##
**Mechanics of finite deformations.
(Mécanique des grandes transformations.)**
*(French)*
Zbl 0889.73001

Mathématiques & Applications (Berlin) 25. Berlin: Springer. x, 404 p. (1997).

This book presents a paradox: its clearly announced purpose is to clarify matters in a field of knowledge that is sometimes marked by confusion and inaccuracies, but it does so in a format and language that are not suitable to many students and researchers in the field, so that, in some sense, it condemns itself to a restricted lectureship. Furthermore, while the subject of finite deformations (contrary to the author, we use the standard vocabulary) is amongst those of interest to scientists studying single-crystal physics, engineers interested in metal forming, and rheologists, the methods and vocabulary used in the exposition are foreign to most of these readers, and this will create an additional difficulty. The reason for that is not so much the geometrical background to whom many applied scientists have become used to in the last decades, but rather is – in our opinion – the unnecessary use of many abbreviations, metaphors, and neologisms (with a more or less English structure, not supported by correctly written French). The general tune of the book is that the author, who claims to have understood better than others, will explain to us what we “clearly” cannot grasp without his aid. This tune is rather unpleasant and is not always appropriate for the presentation which has its own shortcomings and is not as transparent as the author wants us to believe.

To give “lessons” is a difficult art. The book is “geometrical”, as should be largely the case for such a subject, and does present astute remarks from time to time – e.g., on some of the time derivatives. But, in spite of brief excursions in elastoplasticity, it is too much restricted to kinematics (a favorite subject of the author) and offers too few applications, so that the transition to a true “mechanical” or “thermomechanical” study of materials and structures in finite deformation will be difficult to many readers. In our opinion, the author – interested essentially in pedagogy (but clearly aimed at mathematical students only) – has contributed too little to, or shown little interest in, creative continuum mechanics to support his mathematical disquisition by substantial examples. The style of writing may be judged from an example (top of p. 317 translated by this reviewer): “\(\dots\) the easiness of this transport operation is a consequence of the fact that projection onto charts are isomorphisms for those purely set-theoretical properties expressed by these persisting conditions\(\dots\)”). This clearly does not help.

In so far as actual research by others is concerned, the author avoids any citation by using the classical excuse that “it is not possible to quote the whole set of works”, p. 282. This is very convenient indeed. A result of this is a bibliography which emphasizes contributions from a unique school, with citations to several unpublished long reports, and a complete ignorance of the most powerful recent developments such as those relating to the use of \(G\)-structures in the geometrical description of bodies subjected to finite deformations. In contrast, many references appear to be second hand, and the bibliography is rather curiously done – e.g., many references in French (what is natural), but one reference in original Chinese (!). The notation, unfortunately, is not always very spot on – e.g., a “\(C\)” for a stress tensor adding confusion with the already used symbols for some strain tensors – and the framing of formulas in boxes has something childish.

In all, the book will be useful primarily to those who have had the curriculum of pure/applied mathematics in French universities (but this may not always be the best introduction to the subject matter), and to those who are ready to make a special effort on the abstract-geometrical side – in addition to the difficulty of the French reading to non-French readers – to approach a critical reading of this dense and long book. It is difficult to recommend it for teaching as its size and format do not fit any commonly accepted scheme, but it may reasonably belong to the shelf of many researchers in solid mechanics who may have to consult it for specific matters, but with careful caution about the special notation and symbolism.

To give “lessons” is a difficult art. The book is “geometrical”, as should be largely the case for such a subject, and does present astute remarks from time to time – e.g., on some of the time derivatives. But, in spite of brief excursions in elastoplasticity, it is too much restricted to kinematics (a favorite subject of the author) and offers too few applications, so that the transition to a true “mechanical” or “thermomechanical” study of materials and structures in finite deformation will be difficult to many readers. In our opinion, the author – interested essentially in pedagogy (but clearly aimed at mathematical students only) – has contributed too little to, or shown little interest in, creative continuum mechanics to support his mathematical disquisition by substantial examples. The style of writing may be judged from an example (top of p. 317 translated by this reviewer): “\(\dots\) the easiness of this transport operation is a consequence of the fact that projection onto charts are isomorphisms for those purely set-theoretical properties expressed by these persisting conditions\(\dots\)”). This clearly does not help.

In so far as actual research by others is concerned, the author avoids any citation by using the classical excuse that “it is not possible to quote the whole set of works”, p. 282. This is very convenient indeed. A result of this is a bibliography which emphasizes contributions from a unique school, with citations to several unpublished long reports, and a complete ignorance of the most powerful recent developments such as those relating to the use of \(G\)-structures in the geometrical description of bodies subjected to finite deformations. In contrast, many references appear to be second hand, and the bibliography is rather curiously done – e.g., many references in French (what is natural), but one reference in original Chinese (!). The notation, unfortunately, is not always very spot on – e.g., a “\(C\)” for a stress tensor adding confusion with the already used symbols for some strain tensors – and the framing of formulas in boxes has something childish.

In all, the book will be useful primarily to those who have had the curriculum of pure/applied mathematics in French universities (but this may not always be the best introduction to the subject matter), and to those who are ready to make a special effort on the abstract-geometrical side – in addition to the difficulty of the French reading to non-French readers – to approach a critical reading of this dense and long book. It is difficult to recommend it for teaching as its size and format do not fit any commonly accepted scheme, but it may reasonably belong to the shelf of many researchers in solid mechanics who may have to consult it for specific matters, but with careful caution about the special notation and symbolism.

Reviewer: G.A.Maugin (Paris)

### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74C15 | Large-strain, rate-independent theories of plasticity (including nonlinear plasticity) |

74C20 | Large-strain, rate-dependent theories of plasticity |

74B20 | Nonlinear elasticity |

74D10 | Nonlinear constitutive equations for materials with memory |

74A99 | Generalities, axiomatics, foundations of continuum mechanics of solids |

53A45 | Differential geometric aspects in vector and tensor analysis |