Lectures on: Introduction to and fundamentals of discrete dislocations and dislocation dynamics. Theoretical concepts and computational methods

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1 Lectues on: Intoduction to and fundamentals of discete dislocations and dislocation dnamics. Theoetical concepts and computational methods Hussein M. Zi School of Mechanical and Mateials ngineeing Washington State Univesit Pullman W i@wsu.edu Summe school Genealied ontinua and Dislocation Theo Theoetical oncepts omputational Methods nd peimental Veification Jul Intenational ente fo Mechanical Science Udine Ital

2 ontents Lectue : The Theo of Staight Dislocations Zi Lectue : The Theo of uved Dislocations Zi Lectue : Dislocation-Dislocation & Dislocation-Defect Inteactions -Zi Lectue 4: Dislocations in stal Stuctues - Zi Lectue 5: Dislocation Dnamics - I: quation of Motion effective mass - Zi Lectue 6: Dislocation Dnamics - II: omputational Methods - Zi Lectue 7 : Dislocation Dnamics - lasses of Polems Zi

3 Lectue : The Theo of uved Dislocations The Buges Displacement quation Stess Field of a cuved dislocation Voltea Dislocations icula Dislocation Loop ectangula Dislocation Loop The stess filed aout a staight segment dislocation The Somigliana ing Dislocation efeecences I. Demi J.P. Hith and H.H. Zi The Somigliana ing Dislocation J. lasticit haishi T.. Zi H.M. Hith J.P. and de La uia T.D. The stess Field of a Geneal Voltea Dislocation Loop: naltical and Numeical ppoaches Philosophical Magaine haishi T. and Zi H.M. The Displacement Field of a ectangula Voltea Dfislocation Loop Phil Mag

4 Mied Dislocation

5 The self-stess The Peach-oehle equation fo the self-stesses of an cuved closed dislocation loop is given the following line integal see Hith and Lothe 98: G 8 m G 4 im m i imk d G 8 i m im i i d d k jp P Field point dl

6 k j m ijk i k mik i j j m m d d d u The Buges equation fo displacements fo an closed cuved dislocation loop is given in tems of line and aea integals as see Hith and Lothe []:

7 ltenativel Buges equation has the following vecto fom: dl u gad d dl is the solid angle though which the positive side of is seen fom is seen fom and is defined as The displacement field is impotant in inteaction polems etween dislocations and emedded paticles in a softe metallic mati e.g. in metal-mati composites MMs the ounda condition in the polem is that of eo displacement at a set of collocation points on the inteface. The condition is enfoced annulling the undesied displacements caused the cstal o mati dislocations with the displacements coming fom the distiution of ectangula dislocation loops. In ode fo this to happen a quantification of the displacement field of each of the ectangula dislocation loops is necessa and is theefoe developed in this pape. The integation of the Buges equation to otain the displacements tuns out to e athe involved. It is caied out with espect to the gloal sstem. To pefom the integation the indices in ae fist epanded. This gives thee independent equations one fo each displacement component u v and w coesponding to suscipt o inde m values of and espectivel. Note that u is the displacement in the -diection v in the -diections and w in the -diection. Befoe continuing along with the integation few steps ae in ode. Fist note that the elevation of the dislocation loop is fied with espect to the gloal coodinate sstem. This means that the dislocation loop lies in a plane paallel to the planes such that is a constant. Second note that the following holds: /. Thid along segment a and d 0 along segment a and d 0 along segment and d 0 and along segment 4 and d 0.

8 ectangula Dislocation Loop ectangula dislocation loop is composed of fou linea dislocation segments see figue. Fo this wok the Buges vecto is assumed to e constant in magnitude and diection fom one point to anothe along the loop. pt P = dl 4 a O haishi T. and Zi H.M. The Displacement Field of a ectangula Voltea Dfislocation Loop Phil Mag

9 The displacement field of a ectangula dislocation loop u w v u ] [ 8 4 a a a a a a a a 8 a a a a

10 0.06 u a a Figue. Thee-dimensional capet plot of u nomalied the Buges vecto magnitude whee ==0=00. Hee =/ a==00 =0 and =0.

11 Suface pt P B B..: Loop i u=0 ' O pt P O ' ' ' Loop i schematic of a igid fie close to elastic displacement souces e.g. a dislocation loop and/o segment. The coect ounda condition to satisf is eo elastic displacement on the fie's suface. ' igid fie suounded a glide dislocation loop. The fie suface is meshed with ectangula elements epesenting ectangula dislocation loops see inset. Point P is just a field point. ' haishi T. and Zi H.M. Dislocation Dnamics Simulations of the Inteaction Between a Shot igid Fie and a Glide Dislocation Pile-up. omp. Mate. Sci

12 icula Dislocation loop loop loop O O d pt. pt. P

13 Pismatic Dislocation Loops Pismatic dislocation loops with a Buges vecto nomal to the plane of the loop can fom in a mateial suject to iadiation o quenching the pecipitation of vacancies o intestitial atoms. Glide o shea loops can fom eadil e.g. in the glide plane of a Fank- ead souce as it continues to emit popagating dislocations. Hence it is impotant to quantif the stess field of these loops in ode to account fo thei inteaction with othe dislocation cuves o defects in the cstal.

14 The stess field of a cicula Voltea dislocation loop

15 G G G G G G k k k k k k k G k k k k 8 k k k k G k G k 9 G G 4 G G G k k k k 4 k k G k k k k k G 5 k 7 6 k k 8 4 k

16 /G /- The stess field of a cicula loop and staight segments appoimations / N=6 N= N=8 N=4 s anal / a c

17 Inteaction etween a defect Fank sessile loop; defect clustes in iadiated mateials and a glide edge dislocation Inteaction is weak / d= l Fig. 5. a TM pictue showing dislocation pinning the dispesed hadening mechanism. DD pictue showing the same mechanism at wok as captued in the simulations. The dots in the figue epesent defect clustes.

18 Somigliana Dislocation Man intenal defect such as oken fie deonding etc. can e modeled as a set of discontinuities In the continuum esulting in petuations of small o lage magnitudes in an othewise simple elastic field. The Somigliana dislocation can e consides as one of the most geneal kins of such discontinuities an thus can e used to model man impefections. Buges vecto is NOT conseved along the dislocation line

19 The Somigliana ing Dislocation used to model clindical cacks

20 G k k k G d G fo 0 G fo 0 0 pg p g pg g p pg pg g tc

21 . DemiI. HithJ.P. and ZiH.M. "The tended Stess Field ound the lindical ack Using the Theo of Dislocation Pile-ups" Int. J. ngg Sci Zi H.M. Hith J.P. and Demi I. ``The Stess Intensit Facto of lindical acks" Int. J. ngg. Sci Zi H.M. and Demi I. "On the Micomechanics of Defects in Fie-Mati omposites" eview aticle in: Damage in omposite Mateials Voiadjis G.Z. ed. pp. 0- lsevie Science Pulishe N.Y Zi H.M. and Demi. I.``The Inteface ing Dislocation in Fie-Mati omposites: ppoimate naltical Solution" SM Jounal of ngineeing Mateials and Technolog lose S. and Zi H.M. The Stess Intensit Factos and Inteactions Between lindical acks in Fie Mati omposites an invited aticle in: Damage and Intefacial Deonding in omposites ed. VoiadjisG. Z. and llen D.H. lsevie Science The Nethelands 996 pp. -7.

22 Modeling the clindical cack aound a fie as a pileup of Somigliana ing dislocations. DemiI. HithJ.P. and ZiH.M. "The tended Stess Field ound the lindical ack Using the Theo of Dislocation Pile-ups" Int. J. ngg Sci lose S. and Zi H.M. The Stess Intensit Factos and Inteactions Between lindical acks in Fie Mati omposites an invited aticle in: Damage and Intefacial Deonding in omposites ed. VoiadjisG. Z. and llen D.H. lsevie Science The Nethelands 996 pp. -7.

23 cuved dislocation can e appoimated as a set of staight dislocation segments cicula shape a c ita shape J ontinuous cuve J+ Discete segments p N j D j j

24 ita shape j+ j i+ i- i- i j- j v j jp Nodes and collocation points on dislocation loops and cuves j+ v j+ P Field point dl σ αβ p G 8 π G 4 π ν m imα m imk i d β i α β G 8 π δ αβ v Velocit vecto m imβ i i d k d α

25 B p B P ij ij ij Othe foms ae given in Hith and Lothe 98 p. 4 This fom is most convenient to use B The stess field of a dislocation segment

26 /G /- Discete icula Voltea Dislocation Loop N=6 N= N=8 N=4 s anal / a c

27 Stacking-Fault Tetaheda D B SFT Oseved in quenched metals and metals and allos of low stacking-fault eneg. It consists of a tetahedon of intinsic staking faults on {} planes with /6<0> tpe stai-od dislocations along the edges of the tetahedon. dislocation loop of Fank patial dislocation fomed the collapse of vacancies ma dissociate into a low-eneg stai-od dislocation and a Shockl patial on an intesecting slip plane leading to the fomation of a SFT.

28 glide PF-foce N/m.ompaison etween the Stess Fields of Unit Defects dge disloc ation X =0 = SFT FS loop Defect: aloop dissociate into SFT To contast the diffeing stess fields of a FS loop and a SFT we calculate the Peach-oehle foce that oth eet at a point on a staight edge dislocation situated an -distance fom the loop plane o coespondingl fom the tetahedon's ase plane. n edge view of the loop and its possile SFT offsping shown in dashed line is ehiited in this figue along with a nea edge dislocation. In this figue the glide foce in the diection of the dislocation's Buges vecto is plotted vesus the -coodinate of the dislocation at a given. Note that in the foce eeted the SFT is smalle than that caused the FS loop. n eception to this is the case when =9 whee the edge dislocation almost cuts the tip of the tetahedon in question. In such a case the SFT foce is stonge than the loop's. Moeove if the dislocation line collides with a SFT a moe poale event than a loop collision consideing volume diffeences then thei inelastic inteaction is also epected to e stong. In this figue the edge length of the tiangula FS loop o coespondingl the SFT is equal to 0 which is a tpical defect dimension as measued epeimentall. Now if one fies the value and vaies the -coodinate instead it would still e oseved that oveall and as long as elastic inteactions ae concened the FS loop causes a stonge glide foce on the dislocation. The esults demonstate a highe stess field associated with the FS loop which should e in-line with phsical easoning. Since the SFT foms fom a FS loop i.e. it s a pefeed lowe eneg state it is then epected that a moe elaed stain-field accompan such geomet ove that of a loop

29 Dislocation multiplication Fank-ead souces