On the completeness of a method of potentials in elastodynamics
Authors:
Ronald Y. S. Pak and Morteza Eskandari-Ghadi
Journal:
Quart. Appl. Math. 65 (2007), 789-797
MSC (2000):
Primary 74B05, 35Q72; Secondary 35L05
DOI:
https://doi.org/10.1090/S0033-569X-07-01074-X
Published electronically:
October 17, 2007
MathSciNet review:
2370361
Full-text PDF Free Access
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Additional Information
Abstract: In this paper, the theoretical foundation of a compact scalar potential method in three-dimensional classical elastodynamics is substantiated. Beginning with a derivation of two basic lemmas on the decomposition and integration of wave solutions and vector fields which are apt to be of interest to general mechanics and analysis, the treatment proceeds to a proof of the completeness of the proposed representation as well as its extension to non-zero body forces.
References
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- Kellogg, O. D., Foundation of Potential Theory, Dover, New York, 1953.
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References
- Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, 4th ed., Dover, New York, 1944. MR 0010851 (6:79e)
- Fung, Y. C., Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.
- Pao, Y. H. and Mow, C. C., Diffraction of Elastic Waves and Dynamic Stress Concentrations, Crane Russak, New York, 1971.
- Achenbach, J. D., Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1973.
- Miklowitz, J., The Theory of Elastic Waves and Wave Guides, North-Holland, Amsterdam, 1978. MR 515886 (80d:73001)
- Truesdell, C., Invariant and complete stress functions for general continua. Arch. Ration. Mech. Anal., 4 (1959), 1-29. MR 0122083 (22:12810)
- Gurtin, M. E., The Linear Theory of Elasticity, Mechanics of Solids, Vol. II (ed. Truesdell), Springer-Verlag, Berlin, 1972, 1-295.
- Tran-Cong, T., On the completeness and uniqueness of Papkovich-Neuber and the non-axisymmetric Boussinesq, Love and Burgatti solutions in general cylindrical coordinates. J. Elasticity, 36 (1995), 227-255. MR 1318389 (96a:73020)
- Pak, R. Y. S., Asymmetric wave propagation in an elastic half-space by a method of potentials. J. Appl. Mech., ASME, 54(1) (1987), 121-126.
- Sternberg, E., On the integration of the equation of motion in the classical theory of elasticity. Arch. Ration. Mech. Anal., 6 (1960), 34-50. MR 0119547 (22:10308)
- Stippes, M., Completeness of Papkovich potentials. Quart. Appl. Math., 26 (1969), 477-483. MR 0239801 (39:1158)
- Mindlin, R. D., Note on the Galerkin and Papkovich stress functions. Bull. Amer. Math. Soc., 42 (1963), 373-376.
- Sternberg, E. and Gurtin, M. E., On the completeness of certain stress functions in the linear theory of elasticity. Proc. Fourth U.S. Nat. Congr. Appl. Mech. (1962), pp. 793-797. MR 0159454 (28:2671)
- Gurtin, M. E., On Helmholtz’s theorem and the completeness of the Papkovich-Neuber stress functions in infinite domains. Arch. Ration. Mech. Anal., 9 (1962), 225-233. MR 0187467 (32:4917)
- Freiberger, W., On the solution of the equilibrium equations of elasticity in general curvilinear coordinates. Austral. J. Sci., A2 (1949), 483-492. MR 0038821 (12:457d)
- Millar, R. F., On the completeness of Papkovich potentials. Quart. Appl. Math., 41 (1984), 385-393. MR 724050 (85e:73009)
- Tran-Cong, T., On the completeness of the Papkovich-Neuber solution. Quart. Appl. Math., 47 (1989), 645-659. MR 1031682 (91e:73022)
- Youngdahl, C. K., On the completeness of a set of stress functions appropriate to the solution of elasticity problems in cylindrical coordinates. Int. J. Engrg. Sc., 7 (1969), 61-70. MR 0239803 (39:1160)
- Sternberg E. and Eubanks, R. A., On stress functions for elastokinetics and the integration of the repeated wave equation. Quart. Appl. Math., 15 (1957), 149-153. MR 0091657 (19:1000d)
- Kellogg, O. D., Foundation of Potential Theory, Dover, New York, 1953.
- Weinberger, H. F., A First Course in Partial Differential Equations with Complex Variables and Transform Methods, Wiley, New York, 1965. MR 0180739 (31:4969)
- Apostol, T. M., Mathematical Analysis, Addison-Wesley, Reading, Massachusetts, 1957. MR 0087718 (19:398e)
- Pak, R. Y. S. and Guzina, B. B., Three-dimensional Green’s functions for a multi-layered half-space by displacement potentials. J. Engrg. Mech. ASCE, 128(4) (2002), 449-461.
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Additional Information
Ronald Y. S. Pak
Affiliation:
Department of Civil, Environmental and Architectural Engineering, University of Colorado, Boulder, Colorado 80309-0428
Email:
pak@colorado.edu
Morteza Eskandari-Ghadi
Affiliation:
Civil Engineering Department, University of Science and Technology of Mazandaran, Iran
Email:
ghadi@ustmb.ac.ir
Keywords:
Elasticity,
completeness,
potentials,
mechanics,
wave equation,
vector calculus,
elastodynamics,
Helmholtz,
solenoidal field,
Laplacian
Received by editor(s):
May 24, 2007
Published electronically:
October 17, 2007
Dedicated:
A tribute to Eli Sternberg
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.