Representative volume element approximations in elastoplastic spring networks

TL;DR

This paper develops and analyzes a periodic RVE approximation for a homogenized elastoplastic lattice model, demonstrating convergence and numerical validation of the approach for large-scale behavior prediction.

Contribution

It introduces a periodic RVE approximation for the homogenized elastoplastic system and proves its convergence as RVE size increases, including a numerical scheme for simulations.

Findings

RVE approximation converges as RVE size tends to infinity.

Hysteretic stress-strain relation described by a generalized Prandt-Ishlinskii operator.

RVE error decays at the same rate as in static linear elasticity cases.

Abstract

We study the large-scale behavior of a small-strain lattice model for a network composed of elastoplastic springs with random material properties. We formulate the model as an evolutionary rate independent system (ERIS). In an earlier work we derived a homogenized continuum model, which has the form of linearized elastoplasticity, as evolutionary $\Gamma$-limit as the lattice parameter tends to zero. In the present paper we introduce a periodic representative volume element approximation (RVE) for the homogenized system. As a main result we prove convergence of the RVE approximation as the size of the RVE tends to infinity. We also show that the hysteretic stress-strain relation of the effective system can be described with help of a generalized Prandt-Ishlinskii operator, and we prove convergence of a periodic RVE approximation for that operator. We combine the RVE approximation with a numerical scheme for rate-independent systems and obtain a computational scheme that we use to numerically investigate the homogenized system in the specific case when the original network is given by a two-dimensional lattice model. We simulate the response of the system to cyclic and uniaxial, monotonic loading, and numerically investigate the convergence rate of the periodic RVE approximation. In particular, our simulations show that the RVE error decays with the same rate as the RVE error in the static case of linear elasticity.