A closed mathematical formulation of plane non-stationary contact problems for rigid dies and an elastic half-space with deepened cavities is constructed. Using the dynamic theorem of reciprocity of works, a system of resolving equations is obtained. It includes the basic boundary integral equation arising from the principle of reciprocity and boundary conditions, as well as the equations of translational and rotational motion of the die. The fundamental and singular solutions for the elastic plane are the cores of the main resolving equation. They determine displacements and stresses in the elastic plane from the applied single instantaneous concentrated force. An original solution algorithm based on the method of boundary integrals with an additional iterative procedure that allows one to take into account the partial separation of the boundary surfaces of the die and the half-space in the contact area has been developed and implemented on a computer. In this case, at each moment of time in the half-space, there is a region that is in a perturbed state and outside of which there are no perturbations. The integral operators of the resolving system of equations are replaced by discrete analogs in the spatial variable and in time. A parametric study of the process of unsteady contact of an absolutely rigid rectangular die with a half-space having a recessed cavity is carried out. Analysis of the calculation results revealed the manifestation of a significant effect of the cavity on the process of non-stationary contact interaction. The influence of the cavity begins to show itself from the moment the reflected waves arrive from its boundary. In this case, the nature of the unsteady stress-strain state and displacements changes significantly. The developed calculation algorithm can be used in engineering practice by design and research organizations in the process of designing and calculating buildings and structures under the influence of natural and man-made vibrations propagating in the soil.
Modeling of contact interaction of a heated plane rigid elliptical punch with a transversally isotropic elastic half-space