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.

翻译：本文提出一个综合计算框架，用于研究横观各向同性材料中的热弹性断裂问题。在应力集中区域，经典线性弹性理论无法预测符合物理实际的行为。我们通过采用应变限制弹性理论来解决裂纹尖端处非物理应变奇异性问题。该理论以应力与应变之间的代数非线性本构关系为特征，其本质是对应变张量范数施加限制。这种方法允许在裂纹尖端附近产生预期的高应力，同时确保相应应变保持物理有界。我们建立了控制热机械横观各向同性固体响应的线性与非线性偏微分方程弱耦合系统。基于有限元法开发了鲁棒的数值求解方案，在连续伽辽金框架内采用协调有限元离散化方法求解二维边值问题。应用该模型分析了极端热机械载荷作用下边缘裂纹附近的应力场与应变场。数值结果表明与经典预测存在显著差异：虽然应力在裂纹尖端高度集中，但应变增长速率显著减缓，并在整个域内保持有界。本研究验证了应变限制模型在正则化热弹性裂纹尖端场方面的有效性，为先进各向异性材料中热驱动裂纹萌生与扩展的预测建模奠定了可靠的计算基础。