The nonlinear field theories of mechanics. Edited and with preface by Stuart S. Antman. 3rd edition. (English) Zbl 1068.74002

Berlin: Springer (ISBN 3-540-02779-2/hbk). xxix, 602 p. (2004).
[For the review of the 2nd ed. see Zbl 0779.73004.]
As there is hardly anyone in the continuum mechanics community who does not know, use and appreciate Truesdell and Noll’s “Nonlinear field theories of mechanics”, there is surely no need to review this masterpiece again in this place. What it interesting to note is that Springer Verlag released a third edition, and that there are some changes and additions made by the editor of this particular edition, Stuart Antman. Antman used the personal notes of C. Truesdell, who passed away in 2000, and added them to the text. However, these additions are only few and difficult to find. They are mainly in the part on universal solutions.
The third addition starts with a 6-page paper by Walter Noll “The genesis of the nonlinear field theories of mechanics”, which stems from a lecture given in 2000 in the memory of Truesdell [W. Noll, J. Elasticity 70, No. 1–3, 23–30 (2003; Zbl 1039.01527)]. This text is rather personal and gives some interesting insight into the cooperation of the two authors who started writing this work in the late fifties and which was published in 1965. This first edition was soon sold out, and it was not before 1992 that the second edition with only slight changes was issued.
In his preface to the third edition of 2004, S. Antman gives a collection of fields of “subsequent developments in continuum mechanics in the spirit of Truesdell and Noll”, containing representation theorems, foundations, internal constraints, constitutive restrictions, phases, homogenization, memory, liquid crystals, thermodynamics, non-simple materials, including a list of references. This list, however, is quite personal and limited to mainly American and Italian schools. The famous paper of Noll on a new theory of simple materials [Arch. Ration. Mech. Anal. 48, 1–50 (1972; Zbl 0271.73006)], in which he questions the elder approach by history functionals, and instead advocates for a state space description, is not even mentioned. Also the old paths of entirely ignoring the numerical side of continuum mechanics, which became more and more important over the decades, are not questioned at all.
The main part of the work, however, remained unchanged. Naturally, after four decades of rapid development of nonlinear theories, this book became a historical document, in some parts still up to date, but in others incomplete, if not outdated. Any attempt to write an up-dated version of it, would end up in a new book. (W. Noll recently proposed in a paper “Updating the nonlinear field theories” a radical revision, without suggesting an appropriate author for that challenging work). So, a reprint of the original book in its the second edition seems to be more adequate.


74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74Axx Generalities, axiomatics, foundations of continuum mechanics of solids