A 3D CELLULAR AUTOMATON MODEL OF DISLOCATION MOTION IN FCC CRYSTALS

Transcription

1 A 3D CELLULAR AUTOMATON MODEL OF DISLOCATION MOTION IN FCC CRYSTALS Qizhen Li and Peter M. Anderson Department of Materials Science and Engineering Ohio State University 2041 College Road Columbus, OH Abstract: A 3D Dislocation Cellular Automaton (CA) model is developed to study the evolution of dislocation configurations in FCC single crystals. Crystallographic {111} slip planes with three-fold symmetry are discretized into equilateral triangular patches with sides along <110> directions. These patches slip provided there is a sufficient driving force associated with reduction in system energy. Perfect <110>/{111} dislocations are considered. The resulting variables are the triangular patch size and dislocation cut-off, measured relative to Burgers vector magnitude b. Three examples involving operation of a Frank-Read source are chosen to highlight the benefits and drawbacks of the method. A benefit to discretization is that dislocation evolution may be analyzed via spatial averaging over collections of patches, so that the discrete versus continuum nature of the results may be studied. Further, dislocation reactions and cross slip are accommodated easily and, in principle, Monte-Carlo schemes can be integrated into the evolution formalism. A drawback is that collections of patches do not reflect a smooth variation in configuration so that artificial fluctuations in dislocation line length and direction can result. Overall, the discrete nature of the method is attractive for incorporating the kinetics of thermally activated states and for simplifying the range of geometries and threshold criteria associated with dislocation reactions. 1. Introduction The study of dislocation evolution is challenging due to the multiple length scales involved. On the macroscopic level, work-hardening and precipitation strengthening require accurate computation of system energy on scales as large as several hundred microns and further, an associated criteria for system evolution is needed. Embodied in the energy computation and system evolution is a set of degrees of freedom to accurately describe fluctuations in configuration and energy. The challenge is that these degrees of freedom extend, in principle, to atomistic scales where features of dislocation cores, kink formation, jog formation, and dislocation annihilation dictate local driving forces for system evolution. Larger-scale continuum formulations attempt to replace atomistic contributions to system energy with phenomenological relations that specify dislocation mobility, cross slip, and annihilation events in terms of macroscopic configurational variables and forces. Examples of more macroscopic, continuum approaches include those by Kubin et al [1,2], Zbib et al [3], Chrzan et al [4], and Schwarz et al [5] where macroscopically-curved dislocations are approximated by piecewise-linear configurations of arbitrary orientation and an approach by Ghoniem [6] employing curved, parametric 1/16

2 segments with smoothly varying slope and curvature. These methods can compute system energy based on the theory of elasticity for Volterra dislocations in infinite, elastically homogeneous media so that dislocation cores are represented as planar, discontinuous jumps in slip distribution. In some cases, the finite element method or collocation techniques are employed to incorporate the effect of free surfaces or elastically inhomogeneous media [7]. The phase field method has emerged recently as an intermediate-scale approach that implements a Peierls-Nabarro approach for 3-dimensional dislocation networks [8]. Thus, dislocations are modeled in terms of smoothly varying distributions of shear transformation within thin planar regions. This more detailed formalism relies on atomistic results to supply the generalized stacking fault energy for candidate slip systems. Consequently, highly non-linear interactions between dislocation cores and the detailed, core-dependent structure of defects such as extended nodes can be computed. A consequence of current implementations with a regular 3-dimensional discretization is that the system size cannot exceed 100 to 200 core dimensions. Thus, only small portions of arrays can be modeled. Atomistic approaches such as static and dynamic versions of the embedded atom method have been employed for several years to compute the core structure of single dislocations or the interaction of a dislocation with point, line, or area defects [9-11]. This scale of study is essential to supply activation energies for glide processes such as kinkpair formation, cross-slip processes dependent on jog-pair formation, and dislocation reactions with products. Currently, molecular dynamics approaches may involve on the order of millions of atoms for hundreds of picoseconds, so that only regions on the scale of nm can be considered. This manuscript formulates a cellular automaton (CA) approach that combines a need to model small-scale, discrete, crystallographic events such as kink-pair or jog-pair formation with a computationally efficient Volterra dislocation description typical of large-scale simulations. The CA method requires the spatial discretization of the material into a 3-dimensional grid of patches, each of which is capable of slipping. Thus, dislocation processes occur by a succession of unit slip events. The step-wise change in system energy with each event can be interpreted as kink-pair formation or kink migration or instead, the step-wise changes can be averaged over several events to produce a smoothly varying energy typical of continuum formulations. Thus, the scale of discretization and method of averaging can be varied to reflect a discrete or continuous system. The CA method including computation and averaging of event energies is presented in Section 2. Section 3 presents the results and discussion for three applications. First, the critical stress to nucleate a single patch on a straight segment is studied as a function of patch size and cut-off parameter. Next, a Frank-Read source is simulated in an infinite, isotropic, elastic material with isotropic and also anisotropic line energy. Finally, a Frank-Read source is studied in the vicinity of a fixed dislocation that is parallel to and repels the initial Frank-Read line, so that in some cases, cross-slip is required to operate the source. Overall, the discrete nature produces local fluctuations in system energy associated with the abrupt increase or decrease in dislocation line length as individual patches slip. This feature may be regarded as physically reasonable if the patches correspond to real structures such as kinks or jog pairs. In such cases, the energetics of the model enable 2/16

3 formulation of a Monte-Carlo approach to study thermally activated dislocation processes, similar to the work of Cai et al [16]. However, compared to the rectangular grid in that formulation, the present work uses a 3D space-filling array of triangular patches that reflects the symmetry of a FCC structure, so that arbitrary dislocation reactions between perfect dislocations can be accommodated. At larger length scales, the local fluctuations in system energy are regarded as a nonphysical aspect of the model. Under such conditions, the energetics of the system can be smoothened by averaging over collections of patches. Thus, an outcome of the model is that it may have a discrete or continuous nature. In the latter limit, the discretization approach is observed to require more degrees of freedom and thus is not as computationally efficient as methods that permit arbitrary, grid-free motion of frontal tracking nodes [1-6]. 2. Model Development An important feature of the CA method is the discretization of crystalline material into cells that reflect the crystallographic orientation and symmetry of candidate slip systems. In this section, the CA method is applied to perfect dislocations in FCC crystals. Relevant slip systems and possible reactions between them are presented, followed by details of the change in system energy when a particular patch slips. Rules for system evolution are then presented, followed by various schemes for averaging the response over collections of patches. 2.1 Discretization for a FCC crystal A FCC crystal as shown in Figure 1 is discretized into cubes of edge length u that are aligned to the <100> crystallographic basis. Within each cube, equilateral triangular patches can be constructed with vertices that coincide with three corners of the cube and with edge length L = 2 u. These patches correspond to the {111} family of crystallographic slip planes in FCC crystals. Of the four possible slip plane orientations, triangular patches PBA and CBA show the respective (111) and ( 1 1 1) variants and the larger triangle DEF shows the seamless extension of individual patches to produce a macroscopic (111) slip plane. The perpendicular spacing between parallel slip planes is 2/ 3 u. The four {111} planes are projected onto the (001) plane in Figure 2. The planes are labeled a, b, c, or d and each has three Burgers possible vectors (1, 2, 3) for a total of 12 perfect {111}/<110> slip systems. Table 1 lists these 12 systems using the notation a1, a2, etc. to denote the combination of slip plane and Burgers vector. Also included are 6 {100}/<110> pure edge misfit dislocations that may occur as reaction products between {111}/<110> dislocations. For example, misfit dislocations e1 and e2 in Figure 2 can be formed by the reactions e1 = c3 + a1 = -(a3 + c1) and e2 = b1 + d3 = -(b3 + d1). A relevant limit to the discretization is to set u = a/2, where a is a unit cell dimension. In that case, a {111} plane is discretized so that each equilateral triangle corresponds to the area per atom in a close-packed array. Therefore, slip of an individual triangle corresponds to relative slip of a single atom. At this limit, the hexagonal loop ABCDEF in Figure 3 corresponds to the relative slip of 6 close-packed atoms inside the hexagon. The discrete {110} line directions in this model correspond to directions and positions 3/16

4 along which the Peierls energy for a straight segment is expected to be a minimum. Thus, the advance of the line HJ in Figure 3 is expected to occur via the formation of kink pairs as depicted by slip of triangle 1, and kink-pair expansion occurs by the slip of triangular patches 2, 3, 4, and 5. This discretization also accommodates the geometric nature of cross slip. For example, segment AB in the hexagonal loop in Figure 3 has a pure screw character, so that is possible for the loop to expand by slip of triangle 7 or by cross slip onto a competing {111} plane. If the hexagonal loop lies in a (111) slip plane, the slipped triangle 6 would correspond to FAB in Figure 1, expansion in the parent (111) plane would correspond to slip of triangle PBA, and cross slip onto a (1 1 1) slip plane would correspond to slip of CBA. Cross slip of non-screw segments is not considered here. Table 1. FCC slip systems in the CA model slip plane Burgers vectors n b (1) b (2) b (3) (a) [111] [10 1 ] [ 1 10] [0 1 1] (b) [ 1 11] [01 1 ] [ 1 1 0] [101] (c) [ 1 1 1] [ ] [1 1 0] [011] (d) [1 1 1] [0 1 1 ] [110] [ 1 01] (e) [001] [110] [ 1 10] (f) [010] [101] [10 1 ] (g) [100] [0 1 1] [0 1 1 ] 2.2 Energetics of slipping patches The slip of individual patches changes the energy of a stressed crystal by an amount where the change De = 1 2 Df = De - Dw (1) N f N f Âe kl k=1 l =1 Â /16 N i N i Âe kl k=1 l =1 Â (2) in elastic energy of the isolated dislocation configuration is expressed in terms of the change in the sum of interaction energies e kl between interacting dislocation segments k and l in the final (f) versus initial (i) configurations. Detailed expressions for e kl between parallel, non-parallel, and intersecting dislocation segments are provided in Sections 6-1 and 6-2 of Hirth and Lothe [12]. The self energy of a segment is denoted by setting k = l. The corresponding work done by applied and internal sources of stress other than the dislocation segments is expressed as Dw = Ú As ( ) n i s a ij (x)+s m ij (x) b j dx = ls o nb A s +s m nb A s (3)

5 where n i and b j are the components of the slip plane normal and Burgers vector for the area A s to be slipped, and s ij a and s ij m are the components of applied and microstructural stress that vary spatially in general. s ij m includes stress from particles, misfit dislocations and coherency stress from lattice parameter mismatch in multilayers [12-14]. The righthand portion of eqn. (3) expresses the result in terms of a homogeneous applied stress that is proportional to a reference value, s a o ij ls ij (4) and a microstructural stress s ij m averaged over A s. The subscripts nb simply denote the component of stress that is resolved on the slip plane with normal n i in the direction of slip b j. Combining eqns. (1) and (3), Df = De - ls o nb ba s - s m nb ba s (5a) and for the particular case where the triangular patch PBA in Figure 4 is slipped, De = e AP-AP + e AP-OA + e AP-PB + e AP-BZ +e PB-PB + e PB-OA + e PB-BZ (5b) -e AB-AB - e AB-OA - e AB-BZ Substitution of analytic expressions for the e ij into eqn. (5b) yields a lengthy expression for De as a function of L, L 1, L 2, and r, given in eqn. (A-1) in the Appendix. In general, self-energy terms (e.g., e AP-AP ) and interaction terms between segments sharing a common node (e.g., e AP-OA ) will depend on r, defined as the distance within which two infinitesimal segments dx 1 and dx 2 do not interact. When the expression for De is substituted into eqn. (5a) with zero microstructural stress and in the limit L 1, L 2 Æ so that the parent line OZ is infinitely long, the critical applied stress to nucleate a triangular patch is t n l n s o nb = De n A s b = mb 2pL sin(60 0 ) ln Ê 16L ˆ Á (6) Ë 81er The result in eqn. (6) is a measure of the resolved shear stress to initiate motion of an initially straight, infinitely long dislocation line via nucleation of the first triangle. This 0 stress reaches a maximum of mb/2pl sin(60 ) for L/r = 81e 2 /16 ª 37 and monotonically decreases to 0 when L/r = 81e/16 ª 14. Thus, if L is decreased to the atomic scale, r must be selected so that the nucleation stress has a physically reasonable value. Since the energy of a kink pair is on the order of 1eV and with m ~ 100 GPa and b ~ 0.25 nm, r ~ 0.02b for L ~ b. From Hirth and Lothe, r ~ 0.1 b to b to obtain reasonable values of 5/16

6 dislocation line energy. Therefore, r takes on different values according to atomistic versus larger scale energy criteria. Values of r may be recast in terms of the core cut-off parameter r 0 by noting r = r Ï 1 screw 0 2 A, A = Ô Ì (1-2n )/4(1-n ) Ó Ô e edge (7) These equalities ensure that the energy of a pair of infinitely long, opposite-sign dislocation lines is the same, regardless of whether a formulation based on r or r 0 is used. 2.3 Rules for system evolution Several methods exist to update dislocation configurations as a function of time. In the limit of atomic-scale patches, Df may be on the order of kt so that a Monte-Carlo scheme may be appropriate. In this case, the probability that patch k is slipped during some time interval Dt is proportional to P k (Dt) = exp(-df k /kt). The simulation proceeds by determining the time interval Dt* over which the collective probability S P k (Dt*) over all patches is 1. A random number in the interval [0, 1] is then selected to determine the triangle that slips during Dt*. Thus, the simulation captures the thermal fluctuations of dislocation loops. For example, such methods have been used by Daehn [15] to study creep behavior and Cai et al [16] to modeled dislocation motion in BCC metals. One significant difference is that the current work uses a 3D space-filling array of patches reflecting the symmetry of the crystal, compared to 2D rectangular grids. An approach applicable for larger patch sizes is to follow the path of steepest decent in energy, with the constraint that only one patch is slipped at a time. This can be implemented in two forms. The first requires a prescribed stress history. The patch with the most negative Df is selected, so that a path of steepest decent in energy is followed, subject to the constraint that only one patch is selected in each step. A relaxation test is an example of a prescribed stress history, in which the dislocation loop relaxes to a stable configuration under a constraint of zero applied stress. A second macroscopic approach is to determine the stress history necessary to reversibly change the shape of a loop. Here, an applied reference stress state s o ij is prescribed and the applied stress s ij(k) = l k s o ij to activate each triangle k adjoining the existing loop is determined subject to the condition Df k = 0. The minimum positive value, l min, among all possible slip events is identified so that s ij rev = l min s ij o is the stress needed to reversibly expand the loop during that increment. The method is a physically viable one if l min monotonically increases from one step to the next, implying that the configuration expands in a stable fashion. However, a decrease in the magnitude of l min signifies an instability through which there is no stable configuration of the loop. Here, the first form requiring a stress history must be used to study evolution in unstable regimes. 6/16

7 2.4 Average measures of reversible stress Averaging values of l min is an important processing step that corresponds to the smoothening out of the discrete patch nature of the simulation. At the atomic scale, it may be reasonable to view dislocation line motion in terms of nucleation and growth of kink pairs via the addition of discrete triangles 1, 2, 3, 4, 5 to the line HJ in Figure 3. For larger triangles that correspond to several atomic slip distances, the triangular patches lose their physical connection to an atomic lattice and result in the artificial creation and removal of dislocation line length. For example, the addition of triangle 1 on line OZ in Figure 4 requires a large nucleation stress given by eqn. (6), since the dislocation line length increases by L while slipped area increases only by A s. Typically, the preferred site for the next step is triangle 2 or the symmetric counterpart to the left of triangle 1. This also requires a large stress since dislocation line length again increases by L while slipped area increases only by A s. The preferred site for the next step is triangle 3 or the symmetric counterpart, which decreases line length by L and increases slipped area by A s. Thus, for this three-step process, the values of resolved shear stress to nucleate triangles 1 and 2 are approximately t 1, t 2 ª t n in eqn. (6) while t 3 ª - t n. This pattern of two positive values of t followed by a negative value typically signifies formation of a new kink-pair-like structure on a straight dislocation line. Subsequent expansion of the structure involves the slip of triangle 4, followed by 5 or the symmetric counterpart. The expansion pattern thus has alternating positive-negative values of t. This discussion suggests two-levels at which to average successive t [17]. A smaller-scale, or microstructural process is to average over triplets consisting of positive/positive/negative values so that in the example here, t t = Avg(t 1, t 2, t 3 ), and also to average over pairs of positive/negative values so that t p = Avg(t 4, t 5 ). These two average values (t t, t p ) define the nucleation of a triplet and the expansion of it in a row via nucleation of pairs, respectively. A larger, macroscopic average is over all t from one triplet nucleation to the next. This average (t r ) typically defines the forward motion of a linear portion of a macroscopic loop via the slip of an entire row (r) of triangles. Additional smoothening is achieved by computing T r = 0.25t r t r t r +1 (8) where subscript r denotes the row over which the average has occurred. Thus, unlike methods employing continuous motion of front-tracking nodes, the CA approach described here permits a rich hierarchy of energetics and length scales, within which averages over small clusters or entire dislocation fronts are possible, depending on the physical scale of the triangular patches. 3. Results and Discussion This section first reports the principal results for nucleation of a small triangle on a finite, straight dislocation line, for comparison to eqn. (6). A Frank-Read source in an infinite, elastically homogeneous medium is then simulated and the results are compared 7/16

8 to analytic solutions assuming either isotropic or anisotropic dislocation line energy. Finally, a Frank-Read source is modeled next to an obstacle and the critical stress to operate the loop is compared under conditions where cross slip is either enabled or disabled in the code. 3.1 Triangular patch nucleation on a finite segment An initial test of the 3D dislocation cellular atomaton (CA) model is to compare the CA and analytic results for the critical stress to nucleate a triangular patch with side length L onto a straight screw segment of length L seg = L 1 + L + L 2. The relevant geometry is shown in Figure 4 in terms of the nucleation of triangle 1 onto line OZ, and the corresponding analytic results are obtained by inserting De in eqn. (A-1) into eqn. (5a) o and solving for t n l n s nb with s m nb = 0. An important feature to note is that the analytic solution in the Appendix models OA and BZ in Figure 4 as segments of arbitrary length L 1 and L 2 while the CA approach models these segments as several smaller segments of triangular side length L. As discussed in Section 6-2 of Hirth and Lothe [12], the self energy of a straight dislocation segment such as L 1 is independent of whether L 1 is expressed as a single segment or i = 1 to N multiple segments, so that È e self (b,x,l) = m Í 4p Í b x Î Í ( ( ) 2 + b x ) 2 1- n L ln L er = 1 N N Â Â e ij ( b,x,l i,l j ) (9) 2 i=1 j=1 The CA results reproduce the analytic results over a large range of (L 1 + L 2 )/L and L/r, confirming that the program accurately computes the self and interaction energies of dislocation segments. A comparison of the analytic and CA results confirms that the energy of a configuration can be computed by discretization of longer segments into several shorter ones. Figure 5(a) shows the resolved shear stress t n to nucleate a triangular patch along the midpoint of an infinitely long screw segment where L 1, L 2 Æ. As discussed in Section 2.2 and as shown in eqn. (6), t n increases monotonically with decreasing patch size L until L/r 37, and then monotonically decreases to 0 at L/r ª 14. For r = b and 0.3b, t n is less than 0.5%m and 2%m, respectively, for all values of L. Figure 5(b) shows that the resolved shear stress to nucleate a triangular patch is larger for a finite compared to infinite segment. Further, t n is a minimum for nucleation at the midpoint of a segment (L 1 = L 2 ) compared to nucleation off of the midpoint (L 1 = 3L 2 ). 3.2 Simulation of a Frank-Read source in an infinite, elastically isotropic medium This section compares predictions of the CA code and the corresponding analytic continuum solution for the critical stress to operate a Frank-Read source [18], 8/16

9 t FR = Amb 2pS ln S r 0, A = 1+ n 1- n sin2 q (10) where S is the initial length of the source as shown in Figure 6(a), n is Poisson's ratio, and q is the angle subtended by the dislocation line sense x and Burgers vector b. An initial screw segment is considered so that q = 0, and r = 0.3b is used so that, according to eqn. (7), r 0 = 0.6 b. First, an isotropic analysis is considered for which n = 0, so that dislocation line energy is independent of screw versus edge character. The convergence of the solution with decreasing triangular patch size is studied, using the reversible evolution procedure described in Section 2.3 and macroscopic smoothening defined in eqn. (8). Second, an anisotropic line energy analysis is considered for which n = 0.1, so that line energy is a minimum for pure screw character. A non-equiaxed mode of expansion is highlighted in this case Isotropic dislocation line energy (n = 0) This analysis reveals several features about the sequential slip of triangular patches during operation of a Frank-Read source and the effect of averaging over triangles as discussed in Section 2.4, in order to get a continuum measure of critical operating stress for the source. These features are present in all analyses but are identified most easily by considering a limiting case of isotropic dislocation line energy. Figure 6(a) shows that an initially straight screw dislocation of length S = b begins to bow out by slip of triangles 1, 2, and 3. Triangles 1 and 2 require a large positive stress comparable in magnitude to t n discussed in Section 3.1, and triangle 3 requires a negative stress comparable to -t n, due to the annihilation of dislocation line length. The average stress, t t, over these three triangles is shown by the datum labeled t1 in Figure 6(b), with triangular size L = 15 2 b. Patches 4 and 5 and their symmetrical counterparts 6 and 7 are activated in pairs requiring a resolved shear stress t p given by data p1 and p2 in Figure 6(b). The remaining data extending up to i = 39 show the t p to expand parallel to the initial line via triangular pairs until at i = 39, the entire row r1 of triangles in Figure 6(a) has slipped. The datum t2 in Figure 6(b) corresponds to nucleation of the first three triangles in the middle of row r2, and the subsequent data show values of t p to expand this row to completion. This process of row nucleation and completion continues in the order indicated by the labeled row numbers in Figure 6(a). Comparison to the continuum solution result in eqn. (10) requires macro-averaging to compute the resolved shear stress T r defined in eqn. (8). Figure 6(c) shows that T r increases monotonically to a maximum T FR at row r9, oscillates between rows r10 to r14, and then monotonically decreases. A more detailed description of the expansion sequence is available as a video [19]. Figure 7(a) shows good agreement between T FR computed from the CA model and t FR from the analytic expression (eqn. 10), over the range S ª (106 to 636) b. A fixed triangular size L = 15 2 b is used so that the initial number, N i of segments varies from 5 for the smallest S to 30 for the largest S considered. Figure 7(b) indicates that the error 9/16

10 between the CA approach and analytic solution decreases exponentially with increasing N i Anisotropic dislocation line energy (n 0) This analysis shows that the CA model and continuum results can vary substantially due to system evolution rules. In particular, the case for n = 0.1 and S = b involves an initial expansion as outlined in Figure 6(a), where rows r1 to r5 slip successively by the formation of a triplet of triangles in the middle of the row, followed by row expansion due to triangular pairs. Rather than nucleate a new triangle c in row r6, triangles a and b form and expand as to extend the loop in the ±2 direction. This mode of expansion continues, so that a pair of oppositely signed screw dislocation lines is generated with perpendicular spacing s^ ª 92 b. The mode of expansion is artificially trapped by the system evolution rule to select the triangle with the smallest value of t. In particular, nucleation of triangle c in row r6 generates two mixed segments and removes one screw segment while nucleation of triangle a or b generates one mixed and one screw segment but removes one mixed segment. Thus, for n = 0.1, the smaller self energy of a screw segment favors continued nucleation at a and b and therefore continued expansion along the ±2 direction. The CA model predicts a macroscopic resolved shear stress T r that monotonically increases to m, which compares well with m, obtained if eqn. (10) is used with S = s^ ª 92 b, q = 0, and n = 0.1. However, the CA result is substantially different from m, obtained if eqn. (10) is used with S = b instead. 3.3 Simulation of cross-slip during operation of a Frank-Read source This section shows that the CA model is capable of capturing basic features of cross slip. The particular geometry is shown in Figure 8(a), in which a Frank-Read source involving a pinned screw dislocation OZ with line direction and Burgers vector along [ 101] is positioned a distance s^ from a fixed, infinitely long screw dislocation of the same sign. The two screw dislocations lie in the same (111) parent slip plane so that a local coordinate system (1', 2', 3') can be defined with s^ measured along the 1'-direction. Normal operation of the Frank-Read source involves the bow out of line OZ along the 1'- direction in the (111) parent plane. However, the like-signed fixed dislocation will repel the bow-out. Line OZ is also contained within a (11 1) cross-slip plane so that screw portions of the bow out can circumvent the repulsion from the fixed dislocation by crossslipping. Thus, Figure 8(a) shows that the source may expand by slip on the parent plane (p), slip onto the positive portion of the cross slip plane (+c), or slip onto the negative portion of the cross slip plane (-c). The elastic, applied stress, and microstructural stress terms will compete to determine whether mode p, +c, or -c prevails in the initial bow-out process. A map can be constructed to highlight the trade-off in these terms and provide groundwork for evaluation of the CA results to follow. The stress field produced by the fixed, infinitely long dislocation is viewed as a microstructural stress and takes the form [12] 10/16

11 s m 1'2 ' = s m 2 '1' = - mb 2p x 2 ' x 2 1' + x 2 ' m 2, s 2 '3' = s m 3'2 ' = mb 2p x 1' x 2 1' + x 2 ' 2, all other s i m ' j ' = 0 (11) Eqn. (5a) can be used to solve for the critical values l (p), l (c), l (-c) needed to nucleate a triangular patch along the paths p, +c, -c, respectively. In particular, the condition Df = 0 is applied with De approximated by the energy change to nucleate a triangular patch on an infinitely long screw dislocation, given in eqn. (A-2). The microstructural work term for the triangle is evaluated at a point along the line OZ, s m nb b = n i ' s m i ' j ' (x 1' = -s^,x 2 ' = 0)b j ' = f m mb 2ps^ (12) where f m = 1, 1/3, and -1/3 for paths p, +c, -c, respectively. Eqn. (5a) then yields the critical values of l (k) for nucleation of a triangle along paths k = p, +c, and -c l (k) = m o 2ps nb(k) È b Ê 16L ˆ Í lná + b Î L Ë 81er s^ m f (k) (13) o where s nb(k) = n i(k) s ij o b j(k) is the resolved shear stress to drive slip on path k, due a unit applied stress s ij o. Figure 9 shows the predicted paths for initial triangular slip as a function of s^ and o o the ratio R = s nb(c) /s nb(p ) of applied resolved shear stress on paths c and p. Here, the boundary between regions p and c is determined by setting l (p) = l (c) with r = 0.1 b and L = b and the corresponding construction l (p) = l (-c) is used to determine the boundary between p and -c regions. The results indicate that for large s^, slip propagation is favored on the path with the largest positive applied resolved shear stress. As s^ is decreased, repulsion by the fixed dislocation makes slip on the -c and c paths possible even when path p has the largest applied resolved shear stress (i.e. when R < 1). At sufficiently small s^, cross slip onto the -c path is preferred even for R = 0, due to repulsion from the fixed dislocation. Table 2 displays the corresponding numerical results from the CA model using a Frank-Read source length S = b, and r and L values mentioned above, so that N i = 20. Cases 1, 2, and 3 display the results for s^ =, 0.17 S, and 0.13 S, respectively. The components of the applied reference stress state ares o 11 = 1, s o o 12 = s 21 = 1, and all other s o ij = 0, so that there is zero resolved shear stress on the cross slip paths (R = 0). The prediction in Figure 9 is that path p is preferred initially and, indeed, the CA simulations confirm this. For Cases 1 and 2, the source operates without any cross slip but in Case 2, significant distortion occurs as the source pinches through the narrow constriction of 11/16

12 dimension s^, as shown in Figure 8(b). For Case 3, s^ is sufficiently small so that cross slip on the -c path is favored over the pinching process. The competition between the p, +c, and -c paths can be manipulated by imposing nonzero values of resolved shear stress on the cross slip planes. Case 4 shows the result for s^ = 0.17 S and R = -0.54, which is imposed by selecting nonzero components s o 11 = 0.3 and s o o 12 = s 21 = 1 for the applied stress state. According to Figure 9, application of a negative R shifts the prediction toward the -c mode and, indeed, the CA results show that the source expands by a combination of p and -c paths as shown in Figure 8(c). Application of a positive R = in Case 5 shifts the prediction in Figure 9 toward the c mode and, indeed, the CA results shown in Figure 8(d) confirm expansion by a combination of p and +c slip. Thus, Figure 9 provides the correct trends as a function of R and s^, but it is limited to slip of the first triangle. The results in Table 2 support the outcome that cross slip reduces the operating stress for a nearby source. In particular, the CA results for Cases 3, 4, and 5 show that the macroscopic resolved shear stress T FR to operate the Frank-Read source is larger if the cross slip algorithm in the CA simulations is disabled, so that the loop is constrained to slip in the parent plane. For Cases 1 and 2, parent slip is favored anyway so that disabling the cross slip algorithm does not affect the maximum T FR. Table 2. Effect of obstacle distance and applied stress state on Frank-Read source operation o o Case No. s^ R = s nb(c) /s nb(p ) (T FR /m) * (T FR /m) Evolution path p S ~0.035 p S ~0.026 p and -c S ~0.018 p and -c S ~0.018 p and +c S = b, r = 0.1 b and N i = 20. * cross slip is excluded from happening. 4. Conclusions A 3D dislocation cellular automaton (CA) model is developed in which FCC crystalline material is discretized into a 3D array of triangular patches, each of which can slip. The energy of a dislocated system is computed exactly within the limits of elasticity theory for the self and interaction energies of Volterra dislocation segments in elastically homogeneous material. Additional contributions to the system energy stem from the work done by the applied stress and internal sources of stress due to inclusions or other inhomogeneities. The dislocation configuration is updated by successively slipping triangular patches according to a path of steepest decent in energy, with the constraint that only one triangle slips at a time. Computational parameters include the size of the triangular patch and the interaction or core cut-off used to model Volterra segments. 12/16

13 Applications to simulate operation of a Frank-Read source show that the CA model predictions agree well with the continuum analytic solution for an isolated source with isotropic dislocation line energy, provided that a continuum measure of resolved shear stress is computed as a moving window average of the critical resolved shear stress to activate clusters of triangles. Averages over smaller clusters of triangles show substantial fluctuation associated with the nucleation and growth of small dislocation loops. In the limit of decreasing the triangular patch size down to the atomic scale, these small cluster averages correspond to kink pair nucleation and growth. Computationally, it is observed that the triangular patch side length L normalized by the dislocation cut-off r must be at least 81e 2 /16 ª 37 in order to expand dislocation loops in a stable fashion. This consequence of a Volterra-based dislocation model is well known [5]. Application to a Frank-Read source in the vicinity of a repulsive, parallel dislocation line shows that the loop will expand by a combination of slip in the parent plane and cross slip, provided the source is sufficiently close to the obstacle or that the applied stress produces a sufficient resolved component on the cross slip system. In such cases, cross slip significantly decreases the stress to operate the source, compared to simulations where cross slip is turned off in the simulations. The primary benefit to a CA approach is that complicated evolution of dislocation loops, including cross slip, can be modeled in a geometrically simple manner. Computational complexities associated with the singular interaction between Volterra dislocation segments are avoided using a discrete system. The approach has a microstructural appeal since, in the limit of decreasing patch size, the discretization mimics the nucleation and growth of kink or jog pairs. Thus, the model lends itself to a Monte-Carlo approach for system evolution. The 3D space filling discretization permits kink and jog formation at arbitrary locations along a dislocation line, and even loop formation at arbitrary volume sites. The operation of a Frank-Read source near a barrier highlights the 3D cross slip configurations that evolve using this approach. The limitations are that a uniform 3D array of triangular patches requires that the smallest features of the analysis control the overall scale of the discretization. For example, a long straight dislocation line with a small bow out is modeled with triangles dictated by the size of the bow out. Further, slip of triangular patches produces systematic fluctuations in line length, so that smoothening via averaging over clusters of slipped triangles is necessary to mimic a continuum response. Thus, in geometries with disparate length scale and in cases where a continuum response is desired, the CA approach is not as computationally efficient as those using arbitrary grid-free motion of frontal tracking nodes [1-6]. Acknowledgements The authors gratefully acknowledge the support of the National Science Foundation Mechanics & Materials and Metallic Materials Programs (CMS ) and the Air Force Office of Scientific Research Metallic Materials Program (F ). PMA also acknowledges support from a Matthias Scholar position while on sabbatical at Los Alamos National Laboratory. 13/16

14 References [1] Kubin LP, Canova GR. Scripta Metall. Mater. 27: (1992). [2] Devincre B, Kubin LP, Lemarchand C, Madec R. Mater. Sci. Engin. A : (2001). [3] Zbib HM, Rhee M, Hirth JP. Int. J. Mech. Sci. 40: (1998). [4] Faradjian AK, Friedman LH, Chrzan DC. Modelling Simul. Mater. Sci. Eng. 7: (1999). [5] Schwarz KW. J. Appl. Physics 85: (1999). [6] Ghoniem NM. J. Engin. Mater. Tech. 121: (1999). [7] Sasaki K, Kishida M, Ekida Y. Int. J. Numer. Meth. Engin. 54: (2002). [8] Shen C, Wang Y. Acta Mater. 51: (2003). [9] Rao SI, Hazzledine PM. Phil. Mag. A 80: (2000). [10] Mitchell TE, Baskes MI, Hoagland RG, Misra A. Intermetallics 9: (2001). [11] Wirth BD, Odette GR, Maroudas D, Lucas GE. J. Nucl. Mater. 276: (2000). [12] Hirth JP, Lothe J. Theory of Dislocations (2nd ed), Wiley and Sons, New York (1982). [13] Anderson PM, Foecke T, Hazzledine PM. MRS Bulletin 24: (1999). [14] Li Q, Anderson PM, J. Elasticity 64: (2001). [15] Daehn GS. Acta Mater (2001). [16] Cai W, Bulatov VV, Yip S, Argon AS. Mater. Sci. Engin. A : (2001). [17] Anderson PM, Li Q. Modeling the Performance of Engineering Structural Materials III, (Srivatsan TS, Lesuer DR, Taleff EM, ed) TMS (2002). [18] Foreman A. Phil. Mag. 15: 1011 (1967). [19] Animations for Cases 1-5 in Table 2 are available at Appendix The elastic energy to nucleate a triangular patch PBA on a finite length screw dislocation as shown in Figure 4 is obtained by using a construction outlined in Section 6-4 of Hirth and Lothe [12]: 8pDE mb 2 = 2L ln Ê 8L ˆ Á Ë 81er + 2 È -L i ln Ê Á 3 ˆ ÂÍ Ë 4 i =1Î Ê L 2 i + L 2 ˆ + L i L + L + 0.5L i +L i ln Á Á L Ë i Ê L 2 i + L 2 ˆ + L i L + 0.5L - 0.5L i + ( L i + L) ln Á (A-1) Á L i + L Ë Ê L 2 i + L 2 ˆ + L i L + L i + 0.5L +2L ln Á Á L Ë 14/16

15 [ ( ) + ( L i + L) ln( 2L i + 2L) - L ln( 2L) ]] -2 -L i ln 2L i In the limit L 1, L 2 Æ so that the screw dislocation is infinitely long, the above expression may be reduced to 8pDE mb 2 16L = 2L ln( 81er ) (A-2) 15/16

16 Figure Captions Figure 1: Discretization of a FCC crystal into cubes of edge length u aligned to a <100> crystallographic basis. Figure 2: Projection of the <110>/{111} and <110>/{001} FCC slip systems listed in Table 1. Figure 3: Discretized {111} plane with shaded regions denoting slipped patches. Figure 4: Nucleation and growth of triangular slip on a straight dislocation line. Figure 5: (a) Resolved shear stress t n to nucleate a triangular patch along the midpoint of an infinitely long screw segment where L 1, L 2 Æ in Figure 4; (b) Difference, t n - t n, in resolved shear stress to nucleate a triangular patch on a finite length versus infinite length screw dislocation geometry shown in Figure 4. Figure 6: Operation of a Frank-Read source of length S = b using triangular patches of side length L = 15 2 b, r = 0.3 b, and n = 0, showing (a) the sequence of expansion. Corresponding resolved shear stress as a function of the number i of slipped triangles, computed by averaging over (b) triplets (t t ) and pairs (t p ) of patches and (c) rows of patches (T r ) according to eqn. (8). The critical resolved shear stress to operate the source is denoted by T FR. Figure 7: (a) The critical resolved shear stress t FR to operate a Frank-Read source as a function of source length S, as given by eqn. (10) with n = 0 and r 0 = 0.6 b. Corresponding predictions, T FR, from the CA simulation are shown as points, with numbers in parentheses denoting the initial number of segments, N i ; (b) error in CA model predictions as a function of N i for S = b. Figure 8: (a) Geometry of a Frank-Read source of length S in the vicinity of a parallel screw dislocation. Evolution of the Frank-Read source with applied stress, using parameters listed for (b) Case 2; (c) Case 4; and (d) Case 5 in Table 2. Figure 9: Predicted initial paths of expansion (p, +c, -c) for a Frank-Read source as a function of the ratio R of resolved shear stress on path c to path p and perpendicular distance s^ from a parallel, same-sign screw dislocation as shown in Figure 8(a). The number next to each point denotes the Case number listed in Table 2. 16/16