##
**Solution of variational inequalities in mechanics.**
*(English)*
Zbl 0654.73019

Applied Mathematical Sciences, 66. New York etc.: Springer-Verlag. x, 275 p. DM 84.00 (1988).

This book deals with variational inequalities as they arise in mechanics. It may be considered as an account of the authors’ work over the last years in this area. As an introductory topic, the formulation, existence and uniqueness results, as well as the approximation by finite elements for unilateral (obstacle) problems for scalar functions in the plane are treated.

Further topics worked out in the same manner are contact of elastic bodies with and without friction and plasticity for the two-dimensional situation. Whereas the mechanical background is often only touched on and for the mathematical preliminaries from functional and convex analysis reference is made to the literature - thus keeping the book at a modest size, the theory of the triangular finite element approximation for these problems is worked out in full detail. It includes primal, dual and mixed methods, convergence results and error estimates, covering also the main numerical algorithms for these type of problems.

This will provide a thorough introduction for anyone wanting to study the approximation of variational inequalities as well as a useful reference, containing also an interesting and novel approach to the contact problem with Coulomb friction

Further topics worked out in the same manner are contact of elastic bodies with and without friction and plasticity for the two-dimensional situation. Whereas the mechanical background is often only touched on and for the mathematical preliminaries from functional and convex analysis reference is made to the literature - thus keeping the book at a modest size, the theory of the triangular finite element approximation for these problems is worked out in full detail. It includes primal, dual and mixed methods, convergence results and error estimates, covering also the main numerical algorithms for these type of problems.

This will provide a thorough introduction for anyone wanting to study the approximation of variational inequalities as well as a useful reference, containing also an interesting and novel approach to the contact problem with Coulomb friction

Reviewer: H.Matthies

### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74S05 | Finite element methods applied to problems in solid mechanics |

74A55 | Theories of friction (tribology) |

74M15 | Contact in solid mechanics |

65K10 | Numerical optimization and variational techniques |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

49J40 | Variational inequalities |