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.

A Micromorphic Beam Theory for Beams with Elongated Microstructures

“…It follows from Eqs. (40)(41)(42)(43)(44)(45) that the axial displacement and microstrain of the mid-plane, u 0 and s 0 , are independent of the transverse ones. Therefore, explicit solutions for u 0 and s 0 can be obtained by simultaneously solving Eqs.…”

“…Therefore, explicit solutions for u 0 and s 0 can be obtained by simultaneously solving Eqs. (40) and (43). In this way, eliminating s 0 between Eqs.…”

“…However, their implementations to the mechanics of advanced structures, e.g., beams, are scarce. A few studies used the micromorphic theory for the electroelastic bending of piezoelectric beams 36 , the finite element modeling of micromorphic continua [37][38][39] , the static deformation of geometrically nonlinear micromorphic shells 40,41 , and the static bending of micropolar beams 42,43 .…”

A review of size-dependent continuum mechanics models for micro- and nano-structures

“…For the two presented morphologies, the MMT predicts an accurate structural response. In two recent studies, Norouzzadeh et al [465,466] worked on the large elastic deformation of micromorphic shells in two different parts including variational formulation (Part I) and an IG method (Part II). In Part I, by the novel matrix-vector format utilized for the kinematic model, energy functions, and constitutive relations, an IG method-based solution strategy was proposed, which can readily assess the macro/micro-deformation field components.…”

A non-classical model for first-ordershear deformation circular cylindrical thin shells incorporating microstructure and surface energy effects

“…Higher-order elasticity theories containing microstructure-dependent material constants have been applied to study shell deformations (e.g., [4–7]). For example, stability and free vibrations of circular cylindrical thin shells were investigated in [8,9] using a simplified strain gradient elasticity theory (SSGET) (e.g., [10–13]).…”