Á. Rodríguez-Arós, G. Castiñeira (Universidade da Coruña).
In solid mechanics, the obtention of models for rods, beams, bars, plates and shells is based on a priori hypotheses on the displacement and/or stress fields which, upon substitution in the equilibrium and constitutive equations of three dimensional
elasticity, lead to useful simplifications. Nevertheless, from both constitutive and geometrical point of views, there is a need to justify the validity of most of the models obtained in this way.
For this reason a considerable effort has been made in the past decades by many authors in order to derive new models and justify the existing ones by using the asymptotic expansion method, whose foundations can be found in Lions [1]. Indeed, the first applied results were obtained with the justification of the linearized theory of plate bending by Ciarlet and collaborators [2]. The theories of beam bending and rod stretching also benefited from the extensive use of asymptotic methods and so the justification of the Bernoulli-Navier model for the bending-stretching of elastic thin rods was provided by Bermúdez and Viaño [3]. These pioneer works were followed by the contributions of many authors in a variety of fields, including nonlinear elasticity, elastic shells, piezoelectricity, homogenization, contact, viscoelastic beams,...
Recently we became interested in obtaining models for viscoelastic shells by using the asymptotic expansion method. To do that we follow the methodology developed by Ciarlet for elastic shells [4] and the ideas in [5,6] for viscoelastic beams.
In this talk we show some preliminary results regarding the formal derivation of models for elliptic viscoelastic membranes and general viscoelastic membranes.
We also provide results of existence and uniqueness of the limit models and convergence results which justify the models derived in a formal way. We shall also comment the ongoing work and open problems.
[1] J.L. Lions, Perturbations singulières dans les probèmes aux limites et en contrôle optimal. Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin, 1973.
[2] P.G. Ciarlet, P. Destuynder, A justification of a two dimensional linear plate model. J. Mécanique, 18 (1979), 315--344.
[3] A. Bermúdez, J.M. Viaño, Une justification des équations de la thermoélasticité des poutres à section variable par des méthodes asymptotiques. RAIRO Analyse Numerique, 18 (1984), 347--376.
[4] P.G. Ciarlet, Mathematical Elasticity, Vol. III, Theory of Shells, North-Holland, Amsterdam, 2000.
[5] Á. Rodríguez-Arós, J. M. Viaño, Mathematical justification of viscoelastic beam models by asymptotic methods. J. Math. Anal. Appl., 370 (2010), 607–634.
[6] Á. Rodríguez-Arós, J. M. Viaño, Mathematical Justification of Kelvin-Voigt Beam Models by Asymptotic Methods. Z. Angew. Math. Phys., 63 (2012), no. 3, 529-556.