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

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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.