Direct boundary integral equation method in the theory of elasticity

Author:
S. V. Kuznetsov

Journal:
Quart. Appl. Math. **53** (1995), 1-8

MSC:
Primary 73C35; Secondary 35Q72, 73V10

DOI:
https://doi.org/10.1090/qam/1315444

MathSciNet review:
MR1315444

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Abstract | References | Similar Articles | Additional Information

Abstract: The Direct Boundary Integral Equation Method (BIEM), which leads to second-kind singular integral equations for all types of commonly used boundary-value problems of the theory of elasticity, is formulated. The exposition is applied to anisotropic bodies with arbitrary elastic anisotropy.

**[1]**C. A. Brebbia,*Topics in Boundary Element Research. Vol*. 1 .*Basic Principles and Applications*, Springer-Verlag, Berlin, 1984**[2]**V. D. Kupradze and M. A. Alexidze,*On one applied method for solution of some boundary-value problems*, Soobshch. Akad. Nauk Gruz in SSR**30**, 529-536 (1963) (Russian)**[3]**V. D. Kupradze and M. A. Alexidze,*Method of functional equations for approximate solutions of some boundary-value problems*, Zh. Vychisl. Mat. i Mat. Fiz.**4**, 683-715 (1964) (Russian)**[4]**V. D. Kupradze,*On approximate solution for problems of mathematical physics*, Uspekhi Mat. Nauk (2)**22**, 59-107 (1967) (Russian)**[5]**M. A. Alexidze,*On completeness of some functional systems*, Differentsial'nye Uravneniya**3**, 1766-1771 (1967) (Russian)**[6]**A. B. Bakushinsky,*Notes on Kupradze-Alexidze's method*, Differentsial'nye Uravneniya**6**, 1298-1301 (1970) (Russian)**[7]**P. S. Theocaris, N. Karayanopoulos, and G. Tsamasphyros,*A numerical method for the solution of static and dynamic three-dimensional elasticity problems*, Comput. & Structures**16**, 777-784 (1983)**[8]**R. Mathon and R. L. Johnston,*The approximate solution of elliptic boundary-value problems by fundamental solutions*, SIAM J. Numer. Anal.**14**, 638-650 (1977)**[9]**F. J. Rizzo,*An integral equation approach to boundary value problems of classical elastostatics*, Quart. Appl. Math.**25**, 83-95 (1967)**[10]**A. C. Kaya and F. Erdogan,*On the solution of integral equations with strongly singular kernels*, Quart. Appl. Math.**45**, 105-122 (1987)**[11]**P. A. Martin,*End-point behaviour of solutions to hypersingular integral equations*, Proc. Roy. Soc. London Ser. A**432**, 301-320 (1991)**[12]**E. Kroner,*Das Fundamentalintegral der anisotropen elastischen Differentialgleichungen*, Z. Phys.**136**, 402-410 (1953)**[13]**V. D. Kupradze and M. O. Basheleishvili,*New integral equations of the theory of elasticity for anisotropic elastic bodies*, Soobshch. Akad. Nauk Gruzin. SSR**15**, 327-334 (1954) (Russian)**[14]**S. Helgason,*The Radon Transform*, Birkhäuser, Boston, 1980**[15]**I. M. Lifshitz and L. N. Rosenzveig,*On construction of Green's tensor for the main equation of the theory of elasticity in the case of an unbounded elasto-anisotropic medium*, Zh. Eksper. Teoret. Fiz.**17**, 783-791 (1947) (Russian)**[16]**R. Burridge,*The singularity on the plane lids of the wave surface of elastic media with cubic symmetry*, Quart. J. Mech. Appl. Math.**20**, 41-56 (1967)**[17]**J. R. Willis,*A polarization approach to the scattering of elastic waves*, I.*Scattering by a single inclusion*, J. Mech. Phys. Solids**28**, 287-305 (1980)**[18]**R. B. Wilson and T. A. Cruse,*Efficient implementation of anisotropic three dimensional boundary-integral equation stress analysis*, Internat. J. Numer. Methods Engrg.**12**, 1383-1397 (1978)**[19]**S. Bochner,*Harmonic Analysis and the Theory of Probability*, Univ. Calif. Press, Berkeley, 1955**[20]**S. V. Kuznetzov,*Fundamental solutions for Lamé's equations of anisotropic media*, Izv. Akad. Nauk SSSR Mekh. Tver. Tela (4), 50-54 (1989) (Russian)**[21]**S. V. Kuznetzov,*Fundamental solutions for equations of statics in the case of two variables*, Izv. Vyssh. Uchebn. Zaved. Mat. (7), 32-34 (1991) (Russian)**[22]**S. V. Kuznetzov,*Dislocation interaction energy in anisotropic media*, Prikl. Mat. Mekh.**55**, 894-897 (1991) (Russian)**[23]**V. D. Kupradze et al.,*Three-Dimensional Problems of Elasticity and Thermoplasticity*, North-Holland, Amsterdam, 1979**[24]**F. Treves,*Introduction to Pseudodifferential and Fourier Integral Operators. Vol*. 1 , Pseudodifferential Operators, Plenum Press, New York, 1982**[25]**S. V. Kuznetzov,*On discrete spectrum of singular integral operators in the theory of elasticity*, Izv. Vyssh. Uchebn. Zaved. Mat. (5), 26-29 (1991) (Russian)**[26]**S. V. Kuznetzov,*Green and Newmann's tensors in mechanics of anisotropic media*, Prikl. Mekh.**27**, 58-62 (1991) (Russian)**[27]**Pham The Lai,*Potentiels elastiques; tenseurs de Green et de Neumann*, J. Mech.**6**, 211-242 (1967)**[28]**N. M. Kublanovskaya,*Application of the analytical prolongation in numerical analysis by the use of change of variables*, Trudy Mat. Inst. Steklov.**53**, 145-185 (1959) (Russian)

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DOI:
https://doi.org/10.1090/qam/1315444

Article copyright:
© Copyright 1995
American Mathematical Society