Dynamic response of a viscoelastic plate impacted by an elastic rod

Abstract

The problem on low-velocity impact of a long thin elastic rod upon an infinite plate, the viscoelastic features of which are exhibited only within the contact domain and are governed by the fractional derivative standard linear solid model. The part of the plate being out of the contact region is considered to be elastic, and its behavior is described by a set of equations taking the rotary inertia and transverse shear deformation into account. At the moment of impact, shock waves are generated both in the impactor and target, the influence of which on the contact domain is considered via the theory of discontinuities. The contact zone moves like a rigid whole under the action of the contact force and transverse forces applied to the boundary of the contact region, which are obtained on the basis of one-term ray expansions. As this takes place, two cases are considered, namely: (1) the rod with a flat end; and (2) the rod with a rounded end. In the first case, the problem of defining the contact force is a linear one, and the Laplace transform technique is used for its solution. In the second case, the contact force is determined via the Hertz contact theory, and the operators entering in the contact force are decoded with help of the algebra of Rabotnov fractional operators. As a result, a nonlinear integrodifferential equation is obtained in terms of the value of the local bearing of plate and rod's materials, which could be solved numerically. The time-dependence of the contact domain is obtained numerically at different magnitudes of the fractional parameter which defines the order of fractional derivatives.