Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates

The boundary integral equation (BIE) method has been used more and more in the last 20 years for solving various engineering problems. It has important advantages over other techniques for numerical treatment of a wide class of boundary value problems and is now regarded as an indispensable tool for potential problems, electromagnetism problems, heat transfer, fluid flow, elastostatics, stress concentration and fracture problems, geomechanical problems, and steady-state and transient electrodynamics. In this book, the author gives a complete, thorough and detailed survey of the method. It provides the only self-contained description of the method and fills a gap in the literature. No-one seriously interested in eigenvalue problems of elasticity or in the boundary integral equation method can afford not to read this book. Research workers, practising engineers and students will all find much of benefit to them. Contents: Introduction. Part I. Applications of Boundary Integral Equation Methods to Eigenvalue Problems of Elastodynamics. Fundamentals of BIE Methods for Elastodynamics. Formulation of BIEs for Steady-State Elastodynamics. Formulation of Eigenvalue Problems by the BIEs. Analytical Treatment of Integral Equations for Circular and Annular Domains. Numerical Procedures for Eigenvalue Problems. Numerical Analysis of Eigenvalue Problems in Antiplane Elastodynamics. Numerical Analysis of Eigenvalue Problems in Elastodynamics. Appendix: Dominant mode analysis around caverns in a semi-infinite domain. Part II. Applications of BIE Methods to Eigenvalue Problems of Thin Plates. Fundamentals of BIE Methods for Thin Plates. Formulation of BIEs for Thin Plates and Eigenvalue Problems. Numerical Analysis of Eigenvalue Problems in Plate Problems. Indexes.

-PrefaceIntroductionPart I Applications of Boundary Integral Equation Methods to Eigenvalue Problems of Elastodynamics Chapter 1 Fundamentals of Boundary Integral Equation Methods for Elastodynamics 1.1 General Remarks 1.2 Governing Equations of Elastodynamics 1.3 Boundary Value Problems and Radiation Conditions 1.4 Fundamental Solutions 1.5 Elastic Potentials and Their Properties 1.6 Green's Formula (Somigliana's Formula) 1.7 Conclusions References Chapter 2 Formulation of Boundary Integral Equations for Steady-State Elastodynamics 2.1 General Remarks 2.2 Boundary Integral Equations for the First (Displacement) Problem 2.3 Boundary Integral Equations for the Second (Traction) Problem 2.4 Boundary Integral Equations for the Third (Mixed) Problem 2.5 Formal Relations among Integral Equations 2.6 Conclusions References Chapter 3 Formulation of Eigenvalue Problems by the Boundary Integral Equations 3.1 General Remarks 3.2 Equivalence of Eigenvalues of the Integral Equations and Those of the Original Boundary Value Problems 3.3 Integral Equations for Determining Eigenvalues 3.4 A Note for the Integral Equations of Exterior Problems 3.5 Conclusions References Chapter 4 Analytical Treatment of Integral Equations for Circular and Annular Domains 4.1 General Remarks 4.2 Eigenequations and Green's Functions for Antiplane Elastodynamics 4.2.1 Integration of the Simple Layer Potential 4.2.2 Integration of the Double Layer Potential 4.2.3 Jump Conditions of Simple and Double Layer Potentials 4.2.4 Green's Formula 4.2.5 Green's Functions by the Layer Potentials and Green's Formula 4.3 Eigenequations and Green's Tensors for Plane Elastodynamics 4.3.1 Integration of the Simple Layer Potential 4.3.2 Tractions by the Simple Layer Potential 4.3.3 Integration of the Double Layer Potential 4.3.4 Tractions by the Double Layer Potential 4.3.5 Jump Conditions of Simple and Double Layer Potentials 4.3.6 Greeen's Formula 4.3.7 Green's Tensors by the Layer Potentials and Green's Formula 4.4 Eigenequations for Plane Elastodynamics with an Annular Boundary 4.4.1 Eigenequation of the Fixed-Fixed Annulus 4.4.2 Eigenequation of the Free-Free Annulus 4.4.3 Eigenequation of the Free (Outer)-Fixed (Inner) Annulus 4.4.4 Eigenequation of the Fixed (Outer)-Free (Inner) Annulus 4.4.5 A Note for Irrelevant Eigenvalues 4.5 Used Formulae 4.6 Conclusions References Chapter 5 Numerical Procedures for Eigenvalue Problems 5.1 General Remarks 5.2 Eigenvalue Problems 5.3 Evaluation of Eigenvalues by the Complex Determinant 5.4 Numerical Treatment of the Boundary Integral Equations 5.5 Fundamental Kernels in the Two-Dimensional Elastodynamics 5.6 Conclusions References Chapter 6 Numerical Analysis of Eigenvalue Problems in Antiplane Elastodynamics 6.1 General Remarks 6.2 Accuracy of Eigenvalues by the Boundary Integral Equation Method 6.3 Refinement of the Boundary Approximation by an Arc Element 6.4 Accuracy of Eigendensities and Eigenmodes 6.5 Analysis of Eigenvalue Problems of a Triangular Shaped Domain 6.6 Conclusions References Appendix for Chapter 6 Analysis of Resonance Phenomena of an Inhomogeneous Protrusion on a Stratum 6.A.1 General Remarks 6.A.2 Governing Equations and Boundary Conditions 6.A.3 Formulation of an Integral Equation 6.A.4 Numerical Procedures 6.A.5 Numerical Examples 6.A.