Numerical integration of rate constitutive equations under finite deformation

Abstract

An objective stress rate must be used in the constitutive law to accurately describe material response when large rigid body translations and rotations are present. The choice of a suitable objective stress rate has been the subject of considerable debate. For example, a basic flaw in the well known Jaumann stress rate (J-rate) has been demonstrated using the simple shear problem. A number of alternative objective stress rates have been suggested and developed. Among these is the Eulerian or E-rate in which rate constitutive calculations are performed on material axes coincident with the principal axes. The use of the principal axes technique greatly simplifies the constitutive calculations and resolves the difficulties surrounding the question of objectivity and frame invariance which arise in finite deformation problems. A numerical algorithm is presented which incorporates the principal axes technique for the Eulerian stress rate. The algorithm is intended for use in general purpose two or three dimensional solid mechanics software using the incremental or rate form of constitutive modelling. The algorithm focuses on an efficient method of integrating the material rotation tensor relating the principal axes to the fixed or global axes. The implementation of this algorithm into a general purpose non-linear transient dynamic finite element computer code is included. A number of numerical examples ranging from the simple shear problem to a large scale impact simulation demonstrates the accuracy and salient features of the algorithm.