Phase field theory for fracture at large strains including surface stresses

TL;DR

This paper develops a phase field theory for fracture at large strains that accurately incorporates surface stresses, linking geometric nonlinearities with damage evolution and validated through first principles simulations.

Contribution

It introduces a novel formulation of surface stresses in phase field fracture theory at large strains, consistent with sharp surface approaches and validated with first principles data.

Findings

Surface stresses arise from geometric nonlinearities even at small strains.

The model links material parameters with fracture behavior through analytical solutions.

Validation with first principles simulations confirms the model's accuracy.

Abstract

Phase field theory for fracture is developed at large strains with an emphasis on a correct introduction of surface stresses. This is achieved by multiplying the cohesion and gradient energies by the local ratio of the crack surface areas in the deformed and undeformed configurations and with the gradient energy in terms of the gradient of the order parameter in the reference configuration. This results in an expression for the surface stresses which is consistent with the sharp surface approach. Namely, the structural part of the Cauchy surface stress represents an isotropic biaxial tension, with the magnitude of a force per unit length equal to the surface energy. The surface stresses are a result of the geometric nonlinearities, even when strains are infinitesimal. They make multiple contributions to the Ginzburg-Landau equation for damage evolution, both in the deformed and undeformed configurations. Important connections between material parameters are obtained using an analytical solution for two separating surfaces, as well as an analysis of the stress-strain curves for homogeneous tension for different degradation and interpolation functions. A complete system of equations is presented in the undeformed and deformed configurations. All the phase field parameters are obtained utilizing the existing first principle simulations for the uniaxial tension of Si crystal in the [100] and [111] directions.