Cinematics
===========

For the solid shell elements :math:`\Omega`, a reference surface :math:`\omega`, or average, left surface is defined (with curvilinear coordinates :math:`{\xi }_{1}{\xi }_{2}` for example) and a thickness :math:`h({\xi }_{1}{\xi }_{2})` measured according to the normal to the average surface. The position of the points on the shell is given by the curvilinear coordinates :math:`({\xi }_{1},{\xi }_{2})` of the mean surface :math:`\omega` and the elevation :math:`{\xi }_{3}` with respect to this surface.

We recall the great transformation undergone by the shell:

:math:`{\omega }^{\varphi }` (set of points :math:`{P}^{\varphi }` to :math:`{\xi }_{3}=0`) is the transform of the initial mean surface :math:`\omega` (set of points :math:`Pà{\xi }_{3}=0`).

The position of point :math:`{P}^{\varphi }` on the deformed configuration can be established based on the position of the initial point :math:`P` as follows:

:math:`{x}_{P}^{\varphi }({\xi }_{1},{\xi }_{2})={x}_{P}({\xi }_{1},{\xi }_{2})+{u}_{P}({\xi }_{1},{\xi }_{2})`.


.. image:: images/1000569C000069BB0000651456FD8FE196694F41.svg
    :width: 631
    :height: 602


.. _RefImage_1000569C000069BB0000651456FD8FE196694F41.svg:

**Figure 2-a: Volumic shell.Major transformations of a fiber that was initially normal to the average surface**




Parameterization of the mean surface transform
--------------------------------------------------------

The :math:`{w}^{j}` transform can be parameterized in a manner similar to the parametrization of the initial surface. So we can define the infinitesimal tangent vector element to :math:`{w}^{j}`:

:math:`\begin{array}{}{\text{dx}}_{P}^{\varphi }({\xi }_{1},{\xi }_{2})=\frac{\partial {x}_{P}^{\varphi }}{\partial {\xi }_{1}}d{\xi }_{1}+\frac{\partial {x}_{P}^{\varphi }}{\partial {x}_{1}}d{\xi }_{2}\\ {\text{dx}}_{P}^{\varphi }({\xi }_{1},{\xi }_{2})=d{\xi }_{1}{a}_{1}^{\varphi }({x}_{1},{x}_{2})+d{\xi }_{2}{a}_{2}^{\varphi }({\xi }_{1},{\xi }_{2})\end{array}`

where :math:`\left[{a}_{1}^{\varphi }({\xi }_{1},{\xi }_{2});{a}_{2}^{\varphi }({\xi }_{1},{\xi }_{2})\right]` represents a natural base that is not orthogonal :math:`({a}_{1}^{\varphi }\text{.}{a}_{2}^{\varphi }\ne 0)` and is not standardized :math:`(\parallel {a}_{1}^{\varphi }\parallel \ne 1;\parallel {a}_{1}^{\varphi }\parallel \ne 1)` tangent to the surface :math:`{\omega }^{\varphi }`. The two basic vectors can be linked to the movements using the following formula:

:math:`\begin{array}{}{a}_{1}^{\varphi }({\xi }_{1},{\xi }_{2})=\frac{\partial {x}_{P}^{\varphi }}{\partial {\xi }_{1}}=\frac{\partial ({x}_{p}+{u}_{p})}{\partial {\xi }_{1}}\\ {a}_{2}^{\varphi }({\xi }_{1},{\xi }_{2})=\frac{\partial {x}_{P}^{\varphi }}{\partial {\xi }_{2}}=\frac{\partial ({x}_{p}+{u}_{p})}{\partial {\xi }_{2}}\end{array}`

This allows them to be linked to the vectors of the natural base linked to the initial surface :math:`w` by the relationships:

:math:`\begin{array}{}{a}_{1}^{\varphi }({\xi }_{1},{\xi }_{2})={a}_{1}({\xi }_{1},{\xi }_{2})+\frac{\partial {u}_{p}}{\partial {\xi }_{1}}\\ {a}_{2}^{\varphi }({\xi }_{1},{\xi }_{2})={a}_{2}({\xi }_{1},{\xi }_{2})+\frac{\partial {u}_{p}}{\partial {\xi }_{2}}\end{array}`

It is important to note that these vectors are distinct from the vectors obtained by the large rotation :math:`\Lambda` of the vectors :math:`{a}_{1}({\xi }_{1},{\xi }_{2});{a}_{2}({\xi }_{1},{\xi }_{2})`:

:math:`\begin{array}{}{a}_{1}^{\varphi }({\xi }_{1},{\xi }_{2})\ne \Lambda ({\xi }_{1},{\xi }_{2}){a}_{1}({\xi }_{1},{\xi }_{2})\\ {a}_{2}^{\varphi }({\xi }_{1},{\xi }_{2})\ne \Lambda ({\xi }_{1},{\xi }_{2}){a}_{2}({\xi }_{1},{\xi }_{2})\end{array}`

In fact, due to the deformation due to transverse shear, the rotated vectors are no longer tangent to :math:`{\omega }^{\varphi }`. The illustration of this is given by the [:ref:`Figure 3.1-a <Figure 3.1-a>`].

With this parametrization, the infinitesimal vector surface element that is perpendicular to :math:`{\omega }^{\varphi }` can be written as:

:math:`d{\omega }^{\varphi }({\xi }_{1},{\xi }_{2})={a}_{1}^{\varphi }({\xi }_{1},{\xi }_{2})\times {a}_{2}^{\varphi }({\xi }_{1},{\xi }_{2})d{\xi }_{1}d{\xi }_{2}`