×

Application of Lie groups to the theory of shells and rods. (English) Zbl 0903.73040

The paper is concerned with the application of Lie groups to the theory of shells and rods. This includes plates as a special kind of shells. In doing so, the group properties of a class of linear forth-order partial differential equations used in plate theory are examined, and the invariance properties of the field equations of nonlinear Donnell-Mushtari-Vlasov theory for large deflections of isotropic thin elastic shells are studied in detail.
Reviewer: W.Becker (Siegen)

MSC:

74K15 Membranes
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35A30 Geometric theory, characteristics, transformations in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Olver, P. J., Applications of Lie Groups to Differential Equations, (Graduate Texts in Mathematics, Vol. 107 (1993), Springer-Verlag: Springer-Verlag New York) · Zbl 0591.73024
[2] (Ames, W. F., Group Analysis of Differential Equations (1982), Springer-Verlag: Springer-Verlag New York), English transl. · Zbl 0485.58002
[3] Ibragimov, N. Kh., Transformation Groups to Mathematical Physics (1985), Reidel: Reidel Boston, English transl. · Zbl 0529.53014
[4] Vassilev, V. M., Group properties of a class of fourth-order partial differential equations, Annuaire Univ. Sofia Fac. Math. Inform., 82, 163-178 (1988)
[5] Vassilev, V. M., Group Analysis of a Class of Equations of the Theory of Thin Elastic Plates and Shells, (Ph.D. thesis (1991), Bulgarian Academy of Sciences)
[6] Niordson, F. L., Shell Theory (1985), Nord-Holland: Nord-Holland Amsterdam
[7] Galimov, K. Z., Foundations of the Nonlinear Theory of Shells (1975), Kazan’ University Press: Kazan’ University Press Kazan’ · Zbl 0473.73102
[8] Marguerre, K., Zur Theoree der gekriimmten Platte groβer Formanderung, (Proc. Fifth Intern. Congr.. Proc. Fifth Intern. Congr., Appl. Mech., 93 (1938)), Cambridge, Massachusetts
[9] Yu, Y. Y., On equations for large deflections of elastic plates and shallow shells, Mech. Res. Commun., 18, 373-384 (1991) · Zbl 0763.73033
[10] Nakazawa, M.; Iwakuma, T.; Kuranishi, S.; Hudaka, M., Instability phenomena of a rectangular elastic plate under bending and shear, Int. J. Solids Structures, 30, 2729-2741 (1993)
[11] Ciarlet, Ph.; Rabier, P., Les Equations de von Kármán, (Lecture Notes in Math., Vol. 826 (1980), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York) · Zbl 0433.73019
[12] Karman, Th.v., Festigkeitesprobleme im Maschinebau, (Encyklopadie der Mathematischen Wissenschaften, Bd IV, 311 (1910), Taubner: Taubner Leipzig)
[13] Vassilev, V. M., Symmetry groups and equivalence transformations in the nonlinear Donnell-MushtariVlasov theory for shallow shells, J. Theoret. and Appl. Mech., 26, No.4 (1996)
[14] Shwarz, F., Lie symmetries of the von Karman equations, Comput. Phys. Commun., 31, 113-114 (1984)
[15] Ames, K. A.; Ames, W. F., On group analysis of the von Karman equations, Nonlinear Analysis, 6, 845-883 (1982) · Zbl 0511.35017
[16] Saccomandi, G.; Salvatori, M. C., Conservation laws for the von Karman equations of a thin plate, Rendiconti di Matematica, 11, Serie VII, 283-294 (1991), Roma · Zbl 0749.35022
[17] Djondjorov, P.; Vassilev, V., Conservation laws and group-invariant solutions of the von Karman equations, Int. J. Non-Linear Mechanics, 31, 73-87 (1996) · Zbl 0860.73027
[18] Frydrychowicz, W.; Singh, M. C., Discontinuous solutions of nonlinear wave phenomena by similarity approach, Appl. Math. Modelling, 15, 2-13 (1991) · Zbl 0728.35067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.