From nonlinear micromorphic to nonlinear micropolar shell theory

Abstract

Introduced in this paper is the complete nonlinear model of the micropolar theory (MPT) for shell-type materials. Considering the three-dimensional kinematic model in a convected curvilinear coordinate system, the Lagrangian description of micromorphic theory (MMT) is formulated first. Due to certain assumptions, i.e. skew-symmetricity of micro-displacement and orthogonality of micro-deformation tensors, the highly nonlinear micropolar continuum theory is obtained. Unlike the conventional knowledge in the literature, it is found that not only MPT is not a simple version of MMT but also the present micropolar formulation can be reduced to the corresponding micromorphic shell elasticity. Also, unlike the Kafadar-Eringen's nonlinear micropolar model which only predicts the large micro-deformations, the developed micropolar theory is able to investigate the physical behavior in case of large elastic macro-deformations, for the first time. The isogeometric analysis (IGA) solution method with standard base functions is used to prevent the locking issues arising in the low-order finite element analysis (FEA) of thin shells. Accordingly, the proposed IGA micropolar shell model possesses 10-DOFs (degrees of freedom) standing for 7 macro-displacements and 3 micro-rotations. Size / inhomogeneity effects of small-scale / micropolar materials are investigated in the benchmark problems of geometrically nonlinear shells. As a result, it is revealed that the paper contributes to literature by developing the nonlinear micropolar shell theory which is able of capturing large macro- and micro-deformations.