## Table of Contents

Table of Contents

Preface

Chapter 1 – Introduction

Chapter 2 – Computational Recipes of Finite-Part Singular Integral Equations

Chapter 3 – Elasticity and Fracture Mechanics by Finite-Part Singular Integral Equations

Chapter 4 – Aerodynamics by Singular Integral Equation

Chapter 5 – Computational Recipes of Multidimensional Singular Integral Equations

Chapter 6 – Elasticity and Fracture Mechanics of Isotropic Solids by Multidimensional Singular Integral Equations

Chapter 7 – Computational Recipes for the Universal Equation of Elasticity and Thermo-Elasticity

Chapter 8 – Elasticity and Fracture Mechanics of Anisotropic Solids by Multidimensional Singular Integral Equations

Chapter 9 – Plasticity of Isotropic Solids by Multidimensional Singular Integral Equations

Appendix – Mathematical Definitions

Index

Abstract

The present book deals with the computational recipes of the finite-part singular integral equations, the multidimensional singular integral equations and the non-linear singular integral equations, which are widely used in many fields of engineering mechanics and mathematical physics with an applied character, like elasticity, plasticity, thermoelastoplasticity, viscoelasticity, viscoplasticity, fracture mechanics, structural analysis, elastodynamics, fluid mechanics, hydraulics, potential flows and aerodynamics. Such types of linear and non-linear singular integral equations form the latest high technology on the solution of very important problems of solid and fluid mechanics and therefore special attention should be given by the reader of the present book, who is interested for the new computational technology of the twentieth-one century.

Chapter 1 deals with a historical report and an extended outline of References, for the numerical evaluation methods for the finite-part singular integral equations, the multidimensional singular integral equations and the non-linear singular integral equations. Chapter 2 is devoted with the computational recipes for the solution of the finite-part singular integral equations defined in Banach spaces and in general Hilbert spaces. In the same Chapter are proposed and investigated all possible approximation methods for the numerical evaluation of the finite-part singular integral equations, as closed form solutions for the above types of integral equations are available only in simple cases. Also, Chapter 2 provides further several computational integration rules for the solution of the finite-part singular integral equations.

Furthermore, Chapter 3 deals with the application of the finite-part singular integral equations in fracture mechanics and elasticity, by calculating the stress intensity factors in several crack problems which are reduced to the solution of such a type (or systems) of integral equations. Chapter 4 provides further the application of singular integral equations in aerodynamics, by studying planar airfoils in two-dimensional aerodynamics. In Chapter 5 the Singular Integral Operators Method (S.I.O.M.) is introduced and investigated for the numerical evaluation of the multidimensional singular integral equations. This approximation method in many cases offers important advantages over “domain” type solutions, like finite elements and finite difference, as well as analytical methods such as complex variable methods.

Chapter 6 is devoted with the application of the multidimensional singular integral equations in elasticity, viscoelasticity and fracture mechanics of isotropic solids, by considering several two- and three-dimensional elastic stress analysis methods and crack problems. On the other hand, in Chapter 7 is being investigated a special field of applied mechanics, named as Relativistic Mechanics, which is a combination of the classical theory of elasticity and general relativity. Relativistic Mechanics has two main branches Relativistic Elasticity and Relativistic Thermo-Elasticity and according to the above theory, the relative stress tensor for moving structures has been formulated and a formula has been given between the relative stress tensor and the absolute stress tensor of the stationary frame. This leads to the Universal Equation of Elasticity and the Universal Equation of Thermo-Elasticity.

Beyond the above, Chapter 8 deals with the application of the multidimensional singular integral equations in elasticity and fracture mechanics of anisotropic solids, by considering two- and three-dimensional elastic stress analysis. In this case the fundamental solutions of anisotropic stress field analysis are being investigated. Also, Chapter 9 provides further the application of the multidimensional singular integral equations in plasticity of isotropic solids by proposing and studying several applications of two- and three-dimensional plasticity and thermoelastoplasticity.