The Isotropic Cosserat Shell Model Including Terms up to $O(h^{5})$. Part II: Existence of Minimizers

Abstract

In this paper we derive the linear elastic Cosserat shell model incorporating effects up to order O(h 5 ) in the shell thickness h as a particular case of the recently introduced geometrically nonlinear elastic Cosserat shell model. The existence and uniqueness of the solution is proven in suitable admissible sets. To this end, inequalities of Korn-type for shells are established which allow to show coercivity in the Lax-Milgram theorem. We are also showing an existence and uniqueness result for a truncated O(h 3 ) model. Main issue is the suitable treatment of the curved reference configuration of the shell. Some connections to the classical Koiter membrane-bending model are highlighted.

Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

“…Under certain conditions, one can show that the obtained strain energy density (119) is coercive. This feature has been proved for related Cosserat shell models in [19], using the matrix formulation. Then, applying the general existence results for 6-parameter shells presented in [8], one can prove the existence of minimizers for the nonlinear shell model derived in the present work.…”

A constrained Cosserat shell model up to order $O(h^5)$: Modelling, existence of minimizers, relations to classical shell models and scaling invariance of the bending tensor

“…For models which coincide to leading order with the classical Koiter model for small enough thickness [4,5], the existence of the solution is proven under the conditions (1.18). For the geometrically nonlinear Cosserat shell model including terms up to order O(h 5 ) [20], we have shown the existence of the solution [21] for the theory including O(h 5 ) terms, as well as the existence of the solution for the theory including terms up to order O(h 3 ). In Appendix A.3.1 we show that condition (1.16) on the thickness, under which the existence result presented in [21] was shown, can be weakened, but the new condition still remains more restrictive than (1.17).…”

“…For the geometrically nonlinear Cosserat shell model including terms up to order O(h 5 ) [20], we have shown the existence of the solution [21] for the theory including O(h 5 ) terms, as well as the existence of the solution for the theory including terms up to order O(h 3 ). In Appendix A.3.1 we show that condition (1.16) on the thickness, under which the existence result presented in [21] was shown, can be weakened, but the new condition still remains more restrictive than (1.17). We noted that, in order to prove the existence of the solution, while in the theory including O(h 5 ) the condition on the thickness h is similar to that originally considered in the modelling process, in the sense that it is independent of the constitutive parameters, in the O(h 3 )-case the coercivity is proven under more restrictive conditions on the thickness h which are depending on the constitutive parameters.…”

On in-plane drill rotations for Cosserat surfaces

“…The extension of the planar shell model to initially curved shells has been recently given in [18,19,20]. The planar shell model (1.9) has been used to successfully predict the wrinkling behavior of very thin elastic sheets [37].…”