GIUSEPPE BUTTAZZO Istituto di Matematiche Applicate ~ U. Dini ~ Via Bonanno, 25/B - 56126 Pisa (Italy) THIN STRUCTURES IN ELASTICITY: THE VARIATIONAL METHOD (Conferenza tenuta il ~6 giugno 1989) ABSTRACT. -- The behaviour of two-dimensional and one-dimensional thin elastic structures as membranes, shells, strings, beams, is deduced starting from three-dimensional elastic models and passing to the limit when one or two dimensions go to zero. The variational approach and the F-convergence theory are used. 1. - INTRODUCTION. This paper is a survey on some recent results concerning the mathematical justification of linear and nonlinear models for thin structures in elasticity by a variational approach (see Acerbi & But- tazzo & Percivale [1], Acerbi & Buttazzo & Percivale [2], and Percivale [10]). More precisely, starting from a given model of three-dimensional elasticity, that is from a given stored energy func- tional of the form (1.1) f W(x, Du) dx, and passing to the limit when one or two dimensions of Y2 go to zero, we shall obtain the expression of free energies of two-dimen- sional elastic bodies as membranes, plates, shells, or one-dimensional elastic bodies as strings, rods, beams. In (1.1) the reference configuration ~ is a bounded open subset of R ~, the function u:12--> ~,~ is the displacement field (or the position field in the nonlinear case), and the function W : ~ X ~,9_~ [0, ~- ~] is the strain energy density. For instance, if ~ c ~8 is a smooth two-dimensional manifold and 3emina~o Matematir o Fi~rir . I0