An Introduction to the Theory of Elasticity

R. J. Atkin, N. Fox

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486150994
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Thanks to intense research activity in the field of continuum mechanics, the teaching of subjects such as elasticity theory has attained a high degree of clarity and simplicity. This introductory volume offers upper-level undergraduates a perspective based on modern developments that also takes into account the limited mathematical tools they are likely to have at their disposal. It also places special emphasis on areas that students often find difficult upon first encounter. An Introduction to the Theory of Elasticity provides an accessible guide to the subject in a form that will instill a firm foundation for more advanced study.
The topics covered include a general discussion of deformation and stress, the derivation of the equations of finite elasticity with some exact solutions, and the formulation of infinitesimal elasticity with application to some two- and three-dimensional static problems and elastic waves. Answers to examples appear at the end of the book.

1

Deformation and stress

In our discussion of the macroscopic behaviour of materials we disregard their microscopic structure. We think of the material as being continuously distributed throughout some region of space. At any instant of time, every point in the region is the location of what we refer to as a particle of the material. In this chapter we discuss how the position of each particle may be specified at each instant, and we introduce certain measures of the change of shape and size of infinitesimal elements of the material. These measures are known as strains, and they are used later in the derivation of the equations of elasticity. We also consider the nature of the forces acting on arbitrary portions of the body and this leads us into the concept of stress.

1.1 Motion. Material and spatial coordinates

We wish to discuss the mechanics of bodies composed of various materials. We idealize the concept of a body by supposing that it is composed of a set of particles such that, at each instant of time t, each particle of the set is assigned to a unique point of a closed region

of three-dimensional Euclidean space, and that each point of

is occupied by just one particle. We call

the configuration of the body at time t.

To describe the motion of the body, that is, to specify the position of each particle at each instant, we require some convenient method of labelling the particles. To do this, we select one particular configuration

and call this the reference configuration. The set of coordinates (X₁, X₂, X₃), or position vector X, referred to fixed Cartesian axes, of a point of

uniquely determines a particle of the body and may be regarded as a label by which the particle can be identified for all time. We often refer to such a particle as the particle X. In choosing

we are not restricted to those configurations occupied by the body during its actual motion, although it is often convenient to take

to be the configuration

occupied by the body at some instant...

Table of contents

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Deformation and stress
• 2 - Finite elasticity: constitutive theory
• 3 - Exact solutions
• 4 - Infinitesimal theory
• 5 - Anti-plane strain, plane strain, and generalised plane stress
• 6 - Extension, torsion, and bending
• 7 - Elastic waves
• Answers
• References and suggestions for further reading
• Index
• CATALOG OF DOVER BOOKS