The Rothe method for variational-hemivariational inequalities with applications to contact mechanics (Q2790398)

This paper is twofold interesting, on the one hand in reflexive Banach spaces unique existence for a solution of an evolutionary variational-hemivariational inequality is shown using the Rothe method (discretization of the time interval), on the other hand a new kind of time-dependent contact problem is studied leading just to such an evolutionary inequality and can be solved applying the proved existence theorem. To give an impression, let's have a look at the contact problem. It is a quasi-static (time interval \([0,T]\)) frictionless viscoelastic one: given a viscoelastic body \(\Omega\), whose boundary \(\partial \Omega\) is partitioned into three disjoint measurable parts \(\Gamma_i, i=1,2,3\). Along \(\Gamma_1\) the body is clamped, so the displacements vanish. Along \(\Gamma_2\) there are time-dependent surface tractions of density \(f_2\). Along \(\Gamma_3\) volume forces \(f_0\) act on \(\Omega\) and the body is in frictionless contact with a special obstacle, and this contact is modeled with a nonmonotone compliance condition and a unilateral (Signorini) contact condition. The mathematical formulation of the contact problem is to find a displacemant field \(u\) and a stress field \(\sigma\) such that the Kelvin-Voigt viscoelastic constitution law (\(\sigma = c \epsilon(\dot{u})+G \epsilon(u)\)), the equation of equilibrium \(\mathrm{Div } \sigma + f_0 = 0\), the boundary condition (\(u=0\) on \(\Gamma_1 \times (0,T), \sigma_{\nu} = f_2\) on \(\Gamma_2 \times (0,T)\)), the frictionless and initial conditions and especially the contact conditions (as the authors write: the main trait of novelty of the mechanic model in this paper) hold. Those contact conditions describe the rigidity of the obstacle and its deformability. Interestingly, the Clarke subdifferential of a nonconvex potential is used. Clearly, after having explained the contact problem, the authors give a variational formulation for the displacements and now they can apply their new theorem of unique existence of a solution and the result is: there is a unique weak solution for the displacement field \(u\). This solution \(u\) has \(H^1\)-regularity and the stress field \(\sigma\) has \(L^2\)-regularity.