4. Conclusion#

The finite elements that we propose were chosen for a very particular purpose: calculation of axisymmetric thin structures with the aim of obtaining good precision on the membrane and flexural solution while having an element that is simple to implement and not too expensive.

The choice of degrees of freedom allows a good representation of boundary conditions. In addition, this formulation of displacement and rotation leads to elements of lower degree: the elements are \(\mathit{P2}\) in membrane and \(\mathit{P2}\) in flexure. It appears that they are easy to handle and that their formulation makes it possible to use a simple pre- and post-processor structure, a significant advantage for performing fairly fine meshes (one-dimensional) and for easily visualizing the results (on a simple curve). The kinematics chosen: formulation of HENCKY - MINDLIN - NAGHDI, in movements and rotations of the mean surface, allows the intervention of transverse shear energy (interesting for shells of average thickness).

This energy can be affected by a correction factor \(k\): if we want to put ourselves in theory of REISSNER, all we have to do is choose \(k=5/6\) instead of 1 (but of course, the arrow \(W\) and the rotations \(\beta\) are in this theory only weighted averages in thickness). In addition, the shell formulation of LOVE - KIRCHHOFF (for very thin structures) can be simulated by penalizing the condition of nullity of the transverse distortion, by choosing a factor \(k\mathrm{=}{\text{10}}^{6}\mathrm{\times }\), \(h\) being the thickness and \(L\) a characteristic distance (radius of curvature, area of application of the loads, etc.).

Nonlinear behaviors under plane constraints are available for these elements. However, it is pointed out that the stresses generated by the transverse distortion are treated elastically, for lack of better. In fact, taking into account a constant transverse shear that is not zero on the thickness and the determination of the associated correction on the shear stiffness with respect to a model satisfying the boundary conditions are not possible and therefore make the use of these elements, when the transverse shear is non-zero, strictly impossible in plasticity. Strictly speaking, for non-linear behaviors, these elements should therefore be used within the framework of Love-Kirchhoff theory.

Elements corresponding to mechanical elements exist in thermal engineering; thermomechanical linkages are therefore available with finite elements of thermal shells with three nodes described in [R3.11.01] in its axisymmetric version.

In the test cases treated, the blocking phenomena did not occur. The decomposition of the deformation energy will make it possible, if necessary, to selectively integrate the terms responsible for the blockage, such a modification not having to pose any particular difficulties. A more detailed study must of course be carried out on this subject, as to the numerical methods to be used to avoid this blockage when the thickness becomes low.

Possible developments are:

  • anisotropy in order to be able to treat multilayer shells,

  • buckling problems,

  • the serial decomposition of FOURIER to study non-axisymmetric problems of shells of revolution,

  • taking into account a variable thickness…