The Mechanics of a viscoelastic collision between a solid body and a semi-space at impact at an arbitrary angle of attack have been examined in this paper. The radius of the contact area has been determined, using a geometry of mutual approach between two spherical bodies. It was proposed that the forces of viscosity and the forces of elasticity can be found by integration of the specific forces acting inside an elementary volume of the contact zone, and on this basis the principally new method of defining the viscoelastic forces has been developed. In this method, the volume of deformation is considered as a system comprising an infinitely large number of infinitesimal discrete elements connected to each other in a definite way. It is shown here that this method allows finding the viscoelastic forces for any theoretical or experimental dependencies between the distance of mutual approach of two solid bodies and the diameter of the contact area. Also the differential equations of the displacement (the movement) of the centre of mass of the body have been obtained. Equations for the calculation of work and energy in the compression and in the restitution phases, and also in the rolling shear phase have been derived. Approximate solutions for the differential equations of movement (displacement) by using the method of equivalent work have been derived. Equations for the normal contact stresses have been obtained. Also, equations for kinematic and dynamic parameters of the viscoelastic collision have been derived in this article. Examples of the comparison of theoretical results and conclusions have been given in the paper.

Solutions of the problems of a viscoelastic dynamic contact between smooth curvilinear surfaces of two solid bodies by the application of the "Method of the specific forces" have been given in the article, and the new conception for the definition of the elastic and the viscous forces in the common case of dynamics of a viscoelastic contact is proposed here by the further development of this method. Essence of this method is that, the forces of viscosity and the forces of elasticity can be found by integration of the specific forces acting inside an elementary volume of the contact zone. It is shown here, that this method allows finding the viscoelastic forces for any theoretical or experimental dependencies between the distance of mutual approach of two solid bodies and the diameter of the contact area. Also, the derivation of the integral equations of the viscoelastic forces, the equations for pressure in the contact is presented. Viscoelastic dynamic contacts between two spherical bodies, and between a spherical solid body and a semi-space at impact have been examined. Work and Energy in the phases of compression and restitution, and at the rolling shear have been derived. Approximate solutions for the differential equations of movement (displacement) by using the method of equivalent work have been derived. Equations for the normal contact stresses have been obtained. Also, equations for kinematic and dynamic parameters of the viscoelastic collision have been derived in this article. Examples of the comparison of theoretical results and conclusions have been given in the paper.

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