A contact problem for a beam under the conditions of plasticity and creep

A boundary value problem that describes a contact between a beam and a rigid stamp is considered in a precise statement. Here, the equation of state is most general in a sense and involves such properties of material of the beam as elasticity, plasticity, and creep. The presence of two constraints in the form of inequalities imposed on a solution determines the main difficulty in studying the problem. The first constraint has geometric character and presents the impermeability condition $\omega-v\varphi_x\geqslant\varphi$, where $v$ and $\omega$ are the tangent and normal displacements of the points of the beam and the function $\varphi$ describes the shape of the stamp. The second constraint reveals mechanical nature and presents the plasticity condition $|m|\leqslant k$, where m is the bending moment. In this connection, the boundary value problem is formulated as a variational inequality subject to the above-indicated constraints. The main result consists in proving an existence theorem for solutions to the problem in question.