History-Dependent Problems with Applications to Contact Models for Elastic Beams

Krzysztof Bartosz (1) , Piotr Kalita (1) , Stanisław Migórski (1) , Anna Ochal (1) , Mircea Sofonea (2)

1 Institute of Computer Science [Krakow]
2 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations

We prove an existence and uniqueness result for a class of subdifferential inclusions which involve a history-dependent operator. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Problems of such kind arise in a large number of mathematical models which describe quasistatic processes of contact. To provide an example we consider an elastic beam in contact with a reactive obstacle. The contact is modeled with a new and nonstandard condition which involves both the subdifferential of a nonconvex and nonsmooth function and a Volterra-type integral term. We derive a variational formulation of the problem which is in the form of a history-dependent hemivariational inequality for the displacement field. Then, we use our abstract result to prove its unique weak solvability. Finally, we consider a numerical approximation of the model, solve effectively the approximate problems and provide numerical simulations.

Mots clés: Finite element simulations; Hemivariational inequality; Euler–Bernoulli beam; Nonlinear inclusion

Domaines: Mathématiques [math]

Krzysztof Bartosz, Piotr Kalita, Stanisław Migórski, Anna Ochal, Mircea Sofonea. History-Dependent Problems with Applications to Contact Models for Elastic Beams. Applied Mathematics and Optimization, 2016, 73 (1), pp.71-98. ⟨10.1007/s00245-015-9292-6⟩. ⟨hal-01142664⟩