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**Shell theory (Transl. from Danish lecture notes, with changes and additions).**
*(English)*
Zbl 0566.73053

North-Holland Series in Applied Mathematics and Mechanics, 29. Amsterdam-New York-Oxford: North-Holland. XIV, 408 p. $ 49.00; Dfl. 160.00 (1985).

This book gives an introductory course on modern shell theory on a graduate level. It follows from a set of lecture notes, given by the author to students of mechanical engineering at the Technical University of Denmark. The shell theory presented in this book is based on three- dimensional continuum theory. The complete nonlinear equations are derived.

The first two chapters contain a short introduction to tensor analysis and the geometry of the middle-surface. Chapter 3 deals with the deformations of the middle-surface as displacements, stretching, rotation and bending. Finally the equations of compatibility are given.

In the next three chapters the dimensional reduction is achieved by using two-dimensional stresses and moments, reducing the principle of virtual work to two dimensions and introducing well-defined approximations for the strain-energy. Chapter 8 shows the static-geometric analogy which leads to the useful concept of stress-functions.

In the next six chapters the theory is applied to plates and shells. Different approximations as membrane theory and special geometries as circular cylindrical, spherical shell and shells of revolution are discussed. Morley-Koiter and Reissner-Meissner-equations are derived and solved. Specific results for conical and toroidal shells are given. The approximation of Geckeler is analysed. In one chapter the special theory for shallow shells, connected with the names Donnell-Mushtari-Vlasov is given. This theory is then extended to large deflections of plates including large amplitude vibrations. The book closes with a short introduction in stability problems.

This book is of great importance as well for students as for engineers interested in shell problems. The reader needs only the basic knowledges in mathematics and mechanics which he should have at the beginning of post-graduate studies. All basic equations are derived detailed, the necessary approximations are explained carefully, different possible solutions are discussed. A lot of references are given. The outfit of the book is excellent.

The first two chapters contain a short introduction to tensor analysis and the geometry of the middle-surface. Chapter 3 deals with the deformations of the middle-surface as displacements, stretching, rotation and bending. Finally the equations of compatibility are given.

In the next three chapters the dimensional reduction is achieved by using two-dimensional stresses and moments, reducing the principle of virtual work to two dimensions and introducing well-defined approximations for the strain-energy. Chapter 8 shows the static-geometric analogy which leads to the useful concept of stress-functions.

In the next six chapters the theory is applied to plates and shells. Different approximations as membrane theory and special geometries as circular cylindrical, spherical shell and shells of revolution are discussed. Morley-Koiter and Reissner-Meissner-equations are derived and solved. Specific results for conical and toroidal shells are given. The approximation of Geckeler is analysed. In one chapter the special theory for shallow shells, connected with the names Donnell-Mushtari-Vlasov is given. This theory is then extended to large deflections of plates including large amplitude vibrations. The book closes with a short introduction in stability problems.

This book is of great importance as well for students as for engineers interested in shell problems. The reader needs only the basic knowledges in mathematics and mechanics which he should have at the beginning of post-graduate studies. All basic equations are derived detailed, the necessary approximations are explained carefully, different possible solutions are discussed. A lot of references are given. The outfit of the book is excellent.

Reviewer: W.Schnell

### MSC:

74K15 | Membranes |

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

74K25 | Shells |

74B10 | Linear elasticity with initial stresses |

74K20 | Plates |