Introduction

Our aim in this chapter is to understand Newton's laws of motion. From one point of view this is a modest task: Newton's laws are simple to state and involve little mathematical complexity. Their simplicity is deceptive, however. As we shall see, they combine definitions, observations from nature, partly intuitive concepts, and some unexamined assumptions on the properties of space and time. Newton's statement of the laws of motion left many of these points unclear. It was not until two hundred years after Newton that the foundations of classical mechanics were carefully examined, principally by Ernst Mach, and our treatment is very much in the spirit of Mach.

Newton's laws of motion are by no means self-evident. In Aristotle's system of mechanics, a force was thought to be needed to maintain a body in uniform motion. Aristotelian mechanics was accepted for thousands of years because, superficially, it seemed intuitively correct. Careful reasoning from observation and a real effort of thought was needed to break out of the aristotelian mold. Most of us are still not accustomed to thinking in newtonian terms, and it takes both effort and practice to learn to analyze situations from the newtonian point of view. We shall spend a good deal of time in this chapter looking at applications of Newton's laws, for only in this way can we really come to understand them.

The Central Thrust of This Book

A large class of theories in continuum physics takes as its starting point the balance laws for mass, for linear and angular momenta, and for energy, together with an entropy imbalance that represents the second law of thermodynamics. Unfortunately, most engineering curricula teach the momentum balance laws for an array of materials, often without informing students that these laws are actually independent of those materials. Further, while courses do discuss balance of energy, they often fail to mention the second law of thermodynamics, even though its place as a basic law for continua was carefully set forth by Truesdell and Toupin almost half a century ago.

This book presents a unified treatment of continuum mechanics and thermodynamics that emphasizes the universal status of the basic balances and the entropy imbalance. These laws and an hypothesis – the principle of frame-indifference, which asserts that physical theories be independent of the observer (i.e., frame of reference) – are viewed as fundamental building blocks upon which to frame theories of material behavior.

The basic laws and the frame-indifference hypothesis – being independent of material – are common to all bodies that we discuss. On the other hand, particular materials are defined by additional equations in the form of constitutive relations (such as Fourier's law) and constraints (such as incompressibility). Trivially, such constitutive assumptions reflect the fact that two bodies, one made of steel and the other of wood, generally behave differently when subject to prescribed forces – even though the two bodies obey the same basic laws.

Throughout this book:

1. (i) Lightface Latin and Greek letters generally denote scalars.

2. (ii) Boldface lowercase Latin and Greek letters generally denote vectors, but the letters o, x, y, and z are reserved for points.

3. (iii) Boldface uppercase Latin and Greek letters generally denote tensors, but the letters X, Y, and Z are reserved for points.