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A Hierarchy of Plate Models Derived from Nonliinear Elasticity by T-Convergence
, M.Sc. Thesis Sharif University of Technology ; Bahraini, Alireza (Supervisor)
Abstract
In this thesis the elastic energy of plates is computed. Primarily, some physical related concepts such as stress, strain and strain energy density are defined. Then based on rigidity estimate theorem, the gradient map on bounded Lipschitz domains is approximated by a constant rotation map R.Thanks to this approximation theorem,we prove the precompactness of sequences of finite bending energy, and the limiting energy of two dimensional thin plates is shown to be an elastic energy. This limit innovation is called Γ –convergence introduced in 1993. It will indeed be revealed that the induced elastic energy of a deformable body is the limit of the sequence of stored energy of body under...
General solution to wave propagation in media undergoing arbitrary transient or periodic temporal variations of permittivity
, Article Journal of the Optical Society of America B: Optical Physics ; Volume 35, Issue 11 , 2018 , Pages 29232932- ; 07403224 (ISSN) ; Mehrany, K ; Memarian, M ; Sharif University of Technology
OSA - The Optical Society
2018
Abstract
A novel and general formulation for wave propagation in time-varying media is presented. Unlike previousreports, our formalism is able to solve propagation in media with arbitrary time variations of permittivity orpermeability, for both transient and steady-state periodic variations. The formulation is approximate yet strikingly accurate in most practical cases. The provided closed-form expressions show that the normalized averagepower after the transition of the permittivity does not depend on the details of the transition, while the energydensity does. Some important discussions are made about time-periodic media, and it is shown that there is anaccumulation of energy when the temporal...
Effects of couple stresses on anti-plane problems of piezoelectric media with inhomogeneities
, Article European Journal of Mechanics, A/Solids ; Volume 26, Issue 4 , 2007 , Pages 647-658 ; 09977538 (ISSN) ; Ghazisaeidi, M ; Sharif University of Technology
2007
Abstract
The examination of the effect of couple stresses on anti-plane electro-mechanical behaviour of piezoelectric media is of interest. The constitutive equations of piezoelectricity for a transversely isotropic piezoelectric medium of crystal class C6 v = 6 mm are derived in the context of couple stress elasticity. In this framework, a characteristic length appears in the formulation of anti-plane problems, by which examination of the size effect is possible. Also stemming from this approach is a new elasticity constant defined as the ratio of couple stress to the curvature, which based on the assumption of positive definiteness of the internal energy density, must be positive. For...
Artificial neural network simulator for supercapacitor performance prediction
, Article Computational Materials Science ; Volume 39, Issue 3 , 2007 , Pages 678-683 ; 09270256 (ISSN) ; Gobal, F ; Sharif University of Technology
2007
Abstract
Artificial neural network was used to calculate the performance of a model supercapacitor as signified by the power density, energy density and utilization to the synthetic, intrinsic and operating characteristics. A four-layer neural net having two hidden layers having 6 and 15 nodes was found to be well capable of simulating the capacitor performance with the convergence achieved often a relatively small number of epochs. As for the input parameters, crystal size, surface lattice length, exchange current density of the capacitor active material and the cell current employed while utilization, energy density and power density were the outputs. © 2006 Elsevier B.V. All rights reserved
Hyperelastic materials behavior modeling using consistent strain energy density functions
, Article Acta Mechanica ; Volume 213, Issue 3-4 , 2010 , Pages 235-254 ; 00015970 (ISSN) ; Naghdabadi, R ; Sharif University of Technology
Springer-Verlag Wien
2010
Abstract
Hyperelastic materials have high deformability and nonlinearity in load-deformation behavior. Based on a phenomenological approach, these materials are treated as a continuum, and a strain energy density is considered to describe their hyperelastic behavior. In this paper, the mechanical behavior characterization of these materials is studied from the continuum viewpoint. For this purpose, the strain energy density is expressed as sum of independent functions of the mutual multiple of principal stretches. These functions are determined by applying the governing postulates on the form of the strain energy density. It is observed that a consistent strain energy density is expressible in terms...
Constitutive modeling of rubberlike materials based on consistent strain energy density functions
, Article Polymer Engineering and Science ; Volume 50, Issue 5 , 2010 , Pages 1058-1066 ; 00323888 (ISSN) ; Naghdabadi, R ; Kargarnovin, M. H ; Sharif University of Technology
Abstract
Rubberlike materials are characterized by high deformability and reversibility of deformation. From the continuum viewpoint, a strain energy density function is postulated for modeling the behavior of these materials. In this paper, a general form for the strain energy density of these materials is proposed from a phenomenological point of view. Based on the Valanis-Landel hypothesis, the strain energy density of incompressible materials is expressed as the sum of independent functions of the principal stretches meeting the essential requirements on the form of the strain energy density. It is cleared that the appropriate mathematical expressions for constitutive modeling of these materials...
Hyperelastic materials modelling using a strain measure consistent with the strain energy postulates
, Article Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science ; Volume 224, Issue 3 , 2010 , Pages 591-602 ; 09544062 (ISSN) ; Naghdabadi, R ; Kargarnovin, M. H ; Sharif University of Technology
Abstract
In this article, a strain energy density function of the Saint Venant-Kirchhoff type is expressed in terms of a Lagrangian deformation measure. Applying the governing postulates to the form of the strain energy density, the mathematical expression of this measure is determined. It is observed that this measure, which is consistent with the strain energy postulates, is a strain type with the characteristic function more rational than that of the Seth-Hill strain measures for hyperelastic materials modelling. In addition, the material parameters are calculated using a novel procedure that is based on the correlation between the values of the strain energy density (rather than the stresses)...
"Energy Finite Element Method for High Frequency Random Vibrations Analysis of a Reinforced Beam-Plate with a Cutout and Experimental Validation "
, M.Sc. Thesis Sharif University of Technology ; Haddadpour, Hassan (Supervisor) ; Mohammad Navazi, Hossein (Supervisor)
Abstract
In this thesis, the high frequency random vibrations analysis of a reinforced plate with a cutout is carried out using Energy Finite Element Analysis (EFEA) and the final results are validated with experimental results. EFEA method uses the average energy density as the primary variable to form the governing differential equations of energy flow in the structure. In this method the equations of space and time averaged energy density are derived in the Far Field and the energy intensity is considered as continues variable over the structure. In this work, at first for simple beam and plate an EFEA code is developed. Then the relationships for joints are derived using structural dynamics. Some...
A novel isolated DC/DC converter for fuel cell powered load
, Article 2009 IEEE Electrical Power and Energy Conference, EPEC 2009, 22 October 2009 through 23 October 2009 ; 2009 ; 9781424445080 (ISBN) ; Hoseinnia, S ; Roshandel, R ; Sharif University of Technology
Abstract
Fuel Cell (FC) is one of the promising distributed generation resources which are based on renewable energies. Clean electricity generation, high efficiency and high energy density of the fuel cells make them attractive for lots of engineers. In this paper we discuss about the operational principles of fuel cell generally. Then, a circuit model of them will be introduced. Based on the model, we designed a novel DC/DC converter for supplying the fuel cell's load and simulated the design with using Matlab. The results say that its novel converter could conveniently supply the load and its needed power curve. ©2009 IEEE
Application of base-line correction methods to obtain permanent displacements of a near-source ground motion: The 2003 bam earthquake
, Article Journal of Earthquake Engineering ; Volume 13, Issue 3 , 2009 , Pages 313-327 ; 13632469 (ISSN) ; Khatam, H ; Farahi Jahromi, H ; Sharif University of Technology
2009
Abstract
In this article, two base-line correction methods for correcting the near-source strong motions were studied on the near-source recording of the 2003, Bam earthquake. To obtain permanent ground displacements at Bam station, two methods were applied with some modifications and without using any kind of filtering. Even though it is difficult to calculate true permanent displacements from data attained by present accelerograms, the results of this study show that the rational limit of permanent ground displacements can be successfully achieved by rational choosing of effective parameters. In addition, it is observed that some pre-processing steps suggested in literature have no considerable...
Emergent statistical bubble localization in a Z2 lattice gauge theory
, Article Physical Review B ; Volume 99, Issue 5 , 2019 ; 24699950 (ISSN) ; Mohseni Rajaee, M ; Langari, A ; Sharif University of Technology
American Physical Society
2019
Abstract
We introduce a clean cluster spin chain coupled to fully interacting spinless fermions, forming an unconstrained Z2 lattice gauge theory (LGT), which possesses dynamical proximity effect controlled by the entanglement structure of the initial state. We expand the machinery of interaction-driven localization to the realm of LGTs such that for any starting product state, the matter fields exhibit emergent statistical bubble localization, which is driven solely by the cluster interaction, having no topologically trivial noninteracting counterpart, and thus is of a pure dynamical many-body effect. In this vein, our proposed setting provides possibly the minimal model dropping all the...
A stable and high-energy hybrid supercapacitor using porous Cu2O-Cu1.8S nanowire arrays
, Article Journal of Materials Chemistry A ; Volume 8, Issue 4 , 2020 , Pages 1920-1928 ; Qorbani, M ; Shown, I ; Ojaghi Dogahe, B ; Sharif University of Technology
Royal Society of Chemistry
2020
Abstract
A three-dimensional electrode based on porous Cu2O-Cu1.8S nanowires is prepared by means of a facile fabrication process. In this electrode, nanowires are decorated with Cu1.8S polyhedral nanostructures on the top, which are directly grown on a copper foam, thereby eliminating the need for a polymer binder. As an electrochemical electrode, it exhibits an extrinsic pseudocapacitive charge storage mechanism, which is different from that of battery-like Cu2O-CuO and Cu(OH)2 electrodes. The areal and volumetric capacitances of the Cu2O-Cu1.8S electrode can reach 2.6 F cm-2 and ∼200 F cm-3, respectively, at 2 mA cm-2, which are much higher than those obtained using copper(i, ii) oxide and...
Temperature-dependent multi-scale modeling of surface effects on nano-materials
, Article Mechanics of Materials ; Volume 46 , 2012 , Pages 94-112 ; 01676636 (ISSN) ; Ghahremani, P ; Sharif University of Technology
Abstract
In this paper, a novel temperature-dependent multi-scale method is developed to investigate the role of temperature on surface effects in the analysis of nano-scale materials. In order to evaluate the temperature effect in the micro-scale (atomic) level, the temperature related Cauchy-Born hypothesis is implemented by employing the Helmholtz free energy, as the energy density of equivalent continua relating to the inter-atomic potential. The multi-scale technique is applied in atomistic level (nano-scale) to exhibit the temperature related characteristics. The first Piola-Kirchhoff stress and tangential stiffness tensor are computed, as the first and second derivatives of the free energy...
Kinematic and Constitutive Modeling of Elastic and Thermoelastic Continua with Finite Deformation using Multiplicative Decomposition of Deformation Gradient
, Ph.D. Dissertation Sharif University of Technology ; Naghdabadi, Reza (Supervisor) ; Kargarnovin, Mohammad Hassan (Supervisor)
Abstract
In this thesis, a deformation measure is introduced which leads to a class of strain measures in the Lagrangian and Eulerian descriptions. In order to develop a constitutive equation, a second-order constitutive relation based on these strain measures is considered for modeling the mechanical behavior of solids at finite deformation. For this purpose and performance evaluation of the proposed strains, a Hookean-type constitutive equation is considered and the uniaxial loading as well as simple shear and pure shear tests are examined and the results are compared with the test data. Also, in order to characterize the mechanical behavior of elastic continua, constitutive equations through a...
Measurement of the pseudorapidity and centrality dependence of the transverse energy density in pb-pb collisions at √s NN=2.76TeV
, Article Physical Review Letters ; Volume 109, Issue 15 , 2012 ; 00319007 (ISSN) ; Khachatryan, V ; Sirunyan, A. M ; Tumasyan, A ; Adam, W ; Bergauer, T ; Dragicevic, M ; Erö, J ; Fabjan, C ; Friedl, M ; Frühwirth, R ; Ghete, V. M ; Hammer, J ; Hörmann, N ; Hrubec, J ; Jeitler, M ; Kiesenhofer, W ; Knünz, V ; Krammer, M ; Liko, D ; Mikulec, I ; Pernicka, M ; Rahbaran, B ; Rohringer, C ; Rohringer, H ; Schöfbeck, R ; Strauss, J ; Taurok, A ; Wagner, P ; Waltenberger, W ; Walzel, G ; Widl, E ; Wulz, C. E ; Mossolov, V ; Shumeiko, N ; Suarez Gonzalez, J ; Bansal, S ; Cornelis, T ; De Wolf, E. A ; Janssen, X ; Luyckx, S ; Maes, T ; Mucibello, L ; Ochesanu, S ; Roland, B ; Rougny, R ; Selvaggi, M ; Staykova, Z ; Van Haevermaet, H ; Van Mechelen, P ; Van Remortel, N ; Van Spilbeeck, A ; Blekman, F ; Blyweert, S ; D'hondt, J ; Gonzalez Suarez, R ; Kalogeropoulos, A ; Maes, M ; Olbrechts, A ; Van Doninck, W ; Van Mulders, P ; Van Onsem, G.P ; Villella, I ; Clerbaux, B ; De Lentdecker, G ; Dero, V ; Gay, A. P. R ; Hreus, T ; Léonard, A ; Marage, P.E ; Reis, T ; Thomas, L ; Vander Velde, C ; Vanlaer, P ; Wang, J ; Adler, V ; Beernaert, K ; Cimmino, A ; Costantini, S ; Garcia, G ; Grunewald, M ; Klein, B ; Lellouch, J ; Marinov, A ; Mccartin, J ; Ocampo Rios, A. A ; Ryckbosch, D ; Strobbe, N ; Thyssen, F ; Tytgat, M ; Verwilligen, P ; Walsh, S ; Yazgan, E ; Zaganidis, N ; Basegmez, S ; Bruno, G ; Castello, R ; Ceard, L ; Delaere, C ; Du Pree, T ; Favart, D ; Forthomme, L ; Giammanco, A ; Hollar, J ; Lemaitre, V ; Liao, J ; Militaru, O ; Nuttens, C ; Pagano, D ; Pin, A ; Piotrzkowski, K ; Schul, N ; Vizan Garcia, J. M ; Beliy, N ; Caebergs, T ; Daubie, E ; Hammad, G. H ; Alves, G.A ; Correa Martins Junior, M ; De Jesus Damiao, D ; Martins, T ; Pol, M.E ; Souza, M. H. G ; Aldá Júnior, W. L ; Carvalho, W ; Custódio, A ; Da Costa, E.M ; De Oliveira Martins, C ; Fonseca De Souza, S ; Matos Figueiredo, D ; Mundim, L ; Nogima, H ; Oguri, V ; Prado Da Silva, W. L ; Santoro, A ; Soares Jorge, L ; Sznajder, A ; Bernardes, C. A ; Dias, F. A ; Fernandez Perez Tomei, T. R ; Gregores, E. M ; Lagana, C ; Marinho, F ; Mercadante, P. G ; Novaes, S. F ; Padula, S. S ; Genchev, V ; Iaydjiev, P ; Piperov, S ; Rodozov, M ; Stoykova, S ; Sultanov, G ; Tcholakov, V ; Trayanov, R ; Vutova, M ; Dimitrov, A ; Hadjiiska, R ; Kozhuharov, V ; Litov, L ; Pavlov, B ; Petkov, P ; Bian, J. G ; Chen, G. M ; Chen, H. S ; Jiang, C. H ; Liang, D ; Liang, S ; Meng, X ; Tao, J ; Wang, J ; Wang, X ; Wang, Z ; Xiao, H ; Xu, M ; Zang, J ; Zhang, Z ; Asawatangtrakuldee, C ; Ban, Y ; Guo, S ; Guo, Y ; Li, W ; Liu, S ; Mao, Y ; Qian, S. J ; Teng, H ; Wang, S ; Zhu, B ; Zou, W ; Avila, C ; Gomez, J. P ; Gomez Moreno, B ; Osorio Oliveros, A. F ; Sanabria, J. C ; Godinovic, N ; Lelas, D ; Plestina, R ; Polic, D ; Puljak, I ; Antunovic, Z ; Kovac, M ; Brigljevic, V ; Duric, S ; Kadija, K ; Luetic, J ; Morovic, S ; Attikis, A ; Galanti, M ; Mavromanolakis, G ; Mousa, J ; Nicolaou, C ; Ptochos, F ; Razis, P. A ; Finger, M ; Finger, M ; Assran, Y ; Elgammal, S ; Ellithi Kamel, A ; Khalil, S ; Mahmoud, M. A ; Radi, A ; Kadastik, M ; Müntel, M ; Raidal, M ; Rebane, L ; Tiko, A ; Azzolini, V ; Eerola, P ; Fedi, G ; Voutilainen, M ; Härkönen, J ; Heikkinen, A ; Karimäki, V ; Kinnunen, R ; Kortelainen, M. J ; Lampén, T ; Lassila-Perini, K ; Lehti, S ; Lindén, T ; Luukka, P ; Mäenpää, T ; Peltola, T ; Tuominen, E ; Tuominiemi, J ; Tuovinen, E ; Ungaro, D ; Wendland, L ; Banzuzi, K ; Karjalainen, A ; Korpela, A ; Tuuva, T ; Besancon, M ; Choudhury, S ; Dejardin, M ; Denegri, D ; Fabbro, B ; Faure, J. L ; Ferri, F ; Ganjour, S ; Givernaud, A ; Gras, P ; Hamel De Monchenault, G ; Jarry, P ; Locci, E ; Malcles, J ; Millischer, L ; Nayak, A ; Rander, J ; Rosowsky, A ; Shreyber, I ; Titov, M ; Baffioni, S ; Beaudette, F ; Benhabib, L ; Bianchini, L ; Bluj, M ; Broutin, C ; Busson, P ; Charlot, C ; Daci, N ; Dahms, T ; Dobrzynski, L ; Granier De Cassagnac, R ; Haguenauer, M ; Miné, P ; Mironov, C ; Nguyen, M ; Ochando, C ; Paganini, P ; Sabes, D ; Salerno, R ; Sirois, Y ; Veelken, C ; Zabi, A ; Agram, J. L ; Andrea, J ; Bloch, D ; Bodin, D ; Brom, J. M ; Cardaci, M ; Chabert, E. C ; Collard, C ; Conte, E ; Drouhin, F ; Ferro, C ; Fontaine, J. C ; Gelé, D ; Goerlach, U ; Juillot, P ; Le Bihan, A. C ; Van Hove, P ; Fassi, F ; Mercier, D ; Beauceron, S ; Beaupere, N ; Bondu, O ; Boudoul, G ; Chasserat, J ; Chierici, R ; Contardo, D ; Depasse, P ; El Mamouni, H ; Fay, J ; Gascon, S ; Gouzevitch, M ; Ille, B ; Kurca, T ; Lethuillier, M ; Mirabito, L ; Perries, S ; Sordini, V ; Tosi, S ; Tschudi, Y ; Verdier, P ; Viret, S ; Tsamalaidze, Z ; Anagnostou, G ; Beranek, S ; Edelhoff, M ; Feld, L ; Heracleous, N ; Hindrichs, O ; Jussen, R ; Klein, K ; Merz, J ; Ostapchuk, A ; Perieanu, A ; Raupach, F ; Sammet, J ; Schael, S ; Sprenger, D ; Weber, H ; Wittmer, B ; Zhukov, V ; Ata, M ; Caudron, J ; Dietz-Laursonn, E ; Erdmann, M ; Güth, A ; Hebbeker, T ; Heidemann, C ; Hoepfner, K ; Klingebiel, D ; Kreuzer, P ; Lingemann, J ; Magass, C ; Merschmeyer, M ; Meyer, A ; Olschewski, M ; Papacz, P ; Pieta, H ; Reithler, H ; Schmitz, S. A ; Sonnenschein, L ; Steggemann, J ; Teyssier, D ; Weber, M ; Bontenackels, M ; Cherepanov, V ; Flügge, G ; Geenen, H ; Geisler, M ; Haj Ahmad, W ; Hoehle, F ; Kargoll, B ; Kress, T ; Kuessel, Y ; Nowack, A ; Perchalla, L ; Pooth, O ; Rennefeld, J ; Sauerland, P ; Stahl, A ; Aldaya Martin, M ; Behr, J ; Behrenhoff, W ; Behrens, U ; Bergholz, M ; Bethani, A ; Borras, K ; Burgmeier, A ; Sharif University of Technology
2012
Abstract
The transverse energy (E T) in Pb-Pb collisions at 2.76 TeV nucleon-nucleon center-of-mass energy (√s NN) has been measured over a broad range of pseudorapidity (η) and collision centrality by using the CMS detector at the LHC. The transverse energy density per unit pseudorapidity (dE T/dη) increases faster with collision energy than the charged particle multiplicity. This implies that the mean energy per particle is increasing with collision energy. At all pseudorapidities, the transverse energy per participating nucleon increases with the centrality of the collision. The ratio of transverse energy per unit pseudorapidity in peripheral to central collisions varies significantly as the...
A multi-scale modeling of surface effect via the modified boundary Cauchy-Born model
, Article Materials Science and Engineering C ; Volume 32, Issue 7 , 2012 , Pages 1993-2000 ; 09284931 (ISSN) ; Aramoon, A ; Sharif University of Technology
Elsevier
2012
Abstract
In this paper, a new multi-scale approach is presented based on the modified boundary Cauchy-Born (MBCB) technique to model the surface effects of nano-structures. The salient point of the MBCB model is the definition of radial quadrature used in the surface elements which is an indicator of material behavior. The characteristics of quadrature are derived by interpolating data from atoms laid in a circular support around the quadrature, in a least-square scene. The total-Lagrangian formulation is derived for the equivalent continua by employing the Cauchy-Born hypothesis for calculating the strain energy density function of the continua. The numerical results of the proposed method are...
Anti-plane stress intensity, energy release and energy density at crack tips in a functionally graded strip with linearly varying properties
, Article Theoretical and Applied Fracture Mechanics ; Volume 56, Issue 1 , 2011 , Pages 42-48 ; 01678442 (ISSN) ; Nasirai, C ; Torshizian, M. R ; Sharif University of Technology
Abstract
The fracture problem of a crack in a functionally graded strip with its properties varying in a linear form along the strip thickness under an anti-plane load is considered. The embedded anti-plane crack is located in the middle of strip half way through the thickness. The third mode stress intensity factor is derived using two different methods. In the first method, by employing Fourier integral transforms, the governing equation is converted to a singular integral equation, which is subsequently solved numerically by the collocation method based on Chebyshev polynomials. Then, the problem is solved by means of finite element method in which quadrilateral 8-node singular elements around...
Multi-scale modeling of surface effect via the boundary Cauchy-Born method
, Article International Journal for Numerical Methods in Engineering ; Volume 85, Issue 7 , August , 2011 , Pages 827-846 ; 00295981 (ISSN) ; Aghaei, A ; Khoei, A. R ; Sharif University of Technology
2011
Abstract
In this paper, a novel multi-scale approach is developed for modeling of the surface effect in crystalline nano-structures. The technique is based on the Cauchy-Born hypothesis in which the strain energy density of the equivalent continua is calculated by means of inter-atomic potentials. The notion of introducing the surface effect in the finite element method is based on the intrinsic function of quadratures, called as an indicator of material behavior. The information of quadratures is derived by interpolating the data from probable representative atoms in their proximity. The technique is implemented by the definition of reference boundary CB elements, which enable to capture not only...
Frequency-dependent energy harvesting via magnetic shape memory alloys
, Article Smart Materials and Structures ; Volume 24, Issue 11 , October , 2015 ; 09641726 (ISSN) ; Askari Farsangi, M. A ; Sharif University of Technology
Institute of Physics Publishing
2015
Abstract
This paper is focused on presenting an accurate framework to describe frequency-dependent energy harvesting via magnetic shape memory alloys (MSMAs). Modeling strategy incorporates the phenomenological constitutive model developed formerly together with the magnetic diffusion equation. A hyperbolic hardening function is employed to define reorientation-induced strain hardening in the material, and the diffusion equation is used to add dynamic effects to the model. The MSMA prismatic specimen is surrounded by a pickup coil, and the induced voltage during martensite-variant reorientation is investigated with the help of Faraday's law of magnetic field induction. It has been shown that, in...
An exact analysis for the hoop elasticity and pressure-induced twist of CNT-nanovessels and CNT-nanopipes
, Article Mechanics of Materials ; Volume 82 , 2015 , Pages 47-62A ; 01676636 (ISSN) ; Shodja, H. M ; Sharif University of Technology
Elsevier
2015
Abstract
Carbon nanotubes (CNTs) with and without end caps may be used for fluid storage and transport, respectively, referred to as CNT-nanovessel and CNT-nanopipe. The determination of the stiffness in the hoop (circumferential) and radial directions, ideal hoop strength, and hoop stress-strain curve of such nanostructures is of particular interest. Due to the proposed viewpoint, a chiral free-standing single-walled CNT (SWCNT) has a natural angle of twist and natural extension along the axis of the tube. For example, for the SWCNT (9,3) with diameter of 0.85 nm and chirality angle of 13.9°, the natural angle of twist per unit length is 1.45×10-3 rad/nm. Previously, only Vercosa et al. (2010) who...