ABSTRACT. JIN, YUN Atomistic Simulations of Fracture of 2D Graphene Systems and the Elastic

Transcription

1 ABSTRACT JIN, YUN Atomstc Smulatons of Factue of D Gaphene Systems and the Elastc Popetes of Cabon Nanotubes. (Unde the decton of D. Fuh-Gwo Yuan Ths dssetaton has two man pats. The fst pat s the atomstc smulatons of factue of d gaphene systems. In pncple the macoscopc mechancal popetes of mateals ae detemned by the atoms and the basc laws of physcs. Fo the factue of mateals, atomstc smulatons can povde basc undestandng of the ogns of factue due to the bond uptue. In ths dssetaton the factue mechansm of a nanostuctue mateal has been nvestgated by atomstc smulatons. Macoscopc factue paametes have been examned fom both atomstc smulaton and contnuum models. Methods of calculatng enegy elease ates n atomc systems have been successfully developed. The atomc descptons of the stess feld n font of cack tps wee also obtaned. These smulatons showed that, n macoscopc factue mechancs unde small defomaton, lnea elastc factue mechancs s suffcent fo the descpton of cackng behavo fo ths covalently bonded mateal. The esults mege the dscete (atomstc and contnuum (macoscopc descpton of factue. Meanwhle, a method to calculate J-ntegal n the atomc system has been developed. The numecal esults of J-ntegal ageed vey well wth the enegy elease ate n the lnea elastc condton. Afte a necessay modfcaton on the Tesoff-Benne potental, the ctcal values of J-ntegal, denoted by J c, have been obtaned as the measue of the factue toughness of gaphene sheets.

2 In the second pat of ths dssetaton the mechancal popetes of sngle-walled cabon nanotube (SWNT have been evaluated. The values of the n-plane Young s modulus, otatonal shea modulus, and n-plane Posson s ato ae n the ange of exstng theoetcal and expemental esults. Seveal elastc modul of SWNTs have been obtaned usng molecula dynamc smulatons. It has been shown fom the smulatons that the elastc constants of SWNTs ae nsenstve to the mophology patten such as nanotube adus and thus the effect of cuvatue on the elastc constants can be neglected. Assumpton of the tansvesely sotopc popetes on the cylnde suface of the sngle-walled nanotube has been confmed by numecal calculatons. Besdes the conventonal enegy appoach, a new method, whch denoted as foce appoach n the dssetaton, has been developed to analyze the elastc popetes of cabon nanotubes. The esults fom two appoaches matched vey well. The advantage of the foce appoach s that t can povde moe accuate pedcton than the enegy appoach. Futhemoe, the foce appoach can pedct the nonlnea behavo wthout assumpton of assumed total potental enegy n quadatc fom descbed fo smallstan defomaton n the enegy appoach.

3

4 To my paents

5 Bogaphy Yun Jn was bon on Septembe 8, 975 n Wuwe, Anhu Povnce, People s Republc of Chna. He gaduated fom Wuwe Hgh School, Chna and eceved a Bachelo of Scence degee n Theoetcal and Appled Mechancs fom Unvesty of Scence and Technology of Chna (USTC n July 997. He completed hs Maste of Engneeng poject n Sold Mechancs, Expemental and numecal studes on magneto-heologcal fluds, unde the decton of Pofesso Peqang Zhang n USTC n July. Immedately theeafte, he enteed the Ph.D. pogam of Noth Caolna State Unvesty (NCSU n Mechancal Engneeng unde the decton of D. Fuh-Gwo Yuan. Yun Jn s eseach at NCSU manly focused on the atomstc smulatons of elastc popetes of cabon nanotubes and the factues of nanostuctue mateals.

6 Acknowledgements Fst of all, I would lke to thank my advso, D. F. G. Yuan, fo the gudance he has povded me ove the past fou yeas. My eseach has been benefted geatly fom hs knowledge, expeence, as well as the pusung excellence atttude he has always pessted. Thanks also go to D. S. Yang of the Mas Msson Reseach Cente at NCSU fo hs knowledgeable advce and helpful dscussons on ths dssetaton. I also would lke to thank M. Ek Saethe of the NASA Langley Reseach Cente, Nofolk, VA, fo hs nvaluable advce though the past yeas. Thanks also go to my othe commttee membes, Ds. J. W. Eschen, E. C. Klang, K. Petes, fo the oles they have played n my educaton, both on my commttee and n the classoom. Ths eseach s suppoted by the NASA Langley Reseach Cente. The computng suppot fom Noth Caolna Supecomputng Cente s geatly appecated. I would lke also to ecognze some of my fellow gaduate students fo all sots of help and fendshps. These nclude X. Ln, C. Ln, B. Shpman, J. L, L. Wang, S. Nojavan, L. Lu, and H. Hab. Fnally, but not least, I would lke to thank my famly fo the neve-endng suppot, love and encouagement. v

7 TABLE OF CONTENTS LIST OF TABLES... v LIST OF FIGURES... v NOMENCLATURE AND ABBREVIATIONS...x INTRODUCTION.... LITERATURE REVIEW...3. PLAN OF INVESTIGATION...9 METHODOLOGY OF ATOMISTIC SIMULATIONS.... BASIC CHEMISTRY OF CARBON BOND.... MOLECULAR MECHANICS AND DYNAMICS SIMULATION Intoducton of Molecula Dynamcs Smulaton Integaton Algothm Inteatomc Potental Functon STRESS IN ATOMIC SYSTEM ELASTIC PROPERTIES OF SINGLE-WALLED CARBON NANOTUBES INTRODUCTION STRUCTURE OF CARBON NANOTUBES ELASTIC MODULI Stess-Stan Relatonshp of Cabon Nanotube Calculatons of Elastc Modul Detals of the Numecal Pocedue fo Calculatng Modul RESULTS AND DISCUSSIONS ENERGY RELEASE RATE AND STRESS FIELD IN ATOMIC SYSTEM ATOMIC CONFIGURATIONS Case I: Cack n Sem-nfnte Doman Case II: Cack n the gaphene sheet wth fnte wdth Smulaton Technque...54 v

8 4. ATOMIC STRESS ENERGY RELEASE RATE IN ATOMIC SYSTEM Global Enegy Method Local Foce Method RESULTS AND DISCUSSIONS Numecal Results of Case I Numecal Results of Case II J-INTEGRAL IN ATOMIC SYSTEM J-INTEGRAL DISPLACEMENT GRADIENT J-INTEGRAL IN LINEARITY Case I: Cack n Sem-nfnte Doman Case II: Cack n the gaphene sheet wth fnte wdth J-INTEGRAL IN NONLINEARITY Nonlnea Stess-Stan Relatonshp J c as a measue of factue toughness SUMMARY AND FUTURE WORK SUMMARY FUTURE WORK...88 REFERENCES...89 APPENDIX...95 v

9 LIST OF TABLES Table. Constants n Tesoff-Benne potental..... Table 3. Geomety popetes of sngle-walled nanotubes fom n = 6 to...4 Table 3. The aveage values of E z and Table 3.3 Table 3.4 Table 4. Table 4. E θ of selected cabon nanotubes...4 Compason between the numecal esults and the theoetcal esults deved fom Gθ z = Ez /[ ( + vθz ]...4 Elastc modul of sngle-walled cabon nanotubes...47 Compason of the enegy elease ate obtaned by atomstc smulaton and contnuum theoetcal soluton The numecal esults of G /σ n mode I...67 Table 4.3 Table 4.4 The numecal esults of The numecal esults of G /τ n mode II...67 G /τ n smple shea mode...68 Table 5. Compason of J-ntegal wth enegy elease ate...77 Table 5. The numecal esults of J-ntegal and G n mode I...84 Table 5.3 The numecal esults of J-ntegal and G n mode II...84 Table 5.4 The numecal esults of J-ntegal and G n smple shea mode...85 v

10 LIST OF FIGURES Fgue. Tansmsson electon mcogaph of cacks n slcon (Afte Lawn et al., Fgue. Pseudo-atomc factue model wth cack tp...7 Fgue.3 Quas one-dmensonal chan model...7 Fgue. The llustaton of s and p atomc obtal...7 Fgue. Schematc model of sp 3, sp and sp bondng hybdzaton...8 Fgue.3 Molecula mechansm of a basc hexagonal bondng stuctue...8 Fgue.4 Schematc dagam of atomc stuctue of gaphte...3 Fgue.5 The nteatomc foce and potental enegy vesus bond length obtaned by Tesoff-Benne potental...3 Fgue 3. The gaphte plane of nanotube suface coodnates...38 Fgue 3. Examples of am cha, zgzag, and chal nanotubes...38 Fgue 3.3 E z and E θ dstbuton vesus adus of amcha sngle-walled cabon nanotubes n enegy and foce appoaches...44 Fgue 3.4 Posson s atos v θ z and v zθ vesus adus of amcha sngle-walled cabon nanotubes...45 Fgue 3.5 Shea modulus G θ z vesus adus of amcha sngle-walled cabon nanotubes.45 Fgue 3.6 C 33 vesus adus of amcha sngle-walled cabon nanotubes...46 Fgue 3.7 C 3 vesus adus of amcha sngle-walled cabon nanotubes...46 Fgue 4. Schematc stuctue nea a cack tp n a sem-nfnte gaphene sheet...5 Fgue 4. Two basc factue modes...5 Fgue 4.3 Schematc dagam of a cente-cacked gaphene sheet...54 Fgue 4.4 Smple shea and pue shea...55 Fgue 4.5 Illustaton of the method of calculatng the enegy elease ate...57 Fgue 4.6 A geneal llustaton of vtual wok unde Mode I...58 Fgue 4.7 A geneal llustaton of vtual wok unde Mode II...59 Fgue 4.8 Stess dstbuton of σ n font of a cack unde emote K I feld...6 v

11 Fgue 4.9 Stess dstbuton of τ n font of a cack unde emote K II feld...6 Fgue 4. Close-up vew of undefomed and defomed confguatons nea the cack tp n mode I...6 Fgue 4. Close-up vew of undefomed and defomed confguatons nea the cack tp n mode II...63 Fgue 4. Close-up vew of undefomed and defomed confguatons nea the cack tp n smple shea mode...63 Fgue 4.3 Fgue 4.4 Fgue 4.5 G /σ vesus cack length n mode I...65 G /τ vesus cack length n mode II...65 G /τ vesus cack length n smple shea mode...66 Fgue 4.6 Cack suface pofle unde Mode I fom atomstc smulaton and LEFM 66 Fgue 5. Typcal contou fo the path-ndependent J-ntegal...7 Fgue 5. Contou fo J-ntegal n atomc system...7 Fgue 5.3 Illuson of calculaton of u / x j n contnuum and dscete system...74 Fgue 5.4 One-dmensonal dscete spng model...75 Fgue 5.5 J-ntegal wth dffeent and n mode I...76 Fgue 5.6 J-ntegal wth dffeent and n mode II...76 Fgue 5.7 Fgue 5.8 Fgue 5.9 J-ntegal vesus cack length n mode I...78 J-ntegal vesus cack length n mode II...79 J-ntegal vesus cack length n smple shea mode...79 Fgue 5. Elastc-plastc stess-stan cuve of gaphene sheet...8 Fgue 5. The ctcal values of J ntegal n dffeent cack length...8 x

12 NOMENCLATURE & ABBREVIATIONS Bef defntons of the most fequently used symbols and abbevatons ae lsted as followng. Some symbols necessaly have dffeent defntons n dffeent sectons. In such condtons, defntons ae povded locally and ae used consstently wthn the secton. Geek symbols α β a t ε ε Atom dentty Atom dentty Equlbum nteatomc dstance, m tme step, s vecto of delectc pemttvty ε, F/n. Components of stan tenso κ Paamete fo plane stess o plane stan µ Elastc shea modulus, pa ν σ Posson s ato Components of stess tenso, pa δ constant n Tesoff-Benne potental Roman symbols a a c C kl Half of the cack length, m Constant n Tesoff-Benne potental Constant n Tesoff-Benne potental Components of stffness matx x

13 D e d E F fs Constant n Tesoff-Benne potental, ev Constant n Tesoff-Benne potental Young s modulus, pa Inteatomc foce Femto-second, 5 s G, G I, G II Enegy elease ate, pa m K I, K II k B L p p x, ps e p y, p z Stess ntensty factos Boltzmann constant Length, m p atomc obtal Thee dffeent oentatons n p atomc obtal Pco-second, Radus, m s Constant n Tesoff-Benne potental, m, θ, z Cylndcal coodnates S s t U Constant n Tesoff-Benne potental s atomc obtal wall thckness of nanotube and gaphene sheet, m Potental enegy, J V Volume of a doman, m 3 x, x Catesan coodnates of a plane u, u Dsplacements n the dectons of x, x, espectvely, m x

14 Abbevatons AFM BCC CNT DFT EDI FCC FEA GB LEFM MD MM MWNT PVW SWNT TEM Atomc Foce Mcoscopy Body-Centeed-Cubc Cabon Nanotube Densty-Functonal Theoy Equvalent Doman Integal Faced-Centeed-Cubc Fnte Element Analyss Gound Bounday Lnea Elastc Factue Mechancs Molecula Dynamcs Molecula Mechancs Mult-Walled Nanotube Pncple of Vtual Wok Sngle-Walled Nanotube Tansmsson Electon Mcoscopy x

15 INTRODUCTION Factue mechancs has become an engneeng dscplne that has been studed by both atomstc and contnuum appoaches. In 9, Gffth appoached the factue theoy of bttle mateals fom the fundamental enegy balance, by ntoducng a specfc suface enegy along the cack suface. All the atomc o ntenal stuctual factos ae hdden n the suface enegy. An mpotant pecuso to the Gffth theoy was the elastc felds of an ellptcal cavty based on contnuum theoy. Although Gffth fomulated a factue cteon n tems of macoscopc themodynamc quanttes, a complete descpton equed an evaluaton of events at the molecula level. The maxmum stess at the tp of an equlbum cack must coespond to the theoetcal cohesve stength of the mateal; that s, the lagest possble stess level that the molecula stuctue can sustan by vtue of ts ntnsc bond stength. The global enegy appoach could not be extended easly to many dffeent geometes and loadng condtons encounteed n a vaety of expements and pactcal stuctues. In 957, Iwn used a local foce appoach showed, fo the fst tme, the local stesses aound the cack tp n lnea elastc mateals. Iwn suggested that the sngula stesses could be chaactezed n tems of a sngula paamete, stess ntensty facto. Ths facto combnes the effect of geomety and load. As a consequence, the factue chaactezaton poblem was educed to the calculaton of sngle quantty of unvesal sgnfcance. Futhemoe, Iwn povded a quanttatve elaton between that stan enegy elease ate, a global paamete, and the stess ntensty facto, a local cack-tp paamete. Snce the stess ntensty facto chaactezes the falue at the cack tp as long

16 as pocess and nelastc zones ae small compaed wth the macoscopc dmensons of the solds, so called small scale yeldng, the Iwn theoy can apply to non-bttle mateals. Snce then, the contnuum based factue mechancs has been of geat nteest n applyng to ductle mateals usng othe paametes such as J-ntegal and othe advanced mateals ncludng composte mateals. Vaous factue ctea have been establshed fom a macoscopc pont of vew. A damage toleance phlosophy based on factue mechancs has been developed to povde stuctual ntegty assessments n aeospace ndusty. These contnuum appoaches have successfully descbed many factue mechansms n sold mateals and contnue to be of geat use. Nevetheless, they stll have some shotcomngs, such as the spuous sngulaty of the stess feld at cack tp. In pncple, all the macoscopc mechancal esponses of a mateal ae govened by consttutve atoms and the basc laws of physcs, n patcula factue assocated wth successve falue at atomc level. Hence, atomstc smulaton may open new avenues n studes of mcoscopc ogns of mateal falue behavo. Dect obsevatons fom tansmsson electon mcoscopy (TEM have aleady poven the exstence of shap atomc cack, as can be seen n Fgue. (Lawn et al., 98. The dmensons n the fgues ndcated that the phenomena we ae nteested n ae n the level of nm, especally the scales at cack tp s no moe than nm. Obvously, the undelyng atomc stuctues of the mateal ae defntely not able to be neglected fo the nvestgatons on such knd of small scale. Theefoe the atomstc pedctons would be necessay and enable us to gan moe nsghts on the mateal behavo at the most fundamental level. A mateal studed n the pedcton can be always pefectly chaactezed n tems of ntenal stuctue, composton, defects, etc., makng the esults often less ambguous than n expements. Contnuum mechancs appoach, on the othe hand, s often based on

17 phenomenologcal consttutve models whch can be an dealzaton of obseved defomaton popetes that may not be epesentatve of actual physcal mechansm. In summay, to povde a basc undestandng of the ogns of factue due to the uptue of the nteatomc bonds, nvestgatons on the atomc o molecula levels ae necessay and ctcal. Fg. Tansmsson electon mcogaph of cacks n slcon (Afte Lawn et al., 98. Lteatue evew The fst attempt to explctly analyze the nteatomc foces whch esst cack extenson s by Ellott (947. In hs appoach the elatonshp between nomal stess and the dsplacement along the cack plane s deduced based on the lnea elastc soluton of a cacked body unde tenson. By equatng the aea unde the stess-dsplacement cuve to the suface enegy and the maxmum stess to the ctcal uptue stess fo the mateal, Ellott fomulated a model wth two sem-nfnte solds that attact each othe by the atomstc foce-sepaaton cuve. A vey close elaton wth Gffth theoy has been found. Latte, Cbb and Tomkns (967 poposed a moe dect appoach to evaluate the 3

18 dstbuton of nteatomc foces acoss cack suface. They also deved the coespondng nteatomc foce-dstance cuve by the contnuum theoy of lnea elastcty. The above two woks, though attemptng to usng dscete models to analyze the nteactons and dsplacements aound cack tp, wee ndeed stll completely based on lnea elastc contnuum theoy. One of the fst achevements to develop dect numecal computatons of atomc aspect of factue may be the so called Goode-Kannnen model (966. Goode and Kannnen nvestgated an elastc sold contanng extenal plane cacks. The atomc model epesentng the cack tp stuctue s shown n Fgue.. They assumed that the nteatomc foces bdgng the cack suface wee based on nonlnea foce-sepaaton law, whle all othe nteatomc nteactons ae consdeed to be lnealy elastc. In the model, fou dffeent functons wee used to epesent the pncpal featues of the foce-sepaaton law, whch ae lnea, snusodal, exponental, and nvese powe functons. The Young s modulus was obtaned by calculatng the ntal slope of the foce-sepaaton cuve, and the suface enegy was taken as the aea unde the cuve. Fo those atoms on the cack suface, only dsplacements n the vetcal decton wee consdeed. Plus consdeng the dffeent teatment of the atoms on the cack suface and othe places, these made the whole model actually epesentng two lnealy elastc sem-nfnte bodes connected by a sees of non-lnea spngs. Subsequently, Kannnen and co-wokes (Gehlen and Kannnen, 97; Kannnen and Gehlen, 97; Gehlen, Hahn, and Kannnen, 973 extended the pevous model to teat the atoms nea the cack tp as atomc cystal stuctues, whch leaded to thee degees of feedom fo each atom. Afte detaled studed the atomc aangements of body-centeedcubc (BCC on, an mpotant factue paamete, the ctcal appled stess at whch cack popagates, s estmated. 4

19 Fg.. Pseudo-atomc factue model wth cack tp. Sold black ccles ae the atoms on cack plane; lght black ones ae the bounday atoms. Kannnen and co-wokes appoaches leaded us to a deepe undestandng of the cack pocess, and moe mpotantly they contbuted two basc deas of atomstc modelng: the constucton of an atomc model to eflect the key physcal featues of the system; pope selecton of the nteatomc foce-sepaaton laws to chaacteze fundamentally chemcal bondng. Due to the lack of knowledge of nteatomc potental functons and the lmtaton of compute speed at that tme, the shotcomngs of the models ae also obvous. Fst of all too few numbes of atoms (no moe than 5 has been handled n the numecal computaton whch let the atomc dsplacements actually stll beng domnated by lnea contnuum elastcty. The epesentatons of the atomc nteactons wee ovesmplfed and as the autho ecognzed, the nteactons between the atomc egon and the contnuum egon wee gnoed. Hee we want to menton a long-tme msleadng method n many appoaches n whch the Young s modulus was always evaluated fom the ntal slope of local atomc foce-sepaaton laws. Whle The Young s modulus of the mateal s not necessaly equal to the ntal slope snce the global stan of the whole stuctue can not be taken fo ganted as equvalent to the local stan of the atom pas. 5

20 Almost at the same tme Chang (97 pesented a smla wok on smulatng the atomc confguatons and cohesve foces felds aound cack tp n BCC and FCC (facecenteed-cubc metal. Atom-elaxaton methods, whch wee fst appled to compute theoetcal maxmum stengths of peface cystals n bttle solds by Tyson (966 and subsequently developed by Chang (967, wee employed to ntegate the classcal moton equatons. The basc dea of atom-elaxaton methods s to evaluate the total knetc enegy at each step of the ntegaton of the moton equatons. Wheneve the total knetc enegy got to the maxmum, all the veloctes ae set to zeo. Theefoe the potental enegy wll dop monotoncally towads the mnmum. Cacks wee fomed by emovng a sngle plane of atoms, and extenal tensle stans wee appled nomal to the cack. The esults showed that the cack tp foce felds manly depend on the oentaton of the cack wth espect to the cystallte. Howeve, the concept of the sheang component of the foces mentoned n the pape s vague, and the method of computng the sheang foce was also aguable. Sncla and Lawn (97 a, b, c consdeed atomcally shap cacks n damond, slcon and gemanum. Both contnuum appoach and atomstc appoach wee appled. The esults ndcated that contnuum theoy s capable of gvng emakably accuate pedctons of the cack-tp dsplacement feld, except vey nea to the cack tp. The atomstc appoach was actually couplng the atomc and contnuum egon togethe. It dffeed pmaly fom the pevous wok such as Kannnen s s that, nstead of applyng the gd bounday condton, a flexble bounday condton whch let the oute contnuum egon be able to adjust as the esponse of the moton of the nne atomc zone. The late woks of Sncla (975, 978 extended the calculatons and nvestgated the effect of the knks at the cack font. 6

21 Anothe appoach to model the cack n dscete system s the lattce theoy. Thomson and co-woke (Thomson, Hseh, and Rana, 97; Hseh and Thomson, 973; Fulle and Thomson, 977 developed a dscete quas-one-dmensonal chan model of a cack n lattce statc, as shown n Fgue.3. The model conssts of two sem-nfnte chans of atoms that ae bonded wth two types of nteactons whch wee modeled as spng elements: nonlnea n the cack tp bond and lnea n othe bonds. All dsplacements ae assumed to be vetcal. Latte a two-dmensonal lattce model was poposed. They studed the factue n atomc lattces, especally the fundamental ole of the suface enegy n bttle factue. Ashust et al. (976 descbed the cack popagaton n the two-dmensonal tangula lattce. u u b a Fg..3 Quas one-dmensonal chan model. Wth the fast pogess on hgh-speed computes, eseaches can handle even mult-mllon atoms computatons n molecula dynamcs (MD smulaton usng supecompute and paallel algothms n the last decade. Most of the woks wee manly focused on descbng the phenomena of the dynamc cack popagaton. Holan and Ravelo (995 used MD smulaton to smulaton the cack popagaton unde mode-i tensle defomaton at hgh stan ates. They also dscussed the 7

22 dstncton between bttle and ductle behavo fo dynamcal cacks n the pape. A smla appoach pusued by Zhang et al., (995 to study the bttle and ductle factue n metal cystals led to the esults that geomety of cack and cystal oentaton has stong effect on the cack tp pocesses. Also, geneally t has been assumed that a bttle falue mples that the cack popagates wthout the emsson of dslocatons whle the mateals whee dslocatons ae emtted ae geneally beleved to be ductle mateals. Ths vew mples a descpton of bttle-ductle behavo n tems of the pocesses occung at the cack tp. Whle an atomstc smulaton shows contovesy phenomenon that the emtted dslocatons can actually lead to bttle falue unde cetan condton (Fakas, 998. Thee ae also many othe atcles focused on the smulaton of pofles of cack tp pocess n dffeent atomc models (Ludwg and Gumbsch, 998; Tebn et al., 999; Gall et al., ; Gao et al., ; Rountee et al., ; and etc. Meanwhle, Made and Lu (993 nvestgated the nstabltes n cack popagaton and the cack-font speed n a -dmensonal system conssts of coupled spngs whch would snap beyond a ctcal dsplacement. The esults showed that the local cack banches appea when the cack-font speed exceeds.66 V R whee V R s the Raylegh wave speed n the mateal. Consequently, Omeltchenko et al. (997 smulated the cack popagaton unde tensle mode n a gaphte sheet contanng a mllon cabon atoms by MD smulaton based on Benne s eactve empcal bond-ode potental. The factue pofles and the speeds of the cack popagaton have been obtaned. A latte study povded a mult-scale modelng of bttle-cack popagaton and appled t to the factue gowth n a -dmensonal Ag plate n macoscopc dmensons (Raf-Taba et al., 998. The ctcal cack velocty has been pedcted. 8

23 In ths thess ou nvestgaton of atomc factue s manly pefomed n a - dmensonal gaphene system. Theefoe, ecent woks closely elated to the atomc factue of cabon allotopes ae emphaszed as followng. Shendeova et al. ( poposed an atomstc model of the factue of polycystallne damond. Factue stesses fo two oentated Gan Boundaes (GB s wee evaluated. Also, the cleavage eneges fo GB s wee computed though both densty-functonal theoy (DFT and molecula dynamc smulatons (MD. The esults showed that the theoetcal factue stesses of ndvdual GB s elated manly by GB type, such as oentaton, athe than by the cleavage eneges. Belytschko et al. ( studed the factue of cabon nanotubes by molecula mechancs smulatons. The factue behavo s found to be almost ndependent of the sepaaton enegy and to depend pmaly on the nflecton pont n the nteatomc potental. The numecal esults shows that the factue stan of a zgzag nanotube s %-5% and the ange of factue stesses s GPa. Howeve, the nteatomc potental functon used n the pape s a combnaton of an empcal potental functon, the modfed Mose potental, and a smple unealstc bendng functon. As thee s no moe fundamental poof of the valdty of ths combnaton, t would be questonable how much key physcal featues the potental functon eveals wthout futhe examnaton by physcst and chemst.. Plan of Investgaton Eve-nceasng computatonal powe, advancement n the descpton of atomc nteactons n mateals, and a stong dese n developng lghtweght nano-stuctued mateals, such as nanotubes and nanowes, have evealed the emegence of nteest n pedctng the popetes of mateals n the atomc level befoe they ae syntheszed. Stan enegy elease ate, stess ntensty facto, and J-ntegal ae the epesentatve 9

24 mechancal paametes whch always gve the cteon of cack popagaton n the contnuum mechancs. On the othe hand, when descbng the essental phenomena of factue n atomstc smulaton, chaactestcs ae needed to know to evaluate and pedct quanttatvely the mechancal popetes of the damaged mechancal feld. Howeve, thee ae few studes focused on such poblems n the pevous efeences. Nakatan et al. ( poposed a method to estmate the J-ntegal n amophous metal model unde mode I defomaton n molecula dynamcs smulaton. But dung the computaton of each tem n the J-ntegal, the dsplacement gadents calculated n the pape wee actually dependent on the selecton of a weghted functon, whch s a nomal dstbuton functon n the calculaton. That led to dffeent numecal esults of the J- ntegal when choosng dffeent statstcal dstbutons. Based on the above evews and dscussons, to systematcally nvestgate the macoscopc paametes n the atomc factue and develop elable appoach of the calculaton of those paametes n atomc system become ou pmay nteests. In ths dssetaton macoscopc factue paametes ae examned fom both atomc-scale and macoscopc contnuum model. The followng s a smple stuctue of the dssetaton. Chapte s the methodology of atomstc smulaton. A bef ntoducton of basc chemsty of cabon covalent bond s pesented n the begnnng. Then the atomstc smulaton we used n ou nvestgaton, MD smulaton, s ntoduced, togethe wth the descpton of a many-body empcal nteatomc potental, the socalled Tesoff-Benne potental functon. The last pat n chapte s the method of calculatng the atomc stess n the pevous woks. Smulatons of elastc popetes of sngle-walled cabon nanotubes ae povded n chapte 3. The Young s modulus, Posson s ato and shea modulus s obtaned unde dffeent defomaton modes. Both foce and enegy appoaches that lnk the behavo at the atomc and macoscopc scales

25 of the nanotubes ae used to pedct the selected effectve elastc modul. A compason of the elastc constants obtaned fom MD smulatons wth avalable expemental data s made. Chapte 4 ntoduces the method to calculate the enegy elease ates unde Mode I and II small defomaton. Two methods, global enegy appoach and local foce appoach ae both adopted to compute the enegy elease ates. The enegy elease ates and defomed cack suface pofles ae also calculated fom lnea elastc factue mechancs (LEFM though homogenzed mateal popetes fo compason. Also atomc stess felds nea the cack tp n a sem-nfnte atomc egon ae evaluated by applyng emote Mode I and Mode II defomatons. Chapte 5 pesents the appoach of the calculaton of J-ntegal. The ndependency of the J-ntegal n atomc system s examned fstly. Then the esults ae compaed to the elastc enegy elease ate obtaned fom atomc and contnuum appoaches n lnea condtons. Fnally the nonlnea factue toughness, J c, s estmated n dffeent cack length. Chapte 6 povdes the summay of ou nvestgaton and some ecommendatons fo futue wok.

26 METHODOLOGY OF ATOMISTIC SIMULATIONS Atomstc smulatons model the behavo of molecula system at atomc level. The usefulness and physcal sgnfcance of atomstc smulatons has been demonstated by numeous wok n the last two decades snce the hgh-speed compute came up. Altenatve quantum mechancs methods ae usually moe accuate but also much moe computatonally expensve whch make t not feasble to handle molecula systems consstng of moe than hundeds of atoms. In ths chapte we wll ntoduced some methodologes of atomstc smulaton assocated wth ou eseach. The oganzaton of ths chapte s as follows. In the fst secton the basc chemsty of cabon bondng mechansm wll be befly dscussed. Secton povdes the ntoducton of molecula dynamcs smulaton, the ntegaton algothm and a detaled descpton of nteatomc potental functon ncludng nteatomc foces deved fom t. The last secton s the calculaton method of stess n atomc system.. Basc Chemsty of Cabon Bonds Ths secton s a bef explanaton of the molecula bondng mechansm of cabon atoms. Fst of all let us ntoduce seveal basc chemcal concepts that wll be used n the followng pats. It s well known that evey atom conssts of a postvely chaged nucleus suounded by a chaged cloud of electons. The electons movng aound the nucleus have dffeent enegy levels whch ae nomally dentfed as atomc obtal. As we know, completely dffeent fom the common sense of macoscopc theoy, the exact locaton of an electon n an atom s neve possble to detemne not only by expement obsevatons

27 but also by theoetcal analyss. Theefoe, the atomc obtal actually epesents a theedmensonal volume of space wthn whch an electon wth specfed enegy level would have a cetan pobablty to be thee. Fo cabon atoms thee ae only two knds of atomc obtal: s and p obtal. As shown n Fgue., the shape of s obtal s sphecal whle p obtal has thee oentatons, denoted as p x, p y, and p z, le along thee pependcula axes. The chemcal bond s n fact the exchange and shae of electons between atom pa. When atoms get close enough, the atomc obtal on the ndvdual atoms wll be able to ovelap so that the thee-dmensonal pobablty egons shae a common volume pat. p z p y p x Fg.. The llustaton of s and p atomc obtal. Cabon atom has 6 electons and the electonc gound state confguaton s s s p, whee s and p denote dffeent atomc obtal and the supescpts ndcate the numbe of electon that stay on the obtal, the numbe n font of the obtal epesent dffeent electon shells. Dung the chemcal eacton the atom may eaange ts atomc obtal po to the bond fomaton. Instead of usng the ognal atomc obtal dectly, mxtues of them (hybds ae fomed. Fo cabon the hybdzaton s lmted to mxng 3

28 one s and thee p obtal, whch s sp 3, sp and sp hybdzaton. These tems specfcally efe to the hybdzaton of the atom and ndcate the numbe of p obtal used to fom hybds. The balanced angle sepaatons between the hybd obtal ae 9,.. and 8 fo the sp 3, sp and sp hybd bond espectvely, as shown n Fgue sp 3 sp sp Fg.. Schematc model of sp 3, sp and sp bondng hybdzaton. The clea loops denote sp-hybd obtal and the shaded loops denote pue p obtal left unhybd. A typcal sp 3 hybdzaton mateal n the natual wold s the damond, whle the gaphte and gaphte-elated mateals, such as cabon nanotube, have sp bondng. In ths dssetaton, ou study s manly focused on the gaphte and cabon nanotube. An deal cystal stuctue of gaphte conssts of peodcal hexagonal stuctues. In such stuctue the cabon covalent bonds ae fomed between a cabon atom and ts thee neaest neghbos fom one s-obtal (s and two p-obtal electons (p x, p y. The thee bonds angles ae o to each othe wthn a plane. Ths n-plane bond s efeed to as a σ-bond. The σ-bond s a stong covalent bond, whch s the fundamental eason of the hgh stffness and hgh stength n-plane popetes of gaphte and cabon nanotube. The emanng p-obtal, whch s p z as atomc notaton, s pependcula to the plane of σ- bond and povdes only weak nteplana bondng, whch s named as π-bond (Fg..3 4

29 π σ Fg..3 Molecula mechansm of a basc hexagonal bondng stuctue.. Molecula Mechancs and Dynamcs Smulaton.. Intoducton of Molecula Dynamcs Smulaton Molecula dynamcs smulaton (MD smulaton povdes the methodology fo detaled mcoscopc modelng on the molecula scale. It s detemnstc and based on Newton s second law. Though not consdeng quantum chemsty, the MD smulaton can stll accuately descbe some molecula phenomena by choosng pope nteatomc potental functons (Hnchlffe, 995. Theefoe MD smulaton s always egaded as one of the pncpal tools n the theoetcal study of chemcal molecules system. The basc pncple of MD smulaton s to compute the motons of ndvdual molecules, whch ae always teated as gd body patcles, n models of solds, lquds and gases. It s based on Newton s second law, F = ma, whee F s the foce exeted on atom, m s the mass of atom and a s ts acceleaton. Fom knowledge of the foce on each atom, the acceleaton of each atom n the system can be detemned. 5

30 Numecal ntegaton of the equatons of moton then yelds a tajectoy that descbes the postons, veloctes, and acceleatons of the atoms as they vay wth tme. Fom ths tajectoy, the aveage values of popetes can be estmated. The mutual nteacton foce, F, whch exeted on atom, can be detemned by the gadent of the total potental enegy of the cabon nanotubes, as gven by F = U (. whee U s the total potental enegy of the system. Then the acceleaton of atoms can be expessed by the devatve of the total potental enegy. a = m U (. Theefoe, to detemne the tajectoy of the atoms n the system, only the ntal postons of the atoms and ntal dstbuton of veloctes s needed. The ntal postons can be obtaned fom expemental stuctues. The ntal atoms veloctes, v, ae often chosen andomly fom a Maxwell-Boltzmann dstbuton at a gven tempeatue whch gves the pobablty that an atom has a velocty α v n the decton gven by. p( v α α m m ( / v = exp[ ] (.3 π k T k T B B whee k B s the Boltzmann constant, T s the tempeatue whch can be calculated fom the veloctes of the atoms T = N k B N = m v (.4 6

31 whee N s the total numbe of the atoms n the system. If the themo effect s not consdeed, whch means the tempeatue of the system s set at K, the smulaton s molecula mechancs, whch s known by the aconym MM... Integaton Algothm To have a elable computaton on the atoms tajectoes, appopate numecal method ae equed to be appled to solve those dffeental equatons. A numbe of obust algothms ae used to ntegate govenng moton equatons of atoms, such as Velet s algothm, leapfog algothm, and etc. The fnte-dffeence method that has been used n ou MD smulaton s Velet s algothm. Ths algothm s a combnaton of two Taylo expansons, as gven by: 3 dx d x d x 3 4 x( t + t = x( t + t + t + t + O( t (.5 3 dt dt 6 dt 3 dx d x d x 3 4 x( t t = x( t t + t t + O( t (.6 3 dt dt 6 dt Addng these two equatons gves the Velet algothm, whch s: d x x( t t dt + t = x( t x( t t + (.7 whee the local tuncaton eo s n the ode of 5 ode of femto-second ( s. 4 t. Typcally t s on the And nomally the velocty can be estmated as fst-ode cental dffeence. x( t + t x( t t v( t = (.8 t 7

32 ..3 Inteatomc Potental Functon Developng the nteatomc potental functon s the key ssue of molecula smulaton. Many eseaches have been wokng ove the past two decades to deve functonal foms and paametes fo potental functon of geneal applcablty to oganc molecules. A natual statng s to descbe the total nteatomc potental enegy n tems of thee dffeent mechancal nteactons among atoms, whch ae bond stetchng, bond bendng and bond toson espectvely. Ths potental functon s a geneal functon that can be appled to most atomc stuctues and ae much less elatvely computatonally demandng. Howeve t has seveal lmtatons to eflect the eal chemcal bond eactons. Fo example, analyss of the vbaton spectum shows that eal datomc molecules do not vbate as f they wee smple patcles at the ends of classcal spngs. Anothe lmtaton s due to the fxed set of atom types. In the actual chemcal stuctue the atom popetes always depend on a patcula bond envonment,.e., the cabon atom n ethyne s obvously chemcally dffeent fom a cabon atom n a damond molecule. Howeve, when usng ths potental to descbe the two dffeent chemcal systems lsted above, the bond popetes smulated by ths potental would be exactly the same wth each othe, whch s unphyscal. To avod such lmtatons, a Tesoff-Benne potental enegy functon that can epoduce the ealstc chemcal bonds popetes of hydocabon molecules moe accuately s adopted to nvestgate the factue paametes at the atomc level. A detaled descpton of the Tesoff-Benne potental enegy functon wll be povded n the followng. Tesoff-Benne potental s an empcal many-body potental enegy functon, whch s capable of modelng ntamolecula chemcal bondng of damond and gaphte as well as a numbe of essental hydocabons molecules (Benne, 99. The potental s 8

33 based on Tesoff s covalent-bondng fomalsm wth coectons fo functons of neghbong atoms on local envonment (Tesoff, 988a, b, and c. Fo cabon atoms, the so-called Tesoff-Benne potental enegy functon s gven by: U = j( > [ V ( B V ( ] (.9 R A whee U s the total nteatomc potental enegy of the system. efes to the atom of nteest, j efes to neghbong atoms, and s the dstance between atoms and j. Fgue.4 s a schematc dagam of the atomc stuctue of gaphte whch llustates the atom dstance and neghbong atoms n the Tesoff-Benne potental functon moe clealy. Fom the fgue we can see that an abtay atom has thee neghbong atoms, denoted by j, k, k, espectvely. The functons V and V ae the shot-ange pa potental descbng R ( A ( epulsve and attactve nteactons between atoms and j espectvely gven by V R ( = f ( De e S S β ( e (. V A ( = f ( DeS e S / S β ( e (. whee e s the equlbum dstance of two fee cabon atoms wthout any neghbong atoms. D e, S and β ae constants. 9

34 k k k k j cut-off dstance Fg..4 Schematc dagam of atomc stuctue of gaphte. The cut-off functon f, whch estcts the pa potental to neaest neghbos, s smply taken as: ( f ( = π ( [ + cos( ] < > (. whch has a contnuous value fom to n the ange of and. s the cutoff dstance. The functon B s the ctcal featue of the potental. It epesents a measue of the numbe of bonds between atoms n a mateal and bond angle, and s assumed to be a monotoncally deceasng functon of the coodnaton of atoms and j because the moe neghbos an atom has, the weake the bond to each neghbo s. Snce B can eflect the change of local atomc envonment, the Tesoff-Benne potental then has the capablty to accuately descbe dffeent hydocabon systems wthn the same set of paametes. The fom of B s gven by:

35 B = ( B + B j (.3 whee δ B = + G( θ k f ( k (.4 k (, j δ s the constant. G( θ k, whch s the functon of the angle θ k between bonds -j and -k, s gven by c c G ( θ (.5 = a + d d + ( + cosθ whee a, c, d ae constants. All the constants n the Tesoff-Benne potental ae lsted n Table.: Table. Constants n Tesoff-Benne potental. e =.39 D = 6. S =. β = R =.7 R =. nm ev nm δ =.5 nm nm a =.83 c = 33 d = 3. 5

36 Note that e n the potental functon (Eq.. and. s not the eal equlbum bond length n any abtay system. Only when B =. s e the equlbum bond length. The equlbum bond length s detemned by mnmzng the nteatomc potental wth egad to, that s U / =. Appaently, the equlbum bond length s a functon of B whch epesents the nfluence of neaby neghbong atoms exetng on the bond -j. Theefoe fo dffeent bond condtons, such as sp hybdzaton and sp 3 bond, ths potental can automatcally detemne the equlbum bond length n dffeent systems wthn the same set of paametes n the potental functon. Ths new featue also povdes a way to descbe the phenomenon of bond beakng and fomng. Ths lead to the equlbum nteatomc dstance fo the gaphene sheet equal to =.45 nm. To gve a bef pctue of Tesoff-Benne potental functon, Fgue.5 descbes the nteatomc foce and potental enegy vesus bond length fo a sngle bond between atom and j n a gaphene sheet. In both fgues the bond -j eaches equlbum at pont A; at pont B the bond length equals to and at pont C equals to, whch s the cut-off dstance. In Fg..5a the nteatomc foce nceases damatcally when the bond length eaches.7 nm. Ths awkwad phenomenon esults fom the cut-off functon f n the Tesoff-Benne potental. Thus, a pope adjustment n the potental functon s necessay to avod ths poblem when studyng the mechancal popetes of a bond n the ange of the cut-off dstance. In ths dssetaton, dung the nvestgaton of the elastc popetes of cabon nanotube and the factue popetes of gaphene sheet n lnea condton the extenal defomaton felds appled to the atomc system ae suffcently small so that the ange of local bonds length n the system s always much smalle than. (

37 Fo the evaluaton of the factue behavo n nonlneaty a detaled dscusson on ths ssue s poposed n chapte Inteatomc Foce ( N B 5 A C Bond Length (nm 8 Potental Enegy ( J A B Bond Length C (nm (a Inteatomc foce vesus bond length (b Potental enegy vesus bond length Fg..5 The nteatomc foce and potental enegy vesus bond length obtaned by Tesoff-Benne potental..3 Stess n Atomc System In macoscopc contnuous meda stess denotes the magntude of the foces pe unt aea of the suface on whch they act. At the atomc scale, the concept of stess, whch s elated to the nteatomc foces among neghbong atoms, can also be used as a physcally appopate descpton of the mechancal esponse of the local atomc stuctue. The stess calculatons n the atomstc smulatons have been fomulated ove yeas (e.g., Tsa, 979; Swenson, 983; Vtek and Egam, 987; Lutsko, 988; Cheung and Yp, 99; Come et al., ; Zhou, 3. The atomc stess s nomally based on the val theoem of Clausus (87 and the geneal fom of the atomc stess s gve by σ αβ = m v v F (.6 α β V V β α 3

38 αβ whee σ s the atomc stess tenso n the Catesan coodnates at atom. V s the volume pe atom. m s the mass of atom and v α s the velocty of atom n α α α α decton. = s the dstance between atom and j pojected on α decton. j β F s the β component of the nteatomc foce exeted on atom by atom j. Swenson has gven a devaton of the expesson above fo the atomc stess tenso fom a classcal val theoem (Swenson, 983. To get a bette undestandng of the mcoscopc stess, a bef devaton s lsted n the followng pat. Consde a system of N atoms n a volume V. Fo atom, thee s: α dp F α = (.7 dt whee α F s the α-th component of the foce exet on atom, α p s the α-th component of momentum of atom. Multplyng Eq. (.6 by x β, whch s the β-th component of the coodnate of atom, summng ove all the atoms, and takng a tme aveage gve τ N τ N p p x x p (.8 N β α β α F dt + dt = τ = τ = m τ = β α τ whee m s the mass of atom. The basc assumpton leadng to a classcal val theoem s ethe (a the system s peodc and τ s chosen to be a multple of a peod, o (b the system s bounded and τ s chosen to be lage. In ethe case, the ght-hand sde of Eq. (.8 vanshes. Then: V αβ + K αβ = (.9 4

39 whee V αβ = τ N τ = τ τ = x F dt β α (. N β α αβ p p K = dt (. m Now assume that the foce on atom s due to a wall foce, w, and nteatomc foces, F, exeted by atom j. Thus: V V W + V I αβ αβ αβ = (. whee αβ β α VW = x w dt (.3 τ N τ = αβ β α VI = x F dt (.4 τ N τ = The wall foce pat, αβ V W, s elated to the stess tenso. To see ths assume that on a macoscopc scale the wall foce s vey shot anged so that only atoms vey close to the wall wll contbute to wll contbute to αβ V W. In the vcnty of the wall, all of the atoms n a volume dv αβ V W f dv s defned as the poduct of an element of wall aea ds tmes the ange of the wall foce d. Let n be a unt nomal to the suface aea ds and be the poston vecto of the cente of the volume element. Then fom atoms n dv at some nstant of tme thee s 5

40 δv αβ W = = dv dv x β ( x β w α β w α + β dv w α (.5 On a macoscopc scale x β and appoxmated by dv β ae essentally equal, then Eq. (.5 s αβ β α δ V = w (.6 W Takng the tme aveage, Eq. (.6 tuns to be: τ τ αβ β α δvw dt = w dt (.7 τ τ dv And the facto of β s dentfed wth the stess vecto T, the aveage foce pe unt aea exeted by the wall on the system, whch s, τ τ δv αβ W dt = β T α ds (.8 α Defne the macoscopc stess tenso σ n the usual way, T = σ αβ nβ, Whee epeated ndces ae summed. Thus: αβ V αβ W = β suface σ n ds (.9 αγ γ Usng the dvegence theoem, Eq. (.9 can be expessed as αβ β V W = ( σ αγ dv (.3 γ V Based on the equaton of equlbum: 6

41 σ αγ γ = (.3 Then Eq. (.3 tuns to be: V αβ W β = σ αγ dv = Vσ γ αβ (.3 V (.9 as: Then an expesson fo the macoscopc stess tenso can be deved fom Eq. σ αβ = ( K V = V τ τ αβ N + V p β αβ I m p α dt V τ τ N = j ( x β x β j F α dt (.33 In a statc, peodc model stuctue, Eq. (.33 can be shown to be σ αβ β α = F (.34 V 7

42 3 ELASTIC PROPERTIES OF SINGLE- WALLED CARBON NANOTUBES In ths chapte, selected effectve elastc modul of sngle-walled cabon nanotubes ae smulated numecally. Ths effectve macoscopc behavo s studed usng molecula dynamcs (MD smulatons n whch the dynamc esponse and mutual foce nteacton among atoms of the nanostuctues ae obtaned when subjected to smallstan defomaton. Both foce and enegy appoaches that lnk the behavo at the atomc and macoscopc scales of the nanotubes ae used to pedct the elastc modul unde dffeent defomaton modes. A compason of the elastc constants obtaned fom MD smulaton wth avalable expemental data s made. 3. Intoducton Ima (99 fst epoted the fomaton and obsevaton of cabon nanotubes (CNTs. Snce then, eseach on CNTs has been attactng much attenton to exploe the unusual electonc and mateal popetes (e.g. Desselhaus, 997; Schadle et al., 998; Dekke, 999; Qan et al.,. Both expemental and theoetcal studes ndcated that CNTs ae mateal wth extaodnay hgh stffness and axal stength. Due to the lage aspect atos and small dametes, CNTs have emeged as potentally attactve mateals as enfocng elements n lght-weght and hgh stength stuctual compostes. Studes of the mechancal behavo of CNTs have focused manly on the adopton of dffeent empcal potentals and contnuum models usng elastcty theoy. In 8

43 expemented appoaches, the foce-dsplacement esponse of a nanotube s measued and the axal Young s modulus s obtaned by compason to an equvalent elastc beam. Teacy et al. (996 showed an aveage value.8 TPa (wth lage scatte fo the axal Young s modulus fom the dect measuements wth a tansmsson electon mcoscope of a vaety of mult-walled nanotubes (MWNTs of dffeent nne and oute dametes usng an analogy to the themal vbaton analyss of anchoed tubes. The nanotubes wth the smallest nne damete wee consdeably stffe, wth a Young s modulus of 3.7 TPa. Wong et al. (997 used an atomc foce mcoscopy (AFM to measue focedsplacement elatons fo anchoed MWNTs on a substate. Fom a compason to elastc beam theoy, the Young s modulus can be extacted. An aveage of.8 ±.5 TPa wth lttle dependence of nanotube damete was epoted. Loue and Wagne (998 obtaned the axal Young s modulus fo a sees of tempeatues by mco-raman spectoscopy fom measuements of coolng-nduced compessve defomaton of nanotubes embedded n an epoxy matx. At 8 o K, the expemental esults gave 3 TPa fo sngle-walled nanotubes (SWNTs wth an aveage adus of.7 nm, and.4 TPa fo MWNTs wth an aveage adus of 5 to nm. In theoetcal studes of pedctng elastc popetes of nanotubes usng MD smulatons, Yakobson et al. (996 used a ealstc many-body nteatomc potental (Tesoff-Benne potental wth a contnuum shell model and obtaned the axal SWNT modulus that anged fom.4 to 5.5 TPa. Note that the pape used the value.66 nm as the tube wall thckness whle most othe efeences use.34 nm nstead. Conwell and Wlle (997 epoted a elatvely low modulus of. TPa fo open-ended, fee-standng SWNT by usng quenched molecula dynamcs smulatons usng the Tesoff-Benne potental. Lu (997 used an empcal foce-constant model to detemne seveal elastc modul of sngle- and mult-walled nanotubes and obtaned the Young s modulus of about 9

44 TPa and the otatonal shea modulus of about.5 TPa. The analyss showed that the elastc popetes wee nsenstve to the adus, helcty, and the numbe of walls. Howeve, Yao and Lod (998 used MD smulatons and found that changes n stuctue such as adus and helcty of the SMNTs could affect the Young s modulus because the esults showed that the tosonal potental enegy, whch s the domnant component of total potental enegy, nceased as the quadatc functon of the deceasng tube adus. Snnott et al. (998 estmated the axal Young s modulus fo cabon fbes composed of SWNTs algned n the decton of tubula axs. The esult showed that fbes, whch wee composed of smalle adus cabon nanotubes wth a packng aton of.9, had a modulus of appoxmately.5-.4 TPa. Henandez et al. (998 adopted a tght-bndng method to obtan aveaged values of Young s modulus and Posson s ato of SWNTs wth dffeent chal vectos, whch wee found to be.4 TPa and.6, espectvely. Zhou et al. ( clamed that both Young s modulus and the wall thckness wee ndependent of the adus and the helcty of SWNTs. They appled the stan enegy of SWNTs dectly fom electonc band stuctue wthout ntoducng empcal potentals and contnuum elastcty theoy to descbe the mechancal popetes of SWNTs. The estmated value fo the axal modulus was epoted as 5. TPa, whch s fve tmes lage than the value of MWNTs. In spte of the vaety of theoetcal studes on the macoscopc elastc behavo of CNTs, thee stll eman contovesal ssues egadng the effect of geometc stuctue of CNTs on elastc modul, as evdence by the wde scatte among the elastc modul epoted n the lteatue. Meanwhle, the computatons of elastc modul ae stll not vey completed, such as the numecal esults of bulk modulus wee not conssted wth othe elastc modul obtaned n Lu s smulatons (997. The objectve n ths chapte s to eexamne the elastc behavo of CNTs n detal usng two dffeent appoaches n pola 3

45 coodnate system whch should be moe appopate fo the nvestgaton of the mechancal popetes of CNTs. The oganzaton of ths chapte s as follows. In the next secton, the mophologcal stuctue of cabon nanotubes wll be befly dscussed. In Secton 3.3, seveal elastc modul ae detemned by applyng dffeent small-stan defomaton modes. The elastc modul ae pedcted usng enegy and foce appoaches. Numecal esults and a summay ae gven n Secton Stuctue of Cabon Nanotubes Sngle-walled nanotubes ae fomed by foldng a gaphene sheet to fom a hollow cylnde whch s composed of hexagonal cabon ng unts, whch ae also efeed to as gaphene unts. The fundamental cabon nanotube stuctue can be classfed nto thee categoes: amcha, zgzag, and chal, n tems of the helcty. Fgue 3. shows a segment of a sngle gaphte plane that can be tansfomed nto a cabon nanotube by ollng t up nto a cylnde. To descbe ths stuctue, a chal vecto s defned as OA = n a + ma, whee a and a ae unt vectos fo the honeycomb lattce of the gaphene sheet, n and m ae two nteges, along wth a chal angle θ whch s the angle of the chal vecto wth espect to the x decton shown n Fgue 3.. Ths chal vecto, OA, wll be denoted as (n, m whch wll also specfy the stuctue of the cabon nanotube. Vecto OB s pependcula to the vecto OA. To constuct a CNT, we cut off the quadangles OA B B and oll t nto a cylnde wth OB and A B ovelappng each othe. The elatonshp between the nteges (n, m and the nanotube adus,, and chal angle, θ, s gven by / = 3ac c ( m + mn + n / π (3. θ = tan [ 3m /( m + n] (3. 3

46 whee a s the length of the C-C bond c c y B' B x a a O θ A Fg. 3. The gaphte plane of nanotube suface coodnates. Geneally thee ae thee majo categoes of nanotubes based on the chal angle θ. Fgue 3. s the examples of the thee knds of nanotubes. θ = Zgzag < 3 < θ Chal θ = 3 Am Cha 3

47 Fg. 3. Examples of am cha, zgzag, and chal nanotubes. 3.3 Elastc Modul 3.3. Stess-Stan Relatonshp of Cabon Nanotube The geneal elastc chaactestcs of the cabon nanotubes wll now be dscussed. Because of the hexagonal symmety popetes n the cylndcal suface of nanotube, the nanotube exhbts tansvese sotopy n the θ z plane, whch s defned by fve ndependent elastc constants n the stess-stan elatonshps. σ σ σ θθ zz = C C C 3 3 C C C C C C ε ε θθ zz ε (3.3 33

48 34 z z z G θ θ θ γ τ =, z z z G γ τ =, θ θ θ γ τ G = (3.4 whee ( z z z E G θ θ +ν = Eq. (3.3 can also be expessed n tem of engneeng constants as = zz z z z z z z z z z z z z z z zz E E v E v E v E E v E v E v E σ σ σ ε ε ε θθ θ θ θθ (3.5 In the cylndcal coodnate system the elaton of stan and dsplacement s gven by u = ε θ ε θ θθ + = u u z u z zz = ε u z u z z + = γ θ γ θ θ θ + = u u u z u u z z + = θ θ θ γ (3.6

49 3.3. Calculatons of Elastc Modul The potental enegy expesson unde small-stan defomaton n cabon nanotube s gven by: U = U + VC kl ε ε kl (3.7 whee U s the total potental enegy at equlbum, U s the ntal potental enegy at equlbum befoe defomaton s appled, C kl s the fouth-ode tenso of elastc constants, V s the ntal volume of the cabon nanotube, whch fo a hollow cylnde s gven by V = π tl, whee L s the length, s the adus, and t s the assumed tube wall thckness. In ou smulaton, both a foce and enegy appoach s adopted to pedct the elastc modul of cabon nanotubes. In the foce appoach, based on the classcal val theoem dscussed n the pevous chapte, an expesson of the stess tenso n a macoscopc and bounded system s gven n the tem of atom coodnates and nteatomc foces. Thus the elastc modul can be calculated by dectly computng the aveaged atom coodnates and nteatomc foces n the nanotube. Meanwhle, the enegy appoach, whch uses the second devatve of potental enegy wth espect to stan unde each defomaton mode, s also pefomed to obtan the elastc modul. Equatons (3.8 and (3.9 defne the calculaton of elastc modul based on foce appoach and enegy appoach, espectvely: σ αβ β α = F (3.8 V 35

50 C kl = V U ε ε kl (3.9 In Eqs. (3.8 and (3.9, the epeated subscpts ndcate the sum of all atoms, σ αβ s the aveage atomc-level stess tenso n the Catesan coodnates, F s the nteatomc foces ncludng bondng and nonbondng foces between two atoms and j, Based on the two appoaches, seveal elastc modul ae detemned unde smallstan defomaton n the followng pats by applyng defomaton modes smla wth some pevous elastc constants calculatons (Hashn and Rosen, 964; Yuan, Pagano, and Ca, 99. Calculaton of E z and ν z θ To calculate the axal Young s modulus, atoms ae dsplaced by u z = ε z. The aveage stan and stesses ae: zz E z, and the Posson s ato, ν z θ, the ε zz = ε zz, σ, any othe σ = (3. zz Though MD smulaton, the longtudnal elastc modulus, E z, can be calculated by a foce and enegy appoach usng the followng elatonshps: E z U = Enegy Appoach V ε zz E z σ = Foce Appoach (3. ε zz zz 36

51 The Posson s ato s obtaned as: ν zθ ε = (3. ε θθ zz Calculaton of Eθ and ν θz If a adal dsplacement u ε θθ = s appled to evey atom of the nanotube, then we obtan only a adal stess and can calculate a pue adal stess n whch: ε θθ = ε θθ σ, any othe σ = (3.3 θθ Hence, Eθ and ν θz ae gven by: U E θ = Enegy Appoach V ε θθ σ E θ = Foce Appoach (3.4 ε θθ θθ ν θz ε = (3.5 ε zz θθ 3 Calculaton of the Rotatonal Shea Modulus G θ z The equed dsplacement mode n ths case s u = z whch s appled to all θ γ θ z atoms. The esultng stan and stesses ae then gven by: γ = γ, any othe ε = θz θz 37

52 τ θz, any othe σ = (3.6 Then the otatonal shea modulus G θ z can then be calculated by: G θz U = Enegy Appoach V γ θz G θ z τ = Foce Appoach (3.7 γ θ z θ z 4 Calculaton of C 3 and C 33 Hee the cabon atoms ae unde defomaton ae constaned. The aveage stan state mposed s: u z = ε z and lateal dsplacements zz ε =, any othe ε = (3.8 zz ε zz Based on the elatonshp of stess and stan, C 3 and C 33 ae gven by: C 33 U = Enegy Appoach V ε z z C C 3 33 σ θ θ = ε zz Foce Appoach (3.9 σ z z = ε zz Although C 3 can be calculated usng a foce appoach, ths modulus s small, on the ode of GPa, whch s wthn the numecal eo bounds of the pesent analyss. Theefoe, the detemnaton of C 3 s not attempted. 38

53 3.3.3 Detals of the Numecal Pocedue fo Calculatng Modul In the pesent analyss, focus s placed on the elastc modul of amcha nanotubes angng fom n = 6 to. The vaous geometc dmensons of the nanotubes ae lsted n Table 3.. The aspect ato of the nanotubes, whch s the length-to-adus ato, s set to be geate than to ensue that edge effects can be neglected. A key ssue hee s that the wall thckness of the nanotubes needs to be specfed n the contnuum sense befoe pedctng the values of elastc modul fo the sngle-walled cabon nanotubes. Fo mult-walled nanotubes, the dstance between any two walls s close to that of the ntelaye sepaaton dstance n gaphte, whch s.34 nm. Ths ntelaye sepaaton dstance s used as the effectve wall thckness of SWNTs. The geneal pocess of the smulaton s as followng. Fst the MD smulatons ae pefomed to obtan a mnmum enegy state of the nanotube wthout any appled defomaton. The smulaton un fo appoxmately 5 ps wth a tme step of fs afte whch the enegy attaned a constant mnmum value. Then dffeent types of defomaton modes dscussed above ae appled to the nanotube by scalng atomc postons and applyng bounday condtons along the appopate coodnates. The nanotube eaches an equlbum state ove appoxmately 5 ps. Afte the equlbum a 5 ps equlbum peod s caed out. By ntegatng molecula dynamc equatons, the stesses and total potental enegy ae calculated fom the aveaged esults fom ths equlbum peod. Then the value of elastc modul can be obtaned usng foce and enegy appoach though equatons (3.8 and (3.9, espectvely. In the enegy appoach, the ntal potental enegy, U, s obtaned fom the MD smulaton by mnmzng the enegy of the nanotube wthout any extenally appled stan state. Fo each defomaton mode, thee constant stan values,.5,., and 39

54 .5 ae appled to calculate the total potental eneges. A least-squaes ft fo the fou potental eneges s used to obtan the elastc modul. In the foce appoach, only an appled stan value of. fo each defomaton mode s used to calculate the elastc modul. Note that the stesses calculated fom Eq. (7 need to be tansfomed nto pola coodnates fo calculatng some of the elastc modul n the foce appoach. Table 3. Geomety popetes of sngle-walled nanotubes fom n = 6 to. (n, n Numbe of Atoms Radus (nm Length (nm (6, (7, (8, (9, (, (, (, (3, (4, (5, (6, (7, (8, (9, (,

55 3. 4 Results and Dscussons Fgue 3.3 ae the numecal values of E z and E θ of dffeent amcha snglewalled cabon nanotubes wth ad angng fom.47 to.357 nm obtaned fom the enegy and foce appoaches. It s clealy shown that E z and E θ have lttle dependence on the adus of nanotubes n both the enegy and foce appoaches. The two set of pedctons fom the enegy and foce appoach agee vey well fo all the nanotubes studed. Ths suggests that the cabon nanotube s essentally sotopc n the plane of cylndcal suface. The aveage values and elatve eos of E z and nanotubes nvestgated hee ae lsted n Table 3.. E θ fo all the Table 3. The aveage values of E z and E θ of selected cabon nanotubes Enegy Appoach Foce Appoach E z E θ Aveage.73 ± ±. 3 The above values show that the two appoaches yeld smla pedctons of modul of cabon nanotubes. Fgues 3.4 and 3.5 show the Posson s ato, v θ z and v zθ, and shea modulus G θ z as a functon of cabon nanotube ad, espectvely. Fom Fgue 3.4 the aveage values of Posson s atos ae obtaned as v =. 44 and v θ =. 44. Utlzng the elatonshp fo the shea modulus n a tansvesely sotopc mateal, zθ z 4

56 G θz Ez =, a compason between the dect numecal esults fo G θ z and the ( + v zθ values deved fom the elastc elaton usng E z and v zθ s shown n Table 3.3. The compason shows a close ageement and yelds futhe confmaton of the tansvese sotopc popety of the nanotube. Agan, the esults of shea modulus demonstate to be nsenstve to the adus of the nanotube. Fgue 3.6 depcts the pedctons of the C 33 elastc coeffcents vesus nanotubes ad. The aveage value of C 33 s.65 TPa wth a standad devaton.5% usng the enegy appoach. Usng the foce appoach, C 33 shows an aveage value of.6 TPa wth a standad devaton.%. Fgue 3.7 s the value of C 3, whch has an aveage.7 TPa wth a standad devaton.%, fom foce appoach. All elastc modul pedcted though the smulatons ae lsted n Table 3.4. Table 3.3 Compason between the numecal esults and the theoetcal esults deved fom G θ = E /[ ( + v θ ] z z z Enegy Appoach Foce Appoach G θ z -- Numecal (TPa G θ z -- Theoetcal (TPa Relatve eo 4. %. % Snce the sngle-walled nanotube has a sngle cabon atom n the thckness decton, the engneeng constants that ae elated to the adal decton, E, ν z, and G θ (o G z, ae yet to be clealy defned. It has been shown fom smulatons usng both foce and enegy appoaches that oveall the elastc constants of SWNTs ae nsenstve to 4

57 the mophology patten such as nanotube adus and thus the effect of cuvatue on the elastc constants can be neglected. Ths confmed the expemental esults by Wong (Wong, et al., 997 n whch they found that the Young s modulus had no dependence on tube damete. The aveage numecal esults n ou smulatons ae n the same ange wth the pevous expemental and numecal esults,.e. measuements usng AFM showed a ange of.3.5 TPa of Young s modulus (Yu, et al.,. In the smulaton the esults fom enegy appoach agee emakably well wth the esults fom foce appoach. The foce appoach povdes moe nfomaton than that fom the enegy appoach. In addton, the foce appoach povdes moe accuate pedcton than the enegy appoach snce the enegy appoach s equed to calculate seveal total potental enegy values unde dffeent stan levels and then uses a least-squae ft to obtan the elastc modul. Futhemoe, the foce appoach can pedct the nonlnea behavo wthout assumpton of assumed total potental enegy n quadatc fom descbed n Equaton 3.7 fo small-stan defomaton n the enegy appoach. Assumpton of the tansvesely sotopc popetes on the cylnde suface of the sngle-walled nanotube s also confmed by numecal calculatons. 43

58 E z E θ (a Enegy Appoach E z E θ Fgue 3.3 E z and (b Foce Appoach θ E dstbuton vesus adus of amcha sngle-walled cabon nanotubes n enegy and foce appoaches 44

59 v zθ v θ z Fgue 3.4 Posson s atos v θ z and v zθ vesus adus of amcha sngle-walled cabon nanotubes G θz Fg. 3.5 Shea modulus G θ z vesus adus of amcha sngle-walled nanotubes 45

60 C 33 Fg. 3.6 C 33 vesus adus of amcha sngle-walled cabon nanotubes C 3 Fg. 3.7 C 3 vesus adus of amcha sngle-walled cabon nanotubes 46

61 Table 3.4 Elastc modul of sngle-walled cabon nanotubes (a Elastc modul obtaned fom enegy appoach (n,n E z E θ C 33 G θ z (6, (7, (8, (9, (, (, (, (3, (4, (5, (6, (7, (8, (9, (, Aveage.78 ±..697 ±..65 ±..39 ±. 47

62 (n,n E z (b Elastc modul obtaned fom foce appoach E θ C 33 C 3 G θ z (6, (7, (8, (9, (, (, (, (3, (4, (5, (6, (7, (8, (9, (, Aveage.78 ±..68 ±..6 ±..7 ±..43 ±. 48

63 (c Posson s ato (n,n v zθ v θ z (6, (7, (8,8.4.4 (9, (, (, (, (3, (4,4.4.4 (5,5.4.4 (6, (7, (8, (9, (, Aveage.44 ±..44 ±. * The elastc constants ae n the unts of TPa. * z-decton s the axs decton of the nanotube. 49

64 4 ENERGY RELEASE RATE AND STRESS FIELD IN ATOMIC SYSTEM In ths chapte, macoscopc factue paametes ae examned fom both atomc-scale and macoscopc contnuum model. An atomc model nea a cack tp n a sem-nfnte gaphene sheet s poposed. The enegy elease ates and atomc stess ae nvestgated by applyng emote Mode I and Mode II dsplacement feld that s obtaned fom the lnea elastc factue mechancs (LEFM wth an ntally specfed stess ntensty facto. Also a sample made of gaphte that epesents cacks n a plane wth fnte wdth s used to calculate the elastc enegy elease ates unde Mode I and II small defomaton. In the dect atomstc smulaton, Tesoff-Benne nteatomc potental s used to descbe the atomc foce nteactons among cabons. Two methods, global enegy appoach and local foce appoach ae both adopted to compute the enegy elease ates. The enegy elease ates and defomed cack suface pofle ae also calculated fom lnea elastc factue mechancs though homogenzed mateal popetes fo compason. 4. Atomc Confguatons 4.. Case I: Cack n Sem-nfnte Doman Fst we consde the case that a cack s n a sem-nfnte doman. The atomc system whch conssts of 78 cabon atoms epesents a small zone nea cack tp, as shown n Fgue 4.. The quadangle egon wth the dmenson of 5.74 nm 6.79 nm can be 5

65 egaded as beng cut fom an nfnte gaphene sheet. The cack s modeled by emovng the nteatomc chemcal bond to elmnate the nteacton between atom pas acoss the cack suface. The atomc dsplacement feld obtaned fom LEFM wth an ntally specfed stess ntensty facto s appled to the atoms n the model. Enegy elease ates and the atomc stess feld nea cack tp wll be nvestgated though atomstc smulaton. Ou pmay nteest hee s to use atomstc smulaton to obtan the macoscopc factue paametes and to examne the extent to whch the atomc numecal esults ae consstent wth those obtaned fom the lnea elastc soluton. Only though these compasons can we pove the elablty of atomstc smulaton, and the capablty of chaactezng the atomc factue by macoscopc paametes. x θ x Fg. 4. Schematc stuctue nea a cack tp n a sem-nfnte gaphene sheet. In factue mechancs thee ae thee basc defomaton modes: Mode I, II, and III. Mode I s the openng o tensle mode whee the elatve dsplacements between 5

66 coespondng pas sepaate symmetcally nomal to the cack suface. Mode II, the sldng mode, denotes antsymmetc sepaaton though elatve tangental dsplacements to the cack suface, as shown n Fgue 4.. Because the gaphene sheet studed hee has only one sngle laye, thee s no Mode III n ths stuaton. Mode I Fg. 4. Two basc factue modes. Mode II It s well establshed that the lnea elastc soluton of the stess and dsplacement feld n the vcnty of the cack tp can be expessed as u = K I µ cos π θ [ ( κ + sn θ K II ] + µ θ sn [ ( κ + + cos π θ ] (4. u = K I µ θ sn [ ( κ + cos π θ K II ] + µ θ cos [ ( κ + sn π θ ] (4. K I θ θ 3θ K II θ θ 3θ σ = cos [ sn sn ] sn [ + cos cos ] π π (4.3 5

67 σ K I θ θ 3θ K II θ θ 3θ = cos [ + sn sn ] + sn cos cos (4.4 π π τ = K I θ θ 3θ K II θ θ 3θ cos sn cos + cos [ sn sn ] (4.5 π π Whee µ s the elastc shea modulus of the mateal, κ = ( 3ν /( + v fo plane stess, v s the Posson s ato. K I and K II ae the stess ntensty factos fo mode I and II espectvely. 4.. Case II: Cack n the gaphene sheet wth fnte wdth Case II we poposed n ths chapte s a sample of a gaphene sheet n a zgzag fom wth dmensons 5.68 nm 7.55 nm contanng a system of 984 cabon atoms. Fgue 4.3 s a schematc dagam of the sample. A cack s located n the mddle of the gaphene sheet and s pependcula to some of the covalent bonds. Due to the dscete natue of atomstc modelng, the cack length s evaluated as the length measued fom numbe of boken bonds. Dung the smulaton, the centeed cack of length a vayng fom.6 to 6.53 nm of the gaphene sheet s examned. In the lnea elastc factue mechancs, the Young s modulus (E =.68 TPa s calculated fom an enegy appoach (Jn and Yuan, 3. 53

68 x x Fg. 4.3 Schematc dagam of a cente-cacked gaphene sheet; Symbol o epesents cabon atom Smulaton Technque In the dect atomstc smulaton, each and evey atomstc degee of feedom s explctly accounted fo, and the mnmum enegy confguaton s detemned on the bass of nteatomc nteactons. At the begnnng, the ntal state of the atomc system wthout appled defomaton eaches equlbum fo appoxmately ps wth a tme step of.5 fs. Afte equlbum s eached, a small defomaton s appled to the sample by scalng atomc postons and bounday condtons along the appopate coodnate. In case I the atomc dsplacement feld obtaned fom equaton (4. and (4. wth the ntally specfed stess ntensty facto s appled to each atom. In case II thee defomaton modes ae appled to the gaphene sheet sepaately: the fxed-gp dsplacement u = ε x, ε =. 5, s appled 54

69 on the top and bottom sufaces of the sample n mode I; mode II, u = ε x and u = ε x s pescbed on these sufaces. 3 An smple shea mode, whch s also wdely used to study the factue behavo, especally n the expement nvestgaton, s appled to the system as u = ε and u. An llustaton of smple shea defomaton and pue shea x = defomaton, whch s mode II, s shown n Fgue 4.4. Afte applyng the desgnated defomaton, oughly anothe ps peod s equed to each a mnmum potental enegy of the system. All molecula mechancs smulatons ae caed out at tempeatue K. x x x x Fg. 4.4 Illustaton of smple shea and pue shea. 4. Atomc Stess Based on the descpton of atomc stess n chapte, n ths dscete atomc system the atomc stess at atom can be deved fom the atom coodnates and nteatomc foces. Ths method povdes a contnuum measue of the ntenal mechancal nteactons between atoms. Smlaly the expesson s gven by: σ αβ = V j α F β (4.6 55

70 Whee αβ σ s the atomc stess tenso n the Catesan coodnates at atom. V s the α α α volume pe atom. = s the dstance between atom and j pojected on α decton. j β F s the β component of the nteatomc foce exeted on atom by atom j. 4.3 Enegy Release Rate n Atomc System 4.3. Global Enegy Method Elastc stan enegy elease ates ae calculated fom the atomstc smulatons by ealstc cack extenson models. It s assumed that the cack extenson would not cause the bond econfguaton. Two methods ae poposed to calculate the enegy elease ate. The fst method, global enegy method, s based on the change of total potental enegy of two self-smla gaphene sheets wth same dmensons but two slghtly dffeent cental cack lengths, a and a + a, as gven by U G = a+ a U a t a (4.7 whee U a and U a+ a ae the total potental enegy of the two gaphene sheets unde a gven defomaton mode obtaned fom Tesoff-Benne potental. a s the equlbum nteatomc dstance n the x decton whch s equal to.5 nm (.45 3 nm. t s the thckness of the gaphene sheet. In ths study an ntelaye sepaaton dstance of gaphte, whch s.34 nm, s defned as the effectve thckness. Fgue 4.5 s the llustaton of how to calculate the enegy elease ate. 56

71 u u a a + a Fg. 4.5 Illustaton of the method of calculatng the enegy elease ate Local Foce Method Anothe means of evaluatng the enegy elease ates s the local foce method by detemnng the vtual wok that s equed to close the cack extenson. In ths method, t s not equed to calculate the total potental enegy. Theefoe t leads to less computatonal tme, especally fo those mult-mllon atomc systems, snce only those neghbong atoms nea the cack tp need to be evaluated; whle the change of total potental enegy eques all the atoms n the system have to be nvestgated. Based on the pncple of vtual wok (PVW, the vtual wok s llustated n Fgues 4.6 and 4.7 to close the cack extenson n a gaphene sheet fo mode I and II small defomaton espectvely. The enegy elease ates fo mode I and II, denoted by ae expessed by: G I and G II, 57

72 G ( + I = Fy v v (4.8 a t + + GII = [ Fx ( u u + Fy ( v v ] (4.9 a t whee F x and F y ae the nteatomc foces exeted on atom n the x and y dectons espectvely by atom j and ts neghbong atoms at the cack length a. + u, u, + v, v ae the cack openng dsplacements of atom and j n the x and y dectons at cack length a + a. Theefoe two confguatons have been used to obtan the value of vtual wok. Note that n the dscete atomstc smulaton the cack extenson mples the bond boken n the mmedate vcnty of the cack tp wth length a. a + a a F y Cack Suface j k l F y + v k l v j Fg. 4.6 A geneal llustaton of vtual wok unde Mode I. 58

73 a + a a F y Cack Suface F x j Fx l k F y + v + u u j l k v Fg. 4.7 A geneal llustaton of vtual wok unde Mode II. 4.4 Results and Dscussons Numecal Results of Case I In the smulaton, a emote K I dsplacement feld obtaned by assgnng / K I =.3 MPa m and II = K n Equaton 4. and 4. s appled to the above system. The enegy elease ate s calculated by the local foce method ntoduced n the pevous secton. The numecal esult s then compaed wth contnuum theoetcal soluton whch s G theoy = K / E. Smlaly, a emote K II dsplacement feld can be obtaned by assgnng K = and I K II =. MPa m /. Table 4. s the compason of the enegy elease ate obtaned by atomstc smulaton and contnuum theoetcal soluton. The elatve eos of 59

74 the two esults ae vey small,.8% and.% fo Mode I and Mode II espectvely. The esults demonstated the capablty of usng macoscopc factue paamete to descbe the atomc factue va atomstc smulaton. Table 4. Compason of the enegy elease ate obtaned by atomstc smulaton and contnuum theoetcal soluton. K I =.3 MPa m / K II =. MPa m / G smulaton.35.5 G LEFM Relatve Eo.8%.% Fgue 4.8 gves the dstbuton of atomc stess σ n font of a cack n Mode I wth the compason of contnuum stess fom lnea elastc soluton. Fgue 4.9 s the compason between atomc stess and contnuum stess of τ n Mode II. In these two fgues the atomc stess agees emakably well wth the contnuum stess. At the cack tp the atomc stess dstbuton yelds moe easonable descpton than the spuous sngulaty of the stess feld by LEFM. Theefoe, the macoscopc factue paametes that chaactezed the factue phenomenon ae consstent wth each othe unde small defomaton both n atomcscale and macoscopc contnuum model. Thus, the descpton on the atomstc aspect of factue can be lnked to the descpton of macoscopc factue. 6

75 σ ( Pa K I / π Fg. 4.8 Stess dstbuton of σ n font of a cack unde emote K I feld. τ ( Pa K II / π Fg. 4.9 Stess dstbuton of τ n font of a cack unde emote K II feld. 6

76 4.4. Numecal Results of Case II Fgues 4., 4., and 4. ae the close-up vews of the defomaton confguaton nea the cack tp unde the thee knds of extenal appled felds descbed befoe. All the atoms coodnates n the fgues ae based on the numecal esults but wth pope enlagng atos fo the convenence of vew. The cack tp s located at the cente between atom and j. The sold lne and boken lne efe to the defomed and undefomed confguatons espectvely. Fo mode I the defomaton of the suface of a cack obtaned by LEFM s dsplayed as bold-dotted lne n Fgue 4.. Note that n compason wth the defomed shape of the cack sufaces fom atomstc smulaton the defomed cack sufaces obtaned fom LEFM ae fomed by addng the equlbum atomc sepaaton to the esultng cack openng dsplacements. j Fg. 4. Close-up vew of undefomed and defomed confguatons nea the cack tp n mode I. (Bold-dotted lne s the defomed cack suface fom LEFM 6

77 θ Fg. 4. Close-up vew of undefomed and defomed confguatons nea the cack tp n mode II. j Fg. 4. Close-up vew of undefomed and defomed confguatons nea the cack tp n smple shea mode. 63

78 Fgues 4.3, 4.4 and 4.5 show the nomalzed stan enegy elease ates G /σ and G /τ vesus cack length by thee appoaches, whch ae global enegy appoach, local foce appoach and LEFM espectvely. σ (o τ = F / A, whee F s the extenal foce appled on the system esultng fom the fxed-gp pescbed dsplacements and A s the aea of the coss-secton of gaphene sheet. Fom the fgues, a good ageement s eached between the esults fom atomstc smulaton and those fom LEFM. In Fgue 4.3, the values of enegy elease ate obtaned by local foce method s slghtly less than the othe two cuves, whch mples a possble local gd body otaton occus aound the cack tp to make consdeable numecal eo when calculatng the elatve dsplacements of atoms. The detaled factue paametes ae also tabulated n the followng tables. In Fgue 4.6, the defomed cack suface pofle n mode I fom atomstc smulaton and LEFM s shown fo the ntal cack length a = 6.53 nm. As expected they also match well. These esults ndcate that the factue paametes calculated fom LEFM can be also pedcted fom atomstc smulaton. The atomstc smulaton s not nvolved wth the unphyscal sngulates of the stess feld nea cack tp as LEFM. Moeove, t also shed lght on the pope theoetcal descpton of both the nonlneates n the vcnty of the cack and the bond beakng mechansm between atom pas though atomstc smulaton whle the conventonal contnuum appoach would be ultmately lmted. 64

79 / σ G ( m/pa Fg. 4.3 G /σ vesus cack length n mode I. G ( m /Pa /τ Fg. 4.4 G /τ vesus cack length n mode II. 65

80 /τ G ( m/pa Fg. 4.5 G /τ vesus cack length n smple shea mode. Fg. 4.6 Cack suface pofle unde Mode I fom atomstc smulaton and LEFM.(The cack length s 6.53 nm. 66

81 G /σ Cack (* Length (nm Table 4. The numecal esults of Global Enegy Method Atomstc Smulaton G / σ n mode I. Local Foce Method LEFM Table 4.3 The numecal esults of G /τ n mode II. G / τ Atomstc Smulaton Cack (* Global Enegy Local Foce LEFM Length (nm Method Method

82 Table 4.4 The numecal esults of G /τ n smple shea mode. G / τ Atomstc Smulaton Cack (* Global Enegy Local Foce LEFM Length (nm Method Method

83 5 J-INTEGRAL IN ATOMIC SYSTEM In ths chapte, the appoach of calculatng J-ntegal n a specfed doman s developed. By mechancal molecula smulaton the values of J-ntegal fo both cack n a sem-fnte gaphene sheet and cack n the sheet wth fnte wdth ae obtaned unde lnea elastc condton. The nonlnea stess and stess elatonshp of a gaphene sheet s evaluated and ftted wth a powe law cuve. Afte modfed the cut-off functon n the Tesoff-Benne potental, a sees of the ctcal value of J-ntegal, compaed wth the elated efeences. J c, s calculated. The esults s befly 5. J-Integal Enegy elease ate and stess ntensty facto ae nomally only used n lnea elastc mateal. Fo the nonlnea mateal, altenatve factue paametes ae equed to chaacteze the factue behavo. J-ntegal, whch s equvalent to the stan enegy elease ate unde lnea elastc condton, s one of the most epesentatve paametes to evaluate the cack popetes n the nonlnea contnuum meda (Rce and Rosengen, 968; Hutchnson, 968. As shown n Fgue 5., J-ntegal s a closed contou ntegal of the stan enegy densty and wok done by tactons on the contou, as followng, J u = ( W δ j σ n j dγ (5. Γ x 69

84 whee W s the stan enegy densty, σ and u ae the stess and dsplacement espectvely. Subscpton n the above equaton denotes the decton paallelng to the cack, whch s the x decton hee. n x x Γ Fg. 5. Typcal contou fo the path-ndependent J-ntegal. Γ 3 S Γ Γ Γ 4 Fg. 5. Contou fo the J-ntegal n atomc system. 7