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Inequality problems in mechanics and applications. Convex and nonconvex energy functions. (English) Zbl 0579.73014
Boston - Basel - Stuttgart: Birkhäuser. XIX, 412 p. DM 168.00 (1985).
The theory of variational inequalities has started in the early sixties with the seminal works of G. Fichera, J. L. Lions and G. Stampacchia. These authors were mainly concerned with the mathematical aspects of the theory. However, the deep connection of this new theory with a wide variety of problems in applied mechanics and optimization theory became rapidly clear. As the author of this book rightly points out, all the inequality problems studied in the applications were related to convex ”energy” functionals. Since the gradient of a convex functional is a monotone mapping the inequality problems treated were mainly tied to monotonicity e.g. only monotone stress-strain relations could be treated. Making use of two new notions (the generalized gradient of F. H. Clarke and the derivative container of J. Warga) the author tries to overcome the limitations of convexity. In the case of non convex energy functionals he obtains variational expressions which he calls hemivariational inequalities.
The book is quite selfcontained and the first two chapters deal with the necessary mathematical background of functional analysis. Of particular interest is the part devoted to the theory of von Kármán plates and thermoelasticity. The last two chapters are more engineering oriented and deal with the numerical treatment of inequality problems.
Reviewer: G.Cimatti

74S30 Other numerical methods in solid mechanics (MSC2010)
49J40 Variational inequalities
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74K20 Plates
74F05 Thermal effects in solid mechanics