2. Cinematics

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

We recall the great transformation undergone by the shell:

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

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

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

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

2.1. Parameterization of the mean surface transform

The \({w}^{j}\) 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 \({w}^{j}\):

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

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

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

This allows them to be linked to the vectors of the natural base linked to the initial surface \(w\) by the relationships:

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

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

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

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

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

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