 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| A.V. Andreev, "Method for Determining Power-Type Complex Singularities in Solutions of Singular Integral Equations with Generalized Kernels and Complex Conjugate Unknowns," Mech. Solids. 44 (5), 691-704 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
5 |
Pages |
691-704 |
| DOI |
10.3103/S0025654409050069 |
| Title |
Method for Determining Power-Type Complex Singularities in Solutions of Singular Integral Equations with Generalized Kernels and Complex Conjugate Unknowns |
| Author(s) |
A.V. Andreev (Elektrogorsk Research and Engineering Center for Nuclear Power Plant Safety (FSUE "EREC"), Svyatogo Konstanina 6, Electrogorsk, Moscow Region, 142530 Russia, andreev@erec.ru) |
| Abstract |
We develop a method for determining power-type complex
singularities of solutions for a class of one-dimensional singular integral equations with generalized kernels and complex conjugate unknown functions. By analyzing the characteristic part of a singular integral equation, we reduce the problem of determining the solution singularity exponents at the ends of the integration interval to two independent transcendental equations for these exponents. We show that the distribution of admissible singularity exponents is of continuous character. We present numerical results for a two-dimensional elasticity problem whose mathematical statement leads to a singular integral equation of the class under study. We also reveal the drawbacks of one classical approach to the determination of stress field singularities. |
| Keywords |
singular integral equation, generalized kernel, solution singularity, complex function |
| References |
| 1. | F. E. Erdogan, G. D. Gupta, and T. S. Cook,
"The Numerical Solutions of Singular Integral Equations,"
in Mechanics of Fracture,
Vol. 1: Methods of Analysis and Solutions of Crack Problems (Noordhoff Intern. Publ., Leyden, 1973),
pp. 368-425. |
| 2. | M. P. Savruk,
Two-Dimensional Problems of Elasticity for Bodies with Cracks
(Naukova Dumka, Kiev, 1981)
[in Russian]. |
| 3. | N. I. Muskhelishvili,
Singular Integral Equations
(Nauka, Moscow, 1968)
[in Russian]. |
| 4. | H. F. Bueckner,
"On a Class of Singular Integral Equations,"
J. Math. Anal. Appl.
14 (3), 392-426 (1966). |
| 5. | R. V. Duduchava,
Integral Equations of Convolution with Discontinuous Presymbols, Singular Integral Equations with Fixed Singularities, and Their Applications to Problems in Mechanics
(Metsniereba, Tbilisi, 1979)
[in Russian]. |
| 6. | P. S. Theocaris and N. I. Ioakimidis,
"The V-Notched Elastic Half-Plane Problems,"
Acta Mech.
32 (1-3), 125-140 (1979). |
| 7. | M. P. Savruk, E. Madenci, and S. Shkarayev,
"Singular Integral Equations of the Second Kind with Generalized Cauchy-Type Kernels and Variable Coefficients,"
Int. J. Numer. Meth. Engng
45 (10), 1457-1470 (1999). |
| 8. | A. F. Nikiforov and V. B. Uvarov,
Special Functions of Mathematical Physics
(Nauka, Moscow, 1984)
[in Russian]. |
| 9. | A. V. Andreev,
"Direct Numerical Method for Solving Singular Integral Equations of the First Kind with Generalized Kernels,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 1, 126-146 (2005)
[Mech. Solids (Engl. Transl.)
40 (1), 104-119 (2005)]. |
| 10. | F. D. Gakhov,
Boundary Value Problems
(Fizmatgiz, Moscow, 1958)
[in Russian]. |
| 11. | A. I. Kalandiia,
"Remarks on the Singularity of Elastic Solutions near Corners,"
Prikl. Mat. Mekh.
33 (1), 132-135 (1969)
[J. Appl. Math. Mech. (Engl. Transl.)
33 (1), 127-131 (1969)]. |
| 12. | F. Erdogan and V. Biricikoglu,
"Two Bounded Half Planes with a Crack Going through the Interface,"
Int. J. Engng Sci.
11 (7), 745-766 (1973). |
| 13. | A. M. Lin'kov,
Complex Method of Boundary Integral Equations in Elasticity
(Nauka, St. Petersburg, 1999)
[in Russian]. |
| 14. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,
Numerical Recipes in C. The Art of Scientific Computing
(Cambridge Univ. Press, Cambridge, 1992)
(http://www.nr.com). |
| 15. | D. N. Fenner,
"Stress Singularities in Composite Materials
with an Arbitrary Oriented Crack Meeting an Interface,"
Int. J. Fract.
12 (15), 705-721 (1976). |
| 16. | W. Yong-Li,
"Crack Tip Stress Singularities in a Bimaterial with an Inclined Interface,"
Int. J. Fract.
54, R65-R72 (1992). |
| 17. | M. L. Williams,
"Stress Singularities Resulting from Various Boundary Conditions in Angular Corners
of Plates in Extension,"
J. Appl. Mech.
19 (4), 526-528 (1952). |
| 18. | G. B. Sinclair,
"Stress Singularities in Classical Elasticity - II: Asymptotic Identification,"
Appl. Mech. Rev.
57, 385-439 (2004). |
|
| Received |
07 August 2006 |
| Link to Fulltext |
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