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

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, natzira@yahoo.com Natela

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: 8-10 OCT. 2014, TBILISI

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. 8-10 OCT. 2014, TBILISI

2. Internal problems for a semi-ellipse 8-10 OCTOBER, 2014, TBILISI 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. 8-10 OCT. 2014, TBILISI Natela

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: : 8-10 OCT. 2014, TBILISI Natela

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

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

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

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

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

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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

The sum of the solutions of the I and II problems will be the solutions of problems (1), (2), (14), (15), (16), (17) respectively. 8-10 OCT. 2014, TBILISI

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 8-10 OCT. 2014, TBILISI

From (33), (34) conditions and from (36)conditions we obtain following system of equations : and From (38) we obtain following system From (39 8-10 OCT. 2014, TBILISI

From (40), (37), (21), (22) we obtain 8-10 OCT. 2014, TBILISI

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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: 8-10 OCT. 2014, TBILISI

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) 8-10 OCT. 2014, TBILISI

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The sum of the solutions of the first problem and of the second problem respectively, is the solution of problem (29)-(32) 8-10 OCT. 2014, TBILISI

Thank you for your attention 8-10 OCT. 2014, TBILISI

8-10 OCT. 2014, TBILISI References Мусхелишвили Н. И. Некоторые основные задачи математической теории упругости, изд. 5-ое, “Наука”, М., 1966 Timpe A. Die Airysche Function fur den Ellipsenring. Math Z., Bd. 17, 1923. Каландия Ф.И. Математические методы двумерной упругости. “Наука”, М., 1973. Шереметьев М.П. Упругое равновесие еллиптического колца. Прикл. Матем. И механ., т. XVII., вып.1, 1953. Мухадзе М.Г. Две задачи для еллиптического кольца. Труды грузин. политех ин-та, №5 (14),1960. Блох В.И. Кручение стержней с поперечными сечениями, ограниченными ортогональными дугами эллипсов и гипербол Изд. вузов строит. и архит., №12, 1964. Khomasuridze N., Zirakashvili N. Some two dimensional elastic equilibrium problems of elliptic bodies. Proceedings of I. Vekua Institute of Applied Mathematics, V.49, 1999, p. 39-48. Zirakashvili N. The elliptic semi-ring stress condition. Report of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics. V.16, No2, 2001, p. 70-73. N. Zirakashvili. Some boundary value problems of elasticity for semi-ellipses. Proceeding of I. Vekua Institute of Applied Mathematics, V.52, 2002, p. 49-55. Milan Batista. Stresses in a confocal elliptic ring subject to uniform pressure. The Journal of Strain Analysis for Engineering Design (Impact Factor: 1.01). 05/1999; 34(3):217221. DOI: 10.1243/0309324991513768. P.C. Misra, and A.K. Das. Stress distribution in an anisotropic elliptic ring compressed along the major axis. Indian J. pure appl. Matt., 14(10):1209-1216, 1983. By J. Burbea. A Numerical Determination of the Modulus of Doubly Connected Domains by Using the Bergman Curvature. MATHEMATICS OF COMPUTATION, VOLUME 25, NUMBER 116, OCTOBER, 1971 N. Zirakashvili. The numerical solution of boundary-value problems for an elastic body with an elliptic hole and linear cracks. J Eng Math (2009) 65:111–123 , DOI 10.1007/s10665-009-9269-z. N. Zirakashvili. On the numerical solution of some two-dimensional boundary-contact delocalization problems. Meccanica (2013) 48:1791–1804 DOI 10.1007/s11012-013-9709-8. Zirakashvili N. An applicationof the boundary element method in numerical analysis of stress concentration for elastic body. Report of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics. V.20, No2, 2005, p. 68-71. Zirakashvili N. Numerical solution of some boundary value problems of theory elasticity by boundary element method. Report of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics. V.21, No1, 2005, p. 65-68. Khomasuridze N., Zirakashvili N. Study of stress-strain state of a two-linear elliptic cylinder. Seminar of I. Vekua Institute of Applied Mathematics REPORTS, V.35, 2009, p. 34-38. Khomasuridze N. Representation of solutions of some boundary value problems of elasticity by a sum of the solutions of orher boundary value problems. Georgian Mathematical Journal, V.10, No 2, p. 257-270. Brown, J. W. and Churchill, R. V.: Fourier Series and Boundary value problems, 5th ed. New York: McGraw-Hill (1993) Bitsadze A.V. :Equations of mathematical physics, Mir Publishers (Translated from Ruusian) (1980) 8-10 OCT. 2014, TBILISI