A Hierarchical Quadrature Element Formulation for Fracture Parameter Evaluation in Thermoelastic Crack Problems

This paper introduces a novel computational framework for modeling complex thermo-mechanical fracture behavior. A high-order element formulation is established on the basis of the hierarchical quadrature element method (HQEM) to perform spatial discretization of transient heat conduction problems. Due to the use of hierarchical, high-order shape functions, temperature gradients can be effectively captured even on relatively coarse meshes. The discretized transient heat conduction equation is subsequently solved using an implicit time integration scheme, specifically, the backward difference method. Numerical validation against the commercial software ABAQUS demonstrates the accuracy of the proposed approach in simulating heat transfer and predicting thermoelastic deformation fields under mixed thermal boundary conditions. Furthermore, HQEM is integrated with the virtual crack closure method (VCCM) for the evaluation of fracture parameters in two-dimensional cracked structures subjected to coupled thermo-mechanical loadings. Within the VCCM framework, a unified explicit expression of the virtual crack closure integral is derived for HQEM with arbitrary nodal configurations. The integration of HQEM with VCCM significantly reduces mesh refinement requirements and preprocessing effort compared to conventional FEM. Case studies confirm that the integrated HQEM–VCCM approach yields accurate solutions for fracture parameters of cracked structures under coupled thermo-mechanical conditions.