Modified couple stress theory for three-dimensional elasticity in curvilinear coordinate system: application to micro to
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Modified couple stress theory for three-dimensional elasticity in curvilinear coordinate system: application to micro torus panels I. Karimipour . Y. Tadi Beni
. A. H. Akbarzadeh
Received: 23 January 2020 / Accepted: 22 July 2020 Springer Nature B.V. 2020
Abstract This research considers size effects in the linear three-dimensional elasticity analysis of microtori. The fundamental relations (displacement form) are derived for isotropic toroidal shells in the framework of the modified couple stress theory in the curvilinear coordinate system to predict the mechanical responses. A numerical solution for the displacement field is obtained using the GDQ method. The numerical results are in a close agreement with those found by the finite element and the Galerkin method. Parametric studies are conducted to explore the effect of size-dependency, micro-tori geometry, meridional and circumferential angle, toroidal shell thickness, and different boundary conditions on the distribution of the displacement fields. Numerical results for dis-
placement also show that natural frequencies of microtoroidal shells, predicted by modified couple stress theory, are less than those predicted by the classical theory, due to the significant effect of length scale parameter (related to material microstructures) on the mechanical responses. The use of general curvilinear coordinates in toroidal structures enables us to also study the mechanical behavior of irregular geometries, cap-shaped panel, saddle-shaped panel, and sectorialshaped panel. Keywords Generalized differential quadrature method Modified couple stress theory Toroidal coordinates Torus structure
1 Introduction
I. Karimipour Y. Tadi Beni (&) Department of Mechanical Engineering, Shahrekord University, Shahrekord, Iran e-mail: [email protected] I. Karimipour A. H. Akbarzadeh AM3L Laboratory, Department of Bioresource Engineering, McGill University, Montreal, QC H9X3V9, Canada Y. Tadi Beni Nanotechnology Research Center, Shahrekord University, Shahrekord, Iran
Engineered micro/nanomaterials, e.g., micro-architected lattices, architected carbon nanotubes (CNT), and graphene, have recently attracted significant interest due to their outstanding mechanical, electronic, and optical properties [1–5]. A micro torus is a doubly-curved geometry and one of the most commonly-used structures in micro/nanotechnology. Micro toroidal carbon structures are new materials with excellent properties, e.g., higher strength and thermal and mechanical stability than nanotubes (NT) [1]. Experimental studies have suggested that different
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shapes of nanotubes (NT) can be used as molecular transporters to shuttle biological and medicinal cargoes across the cellular membrane [6]. These applications are mainly due to their capacity to store or convey fluids, in particular aqueous solutions, at nanoscale precision. Compared to NTs, micro-tori structures have an additional curvature produced by the rolling transformation of t
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