A finite element model for thermomechanical stress-strain fields in transversely isotropic strain-limiting materials

TL;DR

This work addresses unphysical crack-tip strain predictions in classical thermoelasticity by adopting a strain-limiting implicit constitutive framework for transversely isotropic media. A two-field, one-way-coupled finite element approach is developed, combining a linear heat equation with a nonlinear momentum balance solved via Picard iteration, and it is implemented in a continuous Galerkin setting. The TI strain-limiting model regularizes strain near cracks while allowing large stresses, with key insights showing that the nonlinearity parameter $b$ mitigates peak strains whereas $a$ can amplify crack-tip fields; results confirm bounded strains and elliptical Mode I-like crack openings under various thermal loads. The methodology provides a robust computational foundation for predictive modeling of thermally driven crack initiation and evolution in anisotropic materials, with potential extensions to phase-field or interface-damage formulations for fracture evolution.

Abstract

This paper presents a comprehensive computational framework for investigating thermo-elastic fracture in transversely isotropic materials, where classical linear elasticity fails to predict physically realistic behavior near stress concentrations. We address the challenge of unphysical strain singularities at crack tips by employing a strain-limiting theory of elasticity. This theory is characterized by an algebraically nonlinear constitutive relationship between stress and strain, which intrinsically enforces a limit on the norm of the strain tensor. This approach allows the development of very large stresses, as expected near a crack tip, while ensuring that the corresponding strains remain physically bounded. A loosely coupled system of linear and nonlinear partial differential equations governing the response of a thermo-mechanical transversely isotropic solid is formulated. We develop a robust numerical solution based on the finite element method, utilizing a conforming finite element discretization within a continuous Galerkin framework to solve the two-dimensional boundary value problem. The model is applied to analyze the stress and strain fields near an edge crack under severe thermo-mechanical loading. Our numerical results reveal a significant departure from classical predictions: while stress concentrates intensely at the crack tip, the strain grows at a substantially slower rate and remains bounded throughout the domain. This work validates the efficacy of the strain-limiting model in regularizing thermo-elastic crack-tip fields and establishes a reliable computational foundation for the predictive modeling of thermally driven crack initiation and evolution in advanced anisotropic materials.

Figures (25)

• Figure 6.1: Computational domain
• Figure 6.2: Temperature Profile, $\theta=100,a=0.5,b=0.02$
• Figure 6.3: Stress and strain for $b$ variation, $\theta=100,a=0.5$
• Figure 6.4: Stress and strain for $a$ variation , $\theta=100,b=0.02$
• Figure 6.5: Stress and strain for $\theta=100,a=0.5,b=0.02$