Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations

Abstract

The paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells with drilling degrees of freedom. This theory establishes a general model for shells, which is characterized by two independent kinematic fields: the translation vector and the rotation tensor. Thus, the kinematical structure of 6-parameter shells is identical to that of Cosserat shells. We show the existence of global minimizers for the geometrically non-linear 2D equations of elastic shells. The proof of the existence theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze separately the particular cases of isotropic shells, orthotropic shells, and composite shells.

Refined dimensional reduction for isotropic elastic Cosserat shells with initial curvature

“…The results of [16,15,19] have been obtained by an 8-parameter ansatz of the deformation through the thickness and consistent analytic integration over the thickness in the case of a flat undeformed shell reference configuration. The mathematical approach developed there also allowed for the first existence proof of minimizers [15,19,1]. In this paper, we extend the modelling from flat shells to initially curved shells.…”

The Rayleigh and Courant variational principles in the six-parameter shell theory

“…where c is the infinitesimal rotation vector (24). The increments of the strain measures are expressed as [6]:…”

Geometrically nonlinear FEM analysis of 6‐parameter resultant shell theory based on 2‐D Cosserat constitutive model

“…Theoretical basis can be traced back to works of Reissner [4] and Libai and Simmonds [5]. Further theoretical developments and aspects of numerical formulation can be found for instance in [6][7][8][9][10][11].…”

“…The resulting kinematic model is formally equivalent to the Cosserat surface with three rigidly rotating directors.Theoretical basis can be traced back to works of Reissner [4] and Libai and Simmonds [5]. Further theoretical developments and aspects of numerical formulation can be found for instance in [6][7][8][9][10][11].In the previous works e.g. [12][13][14][15][16][17] the constitutive relation had in some sense postulated character.…”