Non-singular dislocation fields

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1 Hme Search Cllectins Jurnals Abut Cntact us My IOPscience Nn-singular dislcatin fields This article has been dwnladed frm IOPscience. Please scrll dwn t see the full text article. 009 IOP Cnf. Ser.: Mater. Sci. Eng ( View the table f cntents fr this issue, r g t the jurnal hmepage fr mre Dwnlad details: IP Address: The article was dwnladed n 1/1/010 at 11:14 Please nte that terms and cnditins apply.

2 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/0106 Nn-singular dislcatin fields Elias C. Aifantis abratry f Mechanics and Materials, Faculty f Engineering, Aristtle University f Thessalniki, GR-5414, Thessalniki, Greece and Center fr Mechanics f Materials, Michigan Technlgical University, Hughtn MI 49931, USA mm@mm.gen.auth.gr Abstract. Nn-singular slutins fr dislcatin and disclinatin fields have recently been btained by the authr and his c-wrkers by using a rbust mdel f gradient elasticity thery. These slutins, whse frm is simple and easy t implement, are btained by reducing the gradient elasticity prblem t a crrespnding linear elasticity bundary value prblem thrugh the slutins f an inhmgeneus Helmhltz equatin where the surce term is the classical singular slutin. The aplacian in the Helmhltz equatin, invlving the extra gradient cefficient, prduces a new term in the gradient slutin which asympttically appraches the negative f the classical elasticity slutin n the dislcatin line. Thus, the singularity is eliminated and an arbitrary estimate f the dislcatin cre size intrduced in classical thery, is nt required. These predictins are tested against atmistic calculatins and their implicatins t varius dislcatin related cnfiguratins are discussed. Due t the simple and elegant frm f these slutins, it is hped that they will be useful in discrete dislcatin dynamics simulatins. 1. Intrductin Higher-rder elasticity theries have been intrduced as early as in the 18th Century (Cauchy, Vigt) and later in the 19th Century (brthers Csserat) as discussed briefly in [1]. In the 1960 s-70 s perid, a large number f significant cntributins were made in this field (Tupin, Rivlin, Mindlin, Krner, Kunin, Eringen) with mst ntable amng them, fr the present purpses, the 1965 Mindlin s strain gradient thery. Fr a literature accunt f these early cntributins the reader may cnsult [1] and the references quted therein. These theries, in their mst general frm, included a prhibitively large number f cnstants (smetimes ver 1000) and the slutin f bundary value prblems was nt a realistic task t undertake. Mst f them were wrked ut fr wave prpagatin studies and fr mdeling dispersin effects. Even the famus - and still ppular tday - Mindlin s simplest thery invlved 5 new cnstants which were nt nly difficult t evaluate but they were als cmpeting each ther withut being able, amng ther things, t eliminate singularities frm dislcatin fields. It was nt until 199 that this task was cnveniently accmplished thrugh the authr s gradient elasticity thery (Gradela), invlving nly ne new extra cnstant c with a direct physical interpretatin. Determinatin f this cnstant (cmmnly knwn as gradient cefficient) frm c 009 td 1

3 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/0106 atmistic mdels, as well as experimental data frm wave prpagatin, size effect and finite dislcatin and crack stress/strain field studies is pssible, even thugh much mre wrk in this directin is required. Since the authr s 199 paper [1] (fr a recent review see als []), ver 500 articles have been written in the perid using Gradela as a basis. Nevertheless, the slutin f bundary value prblems remained as a difficult task in mst f these wrks, due t the cmplexity f the bundary cnditins derived frm variatinal frmalisms. A simple methd t use Gradela fr the slutin f bundary value prblems is briefly utlined in the next sectin.. Gradela s Rbust Versin Fllwing [], we utline here the basics f Gradela as fllws: It turns ut that the stress and strain fields (, ) can be determined frm the inhmgeneus Helmhltz equatins c ; c, (1) where (, ) are the macrscpic fields and (, ) are the lcal gradient-dependent fields. The macrscpic fields (, ) are determined frm the classical cnstitutive equatin f Hke s law tr 1 G, () where (,G ) are the amé cnstants. The derivatin f Eq. (1) frm the general gradient elasticity thery is established by adpting a prcedure similar t that f Ru and Aifantis [3] but a mre rbust and direct prcedure is pssible as discussed elsewhere. The slutin f bundary value prblems nw prceeds as fllws: First we determine (, ) frm classical elasticity and the crrespnding bundary value prblem. Next we determine (, ) frm the inhmgeneus Helmhltz equatin with self-cnsistent bundary cnditins; a task which will als be discussed, in general, elsewhere. In the next sectin we give a direct applicatin f the methd t determine the stress and strain field f a screw dislcatin. 3. Screw Dislcatin On the basis f Eq. (1) we write the crrespnding expressin fr yz cmpnent f the strain tensr fr a screw dislcatin within the Gradela framewrk, i.e. 0 c, (3) yz yz yz 0 extra 0 and express, furthermre, the slutin yz as the sum yz yz yz. The classical term yz is fund frm Hke s law f Eq. () and the crrespnding elastic bundary value prblem, as 0 yz (b 4 )(cs r) (b 4 )(x r ), where b dentes Burgers vectr magnitude and (r,θ) are the usual plar crdinates. Upn substitutin f the preceding tw expressins in Eq. (3) we btain extra extra yz cyz 0. (4) In general, the abve (hmgeneus) Helmhltz equatin admits slutins, thrugh the separatin f variables technique, f the frm extra r r yz Acs nbsin ni K n n. (5) c c where (I,K) dente mdified Bessel functins f the first and secnd kind, respectively, and (Α,Β,Γ,Δ) extra are cnstants. By impsing the cnditin lim 0, we have 0. Mrever, in view f the r yz symmetries invlved in the present cnfiguratin f a screw dislcatin, we may set B=0 and n=1. Next we nte that the cnditin lim yz 0 implies A b 4 c r0 extra Thus, we find that [ b (4 c)]k (r c)cs and, therefre, yz 1 lim K r c c r., since 1 r0

4 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/0106 b x x r bcs 1 1 r yz K 1 r yz K1 4 r r c c 4 r. (6) c c We nte again that fr r 0, K1 r c c r and, thus, yz vanishes n the dislcatin line. Similar expressins we btain easily fr xz and the crrespnding stress cmpnents, as well as fr the strain energy. 4. Dislcatin Self-Energies Partial dislcatins in Wurtzite GaN were cnsidered [4] thugh cntinuum and atmistic calculatins by using respectively gradient elasticity and a mdified Stillinger-Weber ptential f the frm 4 r ij d rij 1 k e, rij d V r ij 0, rij d ; bdy term, (7) and V r,r, 1 1 rij d rik d ke cs ; 3 bdy term, (8) 3 ij ik ijk ijk 0 where the varius cefficients are cnstants and the rest f the symbls have their usual meaning (see, fr example, [4c] fr details f atmistics). Figure 1a shws a partial be edge dislcatin in GaN. The GaN structure invlves hcp superlattices with lattice cnstants a 3.A, and a 5.A. The far displacement field was cmputed by using anistrpic elasticity and i the calculatin f energies was established first fr each atm ( V ) and then fr a supercell S i ( V V ) invlving atms. The quench mlecular dynamics methd f Verlet [5] was emplyed with peridic bundary cnditins and the defect energy was calculated frm the frmula S S Wd VDislcated VPerfect. Figure 1b shws the simulatin results fr the self-energy W stred in a regin bunded by a cylinder f radius R fr an edge partial dislcatin delineating a I 1 stacking fault frmed by a precipitated interstitial lp. [arge symbls dente N-plarity, whereas small symbls dente Ga-plarity.] Stacking Fault (SF) simulatin identifying the defect SF sequence G S K J H I GaN bnds Burgers Circuit We (ev/a) a) b) lnr Figure 1: a) GaN dislcated supercell; b) Self-energy W stred in a cylindrical regin f radius R fr edge partial dislcatins 3

5 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/0106 Figure shws the plts f the crrespnding analytical results btained frm Gradela based n the relevant strain energy expressin. This expressin is nn-singular and reads Gb R E 1 R c R c W ln K0 K 1, 4 1 c c R (9) c R Gb R E 1 which fr R reduces t the simpler expressin W ln 4 1 c, where E dentes the Euler s cnstant. [It shuld be nted that the term 1 in this last expressin and a crrespnding term in Eq. (9) shuld be replaced, in general, with a term which depends n the Pissn s rati. This term is apprximately 1 fr the mdel f Eq. (1) fr an edge dislcatin in the limiting case f R.] We (ev/a) Atmistic Gradela Elasticity R We (ev/a) We (ev/a) R R Figure : Cmparisn f self-energy calculated frm Gradela and classical elasticity fr three edge partial dislcatin cnfiguratins Frm these results we nte that the gradient cefficient c (r internal length cre 0 c ) varies in the range c0.. A, and the fllwing invariant relatins als hld: W c r ev A, g W (b) c b ev A. [Mre details n experimental and atmistic simulatin cnsideratins cntained in this and the next sectin can be fund in the articles listed in [4] as btained by the electrn micrscpy grup f Aristtle University.] 5. Dislcatin Cres and Dislcatin Density Tensr Recently, high reslutin transmissin electrn micrscpy and image prcessing, alng with circuit mapping techniques were used t btain, thrugh a Gemetric Phase Analysis (GPA), dislcatin cre cnfiguratin. GPA enables t recrd the displacement fields thrugh which we can calculate the 4

6 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/0106 crrespnding defrmatin/distrtin fields and, therefre, t cmpare the relevant strain cmpnents and the Dislcatin Density Tensr (DDT) thrugh the familiar expressin: 0 utside cre, ; (10) 0 including cre. It is well knwn that elasticity thery leads again t singularities. Fr example, fr a screw b x y, dislcatin, the apprpriate cmpnent f the dislcatin density tensr reads where b z is the Burgers vectr. Fr the same dislcatin cnfiguratin, applicatin f Gradela in the frm 1 ; 1 c, gives z b z ( 1)K 0(r 1) ; which is still singular, but smther than the -functin. T btain results enabling cmparisn with experiments we use a mdified higher-rder Gradela mdel which reads (11) The crrespnding slutin fr the apprpriate cmpnent f the dislcatin density tensr is nw btained as 4 (b )(1 c c )[K (r c ) K (r c )] ; c c, c c. (1) z z It then turns ut that fr c1 c, we have bz r r z K 3 1. (13) This suggests that fr r0z b z (4 ), i.e. the relevant cmpnent f the dislcatin density tensr is finite n the dislcatin line. Figure 3 is brrwed frm recent wrk f the authr and his cwrkers [4b]. It shws a gd cmparisn between the predictins f Gradela and the experimental measurements, as depicted in the cmpanin Table. z z a) b) Figure 3: a) HRTEM experimental image f an edge partial dislcatin bunding a I1 SF (110 prjectin). The intensity peaks crrespnding t the psitins f atmic clumns have been marked with dts. The dislcatin cre (5/7- r 1-atm rings cnfiguratin) which has been identified by bth GPA and peak finding is indicated; the stacking sequence... ABABCAC...acrss the SF is als indicated. b) Crrespnding dislcatin density tensr cmpnents a x and a y (3D representatins and in-plane prjectins) btained by GPA with g = 1010 and g = 000, respectively, using masks f radius equal t g/4 arund the Bragg spts in Furier space 5

7 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/0106 Table: DDT Measurements - Dislcatin cre 5/7 ring (nm) 8 ring (nm) Experimental Calculated X-ray ine Prfile Analysis The Furier Transfrm (FT) f the line prfile, designated by A(), is given, accrding t Warren and Averbach [6], by the expressin lg A() lg A () g, (14) s where is the Furier variable, g is the abslute value f the diffractin vectr and A s is the size Furier transfrm, while dentes as usual the mean square strain. Fr randm atmic displacements is cnstant. Fr randmly distributed dislcatins Krivglaz and Ryabshapka [7] btained the expressin b ClgD, (15) i.e. the K-R frmula valid fr small values nly (D is the crystallite size, is the dislcatin density). Wilkens [8] imprved the K-R frmula by calculating the mean strain fr the entire range. The same lgarithmic term as in the K-R frmula hlds but it des nt diverge with crystallite size, * depending n a crrelatin length parameter R e utsize f which there is n dislcatin interactin * ( R e dentes the radius f a tube with dislcatin density ). Accrding t the cntinuum thery fr linear defrmatins, the lngitudinal strain parallel t the directin f g is 1 g g g g 1 g g g g, r u r u r rs ds g g g g g g g where u dentes displacement and strain, while the mean square strain is fund frm the relatin V, g g, r dv dv. (17) In the case f the (100) reflectin, we btain 1 g, r xx rsexds fr any 0, and g r r 0, fr = 0. xx V The strain functin fr edge dislcatins can be cmputed by using Gradela (e.g. []). It turns ut, in particular, that the xx cmpnent f the strain tensr crrespnding t an edge dislcatin with Burgers vectr b b e is x b 1 r x b xx y y 4 y r 13x y, (19) 4 1 r 1 (16) (18) 6

8 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/ c where 1 K1r c, K 4 r c r 3 c r r, r x y. Figure 4 depicts the variatin f with respect t lg as btained thrugh the classical mdel and a lgarithmic apprximatin, thrugh the gradient mdel, and thrugh measurements fr plycrystalline Cu. The crssing f the experimental curve ver the Gradela curve is mst likely due t the experimental uncertainty. The mre accurate data shwn in Figure 5 fr a Cu single crystal result t an experimental curve which seems t have the same trends as the Gradela theretical curve. The X-ray line prfile fr the defrmed Cu single crystal given in Figure 5a depicts the measured line prfile f the (111) reflectin f a defrmed single crystal Cu sample: the (cunt) intensity I is pltted as a functin f K K0, where K sin and K 0 is the K value at the exact Bragg psitin. The intensity scale is lgarithmic. The crrespnding predictin frm Gradela is given in Figure 5b, i.e. the mean square strain as a functin f lg, as determined experimentally fr defrmed Cu single crystal by FT. It is nted that btained this way is nt singular, but it tends t a finite value fr 0. These and the rest f the results listed in this sectin, which are als reviewed in a recent article by the authr and c-wrkers [9], will be discussed in detail in [10]. lg Figure 4: Variatins f with respect t lg Intensity I a) b) K K 0 lg Figure 5: (a) X-ray line prfile fr a defrmed Cu single crystal determined experimentally; (b) Crrespnding graph determined theretically frm Gradela 7

9 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/0106 The results depicted in Figures 4 and 5 suggest that careful measurements seem t validate the predictins f Gradela. T supprt this further we prvide in Figures 6 and 7 results pertaining t plycrystals with ultrafine grain size prduced by the ECAP methd. As already mentined, sme f these results are included in a recent article n applicatins f gradient thery [9], but an extended discussin fcussing n the applicatin f Gradela t X-ray line prfile analysis is frthcming [10] where details n the relevance f Figures 6 and 7 will be given. Intensity I Mean square strain a) K K 0 b) lg Figure 6: a) X-ray line prfile fr ECAP Cu plycrystal; b) Crrespnding graph Intensity I bg crrect + bg raised x bg lwered Mean square strain bg crrect + bg raised x bg lwered K K 0 Figure 7: The effect f Backgrund (bg) I vs. K K0 data n the lg vs. lg plts 7. Revisiting Dislcatin Thery 7.1. Other Nn-singular Dislcatin Mdels It is pinted ut that nn-singular dislcatin mdels have been prpsed in the literature befre. Reference is made, in particular, t the classical Peirls-Nabarr atmistic mdel, t i s hllw dislcatin mdel, and t a mre recent Cai/Arsenlis/Weinberger/Bulatv dislcatin cre-spreading (r b-spreading) mdel [11]. 8

10 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/0106 It turns ut that the shear stress σ (x,0) Gb (1 ν) same peak stress fr these mdels varies accrding t x x (a 4(1ν) ) fr the Peirls-Nabarr; 3 3 1x r0 x xy n the glide plane under the assumptin f the fllwing distributins: fr i s; 1x4c x 3 (x)k x c fr Gradela; x(x α ) fr Cai/Arsenlis/Weinberger/Bulatv (CAWB) mdels (with r 0.6a, 0 α 0.76a where a dentes the lattice parameter). A cmparisn between the afrementined nnsingular dislcatin mdels is given in Figure 8a. xy x,0 Gb a 1 Hllw Dislcatin (i) Peierls-Nabarr Gradela cre-spreading (CAWB) /G d* ~ 9 nm a) xa b) d/b Figure 8: a) Cmparisn f varius nn-singular dislcatin mdels (fr the same maximum stress); b) Qualitative plt f the dimensinless stress quantity G ver the dimensinless grain size quantity db 7.. Image Frce Inverse Hall Petch Behavir By fllwing the same prcedure as in Sectin 4, the self-energy per unit length f a screw dislcatin within the gradient thery f micr/nanelasticity turns ut t be given by the expressin Gb R E R W ln K0 4, (0) c c where R is the radial crdinate defining the material vlume surrunding the dislcatin line E cnsidered, is the Euler cnstant, K 0 dentes the apprpriate Bessel functin, and the rest f the symbls have their usual meaning. It is emphasized again that the self-energy is nt singular as in classical elasticity and the necessity f intrducing an arbitrary dislcatin cre parameter fr dispensing with such singularity is eliminated. In fact, the gradient cefficient c (its square rt dentes the relevant material internal length) prvides a new pssibility t accunt fr dislcatin cre effects as discussed in previus sectins, as well as in [] and the related bibligraphy listed therein. On using the abve gradient mdificatin f the self-energy in cnjunctin with a standard image frce argument advanced in [1] (see als [13]) based n classical elasticity thery, we btain the fllwing relatin Gb 1 1 b K d 1 d, (1) c c fr a dislcatin sitting at the center f a grain f diameter d; the secnd term f the l.h.s. invlving the Bessel functin is due t the gradient elasticity effect. The (qualitative) plt f the dimensinless stress quantity G ver the dimensinless grain size quantity db (Figure 8b) has a rising and a descending branch with a maximum ccurring at a critical grain size at the nanmeter regime. Thus, in principle, this plt may be used fr establishing anther 9

11 Dislcatins 008 IOP Cnf. Series: Materials Science and Engineering 3 (009) 0106 di: / x/3/1/0106 interesting interpretatin f the standard and inverse Hall-Petch behavir as it will be discussed elsewhere. Acknwledgements The supprt f EC under RTN DEFINO HPRN-CT and f the US Natinal Science Fundatin under NIRT Prgram (NIRT Grant DMI-05330) is acknwledged. The supprt f the Greek Gvernment under the PENED and PYTHAGORAS prgrams is als acknwledged. Useful discussins and the generus cperatin with the Electrn Micrscpy abratry f Aristtle University headed by Prfessr Th. Karakstas is gratefully acknwledged. The same hlds fr the X- ray line prfile analysis grup f Etvs University headed by Prfessr T. Ungar. Useful discussins with I. Knstantpuls, J. Kiseglu and G.Ribarik (all pst-graduate researchers supprted by the afrementined prjects) are gratefully acknwledged, alng with the assistance f my clleague A. Knstantinidis wh checked the manuscript and made helpful suggestins. References [1] E.C. Aifantis, On the rle f gradients in the lcalizatin f defrmatin and fracture, Int. J. Engrg. Sci. 30 (199) [] E.C. Aifantis, Update n a class f gradient theries, Mech. Mater. 35 (003) [3] C.Q. Ru and E.C. Aifantis, A simple apprach t slve bundary value prblems in gradient elasticity, Acta Mechanica 101 (1993) (1993). [4] J. Kiseglu, G.P. Dimitrakpuls, Ph. Kmninu, Th. Karakstas, I. Knstantpuls, M. Avlnitis and E.C. Aifantis, Analysis f partial dislcatins in wurtzite GaN using gradient elasticity, Phys. Status Slidi A 03 (006) [see als: J. Kiseglu, G.P. Dimitrakpuls, Ph. Kmninu, Th. Karakstas and E.C. Aifantis, Dislcatin cre investigatin by gemetric phase analysis and the dislcatin density tensr, J. Physics D: Appl. Physics 41 (008) /1-8; J. Kiseglu, G.P. Dimitrakpuls, Ph. Kmninu and Th. Karakstas, Atmic structures and energies f partial dislcatins in wurtzite GaN, Phys. Rev. B 70 (004) /1-1.] [5]. Verlet, Cmputer experiments n classical fluids. I. Thermdynamical prperties f ennard-jnes mlecules, Phys.Rev. 159 (1967) [6] B.E. Warren and B.. Averbach, The Separatin f cld-wrk distrtin and particle size bradening in X-ray patterns J. Appl. Phys. 3 (195) [7] M.A. Krivglaz and K.P. Ryabshapka, Thery f scattering f X-rays and thermal neutrns by real crystals, Fizika Metall. 15 (1963) [8] M. Wilkens, Theretical aspects f kinematical X-ray diffractin prfiles frm crystals cntaining dislcatin distributins, in: Fundamental Aspects f Dislcatin Thery, J.A. Simmns, R. dewit and R. Bullugh (Eds.), Vl. II, pp , Nat. Bur. Stand. (USA) Spec. Publ. N. 317., Washingtn, DC, USA (1970). [9] J. Kiseglu, I. Knstantpuls, G. Ribarik, G.P. Dimitrakpuls and E.C. Aifantis, Nnsingular dislcatin and crack fields: Implicatins t small vlumes, Micrrsyst. Technl. 15 (009) [10] G. Ribarik, T. Ungar and E.C. Aifantis, X-ray line prfile analysis thrugh gradient elasticity, frthcming. [11] W. Cai, A. Arsenlis, C.R. Weinberger and V.V. Bulatv, A nn-singular cntinuum thery f dislcatins, J. Mech. Phys. Slids 54 (006) [1] E.C. Aifantis, W.W. Milligan and S.A. Hackney, Final AFOSR Reprt (F & F ), 000. [13] S. Hackney, M. Ke, W.W. Milligan and E.C. Aifantis, Grain size and strain rate effects n the mechanisms f defrmatin and fracture in nanstructured metals, in: Prcessing and Prperties f Nancrystalline Materials, C. Suryanarayana et al (Eds.), pp , TMS, Warrendale, PA (1996). 10