Plates, Laminates And Shells: Asymptotic Analysis And Homogenization: (52 Series on Advances in Mathematics for Applied Sciences)

This book gives a systematic and comprehensive presentation of the 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 [00n/900m]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.

Table of Contents:
Mathematical preliminaries: function spaces, convex analysis, variational convergence. Elastic plates: three-dimensional analysis and effective models of composite plates; thin plates in bending and stretching; non-linear behaviour of plates; moderately thick transversely symmetric plates; sandwich plates with soft core. Elastic plates with cracks: unilateral cracks in thin plates; unilateral cracks in plates with transverse shear deformation; part-through the thickness cracks; stiffness loss of cracked laminates; comments and bibliographical notes. Elastic-perfectly plastic plates: mathematical complements homogenisation of functional with linear growth; homogenisation of plates loaded by forces and moments; comments and bibliographical notes. Elastic and plastic shells: linear and non-linear models of elastic shells; homogenisation and stiffnesses of thin periodic elastic shells; linear approach; homogenised properties of thin periodic elastic shells undergoing moderately large relations around tangents; perfectly plastic shells. Application of homogenisation methods in optimum design of plates and shells: mathematical complements; two-phase plate in bending, Hashin-Shtrikamn bounds; two-phase plate Hashin-Shtrikman bounds for the in-plane problem; explicit formulae for effective bending stiffnesses and compliances of ribbed plates; explicit formulae for effective membrane stiffnesses and compliance s of ribbed plates; thin bending two-phase plates of minimum compliance; minimum compliance problem for thin plates of varying thickness -application of young measures; thin shells of minimum compliances; Truss-like Michell continua.