Grabovsky Yury. Composite Materials: Mathematical theory and exact relations

IOP Publishing Ltd, 2016. — 225 p. — ISBN 978-0-7503-1049-9.Since the appearance of the general theory of exact relations in 1998, all micro structure-independent formulas and all links between pairs of composites with the same microstructure could theoretically be obtained in a systematic and largely mechanical way. However, actual derivations involve quite a bit of effort and, in some cases, a massive amount of effort. Over the intervening 18 years the author, sometimes alone, and often with the help of a group of bright undergraduates or a graduate student, has been steadily computing complete lists of exact relations and links for conducting, elastic, piezoelectric, thermoelectric and thermoelastic composites. This book contains a compilation of all the results obtained through all these years of work. The book also includes mathematically rigorous and self contained development of the general theory of exact relations and links, which is based on the theory of homogenization. Traditionally, homogenization theorems are proved separately in each physical context. The novelty of the approach taken in this book is that the development occurs in the general L2 framework, common to all
types of composites. This level of generality has made the theory cleaner, but also raised several interesting questions that were previously obscured by the particulars of each physical context. Therefore, this book can be read on three different levels: as a single source for all exact relations and links for effective properties of composite materials (part 3); as the definitive exposition of the general theory of exact relations (part 2); and as a new streamlined development of a homogenization based mathematical theory of composite materials (part 1).Preface
Introduction
Part I Mathematical theory of composite materials
Material properties and governing equations
Composite materials
Part II General theory of exact relations and links
Exact relations
Links
Computing exact relations and links
Part III Case studies
Introduction
Conductivity with the Hall effect
Elasticity
Piezoelectricity
Thermoelasticity
Three-dimensional thermoelectricity
Part IV Appendices
A E- and J -regularity for conductivity and elasticity
B A polycrystalline L-relation that is not exact
C Multiplication of SO(3) irreps in endomorphism algebras