Abstract

The Arruda-Boyce viscoplastic model has been the cornerstone for the development of sophisticated constitutive models for polymers and soft tissues. A simple implicit finite element implementation for the Arruda-Boyce viscoplastic model, which is coded as user material routine in ABAQUS-Standard, is presented in this work. The implementation uses the Backard-Euler method and standard stress-update procedures in the relaxed configuration; it has no iterative procedures; it completely avoids the use of eigen-decompositions, polar decompositions and rotation matrices while square roots, exponentials and natural logarithms of second order tensors are computed using high-order Padé approximants. The tangent-stiffness matrix is computed using a finite difference scheme. As a result, the stress-update algorithm turns out to be very simple to code and to implement, but still very accurate and quite efficient in computational terms. The precision, stability and efficiency of the proposed formulation are assessed and demonstrated by means of a number of examples. As a result, a number of recommendations are given to best set the size of the inelastic strain increment and the residuals for the Newton-Raphson procedures; and to select the most effective algorithms for the evaluation of the tangent stiffness matrix and to solve the system of equations.

Abstract

Two algorithms for the stress update (i.e., time-integration of the constitutive equation) in large strain solid mechanics are discussed, with particular emphasis on two issues: the incremental objectivity and the implementation aspects. It is shown that both algorithms are incrementally objective (i.e., they treat rigid rotations properly) and that they can be employed to add large strain capabilities to a small large deformation paths (rigid rotation, simple shear, extension, extension and compression, dilatation, extension and rotation), have been used to test and compare the two algorithms, both for elastic and plastic analysis. These test evidence different time integration accuracy for each algorithm. However, it is demonstrated that the, in general, less accurate algorithm gives exact results for shear-free deformation paths.