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Mathematical elasticity. Vol. 2: Theory of plates. (English) Zbl 0888.73001
Amsterdam: Elsevier. lxi, 497 p. (1997).
As volume II in a series on Mathematical elasticity ([for Vol. I see the author, Mathematical elasticity. Vol. I: Three-dimensional elasticity. Studies in Mathematics and its Applications 20, Amsterdam etc.: North-Holland (1988; Zbl 0648.73014)], and Vol. III: Theory of shells), this book treats the geometrically linear and nonlinear theories of clamped plates and shallow shells. The preface comprises a compilation of main notations and definitions followed by the plate and shallow shell equations at a glance. Each chapter begins with a short introduction sketching the procedure followed in the text and ends with useful exercises for strengthening the acquired knowledge. Throughout the book homogeneous isotropy and linear elasticity are assumed. Starting out with linear theories, a scaled formulation allows to establish convergence of the three-dimensional displacements to the solution of two-dimensional Kirchhoff-Love theory when reducing the plate thickness. This includes the clamped boundaries. For an asymptotic analysis of plates inserted in a three-dimensional elastic body, both parts are scaled independently, and junction conditions are satisfied after passing to the limit. Since the equations for shallow shells specified in Cartesian coordinates closely follow those of the plate theory, they are included in this book. Besides a rigorous definition of shallowness, the asymptotic analysis needs the proof of a generalized Korn’s inequality. After appropriate scaling, a formal asymptotic expansion reveals that – also for geometric nonlinearity – the classical Kirchhoff-Love theory evolves as the two-dimensional limit. The same procedure is successfully applied to justify the von Kármán equations for nonlinear plates and the Marguerre-von Kármán equations for shallow shells.

##### MSC:
 74-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids 74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids 74A99 Generalities, axiomatics, foundations of continuum mechanics of solids 74K15 Membranes 74K20 Plates