3. Variational formulation

3.1. Virtual work

Figure 3.1-a: Volumic shell.Subsequent pressure on the initial mean surface and its transforme

The virtual work of a follower \(p\) pressure (i.e. acting on the transformed mean surface and moving with it) can be expressed in the form:

\(\delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=-{\int }_{{\omega }^{\varphi }}\delta {u}_{p}\text{.}pd{\omega }^{\varphi }\)

If we use the isoparametric surface element corresponding to our solid shell modeling, the area \(d{\omega }^{\varphi }\) is expressed directly in terms of the isoparametric coordinates \(d{\xi }_{1}d{\xi }_{2}\) and we get the following simple form of the equation above:

\(\delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=-{\int }_{\left[-\mathrm{1,}+1\right]\times \left[-\mathrm{1,}+1\right]}\delta {u}_{p}\text{.}p({\xi }_{1},{\xi }_{2}){a}_{1}^{\varphi }({\xi }_{1},{\xi }_{2})\times {a}_{2}^{\varphi }({\xi }_{1},{\xi }_{2})d{\xi }_{1}d{\xi }_{2}\)

3.2. Tangent operator

As the virtual work of the follower pressure depends on the current configuration, its linear variation \(\Delta\) is not zero and must be taken into account. The tangent operator associated with this virtual work is written in iteration \((i+1)\) in the form:

\(L\left[\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{(i+1)}\right]=\delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{(i)}+\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{(i)}\)

where \(\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{(i)}\) is the increment between two iterations of the virtual work of the subsequent pressure. If the pressure is given in the form:

\(p=\lambda {p}_{0}\)

\(\lambda\) being the load level that is fixed during the iterations (load control \(\Delta \lambda =0\)), we can write:

\(\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=-{\int }_{\left[-\mathrm{1,}+1\right]\times \left[-\mathrm{1,}+1\right]}\delta {u}_{p}\text{.}p({a}_{1}^{\varphi }\times \Delta {a}_{2}^{\varphi }-{a}_{2}^{\varphi }\times \Delta {a}_{1}^{\varphi })d{\xi }_{1}d{\xi }_{2}\)

The incremental variations of the vectors of the local base tangent to the transform of the mean surface are given by:

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

since the initial mean surface doesn’t « move » during iterations which results in \(\Delta {x}_{P}=0\).

These calculations finally make it possible to establish the expression of the increment of the virtual work of next pressure in the form:

\(\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=-{\int }_{\left[-\mathrm{1,}+1\right]\times \left[-\mathrm{1,}+1\right]}\delta {u}_{P}\cdot P(\left[{a}_{1}^{\varphi }\times \right]\frac{\partial }{\partial {\xi }_{2}}\Delta {u}_{P}-\left[{a}_{2}^{\varphi }\times \right]\frac{\partial }{\partial {\xi }_{2}}\Delta {u}_{P})d{\xi }_{1}d{\xi }_{2}\)

where \(\left[{a}_{1}^{\varphi }\times \right]\mathrm{et}\left[{a}_{2}^{\varphi }\times \right]\) are respectively the antisymmetric matrices of the tangent vectors \({a}_{1}^{\varphi }\mathrm{et}{a}_{2}^{\varphi }\) respectively.

Note:

In the reference [bib2], one integration per part is undertaken on the expression above. It is shown that the tangent matrix can be decomposed into a symmetric part resulting from integration on the domain and an anti-symmetric part resulting from integration on the contour. It is also shown that the assembly of the antisymmetric parts of the elementary tangent matrices leads to a zero matrix when the pressure is continuous from one finite element to another, due to the existence of a potential associated with the work of the pressure in this case.