Stability of dislocation structures in copper towards stress relaxation investigated by high angular resolution 3D X-ray diffraction

Transcription

1 Phys. Sttus Solidi A 206, No. 1, (2009) / DOI / Stbility of disloction structures in copper towrds stress relxtion investigted by high ngulr resolution 3D X-ry diffrction physic Bo Jkobsen 1, 2, Henning Friis Poulsen 1, Ulrich Lienert 3, Joel Bernier 3, 4, Crsten Gundlch 1, 5, nd Wolfgng Pntleon *, 1 pplictions nd mterils science 1 Center for Fundmentl Reserch: Metl Structures in Four Dimensions, Mterils Reserch Deprtment, Risø Ntionl Lbortory for Sustinble Energy, Technicl University of Denmrk, Frederiksborgvej 399, 4000 Roskilde, Denmrk 2 Center for Fundmentl Reserch: Glss nd Time, Deprtment of Science, Systems nd Models, Roskilde University, Universitetsvej 1, 4000 Roskilde, Denmrk 3 Argonne Ntionl Lbortory, APS, 9700 South Css Ave., Argonne, IL 60439, USA 4 Engineering Technologies Division, Lwrence Livermore Ntionl Lbortory, 7000 Est Avenue, Livermore, CA 94551, USA 5 Europen Synchrotron Reserch Fcility, 6 rue Jules Horowitz, Grenoble Cedex, Frnce Received 24 June 2008, revised 13 August 2008, ccepted 20 August 2008 Published online 25 September 2008 PACS cp, Lk, F, i, Bt * Corresponding uthor: e-mil wolfgng.pntleon@risoe.dk, Phone: , Fx: A 300 µm thick tensile specimen of OFHC copper is subjected to tensile loding sequence nd deformed to mximl strin of 3.11%. Using the novel three-dimensionl X-ry diffrction method High ngulr resolution 3DXRD, the evolution of the microstructure within deeply embedded grin is chrcterised in-situ by the behviour of individul subgrins. The loding sequence consists of three continuous deformtion stges with strin rtes of s 1 nd s 1, in ech cse followed by period of extended stress relxtion t fixed motor positions, s well s n unloding step. In contrst to the deformtion stges, during ech stress relxtion stge, number, size nd orienttion of subgrins re found to be constnt, while minor mount of clen-up of the microstructure is observed s nrrowing of the rdil X-ry diffrction line profile. The ssocited decrese in the width of the strin distribution indictes homogeniztion of the elstic strins present in the deformtion structure. During reloding, the subgrin structure seemingly strts to develop further when the entire disloction structure is deforming plsticlly. Upon unloding of the smple, the verge bckwrd strin of the subgrins increses. Projections of reciprocl spce mps obtined just before (top) nd fter (middle) stopping the trction nd fter hlf n hour of stress relxtion (bottom). No mjor chnges in the microstructure occur, only minor djustments re observed. 1 Introduction Understnding the dynmics of disloction structures is key problem for modelling crystl plsticity. A generlly ccepted theoreticl frmework is lcking, s number of bsic questions re still unsettled, such s the nture of the formtion of ordered disloction structures, their evolution (by continued subdivision) nd their stbility. Prt of the problem hs been the lck of suitble techniques for chrcterising structurl dynmics. Trnsmission electron microscopy provides very comprehensive sttic informtion, but such studies re intrinsiclly restricted to thin foils nd hence even in-situ studies will in generl not be representtive of bulk dynmics (see e.g. [1 3]). On the other hnd, X-ry diffrction techniques

2 physic 22 B. Jkobsen et l.: Stbility of disloction structures in copper towrds stress relxtion such s line profile nd rocking curve nlysis (e.g. [4 7]) cn probe bulk dynmics, but the results re highly indirect s they represent verges over mny subgrins or grins with different sizes, orienttions nd neighbouring environments. Hence models re required for n interprettion of the dt; the use of which introduces ssumptions which re often hrd to verify by independent mesurements. Recently two synchrotron bsed methods hve been developed tht enble three dimensionl chrcteristion on the micrometer scle: three-dimensionl X-ry diffrction (3DXRD) microscopy [8, 9] nd 3D X-ry crystl microscopy [10]. However, t present both methods hve limittions: the sptil nd ngulr resolution of the 3DXRD microscope does not llow for direct observtion of the deformtion microstructure, wheres the sptil scnning technique of the 3D X-ry crystl microscope mkes dt cquisition slow fr from idel for timeresolved studies. To overcome these issues, we hve developed n insitu chrcteristion technique High ngulr resolution 3DXRD [11, 12]. This method enbles cquisition of 3D reciprocl spce mps with high resolution (of the order of Å 1 ). From such mps, quntittive informtion cn be obtined on size, orienttion, men elstic strin, nd internl elstic strin distribution of set of individul subgrins, ll deeply embedded within thick polycrystlline specimen. Moreover, resonble time resolution of the order of minutes hs been chieved for mpping of n entire grin. In first study of the microstructurl evolution in bulk, copper tensile specimen ws loded incrementlly from 3% to 4.2% elongtion nd chrcterised t ech lod increment. The observed evolution of the subgrins is contrry to conventionl expecttions nd hs led to the suggestion tht the disloction structure follows n intermittent dynmics [11]. In the present work, we hve improved the instrumentl design, such tht the microstructure evolution cn be studied during continuous deformtion. This enbles us to ddress longstnding issues relted to the response of the subgrin structure to continuous deformtion, strin relxtion nd unloding. Lbortory, USA). The X-ry bem is delivered by combintion of opticl elements comprising 6-bounce monochromtor nd verticl focusing by refrctive lenses (for detils see [11]). The 52 kev X-ry bem hs verticl divergence of 17 µrd nd reltive energy spred of The bem size is defined by slits to be 30 µm horizontlly nd 20 µm verticlly. The smple is plced in custom mde lod frme which is positioned on trnsltion stges mounted on Huber 3-xes Eulerin crdle. The lod frme is screw driven nd operted in position-control vi the screw driving motor. The mcroscopic strin in the guge section of the specimen is mesured by single surfce-mounted liner strin guge coupled to Whetstone bridge. The mcroscopic stress is determined by lod cell integrted in the lod frme. Initilly diffrction ptterns re recorded in the undeformed stte using conventionl CCD re detector positioned t distnce of 253 mm behind the smple. By use of 3DXRD principles (e.g. [9]) bulk grin is identified. The selected grin hs 400 reciprocl lttice vector ligned within 9 of the tensile direction. The grin is scnned through the bem nd found to hve dimensions of 15 µm nd 13 µm in horizontl nd verticl direction, respectively. Next, the specimen is rotted such tht the respective 400 reflection is observble in the verticl diffrction plne. The geometry of the setup is sketched in Fig. 1. For the high resolution mpping, MrCCD 165 re detector is used in pixel mode, exhibiting µm pixel size. This detector is positioned t horizontl distnce L of 3737 mm behind the smple nd on the verticl diffrction plne such tht the selected 400 reflection is centred on the detector. Dt cquisition consists of rocking the smple in equidistnt steps of Δω = round the horizontl ω-xis, perpendiculr to the direction of the incoming bem. Continuous smpling is chieved by rocking the smple (rotting with constnt speed) through the intervl Δω during ech exposure. The exposure time is 10 seconds. 2 Experimentl 2.1 Smple The smple mteril is 99.99% pure OFHC polycrystlline copper. It ws initilly cold-rolled to reduction of 80% with resulting thickness of 300 µm nd then fully recrystllized by nneling for 120 minutes t 450 C. With electron bcksctter diffrction the verge grin size (ignoring twin boundries) is determined to bout 30 µm. Dog bone shped tensile smples with guge length of 8 mm nd guge width of 3 mm re prepred by sprk cutting. The investigtion reported here is performed t room temperture on single specimen DXRD experiment The experiment is performed t the X-ry Opertions nd Reserch bemline 1-ID t the Advnced Photon Source (Argonne Ntionl Figure 1 Sketch of the diffrction geometry nd the reciprocl spce coordinte system used (from [12]).

3 Originl Pper Phys. Sttus Solidi A 206, No. 1 (2009) 23 3D reciprocl spce mps of the 400 reflection re generted from such sets of consecutive ω-intervls (s described in detil in [11, 12]). The mps re prmeterised by (q x, q y, q z ), with q x prllel to the horizontl xis of the detector, q y the rdil direction (i.e. prllel to the scttering vector) nd q z perpendiculr to q x nd q y (see Fig. 1). In this mnner, vritions in q y indicte elstic lttice strins qy,0 - qy ε =, (1) q y wheres differences in q x nd q z represent orienttion vritions. The instrumentl resolution in q x nd q y re both Å 1 predominntly limited by detector resolution while the resolution in q z is Å 1, s determined by the step size Δω. During the experiment, reciprocl spce mps re cquired repetedly. The mps re truncted in q z by limiting the number of ω-intervls to 15, thereby improving the time resolution to 3.6 min. To provide better overview, t selected time steps lrger ω-rnges re gthered. During loding the lod frme is trnslted prllel to the lod xis to compenste the smple elongtion, such tht the bem t ll times fully illumintes the selected grin. To estimte the ccurcy of this position correction procedure, nd check for drifts in generl, t selected times the grin is scnned with respect to the bem. From these tests it is estimted tht the positionl fluctutions re t most 2 µm. Given the size of the grin nd the bem, this implies tht ll prts of the grin re completely illuminted during ll cquisition intervls. 2.3 Loding sequence The sequence is illustrted in Fig. 2 by the displcement of the cross-hed driving tension motor s function of time nd the resulting stress strin curve obtined from the strin guges nd the lod cell. The loding sequence comprises the following steps: Preloding The smple is preloded in smll steps to tensile strin of 1.82% (i.e. the mcroscopic strin monitored by the strin guge). By following the movement of the grin during this strining, clibrtion curve is produced relting the displcement of the tension motor to the resulting mcroscopic strin nd the sptil grin movement. First slow loding nd hold The smple is strined continuously with strin rte of s 1 to tensile strin of 2.315%. During the slow loding 20 reciprocl spce mps re gthered. After tht, the tension motor is stopped nd the smple held t constnt position of the tension motor for 41 hours, during which the tensile strin increses to 2.365%. During the first hour nother 19 reciprocl spce mps re gthered. Unfortuntely, for the reminder of the time diffrction dt re unvilble due to synchrotron brek-down. Due to unvoidble drifts of the monochromtor, direct comprison of dt obtined before nd fter the long brek is questionble nd not performed here. Second slow loding nd hold After re-centring of the grin with respect to the bem, the smple is strined continuously with the sme strin rte of s 1 to tensile strin of 2.682%. During this period 14 reciprocl spce mps re gthered. The smple is then held t constnt motor position for 2.2 hours, while gthering 10 mps during the first 45 minutes. At the end of the hold, the strin hs incresed to 2.695%. Figure 2 Loding sequence. (top) Displcement of tension motor versus time, (bottom) stress strin curve determined from lod cell nd strin guge. Verticl lines seprte the different stges of the sequence. The rte of the lst loding is too fst to collect stress nd strin dt during the loding.

4 physic 24 B. Jkobsen et l.: Stbility of disloction structures in copper towrds stress relxtion Fst loding nd hold Next, the smple is strined to 3.110% s fst s possible with the present setup. The estimted strin rte is s 1. Following this, the smple is held t constnt tension motor displcement for 2.9 h, fter which the tensile strin of the specimen hs incresed to 3.137%. A totl of 12 reciprocl spce mps re gthered. The cquisition of the first mp is strted within one second fter stopping the loding. The 10 first mps re tken continuously during the first 35 minutes fter fst strining. The lst two re tken fter totl time of 53 nd 131 minutes fter strining, respectively. Unloding Finlly, the smple is unloded during 221 seconds from mcroscopic stress of MP to 49.7 MP. Rther lrge reciprocl spce mps (comprising 60 ω-intervls) re cquired before nd fter unloding. 3 Results 3.1 Mechnicl behviour As obvious from the stress strin curve in Fig. 2 the flow stress increses during the loding stges nd the mteril work-hrdens. (The vlues for the flow stress re in greement with tensile tests on OFHC copper of comprble grin size t room temperture nd similr strin rte [13].) When the motion of the tension motor is stopped, the guge section of the specimen continues to deform plsticlly in tension due to the pplied tensile lod (cf. [14, 15]). As the totl length of specimen nd mounting ger is fixed, n extension of the guge section of the specimen must be compensted by elstic unloding of other prts of the specimen nd the mounting ger. Any plstic strining in the specimen leds to lowering of the elstic strins in the system ccompnied by reduction of the pplied stresses. In this mnner, the pplied stress is relxed despite the fixed totl elongtion. During stress relxtion under fixed motor displcements, the stress decreses linerly with the totl strin Δε in the guge section of the specimen (e.g. [14, 15]) Δσ = σ - σ = - E Δ ε, (2) 0 m from the vlue σ 0 when stopping the tension motor ( similr reltion holds for the plstic strin in the guge section; only the pplicble, effective mchine modulus E m will be different). Such liner dependence is nicely seen in the stress strin curve in Fig. 2 for the extended first holding stge from 2.315% to 2.365%. Stress relxtion t constnt temperture T is commonly described (e.g. [14 16]) by logrithmic time dependence of the stress drop kt t Δσ =- ln Ê 1 + ˆ, (3) V Ë τ with Boltzmnn constnt k, chrcteristic relxtion time τ nd n pprent ctivtion volume V. Such logrithmic dependence cn be derived from quite generl considertions [17, 18]; more elborte nlyticl expressions proposed s well (cf. [14, 19]) will not be considered here. Figure 3 (online colour t: Stress relxtion during first holding stge. A logrithmic decrese of the stress is fitted to the experimentl dt. As shown in Fig. 3 logrithmic dependence with τ = 510 s nd V = 100b 3 (with Burgers vector b = nm) describes the experimentl dt ccurtely during 37 h. Both ctivtion prmeters re not constnt during relxtion; in prticulr the dt from the first 20 min re better described by shorter relxtion time (150 s) nd lrger ctivtion volume (150b 3 ). In the second tensile loding stge with the sme strin rte s in the first one, plstic flow is observed for stresses bove the initil yield stress, but before the mximl flow stress of the first loding stge is reched. The workhrdening rte is slightly higher in the second loding stge thn in the first. 3.2 Reciprocl spce mps The min results of the experiment re the gthered reciprocl spce mps which cn be combined into three-dimensionl movie. Here selected snpshots from the time series re presented where for ech point in time the three-dimensionl reciprocl spce is projected onto two-dimensionl subspce. Two projections re used: one onto the (q x, q z )-plne (i.e. the zimuthl plne) nd one onto the (q x, q y )-plne (similr to the rw detector imges, slightly corrected for distortion). The first projection corresponds to integrting over the elstic strin degree of freedom nd visulising the mosic spred in other words zoom on smll prt of the 400 pole figure for the selected grin. The second projection represents integrtion over the mosic spred in the direction of the truncted q-rnge (rocking direction) nd enbles convenient visulistion of the evolution of lttice strins. As n exmple, Fig. 4 provides tbleu of both types of projections for prt of the initil loding from 1.817% to 2.010%. Evidently, the reciprocl spce mp is chrcterised by set of shrp peks on top of diffuse cloud of enhnced intensity. Distinct peks in such mp originte from single nd nerly disloction free subgrins in the grin of interest [11, 20]. More extended islnds like the one to the left in the imges in the left column of Fig. 4 re likely to be compound peks, comprising contributions

5 Originl Pper Phys. Sttus Solidi A 206, No. 1 (2009) 25 Figure 4 (online colour t: Projections of eight 3D reciprocl spce mps obtined during the first loding. The projections ech cover the sme rnge [ 0.04 Å 1, 0.03 Å 1 ], [6.925 Å 1, Å 1 ], nd [0.027 Å 1, Å 1 ] in q x, q y, nd q z, respectively. They re plotted with xis of equl scle. The mps shown re truncted in q x direction with respect to the full dtset. The colour br is in rbitrry units of integrted intensity. from severl subgrins. The diffuse cloud is interpreted s rising from the disordered boundry regions of high disloction density (dense disloction wlls) between the subgrins [11, 21]. Comprehensive informtion on prticulr subgrin cn be derived from ech distinct pek [12]. More specificlly, the integrted intensity of shrp pek is directly proportionl to the volume of the subgrin, while the centreof-mss-position of the pek in reciprocl spce ((q x, q z ) nd q y ) reltes to verge orienttion nd verge lttice strin (long the scttering vector) of the subgrin, respectively. From the pek shpe the disloction density inside the subgrin cn be inferred nd is found to be rther smll [12]. Bsed on this interprettion the time evolution of the peks in Fig. 4 cn be rtionlized in terms of rottion, strining nd growth or shrinkge of subgrins. While the informtion is incomplete in the sense tht not ll sub- Figure 5 Averge vlue ( q y) nd width (Δq y ) of the rdil pek profiles (i.e. zimuthlly integrted pek profiles) s function of time. The time in ech prt is mesured with respect to the strt of the deformtion. First slow loding (open squre) nd holding (filled squre), second slow loding (open dimond) nd holding (filled dimond), holding fter fst loding (tringles pointing up), nd unloding (circles). Note tht the first point for the fst loding is even before the loding strts, hence t negtive time. grins in the grin re probed (s they give rise to peks outside the window of observtion in q z ) nd contributions from some of the subgrins cnnot be seprted, this type of dt is very useful towrds ddressing whether the geometry of the subgrins nd their elstic strin stte chnges or not. The instrumentl resolution in the zimuthl directions (q x, q z ) corresponds to rottions of nd respectively. The instrumentl resolution in the rdil direction (q y ) corresponds to strin resolution of Shifts of the peks re detectble down to (nd even beyond) this limit. Additionl insight is gined from trditionl line profiles, i.e. one-dimensionl profiles, where the reciprocl spce mp is integrted over both zimuthl directions (q x

6 physic 26 B. Jkobsen et l.: Stbility of disloction structures in copper towrds stress relxtion nd q z ). From such rdil pek profiles two quntities of interest cn be derived (e.g. [7]): the verge elstic strin (long the diffrction vector) with respect to the initil unloded stte, nd the width of the strin distribution. These re determined from fitting pseudo Voigt function to the rdil pek profile s its centre (identicl to the verge) nd its full width t hlf of the mximum intensity (FWHM). Both quntities re plotted in Fig. 5 s function of time covering the complete loding sequence. A closer inspection of the rdil pek profile revels slower decy of the intensities t higher q y vlues thn for smller q y. If split pseudo Voigt functions re used to quntify the symmetry of the rdil pek profiles, fitting resolves lrger width t hlf of the mximum intensity for the tils t higher q y. The reltive symmetry increses slightly during stress relxtion nd more strongly upon unloding. In the following, the individul stges of the loding sequence re discussed. Selected zimuthl distributions in the (q x, q z ) plne for the individul stges re presented in Figs. 4 nd 6 to First nd second slow loding During the first loding stge significnt chnges in the subgrin structure re observble in the reciprocl spce mps. In Fig. 4 showing only prt of the entire first loding stge the temporry formtion of new subgrins nd the disppernce of others cn be recognized. During the entire tensile loding subgrins of different orienttions nd elstic strins pper nd dispper intermittently (confirming erlier observtions [11]). Similr observtions re mde in the second loding stge. 3.4 Holds fter first nd second slow loding The results during the two holding stges re quite similr to ech other. In both cses, there is no evidence from visul comprison of the reciprocl spce mps (cf. zimuthl projections of the second holding stge in Fig. 6) for ny microstructurl chnges on the subgrin level within the pproximtely first hour of dt tking fter stopping the tension motor. Extrpoltion of the evolution during holding to times prior to the moment of stopping nd comprison with the lst reciprocl spce mp during loding even rules out ny hypotheticl structurl chnges t the subgrin level, which my occur t the exct time of the stop nd re completed before the first mesurement of the scn. From the informtion of the rdil pek profiles summrized in Fig. 5 it is found tht during the loding stges the men position of the rdil profile decreses nd hence (ccording to Eq. (1)) the verge elstic strin in the illuminted grin increses with incresing stress. During the holding period the elstic strins re decresing due to stress relxtion. The relxtion of the elstic strin seemingly comprises two components (in prticulr fter the first loding), fst initil trnsient, followed by much slower response. Figure 6 (online colour t: Azimuthl projection of every second of the 10 reciprocl spce mps obtined during the second holding, nd of the lst mp obtined while still strining. Time is with respect to the moment where the tension motor is stopped, nd for the centre imge of ech dtset. The projections ech cover [0 Å 1, 0.12 Å 1 ] nd [0.027 Å 1, Å 1 ] in q x nd q z, respectively. The colour br is s in Fig. 4. Correspondingly, the width of the internl strin distribution (s observed from the width of the rdil profiles) increses during tensile loding, nd decreses lmost linerly with time during the first hour fter stopping the tension motor. 3.5 Fst loding nd hold Similrly to the holding stges fter the slow lodings, from visul inspection of the reciprocl spce mps (including the zimuthl projections in Fig. 7) there is no evidence of ny structurl evolution t the scle of subgrins upon stopping the tensile motor fter fst loding. Likewise, the relxtion of the verge elstic strin (s seen from the men position of the rdil profile in Fig. 5) initilly occurs fst nd slows down lter. The width of the men strin distribution decreses with time fter interrupting the tensile motion (initilly linerly, but with reduced rte t lter times). Figure 7 (online colour t: Azimuthl projection of every second of the 10 reciprocl spce mps obtined right fter the fst loding, nd of the two obtined t lter times. Time is with respect to the moment where the tensile motor is stopped, nd for the centre imge of ech dtset. The projections ech cover [0.02 Å 1, 0.11 Å 1 ], nd [0.027 Å 1, Å 1 ] in q x nd q z, respectively. The colour br is s in Fig. 4.

7 Originl Pper Phys. Sttus Solidi A 206, No. 1 (2009) 27 As no observtions of the microstructure re vilble during the fst loding stge, structurl relxtion could hve tken plce in the time intervl between holding nd tking of first dt. By noting tht the dt from the first ω-intervl in the first reciprocl spce mp obtined fter stopping the tension motor re identicl to the dt of the sme intervl in consecutive mps, it is concluded tht such hypotheticl rerrngements would hve to hve relx completely within the first 5 seconds fter the hold. 3.6 Unloding By inspection it is found tht the two reciprocl spce mps obtined just prior to nd fter unloding (with zimuthl projections shown in Fig. 8) re nerly identicl but shifted with respect to ech other in reciprocl spce. Seven individul well seprted peks re identified nd nlyzed in detil. Three independent pseudo Voigt functions re fitted to ech pek, one in ech of the three reciprocl spce directions, giving the position of the mximum of the pek (the single pek nlysis technique is described in detil in [12]). Bsed on these seven peks it is found tht the two mps in Fig. 8 on verge re shifted with respect to ech other by Å 1, nd Å 1 in q x, nd q z, respectively. Such n overll offset is redily explined by rottion of the whole smple or rottion of the entire grin driven by the comptibility with neighbouring grins during unloding. However, smll differences between the shifts of individul subgrins re observed. The stndrd devitions for the subgrin shifts in both zimuthl directions re 13% nd 20% of the overll offset in q x, nd q z, respectively. These differences between the rottions of individul subgrins indicte heterogeneity of the elstic unloding within the grin either cused by its neighbours or due to the formtion of domins of slightly different orienttion within grin during tensile loding. The rdil (q y ) position of n individul pek represents the men elstic strin long the scttering vector of the corresponding subgrin. For the identified seven subgrins the elstic strin of ech subgrin is less thn the men elstic strin of the grin (determined from q y by Figure 8 (online colour t: Azimuthl projections of reciprocl spce mps obtined before nd fter unloding the smple. The colour br is s in Fig. 4. Figure 9 (online colour t: Bckwrd strins (compressive stresses with respect to the grin verge) of seven selected subgrins before (strs) nd fter (squres) unloding of the smple. Their verge vlue is indicted by the full nd dshed lines for the loded nd unloded stte, respectively. Eq. (1)). With respect to the grin verge compressive strin qy - qy δε =- < 0, (4) qy (clled bckwrds strin) is present in ech subgrin s illustrted in Fig. 9. The differences between the elstic strins of the seven individul subgrins re rther substntil (but due to the smll number of subgrins, their vrince shll not be nlysed further). Both findings re consistent with the observtions on the lrger set of subgrins reported in [12]. On unloding the mximum of the integrted rdil pek profile shifts to higher bsolute q y vlue (see Fig. 5). The shift corresponds to decrese in the men tensile elstic strin of the grin of 0.059%. This is comprble to the mcroscopic strin difference of 0.067% detected by the strin guges during unloding (nd the differences cn be explined t lest prtilly by the ngle between the tensile xis nd the 400 direction). The width of the rdil pek profile decreses by 3% corresponding to nrrowing of the internl strin distribution by the sme mount. On the other hnd, the compressive strins (with respect to the grin verge) of the seven identified subgrins increse during unloding s seen in Fig. 9 in verge by 14% from 0.037% to 0.043%. 4 Discussion 4.1 Generl All observtions re consistent with the following interprettion: The microstructure s described in terms of the number of subgrins, their orienttion nd extension in reciprocl spce develops mrkedly nd in n intermittent mnner during continuous loding, but becomes frozen t the instnt of interrupting the trction. Within the observtion time, individul disloctions re ble to migrte under the

8 physic 28 B. Jkobsen et l.: Stbility of disloction structures in copper towrds stress relxtion pplied lod further in forwrd direction. Their motion cuses dditionl plstic deformtion of the guge section of the specimen nd hence specimen elongtion. Consequently, the fixed displcement of the motor position requires less elstic deformtion (of ll prts) nd the verge elstic strin decreses. The logrithmic time dependence of the stress relxtion with n pprent ctivtion volume of the order of 100b 3 is consistent with therml ctivted motion of individul disloctions surmounting obstcles. The nture of the obstcles or single rte limiting process (jog drgging, cross slip, cutting of forest disloctions) cnnot be deduced from the dt. Severl different stress relxtion tests t different tempertures would hve to be nlysed [15]. Comprble vlues for the pprent ctivtion volume during stress relxtion hve been reported for OFHC copper [14], but significntly lrges vlues hve been obtined on pure copper s well [16, 22] depending on both the pplied stress nd the used equipment; proper ssessment of the lter effect requires thorough determintion of the mchine stiffness (cf. [22]). Disloctions hving moved long certin pth my find n nnihiltion prtner nd nnihilte. Hence, the forwrd motion of some disloctions is ccompnied by reduction of the disloction density nd both effects cuse nrrowing of the elstic strin distribution. The observed decrese in the width of the rdil pek profile is consequence of such clening-up of the disloction structure. A similrly frozen subgrin structure persists during unloding gin ccompnied by certin mount of clening-up evidenced by reduction in the width of the rdil pek profile. In this cse, however, piled-up disloctions my move bckwrds, disloction loops shrink nd dispper reducing the totl disloction density. Differences in rottions nd elstic strining between individul subgrins indicte loclly heterogeneous chnges in the elstic stress sttes of the individul subgrins during unloding. Interestingly the chnges occurring during unloding re less prominent thn the chnges during stress relxtion (in prticulr during the first hour of the first holding stge, cf. Fig. 5). This is due to the still pplied stress nd the ongoing thermlly ctivted plstic deformtion in forwrd direction under stress relxtion, wheres unloding llows only moving bck of individul mobile disloctions which hve been hindered in their forwrd motion by obstcles nd stored only dynmiclly under the pplied lod. 4.2 Reltion to composite model The bckwrd strins of individul subgrins (compressive strins compred to the grin verge) cn be rtionlized in terms of modified composite model. Anlogous to the originl composite model [23, 24] for n inhomogeneous deformtion structure, comptibility of deformtion requires lrger locl stresses for the disloction-dense boundries (s higher disloction density cuses lrger flow stress) thn for the disloction-depleted regions. The former develop forwrd stresses, wheres the nerly disloction-free subgrins experience bckwrd stresses. The mount of bckwrd stress nd strin in ech individul subgrins, however, is different from subgrin to subgrin s discussed in [11, 12] nd shown in Fig. 9. During stress relxtion nd the ssocited plstic deformtion cused by forwrd motion of disloctions, the decresing width of the rdil profile indictes decrese in the vritions of the elstic strins within the grin nd hence lower totl disloction density. The simultneous increse in symmetry might be chieved by shrpening of the strin distribution between the subgrins leding to more even strin level of the individul subgrins. Upon unloding the verge bckwrd strin of the seven individul subgrins increses, explining instntly the observed increse in the pek symmetry. A more detiled quntifiction of the chnges in the distribution of the bckwrd strins of the subgrins requires dt on lrger number of subgrins nd is in preprtion. 4.3 Reloding nd forwrd Buschinger effect During the 41 hour long holding period following the first loding stge the stress is relxed significntly, before the specimen is reloded during the second loding period. For instnce, the rdil pek width dropped below the vlue t the beginning of the first loding stge. The reloding behviour is investigted in more detil for the clerly seprble pek observed t the left in the top frme of Fig. 6. The pek is found to rotte with respect to the right prt of the mp during the second deformtion stge. Figure 10 (online colour t: Second loding stge: nlysis of the correltion between mcroscopic stress (red line) nd strin (blue line) in the guge section of specimen, men elstic strin (crosses) of the selected grin, nd orienttion of the selected subgrin in q x direction (squres). The curves re scled to obtin mximum correspondence t beginning nd end. The verticl line indictes the point where the mcroscopic stress reches the previous mximum flow stress (t the end) of the first loding stge.

9 Originl Pper Phys. Sttus Solidi A 206, No. 1 (2009) 29 In Fig. 10 different informtion gthered during the entire reloding stge is summrized: the mcroscopic stress nd strin re plotted together with the men elstic strin of the grin (obtined from the men q y vlue of the rdil pek profile integrted zimuthlly over ll relevnt reciprocl spce mps), s well s the zimuthl position (represented by q x ) of the prticulr pek. All curves re scled to coincide t strt nd end of the deformtion stge. Following the constnt speed of the tension motor, the mcroscopic strin nd stress of the guge section of the specimen both increse with time; the mcroscopic strin lmost linerly during the entire deformtion. The mcroscopic stress increses only t the beginning linerly with time indicting pure elstic behviour; t certin instnt in time yielding occurs, even before the pplied flow stress reches the mximl flow stress of the first loding stge. In this mnner, forwrd Buschinger effect is cused by stress relxtion due to the clening-up of the disloction structure nd the reduction of the disloction density. After yielding, deformtion proceeds with continuously decresing work-hrdening rte. The men elstic strin of the grin follows nicely the pplied mcroscopic stress during the entire deformtion stge s expected from elsticity. On the other hnd, the rottion of the individul pek (represented by q x ) increses initilly linerly with time nd proportionl to the totl mcroscopic strin. Their proportionlity is not restricted to the elstic regime; cler devition from linerity is observed first fter the mteril hs strted to deform plsticlly, but before the pplied stress reches the previous mximum flow stress. The observtion tht the onset of devition in the rottion from liner dependence on the mcroscopic strin is delyed with respect to the onset of plstic yielding under reloding cn be rtionlized in terms of the modified composite model: During reloding (fter stress relxtion) initilly only the disloction-free subgrins deform plsticlly (or t lest some of them). In this micro-yielding regime, the reltion between strin nd rottion still holds. Only when yielding of the hrder disloction-rich boundries occurs s well nd the entire disloction structure deforms plsticlly, the lttice rottion of n individul subgrin becomes independent of the mcroscopic strin nd will be controlled by the prticulr behviour of its djcent boundries. 5 Conclusion The experimentl dt re ll consistent with the notion tht the evolution of the microstructure is driven by plstic deformtion of the entire disloction structure: The subgrin structure develops continuously nd intermittently during tensile testing. When the trction is terminted nd the tension motor position fixed, the deformtion structure freezes, nd clen-up process gives rise to stress relxtion. The subgrin structure is unchnged when unloding the smple, but the compressive strins of the subgrins (with respect to the verge of the grin) increse in verge during unloding. The results for the prolonged holding periods in this study demonstrte the long-term stbility of the high ngulr resolution 3DXRD method. Thereby they serve s n extr vlidtion of the previous studies [11, 12]. The present findings lso dd dditionl insight to the reported intermittent dynmics [11]. One could hve speculted tht the observed intermittence is signture of quite unstble configurtions, which re esily perturbed by chnges in the externl elstic field. The results presented here showing stbility of the subgrin structure towrds stress relxtion contrdict such n interprettion nd limit the possible mount of instbility. Acknowledgments The uthors wish to thnk J. Almer, A. Mshyekhi, nd S. D. Shstri for help during the experiments, nd P. Nielsen, P. Olesen, nd L. Lorentzen for smple preprtion. This work ws supported by the Dnish Ntionl Reserch Foundtion nd the Dnish Nturl Science Reserch Council. Use of the Advnced Photon Source ws supported by the U.S. Deprtment of Energy, Office of Science, Office of Bsic Energy Sciences, under Contrct No. DE-AC02-06CH References [1] U. Essmnn, phys. stt. sol. 3, 932 (1963). [2] H. Mughrbi, Philos. Mg. 23, 869 (1971). [3] M. M. Myshlyev, D. Cillrd, nd J. L. Mrtin, Scr. Metll. 12, 157 (1978). [4] M. Wilkens, phys. stt. sol. () 2, 359 (1970). [5] M. A. Krivoglz, X-ry nd Neutron Diffrction in Nonidel Crystls (Springer Verlg, Berlin, 1996). [6] D. Breuer, P. Klimnek, nd W. Pntleon, J. Appl. Crystllogr. 33, 1284 (2000). [7] R. Kuzel, Z. Kristllogr. 222, 136 (2007). [8] E. M. Luridsen, S. Schmidt, R. M. Suter, nd H. F. Poulsen, J. Appl. Crystllogr. 34, 744 (2001). [9] H. F. Poulsen, S. F. Nielsen, E. M. 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