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Plates, laminates and shells. Asymptotic analysis and homogenization. (English) Zbl 0965.74003
Series on Advances in Mathematics for Applied Sciences. 52. Singapore: World Scientific. 768 p. (2000).
Publisher’s description (no review copy received): This book gives a systematic and comprehensive presentation of results concerning effective behavior of elastic and plastic plates with periodic or quasiperiodic structure. One of the chapters covers the hitherto available results concerning the averaging problems in the linear and nonlinear shell models. A unified approach to the problems studied is based on modern variational and asymptotic methods, including the methods of variational inequalities as well as homogenization techniques. Duality arguments are also exploited. A significant part of the book deals with problems important for engineering practice, such as: statical analysis of highly nonhomogeneous plates and shells for which common discretization techniques fail to be efficient, assessing stiffness reduction of cracked $$[0_n^o / 90_m^o]_s$$ laminates, and assessing ultimate loads for perfectly plastic plates and shells composed of repeated segments. When possible, the homogenization formulas are cast in closed form expressions. The formulas presented in this manner are then used in constructing regularized formulations of the fundamental optimization problems for plates and shells, since the regularization concepts are based on introducing the composite regions for which microstructural properties play the role of new design variables.

##### MSC:
 74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids 74K20 Plates 74Qxx Homogenization, determination of effective properties in solid mechanics 74K25 Shells 74Pxx Optimization problems in solid mechanics 74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics