North-Holland Series in Applied Mathematics and Mechanics – Serie

In this second edition several new topics of technological interest have been added. These include: coupled mechanical and nonmechanical overall properties of heterogeneous piezoelectric materials, new upper and lower bounds for these coupled properties, a systematic comparison between the average-field theory and the results obtained using multi-scale perturbation theory, an account of the uniform-field theory, improveable bounds on overall moduli of heterogeneous materials which remain finite even when isolated cavities and rigid inclusions are present, and a brief account of a fundamental duality principle in anisotropic elasticity. In addition, better explanations of a number of topics are given, more recent references are added, the Subject Index has been expanded and printing and typographical errors have been corrected.The material is organized into two parts preceded by a précis. Part 1 consists of four chapters which are organized into fourteen sections and four appendixes. It deals with materials with microdefects such as cavities, cracks, and inclusions, as well as with elastic composites. Part 2 consists of two chapters which are divided into seven sections. It provides an introduction to the theory of linear elasticity, added to make the book self-contained, since linear elasticity serves as the basis of the development of small-deformation micromechanics.Part 2 mainly contains part of the lecture notes on elasticity which the first author wrote in the late 1960's. The material is mostly standard, given for background information.

Most of the natural and biological phenomena such as solute transport in porous media exhibit variability which can not be modeled by using deterministic approaches. There is evidence in natural phenomena to suggest that some of the observations can not be explained by using the models which give deterministic solutions. Stochastic processes have a rich repository of objects which can be used to express the randomness inherent in the system and the evolution of the system over time. The attractiveness of the stochastic differential equations (SDE) and stochastic partial differential equations (SPDE) come from the fact that we can integrate the variability of the system along with the scientific knowledge pertaining to the system. One of the aims of this book is to explaim some useufl concepts in stochastic dynamics so that the scientists and engineers with a background in undergraduate differential calculus could appreciate the applicability and appropriateness of these developments in mathematics. The ideas are explained in an intuitive manner wherever possible with out compromising rigor.The solute transport problem in porous media saturated with water had been used as a natural setting to discuss the approaches based on stochastic dynamics. The work is also motivated by the need to have more sophisticated mathematical and computational frameworks to model the variability one encounters in natural and industrial systems. This book presents the ideas, models and computational solutions pertaining to a single problem: stochastic flow of contaminant transport in the saturated porous media such as that we find in underground aquifers. In attempting to solve this problem using stochastic concepts, different ideas and new concepts have been explored, and mathematical and computational frameworks have been developed in the process. Some of these concepts, arguments and mathematical and computational constructs are discussed in an intuititve manner in this book.

This book deals with the impact of uncertainty in input data on the outputs of mathematical models. Uncertain inputs as scalars, tensors, functions, or domain boundaries are considered. In practical terms, material parameters or constitutive laws, for instance, are uncertain, and quantities as local temperature, local mechanical stress, or local displacement are monitored. The goal of the worst scenario method is to extremize the quantity over the set of uncertain input data.A general mathematical scheme of the worst scenario method, including approximation by finite element methods, is presented, and then applied to various state problems modeled by differential equations or variational inequalities: nonlinear heat flow, Timoshenko beam vibration and buckling, plate buckling, contact problems in elasticity and thermoelasticity with and without friction, and various models of plastic deformation, to list some of the topics. Dozens of examples, figures, and tables are included.Although the book concentrates on the mathematical aspects of the subject, a substantial part is written in an accessible style and is devoted to various facets of uncertainty in modeling and to the state of the art techniques proposed to deal with uncertain input data.A chapter on sensitivity analysis and on functional and convex analysis is included for the reader's convenience.�Rigorous theory is established for the treatment of uncertainty in modeling� Uncertainty is considered in complex models based on partial differential equations or variational inequalities � Applications to nonlinear and linear problems with uncertain data are presented in detail: quasilinear steady heat flow, buckling of beams and plates, vibration of beams, frictional contact of bodies, several models of plastic deformation, and more �Although emphasis is put on theoretical analysis and approximation techniques, numerical examples are also present�Main ideas and approaches used today to handle uncertainties in modeling are described in an accessible form�Fairly self-contained book