The dynamic interaction (contact) Introduction. The numerical solution of the given dynamic three-dimensional problem by the finiteelement, boundary-element, and other methods generally involves the principal difficulty associated with degeneration of the contact region into a set of points. Therefore, to solve the problem, we use asymptotic methods.The integral equation of the problem is constructed using the closed form of the fundamental solution of the nonstationary half-space problem (Lamb's problem) [1]. The asymptotic simplification of this equation yields a problem of smaller dimensionality that is solved numerically.As an example, we consider the vertical motion of several smooth round punches with a plane base. Some results have already been reported at conferences [2][3][4].The asymptotic approach to the corresponding static problem was used for the first time by Galin [5]. Argatov and Nazarov [6] obtained a rigorous asymptotic solution for static loading of an elastic body resting on several small supports. In the 1970s, attempts were undertaken to solve dynamic problems with a small parameter, but these solutions contained assumptions [7][8][9]. The bibliography of studies devoted to numerical solutions can be found in [10][11][12][13][14][15][16].1. Fundamental Solution of Lamb's Problem. We use the closed form of the fundamental solution G(t, r) of the nonstationary Lamb's problem [1]. Being caused by a vertical force suddenly applied to the boundary at the moment t = 0, whose magnitude does not change at subsequent times, the vertical displacement of the points at the plane boundary z --0 of the elastic half-space z ~ 0 has the form
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