We consider a mathematical model which describes the contact between a deformable body and an obstacle, the so-called foundation. The body is assumed to have a viscoelastic behavior that we model with the Kelvin-Voigt constitutive law. The contact is frictionless and is modeled with the well-known Signorini condition in a form with a zero gap function. We present two alternative yet equivalent weak formulations of the problem and establish existence and uniqueness results for both formulations. The proofs are based on a general result on evolution equations with maximal monotone operators. We then study a semi-discrete numerical scheme for the problem, in terms of displacements. The numerical scheme has a unique solution. We show the convergence of the scheme under the basic solution regularity. Under appropriate regularity assumptions on the solution, we also provide optimal order error estimates.
Mikäel Barboteu. Weimin Han. Mircea Sofonea. "A frictionless contact problem for viscoelastic materials." J. Appl. Math. 2 (1) 1 - 21, 25 March 2002. https://doi.org/10.1155/S1110757X02000219