Previous |  Up |  Next

Article

Title: On integration of differential equations in elastostatics through determination of the mean stress (English)
Author: Golecki, Joseph J.
Language: English
Journal: Aplikace matematiky
ISSN: 0373-6725
Volume: 19
Issue: 5
Year: 1974
Pages: 293-306
Summary lang: English
Summary lang: Czech
.
Category: math
.
Summary: The presented method of integration of differential equations in elastostatics - the so-called menas-stress approach - yields a solution dependent on the elastic parameters and the topology of the body, and accordingly directly affected by Poisson's ratio: for example, the assumption of incompressibility $(v=\frac{1}{2})$ transforms its component Poisson's equation into a harmonic equation. Moreover, the solution for a multiply-connected region has to satisfy additional conditions depending inter alia on the geometry of the latter. These conditions ensure a single-valued mean normal stress. ()
MSC: 35Q99
MSC: 74B99
idZBL: Zbl 0314.73017
idMR: MR0366160
.
Date available: 2008-05-20T17:59:36Z
Last updated: 2015-08-05
Stable URL: http://hdl.handle.net/10338.dmlcz/103546
.
Reference: [1] Almansi E.: Sull'integrazione dell' equazione differenziale $\nabla^{2n} =0.Ann. Mat. Рurа Appl. (3), 2, pp. 1-51, 1899.
Reference: [2] Biderman V. L.: Design of rubber and rubber-cord elements.(in Russian). Strength Analysis in Machine Design, 2nd ed., S. D. Ponomarev (Ed.), Vol. 2, 7, pp. 487-491. Mashgiz, Moskva, 1958.
Reference: [3] Dubnov J. S.: Foundations of vector calculus.Vol. 2 (in Russian), Gos. Izd. Teh.-Teor. Lit., Moskva, 1952.
Reference: [4] Dyminkov S. J., Pribylov T. A.: Variational equations of the theory of elasticity for an incompressible medium.(in Russian). Editorial Committee Zh. ,,Mekhanika Polimerov", Riga, 1969.
Reference: [5] Föppl A., and Föppl L.: Drang und Zwang, Eine höhere Festigkeitslehre für Ingenieure.2nd Ed. Vol. 1, München und Berlin, 1924 (1st Ed. Oldenbourg, Munich and Berlin 1920).
Reference: [6] Fuka J.: Das zweite Problem der ebenen Elastizitätstheorie für inkompressible Körper.Aplikace Matematiky 7, 1, pp. 21 - 36, 1962. Zbl 0112.39102, MR 0141270
Reference: [7] Fung Y. C.: Foundations of solid mechanics.Prentice-Hall, Englewood Cliffs, New Jersey, 1965.
Reference: [8] Gehman S. D.: Elasticity theory for identation of rubber plate.Rubber Chemistry and Technology, 41, 5, pp. 1122-1131, 1964. 10.5254/1.3539177
Reference: [9] Golecki J.: On the fundamentals of the theory of elasticity of plane incompressible non-homogeneous media.Lecture to I.U.T.A.M. Symposium on Non-homogeneity in Elasticity and Plasticity, Warsaw, 1958: Abstract in Bull. Acad. Pol. Scien., Ser. Scien. Tech. (Polska Akad. Nauk-Warsaw), 7, 2-3, pp. 85-88, 1959.
Reference: [10] Golecki J.: On the foundations of the theory of elasticity of plane incompressible nonhomogeneous bodies.Arch. Mechaniki Stos. (Appl. Mechan. Arch.), 9, 4, pp. 383 - 398, 1959; also in Non-homogeneity in Elasticity and Plasticity, Olszak, W. (Ed.), pp. 39-51, Pergamon Press (printed in Poland), 1959. MR 0108075
Reference: [11] Golecki J.: The stress function for a two-dimensional incompressible non-homogeneous body in the case of plane stress.Bull. Acad. Pol. Scien. Ser. Scien. Tech. (Polska Akad. Nauk - Warsaw), 7, 6, pp. 371-375, 1959.
Reference: [12] Golecki J.: On the assumption of incompressibility in plane problems of the theory of elasticity.Arch. Mech. Stos. (Appl. Mech. Arch.), 9, 3, pp. 297-301, 1959. Zbl 0086.18605, MR 0105875
Reference: [13] Golecki J.: Displacement functions for an isotropic, incompressible elastic solid.Bull. Acad. Pol. Scien., Ser. Scien. Tech. (Polska Akad. Nauk- Warsaw), 7, 4, pp. 265-271, 1959.
Reference: [14] Golecki J.: On a certain form of solution of static elasticity theory.Bull. Acad. Pol. Scien., Ser. Scien. Tech. (Polska Akad. Nauk - Warsaw), 9, 3, pp. 139-143, 1961.
Reference: [15] Golecki J.: Statics of an isotropic, incompressible, elastic solid.Arch. Mech. Stos., (Appl. Mech. Arch.), 14, 1, pp. 35-46, 1962; Bull. Inst. Politechnic Iaşi, Ser. Nouă, 7, (11), 3 - 4, pp. 43-50, 1961. Zbl 0112.16702
Reference: [16] Golecki J.: On a certain form of solution of the equations of dynamic theory of elasticity.Bull. Acad. Pol. Scien., Ser. Scien. Tech. (Polska Akad. Nauk - Warsaw), 10, 1, pp. 7-16, 1962. Zbl 0166.21303
Reference: [17] Golecki J.: Elastic half-plane with variable Poisson's ratio.Displacement boundary problems. Bull. Acad. Pol. Scien., Ser. Scien. Tech. (Polska Akad. Nauk - Warsaw), 16, 4, pp. 175-182, 1968. Zbl 0188.57101
Reference: [18] Golecki J.: Traction boundary problem for elastic half-plane with variable Poisson's ratio.Israel Journal of Technology, 7, 4, pp. 291-296, 1969. Zbl 0194.26203
Reference: [19] Golecki J.: On non-radial stress distribution in nonhomogeneous elastic half-plane under concentrated load.Meccanica, 6, 3, pp. 147-156, 1971; Report TDM 71 - 03, Department of Mechanics, Technion - Israel Institute of Technology, Haifa, Israel, 1971.
Reference: [20] Golecki J.: Integration of differential equations in elastostatics by determination of the mean stress (I).Report TDM 72 - 06. Department of Mechanics, Technion - Israel Institute of Technology, Haifa, Israel, 1972.
Reference: [21] Golecki J.: Mean-stress approach in two-dimensional elasticity.(normal regions). Report TDM 73 - 06, Department of Mechanics, Technion - Israel Institute of Technology, Haifa, Israel, 1973.
Reference: [22] Golecki J.: Mean-stress approach in two-dimensional elasticity.(star-shaped regions). Report TDM 73 - 09. Department of Mechanics, Technion- Israel Institute of Technology, Haifa, Israel, 1973.
Reference: [23] Golecki J.: Integration of differential equations in elastostatics by determination of the mean stress (II).Report TDM 73 - 27, Department of Mechanics, Technion - Israel Institute of Technology, Haifa, Israel, 1973.
Reference: [24] Golecki J.: On many-valuedness of the Neuber-Papkovich solution in two-dimensional elasticity.Mechanics Research Communications, No. 2, 1974 (in print). Zbl 0356.73019
Reference: [25] Golecki J., and Jeffrey A.: Two-dimensional dynamical problems for incompressible isotropic linear elastic solids with time dependent moduli and variable density.Acta Mechanica, 5, pp. 118-130. 1968. 10.1007/BF01178827
Reference: [26] Goodier J. N.: Slow viscous flow and elastic deformation.Phil. Mag. (7), 22, pp. 678 - 681, 1936. 10.1080/14786443608561718
Reference: [27] Gurtin M. E.: On Helmholtz's theorem and completeness of the Papkovich-Neuber stress functions for infinite domains.Archive Rational Mech. and Analysis, 9, pp. 225-233, 1962. MR 0187467
Reference: [28] Herrmann L. R.: Elasticity equations for incompressible and nearly incompressible materials by a variational theorem.A.I.A.A. J1 3, pp. 1896-1900, 1965. MR 0184477
Reference: [29] Herrmann L. R., Toms R. M.: A reformulation of the elastic field equation in terms of displacements, valid for all admissible values of Poisson's ratio.J. Appl. Mech. 31 E, pp. 140-141, 1964. 10.1115/1.3629536
Reference: [30] Landau L. D., Lifshitz E. M.: Theory of elasticity.Pergamon Press, London, 1959. MR 0106584
Reference: [31] Lavendel E. E.: Solution of elasticity problems for an incompressible medium by displacements.(in Russian), Rigas Politehniska Instituta Zinatniskie Raksti, Vol. 1, pp. 125-140, 1959.
Reference: [32] Lavendel E. E.: Design of cylindrical rubber-metal compression absorbers.(in Russian). Izv. Akad. Nauk Latv. SSR, No. 4, Riga, 1960; and other papers by the author on design of rubber elements.
Reference: [33] Lavendel E. E.: General solutions of elasticity problems for an incompressible medium.(in Russian). Dynamics arid Strength Problems (Collected Papers), No. 7, pp. 107-120, Zvaigzne, Riga, 1961.
Reference: [34] Lavendel E. E.: Application of variational methods to an incompressible medium.(in Russian). Dynamics and Strength Problems (Collected Papers), No. 8, 103 - 118, Riga, 1962.
Reference: [35] Lavendel E. E.: Comparison of some variants of approximate solutions of elasticity problems for an incompressible medium.(in Russian). Dynamics and Strength Problems (Collected Papers), No. 13, pp. 141-148, Riga, 1967.
Reference: [36] Love A. E. H.: A treatise on the mathematical theory of elasticity.4th ed., Cambridge University Press, Cambridge, 1927; (1st ed. Cambridge 1892-1893).
Reference: [37] Mason M., Weaver W.: The electromagnetic field.Dover Publications, New York, 1929. MR 0048304
Reference: [38] Mindlin R. D.: Note on the Galerkin and Papkovitch stress functions.Bull. Arner. Math. Soc., 42, pp. 373-376, 1936. MR 1563303, 10.1090/S0002-9904-1936-06304-4
Reference: [39] Phillips H. B.: Vector analysis.John Wiley & Sons, New York, 1933.
Reference: [40] Sokolnikoff I. S.: Mathematical theory of elasticity.2nd ed., McGraw-Hill, New York, 1956. Zbl 0070.41104, MR 0075755
Reference: [41] Teoderescu P. P.: One hundred years of investigations in the plane problem of the theory of elasticity.Appl. Mech. Reviews, 17, 3, pp. 175-186, 1964,
Reference: [42] Thomson Sir W., (Lord Kelvin): Dynamical problems regarding elastic spheroidal shells and spheroids of incompressible liquid.Philosophical Transactions, Vol. 353, pp. 583 - 616, London, 1964.
Reference: [43] Thomson Sir W., (Lord Kelvin), Tait P. G.: Treatise on natural philosophy.Cambridge, 1879. Reprinted by Dover Publications, as: Principles of Mechanics, New York, 1962.
Reference: [44] Todhunter I., Pearson K.: A history of the theory of elasticity and of the strength of materials.Vol. II, Cambridge, 1886. Reprinted by Dover Publications, New York, 1961. MR 0118011
Reference: [45] Wallis F. R.: Stress analysis of incompressible solids of revolution by point matching.Journal of Spacecraft and Rockets, 6, pp. 86-87, 1969. 10.2514/3.29540
Reference: [46] Walek Z.: Analysis of investigation results on Almansi's problem.(in Polish). Report, Department of Strength of Materials, Academy of Mining and Metallurgy in Cracow, 1968.
Reference: [47] Westergaard H. M.: Effects of a change of Poisson's ratio analyzed by twinned gradients.J. Applied Mechanics, pp. A-113 - A-116, 1940. Zbl 0063.08221, MR 0003159
Reference: [48] Westergaard H. M.: Theory of elasticity and plasticity.Harvard University Press, Cambridge, Mass., 1952. Zbl 0048.42103, MR 0051675
Reference: [49] De Wit R.: The continuum theory of stationary dislocations.Solid State Physics, F. Seitz and D. Turbull (Eds.), pp. 249-292, Academic Press, New York, London, 1960.
Reference: [50] Yeh, Gordon C. K.: A comparison of various elasticity formulations valid for admissible value of Poisson's ratio.Astronautica Acta, 14, pp. 317-326, 1969.
.

Files

Files Size Format View
AplMat_19-1974-5_2.pdf 2.340Mb application/pdf View/Open
Back to standard record
Partner of
EuDML logo