UK Nonlinear News, August 1999

Published by Cambridge University Press, 1998
Paperback: ISBN 0 521 58746 8 (cost: £29.95)
Hardback: ISBN 0 521 58313 6 (cost: £80.00)

This book is described as a comprehensive introductory text on stress waves in nonlinear materials (both fluids and solids), and is aimed at advanced undergraduate or graduate students (and also as a reference for engineers and applied physicists). It contains five chapters and two appendices. Chapter 0 gives an overview of the rest of the book, and also describes two classical one dimensional (1D) experiments - the shock tube and flyer plate - that introduce the concepts of shock, rarefaction and structured waves. These experiments are revisited at points throughout the book.

Chapter 1 is devoted to kinematics and balance laws. Both the material (Lagrangian) and spatial (Eulerian) descriptions are used to derive the conservation laws of mass and momentum, firstly in 1D and then in 3D. In Chapter 2 a purely mechanical theory of waves in elastic materials is developed. The chapter opens with definitions and properties of elastic materials, and then goes on to discuss 1D nonlinear elastic wave equations (briefly) and 1D linear elastic waves (in detail). The theory of Riemann invariants is introduced to solve the linear wave equations (method of characteristics), and this is then generalised to Riemann integral solutions of the nonlinear equations. The chapter continues with sections on 1D structured (continuous) and shock (discontinuous) acoustic waves in nonlinear materials and wave-wave interactions. It concludes with a discussion of the book's key concept, namely that although smooth compressive acoustic waves in nonlinear elastic materials (mathematically) develop shocks, in fact shock waves do not exist in real materials. This apparent paradox is overcome by using thermoelastic rather than purely elastic constitutive models, and the next chapter is devoted to developing the theory of thermodynamics.

At the start of Chapter 3 Drumheller explains that when accurate experimental measurements are taken, waves in real elastic materials that appear to be shocks are found in fact to be continuous. This is because the waves cause changes in temperature as they pass through the material (as well as altering the stress) and some of the wave energy is absorbed by the material. This chapter mainly considers 1D deformations (although the 3D energy balance equation is also derived), and contains derivations of 1D constitutive laws for viscous, thermoelastic and thermoviscous materials using the second law of thermodynamics or Clausius-Duhem inequality. Types of solutions of the resulting equations that preserve various quantities (entropy, temperature, pressure, etc.) are discussed in detail.

The final chapter considers some more specific constitutive models (ideal gases and Mie-Grueneisen solids), and also considers elasto-plastic materials, porous solids, detonation and phase transformations. The appendices are devoted to numerical methods for solving linear and nonlinear wave equations (A), and tables of material constants (B).

As a non-thermodynamicist who has taught an honours course on continuum mechanics, I found aspects of this book interesting. I had not appreciated that shock waves are a mathematical approximation to what happens in practice, caused by using an incomplete set of constitutive equations. (Or equivalently that elastic waves cause oscillations in temperature as well as stress.) I also enjoyed some of the author's historical observations, and applaud his decision to include a modern derivation of various constitutive laws in detail using the second law of thermodynamics. I would however have some reservations (detailed below) both in using this book as a basis for teaching a course or in recommending it as a professional reference.

• Notation
  This is both complicated and somewhat imprecise. The "summation convention" (that repeated indices in an expression are summed) is used when dealing with 3D motions, which is an elegant and precise way of presenting the equations. However the author uses it in combination with a componentwise notation for vectors and tensors which can be misleading. For example, instead of showing the dependence of the deformation x on the material point X and time t by writing x = x(X,t), the author instead uses the notation x_k = x_k(X_m,t). This is not technically correct since the component x_k of the deformation depends on all components of X and not just on the component X_m.
• Definitions
  Imprecision also extends to a few of the basic definitions in the book, in a way that might be confusing to those who have not encountered the concepts before. For example, a distinction is not made between independent and dependent variables in the definition of a linear equation, and an elastic material is defined to be one for which the stress is only a function of the deformation gradient, when this is actually a homogeneous elastic material (the stress in an elastic material can also depend on the material point X).
• 3D vs. 1D
  Although I can understand why the author may have decided to restrict the text to mainly 1D deformations and wave solutions I think that his way of doing so could be misleading. I have particular problems with Chapter 3 in which the 1D character of the solution is essentially imposed as a constitutive requirement in the derivation of constitutive laws for viscous and thermal materials, and think that the text would be much improved by considering 3D constitutive laws and balance equations and then reducing them to 1D in order to look at wave solutions. A full 3D derivation is fundamentally more complicated than the material presented here (it would involve regarding the principle of frame indifference as an identity as well as the Clausius-Duhem inequality), but to my mind the resulting theory is clearer and much more satisfying (see e.g. the derivation of purely mechanical constitutive laws in [1]).
• References
  One of the main drawbacks in the use of this book by professionals is its extremely short reference section. Only 31 references are listed, and of these fewer than 10 date since 1980 (although some additional references are given as footnotes scattered throughout the text). The book would greatly benefit by having many more pointers to related books and papers, so the interested reader could use it to navigate his/her way through some of the more modern developments in rational mechanics and thermodynamics. As an example, at one point in Chapter 3 Drumheller says that the reader would find it rewarding to consult some of the work on entropy in the kinetic theory of gases, without listing a single source! Other obvious references that should have been included are [2] in the section on Riemann invariants and some good numerical analysis texts (see below).
• Numerical solution
  Appendix A (Numerical methods) contains a description of some common schemes (basically all leapfrog methods, although they're not described as such) for solving the linear and nonlinear elastic wave equations. I found this section quite confusing -- the emphasis on the underlying physics often obscured the detail of the numerical approximations. To my mind it would have been better to restrict this section to writing the wave equation in standard first order form, listing some popular solution schemes (by name), and giving some good numerical analysis references both for students and practising engineers or scientists (like [3-5]). As it is, fundamentally important concepts like stability are touched on but not described very clearly, and someone reading about numerical methods for the first time is likely to emerge from this section thinking that the CFL condition and von Neumann stability are one and the same.

References

1. M.E. Gurtin, An Introduction to Continuum Mechanics (Academic Press, 1981).
2. P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (SIAM, 1973).
3. R.J. Leveque, Numerical methods for conservation laws (2nd ed., Birkhauser, 1992).
4. A.R. Mitchell & D.F. Griffiths, The Finite Difference Method in Partial Differential Equations (Wiley, 1980).
5. K.W. Morton & D.F. Mayers, Numerical solution of partial differential equations (CUP, 1994).

UK Nonlinear News thanks Cambridge University Press for providing a copy of this book for review.