Large elastic deformation of micromorphic shells. Part I: Variational formulation

Abstract

We aimed to study the static deformation of geometrically nonlinear shell-type structures on the basis of micromorphic theory. Employing the most comprehensive model in the micro-continuum field, shells in low-dimensions and made of inhomogeneous materials are precisely investigated. The seven-parameter two-dimensional (2D) kinematic model is used which satisfies three-dimensional (3D) constitutive relations and represents the macro-deformation components in mid-surface area of the shell. Also, in the framework of micromorphic continua with three deformable director vectors, nine micro-deformation degrees of freedom, including micro-scale rotations, shears and stretches, are taken into account. Utilizing the energy approach in the convected curvilinear coordinate system leads to the general derivation of the variational formulations in Lagrangian description. High-order stress–strain relations are obtained via introducing the size-dependent as well as size-independent elasticity tensors for the isotropic micromorphic solid. Finally, an equivalent matrix–vector form of representation is proposed to facilitate the solution procedure of the extracted tensor-based formulation. Determining the kinetic and kinematic fields in terms of 16 macro and micro-deformation components, provides the opportunity to directly implement the interpolation-based solution methodologies, such as the finite element isogeometric analysis presented in Part II of this study. Two parts of the article, that are organized to be independent, contribute to the literature respectively from theoretical and computational perspectives. © The Author(s) 2019.