Mathematical Sciences: Thermal Effects in Fully-Coupled Transient Models for Dynamics Quasi-Brittle Fracture

实验表明，在快速(动态)加载下，固体材料的准脆性断裂伴随着温度的升高。这些增加被限制在裂缝边缘附近的非常小的区域，但足以降低固体的断裂阻力。对这一现象的研究必须处理断裂力学的混合边值/初值问题和断裂准则，以及经常耦合双曲型和抛物型偏微分方程组的瞬时热弹性和小规模屈服理论。这里提出的研究将通过获得这些问题的解析解、渐近表达式和数值结果来进行这样的研究。两个重要的变化--允许热波的第二声原理和裂纹附近的晶体缺陷(位错)运动--也被考虑在内。所得到的结果将为考虑经常被忽视的温度效应而对固体快速断裂这一关键问题提供新的启示。为此目的所必需的方程式和数值结果的发展将提供对通常单独检查的方程式组合的洞察。最后，开发过程中遇到的困难将导致提高应用各种分析和计算解决方案技术的能力。

Experiments show that quasi-brittle fracture in solids under rapid (dynamic) loading is accompanied by temperature increases. These increases are confined to very small regions near the crack edge, yet are substantial enough to reduce the solid's resistance to fracture. The study of this phenomenon must deal with the mixed boundary/initial value problems and fracture criteria of fracture mechanics and transient thermoelastic and small-scale yielding theories that often couple otherwise hyperbolic and parabolic partial differential equations. The research proposed here will conduct just such a study by obtaining analytical solutions, asymptotic expressions and numerical results for these problems. Two important variations -- the principle of second sound, which allows thermal waves, and crystal defect (dislocation) motion near cracks -- are also considered. The results obtained will shed new light on the critical problem of rapid fracture in solids by considering the oft- neglected temperature effect. The development of the equations and numerical results necessary for this purpose will provide insights into combinations of equations that are usually examined separately. Finally, the difficulties encountered in the development process will lead to an enhanced capability to apply a variety of analytical and computational solution techniques.