We consider a space filled with a linearly elastic Cosserat medium with a spherical cavity under given nonstationary antisymmetric surface perturbations, which are understood as the corresponding analogue of classical antiplane deformations. The motion of a medium is described by a system of three equations with respect to nonzero components of the displacement vector and potentials of the rotation field, written in a spherical coordinate system with the origin at its center of the cavity. The initial conditions are assumed to be zero. To solve the problem, we use decomposition of functions to Legendre and Gegenbauer polynomials, as well as the Laplace transform in time. As a result, the problem is reduced to independent systems of ordinary differential equations with the Laplace operator for the coefficients of the series. A statement about the structure of the general solution of this system is formulated. Images of the series coefficients are presented in the form of linear combinations of boundary conditions with coefficients - transformants of surface influence functions, the explicit formulas for which include the Bessel functions of a half-integer index. Due to the complexity of these expressions, to determine the originals in the linear approximation, the method of a small parameter is used, which is taken as a coefficient characterizing the relationship between the displacement and rotation fields. Then, taking into account the connection between the Bessel functions and elementary functions, the images are written in the form of linear combinations of exponentials with coefficients - rational functions of the transformation parameter. The further procedure for inverting the Laplace transform is carried out using residues. It is shown that there are three wave fronts corresponding to a shear wave modified with allowance for free rotation and two rotation waves. Examples of calculations for a granular composite of aluminum shot in an epoxy matrix are presented.
Propagation of non-stationary antisymmetric kinematic perturbations from a spherical cavity in Cosserat medium
