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

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.


INTRODUCTION
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]. This led to the feasibility of developing other mathematical approaches and models. In [9, 11 -14], a new approach to solving the problems of impact and nonstationary interaction in the elas-toplastic mathematical formulation [15 -19] was developed. In non-stationary problems, the action of the striker is replaced by a distributed load in the contact area, which changes according to a linear law [20 -22]. 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 [23 -26], elastic half-space [27 -29], elastic layer [30], elastic rod [31,32] were developed using method of the influence functions [33]. In [23] 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 halfspace. 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 [33].
In this paper, we investigate the approach [3 -6] 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 [1 -4], after the reduction of the infinite system of Volterra integral equations of the second kind [5 -7, 10] 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 spherical shell, moving perpendicular to the surface of the elastic halfspace 0 z  , reaches this surface at time t=0.
We associate with the shell, as shown in Fig. 1, a movable spherical coordinate system φθ r , where φis the longitude of the radius vector r, θis the polar angle. The shell penetrates into the elastic medium at a speed , Tthe 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) 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, φis 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   2  0   ,  ,  ,  ,   σ  , , σ, is the vector of movement of points of the environment; σ , σ zz rznonzero components of the stress tensor of the medium; Mis the shell running mass; () T vt , () T wtspeed and movement of the shell as a solid. 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 spherical shells and considering the shear and inertia of rotation of the transverse section, due to (1), take the following form [34, pp. 297, 307]: where Фangle of rotation of the normal section to the middle surface, k sshear ratio, Dcylindrical 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 [3 − 6], 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 [1,2,7,8,10]: 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. 1, the projections of the functions 0 u , 0 w , p and q on the or and oz axes will be equal:  wtdisplacement of the shell as a rigid body, the function f(x) describes the shell profile, * 2θ as can be seen from Figure 1, the size of the shell sector in contact with the halfspace. In the case of a spherical shell: The kinematic condition that determines the half-size of the contact area * () xt is written as follows: 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 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.
Since the impact process is short-term, the perturbation region at each moment of time t is finite. Restricting ourselves to a finite interval Thus, for times ( ) 0 tT  , the considered problem is reduced to a nonstationary problem for a half-cylinder with mixed boundary conditions at its end. To represent the displacement vector as: gradφ rotψ, divψ 0, u = + = on the lateral surface of the half-cylinder, we select, for example, the conditions for sliding termination: Consider the initial -boundary value problem (2)        Just as in [1 -5], the dependence between the harmonics of the vertical component of the velocity and normal stresses on the surface of the half-space is determined [6 -8, 10]:  Substituting (22) and (23) into (39) with allowance for sin θ r = , arising from geometric considerations in the zone of the contact region, and representing both parts of (39) in the form of series in 0 (λ) n Jr , 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 ( 0, ) n = : (4) 0 0 To solve the problem, when the shell penetration velocity () T vt is a predetermined function, it is sufficient to numerically implement equations (40).
The expression for the reaction force of the elastic half-space (12), using (38), can be rewritten as: The equation of motion of the shell (10) with the initial conditions takes the form: To solve the problem of impact with an initial velocity 0 V , the system of equations (40) must be supplemented with the equation of motion (41).
The contact area is determined considering the rise of the medium from the condition: 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 (42), 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.
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 (41) was solved by the Adams method (closed-type formulas) [1 -5]  () m Ot +  [6 -8, 10]. 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 ( , ) Q n t , 22 ( , ) Q n t in (32), (33) have asymptotic exp( ( )) On in the parameter n, 11  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 shells of the S.P. Timoshenko type and elastic bodies on an elastic foundation [6 -8, 10]. 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 axisymmetric problem of the impact of a spherical fine shell 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 revealed. This technique does not allow solving plane and axisymmetric problems of dynamics for refined shells of the S.P. Timoshenko type and elastic bodies.
To solve [9, 11 -14] the problems of impact and nonstationary interaction [15 -19], the elastoplastic formulation [20 -22] can be used. It should be noted that to calibrate the computational [1] process in the elastoplastic formulation at the elastic stage, it is convenient and expedient to use the technique [1 -5] for solving the problems of dynamics, developed in [6 -8, 10].