# Ductile fracture modeling by phase field, Hencky strain elasticity and finite J2 plasticity using nonlocal operator method

## TL;DR

This paper introduces a novel phase field model for ductile fracture that integrates Hencky strain, finite J2 plasticity, and the nonlocal operator method, providing a comprehensive approach to simulate ductile failure.

## Contribution

It develops a variational phase field model incorporating advanced plasticity and strain measures with a spectral decomposition algorithm, enhancing simulation accuracy for ductile fracture.

## Key findings

- Validated with numerical examples including notched plates and necking bars.
- Demonstrated the effectiveness of the model in capturing ductile fracture behavior.
- Provided a consistent computational framework with derived stiffness matrices.

## Abstract

A phase field model for ductile fracture considering Hencky strain and finite J2 plasticity is presented using the nonlocal operator method. A variational derivation of J2 plasticity at finite strain with a phase field model is performed. The method includes a logarithmic strain tensor and an exponential mapping in the plasticity evolution. A spectral decomposition based algorithm for computing the first and second order derivatives of the composite matrix function is implemented. A consistent tangential stiffness matrix is derived and used in Newton-Raphson iterations. Several numerical examples are performed to validate the method, including notched single-edged plates with brittle fracture or ductile fracture and necking of a bar with/without phase field model.