Rheology Bulletin volume 46 (2)
Review by J. R. A. Pearson
Those familiar with Arthur Lodge's two other books, Elastic Liquids (1964) and Body Tensor Fields in Continuum Mechanics (1974) - both published by Academic Press - will recognise his style in this extended version of a well-worked lecture course. It is didactic in the best sense, in that it insists on a rigorous mathematical formulation of the subject, provides a careful verbal explanation of the physical ideas and assumptions involved and sets much of the detailed mathematical analysis as exercises for the student, with solutions given at length in a final chapter. The author is to be forgiven for some typographical and internal referencing errors caused by inadequate proof reading: these slight blemishes are more than outweighed by the quality of his writing and thought behind it. Younger readers brought up on lavishly illustrated textbooks, Nature or Scientific American, may miss the colourful diagrams, boxed synopses and governing equations that go with modern multi-media teaching, but can be reassured that all the essential ideas are properly and fully presented.
The object of the text is, in the author's words suitably paraphrased, to set out "what least must be read to acquire a detailed understanding of some molecular theory of condensed phase macroscopic behaviour". It goes beyond the continuum mechanics of Truesdell, Coleman and Noll in that it builds directly on the statistical mechanical techniques of chemical physics to pass from a discrete molecular view to a continuum one, without in any sense losing any of the rigour or beauty of classical continuum mechanics and thermodynamics. The material chosen, a cross-linked rubber, has been exhaustively studied and closely approximates an ideal elastic solid and so comparison with experiment provides a satisfactory test of the theoretical arguments advanced to yield a "constitutive equation". Treloar's 1975 book The Physics of Rubber Elasticity can be recommended as a perfect companion volume; the insights to be gained by reading both far exceed the sum of those available by reading each one separately.
I found the 11 pages of the Preface and Introduction the most rewarding of the whole text. As expected, the case for using body tensors and associated vector and tensor fields rather than field variables defined in Cartesian space is strongly advanced, justifiably so when only homogeneous deformations are considered. Less expected was a statement of the "standard mathematical sequence: undefined elements, definitions, axioms, and deductions." A clear understanding of this sequence is so important in science that all lecturers should start and continue their courses using Lodge's approach. A lack of such understanding is often at the heart of much confusion in poorly written papers in science and engineering.
Short chapters on macroscopic continuum (mechanics), thermodynamics and statistical mechanics are not quite detailed and comprehensive enough to be sufficient for those wholly unfamiliar with the subjects. However, they will be very helpful to those with rather sketchy recollections of and insights into their earlier courses or reading. Two chapters on network theory represent the heart of the book, and are firmly based on statistical mechanics. The derivation of a full set of thermodynamical functions, from which the constitutive thermomechanical behaviour of ideal rubbers is deduced, is classical theory and one which remains the brightest gem of theoretical rheology. The relevance of the theory to real materials is covered in a short chapter on comparison with experiment.
J R A Pearson