Nonsmooth Mechanics of Solids
October 4, 2004 — October 8, 2004
- Jaroslav Haslinger (Matematicko Fyzik. Fakulta, Praha, Czech Rep.)
- Georgios Stavroulakis (University of Ioannina, Ioannina, Greece)
In many real-life problems coming from engineering or economics one can encounter nondifferentiable or discontinuous functions and set-valued mappings. A deep study of the properties of these maps including a certain generalized differential calculus is the subject of nonsmooth analysis. We shall focus on some problems in mechanics of solids which lead to such models.
The classical mechanics (statics and dynamics) of solids provide a large number of nonsmooth effects: contact problems, collisions, stick-slip motions connected with friction, delaminations in composites. All these effects can be mathematically described by means of differential inclusions. The mathematical research in this area began in the sixties assuming multivalued parts to be represented firstly by maximal monotone mappings, i.e. the case leading to variational inequalities. The monotonicity assumption however turns out to be very restrictive. In practice, we meet a lot of problems whose basic constitutive laws are no longer monotone. At the beginning of eighties Prof. P.D. Panagiotopoulos used tools of nonsmooth analysis and introduced what he called hemivariational inequalities (HE). HE’s represent an appropriate mathematical tool enabling us to involve nonmonotone multivalued relations into the model. Due to HE’s, the range of problems which can be now rigorously treated is enlarged.
The goal of this course is to illustrate the potential of nonsmooth analysis in modelling of various problems in mechanics of solids. The emphasis will be laid on the completeness and mathematical correctness of the presentation, although several industrial applications will be presented. It will cover the following topics: nonsmooth modelling of problems in mechanics of solids, the mathematical theory of variational and hemivariational inequalities, approximation of variational and hemivariational inequalities by finite element and boundary element methods, the numerical realization (including smoothing and regularisation techniques), algorithms and applications from civil and mechanical engineering and related optimal design and identification problems.
A number of well-known experts and active researchers in the field, including mathematicians and engineers, will report on classical and new results covering all the above mentioned topics. The presentation of all these topics will be carefully balanced between theory, numerical methods and applications.
The summer school is addressed to graduate students, PhD candidates and young faculty members in mathematics, physical sciences and engineering.
Engineers working on advanced applications of computational mechanics and modelling of highly nonlinear and nonsmooth effects such as contact and friction problems in industry (civil, aerospace, automotive) as well as applied mathematicians and computer scientists (dealing with nonsmooth analysis, optimisation, calculus of variations, computational mechanics) will benefit from the course.