns or degrees of applied loading and deformation freedom. of the structure. 3 4 1 2. Modify the structure such that it fixed-end forces are vectorially is kinematically determinate or added at the nodes/joints to restrained, i.e., the identified produce the equivalent fixed-end displacements in step 1 all structure forces, which are equal zero. -
20. Rheology & Linear Elasticity
20. Rheology & Linear Elasticity I Main Topics A Rheology: Macroscopic deformation behavior B Linear elasticity for homogeneous isotropic materials 10/29/18 GG303 1 20. Rheology & Linear Elasticity Viscous (fluid) Behavior http://manoa.hawaii.edu/graduate/content/slide-lava 10/29/18 GG303 2 20. Rheology & Linear Elasticity Ductile (plastic) Behavior http://www.hilo.hawaii.edu/~csav/gallery/scientists/LavaHammerL.jpg http://hvo.wr.usgs.gov/kilauea/update/images.html 10/29/18 GG303 3 http://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg 20. Rheology & Linear Elasticity Elastic Behavior https://thegeosphere.pbworks.com/w/page/24663884/Sumatra http://www.earth.ox.ac.uk/__Data/assets/image/0006/3021/seismic_hammer.jpg 10/29/18 GG303 4 20. Rheology & Linear Elasticity Brittle Behavior (fracture) 10/29/18 GG303 5 http://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, http://www.fordogtrainers.com rubber ball 10/29/18 GG303 6 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, rubber ball 10/29/18 GG303 7 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior B Viscosity 1 Deformation is irreversible when load is removed 2 Stress (σ) is related to strain rate (ε ! ) 3 Deformation is time dependent if load is constant 4 Examples: Lava flows, corn syrup http://wholefoodrecipes.net 10/29/18 GG303 8 20. -
Design of a Meta-Material with Targeted Nonlinear Deformation Response Zachary Satterfield Clemson University, [email protected]
Clemson University TigerPrints All Theses Theses 12-2015 Design of a Meta-Material with Targeted Nonlinear Deformation Response Zachary Satterfield Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_theses Part of the Mechanical Engineering Commons Recommended Citation Satterfield, Zachary, "Design of a Meta-Material with Targeted Nonlinear Deformation Response" (2015). All Theses. 2245. https://tigerprints.clemson.edu/all_theses/2245 This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. DESIGN OF A META-MATERIAL WITH TARGETED NONLINEAR DEFORMATION RESPONSE A Thesis Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Master of Science Mechanical Engineering by Zachary Tyler Satterfield December 2015 Accepted by: Dr. Georges Fadel, Committee Chair Dr. Nicole Coutris, Committee Member Dr. Gang Li, Committee Member ABSTRACT The M1 Abrams tank contains track pads consist of a high density rubber. This rubber fails prematurely due to heat buildup caused by the hysteretic nature of elastomers. It is therefore desired to replace this elastomer by a meta-material that has equivalent nonlinear deformation characteristics without this primary failure mode. A meta-material is an artificial material in the form of a periodic structure that exhibits behavior that differs from its constitutive material. After a thorough literature review, topology optimization was found as the only method used to design meta-materials. Further investigation determined topology optimization as an infeasible method to design meta-materials with the targeted nonlinear deformation characteristics. -
Geometric Charges and Nonlinear Elasticity of Two-Dimensional Elastic Metamaterials
Geometric charges and nonlinear elasticity of two-dimensional elastic metamaterials Yohai Bar-Sinai ( )a , Gabriele Librandia , Katia Bertoldia, and Michael Moshe ( )b,1 aSchool of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; and bRacah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel 91904 Edited by John A. Rogers, Northwestern University, Evanston, IL, and approved March 23, 2020 (received for review November 17, 2019) Problems of flexible mechanical metamaterials, and highly A theoretical analysis of the elastic problem requires solv- deformable porous solids in general, are rich and complex due ing the nonlinear equations of elasticity while satisfying the to their nonlinear mechanics and the presence of nontrivial geo- multiple free boundary conditions on the holes’ edges—a seem- metrical effects. While numeric approaches are successful, ana- ingly hopeless task from an analytic perspective. However, direct lytic tools and conceptual frameworks are largely lacking. Using solutions of the fully nonlinear elastic equations are accessi- an analogy with electrostatics, and building on recent devel- ble using finite-element models, which accurately reproduce the opments in a nonlinear geometric formulation of elasticity, we deformation fields, the critical strain, and the effective elastic develop a formalism that maps the two-dimensional (2D) elas- coefficients, etc. (6). The success of finite-element (FE) simula- tic problem into that of nonlinear interaction of elastic charges. tions in predicting the mechanics of perforated elastic materials This approach offers an intuitive conceptual framework, qual- confirms that nonlinear elasticity theory is a valid description, itatively explaining the linear response, the onset of mechan- but emphasizes the lack of insightful analytical solutions to ical instability, and aspects of the postinstability state. -
On the Path-Dependence of the J-Integral Near a Stationary Crack in an Elastic-Plastic Material
On the Path-Dependence of the J-integral Near a Stationary Crack in an Elastic-Plastic Material Dorinamaria Carka and Chad M. Landis∗ The University of Texas at Austin, Department of Aerospace Engineering and Engineering Mechanics, 210 East 24th Street, C0600, Austin, TX 78712-0235 Abstract The path-dependence of the J-integral is investigated numerically, via the finite element method, for a range of loadings, Poisson's ratios, and hardening exponents within the context of J2-flow plasticity. Small-scale yielding assumptions are employed using Dirichlet-to-Neumann map boundary conditions on a circular boundary that encloses the plastic zone. This construct allows for a dense finite element mesh within the plastic zone and accurate far-field boundary conditions. Details of the crack tip field that have been computed previously by others, including the existence of an elastic sector in Mode I loading, are confirmed. The somewhat unexpected result is that J for a contour approaching zero radius around the crack tip is approximately 18% lower than the far-field value for Mode I loading for Poisson’s ratios characteristic of metals. In contrast, practically no path-dependence is found for Mode II. The applications of T or S stresses, whether applied proportionally with the K-field or prior to K, have only a modest effect on the path-dependence. Keywords: elasto-plastic fracture mechanics, small scale yielding, path-dependence of the J-integral, finite element methods 1. Introduction The J-integral as introduced by Eshelby [1,2] and Rice [3] is perhaps the most useful quantity for the analysis of the mechanical fields near crack tips in both linear elastic and non-linear elastic materials.