A thermodynamically consistent phase field model for brittle fracture in graded coatings under thermo-mechanical loading

Abstract

This paper presents a thermodynamically consistent phase field formulation designed for investigating quasi-static brittle fracture under thermo-mechanical loading conditions in functionally graded materials. The governing equations for linear momentum and crack evolution are derived by minimizing the free energy functional, which includes elastic strain energy and surface energy, with respect to displacement field and the phase field variable, respectively. A thermodynamically consistent formulation is employed to establish the governing equation for heat conduction. The proposed formulation is converted into a set of finite element equations using the Ritz method and implemented in COMSOL Multiphysics, a commercial finite element software which allows for writing governing equations in their strong form. A segregated solver is utilized to solve iteratively the coupled problem. The proposed model is verified against existing mechanical and thermo-mechanical phase field models, demonstrating good agreement with the literature. To explore the model’s capabilities, a graded layer composed of Zirconia and a Titanium alloy with a surface crack undergoing thermo-mechanical loading is considered as a case study for this investigation. Additionally, the model incorporates a thermal conductivity degradation proposed. A parametric study is performed to investigate the effect of varying parameters like volume fraction, thermal loading rate, and the thermal conductivity degradation on the load–displacement curve of the graded layer. These findings contribute to the understanding of thermo-mechanical behavior of surface graded layers and serve as a foundation for addressing more complex thermo-mechanical problems.