Abstract
Refined model of S.P. Timoshenko makes it possible to consider the shear and the inertia rotation of the transverse section of the shell. Disturbances spread in the shells of S.P. Timoshenko type with finite speed. Therefore, to study the dynamics of propagation of wave processes in the fine shells of S.P. Timoshenko type is an important aspect as well as it is important to investigate a wave processes of the impact, shock in elastic foundation in which a striker is penetrating. The method of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind and the convergence of this solution are well studied. Such approach has been successfully used for cases of the investigation of problems of the impact a hard bodies and an elastic fine shells of the Kirchhoff-Love type on elastic a half-space and a layer. In this paper an attempt is made to solve the axisymmetric problem of the impact of an elastic fine spheric shell of the S.P. Timoshenko type on an elastic half-space using the method of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind. It is shown that this approach is not acceptable for investigated in this paper axisymmetric problem. The discretization using the Gregory methods for numerical integration and Adams for solving the Cauchy problem of the reduced infinite system of Volterra equations of the second kind results in a poorly defined system of linear algebraic equations: as the size of reduction increases the determinant of such a system to aim at infinity. This technique does not allow to solve plane and axisymmetric problems of dynamics for fine shells of the S.P. Timoshenko type and elastic bodies. This shows the limitations of this approach and leads to the feasibility of developing other mathematical approaches and models. It should be noted that to calibrate the computational process in the elastoplastic formulation at the elastic stage, it is convenient and expedient to use the technique of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind.
Highlights
The approach [1 – 5] for solving problems of dynamics, developed in [6 – 8, 10], makes it possible to determine the stress-strain state of elastic half-space and a layer during penetration of absolutely rigid bodies [1, 2, 7, 8, 10] and the stress-strain state of elastic Kirchhoff– Love type fine shells and elastic half-spaces and layers at their collision [3 − 6]
It should be noted that to calibrate the computational process in the elastoplastic formulation at the elastic stage, it is convenient and expedient to use the technique of the outcoming dynamics problems to solve an infinite system of integral equations Volterra of the second kind
We investigate the approach [3 – 6] for solving the axisymmetric problem of the impact of a spherical fine shell of the S.P
Summary
The approach [1 – 5] for solving problems of dynamics, developed in [6 – 8, 10], makes it possible to determine the stress-strain state of elastic half-space and a layer during penetration of absolutely rigid bodies [1, 2, 7, 8, 10] and the stress-strain state of elastic Kirchhoff– Love type fine shells and elastic half-spaces and layers at their collision [3 − 6]. In (37) An (s) and Bn (s) are determined from the boundary conditions It follows from representations (37) and relations (4) that the sought-for functions on the surface of a halfspace are represented as series in the system of eigenfunctions of the problem. Substituting (22) and (23) into (39) with allowance for r = sin θ , arising from geometric considerations in the zone of the contact region, and representing both parts of (39) in the form of series in J0 (λnr) , we obtain an infinite system of Volterra integral equations (ISVIE) of the second kind regarding to unknown harmonics velocity on the surface of the half-space (n = 0, ) : Vn (t) +
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