1. V ANNUAL MEETING OF THE GEORGIAN MECHANICAL UNION
8-10 OCTOBER, 2014, TBILISI
8-10 OCTOBER, 2014
V ANNUAL MEETING OF THE GEORGIAN MECHANICAL UNION
I.Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University, 2 University St., Tbilisi, Georgia
8-10 OCTOBER, 2014, TBILISI
Natela

2. Nuri Khomasuridze, Natela Zirakashvili
8-10 OCTOBER, 2014, TBILISI
8-10 OCTOBER, 2014
ON A METHOD FOR THE ELASTIC EQUILIBRIUM DETERMINATION OF A CONFOCAL ELLIPTIC RING
Nuri Khomasuridze, Natela Zirakashvili
Iv. Javakhishvili Tbilisi State University, I.Vekua Institute of Applied Mathematics, Tbilisi, Georgia,
Natela

3. 1. Equilibrium equations, physical law, setting the boundary value problems in the elliptic coordinates
In the elliptic coordinates
let us write down the equilibrium equations as follows
where
and
are the components of the
Displacement vector along the normal and tangetial of the curve
are Lamé coefficients,
is the Poisson’s ratio and
is the modulus of elasticity.
The physical equations will be written down as follows:
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4. Let us set the boundary-value problems for a confocal elliptic semi-ring, i.e. in area
Let us find the solution of the system of equations (1) , which meets the following boundary conditions:
Fig. 1. Confocal elliptic semi-ring.
The problems for one-fourth and three-fourth elliptic rings will be set similarly.
These problems are solved by means of the analytical solutions of the internal and external problems for the elliptic body or its parts, which are gained by the method of separation of variables.
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5. 2. Internal problems for a semi-ellipse
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8-10 OCTOBER, 2014
2. Internal problems for a semi-ellipse
Let us solve the internal boundary value problems for the semi-ellipse (Fig. 2) with (3), (4), (6) and the following boundary conditions:
The solution of these problems is as follows:
Fig. 2. Semi-ellipse.
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Natela

6. 8-10 OCTOBER, 2014, TBILISI
8-10 OCTOBER, 2014
The conditions (6) for are expedient to write down in the following equivalent form:
where
and
are harmonic functions, which, by using the method of separation of variables, we
present as follows: :
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Natela

7. fig.3. Image of the main matrix.
After inserting the harmonic functions selected on conditions of (3), (4), (7) from (10) in (9), an infinite system of linear algebraic equations to unknown values and is gained, with its main matrix having a block-diagonal form (Fig. 3).
fig.3. Image of the main matrix.
In all problems the dimension of the matrix
and
and when
where
is a finite number.
Let us insert and gained after solving the system in the selected and ; after inserting these functions in the expressions of displacements and stresses, it is possible to calculate (8) displacements and (2) stresses at any point of the body.
8-10 OCT. 2014, TBILISI

8. 3. External problems for the semi-ellipse
In infinite area (Fig. 4), let us solve external boundary-value problems with boundary conditions (3), (4), (5) . The solution of these problems is as follows :
Fig. 4. Infinite area with a semi-elliptic hole.
8-10 OCT. 2014, TBILISI

9. If and are given for , then we take and when and is given for then
8-10 OCTOBER, 2014, TBILISI
8-10 OCTOBER, 2014
The conditions of (5) for are expedient to write down in the following equivalent form:
If and are given for , then we take and when and is given for then
Harmonic functions and by using the method of separation of variables, are presented as follows:
After inserting and gained after solving the system in the selected and and after inserting these functions in the expressions of displacements and stresses, it is possible to calculate (12) displacements and (2) stresses at any point of the body
8-10 OCT. 2014, TBILISI
Natela

10. ნახ. 5 ელიფსური ნახევარრგოლი
8-10 OCTOBER, 2014, TBILISI
8-10 OCTOBER, 2014
4. Method of solving the boundary value problems
We consider the easy case, when Thus, we solve boundary value problems at
area (Fig.5) with following conditions:
ნახ. 5 ელიფსური ნახევარრგოლი
The essence of the solution of this problems is that the solution of each problem is reduced to the solutions of two problems (relevant external and internal problems).
8-10 OCT. 2014, TBILISI
Natela

11. ნახ. 6 უსასრულო არე, რომელზეც I ამოცანაა განხილული
8-10 OCTOBER, 2014, TBILISI
8-10 OCTOBER, 2014
The I problem is considered on area shown in Figure 6 with the following boundary conditions:
ნახ. 6 უსასრულო არე, რომელზეც I ამოცანაა განხილული
when , then
8-10 OCT. 2014, TBILISI
Natela

12. 8-10 OCT. 2014, TBILISI

13. are the solutions of the I problem.
The II problem is considered on area shown in Figure 7 with the following boundary conditions:
ნახ. 7 არე, რომელზეც II ამოცანაა განხილული
Here
where ,
and
are the solutions of the I problem.
The components of the displacement vector and the components of the tension tensor will be written down by the following formulae:
8-10 OCT. 2014, TBILISI

14. The sum of the solutions of the I and II problems will be the solutions of problems (1), (2), (14), (15), (16), (17) respectively.
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15. 5. To solve a particular problem
let us solve the boundary value problem in area (Fig. 5) with the following boundary conditions:
The solution of this problem is reduced to the solution of the two following problems.
The first problem (external problem) is considered on area shown in Figure 6 with the following boundary conditions:
Conditions (35) by (12b) we write to the following form
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16. From (33), (34) conditions
and from (36)conditions we obtain following system of equations :
and
From (38) we obtain following system
From (39
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17. From (40), (37), (21), (22) we obtain
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18. 8-10 OCT. 2014, TBILISI

19. The second problem (internal problem) is considered on area shown in Figure 6 with the following boundary conditions:
Here where and is the solution of the first problem.
By using (9ბ), the conditions (46) will be written down as follows:
From (43), (44) and (45)
After inserting (48) in (47), we gain:
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20. where and are the coefficients of expansion into Fourier series
and functions, respectively.
After equating the expressions at the same trigonometric functions in both sides of (49) equations, an infinite system of linear algebraic equations to and unknown values is gained.
From (50), we find and insert them in (48), by inserting the gained in ((27), (28), we gain the solution of the second problem (i.e. displacements and stresses at any point of the body)
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21. 8-10 OCT. 2014, TBILISI

22. The sum of the solutions of the first problem and of the second problem respectively, is the solution of problem
(29)-(32)
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23. Thank you for your attention
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24. 8-10 OCT. 2014, TBILISI References
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