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

数学科学：动态弹塑性裂纹扩展的渐近/奇异摄动分析

• 批准号: 9404492
• 负责人: Walter Drugan
• 金额: $ 4.5万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-09-01 至 1997-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404492&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Asymptotic; Singular; Perturbation

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.

9404492 Drugan 之前尝试获得弹塑性材料中动态传播裂纹附近的应力和变形场的解析解，采用了强有力的假设，即场允许极坐标 r、theta 中以提示。 将这些结果与模式 III 的精确裂纹线解以及详细的数值有限元解进行比较表明，它们的有效半径很小，以至于在物理上不适用。提出了一种新的分析方法，避免了过度限制的可分离性假设。该方法将距裂纹尖端距离 r 的控制方程的渐近分析与利用坐标应变的裂纹马赫数（裂纹扩展速度/弹性剪切波速度）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决方案的基础上，因此它将利用最近推导的平面应变近尖端解决方案的分析系列，以及提出者获得的对非常大的有效区域的分析扩展。虽然提出的研究涉及分析用于初步研究的最简单的物理真实弹塑性材料模型之一，但预计这项工作将提供一种新的数学工具，用于推导更复杂的材料模型中动态弹塑性裂纹扩展的分析解决方案拟议的研究旨在开发在物理上有意义的尺寸范围内有效的分析解决方案，用于弹塑性材料（例如延展性金属）中快速扩展的裂纹尖端附近的应力和变形场。更一般地说，我们寻求开发数学方法来寻找一般材料类型中快速裂纹扩展的解决方案。 这些解决方案将提供对快速扩展裂纹尖端附近的材料如何表现的基本了解，并且由于该近尖端材料区域控制裂纹是否以及如何扩展，因此所寻求的解决方案将作为分析裂纹扩展和稳定性的基础标准。我们试图预测裂纹扩展速度如何影响材料的裂纹扩展阻力。当破裂的结构部件的载荷超过稳定性水平时，或者当它受到冲击载荷时，或者当它经历热冲击时，就会发生动态断裂。 这项研究的预期结果促进了对随后的快速裂纹扩展的预测，这对于准确的设计和安全评估至关重要。