Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation, 2012 Interdisciplinary Applied Mathematics Series, Vol. 37

This introductory and self-contained book gathers as much explicit mathematical results on the linear-elastic and heat-conduction  solutions in the neighborhood of singular points in two-dimensional domains, and  singular edges and vertices in three-dimensional domains. These are presented in an  engineering terminology for practical usage. The author treats the mathematical   formulations from an engineering viewpoint and presents high-order finite-element  methods for the computation of singular solutions in isotropic and anisotropic materials,  and multi-material interfaces.  The proper interpretation of the results in engineering practice  is advocated, so that the computed data can be correlated to experimental observations.

The book is divided into fourteen chapters, each containing several sections.

Most of it (the first nine Chapters) addresses two-dimensional domains, where

only singular points exist. The solution in a vicinity of these points admits an asymptotic expansion composed of eigenpairs and associated generalized flux/stress intensity factors (GFIFs/GSIFs), which are being computed analytically when possible or by finite element methods otherwise. Singular points associated with weakly coupled thermoelasticity in the vicinity of singularities are also addressed and thermal GSIFs are computed. The computed data is important in engineering practice for predicting failure initiation in brittle material on a daily basis.  Several failure laws for two-dimensional domains with V-notches are presented and their validity is examined by comparison to experimental observations. A sufficient simple and reliable condition for predicting failure initiation (crack formation) in micron level electronic devices, involving singular points, isstill a topic of active research and interest, and is addressed herein.

Explicit singular solutions in the vicinity of vertices and edges in three-dimensional domains are provided in the remaining five chapters. New methods for the computation of generalized edge flux/stress intensity functions along  singular edges are presented and demonstrated by several example  problems from the field of fracture mechanics; including anisotropic domains and bimaterial interfaces. Circular edges are also presented and the author concludes with  some remarks on open questions.

This well illustrated book will appeal to both applied  mathematicians and engineers working in the field of fracture mechanics and  singularities.

Preface.- Introduction.-An Introduction to the p- and hp-versions of the Finite Element Method.-Eigen-pairs Computation for Two-Dimensional Heat Conduction Singularities.-Computation of GFIFs for Two-Dimensional Heat Conduction Problems.-Eigen-pairs for two-dimensional elasticity.-Computing Generalized Stress Intensity Factors.-Thermal Generalized Stress Intensity Factors in 2-D Domains.-Failure Criteria for Brittle Elastic Materials.-Thermo-Mechanical Failure Criterion at the Micron Scale in Electronic Devices.-Singular solutions of the heat conduction equation in polyhedra domains.-Computation of the Edge Flux Intensity Functions associated with polyhedra domains.-Vertex singularities associated with conical points for the 3-D Laplace equation.-Edge eigen-pairs and ESIFs of 3-D elastic problems.-Summary and Open Questions.