Numerical Issues in Finite Elasticity and Viscoelasticity

This chapter starts with an introduction to the notion of finite element methods in solid mechanics. After an overview of basic concepts in l-D, the multi-D problem is investigated within the context of finite elasticity. Particular attention is paid to the socalled mixed variational principles that are needed for the solution of nearly incompressible solids. Special forms for elasticity in principal stretches are also given. Details of commonly used notations for actual implementations are covered. The chapter closes with several sections covering important special cases that arise in the modeling of elastomeric solids. First, numerical issues in the solution linear and nonlinear finite deformation viscoelasticity are discussed. This includes both convolution form models and multiplicative split type models. This is followed by the application of finite element methods to the solution of inverse design problems of the type that arise in mold design. Next, the general numerical methods are applied to the case of steady state spinning which is important in the analysis of automotive tires. The last two sections deal with the special cases of damage models and energy release rate computations in elastomers.

[1]  R. Taylor,et al.  Thermomechanical analysis of viscoelastic solids , 1970 .

[2]  Paul Steinmann,et al.  Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting , 2000 .

[3]  P. Le Tallec,et al.  Numerical models of steady rolling for non‐linear viscoelastic structures in finite deformations , 1994 .

[4]  J. Z. Zhu,et al.  The finite element method , 1977 .

[5]  J. D. Eshelby.  The elastic energy-momentum tensor , 1975 .

[6]  Sanjay Govindjee,et al.  A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating Mullins' effect , 1991 .

[7]  J. D. Eshelby.  The Continuum Theory of Lattice Defects , 1956 .

[8]  J. C. Simo,et al.  Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory , 1992 .

[9]  Klaus-Jürgen Bathe,et al.  A hyperelastic‐based large strain elasto‐plastic constitutive formulation with combined isotropic‐kinematic hardening using the logarithmic stress and strain measures , 1990 .

[10]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[11]  Sanjay Govindjee,et al.  Inverse Design Methodology of a Tire , 2001 .

[12]  Mary C. Boyce,et al.  Constitutive modeling of the large strain time-dependent behavior of elastomers , 1998 .

[13]  Stefanie Reese,et al.  A Presentation and Comparison of Two Large Deformation Viscoelasticity Models , 1997 .

[14]  J. Rice.  A path-independent integral and the approximate analysis of strain , 1968 .

[15]  R. Christensen.  A Nonlinear Theory of Viscoelasticity for Application to Elastomers , 1980 .

[16]  Francis De S. Lynch.  A finite element method of viscoelastic stress analysis with application to rolling contact problems , 1969 .

[17]  C. Inglis.  Stresses in a plate due to the presence of cracks and sharp corners , 1913 .

[18]  Henri Poincaré,et al.  The foundations of science , 2022 .

[19]  Stefanie Reese,et al.  Theoretical and Numerical Aspects in the Thermo-Viscoelastic Material Behaviour of Rubber-Like Polymers , 1997 .

[20]  Sanjay Govindjee,et al.  Transition from micro-mechanics to computationally efficient phenomenology: Carbon black filled rubbers incorporating mullins' effect , 1992 .

[21]  P. Chadwick,et al.  Applications of an energy-momentum tensor in non-linear elastostatics , 1975 .

[22]  I. Fried.  Finite element analysis of incompressible material by residual energy balancing , 1974 .

[23]  J. Rice,et al.  Conservation Laws and Energy-Release Rates , 1973 .

[24]  R. J. Knops,et al.  Nonlinearanalysis and Mechanics : Heriot-Watt Symposium , 1978 .

[25]  Lallit Anand,et al.  Constitutive Equations and a Time Integration Procedure for Isotropic Hyperelastic-Viscoplastic Solids , 1989 .

[26]  J. M. Bass.  Three-dimensional finite deformation, rolling contact of a hyperelastic cylinder: Formulation of the problem and computational results , 1987 .

[27]  Gérard A. Maugin,et al.  Pseudomomentum and material forces in nonlinear elasticity: variational formulations and application to brittle fracture , 1992 .

[28]  R. Ogden,et al.  A pseudo–elastic model for the Mullins effect in filled rubber , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[29]  M. Gurtin,et al.  An introduction to continuum mechanics , 1981 .

[30]  Sanjay Govindjee,et al.  Mullins' effect and the strain amplitude dependence of the storage modulus , 1992 .

[31]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[32]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[33]  S. Govindjee,et al.  Computational methods for inverse finite elastostatics , 1996 .

[34]  S. Reese,et al.  A theory of finite viscoelasticity and numerical aspects , 1998 .

[35]  F. B. Ellerby,et al.  Numerical solutions of partial differential equations by the finite element method , by C. Johnson. Pp 278. £40 (hardback), £15 (paperback). 1988. ISBN 0-521-34514-6, 34758-0 (Cambridge University Press) , 1989, The Mathematical Gazette.

[36]  R. Rivlin,et al.  Rupture of rubber. I. Characteristic energy for tearing , 1953 .

[37]  J. Padovan,et al.  Finite element analysis of steady and transiently moving/rolling nonlinear viscoelastic structure—I. Theory , 1987 .

[38]  Gene H. Golub,et al.  Matrix computations , 1983 .

[39]  P. Chadwick,et al.  Thermo-mechanics of rubberlike materials , 1974, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[40]  Richard T. Schield.  Inverse deformation results in finite elasticity , 1967 .

[41]  A. Needleman,et al.  A COMPARISON OF METHODS FOR CALCULATING ENERGY RELEASE RATES , 1985 .

[42]  A. A. Griffith.  The Phenomena of Rupture and Flow in Solids , 1921 .

[43]  J. C. Simo,et al.  Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms , 1991 .

[44]  R. Taylor.  The Finite Element Method, the Basis , 2000 .

[45]  R. H. Kennedy,et al.  Finite element analysis of steady and transiently moving/rolling nonlinear viscoelastic structure. II: Shell and three-dimensional simulations , 1987 .

[46]  P. Flory,et al.  Thermodynamic relations for high elastic materials , 1961 .

[47]  L. Anand,et al.  Finite deformation constitutive equations and a time integrated procedure for isotropic hyperelastic—viscoplastic solids , 1990 .

[48]  Joe Padovan,et al.  TIRE MODELLING AND CONTACT PROBLEMS - TRANSIENT AND STEADY STATE VISCOELASTIC ROLLING CONTACT , 1985 .

[49]  J. C. Simo,et al.  On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects , 1987 .

[50]  J. Oden,et al.  On the general rolling contact problem for finite deformations of a viscoelastic cylinder , 1986 .

[51]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[52]  Paul Steinmann,et al.  Application of material forces to hyperelastostatic fracture mechanics. II. Computational setting , 2001 .

[53]  S. Govindjee,et al.  Computational methods for inverse de-formations in quasi-incompressible nite elasticity , 1998 .