Interactions of glide dislocations in a channel of a persistent slip band

Transcription

1 Interactions of glide dislocations in a channel of a persistent slip band JOSEF KŘIŠŤAN JAN KRATOCHVÍL Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83, Prague, Czech Republic Czech Technical University, Faculty of Civil Engineering, Department of Physics, Thákurova 7, Prague, Czech Republic Abstract Two unlike dislocations gliding in parallel slip planes in a channel of a persistent slip band are considered. Initially they are kept apart in straight screw positions. As the dislocations are pushed by the applied stress between two walls in the opposite directions, they bow out and attract one another forming a dipole. With the increasing stress the dislocations become more and more curved, until they separate. The walls of the channel are represented by elastic fields of rigid edge dipoles. The dislocations are modeled as planar curves and their glide is simulated using the flowing volume method and the method of lines. The objective of the simulations is to determine the escape stress in the channel needed for the dislocations to escape one another. The stress and strain controlled regimes considered provide upper and lower estimates of the escape stress. The results are compared with the studies by Mughrabi & Pschenitzka (Phil. Mag (2005)) and Brown (Phil. Mag (2006)) and the recent dislocation dynamics estimates. Problems encountered in the dislocation dynamics evaluation of the escape stress are analysed. Keywords: plasticity, dislocations, line-tension 1 Introduction In conventional fatigue tests, the endurance limit is defined as the maximum stress achieved in saturated hysteresis stress-strain curves. Brown [2] has summarized four Corresponding author. kristan1@ .cz As shown by Mughrabi and Wang [1] the saturation stress lies actually below the endurance limit as the latter requires additionally a crack formation and their propagation. 1

2 possible contributions to the endurance limit: (i) the stress required to make screw dislocations of opposite sign pass one another during their shuttle in a channel of a persistent slip band (PSB); (ii) the stress required to bow-out the screw dislocations between the PSB walls; (iii) the internal stress resulting from inhomogeneous plastic deformation, caused by the resistance to plastic flow of the walls; (iv) the friction stress which might come from dislocation debris left between the walls by the shuttling glide dislocations. The model proposed in the present paper is used to simulate the escape stress consisting of the contributions (i) and (ii), as well as to estimate the influence of boundary conditions partly addressed in item (iii). In the current version of the model, the contribution (iv) is neglected. In reality, the contributions strongly influence each other. From theoretical and computer simulation points of view, the determination of the endurance limit means to formulate and solve a boundary value problem. In this context, the endurance limit is related to occurrence of the irreversible process leading to formation and operation of matured PSBs. In the fatigue test this situation corresponds to the saturated hysteresis loops. The boundary value problem means to consider a PSB sandwiched between two elastic halfspaces which approximate the parts of the crystal filled with a vein structure. Far from the interfaces between the PSB and the vein structure, the half-spaces can be thought of as being exposed to loading conditions of the specimen. The PSB has a composite structure of walls and channels. The plastic properties of the relatively rigid walls consisting mainly of edge dipolar loops are controlled by polarization of the loops, absorption of dislocations deposited at the walls, and the loop and dislocation annihilation. Much softer plastic properties of the channels are determined by dislocation glide. The evolving positions and shapes of the glide dislocations are governed by the equation of motion where each infinitesimal dislocation segment is subjected to the self-force, the interactions with segments of other dislocations, the short-range interaction with the edge dipolar loops of the walls, and the stress field in the channel imposed by the loading conditions. The stress field itself is strongly influenced by the dislocation motion. Such boundary value problem could be understood as an advance version of the very successful composite model of PSB. However, for a computer simulation the specification, formulation, and solution of the PSB boundary value problem presents a challenging task. In the present paper just a partial problem is studied: an encounter of two dislocations of opposite sign in a channel of matured PSB and the needed escape stress. Approximate analytical solutions of that problem involving the coupled contributions (i)-(iii) were presented in the recent papers by Brown [2, 3] and Mughrabi & Pschenitzka [4]. Similarly as in [5, 6], Brown [3] assumed that the bowing stress is equal to the passing stress, and that the two contributions to the escape stress should be added together; the contribution (iii) was neglected. To evaluate the escape stress, Brown [3] considered rigid screw dislocations of opposite signs that must escape from one another while at the same time they deposit edge dislocations in the walls between which they are confined. The escape of the rigid screw dislocations from one another is treated as splitting of an infinitely long screw dislocation dipole of height h. The minimum value of the dipole height, h c,which equals the maximum distance at which the unlike screw dislocations can annihilate one 2

3 another by cross-slip, determines the maximum stress achievable. In the present version of the simulation cross-slip is excluded. The rigid dislocation approximation has been modified by Mughrabi & Pschenitzka [4]. They considered two unlike elliptical dislocation segments stretched between two walls. The dislocations are pushed by an applied stress in the opposite directions till they attract one another forming a dipole. As the stress is increased from zero, the curvature of the elliptical segments grows, until the segments become tangent to the walls, whereupon they escape one another. The stress needed to accomplish this is interpreted as the escape stress. As a result, the bowing stress and passing stress do not add together, but only about 20% of the bowing stress adds to the passing stress. The results of Mughrabi & Pschenitzka [4] motivated Brown to further improve the model [2]. He considered the dislocation shape changes in the vicinity of the walls more carefully. He determined values of the radius of curvature of the dislocations near the walls independently of the curvature in the centers of the elliptical segments. Further on, he refined the model of the walls by introducing their dipolar structure. The elaborate evaluation of these refinements has led to the conclusion that about 50% of the bowing resistance adds to the passing stress, i. e. Brown reached a value higher than the 20% estimated by Mughrabi & Pschenitzka [4], but much lower than the original Brown s 100% adding assumption used in [3]. In Mughrabi & Pschenitzka s and Brown s studies [2, 3, 4], the escape stress has been estimated from the forces between dislocations of a prescribed geometrical configuration. In this paper, which is based on a discrete dislocation dynamics simulation, we follow the process of the encounter of the dislocations of changing shape where all forces exerted on dislocation segments have to be explicitly specified. However, instead of solving the full PSB boundary value problem mentioned above, we evaluate two limiting cases: (a) the so-called stress controlled regime where the stress in the channel induced by boundary conditions is assumed to be uniform, (b) the total-strain controlled regime where the sum of elastic and plastic strain is uniform in the channel. As will be shown in Section 2.3, the stress control provides an upper estimate of the escape stress, whereas the strain control yields a lower estimate. The escape stress in a PSB channel under the stress control was recently evaluated by de Sansal & Devincre [7], Schwarz & Mugrabi [8], and El-Awady et al [9]. In the detailed comparison of the papers, including the present one, El-Awady et al concluded that all the simulation results are comparable. The papers differ in the technical details of the dislocation dynamics simulations, in the introduced approximations, in the ranges of the model parameters explored, and in the emphases of the interpretation. In [7] and [8] the walls are assumed non-penetrable, in [9] and the present paper the influence of the dipole structure of the PSB walls is partly explored. One of the most important questions of the interpretation presented by Schwarz and Mughrabi [8] is the relation between the evaluated escape stress and the flow stress in PSB channel. The unexpected agreement between of the escape stress predicted by the To the authors of [9] the preliminary version of the present paper referred in [9] as reference [9] was available. 3

4 simulations with experimental values of the saturated macroscopic flow stress observed in the presence PSBs is a puzzle, since it is well known that the flow stress in the soft-channel part of the inhomogeneous PSB composite structure is significantly below the saturated macroscopic flow stress. Schwarz and Mughrabi [8] suggested that the statistical treatment including the effects of annihilation and of dislocations interacting over a range of slip plane separations can account for the discrepancy. The puzzle opened the question of the meaning of the evaluated escape stress for the operation of PSBs. In the present paper the application of the dislocation dynamics simulation to the evaluation of the escape stress exposed other problems. The already mentioned problem of the applied stress distribution in the channel described in Section 2.3 lead us to consider both limit cases of the stress and strain control through out the paper. Another problem is that the concepts of the passing, bowing, and escape stresses in the meaning of the papers [2, 3, 4] are global quantities relevant to the channel as a whole. On the other hand in the simulation the passing, bowing, and applied stresses are local quantities with values in the channel from place to place different. As shown in Section 4 a consequence is that the values of the derived global quantities depend on the method used for their evaluation. Moreover, as seen in Section 2.2 considering the dipolar structure of the walls to define the channel width is not straightforward. 2 The model The geometry used in the dislocation dynamics simulations is schematically illustrated in Fig. 1. We consider two perfect dislocations, Γ 1 and Γ 2, of opposite signs in a crystal with Burgers vector b =(b, 0, 0), which is oriented in the x-direction of the x, y, z coordinate system. In a PSB channel of width d c, the dislocations glide in two planes parallel to xoz-plane; the spacing between the slip planes is h. The glide of a planar dislocation is governed by the linear viscous law in the form of the mean curvature flow equation, B v =[Tκ+ bτ eff ] n, (1) where B is the drag coefficient, v( X,t) is the velocity of a dislocation segment at position X and time t. The first term on the right-hand side, which represents the self force, is expressed in the line tension approximation as the product of the line tension T and local curvature κ( X,t); τ eff ( X,t) represents the local resolved shear stress acting on the dislocation segment; n( X,t) is the unit normal of the segment in the slip plane. We accept orientation dependent line tension T (α) =E edge (1 2ν +3cos 2 (α)), where α( X,t)isthe angle between the tangent of the dislocation segment and the Burgers vector and E edge = μb2 ln(d w /r 0 ) 4π(1 ν) is the energy of an edge dislocation per unit length, ν is the Poisson ratio, μ shear modulus, b is the magnitude of Burgers vector, d w width of dislocation walls; the core radius r 0 is taken to be the Burgers vector length. 4

5 Figure 1: Geometry used in the dislocation dynamics simulations: two dislocations moving in parallel slip planes constrained in PSB channel. Three contributions to the resolved shear stress are considered, τ eff = τ disl + τ wall + τ app. (2) Elastic interactions between the dislocations τ disl corresponds to the contribution (i) of the Brown s list; τ wall represents the influence of the walls simulated as elastic fields of rigid edge dipoles; it leads actually to the contribution (iii). The Brown s contribution (ii) enters the equation of motion (1) as the line tension term Tκ. The applied stress τ app approximates the stress in the channel determined by the boundary conditions. As in [2, 3, 4], the influence of a friction stress and dislocation debris is neglected. 2.1 Interaction stress The resolved shear stress in the channel at X caused by the elastic field of curved dislocation Γ can be expressed as (in the following formulae the dependence on time t is omitted) τ disl ( X)= τ d ( X,s)ds. (3) Γ τ d ( X,s) is the resolved shear stress exerted by a dislocation segment ds of Γ, where s is the arc length parameter (details of the arc length description of a dislocation are given in Section 3). The integral is taken along the curve Γ at time t. A general shape of the dislocation with no symmetry constraints is considered. As in the numerical implementation of the model the dislocation is represented by a moving polygon, we employ for evaluation of τ d ( X,s) de Wit s formula [10] for the stress field of finite straight dislocation segment AB, τ d ( X,s) ( ) ( ) τ xy (AB) = τ xy X XB τ xy X XA, (4) where X A and X B are the end points of the straight segment of the polygon at s. According to de Wit [10], the stress tensor components τ ij generated at a point X by the semi-infinite 5

6 Figure 2: Geometrical meaning of the vectors entering the formula (5) for the stress field components τ ij at point X produced by a semi-infinite straight dislocation with the end at X 0. straight dislocation of Burgers vector b,withtheendpointatx 0, and of line direction ξ, are given by, i, j =1, 2, 3 (the geometrical meaning of the symbols is shown in Fig. 2), τ ij ( X X 0 ) = μ { 1 ( b ) ( ) P ξ j + b P ξ i 1 (( ) b ξ P j + 4π R(R + L) i j 1 ν i ( ) ( ) ) b η ξ [ + b ξ P i δ ij + ξ i ξ j +(η i ξ j + η j ξ i + Lξ i ξ j ) R + L + j 1 ν R ]} 2 2R + L + η i η j, (5) R 2 (R + L) where R is the magnitude of the vector R = X X 0, η is the magnitude of the distance vector η from the dislocation, η = R L ξ and L = R ξ is the projection of vector R to the dislocation line. Dot stands for a scalar product of vectors and meansacross product. P i and P j are components of the vector P = R R ξ, ξ i and ξ j components of the vector ξ,andη i,η j components of the vector η; δ ij is Kronecker s delta. 2.2 Wall interaction A long range stress caused by the different deformability of the walls and the channel is incorporated in the applied stress τ app in the channel. However, there is a short range interaction between the dipolar loops clustered in the walls and parts of the dislocations close to the walls. Following Brown [2], these short range interactions are incorporated in the model as elastic fields of fixed rigid infinite edge dipoles located at the surface of the walls. Segments of the dislocations deposited at the walls are trapped in the elastic 6

7 Figure 3: Geometry of dipolar dislocation walls substituted by edge dislocation dipoles. potential valleys of the dipoles parallel to the walls (eventually a sufficiently strong stress can bow out a dislocation segment from a wall into the channel as simulated in [11]). Height of the dipoles h dipole controls the distance of the valleys from the walls and their strength. Fig. 3 indicates that a specification of the width of the channel d c is not straightforward. In [2, 3, 4], d c is taken as the distance between the surfaces of the neighboring walls which is identical with the distance between two edge parts of a dislocation deposited at these walls. In the present paper, the surfaces of the walls are represented by the potential valleys of the edge dipoles. Initially, the ends of the dislocations in the screw positions are deposited in the bottoms of the valleys. The distance between these ends is taken as d c. However, due to the stress changes in the loading process, the edge parts deposited in the valleys are pushed to the sides and the distance between these parts is increasing. In reality, the distance changes in the range of nm, nevertheless the difference in specification of d c has to be considered when one compares the results of the present simulations with the results of the previous papers [2, 4, 7, 8, 9]. In the present paper the distance between the centers of the rigid dipoles representing the walls is 1 μm andtheheight h dipole = 20 nm. For this arrangement the distance between the bottoms of the potential valleys d c =0.96 μm. The arrangement of the dipoles in the walls is shown in Fig. 3. The centers of the dipoles are placed in the slip planes of the dislocations. The resolved shear stress of an 7

8 edge dipole composed of the fields of edge dislocations is τ wall = μb 2π 1 1 ν ( x1 (x 2 1 y2 1 ) (x x 2 (x 2 2 y2 2 ) y2 1 )2 (x y2 2 )2 ), (6) where x 1 and y 1, respective x 2 and y 2, are the coordinates of a point in the channel relative to the dislocation E 1, respective E 2 ; in the case represented by equation (6), Burgers vector of dislocation E 1 is positive, for dislocation E 2 it is negative. 2.3 Applied stress distribution in the channel As already mentioned in the Introduction, instead of solving the full boundary value problem of the stress distribution in the channel two simplified limit cases are considered: the stress controlled regime, where the applied stress in the channel is kept uniform (the same assumption was employed in the original composite model [6, 12]), and the total strain controlled regime, where the total strain remains uniform. The reality is between these two limits. In the stress control, the elastic strain, coupled through the Hooke law to the stress, remains uniform. Therefore, it cannot adjust to the generally nonuniform plastic strain produced by the dislocation glide (the compatibility of total strain in the channel is violated). Such artificial rigidity causes the stress level to be higher than in reality. In this sense, stress control provides the upper estimate of the escape stress. Applied stress τ app, which is the same in each point of the dislocation line, can be an arbitrary function of time. In the numerical simulations in Section 4 we explore a special case where the uniform applied stress τ app is constant in time: τ app =const. (7) In the total strain control, on the other hand, the total shear strain ε tot, being a sum of the elastic and plastic parts, ε tot = τ app + p, (8) μ is assumed to be uniform in the channel. It is not required that the stress in the channel satisfies stress equilibrium (only the equilibrium of the forces exerted on the dislocation lines are guaranteed by the equation of motion (1)). Accordingly, the resulting applied stress is smaller than in reality. This means that the total strain control provides a lower estimate of the escape stress. To estimate the plastic strain p the considered dislocations are taken as representatives of glide dislocations in the channel. The plastic strain carried by a dislocation segment is t p(s, t) =ϱb v(s, t)dt, (9) t 0 where the integral represents the part of the slip plane slipped by the segment s of a unit length, v is the segment speed; the amount of slip is b; initially p(s, t 0 ) = 0. The average 8

9 scalar density ϱ of the glide dislocations in the channel represents in (9) the number of segments piercing a unit area perpendicular to the segment s. In the numerical simulations, we explore the case where the total shear strain ε tot is uniform and increases linearly with time at the rate ε, In the total strain control we get from eqs. (8) (10) that ε tot (t) = εt. (10) τ app (s, t) =μ[ εt ϱb v(s, t)dt]. (11) t 0 The ellipticity of the dislocations assumed in [4] implicitly means that the applied stress in the channel is inhomogeneous (see also [13, 14]). An additional strong inhomogeneity was introduced in [2] by the higher dislocation curvature near the walls represented by a circular dislocation shape of a small radius. 3 Mathematical model of a dislocation A detailed description of the method of the numerical solution of the problem is presented in [11]; here we give only a rough outline. A solution to the problem of the interacting dislocations requires a description of evolving curves. In the parametric method [15, 16], the gliding dislocation curve Γ(t) at time t is described by a smooth time dependent position vector function X(u, t), where u is a parameter from a fixed interval u [U 1,U 2 ]. It means that for any time t, the dislocation curve is given as t Γ(t) = Image( X(,t)) = { X(u, t),u [U 1,U 2 ]}. (12) The mapping is shown schematically in Fig. 4. Since the dislocations move along crystallographic planes, we assume that the set { X(u, t),u [U 1,U 2 ]} is a subset of the corresponding plane in the Cartesian coordinate system for any time t. The vectors tangent and normal to the dislocation line in the glide plane are X u ux and X u, respectively; the outward normal vector X u is determined by the condition det( X u, X u )=1. Consider a smooth curve, X u > 0, for all u [U 1,U 2 ]andforanytimet. Here X u denotes the Euclidean norm of the vector X u. A unit arc length parametrization will be denoted by s. Then, the line element ds of the curve Γ(t) isds = X u du. Letu 0 [U 1,U 2 ] be an arbitrary fixed point in the interval [U 1,U 2 ]. Distance s(u, t) measured along the curve from the point X(u 0,t)topointX(u, t), u>u 0,is u s(u, t) = X u (u,t) du. (13) u 0 As the distance s is a growing function of u, there exists an inverse function u = u(s). The natural parameterization of the dislocation X(u, t) by the distance s(u, t) 0,L(t) 9

10 Figure 4: The mapping Γ(t) = Image( X(,t)) of the interval [U 1,U 2 ] into the glide plane xoz. is X(s, t) X(u(s),t). We denote L(t) the total length of the dislocation curve Γ(t) at the time t. Note that X s = ξ and X s = n represent unit tangent and normal vectors, respectively, introduced in Section 2: ξ = X s = X du u ds = X u / X u, n = X s = X u / X u. (14) The curvature κ = κ(s, t) ofthecurveats is defined through the Frenet s formula, X ss sxs = κx s. (15) Using eqs. (14), (15), and the fact that v = t X, the equation of dislocation glide (1) can be written in the form of an intrinsic diffusion equation, B X t = T X ss + bτ eff X s, (16) for the dislocation position vector X(s, t). Equations in the form (16) are used for description of dislocations Γ 1 and Γ 2 moving in the PSB channel. The equations are supplemented by initial and boundary conditions. Initially, at t = 0, dislocations Γ 1 and Γ 2 of the opposite signs are stretched across the channel in straight screw positions. The z-coordinates of the end points are kept fixed, i. e. the boundary conditions restrict the motion of the end points along the walls, but they allow their displacement in the perpendicular direction. For simulations we use the flowing finite volume method [17] in space and the method of lines [18] in time. In general, each discrete solution is represented by a moving plane polygon. Points of the dislocation curve discretized in the arc length parameterization s are denoted by subindex i, X i = X i (t) = X(s i,t), i =0,...,M, 0 = s 0 <s 1 <...<s M = L(t), 10

11 where L(t) is the total length of the dislocation curve Γ(t) atthetimet. M +1isthe number of points on the curve. The smooth curve is approximated by M linear segments [ X i 1, X i ] called flowing finite volumes (i = 1,...,M). The discretization employed in eq. (16) for the interacting dislocations Γ 1 and Γ 2 provides a system of ordinary differential equations which govern their motion. To solve this system, we use the Runge-Kutta method of the 4 th degree. distance of wall dipoles 1050 nm height of wall dipoles h dipole =20nm width of channel d c = 960 nm width of dislocation walls d w = 150 nm spacing between slip planes h = 55nm magnitude of the Burgers vector b =0.25 nm shear modulus μ = 42.1GPa Poisson ratio ν = 0.43 density of glide dislocations ϱ = m 2 drag coefficient B = Pa s energy of edge dislocation E edge =2.35 nj m 1 Table 1: Parameters used in the simulations. 4 Results and discussion The material parameters used in the simulations are summarized in Table 1. A series of subsequent positions of dislocations Γ 1 and Γ 2 in the geometry of the PSB channel shown in Fig. 1 is presented in Fig. 5 for the stress control at τ app =44MPa. Fig.5adisplays the shape changes from the initial straight screw position through the dislocation bowing till the formation of the screw dipole. The edge parts are deposited in the potential valleys parallel to the walls. Fig. 5b shows the subsequent series of the separation of the dislocations from the dipolar configuration. For the total strain control, the sequence of the shapes is similar; the main difference is that the central parts of the dislocations are flatter and their outer parts are pushed closer to the walls. The reason is that in the middle of the channel the applied stress is lower than at the walls. A typical difference is seen in Fig. 6, where there are compared the shapes of the dislocation Γ 1 for the stress control at τ app =44MPaandtimet =0.545 s and for the total strain control at the strain rate ε = s 1 at t =1.742 s. In the stress control, a direct method to estimate the escape stress τ esc in the channel is to determine the minimum of τ app needed for the separation of the dislocations as shown in Fig. 5b, i. e. τ esc =minτ app. A sensitive measure of the separation is the rate of change The definition of the channel width and its value d c are introduced and commented in Section2.2, values of parameters d w and h are taken from [4], b, μ and ν from [2]. 11

12 in the area S swept by the dislocation (due to the symmetry seen in Fig. 5 the swept area of only one of the dislocations is calculated), Ṡ(t) = ds(t) = v(s, t)ds. (17) dt Despite strong numerical instabilities seen in Fig. 7, the method provides a rough estimate of τ esc. If the applied stress τ app is smaller than the sum of the other stresses exerted on the dislocation, the motion stops after certain time, the rate Ṡ(t) reaches zero. τ esc is identified with the lowest τ app at which Ṡ(t) remainspositive, Fig. 7a. The upper limit of the escape stress estimated in this way gives τ esc 32 MPa. As an alternative to the direct method, we employed a procedure akin to the determination of the escape stress proposed by Mughrabi & Pschenitzka [4]. They derived the escape stress as the maximum of the overall stress (the spatially varying bowing and dipolar interaction stress contributions) as a function of the spacing between the two dislocations measured in the middle of the channel. The modification of Mughrabi & Pschenitzka s method is based on the equation of dislocation glide (1) written in the scalar form for a point u of the dislocation; u is the parameter introduced in Section 3, where the interval [U 1,U 2 ] is identified with the initial screw dislocation position of length d c, Bv(u, t) bτ app (u, t) =T (u, t)κ(u, t)+bτ disl (u, t)+bτ wall (u, t) bτ(u, t), (18) where τ is the overall stress. In the present model, the overall stress consists of the bowing stress Tκ, of the interaction stress represented by τ disl, and of the wall interaction stress τ wall. The left hand side of eq. (18) can be evaluated for any point u from the data of the simulation and in that way τ can be determined. For a given point u we get the overall stress τ u (t) τ(u, t) u as a function of the length l u (t) of the trajectory of u in the parametric form with time t as the parameter, τ u (t) =Bv u (t)/b τ u app (t), lu (t) = Γ t 0 v u (t)dt. (19) In the stress control, τapp u (t) is a prescribed constant stress, in the total strain control τapp(t) u is determined from eq. (11). Escape stress τ esc is interpreted as the maximum of the function τ u (l), i. e. τ esc =maxτ u (l). The dependence of the overall stress τ on the length of the trajectory l for the dislocation point moving in the middle of the channel, i. e. the point u =0.5d c,isshownin Fig. 8. In the stress control, τ(l) is determined for three values of the applied stress τ app, in the case of the total strain control the curves are shown for three values of the total strain rate ε. As seen from Fig. 8, the variations of the loading conditions represented by the three curves for the stress control and the three curves for total strain control do not have a substantial influence on the estimate. An attempt to adapt the method to the total strain controlled case failed due to even stronger instabilities. 12

13 The first maximum on the curves τ(l) in Fig. 8 corresponds to the bowing of the dislocations from the initial straight screw positions. The subsequent minimum is caused by the attractive force between the approaching dislocations. The following sharp increase represents the stress needed for the separation of the dislocations from the dipolar configuration. A steady state motion is then reached. The escape stress τ esc for the middle point u =0.5d c can be identified with the highest maximum of the curves in the particular loading regime. From the graphs we get the estimate of the upper and lower limits, 26 MPa< τ esc < 29 MPa. The dependence of τ esc on the choice of a representative dislocation point u is demonstrated in Fig. 9, where the curves τ u (l) are compared for u = cd c, c =0.2, 0.4, 0.5; cd c means the parametrization according to the initial position of the representative point. For u =0.2d c, the strong influence of the edge wall dipoles is evident. This point was moved to the wall and trapped there. In Fig. 10 the representative curves τ(l) from Fig. 8 are compared with the curve obtained by Mughrabi & Pschenitzka [4] chosen for comparable data in Table 1. Mughrabi & Pschenitzka s curve traces the separation path of the dislocation middle point starting from the dipolar alignment. The approaches mentioned in the present paper can be compared using the Brown s formula [2] consisting of the bowing part 2E edge /(bd c ) and the passing part reduced by bowing μb/(4πh c ) (1 λ)2e edge /(bd c ), i.e. τ esc = 2E edge bd c + μb 4πh c (1 λ) 2E edge bd c = μb 4πh c + λ 2E edge bd c. (20) Brown s formula expresses how much in the escape stress the bowing resistance 2E edge /(bd c ) adds to the dipolar passing stress μb/(4πh c ). As was already noted in the Introduction, Brown s adding assumption [3] corresponds to λ = 1, the estimate of Mughrabi & Pschenitzka [4] reaches λ 0.2 and that of Brown [2] 0.4 <λ<0.7. For the data in Table 1 it can be deduced from the graphs in Fig. 10 that 0.5 <λ< Conclusions 1. The objective of the present paper was to estimate the escape stress in a PSB channel. Discrete dislocation dynamics simulations were used as an alternative to the recent studies of the same problem by Brown [2, 3] and Mughrabi & Pschenitzka [4]. In the simulations, two unlike dislocations in parallel slip planes are pushed by applied stress in the opposite directions till they attract one another forming a dipole. As the stress is increased, the dislocations become more and more curved, until they escape one another. The stress needed to accomplished this is interpreted as the escape stress. 2. The evolving positions and shapes of the glide dislocations are governed by the equation of motion where each infinitesimal dislocation segment is subjected to the line tension, the interactions with segments of other dislocation, the short-range 13

14 interaction with the edge dipoles substituting the walls of the channel, and the stress field in the channel imposed by the loading conditions. 3. Two loading regimes were considered: (a) the so-called stress controlled regime where the stress in the channel induced by the boundary conditions is assumed to be uniform, (b) the total strain controlled regime where the sum of the elastic and plastic strain is uniform in the channel. As shown in Section 2.3, the stress control provides an upper estimate of the escape stress, whereas the total strain control yields a lower estimate. 4. A series of subsequent positions of the dislocations is presented in Fig. 5. To determine the escape stress from the simulations, two methods were employed: (a) In the direct method at the stress control, we determined the upper limit of the escape stress as the minimum of the applied stress needed for the separation of the dislocations from the screw dipolar alignment. The method provided the rough estimate of the upper limit τ esc 32 MPa. (b) As an alternative to the direct method, we employed a procedure akin to the determination of the escape stress proposed by Mughrabi & Pschenitzka [4]. They derived the escape stress as the maximum of the overall stress (the spatially varying bowing and dipolar interaction stress contributions) as a function of the spacing between the two dislocations measured in the middle of the channel. The present version of this method predicts for the escape stress τ esc the upper and lower estimate 26 MPa< τ esc < 29 MPa. 5. The application of the dislocation dynamics to the evaluation of the escape stress exposed the following problems: (a) The problem of the applied stress distribution in the channel described in Section 2.3 as a rough substitute of a solution of PSB boundary value problem lead us to consider both limit cases of the stress and strain control through out the paper. (b) The concepts of the passing, bowing, and escape stresses in the meaning of the papers [2, 3, 4] are global quantities relevant to the channel as a whole. On the other hand in the simulation the passing, bowing, and applied stresses are local quantities with values in the channel from place to place different. As a consequence the estimated values of the derived global quantities depend on the method used for their evaluation, as seen in item 4. (c) Considering the dipolar structure of the walls required an alternative definition of the channel width. It is represented by the distance between the bottoms of the elastic potential valleys of the rigid edge dipoles substituting the walls. 14

15 Acknowledgements We are grateful to Prof. L.M. Brown, Prof. H. Mughrabi and Dr. R. Sedláček for the inspiring discussions, valuable comments and critical reading of the manuscript. We like to express our thanks to Dr. M. Beneš and Ing. V. Minárik for providing us with the basic computer program and generous help. The first author was partly supported by Nečas Center for Mathematical Modeling, project LC06052 of the Ministry of Education of the Czech Republic. The research has been supported by grants VZ-MŠMT and KONTAKT ME654. References [1] H. Mughrabi and R. Wang, in Deformation of Polycrystals: Mechanisms and Microstructures, edited by N. Hansen et al. (Risoe National Laboratory, Denmark, 1981), pp [2] L.M. Brown, Phil. Mag (2006). [3] L.M.Brown,Mater.Sci.Eng.A (2000). [4] H.Mughrabi and F. Pschenitzka, Phil. Mag (2005). [5] J.C. Grosskreutz and H. Mughrabi, in Constitutive Equations in Plasticity, edited by A.S. Argon, (MIT Press, Cambridge, 1975), p [6] H. Mughrabi, in Continuum Models of Discrete Systems 4, edited by O. Brulin and R.K.T. Hsieh (North Holland, The Hague, 1981), p.241. [7] C. de Sansal and B. Devincre (private communication, 2006). [8] K.W. Schwarz and H. Mughrabi, Phil. Mag (2006). [9] J.A. El-Awady, N.M. Ghoniem and H. Mughrabi, in Dislocation modelling of localized plasticity in persistent slip bands, edited by A. Garmestani (TMS, in press, 2007). [10] R. de Wit, Phys. Stat. Sol (1967). [11] J. Křišt an, V. Minárik, M. Beneš and J. Kratochvíl, Comput. Mater. Sci. (in preparation, 2006). [12] H. Mughrabi, Acta Metall (1983). [13] R. Sedláček, Phys. Stat. Sol. (a) (1995). [14] R. Sedláček, Scripta Metall. Mater (1995). [15] M. Gage and R.S. Hamilton, J. Diff. Geom (1986). 15

16 [16] R. Sedláček, Phil. Mag. Letters (1997). [17] K. Mikula and D. Ševčovič, J. Appl. Math. (SIAM) (2001). [18] G. Dziuk, in Mathematical Models and Methods in Applied Sciences (1994). 16

17 (a) z (nm) t = 0.55 s t = 0.27 s t = 0 s x (nm) (b) z (nm) t = 1.25 s t = 0.90 s t = 0.66 s x (nm) Figure 5: (a) Shape changes from the initial straight screw position to the formation the screw dipole. (b) Subsequent series of the separation from the dipolar configuration. 17

18 stress controlled regime strain controlled regime z (nm) x (nm) Figure 6: Comparison of the shapes of dislocation for two regimes. 18

19 τ app = 32 MPa (a) ds/dt (nm 2 s -1 ) ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 time (s) τ app = 40 MPa (b) ds/dt (nm 2 s -1 ) ,0 0,2 0,4 0,6 0,8 1,0 1,2 time (s) τ app = 44 MPa (c) ds/dt (nm 2 s -1 ) ,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 time (s) Figure 7: The rate Ṡ(t) in the stress controlled regime, at (a) τ app τ app =40MPa,(c)τ app =44MPa. 19 =32MPa,(b)

20 30 (a) overall stress τ (MPa) stress controlled regime: τ app = 36 MPa τ app = 40 MPa τ app = 44 MPa position curve l (nm) (from initial configuration) 30 (b) overall stress τ (MPa) strain controlled regime: dε tot /dt = 1.90*10-3 s -1 dε tot /dt = 2.85*10-3 s -1 dε tot /dt = 3.80*10-3 s position curve l (nm) (from initial configuration) Figure 8: The dependence of overall stress τ on the length of the trajectory l for the middle point of the dislocation; (a) stress controlled regime, (b) strain controlled regime. 20

21 35 30 overall stress τ u (MPa) u = 0.5d c u = 0.4d c u = 0.2d c position curve l u (nm) (from initial configuration) Figure 9: The dependence of the overall stress τ u on the length of the trajectory l u for different settings of the parameter u. The dashed curve indicates the trapping of the representative point u =0.2d c by the dipole of the wall which prevents its further motion. 21

22 30 25 overall stress τ (MPa) stress controlled: τ app = 44 MPa total strain controlled: dε tot /dt = 2.85*10-3 s -1 Mughrabi&Pschenitzka (2005) position curve l (nm) (from initial configuration) Figure 10: The dependence of the overall stress on the length of the trajectory l in the two regimes. The results obtained by Mughrabi & Pschenitzka [4] are added. 22