ON FEW SHELL MODELS IN NONLINEAR ELASTICITY AND EXISTENCE OF SOLUTIONS

Abstract. The existence of stable solutions for geometrically nonlinear theory of shells has been widely discussed by mechanicians during the last century. But, certainly because of the difficulty met in the classical three dimensional nonlinear elasticity, few mathemat-ical results have been obtained. A possibility is to apply the polyconvexity introduced in nonlinear elasticity by J. Ball [1] to ad’hoc shell theories. But unfortunately the positive results are restricted to a special class of materials. Another approach consists in using the so-called Γ-convergence. This theory has been suggested by the italian school (E. De Giorgi and G. Dal Maso, [9]), and an application to shell models has been given by H. Ledret and A. Raoult [19]. But the main drawback, in our opinion, is that the solution transgresses the equilibrium equations and the difficulty is to make sense to the model ob-tained. Therefore, it is not yet possible to use these results for the physical problem to be solved. In this paper, we suggest another theory based on some nonlinear mathematical tools which have already been used for particular shell models in [12]. The first part gives a formu-lation of a general geometrically nonlinear shell model based on a full description of the large kinematical movement induced by a Kirchhoff-Love field. The objectivity property is checked for a class of materials (energy invariance under the only effect of a nonlinear-rigid body motion). Then the existence of stable solutions is proved using minimizing sequences and some mathematical tricks based on compactness of few nonlinear terms