Large elastic deformation of micromorphic shells. Part II. Isogeometric analysis

Abstract

In Part I of this study, a variational formulation was presented for the large elastic deformation problem of micromorphic shells. Using the novel matrix-vector format presented for the kinematic model, constitutive relations, and energy functions, an isogeometric analysis (IGA)-based solution strategy is developed, which appropriately estimates the macro- and micro-deformation field components. Due to the capability of constructing exact geometries and the powerful mesh refinement tools, IGA can be successfully applied to solve the equilibrium equations with dominant nonlinear terms. It is known that different types of locking phenomena take place in the conventional finite element analysis of thin shells based on low-order elements. Non-standard finite element models with mixed interpolation schemes and additional degrees of freedom (DOFs) or the ones used the high-order Lagrangian shell elements which require high computational costs, are the available solutions to tackle locking issues. The present 16-DOFs IGA is found to be efficient because of possessing a good rate of convergence and providing locking-free stable responses for micromorphic shells. Such a conclusion is found from several comparative studies with available data in the well-known macro-scale benchmark problems based on the classical elasticity as well as the corresponding numerical examples studied in nano-scale beam-, plate-, cylindrical shell- and spherical shell-type structures on the basis of the micromorphic continuum theory.

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.…”

“…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 new isogeometric Timoshenko beam model incorporating microstructures and surface energy effects

“…Besides, IGA inherently has many advantageous features, including the exact geometry representation, high accuracy, simple meshing, mesh refinement, etc. Due to its significant advantages, IGA has received much attention in a variety of engineering applications, such as beam, plate and shell analysis [35][36][37][38][39], contact analysis [40], damage and fracture mechanics [41][42][43], electromagnetic analysis [44], fluid mechanics [45], fluid-structure interaction [46], structure optimization [47,48] and so on. More recently, IGA has been successfully applied to investigate the size-dependent behavior of microbeams and microplates based on the non-local elasticity theory [49], MCST [50][51][52] and the modified strain gradient elasticity theory [53].…”