Introduction

The developments in the preceding chapter assumed nothing about the constitutive response of the continuum; we now restrict attention to a special class of constitutive laws – the so-called thermoelastic materials introduced previously in Chapter 5. Our discussion in Chapter 5 was focused entirely on the energy wells of the characterizing energy potential. Here we discuss thermoelastic materials and nonlinear thermoelasticity in more detail.

In Section 7.2 we state the constitutive law of nonlinear thermoelasticity, in which stress and specific entropy are specified as functions of deformation gradient and absolute temperature through the Helmholtz free energy potential. An equivalent alternate form of the constitutive law, in which stress and temperature are given in terms of deformation gradient and specific entropy by means of the internal energy potential, is also discussed. The expression for the driving force is then specialized to this setting. Next we state the heat conduction law, and in Section 7.2.3, we write out the full theory in the form of four scalar partial differential equations involving the three components of displacement and temperature. The accompanying jump conditions are also laid out. In the final subsection we specialize the results to a state of thermomechanical equilibrium.

Introduction

In the present chapter we study the equilibrium and quasistatic response of a thin bar composed of a material modeled by the stress–strain curve shown in Figure 2.2 of the preceding chapter. There is an extensive experimental literature devoted to tensile loading and unloading of bars made of materials that are capable of undergoing displacive phase transitions; often the materials studied are technologically important shape-memory alloys such as nickel–titanium. For a small sample of this literature, the reader might consult the papers of Krishnan and Brown [13], Nakanishi [17], Shaw and Kyriakides [19], and Lin et al. [14], as well as the references cited there. The loading in such experiments is slow, in the sense that inertia is insignificant. The objective is typically the determination of the relation between the applied stress and the overall elongation of the bar, though in some studies, such as that of Shaw and Kyriakides [19], local strain and temperature measurements are made as well. The stress–elongation relation that is observed in such experiments exhibits hysteresis, the phase transition being the primary mechanism responsible for such dissipative behavior. For a given material, the size and other qualitative features of the hysteresis loops depend on the loading rate and the temperature at which the test takes place. In the model to be discussed in this chapter, thermal effects are omitted; they will be accounted for in later chapters.

KEY FEATURES

The key features of this chapter are the energy principle for a multi-particle system, the potential energies arising from external and internal forces, and energy conservation.

This is the first of three chapters in which we study the mechanics of multi-particle systems. This is an important development which greatly increases the range of problems that we can solve. In particular, multi-particle mechanics is needed to solve problems involving the rotation of rigid bodies.

The chapter begins by obtaining the energy principle for a multi-particle system. This is the first of the three great principles of multi-particle mechanics that apply to every mechanical system without restriction. We then show that, under appropriate conditions, the total energy of the system is conserved. We apply this energy conservation principle to a wide variety of systems. When the system has just one degree of freedom, the energy conservation equation is sufficient to determine the whole motion.

CONFIGURATIONS AND DEGREES OF FREEDOM

A multi-particle systemS may consist of any number of particles P1, P2, …, PN, with masses m1, m2 …, mN respectively. A possible ‘position’ of the system is called a configuration. More precisely, if the particles P1, P2, …, PN of a system have position vectors r1, r2, …, rN, then any geometrically possible set of values for the position vectors {ri} is a configuration of the system.

KEY FEATURES

The key features of this chapter are the transformation of velocity and acceleration between frames in general relative motion, and the dynamical effects of the Earth's rotation.

So far we have viewed the motion of mechanical systems from an inertial reference frame. The reason for this is simple; the Second Law, in its standard form, applies only in inertial frames. However, circumstances arise in which it is convenient to view the motion from a non-inertial frame. The most important instance of this occurs when the motion takes place near the surface of the Earth. Previously we have argued that the dynamical effects of the Earth's rotation are small enough to be neglected. While this is usually true, there are circumstances in which it has a significant effect. In long range artillery, the Earth's rotation gives rise to an important correction, and, in the hydrodynamics of the atmosphere and oceans, the Earth's rotation can have a dominant effect. If we wish to calculate such effects (as seen by an observer on the Earth), we must take our reference frame fixed to the Earth, thus making it a non-inertial frame. The downside of this choice is that the Second Law does not hold and must be replaced by a considerably more complicated equation.

In addition to applications involving the Earth's rotation, there are instances where the motion of a system looks much simpler when viewed from a suitably chosen rotating frame.

Information for readers

What is this book about and who is it for?

This is a book on classical mechanics for university undergraduates. It aims to cover all the material normally taught in classical mechanics courses from Newton's laws to Hamilton's equations. If you are attending such a course, you will be unlucky not to find the course material in this book.

What prerequisites are needed to read this book?

It is expected that the reader will have attended an elementary calculus course and an elementary course on differential equations (ODEs). A previous course in mechanics is helpful but not essential. This book is self-contained in the sense that it starts from the beginning and assumes no prior knowledge of mechanics. However, in a general text such as this, the early material is presented at a brisker pace than in books that are specifically aimed at the beginner.

What is the style of the book?

The book is written in a crisp, no nonsense style; in short, there is no waffle! The object is to get the reader to the important points as quickly and easily as possible, consistent with good understanding.

Are there plenty of examples with full solutions?

Yes there are. Every new concept and technique is reinforced by fully worked examples. The author's advice is that the reader should think how he or she would do each worked example before reading the solution; much more will be learned this way!

KEY FEATURES

The key features of this chapter are Newton's laws of motion, the definitions of mass and force, the law of gravitation, the principle of equivalence, and gravitation by spheres.

This chapter is concerned with the foundations of dynamics and gravitation. Kinematics is concerned purely with geometry of motion, but dynamics seeks to answer the question as to what motion will actually occur when specified forces act on a body. The rules that allow one to make this connection are Newton's laws of motion. These are laws of physics that are founded upon experimental evidence and stand or fall according to the accuracy of their predictions. In fact, Newton's formulation of mechanics has been astonishingly successful in its accuracy and breadth of application, and has survived, essentially intact, for more than three centuries. The same is true for Newton's universal law of gravitation which specifies the forces that all masses exert upon each other.

Taken together, these laws represent virtually the entire foundation of classical mechanics and provide an accurate explanation for a vast range of motions from large molecules to entire galaxies.

NEWTON'S LAWS OF MOTION

Isaac Newton's three famous laws of motion were laid down in Principia, written in Latin and published in 1687. These laws set out the founding principles of mechanics and have survived, essentially unchanged, to the present day. Even when translated into English, Newton's original words are hard to understand, mainly because the terminology of the seventeenth century is now archaic.

KEY FEATURES

The key features of this chapter are integral functionals and the functions that make them stationary, the Euler–Lagrange equation and extremals, and the importance of variational principles.

The notion that physical processes are governed by minimum principles is older than most of science. It is based on the long held belief that nature arranges itself in the most ‘economical’ way. Actually, many ‘minimum’ principles have, on closer inspection, turned out to make their designated quantity stationary, but not necessarily a minimum. As a result, they are now known to be variational principles, but they are no less important because of this. A good example of a variational principle is Fermat's principle of geometrical optics, which was proposed in 1657 as Fermat's principle of least time in the form:

Of all the possible paths that a light ray might take between two fixed points, the actual path is the one that minimises the travel time of the ray.

Fermat showed that the laws of reflection and refraction could be derived from his principle, and proposed that the principle was true in general. Not only did Fermat's principle ‘explain’ the known laws of optics, it was simple and elegant, and was capable of extending the laws of optics far beyond the results that led to its conception. This example explains why variational principles continue to be sought; it is because of their innate simplicity and elegance, and the generality of their application.

KEY FEATURES

The key features of this chapter are the transformation formulae for the components of tensors and tensor algebra; the inertia tensor and the calculation of the angular momentum and kinetic energy of a rigid body; and the principal axes and principal moments of inertia of a rigid body.

We have previously regarded a vector as a quantity that has magnitude and direction. Picturing a vector as a line segment with an arrow on it has helped us to understand essentially difficult concepts, such as the acceleration of a particle and the angular momentum of a rigid body. Neither of these quantities has any direct connection with line segments, but the picture is a valuable aid nonetheless. Useful though this notion of vectors is, it is not capable of generalisation and this is the main reason why we will now look at vectors from a different perspective. In reality, what we actually observe are the three components of a vector, and the values of these components will depend on the coordinate system in which they are measured. However, a general triple of real numbers {υ1, υ2, υ3}, defined in each coordinate system, does not neccessarily constitute a vector. The reason is that the components of a vector in different coordinate systems are related to each other (in a way that we shall determine) and, unless the quantities {υ1, υ2, υ3} satisfy this transformation formula, they are not the components of a vector.