8. Conclusion

The curved shell finite elements that we describe here are used in the calculations of curved thin structures whose characteristic thickness-to-length ratio is less than 1/10. Two finite volume shell elements based on quadrangular and triangular meshes have been introduced in*Code_Aster*. They were chosen for a very particular purpose: to be able to represent the complete behavior of curved structures whereas until now it was only possible to use elements with plane facets that induced parasitic flexions and required the meshes to be refined.

They are elements where the deformations and stresses in the plane of the element vary linearly with the thickness of the shell. The kinematics chosen is a shell kinematics of the Hencky-Mindlin-Naghdi type making it possible to use transverse shear energy. The distortion associated with transverse shear is constant over the thickness of the element. The variable correction on the transverse shear coefficient \(k\) offers flexibility of use making it possible to go from the theory of HENCKY - MINDLIN - NAGHDI for \(k\mathrm{=}1\), to that of REISSNER for \(k\mathrm{=}5\mathrm{/}6\) and to that of LOVE_KIRCHHOFF (for very thin structures) if we choose a value of \(k\) equal to \({\text{10}}^{6}\mathrm{\times }h\mathrm{/}L\), \(h\) being the thickness and \(L\) being a characteristic distance (mean radius of curvature, load application area…). As in the latter case, we use a penalization method to make the transverse shear terms small, we can, if we take an excessively large value of \(k\), we can make the numerical system singular. In this case, you need to reduce the value of \(k\).

The default value for \(k\) is \(5\mathrm{/}6\). It is generally used when the structure to be meshed has a characteristic thickness-to-length ratio between \(1\mathrm{/}20\) and \(1\mathrm{/}10\). For smaller thicknesses where the transverse distortion becomes low, we may want to use a value of \(k\mathrm{=}{\text{10}}^{6}\mathrm{\times }h\mathrm{/}L\) (to be able to make comparisons with plate elements DKT for example). When the transverse distortion is non-zero, the shell elements do not satisfy the 3D equilibrium conditions and the boundary conditions on the nullity of the transverse shear stresses on the upper and lower shell faces, compatible with a constant transverse distortion in the thickness of the shell. As a result, at the level of behavior, a coefficient of 5/6 for a homogeneous shell corrects the usual relationship between the stresses and the transverse distortion in order to ensure equality between the shear energies of the 3D model and of the shell model with constant distortion. In this case, the interpretation of arrow \({\tilde{u}}_{3}\) is the mean transverse displacement in the thickness of the shell and not the displacement of the mean surface area of the shell.

For thin structures in order to avoid blocking phenomena, reduced underintegration is used for the parts of the membrane and shear of the stiffness matrix. The choice on finite elements was based on the quadrangle elements Heterosis Q9H and triangle T7H. In fact, among the finite elements with quadratic interpolation functions, the performance of the Heterosis Q9H element is known. In particular, it is greater than that of the Q9S Serendip elements or the Q9 Lagrange elements. However, this performance is based on the selective integration of the element with reduced integration of membrane and shear terms on the one hand, and normal integration of flexure terms on the other hand. By analogy with Q9H, the finite element T7H was taken as a triangular element. However, as far as possible, we will use the Q9H rather than the T7H which is much less efficient.

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, the rigorous consideration of a constant non-zero transverse shear on the thickness and the determination of the associated correction on the shear stiffness with respect to a model satisfying the equilibrium conditions and 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. Rigorously, 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; thermo-mechanical linkages are therefore available with finite elements of thermal shells with 7 and 9 knots. Extensions of the previous formulation presented in the appendix also allow the anisotropy of materials and kinematic nonlinearity to be taken into account. This second extension is operational in Code_Aster and is the subject of reference documentation [R3.07.05].