Shell Theory
This page is a work in progress.
Geometry
We identify the shell with a surface $\mathcal{S}$, i.e., a two-dimensional submanifold of the Euclidean point space $\mathcal{E}$. We suppose $\mathcal{S}$ to be orientable, at least locally, and we shall denote by $\boldsymbol{n}(x)$ the orthogonal unit vector at $x$.
Kirchhoff-Love Kinematics
For a Kirchhoff-Love plate, the director $\boldsymbol{d}$ coincides with the normal $\boldsymbol{n}$. Hence:
\[\boldsymbol{\varphi} = d \boldsymbol{n}\]Taking the variation of the identity $\boldsymbol{\mathsf{P}}\boldsymbol{n} = \boldsymbol{0}$ and recalling that $d\boldsymbol{n}$ is a tangential vector, we obtain:
\[\begin{aligned} \boldsymbol{\varphi} &= d\boldsymbol{n} = -(d\boldsymbol{\mathsf{P}})\boldsymbol{n} \\ &= (d\boldsymbol{a}^\alpha \otimes \boldsymbol{a}_\alpha + \boldsymbol{a}^\alpha \otimes d\boldsymbol{a}_\alpha)\boldsymbol{n} \\ &= -\boldsymbol{a}^\alpha \otimes d\boldsymbol{a}_\alpha \boldsymbol{n} \\ &= -\boldsymbol{a}^\alpha \otimes \boldsymbol{u}_{,\alpha} \boldsymbol{n} \\ &= -\nabla\boldsymbol{u}^T \boldsymbol{n} \\ &= -\nabla(\boldsymbol{u} \cdot \boldsymbol{n}) + \nabla\boldsymbol{n}^T \boldsymbol{u} \end{aligned}\]More details coming soon.