热致破裂相场模型及其变分结构
Phase field models for thermal fracturing and their variational structures
摘要：
经常观察到，热应力增强材料中的裂纹扩展，并且相反地，裂纹扩展可有助于材料中的温度变化。在这项研究中，我们首先考虑M。A. Biot（1956）的方法，并研究了其耗能性能。Biot热弹性模型考虑了以下影响。热膨胀和收缩是由温度变化引起的，相反，温度在膨胀区域降低，但在收缩区域增加。此外，我们还通过几个数值例子研究了它的热机械性能，并观察到热弹性效应增强了奇点附近的应力。在第二部分中，我们将裂纹扩展的相场模型和Biot热弹性模型耦合起来，提出了两种热应力作用下的裂纹扩展模型，并给出了它们的变分结构。在我们的数值实验中，我们调查如何热耦合影响裂纹的速度和形状。特别地，我们观察到最低温度出现在裂纹尖端附近，并且裂纹扩展被增强的热应力加速。
英文摘要：
It is often observed that thermal stress enhances crack propagation in materials, and conversely, crack propagation can contribute to temperature shifts in materials. In this study, we first consider the thermoelasticity model proposed by M. A. Biot (1956) and study its energy dissipation property. The Biot thermoelasticity model takes into account the following effects. Thermal expansion and contraction are caused by temperature changes, and conversely, temperatures decrease in expanding areas but increase in contracting areas. In addition, we examine its thermomechanical properties through several numerical examples and observe that the stress near a singular point is enhanced by the thermoelastic effect. In the second part, we propose two crack propagation models under thermal stress by coupling a phase field model for crack propagation and the Biot thermoelasticity model and show their variational structures. In our numerical experiments, we investigate how thermal coupling affects the crack speed and shape. In particular, we observe that the lowest temperature appears near the crack tip, and the crack propagation is accelerated by the enhanced thermal stress.
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