Nonsmooth Mechanics of Solids

herausgegeben von: Jaroslav Haslinger, Georgios E. Stavroulakis

Verlag: Springer Vienna

Buchreihe : CISM International Centre for Mechanical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Mechanics have played an important role in mathematics, from infinitesimal calculus, calculus of variations, partial differential equations and numerical methods (finite elements). Originally, mechanics treated smooth objects. Technological progress has evoked the necessity to model and solve more complicated problems, like unilateral contact and friction, plasticity, delamination and adhesion, advanced materials, etc. The new tools include convex analysis, differential calculus for convex functions, and subgradients of convex functions and extensions for nonconvex problems. Nonsmooth mechanics is a relatively complex field, and requires a good knowledge of mechanics and a good background in some parts of modern mathematics. The present volume of lecture notes follows a very successful advanced school, with the aim to cover as much as possible all these aspects. Therefore the contributions cover mechanical aspects as well as the mathematical and numerical treatment.

Inhaltsverzeichnis

Frontmatter

Collisions. Thermal effects. Collisions of deformable solids and collisions of solids and fluids

Abstract

These notes report lectures devoted to predictive theories of collisions, given at the Centre International des Sciences Mécaniques.

Michel Frémond

An Introduction to Impacts

Abstract

Different methods to model and solve multi-contact collisions are presented in this report. The standard impact constitutive laws from non-smooth dynamics are reviewed for planar frictional collisions and formulated in terms of set-valued maps and linear complementarity. For the frictionless case, a geometric concept based on kinematic, kinetic and energetic compatibility is developed, which provides access to non-standard impact events as in Newton’s cradle. Within this context, Moreau’s impact law is reviewed and stated in various ways, providing even access to the collision problem at re-entrant corner points as an extension. Based on Moreau’s law, a geometric classification of impacts is proposed. Several examples are presented, such as the frictional reversible impact at a super ball, Newton’s cradle and the rocking rod.

Christoph Glocker

Approximation of variational and hemivariational inequalities of elliptic type. Applications to contact problems with friction

Abstract

This article deals with approximations of differential inclusions of elliptic type. We start with problems which involve only monotone mappings, i.e. with classical variational inequalities of the first and the second kind. Next we show how to approximate a class of inclusion problems called hemivariational inequalities with nonmonotone multivalued mappings. These results are then used for the approximation and the numerical realization of contact problems with different models of friction.

Jaroslav Haslinger

Semicoercive Hemivariational Inequalities, Regularization Methods, Applications on Mechanics

Abstract

An approach to semicoercive variational-hemivariational or hemivariational inequalities based on a recession technique introduced in (Naniewicz 2003), is developed. First, problems defined on vector-valued function spaces are considered under unilateral growth conditions imposed on nonlinear parts by making use of the Galerkin method. Second, a minimax method relying on Chang’s version of Mountain Pass Theorem for locally Lipschitz functionals (Chang 1981) is applied to study semicoercive hemivariational inequalities on vector valued function spaces. Third, the resonant problem governed by the p-Laplacian involving the unilateral growth condition is discussed. Some mechanical problems as exemplifications of the presented approach are shown.

Zdzisław Naniewicz

Mathematical Programs with Equilibrium Constraints: Theory and Numerical Methods

Abstract

The lecture notes deal with optimization problems, where a generalized equation (modeling an equilibrium) arises among the constraints. The main attention is paid to necessary optimality conditions and methods to the numerical solution of such problems. The applications come from continuum mechanics.

Jiří V. Outrata

Applied Nonsmooth Mechanics of Deformable Bodies

Abstract

In this article the interconnections between nonsmooth optimization, nonsmooth analysis and complementarity problems, from the one side, and modern computational mechanics, from the other side, are outlined. The arising problems are, in general, variational and hemivariational inequalities. A short discussion of suitable numerical algorithms for the approximation of their solution, in connection with finite element or boundary element techniques, follows. Related topics of existence, stability and path-following of the solution are briebly discussed.

Georgios E. Stavroulakis

Titel: Nonsmooth Mechanics of Solids
herausgegeben von: Jaroslav Haslinger
Georgios E. Stavroulakis
Verlag: Springer Vienna
Electronic ISBN: 978-3-211-48243-8
Print ISBN: 978-3-211-48241-4
DOI: https://doi.org/10.1007/978-3-211-48243-8