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About this book

This book deals in a modern manner with a family of named problems from an old and mature subject, classical elasticity. These problems are formulated over either a half or the whole of a linearly elastic and isotropic two- or three-dimensional space, subject to loads concentrated at points or lines. The discussion of each problem begins with a careful examination of the prevailing symmetries, and proceeds with inverting the canonical order, in that it moves from a search for balanced stress fields to the associated strain and displacement fields.

The book, although slim, is fairly well self-contained; the only prerequisite is a reasonable familiarity with linear algebra (in particular, manipulation of vectors and tensors) and with the usual differential operators of mathematical physics (gradient, divergence, curl, and Laplacian); the few nonstandard notions are introduced with care. Support material for all parts of the book is found in the final Appendix.

Table of Contents

Frontmatter

Chapter 1. One-Dimensional Paradigms

Abstract
In this introductory chapter, we work in a one-dimensional (1-D) setting. Firstly, we exemplify the nonstandard integration method we are going to use systematically in Part II. Secondly, we exemplify the Green-kernel integration method to be exploited, in particular, for the problems collected in Part III. Finally, we use these two integration methods to solve the 1-D versions of Kelvin’s and Mindlin’s problems.
P. Podio-Guidugli, A. Favata

Preliminaries

Frontmatter

Chapter 2. Elements of Linear Elasticity

Abstract
In this chapter we give a short and yet fairly complete exposition of the elemental features of classic elasticity having relevance to our subject matters.
P. Podio-Guidugli, A. Favata

Chapter 3. Geometric and Analytic Tools

Abstract
The problems in classical elasticity we tackle have intrinsic symmetries that are best exploited with the use of ad hoc coordinate systems, because the associated vector and tensor bases allow for convenient representations of the fields of interest and their transformations under the action of differential operators. In this chapter we collect a modicum of basic material from differential geometry and analysis.
P. Podio-Guidugli, A. Favata

Three Classical Problems: Flamant’s, Boussinesq’s, and Kelvin’s

Frontmatter

Chapter 4. The Flamant Problem

Abstract
In 1892, the French mechanist Alfred-Aimé Flamant (1839–1914) posed and solved the equilibrium problem of a linearly elastic, isotropic and homogeneous body occupying a half-space acted upon by a perpendicular line load of constant magnitude per unit length and infinitely long support. In this chapter, we solve the Flamant Problem by a method different from his.
P. Podio-Guidugli, A. Favata

Chapter 5. The Boussinesq Problem

Abstract
The Boussinesq Problem (Joseph Valentin B., 1842-1929) consists in finding the elastic state in a linearly elastic isotropic half-space, subject to a concentrated load applied in a point of its boundary plane and perpendicular to it. This problem has wide geotechnical applications.
P. Podio-Guidugli, A. Favata

Chapter 6. The Kelvin Problem

Abstract
Lord Kelvin (William Thompson, 1824–1907) solved the problem that was later named after him in 1848. The problem consists in finding the equilibrium state of a linearly elastic, isotropic material body occupying the whole space and being subject to a point load.
P. Podio-Guidugli, A. Favata

Three Other Problems: Melan’s, Mindlin’s, and Cerruti’s

Frontmatter

Chapter 7. The Melan and Mindlin Problems

Abstract
This chapter is devoted to solve the equilibrium problem of a linearly elastic isotropic half-space, subject to a load concentrated at an interior point. The two-dimensional version is named after Ernst Melan (1890–1963), who solved it in 1932; the three-dimensional version was studied and solved in 1936 by Raymond D. Mindlin (1906–1987), who returned to it some years later. We concentrate of the case of paramount interest in geomechanics, when the load is directed orthogonally to the boundary plane.
P. Podio-Guidugli, A. Favata

Chapter 8. The Cerruti Problem

Abstract
The equilibrium problem solved by Valentino Cerruti (1850-1909) concerns a linearly elastic isotropic half-space acted upon by a concentrated  load, tangent  to the boundary plane. We take up the version of the Cerruti Problem, where a diffused  tangent load is applied, with constant magnitude per unit length and infinitely long support.
P. Podio-Guidugli, A. Favata

Backmatter

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