Thermal-conductivity degradation across cracks in coupled thermo-mechanical systems modeled by the phase-field fracture method

Abstract

Dynamic loading of polycrystalline metallic materials can result in brittle or ductile fracture depending on the loading rates, geometry and material type. At high strain rates, mechanical energy due to plastic deformation may lead to significant temperature rise and shear localization due to thermal softening. These shear bands reduce the stress bearing capacity of the material and act as a precursor to ductile fracture (e.g. cracks that develop rapidly on top of a shear band).Understanding the heat transfer physics in thermo-mechanical problems when cracks are developed, is a fundamental science that has not been rigorously addressed. In particular, reliable damage models, that degrade the thermal-conductivity in continuum crack descriptions, are necessary to capture the correct heat transfer physics across fracture surfaces but are currently absent.In this paper, a novel set of isotropic thermal-conductivity degradation functions is derived based on a micro-mechanics void extension model of Laplace’s equation. The key idea is to employ an analytical homogenization process to find the effective thermal-conductivity of an equivalent sphere with expanding spherical void. The closed form solution is obtained by minimization of the flux differences at the outer surfaces of the two problems, which can be achieved using the analytical solution of Laplace’s equations, so called spherical-harmonics.A unified model, which has been developed in McAuliffe and Waisman (2015) to account for the simultaneous formation of shear bands and cracks is used herein as a numerical tool to investigate the behavior of the aforementioned thermal-conductivity model. In this unified model, the phase-field method is used to model crack initiation and propagation and is coupled to a temperature dependent visco-plastic model that captures shear bands. Two benchmark problems are presented to show the necessity of such physics-based degradation function in dynamic fracture problems.