Fundamental and singular solutions of Lamé equations for media with arbitrary elastic anisotropy
Author:
Sergey V. Kuznetsov
Journal:
Quart. Appl. Math. 63 (2005), 455-467
MSC (2000):
Primary 35E05, 74S15
DOI:
https://doi.org/10.1090/S0033-569X-05-00969-X
Published electronically:
August 17, 2005
MathSciNet review:
2169028
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Abstract: Fundamental and singular solutions of Lamé equations for media with arbitrary elastic anisotropy are constructed on the basis of multipolar expansions (expansions in spherical harmonics) of symbols and the corresponding operators. Theorems of convergence are formulated. A posteriori error estimates are presented.
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14 I. Fredholm, Sur les équations de l’équilibre d’un corps solide élastique, Acta. Math. 23, 1–42 (1900).
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26 V. D. Kupradze (editor), T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-dimensional Problems of Mathematical Theory of Elasticity and Thermo-Elasticity (in Russian), Nauka, Moscow, 1976.
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32 S. V. Kuznetsov, Fundamental solutions for equations of harmonic vibrations in the theory of elasticity, C. R. Acad. Sci. Paris 322 (Ser. IIb), 871–878 (1996a).
33 S. V. Kuznetsov, On the operator of the theory of cracks, C. R. Acad. Sci. Paris 323 (Ser. IIb), 427–432 (1996b).
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51 W. Thomson (Lord Kelvin), On the equations of equilibrium of an elastic solid, Cambr. Dubl. Math. J. 3, 87–89 (1848).
52 F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, vol. I. Plenum Press, New York and London 1982.
53 F. Tricomi, Equazioni integrali contenenti il valor principale di un integrale doppio, Math. Z. 27 (1), 87–133 (1927).
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- J. R. Willis, The elastic interaction energy of dislocation loops in anisotropic media, Quart. J. Mech. Appl. Math. 18 (1965), 419–433. MR 193828, DOI https://doi.org/10.1093/qjmam/18.4.419
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58 N. Zeilon, Das Fundamenlintegral der allgemeinen partiellen linearen Differentialgleichung mit Konstanten Koeffizienten, Ark. Matem., Astron. Fys. 6(38), 1–32 (1911).
1 M. O. Basheleishvili, On fundamental solutions of the differential equations of an anisotropic elastic body (in Russian), Soobsch, Akad. Nauk Gruz. SSR 19, 393–400 (1957).
2 P. Bézier, Sur quelques propriétés des solutions de problèmes de l’élastostatique linéaire, C. R. Acad. Sci. Paris. A265, 365–367 (1967).
3 S. Bochner, Harmonic Analysis and the Theory of Probability, Univ. of California Press, Berkeley and Los Angeles, 1955.
4 H. Bross, Zur Richtungsabhangigkeit physikalischer Eigenschaften in Kristallen mit besonderer Berucksichtung der galvano-und-thermomagnetishen Effekte, Z. Naturforschnung 15a, 859–874 (1960).
5 H. Bross, Theorie der galvanomagnetischen Erscheinungen bei beliebigen Energiefachen und anisotroper Streuung der Leitungselektronen, Phys. Kondens. Materie 3, 349–373 (1965).
6 H. Bross, Das Fundamentalintegral der Elastizitätstheorie für kubische Median, ZAMP 19, 434–446 (1968).
7 T. V. Burchuladze, On some plane boundary-value problems for anisotropic bodies (in Russian), Trudy Tbil. Mat. Inst. 27, 293–332 (1960).
8 T. V. Burchuladze, Some boundary-value problems for a class of elliptic systems (in Russian), Soobsch. Akad. Nauk Gruz. SSR 31, 513–520 (1963).
9 T. V. Burchuladze and T. G. Gegelia, Development of the Potential Method in the Elasticity Theory (in Russian), Metsniereba, Tbilisi, 1985.
10 A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math. 79, 901–921 (1957).
11 A. Deb, D. P. Henry, Jr., and R. B. Wilson, Alternate BEM formulation for $2$- and $3$D anisotropic thermoelasticity, Int. J. Solids Struct. 27, 1721–1738 (1991).
12 R. E. Edwards, Functional Analysis, Holt, Rinehart and Winston, New York, 1965.
13 G. Fichera, Existence Theorems in Elasticity. In: Handbuch der Physik, Bd. Via/2, Springer-Verlag, Berlin-Heidelberg-New York, 1972.
14 I. Fredholm, Sur les équations de l’équilibre d’un corps solide élastique, Acta. Math. 23, 1–42 (1900).
15 G. M. Hatiashvili, Fundamental solutions of equations of equilibrium for two-dimensional stress-state of anisotropic medium (in Russian), Soobsch. AN Gruz. SSR, 108, 509–512 (1982).
16 E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, Cambridge, 1931.
17 L. Jentsch and D. Natroshvili, Non-classical interface problems for piecewise homogeneous anisotropic elastic bodies, Math. Meth. Appl. Sci. 18, 27–49 (1995).
18 L. Jentsch, D. Natroshvili, and W. Wendland, General transmission problems in the theory of elastic oscillations of anisotropic bodies (Basic interface problems), I., J. Math. Anal. Appl. 220, 397–433 (1998).
19 L. Jentsch, D. Natroshvili, and W. Wendland, General transmission problems in the theory of elastic oscillations of anisotropic bodies(Mixed interface problems). II., J. Math. Anal. Appl. 235, 418–434 (1999).
20 F. John, Plane waves and spherical means: Applied to partial differential equations, Springer N.Y., 1955.
21 N. S. Kahniashvili, On a case of elementary representation of fundamental solutions for differential equations of anisotropic elastic body (in Russian), Trudy Tbil. Univer. 64, 123–126 (1957).
22 R. V. Kapanadze, Analysis of boundary value problems for anisotropic bodies by potential methods. I. (in Russian), Soobsch. AN Gruz. SSR 87, 82–113 (1987).
23 N. Kinoshita and T. Mura, On boundary value problem of elasticity, Res. Rep. Fac. Eng. Meiji. Univ. 8, 56–82 (1956).
24 E. Kröner, Das Fundamentalintegral der anisotropen elastischen Differentialgleichungen, Z. Physik, 136, 402–410 (1953).
25 V. D. Kupradze and M. O. Basheleishvili, New integral equations of the theory of elasticity of anisotropic bodies (in Russian), Soobsch. Akad. Nauk Gruz. SSR 15, 327–334 (1954).
26 V. D. Kupradze (editor), T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-dimensional Problems of Mathematical Theory of Elasticity and Thermo-Elasticity (in Russian), Nauka, Moscow, 1976.
27 S. V. Kuznetsov, Fundamental solutions for Lame’s equations in the theory of elasticity (in Russian), Izv. An SSSR. Mech. Tverdogo Tela 4, 50–54 (1989).
28 S. V. Kuznetsov, Fundamental solutions for equations of equilibrium for cylindrically anisotropic axially symmetric bodies (in Russian), Izv. An SSSR. Mech. Tverdogo Tela 2, 98–102 (1990).
29 S. V. Kuznetsov, Construction of the Green and Neumann tensor in the theory of elasticity of an anisotropic elastic body (in Russian), Prikl. Mekh. 27 (7), 58–62 (1991).
30 S. V. Kuznetsov, Direct boundary integral equation method in the theory of elasticity, Quart. Appl. Math. 53, 1–8 (1995a).
31 S. V. Kuznetsov, Energy and singular solutions in anisotropic elasticity, C. R. Acad. Sci. Paris, 321 (Ser. IIb), 309–314 (1995b).
32 S. V. Kuznetsov, Fundamental solutions for equations of harmonic vibrations in the theory of elasticity, C. R. Acad. Sci. Paris 322 (Ser. IIb), 871–878 (1996a).
33 S. V. Kuznetsov, On the operator of the theory of cracks, C. R. Acad. Sci. Paris 323 (Ser. IIb), 427–432 (1996b).
34 G. Leibfrid, Versetzungen in anisotropen Material, Z. Physik 135, 23–43 (1953).
35 I. M. Lifshits and L. N. Rozentsweig, On computation of Green tensor for the main equation of elasticity at the case of infinite elastic-anisotropical medium (in Russian), J. Exper. Theor. Phys. 17, 783–791 (1947).
36 W. Magnus and F. Oberhettinger, Formeln und Sätze für die speziellen Funktionen der mathematischen Physik, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1948.
37 M. Marcus and H. Mink, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 1964.
38 A. J. McConnell, The hypercircle method of approximation for a system of partial differential equations of the second order, Proc. Roy. Irish Acad. 54A, 263–290 (1951).
39 S. G. Mikhlin, Singular Integrals and Integral Equations (in Russian), Fizmatgiz, Moscow, 1962.
40 D. Natroshvili, Mixed interface problems for anisotropic elastic bodies, Georgian Math. J. 2 (6), 631–652 (1995).
41 M. N. Perelmuter and S. V. Kuznetsov, BEM applications to $3$D stress analysis of anisotropic composite structures, In: Second Int. Conf. on Composites Engng., New Orleans, 585–586 (1995).
42 F. J. Rizzio and D. J. Shippey, A method for stress determination in plane anisotropic elastic bodies, J. Comp. Mater. 4, 36–61 (1970).
43 A. W. Sáenz, Uniformly moving dislocations in anisotropic media, J. Ration. Mech. Anal. 2, 83–98 (1953).
44 N. A. Schclar, Anisotropic Analysis using Boundary Elements. Topics in Engineering, vol. 20, Computational Mechanics Publications, Southampton 1994.
45 C. Somigliana, Sopra l’equilibrio di un corpo elastico isotropo, Nuovo Cimento 18, 161–166 (1885)
46 E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.
47 E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.
48 E. Sternberg and R. A. Eubanks, On the concept of concentrated loads and an extension of the uniqueness theorem in the linear theory of elasticity, J. Ration. Mech. Anal. 4, 135–168 (1955).
49 E. Sternberg and S. Al-Khozaie, On Green’s functions and Saint-Venant’s principle in the linear theory of viscoelasticity, Arch. Ration. Mech. Anal. 15, 112–146 (1964).
50 J. L. Synge, The Hypercircle in Mathematical Physics, Cambridge Univ. Press, Cambridge, 1957.
51 W. Thomson (Lord Kelvin), On the equations of equilibrium of an elastic solid, Cambr. Dubl. Math. J. 3, 87–89 (1848).
52 F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, vol. I. Plenum Press, New York and London 1982.
53 F. Tricomi, Equazioni integrali contenenti il valor principale di un integrale doppio, Math. Z. 27 (1), 87–133 (1927).
54 M. J. Turteltaub and E. Sternberg, On concentrated loads and Green’s functions in elastostatics, Arch. Ration. Mech. Anal. 29, 193–240 (1968).
55 S. M. Vogel and F. J. Rizzo, An integral equation formulation of three dimensional anisotropic elastostatic boundary value problems, J. Elast. 3, 203–216 (1973).
56 J. R. Willis, The elastic interaction energy of dislocation loops in anisotropic media, Quart. J. Mech. Appl. Math. 18, 419–433 (1965).
57 R. B. Wilson and T. A. Cruse, Efficient implementation of anisotropic three dimensional boundary-integral equation stress analysis, Int. J. Numer. Methods Engng. 12, 1383–1397 (1978).
58 N. Zeilon, Das Fundamenlintegral der allgemeinen partiellen linearen Differentialgleichung mit Konstanten Koeffizienten, Ark. Matem., Astron. Fys. 6(38), 1–32 (1911).
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Additional Information
Sergey V. Kuznetsov
Affiliation:
Institute for Problems in Mechanics, Moscow 119526, Russia
Keywords:
Fundamental solution,
singular solution,
Lamé equations,
multipolar series,
spherical harmonics
Received by editor(s):
June 20, 2004
Published electronically:
August 17, 2005
Article copyright:
© Copyright 2005
Brown University
