On integration of differential equations in elastostatics through determination of the mean stress

Golecki, Joseph J.

Applications of Mathematics, Tome 19 (1974) no. 5, pp. 293-306

Abstract (VO)

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.

Détail

DOI : 10.21136/AM.1974.103546

Classification : 35Q99, 74B99

Bibliographie

[1] Almansi E.: Sull'integrazione dell' equazione differenziale $\nabla^{2n} =0. Ann. Mat. Рurа Appl. (3), 2, pp. 1-51, 1899.

[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.

[3] Dubnov J. S.: Foundations of vector calculus. Vol. 2 (in Russian), Gos. Izd. Teh.-Teor. Lit., Moskva, 1952.

[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.

[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).

[6] Fuka J.: Das zweite Problem der ebenen Elastizitätstheorie für inkompressible Körper. Aplikace Matematiky 7, 1, pp. 21 - 36, 1962. | MR | Zbl

[7] Fung Y. C.: Foundations of solid mechanics. Prentice-Hall, Englewood Cliffs, New Jersey, 1965.

[8] Gehman S. D.: Elasticity theory for identation of rubber plate. Rubber Chemistry and Technology, 41, 5, pp. 1122-1131, 1964. | DOI

[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.

[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

[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.

[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. | MR | Zbl

[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.

[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.

[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

[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

[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

[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

[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.

[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.

[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.

[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.

[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.

[24] Golecki J.: On many-valuedness of the Neuber-Papkovich solution in two-dimensional elasticity. Mechanics Research Communications, No. 2, 1974 (in print). | Zbl

[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. | DOI

[26] Goodier J. N.: Slow viscous flow and elastic deformation. Phil. Mag. (7), 22, pp. 678 - 681, 1936. | DOI

[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

[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

[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. | DOI

[30] Landau L. D., Lifshitz E. M.: Theory of elasticity. Pergamon Press, London, 1959. | MR

[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.

[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.

[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.

[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.

[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.

[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).

[37] Mason M., Weaver W.: The electromagnetic field. Dover Publications, New York, 1929. | MR

[38] Mindlin R. D.: Note on the Galerkin and Papkovitch stress functions. Bull. Arner. Math. Soc., 42, pp. 373-376, 1936. | DOI | MR

[39] Phillips H. B.: Vector analysis. John Wiley & Sons, New York, 1933.

[40] Sokolnikoff I. S.: Mathematical theory of elasticity. 2nd ed., McGraw-Hill, New York, 1956. | MR | Zbl

[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,

[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.

[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.

[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

[45] Wallis F. R.: Stress analysis of incompressible solids of revolution by point matching. Journal of Spacecraft and Rockets, 6, pp. 86-87, 1969. | DOI

[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.

[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. | MR | Zbl

[48] Westergaard H. M.: Theory of elasticity and plasticity. Harvard University Press, Cambridge, Mass., 1952. | MR | Zbl

[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.

[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.