Orientation-preserving condition and polyconvexity on a surface: application to nonlinear shell theory (Q387983)

The authors address a nonlinear shell theory, different from well-known theories of W.~T. Koiter and P.~M. Naghdi, respectively.NEWLINENEWLINE In the nonlinear Koiter model, the unknown represents the unit normal vector field to the deformed middle surface, while in the case of the nonlinear Naghdi model, the unknown represents the rotated unit normal vector field along the deformed middle surface and is assumed to have a strictly positive normal component at each point of the deformed middle surface.NEWLINENEWLINE When the surface is thought of as the deformed middle surface of a nonlinear elastic shell and a polyconvex function on the surface is its stored energy function, the authors show that this function is coercive in suitable Sobolev norms. The considered function is subjected to a specific constraint which prevents the covariant basis vector of the tangent plane to the deformed middle surface to become linearly dependent at some points.NEWLINENEWLINE This allows the authors, as distinct from the theories of Koiter and Naghdi, to give sufficient conditions for the existence of minimizers over an appropriate set of admissible fields which, in addition to standard boundary conditions, incorporates also some constraints.NEWLINENEWLINE In my opinion, the definition of the polyconvex function on a surface is not new, because it was used in the papers by \textit{J. M. Ball} [Arch. Ration. Mech. Anal. 63, 337--403 (1977; Zbl 0368.73040)] and by \textit{J. M. Ball} et al. [J. Funct. Anal. 41, 135--174 (1981; Zbl 0459.35020)]. Moreover, almost all the issues raised in this paper have been taken into account in the papers by J.~M. Ball.