# Parallel three-dimensional simulations of quasi-static elastoplastic solids

## TL;DR

This paper introduces a parallel 3D simulation method for quasi-static hypo-elastoplastic solids, extending a projection approach based on incompressible Navier-Stokes equations, enabling detailed modeling of shear band formation in complex materials.

## Contribution

It develops a novel 3D parallel geometric multigrid solver for quasi-static hypo-elastoplastic simulations, applicable to realistic materials and boundary conditions.

## Key findings

- Successfully simulated shear band nucleation and growth in 3D.
- Demonstrated applicability to bulk metallic glasses and heterogeneous materials.
- Provided insights into shear band formation under various deformation conditions.

## Abstract

Hypo-elastoplasticity is a flexible framework for modeling the mechanics of many hard materials under small elastic deformation and large plastic deformation. Under typical loading rates, most laboratory tests of these materials happen in the quasi-static limit, but there are few existing numerical methods tailor-made for this physical regime. In this work, we extend to three dimensions a recent projection method for simulating quasi-static hypo-elastoplastic materials. The method is based on a mathematical correspondence to the incompressible Navier-Stokes equations, where the projection method of Chorin (1968) is an established numerical technique. We develop and utilize a three-dimensional parallel geometric multigrid solver employed to solve a linear system for the quasi-static projection. Our method is tested through simulation of three-dimensional shear band nucleation and growth, a precursor to failure in many materials. As an example system, we employ a physical model of a bulk metallic glass based on the shear transformation zone theory, but the method can be applied to any elastoplasticity model. We consider several examples of three-dimensional shear banding, and examine shear band formation in physically realistic materials with heterogeneous initial conditions under both simple shear deformation and boundary conditions inspired by friction welding.