7R1. Mathematical Modeling in Continuum Mechanics. - R Temam (Univ Paris-Sud, Orsay, France) and A Miranville (Univ de Poitiers, France). Cambridge UP, Cambridge, UK. 2001. 288 pp. ISBN 0-521-64362-7, $54.95.
Reviewed by P Gremaud (Dept of Math, N Carolina State Univ, Rayleigh NC 27695-8205).
This textbook is intended as an introduction to Continuum Mechanics at the advanced undergraduate and beginning graduate level. The authors aimed at breaching the gap that too often exists between engineering and example-oriented textbooks on the one hand, and needlessly abstract mathematical formulations on the other. A wide variety of subjects are briefly discussed from both Fluid and Solid Mechanics.
The book is well written and, for obvious reasons, is strongly influenced by the French school of Applied Mechanics. The point of view taken here, rigorous presentation without excessive formalism, is however nonstandard. The approach is somewhat reminiscent of the classic A Mathematical Introduction to Fluid Mechanics, by Alexandre Chorin and Jerrold Marsden (Springer Verlag, 1979), although Temam’s and Miranville’s book is much broader in scope.
The book is divided into four parts: Part One - Fundamental Concepts in Continuum Mechanics (1. Describing the Motion of a System: Geometry and Kinematics, 2. The Fundamental Law of Dynamics, 3. The Cauchy Stress Tensor - Applications, 4. Real and Virtual Powers, 5. Deformation Tensor, Deformation Rate Tensor, Constitutive Laws, 6. Energy Equations and Shock Equations); Part Two - Physics of Fluids (7. General Properties of Newtonian Fluids, 8. Flows of Inviscid Fluids, 9. Viscous Fluids and Thermohydraulics, 10. Magnetohydrodynamics and Inertial Confinement of Plasmas, 11. Combustion, 12. Equations of the Atmosphere and of the Ocean); Part Three - Solid Mechanics. (13. The General Equations of Linear Elasticity, 14. Classical Problems of Elastostatics, 15. Energy Theorems - Duality: Variational Formulations, 16. Introduction to Nonlinear Constitutive Laws and to Homogenization); and Part Four - Introduction to Wave Phenomena (17. Linear Wave Equations in Mechanics, 18. The Soliton Equation: The Korteweg-de Vries Equation, 19. The Nonlinear Schro¨dinger Equation).
It is unusual to include this many topics in an introductory textbook of less than 300 pages. The aim of the authors is to give as comprehensive a presentation of the field of Continuum Mechanics as possible. In that sense, they largely succeed. The book is lively and pleasant to read, and the pace is brisk. The price to pay for the breadth of the book lies in the amount of information that can be provided for each subject. As a typical example, let us for instance consider Chapter 12, Equations of the Atmosphere and of the Ocean, a topic to which one of the authors has made significant contributions. First, the equations are briefly derived using principles introduced in the first two parts of the book. The constitutive relations are given, although not fully discussed. Eventually, a complete system of equations is derived. A couple of useful mathematical facts, such as the description of differential operators on the sphere in this chapter, are also discussed.
The approach provides the potential instructor with a rich selection of material. After having covered the Fundamentals, essentially Part One, he/she should probably focus on a couple of selected topics from the rest of the book. As illustrated by the previous example, each chapter is relatively complete with respect to the derivation of the equations. However, one might have wished for some comments past the pure modeling stage. The above Chapter 12 for instance ends with the resulting system of equations without discussing what can be learned from such a system. Additional references could have been useful, especially with respect to the use of such models.
In conclusion, the authors have succeeded in providing a useful and elegant introductory textbook to Continuum Mechanics. The strengths of Mathematical Modeling in Continuum Mechanics lay in its mathematical rigor without excessive formalism and the broad overview it provides. On the down side, more references could have been given to ease access to complementary material such as pointers to the important numerical and computational aspects of most of the problems discussed.