8. Conclusion

The formulation that we have just described applies to the calculations of thin structures curved in large displacements, whose characteristic thickness-to-length ratio is less than \(1\mathrm{/}10\). It is a direct complement to the finite elements described in the previous reference documentation [R3.07.04] and used in the context of small displacements and deformations. They rely on the same quadrangle and triangle meshes.

Their formulation is based on a 3D continuous medium approach in which Hencky-Mindlin-Naghdi shell kinematics are introduced, under plane constraints, into the weak equilibrium formulation. The deformation measure used is that of Green-Lagrange, energetically combined with Piola-Kirchhoff stresses of the second kind. The equilibrium formulation is therefore total Lagrangian. Transverse distortion is treated in the same way as in [R3.07.04]. The rotations must remain less than \(2\pi\) because of the choice to update large rotations implemented in Code_Aster for which there is no bijection between the total rotation vector and the orthogonal rotation matrix.

Because of the singularity of the tangent stiffness matrix with respect to the component of the rotation around the transform of the normal, a fictional deformation energy associated with this rotation is defined. This rotation is associated with constant torsional stiffness. The internal forces associated with this potential energy are taken into account. This potential energy, which is not zero, does not correspond to a physical deformation. It must therefore remain negligible, which the user can control by imposing a value of COEF_RIGI_DRZ equal to \({10}^{\mathrm{-}3}\) to \({10}^{\mathrm{-}5}\).

For the post-treatment of stresses, we limit ourselves to the framework of small deformations. It was then possible to establish the identity between the Piola-Kirchhoff stress tensor observed in the initial geometry and the Cauchy stress tensor in the deformed geometry. In addition, since the stress state is flat for the Piola-Kirchhoff tensor, this property is found for the Cauchy stress state. It should be noted that in more general contexts, this property is not maintained.

Linear buckling is treated as a special case of the geometric nonlinear problem. It is based on the hypothesis of a linear dependence of the fields of displacements, deformations and stresses on the load level, before the critical load. As a result, the tangent stiffness matrix can be developed linearly with respect to the load level. We then find the geometric part of the general geometric nonlinear matrices obtained by identifying the deformation of the normal to the mean surface and the initial normal, which is consistent with the linearity of the deformations as a function of the load level.