Abstract
In a recent work in the static case, Gratie (Appl. Anal. 81:1107–1126, 2002) has generalized the classical Marguerre-von Kármán equations studied by Ciarlet and Paumier in (Comput. Mech. 1:177–202, 1986), where only a portion of the lateral face of the shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is subjected to boundary conditions of free edge. Then Ciarlet and Gratie (Math. Mech. Solids 11:83–100, 2006) have established an existence theorem for these equations. In Chacha et al. (Rev. ARIMA 13:63–76, 2010), we extended formally these studies to the dynamical case. More precisely, we considered a three-dimensional dynamical model for a nonlinearly elastic shallow shell with a specific class of boundary conditions of generalized Marguerre-von Kármán type. Using technics from formal asymptotic analysis, we showed that the scaled three-dimensional solution still leads to two-dimensional dynamical boundary value problem called the dynamical equations of generalized Marguerre-von Kármán shallow shells. In this paper, we establish the existence of solutions to these equations using a compactness method of Lions (Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969).
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References
Abels, H., Mora, M.G., Müller, S.: The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity. Calc. Var. Partial Differ. Equ. 41(1), 241–259 (2011)
Andreoiu-Banica, G.: Justification of the Marguerre-von Kármán equations in curvilinear coordinates. Asymptot. Anal. 19, 35–55 (1999)
Böhm, M.: Global well posedness of the dynamic von Kármán equations for generalized solutions. Nonlinear Anal. 27, 339–351 (1996)
Chacha, D.A., Ghezal, A., Bensayah, A.: Modélisation asymptotique d’une coque peu-profonde de Marguerre-von Kármán généralisée dans le cas dynamique. Rev. ARIMA 13, 63–76 (2010)
Chueshov, I., Lasiecka, I.: Global attractors for von Kármán evolutions with a non-linear boundary dissipation. J. Differ. Equ. 198, 196–221 (2004)
Chueshov, I., Lasiecka, I.: Long-time dynamics of von Kármán semi-flows with non-linear boundary/interior damping. J. Differ. Equ. 233, 42–86 (2007)
Ciarlet, P.G.: A justification of the von Kármán equations. Arch. Ration. Mech. Anal. 73, 349–389 (1980)
Ciarlet, P.G.: Plates and Junctions in Elastic Multistructures. Masson, Paris (1990)
Ciarlet, P.G.: Mathematical Elasticity. Theory of Plates, vol. II. North-Holland, Amsterdam (1997)
Ciarlet, P.G., Gratie, L.: From the classical to the generalized von Kármán and Marguerre-von Kármán equations. J. Comput. Appl. Math. 190, 470–486 (2006)
Ciarlet, P.G., Gratie, L.: On the existence of solutions to the generalized Marguerre-von Kármán equations. Math. Mech. Solids 11, 83–100 (2006)
Ciarlet, P.G., Paumier, J.C.: A justification of the Marguerre-von Kármán equations. Comput. Mech. 1, 177–202 (1986)
Devdariani, G., Janjgava, R., Gulua, B.: Dirichlet problem for the Marguerre-von Kármán equations system. Bull. TICMI 10, 23–27 (2006)
Duvaut, G., Lions, J.L.: Les Inéquations en Mécanique et en Physique. Dunod, Paris (1972)
Gratie, L.: Generalized Marguerre-von Kármán equations of a nonlinearly elastic shallow shell. Appl. Anal. 81, 1107–1126 (2002)
Kavian, O., Rao, B.P.: Une remarque sur l’existence de solutions non nulles pour les équations de Marguerre-von Kármán. C.R. Acad. Sci. Paris Sér. 1317, 1137–1142 (1993)
Kesavan, S., Srikanth, P.N.: On the Dirichlet problem for the Marguerre equations. Nonlinear Anal. 7(2), 209–216 (1983)
Koch, H., Stahel, A.: Global existence of classical solutions to the dynamical von Kármán equations. Math. Methods Appl. Sci. 16, 581–586 (1993)
Léger, A., Miara, B.: On the multiplicity of solutions to Marguerre-von Kármán membrane equations. J. Math. Pures Appl. 84, 357–374 (2005)
Li, F.: Global existence and uniqueness of weak solution to nonlinear viscoelastic full Marguerre-von Kármán shallow shell equations. Acta Math. Sin., Engl. Ser. Dec. 25(12), 2133–2156 (2009)
Li, F.: Limit behavior of the solution to nonlinear viscoelastic Marguerre-von Kármán shallow shell system. J. Differ. Equ. 249, 1241–1257 (2010)
Li, F., Bai, Y.: Uniform decay rates for nonlinear viscoelastic Marguerre-von Kármán equations. J. Math. Anal. Appl. 351, 522–535 (2009)
Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)
Lions, J.L., Magenes, E.: Problèmes aux Limites Non Homogènes et Applications, vol. I et II. Dunod, Paris (1968)
Marguerre, K.: Zur theorie der gekrummten platte grosser formanderung. In: Proceedings Fifth International Congress for Applied Mechanics, pp. 93–101 (1938)
Puel, J.P., Tucsnak, M.: Global existence for the full von Kármán system. Appl. Math. Optim. 34, 139–160 (1996)
Rao, B.P.: Marguerre-von Kármán equations and membrane model. Nonlinear Anal. 24, 1131–1140 (1995)
Raoult, A.: Construction d’un modèle d’évolution de plaques avec terme d’inertie de rotation. Ann. Mat. Pura Appl. 139, 361–400 (1985)
Tataru, D., Tucsnak, M.: On the Cauchy problem for the full von Kármán system. Nonlinear Differ. Equ. Appl. 4, 325–340 (1997)
von Karman, T.: Festigkeitesprobleme in maschinenbau. encyklopadie der mathematischen wissenschaften. Taubner IV/4, 311–385 (1910)
von Kármán, T., Tsien, H.S.: The buckling of spherical shells by external pressure. J. Aerosol Sci. 7, 43–50 (1939)
Xiao, L.M.: Asymptotic analysis of dynamic problems for linearly elastic shells-justification of equations for dynamic membrane shells. Asymptot. Anal. 17, 121–134 (1998)
Xiao, L.M.: Existence and uniqueness of solutions to the dynamic equations for Koiter shells. Appl. Math. Mech. 20(7), 801–806 (1999)
Xiao, L.M.: Asymptotic analysis of dynamic problems for linearly elastic shells-justification of equations for dynamic flexural shells. Chin. Ann. Math., Ser. B 22(1), 13–22 (2001)
Xiao, L.M.: Asymptotic analysis of dynamic problems for linearly elastic shells-justification of equations for dynamic Koiter shells. Chin. Ann. Math., Ser. B 22(3), 267–274 (2001)
Yan, G.: Asymptotic analysis of linearly elastodynamic shallow shells with variable thickness. Asymptot. Anal. 50, 1–12 (2006)
Ye, J.: Asymptotic analysis of dynamic problem for linearly elastic generalized membrane shells. Asymptot. Anal. 36, 47–62 (2003)
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Chacha, D.A., Ghezal, A. & Bensayah, A. Existence Result for a Dynamical Equations of Generalized Marguerre-von Kármán Shallow Shells. J Elast 111, 265–283 (2013). https://doi.org/10.1007/s10659-012-9402-5
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DOI: https://doi.org/10.1007/s10659-012-9402-5