Dynamic response of a viscoelastic beam impacted by a viscoelastic sphere

In the present paper, we consider the problem on a transverse impact of a viscoelastic sphere upon a viscoelastic Bernoulli-Euler beam, the viscoelastic features of which are defined via the fractional derivative standard linear solid models. As this takes place, only Young's time-dependent operators are preassigned, while the bulk moduli are considered to be constant values, since the bulk relaxation for the majority of materials is far less than the shear relaxation. Beam's displacement subjected to the concentrated contact force is found by the method of expansion in terms of eigen (beam) functions. The contact force driven displacement of the impactor, which is the sum of the beam's displacement at the place of contact and the indentation of the impactor into the target, is defined from the equation of motion of the material point with the mass equal to sphere's mass. Within the contact domain, the contact force is defined by the modified Hertzian contact law with the time-dependent rigidity function. For decoding the viscoelastic operators involving in the problem under consideration, the algebra of Rabotnov's fractional operators is employed. A nonlinear integro-differential equation is obtained either in terms of the contact force or in the local bearing of the target and impactor materials. Using the duration of contact as a small parameter, approximate analytical solutions have been found, which allow one to define the maximal contact force and the duration of contact for colliding bodies.