Unilateral contact problem for aging viscoelastic beams

The frictionless unilateral contact problem of a viscoelastic Bernoulli-Euler beam resting on a viscoelastic soil is studied. The mathematical formulation involves equilibrium equations, compatibility equations, and constitutive laws, with an aging integral-type form. The unilateral nature of the contact is imposed through a compatibility inequality, which allows the determination of the contact imprint at each time. Further, the governing integro-differential equation for the unknown contact pressure is derived. As special cases, the elastic Winkler-type soil and the rigid soil conditions are discussed. A numerical approach is presented, which employs the finite difference method along space and an adaptive step-by-step algorithm along time. The procedure allows for time discontinuities of both external loads and contact pressure. Several selected numerical examples are. presented and the influence of the most important material and geometrical parameters are shown. For the simplest situations, it was possible to compare the results obtained with known analytical solutions.