# Steady-state fracture toughness of elastic-plastic solids: Isotropic versus kinematic hardening

## TL;DR

This study compares the steady-state fracture toughness of elastic-plastic solids under isotropic and kinematic hardening, revealing that kinematic hardening results in higher toughness due to nonproportional loading effects.

## Contribution

It introduces a steady-state framework to compare isotropic and kinematic hardening models in fracture toughness predictions, highlighting the impact of hardening law on crack propagation.

## Key findings

- Kinematic hardening yields higher fracture toughness than isotropic hardening.
- Differences are linked to nonproportional loading and reverse plasticity effects.
- Mode II crack propagation shows minimal difference due to less nonproportional loading.

## Abstract

The fracture toughness for a mode I/II crack propagating in a ductile material has been subject to numerous investigations. However, the influence of the material hardening law has received very limited attention, with isotropic hardening being the default choice if cyclic loads are absent. The present work extends the existing studies of monotonic mode I/II steady-state crack propagation with the goal to compare the predictions from an isotropic hardening model with that of a kinematic hardening model. The work is conducted through a purpose-built steady-state framework that directly delivers the steady-state solution. In order to provide a fracture criterion, a cohesive zone model is adopted and embedded at the crack tip in the steady-state framework, while a control algorithm for the far-field, that significantly reduces the number of equilibrium iterations is employed to couple the far-field loading to the correct crack tip opening. Results show that the steady-state fracture toughness (shielding ratio) obtained for a kinematic hardening material is larger than for the corresponding isotropic hardening case. The difference between the isotropic and kinematic model is tied to the nonproportional loading conditions and reverse plasticity. This also explains the vanishing difference in the shielding ratio when considering mode II crack propagation as the non-proportional loading is less pronounced and the reverse plasticity is absent.