In the present work, the process of interaction of an abrasive particle with the surface of a component is modeled in the framework of dynamic problems of the theory of thermoelasticity, considering plastic deformation, friction and wear of the surface in the contact region. Two boundary contact problems are considered. The first problem deals with the contact interaction of an element of an abrasive particle in the form of a truncated cone and the surface of a part. The circle of smaller diameter of the cone contacts the surface of the part taking into account the friction and plastic deformation of this surface. Kinematic or force boundary conditions are applied to the circle of greater diameter. In the case of kinematic conditions, the normal and tangential displacements of the circle and its rotation are specified. In the case of force conditions, the force and moment are given. In the second task, the hard stamp slides at a constant speed along the flat boundary of the workpiece, the value of the die insertion is set. In the contact area, the sliding friction force is expressed through normal pressure and heating due to the friction and wear. The stress and temperature fields near the contact region are investigated.