Asymptotic justification of dynamical equations for generalized Marguerre–von Kármán anisotropic shallow shells

References

• Von Kármán T. Festigkeitsprobleme im Maschinenbau [Strength problems in engineering]. Encyklopädie der Mathematischen Wissenschaften. Taubner. 1910;IV/4:311–385.
   Google Scholar
• Marguerre K. Zur Theorie der gekrümmten Platte grosser Formänderung [On the theory of the curved plate with large displacements]. Proceedings, Fifth International Congress for Applied Mechanics, Cambridge, Massachusetts; 1938. p. 93–101.
   Google Scholar
• Von Kármán T, Tsien HS. The buckling of spherical shells by external pressure. J. Aeronaut. Sci. 1939;7:43–50.
   Google Scholar
• Ciarlet PG, Paumier JC. A justification of the Marguerre--von Kármán equations. Comput. Mech. 1986;1:177–202.
   Google Scholar
• Gratie L. Generalized Marguerre--von Kármán equations of a nonlinearly elastic shallow shell. Appl. Anal. 2002;81:1107–1126.
   Google Scholar
• Ciarlet PG, Gratie L. On the existence of solutions to the generalized Marguerre--von Kármán equations. Math. Mech. Solids. 2006;11:83–100.
   Web of Science ®Google Scholar
• Ciarlet PG, Gratie L. From the classical to the generalized von Kármán and Marguerre--von Kármán equations. J. Comput. Appl. Math. 2006;190:470–486.
   Web of Science ®Google Scholar
• Xiao LM. Asymptotic analysis of dynamic problems for linearly elastic shells-justification of equations for dynamic membrane shells. Asymptotic Anal. 1998;17:121–134.
   Web of Science ®Google Scholar
• Xiao LM. Asymptotic analysis of dynamic problems for linearly elastic shells-justification of equations for dynamic flexural shells. Chinese Ann. Math. Ser. B. 2001;22:13–22.
   Web of Science ®Google Scholar
• Xiao LM. Asymptotic analysis of dynamic problems for linearly elastic shells-justification of equations for dynamic Koiter shells. Chin. Ann. Math. Ser. B. 2001;22:267–274.
   Web of Science ®Google Scholar
• Xiao LM. Existence and uniqueness of solutions to the dynamic equations for Koiter shells. Appl. Math. Mech. 1999;20:801–806.
   Web of Science ®Google Scholar
• Ji Y. Asymptotic analysis of dynamic problem for linearly elastic generalized membrane shells. Asymptotic Anal. 2003;36:47–62.
   Web of Science ®Google Scholar
• Guan Y. Asymptotic analysis of linearly elastodynamic shallow shells with variable thickness. Asymptotic Anal. 2006;50:1–12.
   Web of Science ®Google Scholar
• Puel JP, Tucsnak M. Global existence for the full von Kármán system. Appl. Math. Optim. 1996;34:139–160.
   Web of Science ®Google Scholar
• Koch H, Stahel A. Global existence of classical solutions to the dynamical von Kármán equations. Math. Methods Appl. Sci. 1993;16:581–586.
   Web of Science ®Google Scholar
• Böhm M. Global well posedness of the dynamic von Kármán equations for generalized solutions. Nonlinear Anal. Theory Methods Appl. 1996;27:339–351.
   Web of Science ®Google Scholar
• Tataru D, Tucsnak M. On the Cauchy problem for the full von Kármán system. Nonlinear Differ. Equ. Appl. 1997;4:325–340.
   Web of Science ®Google Scholar
• Chueshov I, Lasiecka I. Global attractors for von Kármán evolutions with a non-linear boundary dissipation. J. Differ. Equ. 2004;198:196–231.
   Web of Science ®Google Scholar
• Chueshov I, Lasiecka I. Long-time dynamics of von Kármán semi-flows with non-linear boundary/interior damping. J. Differ. Equ. 2007;233:42–86.
   Web of Science ®Google Scholar
• Li F. Global existence and uniqueness of weak solution to nonlinear viscoelastic full Marguerre--von Kármán shallow shell equations. Acta Mathe. Sinica. 2009;25:2133–2156.
   Web of Science ®Google Scholar
• Li F. Limit behavior of the solution to nonlinear viscoelastic Marguerre--von Kármán shallow shell system. J. Differ. Equ. 2010;249:1241–1257.
   Web of Science ®Google Scholar
• Li F, Bai Y. Uniform decay rates for nonlinear viscoelastic Marguerre--von Kármán equations. J. Math. Anal. Appl. 2009;351:522–535.
   Web of Science ®Google Scholar
• Abels H, Mora MG, Müller S. The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity. Calculus Variations Partial Differ. Equ. 2011;41:241–259.
   Web of Science ®Google Scholar
• Destuynder P. Sur une Justification des modèles de plaques et de coques par les méthodes asymptotiques [On a justification of models of plates and shells by the asymptotic methods]. Paris: Université Pierre et Marie Curie; 1980.
   Google Scholar
• Caillerie D. Thin elastic and periodic plates. Math. Meth. Appl. Sci. 1984;6:159–191.
   Google Scholar
• Caillerie D, Sanchez-Palencia E. Elastic thin shells: asymptotic theory in the anisotropic and heterogeneous cases. Math. Models Methods Appl. Sci. 1995;5:473–496.
   Web of Science ®Google Scholar
• Giroud P. Analyse asymptotique de coques inhomogènes en élasticité linéarisée anisotrope [Asymptotic analysis of heterogeneous shells in linearized anisotropic elasticity]. C.R. Acad. Sci. Paris Sér. I. 1998;327:1011–1014.
   Web of Science ®Google Scholar
• Sanchez-Hubert J, Sanchez-Palencia E. Introduction aux Méthodes Asymptotiques et à l’Homogénéisation. Application à la Mécanique des Milieux Continus [Introduction to Asymptotic Methods and Homogenization. Application to Continuum Mechanics]. Paris: Masson; 1992.
   Google Scholar
• Begehr H, Gilbert RP, Lo CY. The two-dimensional nonlinear orthotropic plate. J. Elast. 1991;26:147–167.
   Web of Science ®Google Scholar
• Gilbert RP, Vashakmadze TS. A two-dimensional nonlinear theory of anisotropic plates. Math. Comput. Model. 2000;32:855–875.
   Google Scholar
• Chacha DA. Asymptotic analysis of nonlinear elastic plates. C. R. Mecanique. 2002;330:581–586.
   Web of Science ®Google Scholar
• Chacha DA, Miloudi M. Asymptotic analysis of nonlinearly elastic shells mixed approach. Asymptotic Anal. 2012;80:323–346.
   Web of Science ®Google Scholar
• Chacha DA, Ghezal A, Bensayah A. Modélisation asymptotique d’une coque peu-profonde de Marguerre--von Kármán généralisée dans le cas dynamique [Asymptotic Modelling of generalized Marguerre-von Kármán shallow shell in dynamical case]. Rev. ARIMA. 2010;13:63–76.
   Google Scholar
• Chacha DA, Ghezal A, Bensayah A. Existence result for a dynamical equations of generalized Marguerre--von Kármán shallow shells. J. Elast. 2013;111:265–283.
   Web of Science ®Google Scholar
• Lions JL. Quelques méthodes de résolution des problèmes aux limites non linéaires [Some methods of solving nonlinear boundary value problems]. Paris: Dunod; 1969.
   Google Scholar
• Bensayah A, Chacha DA, Ghezal A. Asymptotic modelling of a Signorini problem of generalized Marguerre--von Kármán shallow shells. Appl. Anal. 2013;92:1848–1862.
   Web of Science ®Google Scholar
• Ghezal A, Chacha DA. Convergence of finite element approximations for generalized Marguerre--von Kármán equations. Springer proceedings in mathematics and statistics. Adv. Appl. Math.. 2014; 87 : 97–106.
   Google Scholar
• Raoult A. Construction d’un modèle d’évolution de plaques avec terme d’inertie de rotation [Construction of evolution model for plates with rotational inertia term]. Annali Matematica Pura Appl. 1985;139:361–400.
   Web of Science ®Google Scholar
• Ciarlet PG. Mathematical elasticity. Vol. II. Theory of plates. Amsterdam: North-Holland; 1997.
   Google Scholar
• Schwartz L. Théorie des distributions [Theory of distributions]. Paris: Hermann; 1966.
   Google Scholar