APPLICATION OF THE BIPOLAR COORDINATE SYSTEM TO MODELING OF TWO COAXIAL CYLINDRICAL SHELLS DYNAMICS IN ELASTIC MEDIUM - Pozhuev, V. I.; PozhuevА. V.; FasoliakА. V. (Computer Science and Applied Mathematics, 2018)
The present paper considers the dynamics of two coaxial cylindrical shells in a linearly elastic homogeneous inertial medium. In this work studies case, when normal impulsive loads act on inner surface of one of the shell and other shell is stresses free. The contact between the shells and the medium was supposed to be gliding. The mutual influence of the shells on the dynamic stress-strain state of the media is under research and it’s scientific novelty of this work. In present paper, it is assumed that loads are evenly distributed on the axial coordinate, i.e. they don’t depend on axial coordinate, so the problem, which is considered, is reduced to plane elasticity theory problem. It’s also accepted, that loads are applied symmetrically to line, which passes the centers of the cavities, so we can cut the area along this line. This incision will be taken into account in boundary conditions. Then the bipolar coordinate system (BCS) is used to solve this problem. The used BCS gives the opportunity to display a half-space with two cavities, which models the medium in the finite rectangle. As the BCS is an orthogonally coordinate system, the motion equation of elasticity theory, Cauchy’s equations and Hooke’s law are in force. So, we substitute deformations expressions by displacements (Cauchy’s equations) for the deformations in Hooke’s law and obtain the expressions for stresses by displacements in the BCS. Then we substitute the expressions for stresses by displacements (in the BCS) in the motion equation of elastic theory, and we obtain motion equations of elastic theory by displacements in the BCS. These equations are used to modeling media motions. The shell motions are described by classic thin shells theory equations, which satisfy Kirchhoff-Love theory. As motion equations of elastic theory by displacements in the BCS have sufficiently difficult coefficients, which depend on space variables, so we use finite differences method on space variables for solution of the problem to be obtained. Dynamic problem can be reduced to iterative sequences of quasi-static problems using Wilson’s θ-method. So, the approach of solving dynamic problem in BCS, based on finite differences method and Wilson’s θ-method has been developed It was established that in cases, when the distance between the shells is more than five times its radii, the influence of the mutual influence of the shells on the stress-strain state of considered mechanical system, is insignificant. The obtained results, and the methods, developed in present work, can be applied in calculations of underground structures, in particular, subway tunnels and pipelines.