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This book is devoted to describing theories for porous media where such pores have an inbuilt macro structure and a micro structure. For example, a double porosity material has pores on a macro scale, but additionally there are cracks or fissures in the solid skeleton. The actual body is allowed to deform and thus the underlying theory is one of elasticity. Various different descriptions are reviewed.

Chapter 1 introduces the classical linear theory of elastodynamics together with uniqueness and continuous dependence results. Chapters 2 and 3 review developments of theories for double and triple porosity using a pressure-displacement structure and also using voids-displacement. Chapter 4 compares various aspects of the pressure-displacement and voids-displacement theories via uniqueness studies and wave motion analysis. Mathematical analyses of double and triple porosity materials are included concentrating on uniqueness and stability studies in chapters 5 to 7. In chapters 8 and 9 the emphasis is on wave motion in double porosity materials with special attention paid to nonlinear waves. The final chapter embraces a novel area where an elastic body with a double porosity structure is analyzed, but the thermodynamics allows for heat to travel as a wave rather than simply by diffusion.

This book will be of value to mathematicians, theoretical engineers and other practitioners who are interested in double or triple porosity elasticity and its relevance to many diverse applications.

### Chapter 1. Introduction

This book is dedicated to describing theories and analysis for the class of elastic materials which display a multiple porosity structure. If one has an elastic body which has pores (holes) in it then one begins with the macro porosity to which there is an associated pressure in the pores.
Brian Straughan

### Chapter 2. Models for Double and Triple Porosity

In this chapter we review several of the mathematical models which have appeared in the literature and are capable of describing the evolutionary behaviour of a double or triple porosity elastic medium, in some appropriate sense.
Brian Straughan

### Chapter 3. Double Porosity and Voids

Our aim in this chapter is to describe a theory of nonlinear thermoelasticity where the body contains a double porosity system of voids. There is a void distribution associated to the macro pores, but additionally there is another void distribution due to the micro pores. This work was developed by Iesan & Quintanilla [? ].
Brian Straughan

### Chapter 4. Comparison of Porosity and Voids Theories

In chapter 2 we discussed several different models for the description of a double porosity elastic medium. Chapter 3 also derives another model for an elastic body containing two levels of porosity, a theory based on two voids distributions. In this chapter we shall compare the Svanadze model of chapter 3, see section 2.3, to that of Iesan & Quintanilla [? ] as discussed in section 3.4.1. The comparison we make is based on the equations of a linearized theory and we use two methods to compare the solution behaviour. We firstly study the movement of an acceleration wave. The second method is to study how uniqueness follows from a logarithmic convexity argument when the elastic coefficients are not required to be sign definite.
Brian Straughan

### Chapter 5. Uniqueness and Stability by Energy Methods

In this chapter we investigate questions of uniqueness, continuous dependence on the initial data, and continuous dependence on the model itself, by employing techniques based on the energy function generated by the system of equations. We concentrate on equations of elasticity with a double porosity structure, or thermoelasticity with a double porosity structure. We firstly deal with a thermoelastic body with a double porosity structure allowing for cross inertia coefficients, equations (2.33), when the body is bounded. After that we examine the equations for isothermal elasticity with double porosity, equations (2.25), (2.26) and (2.27), when the domainΩ is unbounded.
Brian Straughan

### Chapter 6. Uniqueness Without Definiteness Conditions

In this chapter the elastic coefficients ai jkh have the condition of positive definiteness or even positivity relaxed and all we require is that they satisfy the symmetry conditions (1.16). We have already given cogent reasons in the first paragraph of section 1.4.4 why it is important to consider the situation when only symmetry is stipulated and no definiteness is demanded.
Brian Straughan

### Chapter 7. Continuous Dependence in Multi - Porosity Elasticity

We commence this chapter with an analysis of continuous dependence upon the initial data for the isothermal triple porosity system (2.44) and we require only the symmetry conditions (6.1) for the elastic coefficients ai jkh.
Brian Straughan

### Chapter 8. Waves in Double Porosity Elasticity

Rayleigh waves and Love waves have long been used as an important tool in geophysical problems employing linear elasticity theory. Rayleigh waves are studied using the Berryman & Wang [? ] double porosity theory presented in section 2.2, by Dai et al. [? ]. Dai et al.
Brian Straughan

### Chapter 9. Acceleration Waves in Double Voids

In this chapter we develop a fully nonlinear acceleration wave analysis for the double voids elasticity theory of Iesan & Quintanilla [115]. We commence with the isothermal case and so the starting point is equations (3.55), (3.56) and (3.57).
Brian Straughan

### Chapter 10. Double Porosity and Second Sound

Up to this point in this book we have considered thermoelastic bodies with a multiple porosity structure assuming that the heat flux is governed by a Fourier law, i.e. the heat flux is determined directly by the temperature gradient. Materials technology is advancing very rapidly and it is likely that for many future applications a Fourier law may be inadequate. The phenomenon of temperature travelling as a wave is studied in detail in the book by Straughan [? ] and he outlines many examples of classical elasticity where the temperature field has wave like behaviour.
Brian Straughan