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Flow, Deformation and Fracture

G. I. Barenblatt
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Texts in Applied Mathematics
[Reviewed by
Anna Zemlyanova
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The book is dedicated to the exposition of the fundamental ideas of continuum mechanics. It is based on the course on mechanics of continua which has been taught by the author at several institutions of higher education including Berkeley, Stanford, Cambridge, and Moscow State University.

The book is divided into twelve chapters. The topics include the concept of continuous media, dimensional analysis and physical similitude, ideal incompressible fluid approximation, linear elastic solids, brittle and quasi-brittle fracture, Newtonian viscous fluid approximation, advanced similarity methods, ideal gas approximation and turbulent flows. For each of the physical models, the fundamental equations are derived and the degree of applicability of the model is discussed. The classical continuum models are illustrated with examples and historical notes.

The concept of intermediate asymptotics is introduced early on and used widely throughout the book. The author pays special attention to the dimensional analysis and similarity methods. The main ideas behind the use of dimensional analysis are presented and illustrated through a large number of practical examples. Physical similitude and its applications to modeling of physical phenomena is discussed. The author returns to dimensional analysis in chapter nine, where advanced similarity methods are introduced. The limitations of dimensional analysis are shown via the examples of scaling laws which cannot be obtained by means of dimensional analysis. Complete and incomplete similarity of physical phenomena is discussed. Incomplete similarity in fatigue experiments (Paris’s law) and scaling laws in nanomechanics are considered.

The book pays considerable attention to the brittle and quasi-brittle fracture problems in linear elasticity. The conditions of stress finiteness and continuity and smooth closing of the crack contours in a mobile equilibrium state of a crack are postulated. The cohesion crack model pioneered by the author is presented in the book as well. The phenomenon of crack fatigue and Paris’s law is discussed.

Turbulence is a subject of special attention in the book. Scaling laws for wall-bounded shear flows at very large Reynolds numbers are studied. Reynolds-number-dependent power law for velocity distribution which has been developed in a series of works by A. J. Chorin, V. M. Prostokishin and the author is discussed and compared to the universally accepted von Karman-Prandtl logarithmic law. The comparison is made on a large number of experimental data. It is shown in the book that the power law for the velocity distribution is in better agreement with experimental results.

The book touches upon many fundamental continuum models and provides the reader with a unified bird’s eye view of the field. It will make a great supplemental text for anybody interested in continuum mechanics. I am happy to recommend it to students and researchers in applied mathematics at various levels.

Anna Zemlyanova is an Assistant Professor at the Department of Mathematics, Kansas State University, Manhattan, KS, USA.

Foreword Alexandre J. Chorin
1. Idealized continuous media: the basic concepts
2. Dimensional analysis and physical similitude
3. The ideal incompressible fluid approximation. General concepts and relations
4. The ideal incompressible fluid approximation. Analysis and applications
5. The approximation of a linear elastic solid. Basic equations and boundary value problems of linear theory of elasticity
6. Approximation of a linear elastic body. Applications: brittle and quasi-brittle fracture, strength of structures
7. The approximation of Newtonian viscous fluids: general comments and basic relations
8. Approximation of a Newtonian viscous fluid: the boundary layer
9. Advanced similarity methods: complete and incomplete similarity
10. The ideal gas approximation. Sound waves. Shock waves
11. Turbulence. Generalities. Scaling laws for shear flows
12. Turbulence. Mathematical models of turbulent shear flows and of the local structure of turbulent flows at very large Reynolds numbers