Non-linear shell theory

: a new approach

Abstract

We propose a definition of "a polyconvex function on a surface", inspired by the definitions set forth in other contexts by J. Ball in 1977 and by J. Ball, J.C. Curie and P.J. Olver in 1981. The application of this notion will be to nonlinear shell theory, where the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function. In this case, we first show that it is possible to assume in addition that this function satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the "surface analogue" of the orientation-preserving condition in three-dimensional elasticity. We also require the stored energy function to be coercive for appropriate Sobolev norms. We further show that a functional with such a polyconvex integrand satisfying the aforementioned conditions is weakly lower semi-continuous in ad hoc function spaces, which allows to establish the existence of minimizers. We also indicate how this new approach compares with the classical nonlinear shell theories, such as those of W.T. Koiter and P.M. Naghdi. As is well-known, there is to this date no available existence result for the nonlinear Koiter or Naghdi models, and in fact there is little hope of ever obtaining any, if only because there is nothing in these models that prevents the vector fields of the covariant bases along the deformed middle surface to become linearly dependent at some points. As a way of circumventing this difficulty, we give here sufficient conditions which have to be satisfied by the integrand appearing in the considered functional, then more general than those of Koiter or Naghdi's models, but appropriate for guaranteeing the existence of minimizers over an appropriate set of admissible fields, which, in addition to standard boundary conditions incorporates the above surface analogue of the orientation-preserving condition in three-dimensional elasticity.

Date of Award	2 Oct 2013
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Philippe G CIARLET (Supervisor)