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Analysis and approximation of contact problems with adhesion or damage. (English) Zbl 1089.74004

Pure and Applied Mathematics (Boca Raton) 276. Boca Raton, FL: Chapman & Hall/ CRC (ISBN 1-58488-585-8/hbk; 978-1-4200-3483-7/ebook). xviii, 220 p. (2006).
The important and useful monograph is devoted to quasistatic and dynamic contact problems within the infinitesimal strain theory. The contact model (for viscoelastic or viscoplastic materials) is described with Signorini or normal compliance conditions and with the associated frictionless or frictional conditions. The dynamic contact between a viscoelastic or viscoplastic body and an obstacle is studied. The contact is with adhesion, the evolution of which is described by an ordinary differential equation. The adhesion of the contacting surfaces is modelled by a surface variable – the bonding field, whose evolution is described by a nonlinear differential equation. Furthermore, the contact problems with material damage for viscoelastic and viscoplastic materials are studied precisely.
The authors focus on the study of fully discrete schemes where both the spatial and temporal variables are discretized. For each numerical scheme the existence and uniqueness of its solution are proved, and optimal order error estimates are derived under certain regularity assumption on the solution of the continuous problem. The authors give a rigorous analysis of finite element approximations for a class of variational inequalities of elliptic and evolution type. Finite element models are described, and their convergence properties are established. The book includes also a comprehensive treatment of several contact models with material damage. Here links between different contact models are explored, and it is rigorously shown that the Signorini non-penetration condition is a limiting case of a normal compliance contact condition in two of the models.
Basic knowledge of finite element mathematics, elliptic and evolutionary variational inequalities, convex analysis, nonlinear equations with monotone operators or fixed points of operators is needed. The book is self-contained. All necessary results from these disciplines are summarized in the introductory chapter. The work is intended for a wide audience: this would include specialists in contact processes in structural and mechanical systems who wish to know more about the mathematical theory, as well as those with a background in mathematical sciences who seek a self-contained account of the mathematical theory of contact mechanics. The text is suitable for graduate students and researchers in applied mathematics, computational mathematics, and computational mechanics.

MSC:

74-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids
74M15 Contact in solid mechanics
74M10 Friction in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
49J40 Variational inequalities

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