Mathematical modeling in mechanics of granular materials. With a foreword by Holm, Altenbach (Q415941)

This book, written by two authors from Russia, contains ten chapters. The first one is an introduction, very short. The second chapter, presenting the rheological schemes, is necessary to understand the book. It is mainly devoted to cohesive granular materials. Chapter three describes the mathematical apparatus (convex sets and convex functions and their properties, discrete variational inequalities, and subdifferential calculus). Chapter four examines the spatial constitutive relationships, introducing the Coulomb-Mohr and von Mises-Schleicher cones. The limiting equilibrium of a material with load-dependent strength properties is the topic of chapter five, including the static and kinematic theorems, computational algorithm, and examples of plane strain state. Chapter six, devoted to elastic-plastic waves in a loosened material, contains some examples, difficult to understand. The name of Rakhmatulin is not mentioned, though he was one of the first in the world who studied the elastic-plastic waves (see the Russian literature in \textit{H. A. Rakhmatulin} et al. [Two-dimensional problems of non-stationary motion of compressible media. (Russian), Taskent: Verlag 'Fan' (1969; Zbl 0185.53003)]). Also many other Russian workers are not mentioned (see e. g. the names mentioned in \textit{N. D. Cristescu} [Dynamic plasticity. Hackensack, NJ: World Scientific (2007; Zbl 1120.74001)]). The theoretical examples are not compared with experiments. Chapter seven introduces contact interaction of layers. Again, some examples are given, but not compared with experiments. Chapter eight, devoted to results of high-performance computing, presents several examples of two-dimensional and three-dimensional computations, again without comparisons with tests. In Chapter nine, the authors discuss the finite strains of a granular material, the dilatancy effect, viscous materials with rigid particles, the Couette flow, motion on inclined plane, plane-parallel motion, etc. The last chapter describes the rotational degrees of freedom of particles. It starts with Cosserat continuum in elasticity, and gives some examples. The Lamb problem is given in detail, together with finite strains of a medium with rotating particles. The book is rather a mathematical theory of the corresponding problems, without connections with real experiments. The literature is poor, including the Russian.