Material inhomogeneities and their evolution. A geometric approch.

*(English)*Zbl 1130.74001
Interaction of Mechanics and Mathematics. Berlin: Springer (ISBN 978-3-540-72372-1/pbk). xiii, 274 p. (2007).

This book represents an important passage in the general construction of the theory outlined by W. Noll [Arch. Ration. Mech. Anal. 27, 1–32 (1967); Errata. ibid. 31, 401 (1968; Zbl 0168.45701)] and further extended by C. C. Wang [ibid. 27, 33–94 (1967; Zbl 0187.48802)]. Noll and Wang have shown that in a uniform body a material connection can be defined consistent with a given response functional of the body. This material connection becomes the geometrical tool for the analysis of different types of inhomogeneities. The objective of the present book is to present a point of view that emphasizes the differential-geometric aspect of the inhomogeneity theory.

By following the presentation in the preface, the book is divided in three parts, the first of which is devoted to the formulation of the theory in the absence of evolution. It opens with the geometric characterization of material inhomogeneity within the context of simple bodies in Chapter 1, followed by extensions to second-grade and Cosserat media in Chapters 2 and 3. Chapter 4 deals with a novel generalization of the notions of material uniformity and homogeneity to functionally-graded media. Throughout these chapters, the required fundamental differential-geometric constructs are introduced and discussed in a rather intuitive fashion, as much as the inherent difficulty of the topic permits. Thus, concepts such as principal frame bundle, \(G\)-structures and groupoids are motivated within the continuum mechanics context without paying exaggerated attention to mathematical rigour.

The second part of the book deals with the framework underlying various phenomena of material evolution in time, such as anelasticity, growth and remodelling. The role of material forces, in particular the Eshelby and Mandel stresses, is highlighted. In fact, most of Chapter 5 deals with it, and Chapter 6 revolves around the formulation of general principles of the theory of material evolution of simple bodies. Possible applications to material growth and remodelling are discussed. Chapter 7 formulates the theory of material evolution in the realm of second-grade bodies.

The final part of the book is a rigorous compendium of modern differential geometry as necessitated by the mechanical theory presented in the preceding parts. This book is highly recommended to the workers on modern continuum mechanics.

By following the presentation in the preface, the book is divided in three parts, the first of which is devoted to the formulation of the theory in the absence of evolution. It opens with the geometric characterization of material inhomogeneity within the context of simple bodies in Chapter 1, followed by extensions to second-grade and Cosserat media in Chapters 2 and 3. Chapter 4 deals with a novel generalization of the notions of material uniformity and homogeneity to functionally-graded media. Throughout these chapters, the required fundamental differential-geometric constructs are introduced and discussed in a rather intuitive fashion, as much as the inherent difficulty of the topic permits. Thus, concepts such as principal frame bundle, \(G\)-structures and groupoids are motivated within the continuum mechanics context without paying exaggerated attention to mathematical rigour.

The second part of the book deals with the framework underlying various phenomena of material evolution in time, such as anelasticity, growth and remodelling. The role of material forces, in particular the Eshelby and Mandel stresses, is highlighted. In fact, most of Chapter 5 deals with it, and Chapter 6 revolves around the formulation of general principles of the theory of material evolution of simple bodies. Possible applications to material growth and remodelling are discussed. Chapter 7 formulates the theory of material evolution in the realm of second-grade bodies.

The final part of the book is a rigorous compendium of modern differential geometry as necessitated by the mechanical theory presented in the preceding parts. This book is highly recommended to the workers on modern continuum mechanics.

Reviewer: Franco Cardin (Padova)