Thin-Walled Structures with Initial Stresses: Multiscale Analysis of Classical and Non-Classical Problems(Mathematical Engineering)

This book is based on two principles: - the problem for an elastic body with initial stresses is similar to the elasticity theory problem, but the coefficients of the equations relating additional stresses and additional strains do not have all the symmetries incident to the elastic constants. - since the problems of thin-walled structures are asymptotic in nature, asymptotic methods (including homogenization theory if the structure is non-uniform) should be used to analyze the problems. Following these principles, one can derive the theory of inhomogeneous/composite structural elements directly from the elasticity theory without to use any additional/simplified hypothesis. This book treats both homogeneous and non-uniform plates and beams. The chapters on uniform plates and beams may be considered as an alternative method deriving the classical stability equations. The main part of the book is devoted to non-uniform plates and beams. When performing the analysis of the problems, three possible ranges of initial stresses are distinguished, which correspond to: - in-plane forces in a plate and axial force in a beam (this is the case of Euler and Bryan forces); - moments (it happens that the moments of initial stresses influence beams and do not influence plates); - stresses of the order of elastic constants (which arise, for example, as a result of thermal heating), which affect elastic constants of plates and beams. Local stresses are recorded for all the considered cases. It makes possible to apply the results presented in the book both to the problem of stability and the strength analysis of the structural elements with initial tresses.

Table of Contents:
Introduction.- Thin-walled structural elements operating under initial stresses.- Homogenization method in the linear elasticity. Basic approaches.- Homogenization in linear elasticity theory problems with initial stresses.- Uniform plates with initial stresses.- Non-homogeneous plates with initial stresses.- Initial stresses comparable to the elastic constants of the material of the plate.- Uniform beams with initial stresses.- Non-homogeneous beams with initial stresses.- Initial stresses comparable with the elastic constants of the material of the beam.- Comparison of the classical hypothesis and terms of the asymptotic expansions.

About the Author :
Dr Kolpakov worked as professor and independent researchers in Russia, EU, and USA, he affiliated with both universities and research/industrial units. Their researchers are devoted to a wide range of problems from pure mathematics to mechanics of solids and numerical computations. The research papers of Dr Kolpakov were published in the referred internationally recognized journals in mathematics and mechanics.