Investigation of the stress state of the composite in the form of a half-space and a layer with a cylindrical pipe, with displacements specified on the boundary surfaces Текст научной статьи по специальности «Математика»

<<C©yL©qUQUM~J©U©MaL>>#»I)),2(0]9 / TECHNICAL SCIENCE

UDC 539.3

Miroshnikov Vitaly Yuryevich PhD, Associate Professor Kharkiv National University of Construction and Architecture

Kharkov, Ukraine DOI: 10.24411/2520-6990-2019-10677 INVESTIGATION OF THE STRESS STATE OF THE COMPOSITE IN THE FORM OF A HALFSPACE AND A LAYER WITH A CYLINDRICAL PIPE, WITH DISPLACEMENTS SPECIFIED ON

THE BOUNDARY SURFACES

The spatial problem of the theory of elasticity for a composite in the form of an elastic half-space rigidly adhered to a layer in which there is a longitudinal thick-walled circular cylindrical pipe is solved. Layer, halfspace and pipe - elastic homogeneous isotropic materials different from each other.

On the free surface of the layer and the inner surface of the pipe, displacements are specified. At the boundary of the layer and half-space, as well as at the boundary of the layer and the outer surface of the pipe, the matching conditions are specified. It is necessary to evaluate the stress state of a given environment.

Using the generalized Fourier method, a solution is obtained of the spatial problem of the theory of elasticity in cylindrical coordinates associated with the pipe and Cartesian coordinates associated with the layer and halfspace. Satisfying the boundary conditions and conjugation conditions, infinite systems of linear algebraic equations are obtained. The system of equations was solved by the reduction method. As a result, displacements and stresses were obtained at various points of the layer, half-space, and pipe.

Keywords: thick-walled pipe in a layer, composite, conjugation conditions, generalized Fourier method

Introduction.

You have to deal with a composite medium both in the calculations of composite materials, and in the calculations of large geotechnical structures. In all cases, it is important to know the stress state that occurs in each element of the composite medium. The most accurate calculation methods are analytical or numerical-analytical methods.

One of the numerical-analytical methods is the generalized Fourier method, based on which stationary problems of diffraction of elastic waves are solved [13].

In [4], the generalized Fourier method was supplemented by the theorems of addition of basic solutions, which made it possible to solve problems for a spatial body with several boundary surfaces (more than three).

Based on this method, problems are solved for a space with cylindrical cavities and various boundary

It is necessary to find a solution to the Lame equation AUj + (l — 2<J j ^1 'VdivUj = 0, where is the Poisson's ratio of the layer (j = 1), half-space (j = 2) or pipe (j = 3).

At the upper boundary of the layer, displacements Ui (x, z = U° (x, z) are specified, on the inner surface of the pipe displacements U3 (9, z)p=R = U° (9, z), where UY are displacements in the layer; U3

conditions [5], half-spaces with a cylindrical cavity or inclusion [6-11], for a cylinder with cylindrical cavities or inclusions [12], for a layer with a cylindrical cavity, inclusion or the pipe [13-16].

Formulation of the problem.

In a homogeneous elastic layer, a pipe is placed parallel to its boundary, with an outer radius R1 and an inner radius R2. The distance from the center of the pipe

to the upper boundary h, to the lower boundary h . The layer with the lower boundary is rigidly connected with the elastic homogeneous half-space.

We will consider the pipe in a cylindrical coordinate system (p, 9, z), the layer and half-space in the Cartesian coordinate system (x, y, z), which is equally oriented and aligned with the pipe coordinate system.

Gj -

0 j

d

1

- movement in the pipe; FjUj = 2G j —n divU, +—Uf +-(n X rotU j)]; G

E:

E i

1 1 - 2 0 j 1 dn 1 f j j 2(1 + Gj Y j

is

the elastic modulus of the layer (j = 1), half-space (j = 2) or pipe (j = 3);

d (x, z )=ujHe,C)+1/»#+u<%W,

UR („ z ) = )e<2) + U* )e22) + if ^

(1)

TECHNICAL SCIENCE / <<€©LL@(MUM~J©U©MaL>>#21](iqi)),2(

- {k)

known functions; ej , j = 1, 2, 3 -unit vectors of the Cartesian (k = 1) and cylindrical (k = 2) coordinate systems.

On the boundary of the layer and half-space, the conjugation conditions are given

U

y=-h

= U

y=-h

FiUi

y=-h

= F2U2

y=-h

at the boundary between the layer and the pipe, the matching conditions

U1(9 *)p=R = U3 (9 z)p=R ,

FUi(9 z = FU3 (9, z

(2)

(3)

(4)

(5)

where U2 — displacements in half-space.

The specified functions will be considered as rapidly decreasing from the origin of coordinates along the z axis for the cylinder and along the z and x axes for the layer boundaries.

Solution method.

We take the basic solutions of the Lame equation in the form [4]

y,z.

(x, y, z; X, |i) = N(d V(Xz)±Yy ; Rk (p, z; X) = Nk \ (XpV(Xz+m9); 4 ,m (p, 9, z; X) = Nkp )[(sigi X)mKm (X|p). é(Xz+k = 1,2,3;

(6)

N<d) = 1V ; N<d) = 4 (a- 1>f + 1 V( y •) ; N« = { rot(#.); Ni(p) = 1V ; XXX X X

N.

( p )

1 X

V

p

_d_ dp

+ 4(a-l)fv- e3(2 ^

dz

; N(p)= - rote ; 3 X V3

m

Y

= ^/x2 + , -œ<X,^<œ,

where /m {x), Km {x) - are the modified Bessel functions; R^ m, S^ - are, respectively

the

internal and external solutions to the Lamé equation for the cylinder; u( ), u- are the solutions to the

Lamé equation for the layer.

We will present the solution to the problem in the form

^ 3 œ œ ^

U1 = Z j ZBk,mM"Sk,mfoz;^)d^+

k=1-œ m=-œ

3 œ œ, \

+ EJ JH^X n) • 4+)(x,y,z;X,ai) + /~kl)(X,■ 4 )(x,y,z;X,ai) WpdX,

k=1-œ -œ

(7)

lk

3 œ œ

U

3 ^ \ \

2 = E J JHk2)(x,4+)(x,y,z;X,^a2)Jd^dx,

k=1-œ-œ

3 œ œ

U = E J E Ak,m (X) ■ Rk,m (p, 9, z; X) + A~k,m (X) ■ 4,m (p, 9, z; X )dX

(8)

(9)

k=1-œ m=-œ

/ TECHNICAL SCIENCE_

where Sk m {p, z;A), Rk m {p, 9, z; A) wk+){x,y, z; A, |) and uk , y, z; A, |) - the

basic solutions given by formulas (6), and the unknown functions H^^A, |), H^^A, |), Bk m (A),

h|2)(A, Ak m (A) and Afc m (A) must be found from the boundary conditions (1) and the conjugation conditions (2 - 5).

For the transition in basic solutions between coordinate systems, we use the formulas [15].

To fulfill the boundary conditions at the upper boundary of the layer, we equate (7), for y=h, given by

Ufo (x, z), represented through the double Fourier integral. So we get three equations (one for each projection)

with nine unknowns H^ (A, |), H^(A, |) and Bk m (A).

To satisfy the conjugation conditions at the boundary of the layer and half-space in displacements, we substitute in (2), for y= — h , the right-hand sides of (7) and (8). Moreover, writing down the expression U2 (x, z)y=_~, it is necessary to use the formulas for the transition from the solutions Sk m of the cylinder - (_)

to the solutions ' [15, formula (7)]. In a similar way, we can write three additional equations for stresses (3). So we get nine infinite systems of equations with unknown functions H(l)(A, |), //^(A, |),

Hk%, |) and Bk,m (A).

The determinant A of this system has the form A = —16 • y6 • e_3y(h—'• O(y)/ A6, where 0(y) -

is a function, for y> 0, it has only positive values and does not vanish, it follows from this that this system of equations has a unique solution.

Express H(1)(A, |), H(1)(A, |) and H ^2)(A, |) through Bk ,m (A).

To meet the conjugation conditions at the boundary between the layer and the pipe, in (7) we expand the basic solutions U^ and uk ^ according to the solutions Rk m using the transition formulas [15, formula (8)]. The

resulting vector, as well as (9), for p=R\, is substituted into (4). So we get three infinite systems of equations for conjugation of a layer and a pipe in displacements. Similarly, we write three equations for stresses (5).

To fulfill the boundary conditions on the inner surface of the pipe, we equate (9), for p=R2, given by

UR (9, z ), represented by the integral and the Fourier series.

Having received 9 infinite equations, instead of H^1 (A, |) and H(1 (A, |), we substitute the previously expressed functions through Bk m (A) and free ourselves from series in m and integrals in A. As a result, we

obtain a set of nine infinite systems of linear algebraic equations for determining unknown B^ OT(A),

Ak m (A) and Ak m (A). These infinite systems have the properties of equations of the second kind and, as a consequence, the reduction method can be applied to them.

Solving this system of equations, we find the unknowns Ak m (A), Ak m (A) and B^ m (A).

Found from an infinite system of equations B^ m (A), we substitute in the expressions for H^l)(A, |),

H|l)(A, |) and H ¿2)(A, This will identify all unknown tasks. Numerical studies of stress.

The layer material is concrete, Poisson's ratio ct\ = 0.16, elastic modulus E\=3.25 • 104 MPa. Half-space is clay, Poisson's ratio CT2 = 0.15, elastic modulus E2=250 MPa. The pipe is steel, Poisson's ratio CT3 = 0.25, elastic modulus E3=2 105 MPa. The outer radius of the pipe is R\= 25 mm., The inner R2 = 20 mm. The distance from the center

of the pipe to the upper boundary of the layer is h = 60 mm., To the lower boundary of the layer h = 40 mm.

TECHNICAL SCIENCE / «Ш^ШМУМ-ШУГМак»,®»!))^

With the weight of the processing equipment taken into account, on the upper boundary of the layer, the

displacement üf\x, z) = -1Q8-(z2 +102 •(;

X2 + 1Q2 )-2, UX") = UZ") = 0 are given. On the in-

2

r(" )-tt(" )

ner surface of the tube, there are no displacement U^ — U^ ^ — U^ — 0.

A finite system of equations of order m = 6 was solved. The accuracy of the fulfillment of the boundary conditions for the indicated values of geometric parameters was equal to 10-3.

In Fig. 1. stresses at the upper boundary of the layer in MPa are presented. Stresses Cx and Cz differ little from each other, therefore the stresses C z in the figure are presented.

-20

-10

0

10

20

-20

-10

0

10

1000 0

-1000 -2000 -3000 -4000 -5000

1000 0

-1000 -2000 -3000 -4000 -5000

Рис. 1. Stresses at the upper boundary of the layer: а - along the z axis, at x = 0; b - along the x axis, at z = 0; 1 - 7y ; 2 -

3 - /

1 / 2 /

1 \

3 \ — / N

1 J 2 /

1 \

20 x

b

®x; 3 ^xy

The maximum stresses that occur at the upper boundary of the layer in the direction of the x and z axis (Fig. 1) are normal stresses Cy . Stresses Cx have negative values (compression zone), and tangential stresses Txy also have rather large values.

At the lower boundary of the layer, the stresses are shown in Fig. 2. Here the voltages Cx and Cz differ from each other (lines 2, 3).

-40

-20

0

20

40

-40

-20

0

20

25 20 15 10 5 0

p-

ч x

3

25 20 15 10 5 0

40

2

J—- ✓ ч

s * * ✓ ✓ ✓ 4 V 4 4 \ 4

N

Fig. 2. Stresses at the lower boundary of the layer: a — along the z axis, at x = 0; b - along the x axis, at z = 0; 1 - Cy ; 2 - Cx; 3 - Cz

b

a

At the lower boundary of the layer (Fig. 2), normal stresses along the z axis decrease more or less uniformly, along the x axis (across the cylindrical cavity), stresses Cz fall off more slowly from the coordinate center. In fig. Figure 3 shows the stresses on the pipe surfaces along the radii Ri and R2, at z = 0 in MPa.

<<C©yL©qUQUM~J©U©MaL>>#»I)),2(0]9 / TECHNICAL SCIENCE

Fig. 3. Stresses on the pipe surfaces: a - along the radius R1; b - along the radius R2; 1 - 7p ; 2 - 7,

9 ; 3 ; 4

Stresses on the outer and inner surfaces of the pipe are of the same nature and differ only in the magnitude of the stresses (Fig. 3). The highest stresses occur in the upper part of the pipe (tc/2).

Conclusions.

The three-dimensional problem of the theory of elasticity for a multilayer medium consisting of a layer, half-space and a thick-walled pipe, which are interconnected by conjugation conditions, is solved. On the free boundary of the layer and the inner surface of the pipe, displacements are specified.

The proposed solution method is based on the generalized Fourier method and allows determining the stress-strain state of the medium under study with a predetermined accuracy.

Numerical studies were carried out for given nonzero displacements on the surface of the layer and zero on the inner surface of the pipe. The analysis showed that the largest loaded part of the pipe is its upper part.

Further research is relevant for more pipes.

References

1. Guz' A.N., Kubenko V. D. , Cherevko M. A. (1978) Difraktsiya uprugikh voln [Diffraction of elastic waves]. Kiev: Nauk. Dumka, 307 p. (In Russian).

2. Grinchenko V.T., Meleshko V. V. Garmonich-eskiye kolebaniya i volny v uprugikh telakh [Harmonic vibrations and waves in elastic bodies]. - Kiev: Nauk. Dumka. - 1981. - 284 p. (In Russian).

3. Volchkov V. V., Vukolov D. S., Storogev V. I. Difraktsiya voln sdviga na vnutrennikh tunnel'nykh tsilindricheskikh neodnorodnostyakh v vide polosti i vklyucheniya v uprugom sloye so svobodnymi granyami [Diffraction of shear waves by internal tunneling cylindrical non-homogeneities in the form of a cavity and inclusion in an elastic layer with free faces] / Mek-hanika tverdogo tela (Solid mechanics). - 2016. - Vol. 46. - P. 119 - 133. (In Russian).

4. Nikolayev, A. G., Protsenko, V. S. (2011) Obobshchennyy metod Fur'ye v prostranstvennykh zadachakh teorii uprugosti [The generalized Fourier method in spatial problems of the theory of elasticity]. Kharkov: Nats. aerokosm. universitet im. N.Ye. Zhu-kovskogo «KHAI», 344 p. (in Russian).

5. Miroshnikov V. Yu. (2017) On Computation of the Stress-Strain State of a Space Weakened by a System of Parallel Circular Cylindrical Cavities with Different Edge Conditions. Science and Practice: A New Level of Integration in the Modern World. 4th Intern. Conf. Proc. Scope Academic House. Sheffield. P. 7783. UK.

6. Nikolayev, A. G., Orlov, Ye. M., (2012) Resh-eniye pervoy osesimmetrichnoy termouprugoy kra-yevoy zadachi dlya transversal'no-izotropnogo po-luprostranstva so sferoidal'noy polost'yu [Solution of the first axisymmetric thermoelastic boundary value problem for a transversely isotropic half-space with a spheroidal cavity]. Problemy obchyslyuval'noyi mek-haniky i mitsnosti konstruktsiy - Problems of computational mechanics and structural durability, vol. 20, pp. 253-259. (in Russian).

7. Miroshnikov, V. Yu., (2018) First basic elasticity theory problem in a half-space with several parallel round cylindrical cavities. Journal of Mechanical Engineering, vol.21, no. 2, pp. 12 - 18.

8. Protsenko, V., Miroshnikov, V. (2018) Investigating a problem from the theory of elasticity for a halfspace with cylindrical cavities for which boundary conditions of contact type are assigned. Eastern-European Journal of Enterprise Technologies. Applied mechanics, vol. 4, no. 7, pp. 43 - 50. DOI: https://doi.org/10.15587/1729-4061.2018.139567

9. Nikolayev, A. G., Shcherbakova, A. YU., Yu-khno, A. I. (2006) Deystviye sosredotochennoy sily na transversal'no-izotropnoye poluprostranstvo s parabo-loidal'nym vklyucheniyem [Action of Concentrated Force on a Transversely-Isotropic Half-Space with Pa-raboloidal Inclusion]. Voprosy proyektirovaniya i pro-izvodstva konstruktsiy letatel'nykh apparatov - Issues of Design and Production of Structures of Aircraft, vol. 2, pp. 47-51. (in Russian).

10. Protsenko, V. S., Ukrainets, N. A. (2015) Primeneniye obobshchennogo metoda Fur'ye k resh-eniyu pervoy osnovnoy zadachi teorii uprugosti v po-luprostranstve s tsilindricheskoypolost'yu [Application of the generalized Fourier method to the solution of the first main problem of the theory of elasticity in a halfspace with a cylindrical cavity]. Visnyk Zaporiz'koho

TECHNICAL SCIENCE / <<€©LL@(MUM~J©U©MaL>>#»I)),2©]]9

natsional'noho universytetu - Bulletin of the Za-porizhzhyaNational University, vol. 2, pp. 193-202. (in Russian).

11. Protsenko, V. S., Ukrainets, N. A. (2004) The second main boundary value problem of the theory of elasticity for a half-space with a circular cylindrical cavity. Reports of the National Academy of Sciences of Ukraine vol. 12, pp. 52-58.

12. Nikolayev A. G., Tanchik Ye. A. Uprugaya mekhanika mnogokomponentnykh tel [Elastic mechanics of multicomponent bodies]. Khar'kov: Nats.aero-kosm.un-t im. N.Ye. Zhukovskogo «KHAI». 2014. 272 p. (in Russian).

13. Miroshnikov V. Yu. The study of the second main problem of the theory of elasticity for a layer with a cylindrical cavity [text] / V. Yu. Miroshnikov -Strength of Materials and Theory of Structures. - 2019. - №102. - P. 77-90. https://doi.org/10.32347/2410-2547.2019.102.77-90

14. Miroshnikov V. Yu. Determination of the Stress State of the Layer with a Cylindrical Elastic Inclusion [text] / V. Yu. Miroshnikov, A. V. Medvedeva, S. V. Oleshkevich - Materials Science Forum. - 2019. - Vol. 968. - pp. 413-420. https://doi.org/10.4028/www.scien-tific.net/MSF.968.413

15. Miroshnikov V. Investigation of the Stress Strain State of the Layer with a Longitudinal Cylindrical Thick-Walled Tube and the Displacements Given at the Boundaries of the Layer [text] / V. Miroshnikov -Journal of Mechanical Engineering. - 2019. - Vol. 22, N 2. - P. 4452. https://doi.org/10.15407/pmach2019.02.044

16. V. Y. Miroshnikov Third main problem of the theory of elasticity for the layer with a longitudinal thick-walled tube. East European Scientific Journal. -№6 (46). - 2019. - C. 21-27