An existence theorem for a two-dimensional nonlinear shell model of Koiter's type

We propose a minimization problem with a stored energy function that is polyconvex and satisfies all the other assumptions of John Ball's theorem, while being at the same time well adapted for modeling a nonlinearly elastic shell. By restricting the admissible deformations to be specific quadratic polynomials with respect to the transverse variable, we are able to define a new nonlinear shell model for which a satisfactory existence theory is available and that is still two-dimensional, in the sense that minimizing the corresponding total energy amounts to finding three vector fields defined on the closure of a bounded open subset of R².
The most noteworthy feature of our nonlinear shell model is that the "lowest order part" of its stored energy function coincides, at least formally, with the stored energy function found in Koiter's model for a specific class of deformations that are to within the first-order identical to the Kirchhoff-Love deformations considered by W. T. Koiter.