7R1. Foundations and Applications of Mechanics, Volume I: Continuum Mechanics. - CS Jog (Dept of Mech Eng, Indian Inst of Sci, Bangalore, 560 012, India). Narosa Publ, New Delhi, India. Distributed in USA by CRC Press LLC, Boca Raton, FL. 2002. 254 pp. ISBN 0-8493-2414-9. $89.95.
Reviewed by K Hutter (Dept of Mech, Darmstadt Univ of Tech, Hochschulstr 1, Darmstadt, D-64289, Germany).
This is a formal book on introductory continuum mechanics, written primarily for applied mathematicians and theoretical engineers kept in the style of rational continuum mechanics and consisting of seven chapters. The preface provides a brief descriptive coverage of its content. Chapter 1 gives an introduction into tensors at an intermediate level using symbolic and Cartesian tensor notation interchangingly. The spectral properties of symmetric tensors and their polar decomposition, simple isotropic tensor functions as well as differentiation and integral laws are discussed. The chapter ends with an introduction to groups where the maximality of the orthogonal group in the unimodular group is derived. Chapter 2 is devoted to kinematics; the motion function and its spatial and time derivatives are introduced as are various measures of strain and rates of strain as suggested in the Eulerian and Lagrangian description.
Chapter 3 carries the vague title, “Governing Equations,” but is devoted to the conservation laws of mass, linear and angular momenta, and the energy and entropy balances. It also covers a presentation of the properties of the Cauchy stress tensor and introduces into a variational formulation of the mechanical equations. Constitutive relations are dealt with in Chapter 4. Frame indifference is discussed both as invariance under a change of observer as well as one under a superimposed rigid body motion. Simple materials are defined as materials whose stress depends only on the history of the deformation gradient. Material symmetry is discussed in detail, in particular with regard to the characterization of a body being a solid, a fluid, or a fluid crystal. The chapter ends with a presentation of nonlinear and linear elasticity, isotropic and anisotropic, including the energy formulation of a hypoelastic solid.
Chapter 5 is devoted to linear elasticity, its variational formulation, and uniqueness proofs for elastostatic and elastodynamic problems. Chapter 6, devoted to thermodynamics, begins with the derivation of the thermodynamical balance laws of mass and momenta by applying the rules of frame indifference to the first law of thermodynamics. The consequences of the second law of thermodynamics are exclusively dealt with in the context of the Clausius-Duhem inequality and by using the Coleman-Noll approach in exploiting the entropy principle. The constitutive relations for thermoelastic solids and some viscoelastic fluids are reduced to their thermodynamically admissible form and—for the fluid—relations relating caloric and thermal equations of state are presented. The last chapter, 7 is devoted to rigid body dynamics and is seen by this reviewer as being alien to the previous text.
The spirit of the book is formalistic, and the mathematics is clearly presented, but the addressees are more likely theoretically inclined engineers than mathematicians. Books in a similar spirit are those of Chadwick and Gurtin, referenced in this book as [4] and [9], but this reviewer prefers the latter two because they are more thought provoking and less motoric in the development of the concepts. The book can be recommended to upper-level undergraduate and graduate students who wish to learn rational continuum mechanics with a great deal of the demonstration of its technicalities. Problem sets at the end of each chapter support such a desire.
There are, however, also severe weaknesses. In the preface, the author emphasizes the significant role played by Leonhard Euler, and how crucial it is to identify the equations of motion as Euler’s achievement and not Newton’s; but neither Euler’s nor Newton’s works are referenced. Quite generally, the reference list is more than indigent, and references to scientists in the text are often not found in the reference list. The concept of frame indifference is introduced in two different ways, which are conceptually different and yield different results in general, but Fig. 4.3 states both to be equivalent; this distinction is not made clear in the text. The concept of simple materials is confusingly, if not incorrectly, introduced. Similarly, Theorem 4.7.4 states the linearized stress strain relation for an elastic body when referred to a stress-free homogeneous reference configuration, but does not explicitly say so. The theorem is therefore misleading. Most severe, however, is the author’s handling of the second law of thermodynamics involving the Clausius-Duhem inequality. No mention is made about the role played by the balance laws of mass and momentum when exploiting the entropy inequality. This has been made very clear by the founders of this way of deducing results from the entropy principle. This reviewer cannot see how any student could possibly understand this from the authors approach.
In summary, Foundations and Applications of Mechanics, Volume I: Continuum Mechanics can serve as a valuable source for teachers and students of continuum mechanics. It is written in good English and with almost no misprints. It may well be adequate as a class book or for complementary reading.
