Cosserat theories: Shells, rods and points.

*(English)*Zbl 0984.74003
Solid Mechanics and Its Applications. 79. Dordrecht: Kluwer Academic Publishers. xv, 480 p. (2000).

Summary: There is a huge literature on the theory of “thin bodies,” defined as those bodies whose initial configuration possesses one or two dimensions much smaller than the others. The former are called “shells,” the latter “rods.” It is also possible to consider bodies which are thin in all three of their spatial dimensions, and define them as “pointlike structures.” The study of point-like structures is the most recent among the three possibilities.

The purpose of this book is to offer a unified approach to the development of all these theories by use of either suitable kinematic assumptions on the three-dimensional theory or introducing directors. The two procedures are conceptually different and not always equivalent. The introduction of directors is attributed by the author to E. and F. Cosserat in 1909, who exploited a previous idea of P. Duhem (1893). However, for historical precision the date of birth of the notion of one-dimensional polar continuum is the year 1691, when Jakob Bernoulli wrote his first paper on beams enunciating the result in the form of a cryptogram. In any case, the importance of these theories stems from the possibility of treating problems of equilibrium and motion of thin bodies with simple equations, derived by exploiting the geometric properties in order to reduce the number of independent variables.

The book consists essentially of three chapters and two introductions. The first introduction is an outline of tensor calculus by use of the notation introduced by A. F. Green and W. Zerna in their book [Theoretical elasticity. Oxford: At the Clarendon Press, XIII (1954; Zbl 0056.18205)]. There is no general agreement whether this symbolism is the best, because it resides between two extremes (the absolute direct notation and the index notation), but it was popular in the sixties of the 20th century, and was well accepted by engineers. The second introduction is a résumé of principles and equations of continuum mechanics as they has been formalized and thought since the year 1960. The author makes here a commendable effort of writing all the equations in terms of Green and Zerna’s notation. The conversion may appear to be a useless complication, since direct notation is much more elegant and effective, but it has been performed in order to show how to derive the equations of thin structures from their three-dimensional form. In this introduction the author also explains the principle of objectivity in an inductive and original manner: starts from two simple examples, and then writes the general form of the principle. In contrast, the detailed classification of particular constitutive equations and the analysis of special problems of cube and cylinder are hardly relevant.

The main part of the book is concentrated in chapters 4, 5, 6, where the author derives general field equations of shell, rods, and points both from the three-dimensional equations and from the director theory. Additionally, the author gives modified equations when internal constraints are imposed on the fields of displacement and director by the principle of frame indifference and, eventually, by kinematical constraints. This is a remarkable achievement of the book, since, in contrast to many others, it shows how many approximate methods of modelling structures can be regarded as the result of (sometimes not clearly stated) imposition of kinematic constraints on the field of displacement or director. Typical examples of this procedure are Bernoulli and Timoshenko theories of beams. Unfortunately, the general part of each chapter is followed by a tedious and incomplete classification of linear constitutive equations and by few particular examples which are mainly restricted to the linear case, which does not adequately illustrate the potential applications of the theory. The book is completed by a chapter on numerical applications of the above theories, but it is not coherent with the whole program of the work.

By the description of the topics treated in the book, the reviewer feels that a general evaluation of this subject must be a compromise between the innovative elements conveyed to the theory of thin bodies, and some useless redundancies and omissions. To start from the last, it is a pity to notice that no mention is made of von Kármán’s and Koiter’s simplified theories which, though questionable, are classical. Another deficiency is the neglection of recent mathematical progress in treating boundary value problems both in the linear and nonlinear case, either by variational methods or by integral equations. As for the repetitive part, the long list of linear constitutive equations for orthotropic materials represent mere an obstacle in the reading the text than a contribution to completeness. On the other hand, the deduction of balance and constitutive equations from general principles of continuum mechanics is done with extreme precision and clarity. Director theories, in particular, are conceptually easy to be understood, but prohibitively hard to be manipulated, and here the author is able to offer an elegant exposition of them. In contrast to Naghdi’s article in the Handbuch der Physik [S. Flügge (ed.), Handbuch der Physik. Band VIa/2: Festkörpermechanik II. Bandherausgeber: C. Truesdell. Encyclopedia of physics. Vol. VIa/2: Mechanics of solids II. Editor: C. Truesdell. Berlin-Heidelberg-New York: Springer-Verlag. X (1972; Zbl 0277.73001)], which is labyrinthine and chaotic, the present book is organic and consistent.

As a final remark, the book is not intended for engineers eager to find explicit solutions, but rather for researchers willing to become familiar with Cosserat theories for further promising extensions.

The purpose of this book is to offer a unified approach to the development of all these theories by use of either suitable kinematic assumptions on the three-dimensional theory or introducing directors. The two procedures are conceptually different and not always equivalent. The introduction of directors is attributed by the author to E. and F. Cosserat in 1909, who exploited a previous idea of P. Duhem (1893). However, for historical precision the date of birth of the notion of one-dimensional polar continuum is the year 1691, when Jakob Bernoulli wrote his first paper on beams enunciating the result in the form of a cryptogram. In any case, the importance of these theories stems from the possibility of treating problems of equilibrium and motion of thin bodies with simple equations, derived by exploiting the geometric properties in order to reduce the number of independent variables.

The book consists essentially of three chapters and two introductions. The first introduction is an outline of tensor calculus by use of the notation introduced by A. F. Green and W. Zerna in their book [Theoretical elasticity. Oxford: At the Clarendon Press, XIII (1954; Zbl 0056.18205)]. There is no general agreement whether this symbolism is the best, because it resides between two extremes (the absolute direct notation and the index notation), but it was popular in the sixties of the 20th century, and was well accepted by engineers. The second introduction is a résumé of principles and equations of continuum mechanics as they has been formalized and thought since the year 1960. The author makes here a commendable effort of writing all the equations in terms of Green and Zerna’s notation. The conversion may appear to be a useless complication, since direct notation is much more elegant and effective, but it has been performed in order to show how to derive the equations of thin structures from their three-dimensional form. In this introduction the author also explains the principle of objectivity in an inductive and original manner: starts from two simple examples, and then writes the general form of the principle. In contrast, the detailed classification of particular constitutive equations and the analysis of special problems of cube and cylinder are hardly relevant.

The main part of the book is concentrated in chapters 4, 5, 6, where the author derives general field equations of shell, rods, and points both from the three-dimensional equations and from the director theory. Additionally, the author gives modified equations when internal constraints are imposed on the fields of displacement and director by the principle of frame indifference and, eventually, by kinematical constraints. This is a remarkable achievement of the book, since, in contrast to many others, it shows how many approximate methods of modelling structures can be regarded as the result of (sometimes not clearly stated) imposition of kinematic constraints on the field of displacement or director. Typical examples of this procedure are Bernoulli and Timoshenko theories of beams. Unfortunately, the general part of each chapter is followed by a tedious and incomplete classification of linear constitutive equations and by few particular examples which are mainly restricted to the linear case, which does not adequately illustrate the potential applications of the theory. The book is completed by a chapter on numerical applications of the above theories, but it is not coherent with the whole program of the work.

By the description of the topics treated in the book, the reviewer feels that a general evaluation of this subject must be a compromise between the innovative elements conveyed to the theory of thin bodies, and some useless redundancies and omissions. To start from the last, it is a pity to notice that no mention is made of von Kármán’s and Koiter’s simplified theories which, though questionable, are classical. Another deficiency is the neglection of recent mathematical progress in treating boundary value problems both in the linear and nonlinear case, either by variational methods or by integral equations. As for the repetitive part, the long list of linear constitutive equations for orthotropic materials represent mere an obstacle in the reading the text than a contribution to completeness. On the other hand, the deduction of balance and constitutive equations from general principles of continuum mechanics is done with extreme precision and clarity. Director theories, in particular, are conceptually easy to be understood, but prohibitively hard to be manipulated, and here the author is able to offer an elegant exposition of them. In contrast to Naghdi’s article in the Handbuch der Physik [S. Flügge (ed.), Handbuch der Physik. Band VIa/2: Festkörpermechanik II. Bandherausgeber: C. Truesdell. Encyclopedia of physics. Vol. VIa/2: Mechanics of solids II. Editor: C. Truesdell. Berlin-Heidelberg-New York: Springer-Verlag. X (1972; Zbl 0277.73001)], which is labyrinthine and chaotic, the present book is organic and consistent.

As a final remark, the book is not intended for engineers eager to find explicit solutions, but rather for researchers willing to become familiar with Cosserat theories for further promising extensions.

Reviewer: Piero Villaggio (Pisa)

##### MSC:

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |

74A35 | Polar materials |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |