Mathematical aspects of multi-porosity continua
This book contains a review and new elements concerning the porous media with double and triple porosity. The solid skeleton of such media contains craks or fissures and is allowed to deform -- then it can be described by the elasticity theory. Important uniqueness results are obtained by using the energy method and the convex functions properties -- see section 1.3.6. The book contains 10 chapters and a large list of classical and very recent references. The basic elements of linear elasticity and some mathematical methods, used later, are detailed in Chapter 1. Multi-porosity and basic important equations are described in Chapter 2. The void distribution approach is studied in chapter 3, compared with the double porosity and double voids theories given in Chapter 4. The Svanadze and Iesan-Quintanilla models are compared -- see ref. [\textit{D. Iesan} and \textit{R. Quintanilla}, ``On a theory of thermoelastic materials with a double porosity structure'', J. Thermal Stresses 37, 1017--1036 (2014)], [\textit{M. Svanadze}, Acta Appl. Math. 122, No. 1, 461--471 (2012; Zbl 1254.74047); ``Dynamical problems of the theory of elasticity for solids with double porosity'', Proc. Appl. Math. Mech. 10, 309--310 (2010)]. The uniqueness rsults follows from a logarthmic convexity arguments, which hold also for unbunded porous media and for a large number of multi-porous domains -- see Chapter 6. The continuous dependence in terms of the initial data is studied in Chapter 7. The wave motion in double-porosity elastic media is described in Chapters 8 and 9. New elements concerning the effects of temperature waves are given in Chapter 10, where the acceleration waves for double porosity and double voids are studied by using the Cataneo theory. Although some of the first figures describe periodic cracked (or fissured) porous media, the book does not contain references to the homogenization method, which has managed to give some important results in the field. Each chapter contains a large number of exercises, very useful for students and researchers. The book is very useful to those interested in porous media and new materials technology.
scientific article; zbMATH DE number 6808284