Dislocation Multiplication from the FrankRead Source in Atomic Models

Tomotsugu Shimokawa1,+ and Soya Kitada2

1School of Mechanical Engineering, College of Science and Engineering, Kanazawa University, Kanazawa 920-1192, Japan2Division of Mechanical Science and Engineering, Graduate School of Natural Science & Technology, Kanazawa University,Kanazawa 920-1192, Japan

Dislocation multiplication from the FrankRead source is investigated in aluminum by applying atomic models. To express the dislocationbow-out motion and dislocation loop formation, we introduce cylindrical holes as a strong pinning point to the dislocation-bowing segment.The critical configuration for dislocation bow-out in atomic models exhibits an oval shape, which agrees well with the results obtained by theline tension model. The critical shear stress for the dislocation bow-out in atomic models continuously increases with decreasing length L of theFrankRead source (even at the nanometer scale). This is expressed by the function L¹1 lnL, which is obtained by a continuum model based onelasticity theory. The critical shear stresses for the FrankRead source are compared with those for grain boundary dislocation sources, as well asthe ideal shear strength. [doi:10.2320/matertrans.MA201319]

(Received September 3, 2013; Accepted October 15, 2013; Published November 29, 2013)

Keywords: dislocation multiplication, FrankRead source, dislocation bow-out, dislocation source, atomic simulation, mechanical property

1. Introduction

Plastic deformation of crystalline materials at low temper-atures occurs by dislocation movements; hence, the resistanceto dislocation motion controls material strength. For coarse-grained polycrystalline metals with a grain size d larger than1 µm, yield strength increases as grain size decreases. Thisphenomenon is known as the HallPetch relation,1,2) wherethe grain boundaries act as a strong barrier to the dislocationmovement. One of the theoretical models available todescribe this behavior is the dislocation pile-up model,which assumes that a dislocation pile-up will induce a slip inthe adjoining grain if the stress at the spearhead of the pile-upreaches a critical value.3) On the other hand, if the grain sizedecreases below 1µm, the strength cannot be expressed bysimple extrapolation of the conventional HallPetch slope.4)

This result implies that we should consider other size effectsof plastic-deformation phenomena, such as the competitionbetween critical shear stresses for dislocation nucleationfrom intragranular and intergranular dislocation sources,57)

which is believed to be a dominant plastic-deformationmode in nanocrystalline metals (d < 100 nm).8) Atomic-levelresolution is required to investigate grain-boundary-mediatedplastic deformations; therefore, atomic simulations thatsimultaneously treat intra- and intergranular dislocationsources are expected to be a powerful tool to elucidate theunique mechanical properties of ultrafine grained metals.911)

The FrankRead source is the well-known intragranulardislocation source,12) in which a dislocation segment lying onan active slip plane ® and whose ends are strongly pinned bythe other parts of the dislocation lying outside the plane®bows out under an applied stress. At critical stress, thisdislocation segment can generate dislocation loops, as aresult of which dislocation multiplication can occur.Although many studies on the FrankRead source have beenreported to date, including the use of analytical expressionswith polygonal shapes,13) numerical calculations investigat-ing its equilibrium shape under an external stress14) with self-

stress,15) or dislocation dynamics simulations,1618) only afew atomic simulations on this system are currently availablein the literature.19,20) This is due to difficulties in obtainingstrong pinning points in the atomic models for the dislocationbow-out and the subsequent dislocation multiplication.21)

In this study, we attempt to realize the FrankRead source(which can induce stable dislocation multiplication) in atomicmodels and compare the critical shear stress for this sourcewith previously reported values13,14,19,22) to evaluate theirvalidity at the nanometer scale. Furthermore, the criticalshear stresses for dislocation nucleation from the FrankRead source are compared with those from grain boundariesreported in Ref. 7).

2. Methodology

Figure 1 shows our developed FrankRead source model.The embedded atom method for aluminum, proposed byMishin et al.,23) is adopted for the atomic interactions. Thecrystal orientation is set to be the same as that used in themodel reported by de Koning et al.19) However, it shouldbe noted that the Burgers vectors of the two dislocationscreated on the {111} planes and the two dislocations createdon the {112} planes have directions opposite to those givenin the de Koning model because of the different proceduresapplied to make the FrankRead sources. The FrankReadsource in our model is created by introducing an atomic plateinto the {110} atomic layers. The dimensions of the atomicplate are L and H in the y and z directions, respectively.

We performed a preliminary simulation by applying theFrankRead source model developed by de Koning usingH = 4 nm and L = 10 nm. We observed cross slips whichhave their stating point at the pinning arms of {112}dislocations, a migration of the pinning points, and thedislocation bow-out from the dislocation segment on the{112} planes before reaching the critical conditions. Theseresults suggest that no perfect dislocation bow-out motionoccurred when applying the model by de Koning. Thisunexpected phenomenon has also been reported by Liet al.20) and Tsuru et al.11) To obtain strong pinning points,+Corresponding author, E-mail: simokawa@se.kanazawa-u.ac.jp

Materials Transactions, Vol. 55, No. 1 (2014) pp. 58 to 63Special Issue on Strength of Fine Grained Materials ® 60 Years of HallPetch®©2013 The Japan Institute of Metals and Materials

http://dx.doi.org/10.2320/matertrans.MA201319

cylindrical holes with a diameter of 1 nm are introduced byremoving the dislocation core structure on the {112} planes.The dislocation segments on the {111} planes are thusterminated by the free surfaces of the cylindrical holes.Weinberger et al. reported that cylindrical holes have enoughdislocation-pinning potential.21) Furthermore, to prevent across slip, whose starting point is the pinning segment alongthe [111] direction, the x-directional displacements of theatomic clusters located next to the slip plane of thedislocation segments on the {111} planes are controlled.The displacement-controlled regions around the cylindricalholes are shown in Fig. 1(a) as hatching regions and inFig. 1(b) as solid marked atoms. Each FrankRead sourcecorner has four atomic clusters, as shown in Fig. 1(b), and thex-directional displacement of each cluster is taken from theaverage x-directional displacement of each atom in thatcluster. The other directional displacements of the atomicclusters are not controlled.

The dimensions of the analytical models in the x, y and zdirections (lx, ly and lz) are set to 40, 40 and 20 nm,respectively, and periodic boundary conditions are adopted inall directions. H is set to 4, 6 and 8 nm, and L is set to begreater than or equal to H. Each analytical model is relaxedfor 20 ps under no external stress. The analysis temperature iskept at 10K by the velocity scaling method,24) and eachdimension lx, ly and lz is controlled to keep the nominal stressat zero.25) The equilibrium atomic configuration of the FrankRead source is shown in Fig. 1(c), where the dark- and light-gray spheres correspond to atoms in the local hexagonalclose-packed (hcp) configuration and defect structures,

respectively; the atoms in the face-centered cubic structureare not shown. The classification of atomic structures isperformed by common neighbor analysis.26) For each relaxedmodel, shear deformation tests are performed at _£zx ¼ �5�108 1=s at 10K to investigate the dislocation bow-out motionfrom the FrankRead source.

3. Results and Discussions

3.1 Dislocation multiplication from the FrankReadsource

Figure 2 shows the changes in shape of the upperdislocation segment with H = L = 4 nm under externalloading. The lower dislocation segment behaves similar tothe upper one. Figure 3 shows the variations in the number ofdefect atoms (Ndef ), including hcp atoms. The letters in Fig. 3correspond to the shapes of the dislocation bowing-outsegment in Fig. 2. Initially, the value of Ndef increasesgradually, until the dislocation bow-out state in Fig. 2(b) isreached, and after that it continues to grow dramatically.Finally, a perfect dislocation loop is formed, as shown inFig. 2(f ). Because periodic boundary conditions are adoptedin all directions, the dislocation loop is eventually destroyedby neighboring dislocation loops in image cells (dislocationannihilation), and therefore, the defect structures in theanalytical model return to their initial state with plasticdeformation of one of the Burgers vectors. As shown inFig. 3, the dislocation segment can regenerate a dislocationloop under an external loading, which confirms that ourdeveloped FrankRead source has strong pinning points for

Fig. 1 (a) Analytical model for the FrankRead source showing a dislocation segment L terminated by cylindrical holes with height H.(b) Displacement control regions to prevent cross slip of the bowing dislocation segment. (c) Atomic configurations of the FrankReadsource under no external load. The light and dark colored spheres represent defect and hcp atoms, respectively.

Dislocation Multiplication from the FrankRead Source in Atomic Models 59

the dislocation segments and can suppress the cross slips.Thus, our results demonstrate that dislocation multiplicationcan be realized by atomic simulations.

3.2 Critical dislocation bow-out configurationIn situation (b) in Fig. 3, Ndef increases dramatically once

the critical conditions for the dislocation bow-out motion arereached. The stress in situation (b) can therefore be attributedto the critical shear stress, ¸MD.27) Figure 4 shows the shapesof dislocation segments with different L values in the criticalstate. The value of H is fixed to 4 nm in all models. Thewhite broken lines in Fig. 4 represent the critical dislocationconfiguration in an isotropic elastic body, with a Poisson ratioof 0.33, estimated by de Wit et al.28) Because of the differentline tensions between the edge and screw dislocationcomponents, the critical configuration does not show aperfect circular form, as estimated by the constant linetension model, but rather an oval shape with a ratio of theminor axis length to the major axis length of 1 ¹ ¯, where ¯is the Poisson ratio. The critical configuration obtained byatomic simulations agrees well with the results reported by deWit et al. because the anisotropic factor, calculated by c44/(c11 ¹ c12), is 1.25 for aluminum (expressed by the atomicpotential); hence, we can assume that this metal showsisotropic elastic properties. Hatano et al.29) also reported the

same agreement: critical dislocation bow-out shapes, thatends are pinned by nanovoids, in copper calculated by atomicsimulations agree well with the results reported by de Witet al. The obtained results suggest that the influence of thedisplacement control regions on the dislocation bow-outmotion is neglected in our models.

3.3 Critical shear stress for dislocation bow-outFigure 5 shows the relationship between the critical shear

stress ¸MD and the length of the dislocation segment L.The value of ¸MD reported by de Koning et al. (withL = 5 nm and H = 2.8 nm) is also plotted. We also showthree critical shear stresses, namely, ¸OR, ¸HL, and ¸SB, where¸OR is estimated by a constant line tension approximationcommonly known as the Orowan stress30) [eq. (1)], ¸HL isderived from analytical expressions for the energies of thedislocation segments with a semi-regular hexagon config-uration [eq. (2) by Hirth and Lothe13)], and ¸SB represents thevoid-row bypassing stress, void spacing L, and diameter D,and is numerically evaluated by considering a dislocationself-interaction and a dislocation-void interaction in eq. (3)by Scattergood and Bacon.22)

¸OR ¼®b

Lð1Þ

¸HL ¼®b

2³Lð1� ¯Þ 1�¯

2ð3� 4 cos2 ¢Þ

h iln

L

r0� 1þ ¯

2

� �

ð2Þ¸SB ¼

®b

2³Lln

r0D

þ r0L

� ��1þ 1:52

� �ð3Þ

Here, ®, b, r0 and ¢ are the shear modulus, the magnitude ofthe Burgers vector, the dislocation core radius, and the anglebetween the Burgers vector and the original dislocationstraight line, respectively. In this study, r0 and ¢ are set to band 0, respectively, and because eqs. (1) to (3) are valid forisotropic elastic bodies, the shear modulus ® and Poisson’sratio ¯ are estimated by the Voigt approximation31) foranisotropic elastic bodies, expressed by the atomic potential.We thereby obtain ® = 29.2GPa and ¯ = 0.33. In eq. (3),although D originally expresses the void diameter, we assumethat D is ly ¹ L because the FrankRead sources areperiodically arranged in the y direction in Fig. 1.

Generally, considering the dislocation self-interactiondecreases the critical bow-out stress, that is, ¸HL and ¸SB issmaller than ¸OR, as shown in Fig. 5. Moreover, ¸HL shows a

Fig. 2 Dislocation multiplication from the FrankRead source under external loading in the atomic simulations. L = H = 4 nm.

Fig. 3 Relationship between the number of defect atoms Ndef (includinghcp atoms) and the simulation time for the dislocation multiplicationconditions shown in Fig. 2 (the letters represent the different dislocationbow-out states indicated in Fig. 2).

T. Shimokawa and S. Kitada60

non-monotonic dependence on L, as shown in Fig. 5. Thedislocation bow-out from the FrankRead source with thisnon-monotonic dependence is expected to be a dominantmechanism for explaining the inverse HallPetch relationfound for a number of nanocrystalline metals.32) However,the ¸MD value obtained by atomic simulations shows amonotonic dependence on L, which means that ¸MDcontinuously increases with decreasing L. It is interestingthat the ¸MD values obtained by our atomic models fit the ¸SBvalues obtained by continuum model simulations, whichinclude several approximations regarding the dislocation coreradius used (r0 = b) and the dislocation-void interactionconsidered using reasonable surface energy.22) Osetsky et al.also found the same agreement for the motion of an infinite,straight but flexible edge dislocation through a row of voidswith diameters ranging between 0.9 and 4 nm in body-centered cubic iron.33) These results suggest that thecontinuum model based on elasticity theory has enoughpotential to describe dislocation bow-out motions, even in thenanometer region. Our atomic simulations studies also showthat it is difficult to explain the inverse HallPetch relation byusing the dislocation bow-out motion from the FrankReadsource.

3.4 Influence of dislocationdislocation interactions onthe dislocation bow-out

As shown in Fig. 5, the ¸MD values, including the results ofthe pinning arms of the {112} dislocations reported by deKoning et al.,19) decreases with increasing H, which is thedistance between the upper and lower dislocation segments

in the absence of external loading. If we assume that theinfluences of the pinning points of both the cylindrical holesand the {112} dislocations on the dislocation bow-out motionare similar, the interactions between the upper and lowerdislocations will influence the value of ¸MD. Foremansuggests that the critical bow-out stress for an edgedislocation segment is approximately described by:

¸c ¼ A®b

2³Lln

L

r0

� þ B

� �; ð4Þ

where A is almost 1 for the edge and 1.5 for the screwsegments, and the constant B depends on the orientation ofthe side-arms and the presence of local stress fields.14) In thissection, we assume that ¸MD is expressed by eq. (4) andevaluate the influence of dislocationdislocation elasticinteractions by comparing the critical stress ¸FO obtainedby Foreman14) for a dislocation segment L with verticaldislocation arms to the glide plane of the segment using theBrown self-stress method.15) Figure 6 shows the relationshipbetween the critical shear stress ¸MD (in units of ®b/L: ¸OR)and L/r0 (r0 = b). The broken and solid lines represent ¸FOand eq. (4), respectively. The values of the constant B,fitted to each H value, are also shown in Fig. 6. Although theupper and lower dislocation segments in the atomic modelsinteract with other dislocation segments in the image cells(owing to periodic boundary conditions), the relationshipbetween ¸MD/¸OR and L/r0 is found to fit the form eq. (4),with A = 1, for all the studied H values. A similar influenceof the periodic boundary conditions is observed in discretedislocation simulations.18)

Fig. 5 Critical shear stress as a function of the FrankRead source length. Fig. 6 Critical shear stress (in units of ®b/L) as a function of the FrankRead source length (in units of r0).

Fig. 4 Critical configurations of the dislocation bow-out for different source lengths. The white broken lines were obtained by the linetension model.

Dislocation Multiplication from the FrankRead Source in Atomic Models 61

Because the Foreman model contains one dislocationsegment for the bow-out motion, ¸FO is regarded as a result ofH¼ ¨. The constants A and B for ¸FO are approximately 1and 0.66, respectively; hence, we can simply considerthe relationship between B and H. Figure 7 shows a plotof B vs. H¹1 for ¸MD and ¸FO. It is interesting to notethat the value of B increases linearly with H¹1, as shownby the broken line. This relationship is expressed by theequation B = 2.6/H + 0.66; however, it is not easy toanalytically understand. The critical shear stress ¸MD fordislocation multiplication from the FrankRead source inatomic models is described by a form of eq. (4), and thevalue of ¸MD when H¼ ¨ is possibly expressed by ¸FOassuming that the influences of the pinning points of thecylindrical holes and dislocations on the dislocation bow-outmotion are similar.

3.5 Intra- and intergranular dislocation sourcesThe critical shear stresses for dislocation nucleation from

the FrankRead source, ¸FR (= ¸MD), are compared withthose from the h112i symmetrical tilt grain boundaries, aswell as the ideal shear strength. The critical shear stresses ofthe intergranular dislocation sources, ¸GB, are evaluated usingreported values of the critical tensile or compressive stressesfor dislocation nucleation from structural units in the tiltgrain boundaries.7) The ideal shear strength, ¸IS, is estimatedusing the critical shear stress for homogeneous nucleation ofdislocation loops in a perfect crystal structure whose crystalorientation is the same as that used in the FrankRead sourcesimulation. All the critical shear stresses are evaluated usingthe (111)[110] shear stress component. The analysis temper-ature is maintained at 10K for all the simulations, and thestrain rates are _£ ¼ 5� 108 1=s and _¾ ¼ 1� 109 1=s for thehomogeneous dislocation nucleation model and the grainboundary model, respectively. The obtained ideal shearstrength, ¸IS, evaluated by the homogeneous nucleation ofdislocation loops, is 3.65GPa.

We introduce a characteristic length of the dislocationsources S to compare the critical shear stresses for variousdislocation sources. S is determined by the source length L,regular intervals of the structural units h,7) and Burgersvectors b for the FrankRead source, the grain boundarydislocation source, and the homogeneous dislocation nucle-ation source, respectively. It should be noted that S has noconnection with grain size, except for the FrankRead source.Figure 8 shows the relationship between the critical shearstress and S for each dislocation source. ¸FR continuouslyincreases with decreasing length S. The solid line obtained bya continuum model in eq. (4) (with A = 1 and B = 1.64)indicates that ¸FR becomes larger than ¸IS if S is smallerthan 1 nm. However, we performed a FrankRead sourcesimulation using S = 3 nm and H = 4 nm in the atomic

Fig. 7 Constant B in eq. (4) as a function of H¹1. The symbols , ,and © have the same meaning as in Fig. 6, and represents the resultsobtained by Foreman14) when H¼¨.

Fig. 8 Relationship between critical shear stress and characteristic length of the dislocation source. The symbols , ( ) andrepresent the critical shear stresses ¸FR (= ¸MD), ¸GB under tensile loading (compressive loading), and ¸IS, respectively. Grain boundarystructures, in descending order by regular intervals of structural units S (= h), are 73, 77, 15 and 59 for tensile loading, and 105,125, 31, 21 and 35 for compressive loading.

T. Shimokawa and S. Kitada62

model; we observed that dislocation multiplication occurred(¸FR µ 1.81.9GPa), but the critical configuration of thedislocation bow-out was different from that shown in Fig. 4.This is due to the interaction between the bowing-outdislocation and another dislocation nucleation from thecylindrical hole, which is observed when ¸ reaches 1.7GPa.Furthermore, in the case of S = 2 nm and H = 4 nm, an edgedislocation dipole terminated by cylindrical holes, as shownin Fig. 1(c), cannot be constructed after relaxation because ofthe short distance between the two surfaces of the cylindricalholes (i.e., approximately 1 nm). Therefore, the minimumlength of S is 4 nm for our developed FrankRead source, andthe maximum ¸FR value is 1.36GPa.

In the case of the grain boundary dislocation source, ¸GB

increases upon decreasing the regular interval of structuralunits S. Because edge dislocations are nucleated from thestructural units, the elastic interactions between grainboundary dislocations existing at the structural unit7) stronglyinfluence ¸GB. The minimum ¸GB value is approximately1GPa in the case of 105 under compressive loading; thisvalue is smaller than that of ¸FR for S = 5 nm. However,almost all ¸GB values are larger than the maximum ¸FR value,which suggests that the dominant dislocation source is theFrankRead source if a crystal contains FrankRead sourceswith a source length S larger than approximately 5 nm.However, it should be noted that there is a possibility ofdecreasing the value of ¸GB, obtained by atomic simulationsusing bicrystal models under nearly athermal condition,7)

if we consider the thermally activated process under areasonable strain rate used in conventional experiments andstress concentration at grain boundaries and triple junctions.

4. Conclusion

Dislocation multiplication from the FrankRead sourceis realized by atomic models. The obtained critical shapeand stress for the dislocation bow-out motion are comparedwith previously reported results to evaluate them. Our mainfindings are summarized as follows:

(1) Dislocation multiplication from the FrankRead sourceis realized by atomic models by adopting cylindrical holes asa strong pinning point to the dislocation-bowing segment andalso suppressing cross slips, which have their starting pointat the surface of the pinning holes.

(2) The critical configuration for the dislocation bow-outobtained by atomic simulations exhibits an oval shape with aratio 1 ¹ ¯ between the minor and major axis lengths. Theobtained shape agrees well with the results obtained by theline tension model.

(3) The critical shear stress for the dislocation bow-out,determined by atomic models, is found to continuouslyincrease with decreasing values of the FrankRead sourcelength L. This can be expressed by the function L¹1 lnL,which is obtained by a continuum model based on elasticitytheory, even at the nanometer scale.

(4) The minimum FrankRead source length L realized inthe developed atomic model is 4 nm and the maximumcritical shear stress, which is one third of the ideal shearstrength, is larger than that for dislocation nucleation fromh112i tilt grain boundaries with long structural periodicities.

Acknowledgements

This research was supported by the Ministry of Education,Culture, Sports, Science and Technology (MEXT)KAKENHI Grant Number 22102007, the Japan Scienceand Technology Agency (JST) under Collaborative ResearchBased on Industrial Demand “Heterogeneous StructureControl: Towards Innovative Development of MetallicStructural Materials”, and the Japan Society for thePromotion of Science (JSPS) KAKENHI Grant Number24560091.

REFERENCES

1) E. O. Hall: Proc. Phys. Soc. B 64 (1951) 747753.2) N. J. Petch: J. Iron Steel Inst. 174 (1953) 2528.3) J. C. M. Li and Y. T. Chou: Metall. Trans. 1 (1970) 11451159.4) N. Kamikawa, X. Huang, N. Tsuji and N. Hansen: Acta Mater. 57

(2009) 41984208.5) D. E. Spearot, M. A. Tschopp, K. I. Jacob and D. L. McDowell: Acta

Mater. 55 (2007) 705714.6) M. Kato: Mater. Sci. Eng. A 516 (2009) 276282.7) T. Shimokawa: Phys. Rev. B 82 (2010) 174122.8) Y. Estrin, H. S. Kim and F. R. N. Nabarro: Acta Mater. 55 (2007) 6401

6407.9) T. Shimokawa, T. Kinari and S. Shintaku: Phys. Rev. B 75 (2007)

144108.10) T. Shimokawa, M. Tanaka, K. Kinoshita and K. Higashida: Phys. Rev.

B 83 (2011) 214113.11) T. Tsuru, Y. Aoyagi, Y. Kaji and T. Shimokawa: Mater. Trans. 54

(2013) 15801586.12) F. C. Frank and W. T. Read, Jr.: Phys. Rev. 79 (1950) 722723.13) J. P. Hirth and J. Lothe: Theory of Dislocations, 2nd ed., (McGraw-Hill,

New York, 1982) p. 752.14) A. J. E. Foreman: Philos. Mag. 15 (1967) 10111021.15) L. M. Brown: Philos. Mag. 10 (1964) 441466.16) D. Gomez-Garcia, B. Devincre and L. Kubin: J. Comput. Aided Mater.

Des. 6 (1999) 157164.17) T. Ohashi, M. Kawamukai and H. Zbib: Int. J. Plasticity 23 (2007) 897

914.18) S. Soleymani Shishvan, S. Mohammadi and M. Rahimian: Model.

Simul. Mater. Sci. Eng. 16 (2008) 075002.19) M. de Koning, W. Cai and V. V. Bulatov: Phys. Rev. Lett. 91 (2003)

025503.20) X. Y. Li and W. Yang: Phys. Rev. B 74 (2006) 144108.21) C. R. Weinberger and G. J. Tucker: Model. Simul. Mater. Sci. Eng. 20

(2012) 075001.22) R. O. Scattergood and D. J. Bacon: Acta Metall. 30 (1982) 16651677.23) Y. Mishin, D. Farkas, M. J. Mehl and D. A. Papaconstantopoulos: Phys.

Rev. B 59 (1999) 33933407.24) M. P. Allen and D. J. Tildesley: Computer Simulation of Liquids,

(Oxford University Press, New York, 1987).25) M. Parrinello and A. Rahman: Phys. Rev. Lett. 45 (1980) 11961199.26) J. Dana Honeycutt and H. C. Andersen: J. Phys. Chem. 91 (1987)

49504963.27) To estimate the critical state of dislocation bow-out correctly, we also

use values of dNdef/dt.28) G. de Wit and J. S. Koehler: Phys. Rev. 116 (1959) 11131120.29) T. Hatano and H. Matsui: Phys. Rev. B 72 (2005) 094105.30) D. Hull and D. J. Bacon: Introduction to Dislocations, fourth ed.,

(Butterworth Heinemann, 2001) p. 150.31) W. Voigt: Lehrbuch der Kristallphysik, (Teubner, Leipzig, 1928).32) J. Lian, B. Baudelet and A. A. Nazarov: Mater. Sci. Eng. A 172 (1993)

2329.33) Yu. N. Osetsky, D. J. Bacon and V. Mohles: Philos. Mag. 83 (2003)

36233641.

Dislocation Multiplication from the FrankRead Source in Atomic Models 63

http://dx.doi.org/10.1088/0370-1301/64/9/303http://dx.doi.org/10.1016/j.actamat.2009.05.017http://dx.doi.org/10.1016/j.actamat.2009.05.017http://dx.doi.org/10.1016/j.actamat.2006.08.060http://dx.doi.org/10.1016/j.actamat.2006.08.060http://dx.doi.org/10.1016/j.msea.2009.03.035http://dx.doi.org/10.1103/PhysRevB.82.174122http://dx.doi.org/10.1016/j.actamat.2007.07.052http://dx.doi.org/10.1016/j.actamat.2007.07.052http://dx.doi.org/10.1103/PhysRevB.75.144108http://dx.doi.org/10.1103/PhysRevB.75.144108http://dx.doi.org/10.1103/PhysRevB.83.214113http://dx.doi.org/10.1103/PhysRevB.83.214113http://dx.doi.org/10.2320/matertrans.MH201313http://dx.doi.org/10.2320/matertrans.MH201313http://dx.doi.org/10.1103/PhysRev.79.722http://dx.doi.org/10.1080/14786436708221645http://dx.doi.org/10.1080/14786436408224223http://dx.doi.org/10.1023/A:1008730711221http://dx.doi.org/10.1023/A:1008730711221http://dx.doi.org/10.1016/j.ijplas.2006.10.002http://dx.doi.org/10.1016/j.ijplas.2006.10.002http://dx.doi.org/10.1103/PhysRevLett.91.025503http://dx.doi.org/10.1103/PhysRevLett.91.025503http://dx.doi.org/10.1016/0001-6160(82)90188-2http://dx.doi.org/10.1103/PhysRevB.59.3393http://dx.doi.org/10.1103/PhysRevB.59.3393http://dx.doi.org/10.1103/PhysRevLett.45.1196http://dx.doi.org/10.1021/j100303a014http://dx.doi.org/10.1021/j100303a014http://dx.doi.org/10.1103/PhysRev.116.1113http://dx.doi.org/10.1103/PhysRevB.72.094105http://dx.doi.org/10.1016/0921-5093(93)90422-Bhttp://dx.doi.org/10.1016/0921-5093(93)90422-Bhttp://dx.doi.org/10.1080/14786430310001603364http://dx.doi.org/10.1080/14786430310001603364