Impact response of a viscoelastic beam considering the changes of its microstructure in the contact domain

Abstract

The problem on low-velocity impact of a long thin elastic rod with a flat end upon an infinite viscoelastic Timoshenko-type beam, the dynamic behaviour of which is described by a set of equations taking the rotary inertia, transverse shear deformation and extension of the beam’s middle surface into account, is considered. The viscoelastic features of the beam are governed by the standard linear solid model with derivatives of integer order. At the moment of impact, shock waves (surfaces of strong discontinuity) 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 longitudinal and transverse forces applied to the boundary of the contact region, which are obtained on the basis of one-term ray expansions. During the impact process, decrosslinking within the domain of the contact of the beam with the rod occurs, resulting in more free displacements of molecules with respect to each other, and finally in the decrease of the beam material viscosity in the contact zone. This circumstance allows one to describe the behaviour of the beam material within the contact domain by the standard linear solid model involving fractional derivatives, since variation in the fractional parameter (the order of the fractional derivative) enables one to control the viscosity of the beam material. The contact force has been determined analytically via the Laplace transform technique.