# Moment Elastic Half-Plane under the Action of Surface Non-Stationary Normal Displacements

### Аuthors

Tran L. T.1*, Tarlakovsky D. V.2**

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. Institute of Mechanics Lomonosov Moscow State University, 1, Michurinsky prospect, Moscow, 119192, Russia

*e-mail: tranlethaivvk@gmail.com
**e-mail: tdvhome@mail.ru

### Abstract

The article discusses an elastic homogeneous isotropic half-plane filled with a Cosserat medium. At the initial moment of time and at infinity, there are no perturbations. On the border of a half-plane, non-stationary normal displacements are given. All components of the stress-strain state are limited. A complete system of equations of the asymmetric theory of elasticity is presented, which includes linear vector equations of motion in displacements, geometric and physical relations. Using the representation of fields of displacement in the form of potential and solenoidal parts, a system of equations of motion with respect to the scalar, vector displacement potentials and angle of rotation is written.

For the plane problem in question, a rectangular Cartesian coordinate system is used. The resolving system of equations includes three hyperbolic equations for the scalar potential, the nonzero component of the vector potential, and the rotation vector. The solution of the problem is sought in the form of convolutions of a given normal displacement with the corresponding surface Green functions. To construct the latter, we apply the Fourier transform with respect to the coordinate and the Laplace in time. The originals of the images are found using the joint inversion of the Fourier and Laplace transforms.

Examples of the action of various non-stationary loads on the border of a half-plane and examples of calculations for a granular composite of aluminum shot in an epoxy matrix are given.

As results of this article, an analytical solution is constructed for the problem of the propagation of non-stationary surface kinematic perturbations in the moment-elastic half-plane, which allows using quadratures to find stresses for any law of load variation. It is established that the corrections made to the solution, taking into account the moment properties of the medium, have the order of the coefficient relating the fields of displacement and rotation. It is shown that the surface Green’s functions in this case have an integrable singularity at the front of the shear wave.

### Keywords:

cosserat medium, half-plane, superficial influence function, Laplace and Fourier transforms, joint inversion of transform

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