INVESTIGATING A PROBLEM FROM THE THEORY OF ELASTICITY FOR A HALF-SPACE WITH CYLINDRICAL CAVITIES FOR WHICH BOUNDARY CONDITIONS OF CONTACT TYPE ARE

V . P r o t s e n k o Doctor of Physical and Mathematical Sciences, Professor Department of Mathematics and Systems Analysis N. E. Zhukovsky National Aerospace University "Kharkiv Aviation Institute" Chkalovа str., 17, Kharkiv, Ukraine, 61070 Е-mail: k405@khai.edu V . M i r o s h n i k o v PhD, Associate Professor Department of Construction Mechanics Kharkiv National University of Construction and Architecture Sumska str., 40, Kharkiv, Ukraine, 61002 Е-mail: m0672628781@gmail.com При проектуваннi просторових конструкцiй необхiдно знати напружено – деформований стан тiла. Серед таких задач зустрiчаються розрахунковi схеми, в яких є пiвпростiр з цилiндричними порожнинами, на межах яких заданi умови контактного типу. Сегмент таких задач недостатньо дослiджений та потребує уваги. Запропоновано аналiтико-чисельний алгоритм розв’язання просторової задачi теорiї пружностi для пiвпростору з цилiндричними порожнинами. На межах порожнин заданi радiальнi перемiщення та дотичнi напруження, а на межi пiвпростору заданий один iз двох типiв граничних умов – перемiщення або напруження. Проведеними розрахунками встановлено напружено деформований стан пiвпростору. При фiксованих геометричних умовах було проведено чисельний аналiз трьох варiантiв задачi, коли на межi пiвпростору заданi перемiщення та трьох варiантiв задачi, коли на межi пiвпростору заданi напруження. Проведено порiвняльний аналiз варiантiв з рiзними граничними умовами мiж собою. Встановлено, що при рiзних видах заданих крайових умов (напруження або перемiщення), напруження i на межi прикладення таких умов змiнюються на протилежнi, тобто з розтягуючих на стискаючi або навпаки. Також встановлено, що крайовi умови на межi пiвпростору у виглядi напружень мають бiльший вплив на напружений стан нiж крайовi умови у виглядi перемiщень. Цi твердження мають мiсце при заданих на межах цилiндричних порожнин граничних умов контактного типу, якщо задана функцiя перемiщень та задана функцiя напружень однаковi. Наведений аналiз можна використовувати при проектуваннi конструкцiй, в розрахункових схемах яких є межа пiвпростору iз заданими на нiй граничними умовами контактного типу та цилiндричних порожнин, на поверхнi яких заданi перемiщення або напруження Ключовi слова: цилiндричнi порожнини в пiвпросторi, рiвняння Ламе, узагальнений метод Фур’є, нескiнченi системи лiнiйних алгебраїчних рiвнянь


Introduction
In the design of various structures, underground facilities and communications, there is a need in assessing stresses in a half-space with cavities.For this, it is necessary to have a method for calculation of the problems of the theory of elasticity, which makes it possible to find a stressed-strained state of a half-space with cylindrical cavities.We explore the problem, where displacement or stress are assigned at the boundaries of a cavity, and the conditions of a contact type are assigned at the boundaries of a half-space in the form of radial displacement and tangential efforts.Ready calculations of similar problems in the spatial variant were not found, so the problem of calculation of such problems is important.In addition to the proposed algorithm of calculation, an analysis of the stressed state, which enables prediction of the weak points at the stage of design, is presented in this paper.

Literature review and problem statement
To evaluate the stressed-strained state of a half-space with cylindrical cavities, several scientific studies apply the finite element method [1,2], which is an approximated calculation method and does not provide full reliability on the UDC 621 DOI: 10.15587/1729-4061.2018.127345accuracy of calculation when there are the infinite boundaries of an elastic body.
The most common problems for a half-space with cavities are the problems when cavities are perpendicular to the surface of a half-space [3][4][5][6].The calculation of these problems is based on the Weber integral transformation, the method of representations of Papkovich-Neuber, the theory of integral equations of Fredholm, the Neumann series, and the generalized integral equation of Cauchy.However, these methods cannot be used when cylindrical cavities are parallel to the surface of a half-space.
Papers [7][8][9] consider the problems of diffraction of waves in a half-space with a single cylindrical cavity based on the Helmholtz equation.Since article [7] considers all regions to be proportional to e ikx , in fact a flat problem with the use of the transformation of the plane with a circular opening into a concentric ring is solved.Papers [8,9] also consider flat problems, the wave equation of which is solved approximately using the comformal transformation and the collocation scheme of the least square.The described algorithms make it possible to calculate the problems of a half-space with only one cavity.
Articles [10,11] are devoted to determining the stressed state of the final cylinder.The method is based on the superposition of solutions and decomposition into Fourier and Dini series.But the problems for elastic bodies with multiple boundary surfaces cannot be solved within the framework of the classical approach.For such problems it was necessary to create a generalized Fourier method [12], the substantiation of which for spatial problems of the elasticity theory was given in [13].This method was laid as the basis for the approach to the solution of the considered problem.
The first major problem of the theory of elasticity for transversally-isotropic bodies limited by coordinate surfaces in cylindrical and parabolic coordinates of the stationary parabolic cavity is solved in [14] and with actual parabolic inclusion in [15].The applied problem about the effect of a concentrated force on a sandstone half-space with parabolic inclusion was considered in [16].The thermo-elastic boundary problem for a transversally-isotropic half-space with a spherical cavity was considered in [17].In articles [18,19], the first and the second basic problems of the elasticity theory for a half-space with a single cylindrical cavity were explored.The second basic problem of the elasticity theory for a half-space with several cylindrical cavities was solved in [20].All these papers are based on the generalized Fourier method, but the algorithms, presented in them, do not make it possible to directly address the mixed problems with the boundary conditions of the contact type and explore the stressed state of a half-space with such boundary conditions.For this, it is necessary to explore further the possibility of solving the problem with the contact type conditions.
Mixed problems were considered for a space with cylindrical cavities, when displacements are assigned on the boundaries of some cavities, stresses are assigned on the boundaries of other cavities [21].Mixed problems for space were considered when displacements are assigned on some boundaries, stresses are assigned on the other boundaries, and the conditions of the contact type are assigned on the third boundaries [22].These works are based on the generalized Fourier method, but can not be applied directly to the problems in a half-space.
It follows from the above that the problems for a halfspace with cylindrical cavities with the boundary conditions of the contact type need studying.
That is why it is appropriate to develop the analytical-numerical method for solving a mixed problem of the elasticity theory for a half-space with cylindrical cavities and some boundary conditions of the contact type.

The aim and objectives of the study
The aim of this research is the evaluation of the stressedstrained state of a half-space with cylindrical cavities, under the following conditions: radial displacements and tangential stresses are assigned at the boundaries of cavities, and one of the two types of boundary conditions -displacements or stresses -is assigned at the half-space boundary.
To accomplish the aim, the following tasks have been set: -to develop the analytical-numerical algorithm of calculation; -to conduct numerical studies for a half-space and two cylinders and to analyze the influence of the type of boundary conditions on the stress in the zone of the isthmus between the cylinders and the isthmus between the half-space boundary and a cylinder.

1. Problem statement
Elastic homogeneous half-space has N circular cylinder parallel cavities, non-crossing each other and the boundary of a half-space (Fig. 1).
The cavity will be considered in the cylindrical coordinate system ( , p ρ , z where p is the number of the cylinder), a half-space will be considered in Cartesian coordinates (x, y, z), which are equally oriented and combined with the coordinate system of the cylinder with number p=1.The half-space boundary is located at the distance y=h, the equation of the surface of the cylinders S p : ρ p =R p , p=1, 2,…, N. It is necessary to find a solution to the Lame's equation under conditions that at the boundaries of a half-space, one of the two types of boundary conditions -displacement x z is assigned at the boundaries of a half-space, and conditions of the contact type are assigned at the boundaries of the cavities where the right sides of these equalities are the known functions.
All the assigned vectors and functions will be considered descending to zero at long distances from the origin along coordinate z for cylinders and along coordinates x and z for a half-space.

2. The method of solution
Let us select the basic solutions of the Lame's equation for the specified coordinate systems in the form of [12]: ; ; , ; ; , ; ; where M d =(x, y, z), M p =(ρ p , ϕ p , z) are the points of a space, respectively, in Cartesian coordinates and in the cylindrical coordinate system, connected with p -cylinder; ( )  The solution to the problem will be represented in the form of , ; , , For transition between the coordinate systems (Fig. 1), we will use the formulas: -for transition from the coordinates of the cylinder with number p to the coordinates of a half-space, we will generalize formula [18] ( ) ( where -for transition from the coordinates of a half-space to the coordinates of the cylinder p, we will generalize formula [18] ( ) , where , ; x y are the coordinates of cylinder p relative to the cylinder number 1. ( , ; where , e ρ , e φ z e are the orts in the cylindrical coordinate system, -for transition from the coordinates of cylinder p to coordinates of cylinder 1 [12] ( ) ; , , ; where α p1 is the angle between the coordinate axis x 1 and section 1 , For transition from the coordinates of cylinder 1 to coordinates of cylinder p in formula (7), the places of indices should be changed To satisfy the boundary conditions at the boundary of half-space y=h, the left part (4) with the help of transition formula (5) will be re-written in the Cartesian coordinates through the basic solutions ( ) .k u − If the boundary conditions on the boundary y=h are assigned in displacements, the resulting vector (at y=h) will be equaled to the assigned vector but if boundary conditions are assigned in stresses, we will find the stress for the resulting vector and equal it (at y=h) to the assigned vector ( )

B λ
Using the formulas of transition from the Cartesian system to the cylindrical system (6), as well as from one cylinder to the other (7), we will rewrite (4) in the coordinates of the cylinder number p through basic solutions , , k m S If now we find U ρ (ϕ ρ , z) and stresses τ ρϕ , τ ρz , for the right side of equation ( 4) on the surface of each cylindrical cavity and take into consideration boundary conditions (1), we will obtain the system of equations for coefficients ( ) ( ) , , p k m B λ which includes functions H k (, ), for each cavity p.The determinant of this system is not equal to zero, moreover -for ( ) ( ) ( ) -for ( ) ( ) ( )

B λ
For the derived systems, using inequality (8), a definite possibility of solution on conditions of not touching the boundaries was proved.Moreover, these systems can be solved with the truncating method and approximate solutions converge to the exact ones.Functions ( ) ( ) , , p s m B λ found from the infinite system of equations, will be substituted in expressions for H k (, ).This will determine all the unknown problems.

Numerical studies for a half-space and two cylinders
We have two parallel cylindrical cavities in a half-space (Fig. 1), p=2.A half-space is isotropic material, Poisson ratio σ=0.35, the elasticity modulus E=2 kN/cm 2 .The boundary of a half-space is located at the distance h=40 cm, the cylinders, the radii of which are R 1 =R2=10 cm, are located on the horizontal axis (α 12 =0) at the distance of 12 50� = cm.Several variants of the problems with various boundary conditions were calculated -three variants, when displacement was assigned at the boundary of a half-space and three variants when stresses were assigned at the boundary of a half-space.In all variants, conditions of the contact type are assigned at the boundaries of the cylinders.
An infinite system of equations was reduced to the finite by parameter m -the order of the system.The influence of the value of parameter m was studied in [18].The integration boundaries for the assigned boundary functions were taken from -1...1.Calculation of integrals was performed using quadratic formulas of Filon and Simpson.Accuracy of the implementation of the boundary conditions at specified values of geometrical parameters was brought to 10 -3 (m=8).
Variant 1 Displacement is assigned at the boundary of a half-space.At the boundary of cylinder 1, radial displacement is assigned ) and tangential stresses = are assigned at the boundary of cylinder 2, boundary conditions are z ρφ ρ τ = τ = Fig. 2 shows the diagram of normal stresses on the isthmus between cylinder 1 and the boundary of a half-space (Fig. 2, а) and on the isthmus between the cylinders 1 and 2 (Fig. 2, b) in plane z=0.
Variant 3 Displacement ( ) ( ) 0 = is assigned at the boundary of a half-space and tangential stresses are assigned In this variant, the boundary of the half-space with the maximum values in the middle between the cylinders is "loaded", which influenced the stressed state on the isthmuses (Fig. 5).Thus, on the line between cylinder 1 and the half-space boundary (Fig. 5, а) σ ρ =-0.005 kN/сm 2 has an extreme value at the border of cylinder 1, at the same time σ ϕ and σ z have the extreme values between the cylinder and the halfspace boundary (σ ϕ =-0.003 kN/сm 2 , σz=0.002 kN/сm 2 ).On the isthmus between the cylinders (Fig. 5b), due to the vertical pressure of the boundary points of the half-space, stress σ ϕ , which is directed perpendicularly to the halfspace boundary, has extreme compressing values contrary to the maximum assigned displacements of the half-space (σ ϕ =-0.004 kN/сm 2 ).Stress σ ρ =-0.001 kN/сm 2 also has a small increase in compressing forces on the surface of the cylinders.Compared to variant 1, in this case stress is assigned at the boundary of a half-space (diagram of the stressed state is shown in Fig. 6).Fig. 6 shows that a change in boundary conditions at the half-space boundary did not affect the stressed state of the isthmus between the cylinders (Fig. 6, b), but affected the stressed state of the isthmus between cylinder 1 and halfspace boundary (Fig. 6, a).Thus, at the boundary half-space σ ρ =0, σ ϕ and σ z have now not compressing but stretching values (σ ϕ =0.015 kN/сm 2 , σz=0.015 kN/сm 2 ).
Variant 6 At the half-space boundary, stress ( ) ( ) 0 At the boundaries of cylinders 1 and 2, radial displacement ( ) ( ) ( ) ( ) and tangential stresses are assigned In contrast to variant 3, at the half-space boundary, stresses are applied instead of displacements, which influences the stressed state of the isthmuses (Fig. 8).
Load in the form of a single stress, unlike the load in the form of a single displacement, have a greater impact on the stressed state.Thus, the stress on the isthmus between the cylinders (Fig. 8, b) has the same form as in Fig. 5, b, but is larger in values, for example, the stress between the cylinders in variant 6 σ ρ =-0.012 kN/сm 2 , σ ϕ =-0.049 kN сm 2  On the isthmus between cylinder 1 and the half-space boundary, unlike variant 3 (Fig. 5, а), the values of stresses are higher, moreover, σ z in this case is compressing and increases at the half-space boundary (σ z =-0.024 kN/сm 2 ).Stress σ ρ also, unlike variant 3 (Fig. 5, а), at the half-space boundary is compressing (σ ρ =-0.023 kN/сm 2 ).Stress σ ϕ , in comparison with variant 3 (the maximum value σ ϕ = =-0.003kN/сm 2 ), has higher values (the maximum value σ ϕ =-0.034 kN/сm 2 ).

Discussion of the obtained results for the stressed state and the method for solving the problem
In the framework of the accepted linear model of the homogeneous isotropic medium and precise problem statements, the derived results (distribution of stress fields in a multi-link body) are explained by the response of an elastic body to: 1) existence of some flat and curved surfaces that limit the body; 2) selected system of boundary conditions on these surfaces.
For another system of boundary conditions, at other equal factors, the response of an elastic body will be different.
Based on the generalized Fourier method, the analytical-numerical algorithm of calculation of the spatial problem of the elasticity theory was developed.The algorithm implies the following boundary conditions: one of the two types of boundary conditions -displacement or stress -at the halfspace boundary, the contact type conditions in the form of

Fig. 1 .
Fig. 1.Half-space with cylindrical cavities , 2, 3 j = is the orts of the Cartesian (k=1) and cylindrical (k=2) coordinate systems; σ is the Poisson coefficient; I m (x), K m (x) are the modified Bessel functions; , , respectively internal and external solutions to the Lame's equation for a cylinder;( ) , the solutions to the Lame's equation for a half-space.
are the basic solutions, which are assigned by formulas (2) and (3), and it is neces-sary to find the unknown functions H k (, ) and from the boundary conditions.
, ±1, ±2,…; the double Fourier integral.From the resulting equations, we will find the functions H k (, ) As a result, we will obtain the totality from N•3 of non-finite systems of linear algebraic equations for determining unknown ( ) ( )

Fig. 2 .
Fig. 2. Normal stresses in the coordinates of cylinder 1 in plane z=0: а -on straight line x=0 between cylinder 1 and the boundary of a half-space; b -along section O 1 O 2 between cylinders

Fig. 3 .
Fig. 3. Normal stresses in coordinates of cylinder 1 in plane z=0: а -along the straight line x=0 between cylinder 1 and the boundary of the half-space; b -along section O 1 O 2 between cylinders

. 4 .
At the boundary of cylinders 1 and 2, radial displacement

Fig. 4 .
Fig. 4. Function of normal displacement which is set on the surface of the half-space

a b Fig. 6 .
Normal stresses in coordinates of cylinder 1 in plane z=0: а -on straight line x=0 between cylinder 1 and halfspace boundary; b -on section O 1 O 2 between cylinders Variant 5 Stress is assigned at the boundary of the half-space

Fig. 7 .
Fig. 7. Normal stresses in the coordinates of cylinder 1 in the plane z=0: а -along the straight line x=0 between cylinder 1 and half-space boundary; balong section O 1 O 2 between cylinders

Fig. 8 .
Fig. 8. Normal stresses in the coordinates of cylinder 1 in plane z=0: а -along the straight line x=0 between cylinder 1 and the half-space boundary; b -along section O 1 O 2 between the cylinders ;