# Numerical Investigations of Strain-Gradient Plasticity with Reference to Non-Homogeneous Deformations

## TL;DR

This paper numerically investigates strain-gradient plasticity in non-homogeneous materials, revealing how micro-boundary conditions and length scales influence strengthening, hardening, and yield behavior in small strain regimes.

## Contribution

It provides a detailed numerical analysis of strain-gradient plasticity with non-homogeneous deformations, highlighting the effects of boundary conditions and length scales on material response.

## Key findings

- Strong dependence of strengthening on micro-hard boundary conditions.
- Marked dependence of initial yield on dissipative length scale under micro-free conditions.
- Upper-bound approximation of the yield function is effective in predicting initial yield.

## Abstract

In this work, a higher-order irrotational strain gradient plasticity theory is studied in the small strain regime. A detailed numerical study is based on the problem of simple shear of a non-homogeneous block comprising an elastic-plastic material with a stiff? elastic inclusion. Combinations of micro-hard and micro-free boundary conditions are used. The strengthening and hardening behavior is explored in relation to the dissipative and energetic length scales. There is a strong dependence on length scale with the imposition of micro-hard boundary conditions. For micro-free conditions there is marked dependence on dissipative length scale of initial yield, though the differences are small in the post-yield regime. In the case of hardening behavior, the variation with respect to energetic length scale is negligible. A further phenomenon studied numerically relates to the global nature of the yield function for the dissipative problem; this function is given as the least upper bound of a function of plastic strain increment, and cannot be determined analytically. The accuracy of an upper-bound approximation to the yield function is explored, and found to be reasonably sharp in its prediction of initial yield.