Multiscale-Failure Criteria of Carbon Nanotube Systems Under Biaxial Tension–Torsion
Total Page：16
File Type：pdf, Size：1020Kb
IOP PUBLISHING NANOTECHNOLOGY Nanotechnology 18 (2007) 485715 (7pp) doi:10.1088/0957-4484/18/48/485715 Multiscale-failure criteria of carbon nanotube systems under biaxial tension–torsion Byeong-Woo Jeong1,2, Jang-Keun Lim3 and Susan B Sinnott1,4 1 Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611-6400, USA 2 School of Mechanical Engineering, Yeungjin College, Taegu 702-721, Korea 3 Department of Mechanical Design and Production Engineering, Hanyang University, Seoul 133-791, Korea E-mail: [email protected]fl.edu Received 18 July 2007, in final form 8 October 2007 Published 1 November 2007 Online at stacks.iop.org/Nano/18/485715 Abstract The failure criteria for carbon nanotube system fracture under biaxial tensile–torsional loads are developed based on a multiscale approach that adopts continuum mechanics models to describe atomistic predictions of failure from molecular dynamics simulations. The failure strength or envelope of carbon nanotube systems under this type of loading is significantly different from what occurs under uniaxial tensile loading and, importantly, is different from the predictions of failure criteria for macroscopic objects. The failure criteria developed here can be used to design carbon nanotube-based devices and materials, such as nanoelectromechanical systems and nanocomposites, which undergo biaxial tensile–torsional loading. (Some figures in this article are in colour only in the electronic version) 1. Introduction biaxial tensile and torsional loading, especially concerning the details of failure responses or failure criteria. The unique electrical and mechanical properties of carbon When stress is applied to a part in a uniaxial manner, nanotubes (CNTs) have attracted considerable interest and stress and strength can be compared directly to estimate sparked discussion of their potential use in applications such whether or not the part will fail. This comparison is relatively as nanoelectromechanical systems (NEMS) [1] and nanotube simple because there is only one value of stress and strength. composite materials [2]. In these and similar applications, However, the problem becomes more complex when the stress biaxial tensile and torsional loads on the CNTs are widely state is multiaxial. In such cases there are a multitude expected to occur. For instance, CNTs may be used as drive of stresses but only one significant value for strength, and shafts [3], torsion bar springs [4], and torsional actuators [5] this requires that failure be characterized using multiaxial that can experience torsion as well as tension. Thus, strength (or failure) criteria [15]. Under multiaxial loading understanding the mechanical responses of CNTs undergoing conditions, the details of failure at the micromechanical and this type of loading is important in optimizing their use in new nanomechanical levels are so incomplete that the failure materials and devices. While numerous studies have examined process cannot be followed analytically [15]. Thus, failure some aspect of the mechanical responses of CNTs, such criteria for macroscopic objects have evolved from attempts as their strength [6–9], buckling instability [10, 11], elastic to develop analytical or empirical macromechanical models to modulus [12, 13], and twist induced by tension [14] under describe experimental observations of failure under multiaxial uniaxial loading conditions, there is much that is still unknown loading [15]. Such failure criteria use the concept of ‘a failure about other aspects of the mechanical responses of CNTs in surface’ or ‘a failure envelope’ generated by plotting principal 4 Author to whom any correspondence should be addressed. stress components in principal material axes [15]. 0957-4484/07/485715+07$30.00 1 © 2007 IOP Publishing Ltd Printed in the UK Nanotechnology 18 (2007) 485715 B-W Jeong et al [18] Brenner D W, Shenderova O A, Harrison J A, Stuart S J, [23] Tsai D H 1979 J. Chem. Phys. 70 1375 Ni B and Sinnott S B 2002 J. Phys.: Condens. Matter Cheung K S and Yip S 1991 J. Appl. Phys. 70 5688 14 783 Andia P C, Costanzo F and Gray G L 2006 Modelling Simul. [19] Jeong B-W, Lim J-K and Sinnott S B 2007 Appl. Phys. Lett. Mater. Sci. Eng. 14 741 90 023102 Zimmerman J A, Webb E B III, Hoyt J J, Jones R E, Klein P A [20] Jeong B-W, Lim J-K and Sinnott S B 2007 J. Appl. Phys. and Bammann D J 2004 Modelling Simul. Mater. Sci. Eng. 101 084309 12 S319 [21] MielkeSL,TroyaD,ZhangS,LiJ-L,XiaoS,CarR, Zhou M 2003 Proc. R. Soc. A 459 2347 Ruoff R S, Schatz G C and Belytschko T 2004 Chem. Phys. [24] Shenderova O, Brenner D and Ruoff R S 2003 Nano Lett. Lett. 390 413 3 805 [22] Huhtala M, Krasheninnikov A V, Aittoniemi J, Stuart S J, Gall K, Diao J K and Dunn M L 2004 Nano Lett. 4 2431 Nordlund K and Kaski K 2004 Phys. Rev. B 70 045404 Park H S and Zimmerman J A 2005 Phys. Rev. B 72 054106 7 PHYSICAL REVIEW B 76, 094114 ͑2007͒ Atomistic and multiscale analyses of brittle fracture in crystal lattices Sulin Zhang,1,2,* Ting Zhu,3 and Ted Belytschko4 1Department of Mechanical Engineering, University of Arkansas, 204 Mechanical Building, Fayetteville, Arkansas 72701, USA 2Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA 3Woodruff School of Mechanical Engineering, Georgia Institute and Technology, Atlanta, Georgia 30332, USA 4Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60201, USA ͑Received 28 April 2007; revised manuscript received 17 July 2007; published 27 September 2007͒ Applicability of the Griffith criterion ͓A. A. Griffith, Philos. Trans. R. Soc. London, Ser. 221, 163 ͑1920͒; S. Zhang, S. L. Meilke, R. Khare, D. Troya, R. S. Ruoff, G. C. Schatz, and T. Belytschko, Phys. Rev. B 71, 115403 ͑2005͔͒ for predicting the onset of crack extension in crystal lattices is systematically evaluated using atomistic and multiscale simulations with a focus on the effects of crack size and lattice discreteness. An atomistic scheme is developed to determine the true Griffith load defined by the thermodynamic energy balance of crack extension for both finite-sized and semi-infinite crack models. For a model monolayer lattice, we identify a characteristic crack length ͑about ten lattice spacings͒ below which the Griffith fracture stress markedly overestimates the true Griffith load. Through a stability analysis of crack-tip bond separation, the athermal ͑nonthermally activated͒ loads of instantaneous fracture are determined, thereby yielding the esti- mated lattice trapping range. Our simulations show that the strength of lattice trapping depends on the inter- action range of the interatomic force fields. Using the reaction pathway exploration method, we determine the minimum energy paths of bond breaking and healing at a crack tip, giving a more precise estimate of the lattice trapping range. The activation energy barriers governing the rate of kinetic crack extension are extracted from the minimum energy paths. Implications concerning the distinction between the athermal and Griffith fracture loads are discussed. Based on these results, a general criterion is established to predict the onset of crack growth in crystal lattices. In addition to taking into account the lattice trapping effect, this criterion is appli- cable to a large spectrum of crack sizes. DOI: 10.1103/PhysRevB.76.094114 PACS number͑s͒: 61.72.Ϫy, 61.50.Ah, 62.20.Ϫx, 68.65.Ϫk I. INTRODUCTION length 2a embedded in an infinitely large, linear elastic me- dium, subject to remotely applied uniform tension, the Predicting the failure strength of nanostructured materials energy-balance criterion yields a critical stress1 often involves quantum mechanical calculations or atomistic models with empirical force fields. While these numerical Ј = ͱ2Y␥ /a ͑2͒ methods have been useful to elucidate the failure mecha- G s nisms at the atomic level, their prohibitive computational for plane-stress condition, where Y denotes Young’s modu- cost becomes a major concern for specimens of realistic size. lus. To distinguish the critical stress given by Eq. ͑2͒ from In contrast, the fracture criteria1,2 established within the the true Griffith load given by the energy-balance criterion of framework of continuum fracture mechanics have been Eq. ͑1͒, the critical stress determined by Eq. ͑2͒ is hereafter widely used to predict the critical conditions for the onset of referred to as the Griffith fracture stress. crack extension in continua. If such continuum-based frac- The Griffith fracture stress has been widely used to pre- ture criteria were applicable to nanostructured materials, the dict the onset of crack extension in continua, yet it suffers aforementioned computational burden could often be deficiencies when applied to specimens with nanosized avoided. Thus, it is both fundamentally and practically criti- cracks. For extremely short cracks ͑a→0͒, the Griffith frac- cal to evaluate the applicability of these fracture criteria to ture stress may exceed the theoretical strength of the crystal lattices with the consideration of flaw size and lattice th perfect lattice, which, of course, is nonphysical. In address- discreteness. ing this issue, Gao et al.4 suggested that a characteristic The fundamental fracture criterion for brittle continua is crack length can be identified as a* ϰY␥ /2 , below which the energy-balance criterion by Griffith,1,3 which holds when s th the Griffith fracture stress overestimates the true fracture there is no generation or motion of dislocations or other dis- stress and the material becomes flaw insensitive. Pugno and sipation mechanisms, such as void nucleation. The Griffith Ruoff5 developed a quantized fracture mechanics ͑QFM͒ criterion states that a crack meets the critical growth condi- theory, where the classical stress intensity factor is redefined tion when the net change in the total energy of the system by considering an infinitesimal crack extension at the crack ⌬E vanishes upon crack extension by an infinitesimal dis- tip.