A Brief Review of Elasticity and Viscoelasticity
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ME 285A: Foundations of the Theory of Continuous Media
University of California, Berkeley Department of Mechanical Engineering ME 285A: Foundations of the Theory of Continuous Media Syllabus 1. Introduction. The nonlinear field theories. Mathematical preliminaries. Tensors. Fréchet derivative and differential. 2. Kinematics of Deformable Continua. Bodies. Deformations. Polar decomposition theorem. Derivatives of stretch and rotation tensors. Velocity and vorticity fields. Pull-back and push- forward operations. Lie derivative. Material transport of fields. Internal constraints. Relative deformation gradient. Rivlin-Ericksen tensors. 3. Balance Laws. Conservation of mass. Balance of linear and angular momenta for continua. Stress tensors. Balance of energy. A remark on the second law of thermodynamics. 4. General Constitutive Relations (Purely Mechanical Theory). Relationship between stress and history of deformation. Invariance requirements. Simple materials. Symmetry considerations. Cauchy-elastic and Green-elastic (or hyperelastic) materials. Internally constrained elastic materials. Viscous fluids. Materials with memory. Viscoelastic fluids and solids. Reiner-Rivlin fluids. Rivlin-Ericksen fluids. Materials with fading memory. Linear viscoelasticity. Pseudo-Rigid materials. 5. Thermoelastic Materials. Ideal gas. Entropy and Helmholtz free energy functions. Clausius- Duhem inequality. Thermoelasticity. Approach of Coleman and Noll. Approach of Rivlin: construction of entropy function. Internally constrained thermoelatic materials. Linear thermoelasticity. 6. Thermoelastic Materials with Viscosity. Construction of the entropy function for thermoelastic materials with viscosity, and subject to internal constraints. 7. Thermodynamics of Materials with Memory. Materials with fading memory. Instantaneous elasticity. Open questions. 8. Selected Topics. Mixture theory, nonlocal continua. ______________________________________________________________________________ Grading scheme: Homework (~40%), Midterm (~20%), Final Project (~40%). . -
Cohesive Crack with Rate-Dependent Opening and Viscoelasticity: I
International Journal of Fracture 86: 247±265, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Cohesive crack with rate-dependent opening and viscoelasticity: I. mathematical model and scaling ZDENEKÏ P. BAZANTÏ and YUAN-NENG LI Department of Civil Engineering, Northwestern University, Evanston, Illinois 60208, USA Received 14 October 1996; accepted in revised form 18 August 1997 Abstract. The time dependence of fracture has two sources: (1) the viscoelasticity of material behavior in the bulk of the structure, and (2) the rate process of the breakage of bonds in the fracture process zone which causes the softening law for the crack opening to be rate-dependent. The objective of this study is to clarify the differences between these two in¯uences and their role in the size effect on the nominal strength of stucture. Previously developed theories of time-dependent cohesive crack growth in a viscoelastic material with or without aging are extended to a general compliance formulation of the cohesive crack model applicable to structures such as concrete structures, in which the fracture process zone (cohesive zone) is large, i.e., cannot be neglected in comparison to the structure dimensions. To deal with a large process zone interacting with the structure boundaries, a boundary integral formulation of the cohesive crack model in terms of the compliance functions for loads applied anywhere on the crack surfaces is introduced. Since an unopened cohesive crack (crack of zero width) transmits stresses and is equivalent to no crack at all, it is assumed that at the outset there exists such a crack, extending along the entire future crack path (which must be known). -
An Investigation Into the Effects of Viscoelasticity on Cavitation Bubble Dynamics with Applications to Biomedicine
An Investigation into the Effects of Viscoelasticity on Cavitation Bubble Dynamics with Applications to Biomedicine Michael Jason Walters School of Mathematics Cardiff University A thesis submitted for the degree of Doctor of Philosophy 9th April 2015 Summary In this thesis, the dynamics of microbubbles in viscoelastic fluids are investigated nu- merically. By neglecting the bulk viscosity of the fluid, the viscoelastic effects can be introduced through a boundary condition at the bubble surface thus alleviating the need to calculate stresses within the fluid. Assuming the surrounding fluid is incompressible and irrotational, the Rayleigh-Plesset equation is solved to give the motion of a spherically symmetric bubble. For a freely oscillating spherical bubble, the fluid viscosity is shown to dampen oscillations for both a linear Jeffreys and an Oldroyd-B fluid. This model is also modified to consider a spherical encapsulated microbubble (EMB). The fluid rheology affects an EMB in a similar manner to a cavitation bubble, albeit on a smaller scale. To model a cavity near a rigid wall, a new, non-singular formulation of the boundary element method is presented. The non-singular formulation is shown to be significantly more stable than the standard formulation. It is found that the fluid rheology often inhibits the formation of a liquid jet but that the dynamics are governed by a compe- tition between viscous, elastic and inertial forces as well as surface tension. Interesting behaviour such as cusping is observed in some cases. The non-singular boundary element method is also extended to model the bubble tran- sitioning to a toroidal form. -
Analysis of Viscoelastic Flexible Pavements
Analysis of Viscoelastic Flexible Pavements KARL S. PISTER and CARL L. MONISMITH, Associate Professor and Assistant Professor of Civil Engineering, University of California, Berkeley To develop a better understandii^ of flexible pavement behavior it is believed that components of the pavement structure should be considered to be viscoelastic rather than elastic materials. In this paper, a step in this direction is taken by considering the asphalt concrete surface course as a viscoelastic plate on an elastic foundation. Assumptions underlying existing stress and de• formation analyses of pavements are examined. Rep• resentations of the flexible pavement structure, im• pressed wheel loads, and mechanical properties of as• phalt concrete and base materials are discussed. Re• cent results pointing toward the viscoelastic behavior of asphaltic mixtures are presented, including effects of strain rate and temperature. IJsii^ a simplified viscoelastic model for the asphalt concrete surface course, solutions for typical loading problems are given. • THE THEORY of viscoelasticity is concerned with the behavior of materials which exhibit time-dependent stress-strain characteristics. The principles of viscoelasticity have been successfully used to explain the mechanical behavior of high polymers and much basic work U) lias been developed in this area. In recent years the theory of viscoelasticity has been employed to explain the mechanical behavior of asphalts (2, 3, 4) and to a very limited degree the behavior of asphaltic mixtures (5, 6, 7) and soils (8). Because these materials have time-dependent stress-strain characteristics and because they comprise the flexible pavement section, it seems reasonable to analyze the flexible pavement structure using viscoelastic principles. -
Engineering Viscoelasticity
ENGINEERING VISCOELASTICITY David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 October 24, 2001 1 Introduction This document is intended to outline an important aspect of the mechanical response of polymers and polymer-matrix composites: the field of linear viscoelasticity. The topics included here are aimed at providing an instructional introduction to this large and elegant subject, and should not be taken as a thorough or comprehensive treatment. The references appearing either as footnotes to the text or listed separately at the end of the notes should be consulted for more thorough coverage. Viscoelastic response is often used as a probe in polymer science, since it is sensitive to the material’s chemistry and microstructure. The concepts and techniques presented here are important for this purpose, but the principal objective of this document is to demonstrate how linear viscoelasticity can be incorporated into the general theory of mechanics of materials, so that structures containing viscoelastic components can be designed and analyzed. While not all polymers are viscoelastic to any important practical extent, and even fewer are linearly viscoelastic1, this theory provides a usable engineering approximation for many applications in polymer and composites engineering. Even in instances requiring more elaborate treatments, the linear viscoelastic theory is a useful starting point. 2 Molecular Mechanisms When subjected to an applied stress, polymers may deform by either or both of two fundamen- tally different atomistic mechanisms. The lengths and angles of the chemical bonds connecting the atoms may distort, moving the atoms to new positions of greater internal energy. -
Oversimplified Viscoelasticity
Oversimplified Viscoelasticity THE MAXWELL MODEL At time t = 0, suddenly deform to constant displacement Xo. The force F is the same in the spring and the dashpot. F = KeXe = Kv(dXv/dt) (1-20) Xe is the displacement of the spring Xv is the displacement of the dashpot Ke is the linear spring constant (ratio of force and displacement, units N/m) Kv is the linear dashpot constant (ratio of force and velocity, units Ns/m) The total displacement Xo is the sum of the two displacements (Xo is independent of time) Xo = Xe + Xv (1-21) 1 Oversimplified Viscoelasticity THE MAXWELL MODEL (p. 2) Thus: Ke(Xo − Xv) = Kv(dXv/dt) with B. C. Xv = 0 at t = 0 (1-22) (Ke/Kv)dt = dXv/(Xo − Xv) Integrate: (Ke/Kv)t = − ln(Xo − Xv) + C Apply B. C.: Xv = 0 at t = 0 means C = ln(Xo) −(Ke/Kv)t = ln[(Xo − Xv)/Xo] (Xo − Xv)/Xo = exp(−Ket/Kv) Thus: F (t) = KeXo exp(−Ket/Kv) (1-23) The force from our constant stretch experiment decays exponentially with time in the Maxwell Model. The relaxation time is λ ≡ Kv/Ke (units s) The force drops to 1/e of its initial value at the relaxation time λ. Initially the force is F (0) = KeXo, the force in the spring, but eventually the force decays to zero F (∞) = 0. 2 Oversimplified Viscoelasticity THE MAXWELL MODEL (p. 3) Constant Area A means stress σ(t) = F (t)/A σ(0) ≡ σ0 = KeXo/A Maxwell Model Stress Relaxation: σ(t) = σ0 exp(−t/λ) Figure 1: Stress Relaxation of a Maxwell Element 3 Oversimplified Viscoelasticity THE MAXWELL MODEL (p. -
Polymer Rheology: Principles, Techniques and Applications
Polymer Rheology: Principles, Techniques and Applications G. C. Berry Carnegie Mellon University Contents Example Problem............................................................................................. i 1. Introduction ..................................................................................................... 1 2. Principles of Rheometry .................................................................................. 3 3. Linear viscoelastic phenomenology.................................................................. 7 4. Approximations used in linear viscoelasticity................................................... 19 5. The viscosity of dilute of polymer solutions and colloidal dispersions.............. 24 6. Linear viscoelastic behavior of concentrated and undiluted polymeric fluids.... 29 7. Nonlinear viscoelastic behavior of concentrated and undiluted polymeric 35 fluids................................................................................................................ 8. Strain-induced birefringence for concentrated and undiluted polymeric fluids.. 43 9. Linear and nonlinear viscoelastic behavior of colloidal dispersions .................. 55 10. References ....................................................................................................... 59 Table 1 Linear Viscoelastic Function..................................................................... 68 Table 2 Functions and Parameters Used................................................................. 69 Table 3 Relations -
Thermomechanics of Solid Materials with Application to the Gurson-Tvergaard Material Model
VTT PUBLICATIONS 312 F19800095 Thermomechanics of solid materials with application to the Gurson-Tvergaard material model Kari Santaoja VlT Manufacturing Technology . TECHNICAL RESEARCH CENTRE OF FINLAND ESP00 1997 312 Kari Santaoja Thermomechanics of solid materials with application to the Gurson-Tvergaard material model TECHNICAL RESEARCH CENTRE OF FINLAND ESP00 1997 ISBN 951-38-5060-9 (soft back ed.) ISSN 123.54621 (soft back ed.) ISBN 95 1-38-5061-7 (URL: http://www.inf.vtt.fi/pdf/) ISSN 14554849 (URL: http://www.inf.vtt.fi/pdf/) Copyright 0 Valtion teknillinen tutkimuskeskus (VTT) 1997 JULKAISIJA - UTGIVARE - PUBLISHER Valtion teknillinen tutkimuskeskus (VTT), Vuorimiehentie 5, PL 2000, 02044 VTT puh. vaihde (09) 4561, faksi (09) 456 4374 Statens tekniska forskningscentral (VTT), Bergsmansvagen 5, PB 2000,02044 VTT tel. vaxel(O9) 4561, fax (09) 456 4374 Technical Research Centre of Finland (VTT), Vuorimiehentie 5, P.O.Box 2000, FIN42044 VTT, Finland phone internat. + 358 9 4561, fax + 358 9 456 4374 VIT Valmistustekniikka, Ydinvoimalaitosten materiaalitekniikka, Kemistintie 3, PL 1704,02044 VTT puh. vaihde (09) 4561, faksi (09) 456 7002 VTT Tillverkningsteknik, Material och strukturell integritet, Kemistvagen 3, PB 1’704,02044 VTT tel. vaxel(O9) 4561, fax (09) 456 7002 VTT Manufacturing Technology, Maaterials and Structural Integrity, Kemistintie 3, P.O.Box 1704, FIN42044 VTT, Finland phone internat. + 358 9 4561, fax + 358 9 456 7002 Technical editing Kerttu Tirronen VTT OFFSETPAINO. ESP00 1997 Santaoja, Kari. Thermornechanics of solid materials with application to the Gurson-Tvergaard material model. Espoo 1997. Technical Research Centre of Finland, UTPublications 312. 162 p. + app. 14 p. UDC 536.7 Keywords thermomechanical analysis, thermodynamics, plasticity, porous medium, Gurson- Tvergaard model ABSTRACT The elastic-plastic material model for porous material proposed by Gurson and Tvergaard is evaluated. -
Elasticity and Viscoelasticity
CHAPTER 2 Elasticity and Viscoelasticity CHAPTER 2.1 Introduction to Elasticity and Viscoelasticity JEAN LEMAITRE Universite! Paris 6, LMT-Cachan, 61 avenue du President! Wilson, 94235 Cachan Cedex, France For all solid materials there is a domain in stress space in which strains are reversible due to small relative movements of atoms. For many materials like metals, ceramics, concrete, wood and polymers, in a small range of strains, the hypotheses of isotropy and linearity are good enough for many engineering purposes. Then the classical Hooke’s law of elasticity applies. It can be de- rived from a quadratic form of the state potential, depending on two parameters characteristics of each material: the Young’s modulus E and the Poisson’s ratio n. 1 c * ¼ A s s ð1Þ 2r ijklðE;nÞ ij kl @c * 1 þ n n eij ¼ r ¼ sij À skkdij ð2Þ @sij E E Eandn are identified from tensile tests either in statics or dynamics. A great deal of accuracy is needed in the measurement of the longitudinal and transverse strains (de Æ10À6 in absolute value). When structural calculations are performed under the approximation of plane stress (thin sheets) or plane strain (thick sheets), it is convenient to write these conditions in the constitutive equation. Plane stress ðs33 ¼ s13 ¼ s23 ¼ 0Þ: 2 3 1 n 6 À 0 7 2 3 6 E E 72 3 6 7 e11 6 7 s11 6 7 6 1 76 7 4 e22 5 ¼ 6 0 74 s22 5 ð3Þ 6 E 7 6 7 e12 4 5 s12 1 þ n Sym E Handbook of Materials Behavior Models Copyright # 2001 by Academic Press. -
Causal Heat Conduction Contravening the Fading Memory Paradigm
entropy Article Causal Heat Conduction Contravening the Fading Memory Paradigm Luis Herrera Instituto Universitario de Física Fundamental y Matematicas, Universidad de Salamanca, 37007 Salamanca, Spain; [email protected] Received: 1 September 2019; Accepted: 26 September 2019; Published: 28 September 2019 Abstract: We propose a causal heat conduction model based on a heat kernel violating the fading memory paradigm. The resulting transport equation produces an equation for the temperature. The model is applied to the discussion of two important issues such as the thermohaline convection and the nuclear burning (in)stability. In both cases, the behaviour of the system appears to be strongly dependent on the transport equation assumed, bringing out the effects of our specific kernel on the final description of these problems. A possible relativistic version of the obtained transport equation is presented. Keywords: causal dissipative theories; fading memory paradigm; relativistic dissipative theories 1. Introduction In the study of dissipative processes, the order of magnitude of relevant time scales of the system (in particular the thermal relaxation time) play a major role, due to the fact that the interplay between this latter time scale and the time scale of observation, critically affects the obtained pattern of evolution. This applies not only for dissipative processes out of the steady–state regime but also when the observation time is of the order of (or shorter than) the characteristic time of the system under consideration. Besides, as has been stressed before [1], for time scales larger than the relaxation time, transient phenomena affect the future of the system, even for time scales much larger than the relaxation time. -
Viscoelastic Creep Crack Growth: a Review of Fracture Mechanical Analyses
Mechanics of Time-Dependent Materials 1: 241–268, 1998. 241 c 1998 Kluwer Academic Publishers. Printed in the Netherlands. Viscoelastic Creep Crack Growth: A Review of Fracture Mechanical Analyses W. BRADLEY1, W.J. CANTWELL2 and H.H. KAUSCH2 1Department of Mechanical Engineering, Texas A&M University, College Station, TX, U.S.A.; 2Materials Science Department, EPFL, DMX-LP, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland (Received 8 February 1997; accepted in revised form 2 October 1997) Abstract. The study of time dependent crack growth in polymers using a fracture mechanics approach has been reviewed. The time dependence of crack growth in polymers is seen to be the result of the viscoelastic deformation in the process zone, which causes the supply of energy to drive the crack to occur over time rather than instantaneously, as it does in metals. Additional time dependence in the crack growth process can be due to process zone behavior, where both the flow stress and the critical crack tip opening displacement may be dependent on the crack growth rate. Instability leading to slip-stick crack growth has been seen to be the consequence of a decrease in the critical energy release rate with increasing crack growth rate due to adiabatic heating causing a reduction in the process zone flow stress, a decrease in the crack tip opening displacement due to a ductile to brittle transition at higher crack growth rates, or an increase in the rate of fracture work due to more rapid viscoelastic deformation. Finally, various techniques to experimentally characterize the crack growth rate as a function of stress intensity have been critiqued. -
Models for Shape Memory Alloy Behavior: an Overview of Modeling Approaches
INTERNATIONAL JOURNAL OF STRUCTURAL CHANGES IN SOLIDS – Mechanics and Applications Volume 1, Number 1, December 2009, pp. 111-148 Models for Shape Memory Alloy Behavior: An overview of modeling approaches Ashish Khandelwal and Vidyashankar Buravalla* GM R&D, India Science Lab, Bangalore, India Abstract Shape memory alloys (SMA) have been known for over five decades and modeling of their response has attracted much attention over the last two decades. Recent increase in the range of applications of these materials has lead to an increased focus on modeling their thermomechanical response. Various approaches ranging from macroscopic phenomenology to microscopic lattice based methods are being pursued to model SMA behavior. Models ranging from simple tools as design aids to complex thermodynamic based models to understand the characteristic martensitic transformation responsible for the shape memory and superelastic response exist. In this report, following a brief introduction to the SMA behavior and the underlying martensitic transformation, an overview of models for SMAs is provided. This is aimed at providing an understanding of the state of the art and also to identify possible future directions. In this review, models are classified both on the basis of the approach and the level (scale) of continuum they address. General methodology of each type of approach is presented briefly followed by a review of the models that fall under that category. Certain liberties are taken in classifying these models to facilitate understanding and thus the present classification is by no means either unique or completely rigorous. Also, while an attempt is made to cover most of the approaches, review of all the models that exist in literature is practically impossible.