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Symmetry groups and equivalence transformations in the nonlinear Donnell-Mushtari-Vlasov theory for shallow shells. (English) Zbl 0898.73043

Large deflections of isotropic thin elastic shallow shells under transverse loading are considered within the frame of the Donnell-Mushtari-Vlasov’s theory [see F. Niordson, Shell theory, North-Holland, Amsterdam (1985; Zbl 0566.73053)]. This results in a system a coupled nonlinear system of PDE for the displacement and stress’ Airy function, named after Marguerre [K. Marguerre, in: Proc. V. internat. Congr. Appl. Mech. 1938, Cambridge, Mass., 93–101 (1938; JFM 65.0946.01)]. In the paper the point symmetry groups of Marguerre’s equations are described and the appropriate group classification problem is solved. Moreover, an equivalence transformation is suggested which makes Marguerre’s equations equivalent to the well-known von Kármán equations that describe large deflections of plates. Since the latter have been extensively studied in the literature, this equivalence allows to extend to the more general case of shallow shells, e.g., the results of P. A. Djondjorov and V. M. Vassilev [Int. J. Non-Linear Mech. 31, 73–87 (1996; Zbl 0860.73027)] concerning group-invariant solutions to the time-dependent von Kármán equations.
Reviewer: K.Z.Markov

MSC:

74K15 Membranes
35A30 Geometric theory, characteristics, transformations in context of PDEs
35Q72 Other PDE from mechanics (MSC2000)
74K20 Plates
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