Shells without drilling rotations: a representation theorem in the framework of the geometrically nonlinear 6-parameter resultant shell theory

In the framework of the geometrically nonlinear 6-parameter resultant shell theory we give a characterization of the shells without drilling rotations. These are shells for which the strain energy function $W$ is invariant under the superposition of drilling rotations, i.e. $W$ is insensible to the arbitrary local rotations about the third director $\boldsymbol{d}_3\,$. For this type of shells we show that the strain energy density $W$ can be represented as a function of certain combinations of the shell deformation gradient $\boldsymbol{F}$ and the surface gradient of $\boldsymbol{d}_3\,$, namely $W\big(\boldsymbol{F}^{ T}\boldsymbol{F} , \, \boldsymbol{F}^T \boldsymbol{d}_3 \,, \, \boldsymbol{F}^T\mathrm{Grad}_s\boldsymbol{d}_3 \big)$. For the case of isotropic shells we present explicit forms of the strain energy function $W$ having this property.