*(English)*Zbl 0277.73001

This recent volume of one of the great works of mid-twentieth century science aims to present the “state of the art” of the rational mechanics of elastic bodies, supplementing the earlier expositions of the subject in the Encyclopedia. The volume contains five articles: “The linear theory of elasticity” by *Morton E. Gurtin*, “Linear thermoelasticity” by *Donald E. Carlson*, “Existence theorems in elasticity” and “Boundary value problems of elasticity with unilateral constraints” by *Gaetano Fichera*, “The theory of shells and plates” by *Paul M. Naghdi*, and “The theory of rods” by *Stuart S. Antman*. The articles are characterized by the uniformly high quality and thoroughness that the scientific community has come to expect in this series, and while no book of this sort can pretend to completeness, the coverage presented here is in many cases unique, and the gaps are filled by other works.

Gurtin’s article sets forth the linearized theory of elasticity in a mathematically rigorous fashion; the applications of the theory customarily found in texts on the subject are not treated here. The author makes full use of the apparatus of modern mathematics, resulting in an unusually compact and simple presentation of many results. A concomitant and probably inescapable disadvantage is that many theorems are not immediately accessible to the casual reader. A very complete list of symbols together with their places of first occurance helps to mitigate this problem. The article fills a real need for a sound, rigorous, and modern presentation of the linearized theory, and it acknowledges its debt to E. Sternberg, who pioneered this approach to elasticity in the United States. The article concludes with an unusually extensive list of references.

Carlson’s article on linearized thermoelasticity was originally to be written jointly with Gurtin, and although the article does not carry his name, it does share with his treatise a common notation and point of view, which enhances its usefulness. The article begins with a brief exposition of elasticity and thermodynamics from the general point of view, in order that the linearizations may be made precisely. Two chapters then treat equilibrium and dynamic problems within the linearized theory. The treatment exhibits the same degree of rigor and concentration upon theoretical structure as Gurtin’s exposition.

The last two articles on the linearized theory of elasticity, by Fichera, are really treatments of the mathematics which beans upon these areas of mechanics, rather than of the mechanics itself. They are not “watered down” mathematics in any sense, though their scope has been carefully and explicitly limited. The first article is essentially an exposition of the existence theory for strongly elliptic linear systems of partial differential equations. All of the commonly encountered boundary value problems of linearized elasticity can be treated with this apparatus, though some, principally those associated with small deformations superposed upon a finite equilibrium state, may require more general theorems. The author considers both the usual existence theorems for boundary value problems, and also those for the associated propagation and diffusion problems, related to the dynamic problems of the linearized theory. Although the analysis assumes considerable mathematical sophistication on the part of the reader, it is essentially self-contained. Most of the work is valid in an arbitrary number of space dimensions, thus providing results for both plane and three-dimensional elasticity. In a second briefer article, Fichera treats problems involving unilateral constraints, for example the Signorini problem of the deformation of an elastic body supported on a rigid frictionless surface. Though conceptually simple, these problems pose some formidable mathematical difficulties, and Fichera succinctly summarizes the present state of mathematical knowledge concerning them.

The remaining two articles differ from the first four in that they consider bodies of special shape, i.e. rods, plates, and shells, undergoing deformations not necessarily infinitesimal, P. M. Naghdi’s account of the theory of plates and shells considers two separate approaches to the theory. The first, termed by him “direct”, seeks to build a phenomenological model of a shell by considering it as a two-dimensional direct continuum, a Cosserat surface. As could be expected from the author’s considerable accomplishments in this area, the account is usually deep and thorough. The second approach, seeking a physical basis at the expense of some mathematical rigor, treats the shell as a limiting case of a three-dimensional elastic body. Following preliminary kinematic results, Naghdi derives basic field equations for shells using each method. The following chapter discusses elastic and thermoelastic shells from both viewpoints, and the final chapter sets out the linearized theory. The author also briefly discusses the history of the development of field and constitutive equations for shells, and explores the relationship of the results obtained by the two approaches.

The final article by Stuart S. Antman begins by developing the theory of rods from viewpoints analogous to those in Naghdi’s article. The article concludes with existence theorems for planar problems of Cosserat rods, due pricipally to the author. The theorems are particualrly interesting in that they are global theorems, requiring no linearizations. The results are suggestive of what it may be possible to obtain in the analogous problems for two- and three -dimensional bodies, which are still largely unsolved

##### MSC:

74-02 | Research monographs (mechanics of deformable solids) |

74Bxx | Elastic materials |

74A35 | Polar materials |

74Jxx | Waves (solid mechanics) |

74K20 | Plates (solid mechanics) |

74K25 | Shells (solid mechanics) |

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

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

74K15 | Membranes (solid mechanics) |