Characteristic form of dynamics equations of Cosserat medium

In this paper, we construct the characteristic form of the equations of dynamics of the Cosserat medium and the Cosserat pseudocontinuum for bounded bodies. The method of matrix transformations proposed by the author is used for construction and allows obtaining the necessary relations using identical transformations. The obtained equations are compared with those for a symmetrically elastic isotropic homogeneous body. A method is proposed for selecting the necessary equations for computational schemes at the internal and boundary points of the body. A sequence of operations is proposed for iterative calculations of stresses, particle velocities, moment stresses, and angular velocities of particles in a coupled model of the Cosserat medium.

Propagation of non-stationary antisymmetric kinematic perturbations from a spherical cavity in Cosserat medium

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.

DISTRIBUTION OF THE RAYLEIGH WAVE ALONG THE BORDER OF THE HALF-SPACE, DESCRIBED BY THE SIMPLIFIED MODEL OF THE COSSERAT

We consider a simplified (reduced) dynamic model of a Cosserat medium, which occupies an intermediate position between the classical dynamic theory of elasticity and the proper Cosserat medium model, which has asymmetry in the stress tensor and the presence of moment stresses. In contrast to the latter, in the simplified model, three of the six elastic constants are zero and, as a result, there is no moment stress tensor. In the two-dimensional formulation for the model of a reduced medium, the problem of the propagation of an elastic surface wave along the half-space boundary was solved. The solution of the equations was described as the sum of the scalar and vector potentials, and only one component of the vector potential is nonzero. It is shown that such a wave, in contrast to the classical surface Rayleigh wave, has a dispersion. In the plane “phase velocity-frequency” for such waves there are two dispersion branches: the lower (acoustic) and upper (optical). With increasing frequency, the phase velocity of the wave related to the lower dispersion branch decreases. The phase velocity of the wave related to the upper dispersion branch increases with increasing frequency. The phase velocity of the surface wave in the entire frequency range exceeds the phase velocity of the bulk shear wave. The stresses and displacements arising in the zone of propagation of the surface wave are calculated.

Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media

A Lie groupoid, called second-order non-holonomic material Lie groupoid, is associated in a natural way to any Cosserat medium. This groupoid is used to give a new definition of homogeneity which does not depend on a material archetype. The corresponding Lie algebroid, called second-order non-holonomic material Lie algebroid, is used to characterize the homogeneity property of the material. We also relate these results with the previously obtained ones in terms of non-holonomic second-order [Formula: see text]-structures.