The static stress needed to depin a 2D edge dislocation, the lower dynamic
stress needed to keep it moving, its velocity and displacement vector profile
are calculated from first principles. We use a simplified discrete model whose
far field distortion tensor decays algebraically with distance as in the usual
elasticity. An analytical description of dislocation depinning in the strongly
overdamped case (including the effect of fluctuations) is also given. A set of
$N$ parallel edge dislocations whose centers are far from each other can depin
a given one provided $N=O(L)$, where $L$ is the average inter-dislocation
distance divided by the Burgers vector of a single dislocation. Then a limiting
dislocation density can be defined and calculated in simple cases.Comment: 10 pages, 3 eps figures, Revtex 4. Final version, corrected minor
  error

A. Carpio

A. I. Landau

D. Hull

F. R. N. Nabarro

J. P. Hirth

L. L. Bonilla

L. B. Freund

L. D. Landau

N. G. van Kampen

English

arXiv

Crossref

Edge Dislocations in Crystal Structures Considered as Traveling Waves in Discrete Models

The static stress needed to depin a 2D edge dislocation, the lower dynamic stress needed to keep it moving, its velocity, and displacement vector profile are calculated from first principles. We use a simplified discrete model whose far field distortion tensor decays algebraically with distance as in the usual elasticity. Dislocation depinning in the strongly overdamped case (including the effect of fluctuations) is analytically described. N parallel edge dislocations whose average interdislocation distance divided by the Burgers vector of a single dislocation is L≫1 can depin a given one if N=O(L). Then a limiting dislocation density can be defined and calculated in simple cases.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Carpio Rodríguez, Ana María

Bonilla, Luis L.

Docta Complutense

P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2003VOLUME 90, NUMBER 13Edge Dislocations in Crystal Structures Considered as Traveling Waves in Discrete ModelsA. CarpioDepartamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, SpainL. L. BonillaDepartamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid,Avenida de la Universidad 30, 28911 Leganés, Spain(Received 21 May 2002; published 1 April 2003)135502-1The static stress needed to depin a 2D edge dislocation, the lower dynamic stress needed to keep itmoving, its velocity, and displacement vector profile are calculated from first principles. We use asimplified discrete model whose far field distortion tensor decays algebraically with distance as in theusual elasticity. Dislocation depinning in the strongly overdamped case (including the effect offluctuations) is analytically described. N parallel edge dislocations whose average interdislocationdistance divided by the Burgers vector of a single dislocation is L 1 can depin a given one if N OL. Then a limiting dislocation density can be defined and calculated in simple cases.DOI: 10.1103/PhysRevLett.90.135502 PACS numbers: 61.72.Bb, 05.45.–a, 45.05.+x, 82.40.Bjments in the standard treatment [9]. Then a macroscopic elastic interaction between nearest neighbors within theIn many fields, genuinely microscopic phenomena af-fect macroscopic behavior in a way that is difficult toquantify precisely. Typical cases are the motions of dis-locations [1,2], cracks [3], vortices in Josephson arrays[4], or other defects subject to pinning due to the under-lying crystal microstructure. Emerging behavior due tomotion and interaction of defects might explain commonbut poorly understood phenomena such as friction [5].Macroscopic theories consider the continuum mechanicsof these solids subject to forces due to the defects andadditional equations for the densities of defects and prop-erties of their motion [6]. The latter are usually postulatedby phenomenological considerations. An important prob-lem is to derive a consistent macroscopic descriptiontaking into account the microstructure.Here we tackle a simplified problem containing allthe ingredients of the previous description: the pinningand motion of edge dislocations. First, we study a two-dimensional (2D) discrete model [7] describing thedamped displacement of atoms subject to the field gen-erated by a 2D edge dislocation and a constant appliedshear stress of strength F. If jFj<Fcs (Fcs is related tothe static Peierls stress), the stable displacement field isstationary, whereas the dislocation core and its surround-ing displacement field move if jFj > Fcs. A crucial ob-servation is that there exists a stable uniformly movingdislocation with both core and far field advancing at thesame constant velocity. This suggests that a moving edgedislocation is a traveling wave of the discrete model.Our self-consistent calculation [8] based on this picturepredicts the following magnitudes: (i) the critical staticstress needed to depin a static dislocation, (ii) the dy-namic stress Fcd < Fcs below which a moving dislocationstops (in the strongly overdamped case, Fcd  Fcs), and(iii) the dislocation velocity as a function of appliedstress. The latter information has to be taken from experi-0031-9007=03=90(13)=135502(4)$20.00quantity, the dislocation velocity, is obtained from analy-sis of a microscopic model.Second, we consider a distribution of many paralleledge dislocations separated by macroscopic distancescomprising many lattice periods. A dislocation cannotmove under the influence of other dislocations far awayunless the latter have finite density [there are N suchdislocations and the average distance between them is Lwith N  OL as L! 1]. Under the influence of such adistribution, one dislocation may be pinned or movedepending on the dislocation density. The latter is calcu-lated at the critical stress in a simple configuration.A simplified discrete model of edge dislocations.—Consider an infinite three dimensional cubic lattice withsymmetry axes x; y; z. We insert an extra half plane ofatoms parallel to the plane yz. The border of this extrahalf plane is a line (parallel to the z axis), which is calledan edge dislocation in the crystal. The Burgers vector ofthe dislocation points in the x direction and the plane xz isthe glide plane of the dislocation (see [6]). If we apply anexternal stress , the dislocation moves on its glide planeand in the direction of its Burgers vector in response tojust one component of: the stress  resolved on the glideplane in the glide direction [2,6]. All the sections of thelattice by planes parallel to xy look alike. Thus, we reducethe problem to a 2D lattice with an extra half line ofatoms; see Fig. 22 of Ref. [6]. Assuming that glide ispossible only in the x direction, the dynamics of anedge dislocation in a 2D lattice can be described by [7]md2ui;jdt2dui;jdt ui1;j 	 2ui;j  ui	1;j Asinui;j1 	 ui;j sinui;j	1 	 ui;j: (1)The lattice is a collection of chains in the x direction with 2003 The American Physical Society 135502-1-10-50510-5050246yxu i,jFIG. 1. Displacement field profile for the stationary edgedislocation with A  1 and N  50.P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2003VOLUME 90, NUMBER 13same chain and sinusoidal interaction between chains.ui;j=2 is the dimensionless displacement of atomi; j in the x direction measured in units of the Burgersvector length b. A > 0 measures the relative strengths ofthe nonlinear forces exerted by atoms on different planesy  k (constant) and the linear forces exerted within anyplane y  k. The dimensionless parameter A also deter-mines the width of the dislocation core (1=Ap). Finally,the time unit is the ratio between the friction coefficientand the spring constant in the x direction. Then m is thedimensionless ratio between the atomic mass times thespring coefficient and the square of the friction coeffi-cient. In dislocation dynamics, an important case is thatof overdamped dynamics, m  0 [10]. Equation (1) canbe generalized to a vector model having a displacementvector uij; vij and a continuum limit yielding the 2DNavier equations with cubic symmetry [11]. Such a modelhas among its solutions edge dislocations with Burgersvectors in the x or y directions gliding in the directionthereof (which does not have to be assumed as in thepresent simple model). This model can also be solvedusing our methods at the expense of technical complica-tions and high computational cost.In this geometry, the far field of a static 2D edgedislocation is approximately given by the correspondingcontinuum elastic displacement uij  ux; y with x  i,y  j (where   b=L 1, i, j are large, and L is theappropriate mesoscopic length). Then the stationary so-lutions of Eq. (1) satisfy the equations of anisotropiclinear elasticity, uxx  Auyy  uxx  uYY  0 (Y y=Ap), far away from singularities and jumps [7].The solution corresponding to the edge dislocation isthe polar angle x; Y 2 0; 2, measured from thepositive x axis. Continuum approximations break downnear the dislocation core, which should be described bythe discrete model [12]. The advantage of Eq. (1) com-pared to other 2D generalizations [13] of the Frenkel-Kontorova model is that it yields the correct decay forstrains and stresses: r	1 as r2  x2  Y2 ! 1, instead ofexponential decay.Overdamped dynamics and static Peierls stress.—Weshall now study the structure of a static edge disloca-tion of Eq. (1), the critical stress needed to set it in motionand its subsequent speed. We solve numerically Eq. (1)with m  0 on a large lattice jij; jjj  N using Neumannboundary conditions (NBC) corresponding to applying ashear stress of strength F in the x direction. The (far field)continuum elastic displacement for a static 2D edge dis-location subject to such a shear stress is x; y=Ap  Fy.Then the NBC are uN1;j 	 uN;j  AN1;j	AN;j and ui;N1 	 ui;N  Ai;N1 	 Ai;N F, where Ai;j  i; j=Ap with 0; 0  =2. If F  0and the initial condition is the elastic far field Ai;j, thesystem relaxes to a stationary configuration ui;j. Thedislocation is expected to remain stationary for F  0135502-2unless jFj is larger than a critical value FcsA, related tothe so-called static Peierls stress [1,2]. Nonlinear stabilityof the stationary edge dislocation for jFj<Fcs was pro-ven in Ref. [14]. To test this picture, we solve numericallyEq. (1) in a large lattice, using NBC and the staticdislocation obtained for F  0 as initial data. For largetimes and jFj small, the system relaxes to a steady con-figuration ui;j which provides the structure of the core; seeFig. 1. When jFj is large enough, the dislocation is ob-served to glide in the x direction: to the right if F > 0,and to the left if F < 0.To calculate Fcs, we extend the depinning calculationsof Ref. [15] to 2D systems. We redefine ui;j  Ui;j  Fj,insert Ui;j  Ui;jF;A  vi;jt in Eq. (1) with m  0,and expand the resulting equation in powers of vi;j, aboutthe stationary state Ui;jA;F up to cubic terms.Subscripts in the resulting equation can be numberedwith a single one starting from the point i  j  	N:Ui;j  Uk and vi;j  vk, k  i j N2N  1 fori; j  	N; . . . ; N. The resulting equation can be writtenformally as dv=dt  MFvBv; v;F, where the vec-tor v has components vk. The linear stability of the sta-tionary state UkA;F depends on the eigenvalues of thematrix MF. These eigenvalues are all real negative forjFj<Fcs, whereas one of them vanishes at jFj  Fcs.This criterion allows us to numerically determine Fcs asa function of A; see Fig. 2(a). Notice that the critical stressincreases with A. Thus narrow core dislocations (A large)are harder to move.Dislocation velocity.—Let us assume that F > Fcs (thecase F <	Fcs is similar). Then v  !tr (plusterms that decay exponentially fast in time). Theprocedure sketched in Refs. [15,16] for discrete 1Dsystems yields the amplitude equation d!=dt " #!2. Here "  l M0FcsrF	 Fcs=l  r and#  l Br; r;Fcs=l  r, and r are the left and right135502-2-5-4-3-2-10-505246yζu(ζ,j)FIG. 3 (color online). Wave front profiles, ui;jt  u$; j,$  i	 ct, c > 0, near F  Fcs for A  3, m  0 and N  25.0.03 0.035 0.04 0.04500.0050.010.015Fc(F)(b)0 1 200.020.040.060.080.10.120.14AFc(A)(a)FIG. 2. (a) Static (squares, m  0) and dynamic (asterisks,m  0:5) critical stresses Fcs and Fcd versus A. (b) Theoretical(solid line, m  0) and numerical (squares, m  0; asterisks,m  0:5) dislocation velocity vs F (A  1, N  25).P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2003VOLUME 90, NUMBER 13eigenvectors of the matrix MFcs corresponding to itszero eigenvalue (its largest one). From the amplitudeequation, the approximate dislocation velocity is [15]:c"#p=  OjF	 Fcsj1=2. Numerically measuredand theoretically predicted dislocation velocities arecompared in Fig. 2(b). Calculations in lattices of differentsizes yield similar results.How do we calculate numerically the dislocation ve-locity? This is an important point for using the calcu-lated dislocation velocity as a function of stress inmesoscopic theories and a few comments are in order.If we solve numerically Eq. (1) with static NBC forjFj > Fcs, the velocity of the dislocation increases as itmoves towards the boundary. The dislocation acceleratesbecause we are using the far field of a steady dislocationas boundary condition, instead of the (more sensible) farfield of a moving dislocation. However, the latter is inprinciple unknown because we do not know the disloca-tion speed. We will assume nevertheless that the disloca-tion moves at constant speed c once it starts moving, as itwould in a stressed infinite system. Then the correctdislocation far field is i	 ct; j=Ap  Fj. With thisfar field in the NBC, Eq. (1) has traveling wave solutionsui;jt whose velocity can be calculated self-consistently.How? By an iterative procedure that adopts as initial trialvelocity that of a dislocation subject to static NBC as itstarts moving. Near threshold, steplike profiles are ob-served (see Fig. 3) that become smoother as F increases.The profiles have been calculated by following the tra-jectories of points with the same value of i and differentvalues of j for F > FcsA, according to the formulaui;jt  u$; j, $  i	 ct. Notice that the wave frontprofiles are kinks for j < 0 and antikinks for j  0.Influence of fluctuations.—The original discrete modelcontains both damping and fluctuation terms [7]. Fluc-tuation terms are appreciable only near Fcs, and contrib-ute an additive white noise term to the amplitude equa-tion. Because of this term, there is a small probability135502-3for the dislocation to move even if jFj<Fcs and m  0.The resulting average velocity can be estimated by ob-serving that the potential of the corresponding Fokker-Planck equation is cubic and it has a small bar-rier of height proportional to Fcs 	 jFj3=2. Then theexponentially small velocity of the dislocation underthe critical stress is the reciprocal of the mean escapetime from the barrier [17]. Provided Fcs 	 jFj  D(where D measures the noise strength), we have 	 lnc /Fcs 	 jFj3=2=D.Inertia and dynamic Peierls stress.—Inertia changesthe previous picture of dislocation motion in one impor-tant aspect: the dislocations keep moving for an intervalof stresses below the static Peierls stress, Fcd < jFj<Fcs.On this stress interval, stable solutions representing staticand moving dislocations coexist: to depin a static dislo-cation, we need jFj > Fcs. However, if jFj decreasesbelowFcs, a moving dislocation keeps moving until jFj<Fcd; see Fig. 2. Thus Fcd represents the dynamic Peierlsstress of the dislocation [1]. Our theory therefore yieldsthe static and the dynamic Peierls stresses and the veloc-ity of a dislocation.Interaction between edge dislocations.—Let us assumethat there are N static edge dislocations at the pointsxn; yn parallel to one dislocation at x0; y0, and thatall dislocations are separated from each other by dis-tances of order L 1 (measured in units of the Burgersvector length, b). We want to analyze whether the collec-tive influence of the N distant dislocations can move thatat x0; y0. This problem is similar to that of deriving areduced dynamics for the centers of 2D vortices ofGinzburg-Landau equations subject to their mutual in-fluence [18]. In the case of dislocations, the existence ofa pinning threshold implies that the reduced dynamicsis that of a single dislocation subject to the meanfield created by the others. We thus have a reduced fielddynamics, not particle dynamics as in the case of theGinzburg-Landau vortices.135502-3P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2003VOLUME 90, NUMBER 13Displacement vectors pose the problem of definingbranch cuts in the continuum (elastic) limit, which leadsus to consider instead the distortion tensor as a primaryquantity [6]. For our discrete system, the distortion tensorhas nonzero components w1i;j  ui1;j 	 ui;j and w2i;j sinui;j1 	 ui;j that become  @u=@x and  @u=@y,respectively, in the continuum limit   1=L! 0,x	 x0  i, y	 y0  j finite. In the continuum limit,the distortion tensor of an edge dislocation centeredat the origin has nonzero components w1  	Apy=Ax2  y2 and w2  Apx=Ax2  y2. If we have Nedge dislocations at xn; yn, 1  n  N, far from one atx0; y0, the distortion tensor is the sum of individualcontributions. Then the far field distortion tensor seenby the dislocation at x0; y0 isw1i;j  	ApjAi2  j2 F1; w2i;j ApiAi2  j2 F2; (2)F1  	XNn1Apy0 	 ynAx0 	 xn2  y0 	 yn2  . . . ; (3)F2  XNn1Apx0 	 xnAx0 	 xn2  y0 	 yn2  . . . : (4)The dislocation at x0; y0 moves if F2 > FcsA. Thiscannot be achieved as ! 0 unless N  O1=. Thenthe sums in Eqs. (3) and (4) become integrals. We definea static dislocation density )x; y as the limit ofN	1PNn1 *x	 xn*y	 yn as N ! 1. ThenF1  	"Z 1	1Z 1	1Apy0 	 y)x; yAx0 	 x2  y0 	 y2dxdy; (5)F2  "Z 1	1Z 1	1Apx0 	 x)x; yAx0 	 x2  y0 	 y2dxdy; (6)where "  N is the ratio of the total Burgers vector tothe mesoscopic length measuring average interdislocationdistance. As an example, let us assume that y0  yn  0.Then )  )x*y and F1  0. Let us assume that thedislocations are constrained by two obstacles at x0  land subject to the same critical stress. Then the criti-cal dislocation density is )x  1	 ApFcs=" x=l2 	 x2p, provided " >ApFcsl (cf. [6], p. 127).In conclusion, edge dislocations can be characterizedas traveling waves of discrete models. The dislocation farfield moves at a constant velocity equal to that of thedislocation core. Static and dynamic Peierls stresses andthe dislocation velocity as a function of applied stress canbe found numerically (or analytically near critical stressin the overdamped case) and adopted as the basis of a135502-4mesoscopic theory [10]. We have also shown that theinteraction between distant edge dislocations can be de-scribed in terms of a continuous dislocation density.This field-theoretical reduced description greatly con-trasts with the case of interacting point vortices, whichis completely described by the particle dynamics of thevortex centers [18]. Extension to fully vectorial modelsand to other types of dislocation should follow alongsimilar lines.We thank Joe Keller for helpful discussions. This workhas been supported by the MCyT Grant No. BFM2002-04127-C02, by the Third Regional Research Programof the Autonomous Region of Madrid (Strategic GroupsAction), and by the European Union under GrantNo. HPRN-CT-2002-00282.[1] F. R. N. Nabarro, Theory of Crystal Dislocations (OxfordUniversity Press, Oxford, 1967).[2] J. P. Hirth and J. Lothe, Theory of Dislocations (JohnWiley and Sons, New York, 1982).[3] L. B. Freund, Dynamic Fracture Mechanics (CambridgeUniversity Press, Cambridge, U.K., 1990).[4] H. S. J. van der Zant, T. P. Orlando, S. Watanabe, and S. H.Strogatz, Phys. Rev. Lett. 74, 174 (1995).[5] E. Gerde and M. Marder, Nature (London) 413, 285(2001). See also D. A. Kessler, Nature (London) 413,260 (2001).[6] L. D. Landau and E. M. Lifshitz, Theory of Elasticity(Pergamon Press, London, 1986), 3rd ed., Chap. 4.[7] A. I. Landau, A. S. Kovalev, and A. D. Kondratyuk, Phys.Status Solidi B 179, 373 (1993); A. I. Landau, Phys.Status Solidi B 183, 407 (1994).[8] The far field of a dislocation is not spatially homoge-neous, and it should be rigidly and self-consistentlydisplaced at the velocity of the dislocation core.[9] D. Hull and D. J. Bacon, Introduction to Dislocations(Butterworth-Heinemann, Oxford, U.K., 2001), 4th ed.[10] I. Groma and B. Bakó, Phys. Rev. Lett. 84, 1487 (2000).[11] A. Carpio and L. L. Bonilla (unpublished).[12] A. Carpio, S. J. Chapman, S. D. Howison, and J. R.Ockendon, Philos. Trans. R. Soc. London, Ser. A 355,2013 (1997).[13] P. S. Lomdahl and D. J. Srolovitz, Phys. Rev. Lett. 57,2702 (1986).[14] A. Carpio, Appl. Math. Lett. 15, 415 (2002).[15] A. Carpio and L. L. Bonilla, Phys. Rev. Lett. 86, 6034(2001).[16] A. Carpio, L. L. Bonilla, and G. Dell’Acqua, Phys. Rev. E64, 036204 (2001).[17] N. G. van Kampen, Stochastic Processes in Physics andChemistry (North-Holland, Amsterdam, 1981).[18] J. C. Neu, Physica (Amsterdam) 43D, 385 (1990); 43D,407 (1990); 43D, 421 (1990).135502-4

Physical Review Letters

Edge dislocations in crystal structures considered as traveling waves in discrete models

Edge dislocations in crystal structures considered as traveling waves of
  discrete models

Edge dislocations in crystal structures considered as traveling waves of discrete models

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