Application of computationally efficient TR-BDF2 scheme for the finite element implementation of explicit non-linear viscoelastic models for filled elastomers

Abstract

This paper presents the implementation of a nonlinear viscoelastic multi-configurational rate type material model within the general framework of the commercial finite element package Abaqus (Implicit). A user-defined subroutine, UMAT for continuum elements, is employed to implement a model similar to the one developed and extensively validated experimentally by Devendiran et al. One of the major challenges of using an implicit finite element (FE) solver is the description of material Jacobian, which may be difficult to obtain in the form of an analytical equation for multi-configurational models. Hence, numerical approximations are used to implement the consistent material Jacobian used in the global Newton iterations. We have used a Jacobian that is formulated by perturbing the deformation gradient, by extending an idea developed by Miehe and widely implemented in hyperelasticity. In order to determine the multiple intermediate configurations that keep evolving with time, this study also examines the computational efficiency of a fully second order integrator in comparison to traditional first order BDF integrator. A highly efficient TR-BDF2 integrator, originally developed for simulating transients of silicon VLSI devices, is utilised to determine the “current relaxed configuration”, which to the author’s knowledge is the first implementation in literature for rate type viscoelastic constitutive equations. Coupling the numerically approximated Jacobian and TR-BDF2 integrator, the UMAT is generic and can easily be modified for various combinations of stored energy potential functions and rate of dissipation equations. The UMAT is validated against material point solution of the constitutive model in MATLAB; and is found to be exact within a tight tolerance. The implemented UMAT is used to determine the rolling resistance of a Grosch wheel which demonstrates the practical application of such a material model focussed towards the workflows for tyre analysis.