MATHEMATICAL MODELING OF THE DYNAMICS OF AN ELASTIC HALF-MEDIUM WITH A CYLINDRICAL CAVITY REINFORCED BY A SHELL UNDER AXISYMMETRIC LOADS

Abstract

Purpose. The development of a mathematical model of an elastic semi-medium with a cylindrical cavity reinforced by a shell and a plate that reinforces the surface of the semi-medium based on the finite-element approach. The non-stationary dynamic stress-strain state of this mechanical system under axisymmetric surface loads, which depends on time as a unit Heaviside function was analyzed. The solution of the corresponding static problem was obtained. Based on the obtained results, the dynamic coefficient of the considered mechanical system was calculated.
 Research methods. The finite element method and the Wilson - method were used for solving static and the corresponding dynamic matrix equation correspondingly. The last one allows reducing the differential matrix equation to an iterative sequence of quasi-static problems. To obtain an approximate solution of systems of linear algebraic equations that take place in the process of solving static and a sequence of quasi-statistical problems, the conjugate gradients method was used.
 Results. A finite-element model of the mechanical system of an elastic semi-medium with a cylindrical cavity reinforced by a shell and a plate reinforces the surface of the semi-medium was developed. The non-stationary dynamic stress-deformed state of this mechanical system under action of the surface axisymmetric loads was analyzed. The solution of the corresponding static problem was obtained. Based of the results has been obtained, the dynamic coefficient of the considered mechanical system was determined.
 Scientific novelty. The developed approach allows to take into account the mutual influence of a cylindrical cavity reinforced by a shell and a plate that reinforces the surface of the semi-medium under the action of the non-stationary dynamic loads applied to the surface of the plate.
 Practical value. The developed mathematical models and obtained results can be used in the design of underground structures, in particular mine shafts.