The Rothe Method for Variational-Hemivariational Inequalities with Applications to Contact Mechanics

Krzysztof Bartosz (1) , Mircea Sofonea (2)

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

We consider a new class of first order evolutionary variational-hemivariational inequalities for which we prove an existence and uniqueness result. The proof is based on a time-discretization method, also known as the Rothe method. It consists of considering a discrete version of each inequality in the class, proving its unique solvability, and recovering the solution of the continuous problem as the time step converges to zero. Then we introduce a quasi-static frictionless problem for Kelvin--Voigt viscoelastic materials in which the contact is modeled with a nonmonotone normal compliance condition and a unilateral constraint. We prove the variational formulation of the problem cast in the abstract setting of variational-hemivariational inequalities, with a convenient choice of spaces and operators. Further, we apply our abstract result in order to prove the unique weak solvability of the problem.

Keywords: variational-hemivariational inequality; Clarke subdifferential; Rothe method; existence and uniqueness; viscoelastic material; frictionless contact; normal compliance; unilateral constraint; weak solution

Domains: Mathematics [math]

Krzysztof Bartosz, Mircea Sofonea. The Rothe Method for Variational-Hemivariational Inequalities with Applications to Contact Mechanics. SIAM Journal on Mathematical Analysis, 2016, 48 (2), pp.861-883. ⟨10.1137/151005610⟩. ⟨hal-01346147⟩