One approach to the problems of impact of fine shells of the S.P. Timoshenko type on elastic half-space

Shell elements are used in many thinwalled structures. 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. 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. 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 plane and the axisymmetric problems of the impact of an elastic fine cylindric and spheric shells 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. 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. It is shown that this approach is not acceptable for investigated in this paper the plane and the axisymmetric problems. 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 of deformation 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.


INTRODUCTION
The approach [2 -6] for solving problems of dynamics, developed in [7 -9, 11], makes it possible to determine the stress-strain state of elastic half-space and a layer during penetration of absolutely rigid bodies [2,3,8,9,11] and the stress-strain state of elastic Kirchhoff-Love type fine shells and elastic half-spaces and layers at their collision [4 − 7]. This led to the feasibility of developing other mathematical approaches and models. In [10, 12 -15], a new approach to solving the problems of impact and nonstationary interaction in the elastoplastic mathematical formulation [16 -20] was developed. In nonstationary problems, the action of the striker is replaced by a distributed load in the contact area, which changes according to a linear law [21 -23]. The contact area remains constant. The developed elastoplastic formulation makes it possible to solve impact problems when the dynamic change in the boundary of the contact area is considered and based on this the movement of the striker as a solid body with a change in the penetration speed is taken into account. Also, such an elastoplastic formulation makes it possible to consider the hardening of the material in the process of nonstationary and impact interaction.
The solution of problems for elastic shells [24 -27], elastic half-space [28 -30], elastic layer [31], elastic rod [32,33] were developed using method of the influence functions [34]. In [24] the process of non-stationary interaction of an elastic cylindrical shell with an elastic half-space at the so-called "supersonic" stage of interaction is studied. It is characterized by an excess of the expansion rate areas of contact interaction speed of propagation tension-compression waves in elastic half-space. The solution was developed using influence functions corresponding concentrated force or kinematic actions for an elastic isotropic half-space which were found and investigated in [34].
In this paper, we investigate the approach [4 -7] for solving the axisymmetric problem of the impact of a spherical fine shell of the S.P. Timoshenko type on an elastic half-space.
It is shown that the approach [2 -5], after the reduction of the infinite system of Volterra integral equations of the second kind [6 -8, 11] and discretization using the Gregory methods for numerical integration and Adams for solving the Cauchy problem, a poorly defined system of linear algebraic equations is obtained for which the determinant of the matrix of coefficients increases indefinitely with increasing size of reduction.

PROBLEM FORMULATION
A thin elastic cylindrical shell comes into collision with the elastic half-space 0 z  with its lateral surface along the generatrix of the cylinder at the moment of time 0 t = . We associate with the shell, as can be seen in Figure 1, a movable cylindrical coordinate system θ rz  : θ -the polar angle, which is plotted from the positive direction of the oz axis, the oy axis coincides with the cylinder axis. Let us denote by 0 (,θ) ut , 0 (,θ) wt , (,θ) pt , (,θ) qt the tangential and normal displacements of the points of the middle surface of the shell and the radial and tangential components of the distributed external load, which acts on the shell. We associate a fixed Cartesian coordinate system xyz with the half-space, so that the Oz axis is directed deep into the medium, the Ox axis is directed along the surface of the half-space, and the Oy axis is parallel to the generatrix of the cylinder. The shell thickness h is much less than the radius R of the middle surface of the shell ( / 0,05 hR  ). In case of axisymmetric problem, a thin elastic spherical shell, moving perpendicular to the surface of the elastic half-space 0 z  , reaches this surface at time t=0. We associate with the shell, as shown in Fig. 2  The cylindric or spheric shell penetrates into the elastic medium at a speed , T -the time during which the shell interacts with the half-space. The shell thickness h is much less than the radius R of the middle surface of the shell ( / 0,05 hR  ). Let us denote by 0 (,θ) ut , 0 (,θ) wt , (,θ) pt , (,θ) qt the tangential and normal displacements of the points of the middle surface of the shell and the radial and tangential components of the distributed external load, which acts on the shell. With the half-space we associate a fixed cylindrical coordinate system φ rz , the Oz axis is directed deep into the medium, φ -is the polar angle. Angle θ is plotted from the positive direction of the Oz axis. The physical properties of the half-space material are characterized by elastic constants: volumetric expansion module K, shear modulus μ and density ρ . An elastic medium with constants K, μ , ρ will be associated with a hypothetical acoustic medium with the same constants K, ρ , wherein μ0 = . Under  Let's introduce dimensionless variables:   0   00  00  0   1  ,  ,  ,  ,   σ  , , σ In what follows, we will use only dimensionless quantities, so we omit the dash. The elastic half-space and the spheric shell are in a state of axisymmetric deformation.
Differential equations (of the S.P. Timoshenko type) describing the dynamics of cylindrical (2) and spherical (3) shells and considering the shear and inertia of rotation of the transverse section, due to (1), take the following form [35, pp. 297     b -coefficient that considers the distribution of tangential forces in the transverse section of the cylindrical shell, k s -shear ratio of the spherical shell, D -cylindrical stiffness, 0 0 0 ν , ,ρ E -Poisson's ratio, Young's modulus and density of the shell material, p и qrespectively, the radial and tangential components of the distributed load acting on the shell, R − is the shell radius.
The motion of an elastic medium is described by scalar potential φ and non-zero component of vector potential ψ , which satisfy the wave equations [  If the shear modulus μ is set equal to zero μ0 = , then the equations of motion of the elastic medium will be the equations of acoustics.
Let us consider the initial stage of the process of impact of elastic shells on the surface of an elastic half-space [4 − 7], when no plastic deformations occur and the depth of the shell penetration into the medium is small.
The problem of interaction of elastic shells with an elastic half-space is solved in a linear formulation, therefore, we linearize the boundary conditions [2,3,8,9,11]: we transfer the boundary conditions from the perturbed surface to the undisturbed surface of the bodies that are deformed. We assume that there is no friction between the elastic halfspace and the penetrating body, or the slippage condition is valid.
As can be seen from Fig at the surface points of the contact area will be written as: wt -displacement of the shell as a rigid body, the function f() describes the shell profile, * 2θ as can be seen from Figures 1 and 2, the size of the shell sector in contact with the half-space. In the case of the cylindrical and the spherical shells: .
The kinematic condition that determines the half-size of the contact area We assume that the contact area is simply connected region, and this statement is equivalent to the fact that the stresses normal to the contact area are compressive: The initial conditions for potentials andare zero: For the problem of impact of an elastic shell on an elastic half-space, the velocity and displacement of the impacting body are found from the equation of motion by integrating it.
The equation of motion of a shell of mass M for the problem of impact with an initial velocity 0 V has the form: The condition for the absence of disturbances ahead of the front of longitudinal waves and the condition for damping of disturbances at infinity are valid.
We apply to the system of equations (2)           (3) the Laplace transform in the variable t (s is the transformation parameter) and the Fourier method of separation of variables, considering the evenness in x of the potential φ and the oddness of the potential ψ , and require the satisfaction of condition (14) - (15). Then [2 -6], in the space of Laplace transformants, we obtain the following representations for wave potentials [7 -9, 11]:  Using the orthogonality of the trigonometrical functions and the polynomials and the associated Legendre polynomials, we obtain the relations establishing the relationship between the harmonics of the series expansions of the functions p, q and V: Thus, the final form of the resolving ISVIE of the second kind will be as follows: symbol. Index j=1 corresponds to the case when the body penetrates into the medium at a speed varying according to a predetermined law (setting 1); if the velocity of the penetrating body is known only at the initial moment of time 0 t = , and at subsequent moments is determined from the equation of motion (statement 2), then j=2. If we exclude the fourth term in relation (43), then we obtain a condition from which the boundary of the contact region is determined without considering the rise of the medium.

NUMERICAL SOLUTION
The size of reduction N of the ISVIE of the second kind will be chosen from considerations of practical convergence. In case of plane problem To smooth out the oscillations arising from the summation of a finite number of terms of the series, as well as Gibbs phenomena near points of weak discontinuity, the averaging operation was used, defined in [2 -6], which, in the case of a sum of a finite number of terms of the trigonometric series, to memberwise multiplication of the members of the finite sum on σ n -Lanczos multipliers [8,9,11]. The integrals were calculated using the method of mechanical quadratures, in particular, the symmetric Gregory quadrature formula for equidistant nodes. The Cauchy problem for the differential equation (52) was solved by the Adams method (closed-type formulas) [2 -6] of order 1 m with a local truncation error 1 1 () m Ot +  [7 -9, 11]. As a result of discretization, we obtain a system of linear algebraic equations (SLAE). Calculations have shown that with an increase in the reduction size N, the determinant of the SLAE matrix increases indefinitely. The SLAE is poorly defined: as the reduction size N tends to infinity, the value of the determinant of the SLAE matrix also tends to infinity. This is due to the fact that the kernels 11  Methods of Tikhonov regularization and orthogonal polynomials do not work to neutralize such an exponential singularity. The approach [1 -5] for solving problems of dynamics makes it impossible to study the impact of elastic cylindric and spheric shells of the S.P. Timoshenko type and elastic bodies on an elastic foundation [7 -9, 11]. In addition, this approach makes it possible to determine the stress-strain state only on the surface of the medium into which the striker penetrates.

CONCLUSIONS
As a result of an attempt to solve the plane and the axisymmetric problems of the impact of a cylindric and a spheric fine shells of the S.P. Timoshenko type on the surface of an elastic half-space, applying the method of reduction of dynamic problems to infinite systems of Voltaire's equations of the second kind, the limitations of this technique were Математика та статистика 16 ПІДВОДНІ ТЕХНОЛОГІЇ • Вип.11 (2021), 3-18 промислова та цивільна інженерія revealed. This technique does not allow solving plane and axisymmetric [1] problems of dynamics for refined shells of the S.P. Timoshenko type and elastic bodies.
To solve [10, 12 -15] the problems of impact and nonstationary interaction [16 -20], the elastoplastic formulation [21 -23] can be used. It should be noted that to calibrate the computational [2] process in the elastoplastic formulation at the elastic stage, it is convenient and expedient to use the technique [2 -6] for solving the problems of dynamics, developed in [7 -9, 11].