Nonstationary Axisymmetric Motion of an Elastic Momentum Semi-Space under Non-Stationary Normal Surface Movements
Tran Le Thaia* , D.V. Tarlakovskiia;b**
aMoscow Aviation Institute (National Research University), Moscow, 125993 Russia
bResearch Institute of Mechanics, Moscow State University, Moscow, 119192 Russia
E-mail: *tranlethaivvk@gmail.com, **tdvhome@mail.ru
Received February 28, 2017
Abstract
An elastic homogeneous isotropic half-space filled with the Cosserat medium has been considered. The deformed state is characterized by independent displacement and rotation vectors. At the initial instant of time and at infinity, there are no perturbations. On the boundary of a half-space, normal displacements have been given. All components of the stress-strain state are supposed to be limited. A cylindrical coordinate system with an axis directed inward into the half-space has been used. With allowance for axial symmetry, the resolving system of equations includes three hyperbolic equations with respect to the scalar potential and the nonzero components of the vector potential and the rotation vector. The components of displacement vectors, rotation angle, stress tensors, and stress moments are related to the potentials by the known relationships.
The solution of the problem has been sought in the form of generalized convolutions of the given displacement with the corresponding surface influence functions. To construct the latter, Hankel transforms along the radius and Laplace transforms in time have been applied. All images have been presented in three terms. The first of them correspond to the tension-compression wave, and the other two are determined by the associated shear and rotation waves. The originals of the first components have been found accurately by means of successive reversal of the transformations. For the remaining terms, we have used expansion in power series in a small parameter characterizing the connection between the shear and rotation waves. The images of the first two coefficients of these series have been found. The corresponding originals have been determined by the successive inversion of the transformations.
Examples of calculations of the regular components of the influence of a granular composite from an aluminum shot in an epoxy matrix have been given.
Keywords: Cosserat medium, superficial influence function, Laplace and Hankel transforms, small parameter method, relation between flat and axially symmetric problems
Acknowledgments. The study was supported by the Russian Foundation for Basic Research (project no. 15-08-00787).
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For citation: Tran Le Thai, Tarlakovskii D.V. Nonstationary axisymmetric motion of an elastic momentum semi-space under non-stationary normal surface movements. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 2, pp. 231–245. (In Russian)
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