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

9404492德鲁根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，采用了一个强有力的假设，即这些场在极坐标r中允许变量可分离的近尖端解，其中以尖端为中心。将这些结果与III型的精确裂纹线解和详细的数值有限元解进行比较，结果表明它们的有效半径很小，因此在物理上是不适用的。提出了一种新的分析方法，避免了过于严格的可分离性假设。该方法将控制方程在距裂纹尖端的距离r处的渐近分析与利用坐标应变的裂纹马赫数(裂纹扩展速度/弹性剪切波速)的奇异摄动分析相结合。由于该方法建立在准静态裂纹扩展解的基础上，它将利用最近导出的平面应变近尖端解析族，并将其解析扩展到非常大的有效区域。虽然所建议的研究包括分析用于初始研究的最简单的物理现实弹塑性材料模型之一，但预计该工作将提供一种新的数学工具，用于推导在更复杂的材料模型中的动态弹塑性裂纹扩展的解析解。所提议的研究试图针对弹塑性材料(如延性金属)中快速扩展的裂纹尖端附近的应力场和变形场建立解析解，该解析解在物理上具有重要意义。更广泛地说，我们寻求开发数学方法来寻找在一般材料类型中快速裂纹扩展的解决方案。这些解将提供对快速扩展裂纹尖端附近的材料行为的基本了解，并且由于该近尖端材料区域控制裂纹是否以及如何扩展，因此所寻求的解将作为分析裂纹扩展和稳定性标准的基础。我们寻求预测材料的裂纹扩展阻力如何受裂纹扩展速度的影响。当裂纹结构部件的载荷超过稳定水平时，或者当它受到冲击载荷时，或者当它经历热冲击时，可能会发生动态断裂。根据这项研究的预期结果，对随之而来的快速裂纹扩展进行预测，对于准确的设计和安全评估至关重要。

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.