Non-stationary contact of a spherical shell and an elastic half-space

Mathematics. Physics. Mechanics


Mikhailova E. Y.*, Tarlakovsky D. V.**, Fedotenkov G. V.***

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



The stress-strain state of a spherical shell of Timoshenko type and elastic isotropic half-space in process of their impact on an arbitrary time interval of interaction is investigated. In this case the contact area changes dynamically. The movable spherical coordinate system is used for the shell and for the half-space the non movable cylindrical coordinate system is used. Contact occurs in frictionless circumstances.

Statement of the problem involves the equations of motion of the shell and the half-space, the physical and geometrical relations, the coupling equations for displacement and elastic potentials, he coupling equations for stresses and elastic potentials, equation of motion of the shell as a rigid body, boundary and initial conditions.

To solve the problem the influence functions for the half-space and spherical shell is used. First of them received explicitly earlier. Second of them is a normal displacements of the shell as a solution of the initial-boundary value problem on the impact of the normal pressure on the surface of the elastic shell, given as the product of the Dirac delta function depending on the angular coordinate and time. To construct the influence functions the expansions in series of Legendre polynomials and their derivatives are used.

The initial system of dynamics equations of the shell is reduced to the infinite system for the unknown coefficients of expansions, which are depend on time and angular coordinates. With the help of the Laplace transform in time and further treatment the solution of this is system is constructed.

A system of governing equations, the basic equation of which follows from the boundary conditions and integral representations of normal displacements of the shell and the half-space, based on the principle of superposition is developed. The kinematic relation between the radius of the contact area and the depth of penetration of the shell, the equation of motion of the shell as a rigid body written in integral form, and the initial conditions are completing the system.

A numerical-analytical algorithm for solving a system based on the method of quadratures using the formulas of Gauss and Simpson is developed and implemented. In the case of singular integrals the method of weighting coefficients and canonical regularization is used.

As the results of the calculations the graphs of the distribution of the contact pressure and normal displacements are shown. Dependences of contact pressure and normal displacements on time in a front point of the shell are also presented. The analysis of the received results is carried out.

The obtained results can be used in the aerospace industry in the cases of predicting the results of hard landing of the space landers on the ground.


non-stationary contact problems, spherical shell of Timoshenko type, elastic half-space, influence functions, integral equations, quadrature formulas, canonical regularization


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