# Numerical Issues in Finite Elasticity and Viscoelasticity

• Sanjay Govindjee
Chapter
Part of the International Centre for Mechanical Sciences book series (CISM, volume 452)

## Abstract

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.

## Keywords

Boundary Value Problem Weak Form Energy Release Rate Deformation Gradient Discontinuous Galerkin

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