Mathematical Sciences: Asymptotic/Singular Perturbation Analysis of Dynamic Elastic-Plastic Crack Growth

小行星9404492 以前试图获得弹塑性材料中动态扩展裂纹附近的应力场和变形场的解析解时，都采用了强假设，即这些场在以尖端为中心的极坐标r，θ中允许变量可分离的近尖端解。 这些结果与精确的裂纹线的解决方案模式III，并与详细的数值有限元解的比较，表明其有效性半径是如此之小，使他们在物理上不适用。提出了一种新的分析方法，避免了过于严格的可分性假设。该方法结合渐近分析的距离r从裂纹尖端的奇异摄动分析裂纹马赫数（裂纹扩展速度/弹性剪切波速度），利用坐标应变的控制方程。由于这种方法是建立在准静态裂纹扩展解的基础上的，因此它将利用最近导出的 分析家庭的平面应变近尖端的解决方案，以及他们的分析扩展到非常大的区域的有效性，由提议者获得。虽然所提出的研究涉及分析一个最简单的物理现实弹塑性材料模型的初步研究，预计这项工作将提供一个新的数学工具，推导出动态弹塑性裂纹扩展的解析解，在更复杂的材料模型也 拟议的研究旨在开发的分析解决方案，有效的物理上显着的尺寸尺度，在弹塑性材料，如韧性金属的快速扩展裂纹的尖端附近的应力和变形场。更一般地说，我们寻求发展数学方法， 这种解决方案用于一般材料类型中的快速裂纹生长。 这些解决方案将提供一个基本的理解如何快速扩展的裂纹尖端附近的材料的行为，因为这个近端材料区域控制是否和如何裂纹将增长，寻求的解决方案将作为分析裂纹增长和稳定性标准的基础。我们寻求 预测材料对裂纹扩展的抵抗力如何受裂纹扩展速度的影响。当开裂的结构部件承受超过稳定水平的载荷时，或者当其受到冲击载荷时，或者当其经历热冲击时，可能发生动态断裂。 预测随后的快速裂纹扩展，从这项研究中预期的结果，促进准确的设计和安全评估是至关重要的。

9404492 Drugan Previous attempts to obtain analytical solutions for the stress and deformation fields near a dynamically propagating crack in elastic-plastic material have employed the strong assumption that the fields admit variable-separable near-tip solutions in the polar coordinates r, theta centered at the tip. Comparisons of these results with an exact crack-line solution for Mode III, and with detailed numerical finite element solutions, show that their radius of validity is so small as to render them physically inapplicable. A new analytical approach is proposed that avoids the overly restrictive separability assumption. The approach combines asymptotic analysis of the governing equations in distance r from the crack tip with a singular perturbation analysis in the crack Mach number (crack growth speed/elastic shear wave speed) that utilizes coordinate straining. Since the approach thus builds on quasi-static crack growth solutions, it will make use of the recently-derived analytical family of plane strain near-tip solutions, and also their analytical extension to very large regions of validity, obtained by the proposer. While the research proposed involves analyzing one of the simplest physically realistic elastic-plastic material models for initial study, it is anticipated that the work will provide a new mathematical tool for deriving analytical solutions for dynamic elastic-plastic crack growth in more sophisticated materials models also The proposed research seeks to develop analytical solutions, valid over a physically significant size scale, for the stress and deformation fields near the tip of a rapidly propagating crack in elastic-plastic materials such as ductile metals. More generally, we seek to develop mathematical methods for finding such solutions for rapid crack growth in general material types. These solutions will provide a fundamental understanding of how the material near a rapidly propagating crack tip behaves , and since this near-tip material region controls whether and how the crack will grow, the solutions sought will serve as the basis for analytical crack growth and stability criteria. We seek to predict, among other things, how the material's resistance to crack growth is affected by crack propagation speed. Dynamic fracture can occur when a cracked structural component is loaded beyond the stability level, or when it is subjected to impact loading, or when it experiences a thermal shock. Predictions of the ensuing rapid crack growth, facilitated by the results anticipated from this research, are crucial for accurate design and safety assessments.