The role of viscous regularization in dynamical problems, strain localization and mesh dependency

TL;DR

This paper investigates how viscous regularization affects strain localization and mesh dependency in numerical models, deriving new dispersion relations and confirming findings with numerical analyses.

Contribution

It introduces a complex-plane approach to strain-softening models, providing new insights into strain localization and mesh dependence in continuum mechanics.

Findings

Strain localization can occur on a mathematical plane under complex frequency and wave number.

Numerical analyses show mesh-dependent strain and plastic strain rate profiles.

Derived dispersion relation differs from previous models, highlighting new localization conditions.

Abstract

Strain localization is responsible for mesh dependence in numerical analyses concerning a vast variety of fields such as solid mechanics, dynamics, biomechanics and geomechanics. Therefore, numerical methods that regularize strain localization are paramount in the analysis and design of engineering products and systems. In this paper we revisit the elasto-viscoplastic, strain-softening, strain-rate hardening model as a means to avoid strain localization on a mathematical plane in the case of a Cauchy continuum. Going beyond previous works (de Borst and Duretz (2020); Needleman (1988); Sluys and de Borst (1992); Wang et al. (1997)), we assume that both the frequency {\omega} and the wave number k belong to the complex plane. Therefore, a different expression for the dispersion relation is derived. We prove then that under these conditions strain localization on a mathematical plane is possible. The above theoretical results are corroborated by extensive numerical analyses, where the total strain and plastic strain rate profiles exhibit mesh dependent behavior.