The mathematical method of composites has reached a very high level of maturity and developments have increased our understanding of the relationship between the microstructure of composites and their macroscopic behaviour.

This book provides a self-contained unified approach to the mathematical foundation of the theory of composites, leading to the general theory of exact relations. It also provides complete lists of exact relations in many specific physically relevant contexts, such as conductivity, fibre-reinforced elasticity, piezoelectricity, thermoelectricity and more.

The mathematical method of composites has reached a very high level of maturity and developments have increased our understanding of the relationship between the microstructure of composites and their macroscopic behaviour.

This book provides a self-contained unified approach to the mathematical foundation of the theory of composites, leading to the general theory of exact relations. It also provides complete lists of exact relations in many specific physically relevant contexts, such as conductivity, fibre-reinforced elasticity, piezoelectricity, thermoelectricity and more.

Divided into three parts, the book starts with a development of the mathematical theory of composites before progressing to the general theory of exact relationships and concludes with complete lists of exact relations. Written in a two-layer structure, each chapter starts with a non-technical review that is supported by deeper rigorous mathematical proof, this book will be key reading for graduate students and researchers involved with understanding and modelling composite materials. The extensive collection of explicit exact relations will also make it a useful reference for engineers to want to use composites to create materials with specific properties.

I Mathematical theory of composite materials
1 Material properties and governing equations
2 Mathematical definition of a composite
3 H-limits and G-closures
4 G-closed sets and -quasiconvexity
5 Lamination closure and -convexity

II Exact relations and links
6 Exact relations
7 Links
8 Exact relations with volume fraction information
9 Computing exact relations and links
10 Polycrystalline exact relations

III Case studies
11 2D conductivity with Hall effect
12 3D conductivity with Hall effect
13 Fiber-reinforced conducting composites with Hall effect
14 2D thermoelectricity
15 3D thermoelectricity
16 2D elasticity
17 3D elasticity
18 Fiber-reinforced elastic composites
19 2D Piezoelectricity
20 3D Piezoelectricity
21 Arbitrary number of coupled electric fields

Appendices
A Jordan multialgebras
B Ideals and factor-algebras
C Characterization of all subalgebras of Sym(Rn)
D Lamination exact relations that are not closed under homogenization

The mathematical method of composites has reached a very high level of maturity and developments have increased our understanding of the relationship between the microstructure of composites and their macroscopic behaviour.

This book provides a self-contained unified approach to the mathematical foundation of the theory of composites, leading to the general theory of exact relations. It also provides complete lists of exact relations in many specific physically relevant contexts, such as conductivity, fibre-reinforced elasticity, piezoelectricity, thermoelectricity and more.

Divided into three parts, the book starts with a development of the mathematical theory of composites before progressing to the general theory of exact relationships and concludes with complete lists of exact relations. Written in a two-layer structure, each chapter starts with a non-technical review that is supported by deeper rigorous mathematical proof, this book will be key reading for graduate students and researchers involved with understanding and modelling composite materials. The extensive collection of explicit exact relations will also make it a useful reference for engineers to want to use composites to create materials with specific properties.