Continuum Mechanics

Concise Theory and Problems

P. Chadwick

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486401805
• eBook ISBN: 9780486139142
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Written in response to the dearth of practical and meaningful textbooks in the field of fundamental continuum mechanics, this comprehensive treatment offers students and instructors an immensely useful tool. Its 115 solved problems and exercises not only provide essential practice but also systematically advance the understanding of vector and tensor theory, basic kinematics, balance laws, field equations, jump conditions, and constitutive equations.
Readers follow clear, formally precise steps through the central ideas of classical and modern continuum mechanics, expressed in a common, efficient notation that fosters quick comprehension and renders these concepts familiar when they reappear in other contexts. Completion of this brief course results in a unified basis for work in fluid dynamics and the mechanics of solid materials, a foundation of particular value to students of mathematics and physics, those studying continuum mechanics at an intermediate or advanced level, and postgraduate students in the applied sciences. "Should be excellent in its intended function as a problem book to accompany a lecture course." — Quarterly of Applied Math.

Chapter 1

VECTOR AND TENSOR THEORY

The theory of scalar-, vector- and tensor-valued functions defined on subsets of a three-dimensional Euclidean space is a major part of the mathematical framework upon which continuum mechanics is built. This chapter is intended to provide a concise survey of basic results needed in the rest of the book and its contents will be found to be closely integrated into the subsequent text. It is not advisable, however, for the reader to postpone his study of continuum mechanics until the whole of this material has been mastered. Rather he should use Sections 1 to 3, 9 and 10 to refresh, and perhaps reorientate, his knowledge of vector algebra and analysis, and then turn back to the topics discussed in the remaining sections as the need arises.

1VECTOR ALGEBRA

The scalar and vector products, with which the reader is already assumed to be familiar, can be defined by a system of axioms and we follow this approach here as a means of reviewing briefly the essential facts of vector algebra and, at the same time, providing a natural starting point for the development of tensor algebra. Contact with the geometrical viewpoint customarily adopted in elementary treatments of vector theory is made in Section 9.

Let

be a three-dimensional vector space over the field

of real numbers. We say that

is a Euclidean vector space if, to each pair of vectors a, b in

, there corresponds a scalar (in

), denoted by a . b and called the scalar product of a and b, and a vector (in

), written a Λ b and referred to as the vector product of a and b, with the following properties:

The norm (or magnitude), |a|, of a vector a is defined by

and a vector with unit norm is termed a unit vector. Two vectors a and b are said to be orthogonal if a . b = 0.

Problem 1 Prove that a Λ b = 0 if and only if ...

Table of contents

• Cover
• Title page
• Copyright
• Preface
• Contents
• 1 Vector and Tensor Theory
• 2 Basic Kinematics
• 3 Balance Laws, Field Equations and Jump Conditions
• 4 Constitutive Equations
• Hints and Answers to Exercises
• Appendices
• Index