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: 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.
C. A. Brebbia, Topics in Boundary Element Research. Vol. 1 . Basic Principles and Applications, Springer-Verlag, Berlin, 1984
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)
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)
V. D. Kupradze, On approximate solution for problems of mathematical physics, Uspekhi Mat. Nauk (2) 22, 59–107 (1967) (Russian)
M. A. Alexidze, On completeness of some functional systems, Differentsial’nye Uravneniya 3, 1766–1771 (1967) (Russian)
A. B. Bakushinsky, Notes on Kupradze-Alexidze’s method, Differentsial’nye Uravneniya 6, 1298–1301 (1970) (Russian)
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)
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)
F. J. Rizzo, An integral equation approach to boundary value problems of classical elastostatics, Quart. Appl. Math. 25, 83–95 (1967)
A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. 45, 105–122 (1987)
P. A. Martin, End-point behaviour of solutions to hypersingular integral equations, Proc. Roy. Soc. London Ser. A 432, 301–320 (1991)
E. Kroner, Das Fundamentalintegral der anisotropen elastischen Differentialgleichungen, Z. Phys. 136, 402–410 (1953)
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)
S. Helgason, The Radon Transform, Birkhäuser, Boston, 1980
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)
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)
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)
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)
S. Bochner, Harmonic Analysis and the Theory of Probability, Univ. Calif. Press, Berkeley, 1955
S. V. Kuznetzov, Fundamental solutions for Lamé’s equations of anisotropic media, Izv. Akad. Nauk SSSR Mekh. Tver. Tela (4), 50–54 (1989) (Russian)
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)
S. V. Kuznetzov, Dislocation interaction energy in anisotropic media, Prikl. Mat. Mekh. 55, 894–897 (1991) (Russian)
V. D. Kupradze et al., Three-Dimensional Problems of Elasticity and Thermoplasticity, North-Holland, Amsterdam, 1979
F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 1 , Pseudodifferential Operators, Plenum Press, New York, 1982
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)
S. V. Kuznetzov, Green and Newmann’s tensors in mechanics of anisotropic media, Prikl. Mekh. 27, 58–62 (1991) (Russian)
Pham The Lai, Potentiels elastiques; tenseurs de Green et de Neumann, J. Mech. 6, 211–242 (1967)
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)
C. A. Brebbia, Topics in Boundary Element Research. Vol. 1 . Basic Principles and Applications, Springer-Verlag, Berlin, 1984
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)
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)
V. D. Kupradze, On approximate solution for problems of mathematical physics, Uspekhi Mat. Nauk (2) 22, 59–107 (1967) (Russian)
M. A. Alexidze, On completeness of some functional systems, Differentsial’nye Uravneniya 3, 1766–1771 (1967) (Russian)
A. B. Bakushinsky, Notes on Kupradze-Alexidze’s method, Differentsial’nye Uravneniya 6, 1298–1301 (1970) (Russian)
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)
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)
F. J. Rizzo, An integral equation approach to boundary value problems of classical elastostatics, Quart. Appl. Math. 25, 83–95 (1967)
A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. 45, 105–122 (1987)
P. A. Martin, End-point behaviour of solutions to hypersingular integral equations, Proc. Roy. Soc. London Ser. A 432, 301–320 (1991)
E. Kroner, Das Fundamentalintegral der anisotropen elastischen Differentialgleichungen, Z. Phys. 136, 402–410 (1953)
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)
S. Helgason, The Radon Transform, Birkhäuser, Boston, 1980
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)
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)
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)
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)
S. Bochner, Harmonic Analysis and the Theory of Probability, Univ. Calif. Press, Berkeley, 1955
S. V. Kuznetzov, Fundamental solutions for Lamé’s equations of anisotropic media, Izv. Akad. Nauk SSSR Mekh. Tver. Tela (4), 50–54 (1989) (Russian)
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)
S. V. Kuznetzov, Dislocation interaction energy in anisotropic media, Prikl. Mat. Mekh. 55, 894–897 (1991) (Russian)
V. D. Kupradze et al., Three-Dimensional Problems of Elasticity and Thermoplasticity, North-Holland, Amsterdam, 1979
F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 1 , Pseudodifferential Operators, Plenum Press, New York, 1982
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)
S. V. Kuznetzov, Green and Newmann’s tensors in mechanics of anisotropic media, Prikl. Mekh. 27, 58–62 (1991) (Russian)
Pham The Lai, Potentiels elastiques; tenseurs de Green et de Neumann, J. Mech. 6, 211–242 (1967)
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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Article copyright:
© Copyright 1995
American Mathematical Society
