This book is designed for a one- or two-quarter course in continuum mechanics for first-year graduate students and advanced undergraduates in the mathematical and engineering sciences. It was developed, and continually improved, by over four years of teaching of a graduate engineering course (ME 238) at Stanford University, USA, followed by over four years of teaching of an advanced undergraduate mathematics course (MA3G2) at the University of Warwick, UK. The resulting text, we believe, is suitable for use by both applied mathematicians and engineers. Prerequisites include an introductory undergraduate knowledge of linear algebra, multivariable calculus, differential equations and physics.

This book is intended both for use in a classroom and for self-study. Each chapter contains a wealth of exercises, with fully worked solutions to odd-numbered questions. A complete solutions manual is available to instructors upon request. A short bibliography appears at the end of each chapter, pointing to material which underpins, or expands upon, the material discussed here. Throughout the book we have aimed to strike a balance between two classic notational presentations of the subject: coordinate-free notation and index notation. We believe both types of notation are helpful in developing a clear understanding of the subject, and have attempted to use both in the statement, derivation and interpretation of major results. Moreover, we have made a conscious effort to include both types of notation in the exercises.

Chapters 1 and 2 provide necessary background material on tensor algebra and calculus in three-dimensional Euclidean space.

In this chapter we state various axioms which form the basis for a thermo-mechanical theory of continuum bodies. These axioms provide a set of balance laws which describe how the mass, momentum, energy and entropy of a body change in time under prescribed external influences. We first state these laws in global or integral form, then derive various corresponding local statements, primarily in the form of partial differential equations. The balance laws stated here apply to all bodies regardless of their constitution. In Chapters 6–9 these laws are specialized to various classes of bodies with specific material properties, via constitutive models.

The important ideas in this chapter are: (i) the balance laws of mass, momentum, energy and entropy for continuum bodies; (ii) the difference between the integral form of a law and its local Eulerian and Lagrangian forms; (iii) the axiom of material frame-indifference and its role in constitutive modeling; (iv) the idea of a material constraint and its implications for the stress field in a body; (v) the balance laws relevant to the isothermal modeling of continuum bodies.

Motivation

In order to motivate the contents of this chapter it is useful to recall some basic ideas from the mechanics of particle systems. To this end, we consider a system of N particles with masses mi and positions xi as illustrated in Figure 5.1. It will be helpful to think of these particles as the atoms making up a continuum body.

Here we introduce the notion of a continuum body and discuss how to describe its mass properties, and the various types of forces that may act on it. As we will see, the discussion of internal forces in a continuum body will lead to the notion of a stress tensor field – our first example of a second-order tensor field arising in a physical context. We introduce the basic conditions necessary for the mechanical equilibrium of a continuum body and then derive a corresponding statement in terms of differential equations.

The important ideas in this chapter are: (i) the notion of a mass density field, which enables us to define the mass of an arbitrary part of a body; (ii) the notion of body and surface force fields, which enable us to define the resultant force and torque on an arbitrary part of a body; (iii) the notion of a Cauchy stress field and its relation to surface force fields; (iv) the equations of equilibrium for a body.

Continuum Bodies

The most basic assumption we make in our study of any material body, whether it be a solid, liquid or gas, is that the material involved can be modeled as a continuum: we ignore the atomic nature of the material and assume it is infinitely divisible. This assumption can lead to very effective material models at length scales much longer than typical interatomic spacings. However, at length scales comparable to or smaller than this, it is no longer expected to be valid.