Pitfalls in the finite element modeling of the buckling of sandwich shells of revolution

The objective of this work is to answer the following question: Is a multilayer shell model always appropriate for the prediction of the buckling of a thin structure? In order to do that, we chose to analyze the buckling of a three-layer cylindrical tube in uniform axial compression. We compared three approaches for the calculation of the theoretical buckling load: a fully analytical formula, a Fourier series buckling model of a multilayer axisymmetric shell and a calculation based on a mechanical mesh of the axisymmetric medium of the middle layer. This analysis shows that two buckling modes are in competition: a global beam mode and a local skin buckling mode (wrinkling) which develops when the stiffness of the middle layer is quite small. The critical loads can be estimated analytically for both types of modes, leading to a critical length under which the cylinder fails by skin buckling; otherwise, it fails in beam mode. Then the problem was solved by finite elements using two models: first, a multilayer axisymmetric shell model, then a model with two types of elements in each section (an 8-node axisymmetric element for the middle layer, and axisymmetric shell elements for the two skins). We showed that both models can predict the beam mode, but only the latter was capable of predicting the skin buckling mode. Similar results are proposed for the prediction of the buckling of a three-layer cone under internal pressure. We also compared the formulations for geometrically nonlinear problems with nonlinear materials and showed that in such problems the nonlinearities cause even more pronounced differences. This clearly raises the question of the choice of the type of finite element for the prediction of buckling in a multilayer structure in which some layers have only a small relative stiffness. The analytical solution and the analysis of the nonlinear response in the case of the cone confirm that the poor prediction of the skin buckling mode is due to a failure to account for the flexibility perpendicular to the mean surface.