International Journal of Plasticity 69 (2015) 36e53

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International Journal of Plasticity

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Macrosegmentation and strain hardening stages in coppersingle crystals under compression

D.V. Lychagin a, b, S.Yu. Tarasov b, c, *, A.V. Chumaevskii b, d, E.A. Alfyorova b

a National Research Tomsk State University, Lenin Avenue, 36, Tomsk, 634050, Russiab National Research Tomsk Polytechnic University, Lenin Avenue, 30, Tomsk, 634050, Russiac Institute of Strength Physics and Materials Science, SB RAS, pr. Akademicheskii 2/4, Tomsk, 634055, Russiad Tomsk State University of Architecture and Building, pl. Solyanaya 2, Tomsk, 634003, Russia

a r t i c l e i n f o

Article history:Received 24 July 2014Received in revised form 26 January 2015Available online 13 February 2015

Keywords:Single crystalCopperStress-strain curvesSurface deformation patternElectron backscattered diffraction

* Corresponding author. Institute of Strength PhysE-mail addresses: dvl-tomsk@mail.ru (D.V. Lycha

Alfyorova).

http://dx.doi.org/10.1016/j.ijplas.2015.01.0070749-6419/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

The surface deformation-induced pattern (relief) of copper single crystals with theorientation of the compression axis along [111] has been investigated by means of optical,scanning electron and atomic force microscopy. The misorientations between bothmacroscopic and mesoscopic areas in [111]-single crystals have been determined using theelectron back scattering diffraction (EBSD) technique. The macroscopic reorientation hasbeen revealed to rotate the crystalline lattice around the [110] axis. The single crystal hasbeen divided into five macrosegments with their misorientations distributed along thecompression axis in a manner that the deformation axis sequentially coincided with thecrystallographic directions in the order [111]-[221]-[773]-[110]. Shear by unloaded plane(111) has been observed. The macrolevel deformation up to 25% has been developing asfollows: shear by octahedral planes e development of macrobands e reorientation of thecentral zones e shear by unloaded octahedral plane in the reoriented zones. The steps ofthis sequence corresponded to the stages of the stress-strain curves.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

To study texturing in polycrystalline materials we need to define more accurately the already known strain patterns inorder to apply them to an isolated grain. Earlier such researchers as Taylor (1938), Lebensohn and Tome (1993), van Houtteet al. (1999), and Mahesh (2009, 2010) treated each grain of the polycrystalline material as a homogeneously deformed solid.Nevertheless, the results of numerous experimental works showed that the grains deformed inhomogeneously. Therefore, wehave to separate macroscopically induced strain gradients from the micromechanical effect of plastic inhomogeneity withinthe grain scale (Raabe et al., 2001). The important thing in this case is not only the crystallographic orientation of an isolatedgrain with respect to the deformation axis but also an interaction between the neighboring grains. This interaction may addcomplexity to themacroscopically simple strain-stress state. One of the approaches to find a solution to this problem has beenoffered by Raabe and Zhao et al. (2002a), (2002b) in the form of a theory of orientation gradients where these reorientationgradients arise both by internal and external reasons such as the crystallite orientation and effect of neighboring grains,

ics and Materials Science, SB RAS, pr. Akademicheskii 2/4, Tomsk, 634055, Russia. Tel.: þ7 3822 286815.gin), tsy@ispms.ru (S.Yu. Tarasov), Tch7AV@rambler.ru (A.V. Chumaevskii), katerina525@mail.ru (E.A.

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respectively. Such an approach allowed establishing a crystallographic orientation which is prone to increasing the reor-ientation gradient both in fcc and bcc lattices (Raabe et al., 2002a). Moreover, some important points concerning an inter-action between the neighboring grains have also been established by Raabe et al. (2002b). In particular, the essential influenceof both the grain topology andmicrotexturing on the strain inhomogeneity has been demonstrated by Raabe et al. (2003) andZhao et al. (2008). Also the practically important effect of the rolled sheet roughness has been taken into account.

It appears to be important to consider both the surface roughness and complexity of the subsurface stress-strain statewhen performing the compression test. The inhomogeneous triaxial compression primary stress pattern is created in thesubsurface of a sample below the testing machine's platen according to Lychagin (2006). Both vertexes and edges of thetetragonal prism-shaped sample serve as stress concentrators so that stresses dampwith the distance away from them. Thesestresses may create not only the compression strain local zones in accordance with the axial loading but also the small localzones of the tensile strain. The irregularities on both sample's and platens' surfaces add even more complexity to the sub-surface stress pattern at the microscale level and therefore sample's surface has to be polished or coated. The friction forceinfluence is more clearly observed in tribological testing by intense grain subdivision (Tarasov et al., 2012) and developmentof rotational plasticity in the subsurface of real contact area with final generation of submicro e and nanocrystalline struc-tures (Tarasov et al., 2013).

There is a point to be made concerning the boundaries between the zones having different primary stress patterns. Theavailability of these zones and boundaries will facilitate the development of rotational plasticity, cross slip and hinderedshear. Conceptual models describing such situations have been presented elsewhere (Lychagin et al., 2006). Even if specialprecautions were made to provide free displacement of samples in machine's grips the strain pattern in the subsurface ofcontact areas differs from that of the sample's bulk. That such is true has been shown both by the results of our experimentsreported by Lychagin, (2006), Lychagin et al. (2006, 2011) and by the results of other researchers who studied strain in singlecrystals using a digital image correlation technique (Raabe et al., 2001, 2003; Zhao et al., 2008; Roters et al., 2010; Magid et al.,2009; Ha and Kim, 2011; Florando et al., 2007).

The issue of the single crystal reorientation in deformation is closely intertwined with the initial orientation of its faces asshown by Franciosi, Zaoui (1982), Roters et al. (2010), Havner (2007), (2011), Paul et al. (2010). When studying thecompression of high-symmetry orientation single crystals it is reasonable to provide consistency between the symmetry ofequally-loaded slip planes positions and lateral faces. Franciosi and Zaoui (1982) took that into account when they carried outcompression testing of copper single crystals and used trigonal prism-shaped samples with two sets of faces {110}, {112} fororientation [111] and tetragonal prism-shaped ones for all orientations other than [111]. Our own investigations show thatdeformation-induced surface pattern as well as the development of strain inhomogeneity on the sample faces is influencednot only by differences in the primary stress patterns between the subsurface and the central part of the sample but also bythe orientation of the lateral faces (Lychagin, 2006; Lychagin et al., 2011). Geometrical positions of slip planes in the bulk ofthe sample may differ depending on it. Analogously, the slip plane positions are changed with respect to basic stress con-centrators so that single crystal is subdivided into zones within which definite shear systems or shear domains are working.We call such a process as fragmentation or segmentation. The strain inconsistency between these shear domains will result inintense strain localization at the domain boundary (Lychagin et al., 2011).

It is especially important to take the lateral face orientation into account when carrying out a channel die test. Havner(1992, 2007, 2011) gives a review of works done on single crystals in this field. Also he analyzed in details the [110] orien-tation of single crystals (Havner, 2007, 2011, 2014) and stressed the importance of these works by citing Fares et al. (2002) asfollows: “The channel die test occupies an important position within single crystal deformation studies and has been viewedas representing a first approximation to the constraint and loading condition for metal poly-crystals”.

It seems interesting to study these results in connectionwith a stress-strain hardening curve. The orientation dependenceof lateral faces has been studied by Paul et al. (2010). Copper single crystals having the Goss {110}<001> orientation have beenchannel die tested up to reaching the preliminary strain corresponding to the stage IV. For further testing they used both pre-strained samples and those having their orientations changed for Brass {110}<112>, M{110}<111> and Hard {110}<011> ones.The result was that the stress level reduced in the order Hard/M/ Goss/ Brass orientations and in accordance with theTaylor factor changed. However, the higher is the stress level achieved, the higher is the effect of softening in the subsequenttest. Such a specificity can not be explained without extra experimenting. Paul et al. (2010) studied surface deformationpattern and slip band structures at macro- and microscale level, respectively. It was shown then that the rate of softening isrelated both to the macroband generation intensity and the slip band density in them. Accumulation of misorientationsbetween the macrobands is determined by accumulation of misorientations in the microband structures. So these resultsdictate the necessity of studying the deformation at different scale levels to enable description of the stress-strain curvebehavior. This study should involve not only the analysis of general orientation changes with respect to either a whole singlecrystal or its parts, but also the detailed examination of the deformation-induced surface pattern in connection with therotations observed. Such an approach would allow us to reveal mechanisms responsible for hardening and softening in thesingle crystals under consideration.

Classification of structural elements generated and observed in metallic materials under deformation is given elsewhere(Honeycombe, 1984; Jasienski, Piatkowski, 1980; Dillamore, 1980; Hatherly, 1982; Laird, 1996). Those elements include kinkbands, secondary shear or slip bands, accommodation bands, shear bands, persistent slip and deformation bands. Thesample's areas occupied by persistent slip bands generated by fatigue mechanism contain both extrusions and intrusions(Laird, 1996; Man et al., 2009a,b; Mughrabi, 2009). Kumar and Mahesh (2012, 2013) focused on generation and evolution of

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two band types such as (i) regular deformation bands (RDB) and (ii) microscopic deformation bands (MDB). They note that thedeformation inhomogeneity is the reason for grain subdivision into misoriented zones e domains.

Based upon our own results obtained on fcc single crystals we are supportive of the above suggestion since we observedthe generation of those domains by shear þ rotation mechanism (Lychagin, 2006; Lychagin et al., 2006; Lychagin et al., 2011).The rationale behind generation of these domains, which are the basic structural element of the deformed sample, is thestress field inhomogeneity as well as the specificity of a shear system orientation with respect to this stress field. Also thelatter is related to the crystal face's orientation.

Another macroscopic structural elements generated by deformation are themacrobands that also can be of different typesandmany of them arewell known from the literature. Those deformationmacrobands have been obtained from either fatigueor severe active plastic deformation tests when reorientation in the macrobands has been related to the texturing in thecourse of deformation (Dillamore, 1980; Laird, 1996; Higashida et al., 1986; Нarren et al., 1988; Kashchenko et al., 1996;Korduck et al., 1980; Korbel et al., 1982; Duggan, 1978; Hatherly and Malin, 1984). Also the macroband modeling has beendeveloped intensely during recent years (Dmitrieva et al., 2009; Jia et al., 2012a; Jia, 2013), together with studying the effect ofmacroscopic stability of samples on the strain inhomogeneity and local reorientation (Kuroda, 2007, 2013; Jia et al., 2012b).

Another important phenomenon of the deformation processes is folding (wrinkling). Many papers have been devoted tothis subject especially in the field of geology (Williams et al., 1979; Huddleston, 1973; Bastida et al., 1999) while only few suchpapers can be found for metals (Gubernatorov, 1999a; Gubernatorov et al., 1999b; Sokolov et al., 1995). It is regrettablebecause folding is a frequent finding in fcc single crystals having [111] orientation with respect to the compression axis(Lychagin, 2006; Lychagin et al., 2011).

We can state here after doing the literature search that in order to describe texturing in polycrystallinematerials one needscombining different investigation methods such as those focused on (i) isolated grain behavior, (ii) strain transmission ef-ficiency of the grain boundary, and (iii) interaction of neighboring grains.

The (i) situation can be conveniently elucidated using single crystals and analyzing the strain-generated structural ele-ments for developing consistent models of plastic deformation and strain hardening.

Depending upon the problem to solve one can deform the single crystals using either free or channel die compression aswell as some other more complicated type of loading. Simultaneously, it is reasonable to vary the test parameters such asshape of samples, roughness and friction between the sample and the machine's platen. However, we do not pretend here totake into consideration the great variety of cases described above.

The objective of this work is to study subdivision of copper single crystals of unstable orientations under shear þ rotationdeformation conditions and using standard sample preparation and testing procedures as well as to establish a relationbetween deformation-induced surface pattern, misorientation and stages of the strain hardening curve.

2. Materials and methods

The samples for investigation have been made of copper with stacking fault energy 40e60 mJ/m2 which implies that thebasic deformationmechanism is to be the dislocation gliding by close-packed planes. Pure copper is a ductile f.c.c metal of lowelasticity and strength limits and it is not prone to both twinning and slipping by cubic planes at room temperature. Assumingall this we have chosen copper for studying shear by octahedral planes.

For this purposewe used technical grade copper 3.5� 3.5�7 ± 0.1mm single crystals grown using the Bridgemanmethodand having their compression axis orientations coinciding with [111]. The lateral faces were (110) and (112). The singlecrystals' orientations have been controlled by the back scattering patterns to the accuracy 2� on XRD instrument IRIS-3. Moreaccurate ±0.02� orientation has been achieved using DRON-3 diffractometer.

Compression tests have been carried out using Instron ElektroPuls E10000 machine operated at strain rate 3.10�4 s�1 atambient temperature. Prior to testing the samples have been polished and lubricated by graphite. Shear traces on the singlecrystal faces were obtained and then examined using optical microscope LeicaDM 2500P, SEM instrument Tescan Vega II LMUand AFM instrument SOLVER PRO 47H.

3. Results

3.1. Development of octahedral shear in copper single crystals

For copper single crystals with compression axis orientation along [111] there are six equally loaded (Schmid factorm ¼ 0.272) slip systems, including three {111} slip planes each having two <110> slip directions (Honeycombe, 1984). Slipsystems relating to (111) plane stay unloaded with m ¼ 0 as shown in Fig. 1a. Given the single crystal orientation as statedabove, the octahedral plane (111) is symmetrical with respect to plane (110) that intersects the single crystal's center. Alsoplanes (111) and (111) are symmetrical with respect to this plane. All octahedral planes are non-symmetrical relative to (112)plane intersecting the sample's center. Taking into account the slip planes' positions with respect to the single crystal's faces,wemay say that slipping by these planes will bemore expressed on (112)-face as compared to that on (110)-face. So in spite ofhigh order symmetry of both applied stress and crystallographic orientation of the single crystal with respect to the loadingaxis, the shape of the sample (parallelepiped) results in the shear strain asymmetry with respect to one of the lateral faces.

Fig. 1. The crystallographic scheme of shear planes and directions for [111]-single crystals (a) and constraint shear zones (bed).

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The type of orientation analyzed here differs from the other ones found within the corners of the standard stereographictriangle by the specificity of the slip plane positions with respect to the test machine's platen. That is, the slip planesbelonging to rather large single crystal's both top and bottom zones are confined by the platens e Fig. 1(b, c). It is notinconceivable that shear in these zones is constrained since the reverse sign stresses are induced there by the opposite platen.Let us denote these zones as constraint shear zones and notice that they occupy rather large volume of the [111]-orientedsingle crystal.

Another regularity observed on the [111]-oriented single crystals is that their slip planes and directions belonging to theconstraint zone have common transition point in the center of the sample. It will be shown below that surface deformationpattern on the single crystal's lateral faces is interrelated with the specificity of crystallographic orientation for the abovediscussed [111]-single crystals.

Surface deformation pattern generated on the [111]-single crystal's lateral face after ε¼ 4% compression is represented by(i) rectilinear slip bands developing by octahedral planes and denoted as SB in Fig. 2(a, b); (ii) nucleating macro-andmesofolds on (112) faces; (iii) macrobands shown as MB in Fig. 2. The rectilinear slip bands start forming from the veryonset of deformation and up to ε¼ 2e4%whenmacrobandsmake their appearance. The slip lines are found in a great numberbetween the macrobands. Nucleation of the centerline macrobands starts along the constraint shear zone's boundaries. Themajority of macrobands is found outside of these zones. The (111)-plane slip lines are locatedmainly beyond these zones too,whereas it is not definitely the case both for (111) and (111) planes since large slip line systems belonging to these planesmaybe observed inside the constraint shear zones too.

Another specificity relating to the crystallographic orientation is observed on both (110) lateral faces, denoted as b and d.Deformation zones having the greatest number of slip bands and macrobands and observed on the b face can be obtained byrotation of the d face around the compression axis and mirror reflection with respect to single crystal's end face. So thesample's center plays the role of a transition point in forming the deformation surface pattern.

The intense folding has been observed only for (112) faces in the form of small thin lens-shaped or irregular 10e70 mmlength and 20 mm width formations as well as large mesofolds in the form of smooth 0.07e1 mm length and up to 200 mmwidth surface smooth irregularities. It was found that folding is not that pronounced for (110) faces.

Fig. 2. The optical images of copper single crystal's faces (110) b (a, с) and ð110Þd (b, d) after deformation by ε ¼ 4% and ε ¼ 13%, respectively.

Fig. 3. The “dashed” slip bands on d (110) face (a e OM image, b e schematics).

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The deformation surface pattern becomes more complicated when strain increased to ε¼ 13% as can be seen from Fig. 2(c,d). The deformation pattern surface area occupied both by slip bands and macrobands formed by (111), (111) and (111)planes has increased. Shear by (111) plane is negligible and its traces are localized within two zones on the (112) face.

Even greater contribution to the total deformation is provided by (111)-plane thus giving the macrobands on both a and c(112) faces as well as rectilinear slip bands on faces b, d, smaller SB(111) and MB(111) macrobands as shown in Fig. 2(c, d).One may note also that the MB(111) systems shown in Fig. 2(c) (face b, left bottom corner) have been developed from the slipband SB(111) systems formed earlier at ε ¼ 4% and shown in Fig. 2(a). The macroband system MB(111) on the d-face inFig. 2(d) has also been developed from the SB(111) system shown in Fig. 2(b). It provides support for the suggestion that theslip bands precede the macrobands in the deformation pattern development.

The final deformation stage by folding is interrelated with generation of fold systems, which reveal themselves on both a-and c-faces. Three basic fold systems are: (i) curved folds in the middle of the sample and denoted as “fold” in Fig. 2(c, d); (ii)near-top face BBI fold system; and (iii) near-bottom face BBII fold in Fig. 2(c, d). The bending bands are formed basically in thevicinity of sample's end faces but some of them may be found in the middle part of the sample either on a or c faces.

Under the conditions given, the most intense shear deformation occurs by (111)-plane so that both slipe andmacrobandsoccupy up to 70e80% of the b- or d-face total surface areas. The isolated (111)- plane slip band systems are observed on the(110)-faces in the form of different size “dashed” slip bands shown in Fig. 3. These macrobands show up as curved ones at theperiphery of the “dashed” bands so that when found in f.c.c metals such a formation may be associated with kink bandsaccording to Honeycombe (1984).

Also the slip band systems may be clearly seen on the lateral (110) faces b and d in Fig. 2 and Fig. 4. The b and d faces showonly 14% and 5% of their areas occupied by these slip bands, respectively. These slip bands are observed in the areas withcurved (111)-macrobands. Since no slip bands from cubic planes {001} have been found on the lateral faces we can state thatthese slip bands have nothing to do with the cubic slip and are of different origin.

3.2. Surface deformation pattern macrosegmentation at ε ¼ 13%

The deformation of copper [111]-single crystal in the strain range ε ¼ 4% to 13 % is related to two processes: (i) shear byclose-packed planes and (ii) inclination of the single crystal parts with respect to the compression axis. These two processesserve to subdivide the single crystal into isolated segments of different scale size. It is worthwhile to note that sheardeformation segments or domains described elsewhere (Magid et al., 2009; Lychagin, 2006) are not the same as those dis-cussed above and even have nothing to do with them.

Fig. 4. The “new” system slip bands on the (110) b-face.

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The schematics of deformation-induced subdivision in one of the samples is shown in Fig. 5, where the numbers on theright stand for misorientation of the (111) slip band direction from its initial crystallographic direction, i.e. from the directionof the (111)/(110) intersection line [112]. The minimal misorientations are shown for two near-end segments I and V.Somewhat greater ones were found for central segments III and the maximal numbers corresponded to II and IV segments.The slip band densities for both (111)-plane and the “new” one are maximal for these segments too.

The curved area portions of (112)- faces are located in the vicinity of the segment's boundaries shown as B in Fig. 5. Theinterrelation between this factor and folding has been found, i.e. that wide bending bands develop along the face's curvedareas with the increased density of folds.

Also such a macrosubdivision is typical for deformation of many [111]-copper and nickel single crystals, as studied byLychagin (2006), Lychagin et al. (2006, 2011).

3.3. Development of surface deformation pattern at higher strain

The surface deformation pattern on the (110) lateral faces of copper single crystal does not change significantly with theincrease of strain from 13 to 19% (Fig. 6) and is represented by the same structural components as those found on the ε ¼ 13%samples. The maximal densities of both macro- and slip bands on the (110) faces have been found for segments II and IV. Thesize and interior structure of those macrobands is the same without any new components added in spite of the fact that(111)-macrobands deviate more greatly from the (111)/(110) intersection line direction. The sample's shape becomes morecurved and folds become the most intensely developing structural components on the (112)-faces. Such a situation isopposite to that of (110)-faces where both folds and bending bands either develop within some small areas or do not developat all.

In the same way, deformation to ε ¼ 25% results neither in generation of any new macroband nor in any notable devel-opment of those generated earlier. For instance, two optical images of one and the same segment on the (110) b-face areshown in Fig. 7 to demonstrate the difference between ε ¼ 13% and ε ¼ 25% (the 25% strain image is rotated by 18� due tobending of the sample as a result of deformation). We can see from Fig. 7 that the macrobands do not change and what reallychanged is the amount of slip bands between them. These slip bands have been found in the same places even after repo-lishing and reloading the samples. We can state here that in spite of no changes in the macrobands found on the lateral facesduring continuous loading, shear by the same plane as well as further macroband development occurred in the bulk of thesample.

A small area occupied by (111)-slip bands may be found on the surface of a lip-like formation obtained from the sample'sedge on (110) b-face at ε ¼ 25%. This finding suggests on the complexity of the stress state in this area or deviation of thecompression axis from the preset value during the lip growth.

Fig. 5. The scheme of macrosegmentation pattern at ε ¼ 13%.

Fig. 6. The morphology of surface deformation pattern on the (110) faces b (a) and d (b) at ε ¼ 19%.

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Given the detailed consideration of the surface shear deformation pattern both on the b-face and segment II of the ð110Þ d-face, we observed slip bands belonging to none of the close-packed planes studied. Such a slip system has been obtained firston the samples compressed to ε ¼ 13%. Then both the area occupied by these bands and their density increased with furtherdeformation so that more than 50% of the b-(110)-face area became occupied by them at ε ¼ 25%. We can see here thatdeformation is developed by this system but we have not any idea about why and how this system became involved in thedeformation. A new slip system also has been revealed in (111)-slip bands as shown in Fig. 3. However in this case we at leastcould see that these new slip system bands had been formed after the primary (111)-slip “dashed” bands (Fig. 8.). The highestdensity of these slip system bands has been found within the “dashed” slip bands, which in their turn, could be observed inareas II, III, and IV. Later we will discuss this phenomenon again.

3.4. Surface deformation pattern macrosegmentation at ε ¼ 25%

The deformation subdivision pattern has changed for samples compressed at ε¼ 25% (Fig. 9) so that the (111)-macrobanddeviation from their crystallographic orientation increased for all segments as follows from the surface deformation patternon b- and d-faces. These deviation increments for segments II, III, IV were 15e30%,19e20% and 9e10%, respectively. The near-end face segments' deviations remained at <10% for segment I and 16% for segment V. The primary segment boundariescontain secondary (or transitional) slip bands, which differ by their higher deviations in the slip band positions on the faces ascompared to those of the primary ones (see dashed lines in Fig. 9). The surface deformation pattern on the d -face was similarto that discussed above (not shown).

The above-discussedmechanisms for [111]-single crystals have been supported by the EBSD results. The central part of thecrystal experienced the crystalline lattice rotation with respect to [110]-axis so that the deformation axis reoriented in theorder [111]/ [110]. For segments II and IV at ε¼ 13% the deformation axis approached but did not coincide with [110] whilefor near-platen and central I, II, V segments it deviated from [111] too, but only by a <5� angle. Fig. 5 shows the deviationsmeasured from the slip band pattern. The deformation axis misorientation map for the band crossing the face b (110)

Fig. 7. The OM images of b (110) face at ε ¼ 13% (a) and ε ¼ 25% (b).

Fig. 8. The SEM image of “new” slip system bands developed in the “dashed” (1e11)-slip bands.

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downward from segment I to segment V is shown in Fig. 10, a. For ε ¼ 25% the crystalline lattice rotation changed thedeformation axis of the central segment to be close to [773] while for the near-platen segments it is still [111]. The crystallinelattice of the segments II and V continues its rotation around [110] so that their deformation axis changes from [111] to [110].The segment boundaries are soft withmisorientations shown by dashed lines in Fig. 9. The crossover areas between segmentsI, II and IV, V are marked as segments a1 and a2, respectively (Fig. 10a). The reorientation of these crossover segments is 35.2�

and their structure is shown in Fig. 10b, c in more details. The reorientation of these crossover segments is changed in theorder [111]-[221]-[773]-[110] as shown by the pole figure reconstructed from the point orientations in a2 band (Fig.10d). Thecrossover segments between the central and neighboring segments are not so expressed and deformation axis there changesfrom [110] to [773] with misorientation angle 16.9�. Both (111) and (111) slip bands can be seen in Fig. 10(b) with theirdeformation axis being close to [773].

It is worthy to note that the EBSD data are similar to those obtained from the (111) slip band pattern, including the fact thatcrystalline lattice reorientation in segment II is higher as compared to that of segment IV. However, the misorientations foundfrom both the slip band and macroband deviation patterns are somewhat higher than those obtained from the segmentreorientation analysis. Such an inconsistency stems from the fact that the reoriented segment experience extra shear by (111)

Fig. 9. The scheme of macrosegmentation pattern at ε ¼ 25%.

Fig. 10. Schematics of ε ¼ 25% deformation axis rotation in a band crossing downward the (110) face between segments I and V (a) along the line A in Fig. 9, thecrossover а1 (b) and а2 (c) band structures and pole figures reconstructed from the point orientations in a2 band (d).

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planes. The result is that deformation axis of segments II, IV and III becomes close to [110] and [773], respectively and cor-rected to plastic deformation inhomogeneities. Such a reorientation serves to change the Schmid factor value for the slipplanes in a single crystal.

In addition to the fact that shear by (111) becomes feasible, the reorientation serves to increase the Schmid factor for (111).In this connection, the shear carriers emergence on the free crystal faces get changed. The specificity of the slip planes'positions in different segments is shown in Fig. 11 with the exception of planes (111) and (111). Since the Schmid factor valueis growing for (111), the same is true for the slip band number and therefore we can see the maximum slip band densitywithin segments II and IV. These bands are arranged in linear formations of various sizes similar to those shown abovebecause the reorientation starts from these segments. Planes (111) and (111) do not come into the deformation because ofzero Schmid factor.

Fig. 11. The schematics of (111) and (111) slip plane positions in the direction [110] (a) in the crystal and (b) crystallographic diagram of slip planes and directionin segments II and IV.

Fig. 12. AFM image (a) of slip band steps on the surface of (110)-face and (b, c) corresponding cross section profiles at 25%.

D.V. Lychagin et al. / International Journal of Plasticity 69 (2015) 36e53 45

As can be seen from the above description, single crystals with orientation [111] display at least two parallel occurringmacrosegmentation processes under deformation. The first will be macrosegmentation by shear, which divides the crystalinto shear strain domains. The second will be macrosegmentation by the crystalline lattice reorientation with the resultingmisorientation macrosegments.

3.5. Structural components of the surface deformation pattern

As shown and discussed above, different structural units may be found on the [111]-crystal faces under deformation. Theshear surface deformation pattern is generated as a result of deformation processes occurring both inside and at the surface ofthe sample, as dictated by the crystallographic orientation of the sample with respect to loading as well as by the location inthe crystal. Each structural unit denotes the specificity of the deformation occurrence in this specific part of the sample,possessing its own specific stress-strain state. Of no less importance is the fact that shear carrier should be able to emerge onthe free surface. Contributions from structural unit into the single crystal total strain may differ both by time and size, andtherefore, more details are needed to assess plastic deformation specificity under conditions given.

3.5.1. Slip bandsThe smallest component of the surface deformation pattern is a slip band being denoted as SB in Fig. 12. Slip bands

belonging to planes (111), (111), and (111) are generated from the very beginning of deformation. They form 5e200 nmheight steps at the crystals faces (Fig. 12) with the mean height being 8e10 nm. Further deformation generates the reor-ientation areas and (111) slip bands appear at this stage of the process.

These slip bands compose a basic surface structural component at the onset stages of deformation. For higher strain thesebands can be found in areas between macrobands and serve partially for accommodating great macroband displacements.Also slip bands are generated prior to the macrobands which are composed of them as well as the mesobands.

Fig. 13. SEM images of macrobands.

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3.5.2. Macro- and mesobandsLarge macrobands up to 5 mm length are the structural components that appear on the [111] crystal' faces in the

beginning of deformation (see MB in Fig. 13). These macrobands appear as most intensely developing structural surface unitsfor strain in the range ε ¼ 10e13%. However, they do not seem to develop further at higher strain; instead, as noted above theslip bands are generated in the areas between the macrobands (see SB in Fig. 13). The macroband itself is composed of slipbands and at first approximation could be even classified as a pack of slip bands but this is not acceptable because of its morecomplex morphology.

We can see that macrobands are rather large 0.5e5 mm and even 10e20 mm height step-like formations at the crystalsurface, i.e. an order of magnitude higher as compared to that of the slip bands. We can easily estimate that it takes about30e60 thousand dislocations to form themacroband assuming the Burgers vector b¼ 0.3 nm. The cross section profiles in Fig14b, c show themacroband's structural inhomogeneity by its length so that steps A and B are either diminishing (B in Fig. 14b)or disappearing (A in Fig. 14c) with the macroband length.

According to above result we can say that macroband is not simply a pack of slip bands especially if its height is amaximum at one end and almost zero at the other. The macroband may accumulate displacement and increase its heightstepwise (Fig. 14d, e) or first increase and then diminish its rise over the crystal' face level (Fig. 14aec). The resulting effect isgeneration of a surface pattern similar to that composed by slip bands (Fig. 12).

The origin of the macrobands may be related to the existence of a constraint shear zone since no macrobands are found onthe lateral sides of crystals oriented in a manner that excludes formation of such constraint shear zones axis (Lychagin, 2006;Lychagin et al., 2011). In other words, the macroband system is a complex structural unit of the surface deformation pattern,being morphologically similar to the slip band system albeit producing the order of magnitude higher displacements.

Deformation mesobands are not found on [111]-single crystals at low strain but appear as basic structural componentsalong with mesopacks and rectilinear slip bands in the course of reorienting in segments II, III and IV when deformation axisbecomes close to [110]. It is known that all above named structural units are typical findings on the [110]-oriented singlecrystals axis (Lychagin, 2006). The mesobands shown in Fig. 15 are composed of slip band packs inclined with respect to themesoband boundaries. The origin of these mesobands is plausibly determined by the stress distribution throughout thesample so that shear becomes more preferable in a band inclined at some definite angle. Under conditions given themesobands appear along the directions close to those of segments' boundaries.

3.5.3. Folds and lipsWe noted above that in addition to generation of macrobands there are folds on the surface of [111]-single crystals under

deformation. These folds form as a result of bending the single crystal and plastic flow of sample's edges in the vicinity of theend faces contacting with the test machine's platen (generation of a lip). Preferably they form on (112)-face because of

Fig. 14. 3D-images of macroband systems (a, d) and their cross section profiles (b, c, e).

Fig. 15. SEM image of mesobands belonging to segment III at ε ¼ 25%.

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preferable sample's bending with respect to these faces. The deformation induced folds form as a system of folds fromwhichthe bending bands possess the highest density of folds (Fig. 16a) and may be found on the segments' boundaries (segments IIand III in Fig. 16a). Fold systems of lower density are found inside the segments' area (Fig. 16b).

In spite of the fact that folds on metallic surfaces are very interesting details of the surface deformation pattern, they didnot attract much attention and were studied basically in relation to complex loading methods such as rolling (Gubernatorovet al., 1999a, 1999b, 2001; Sokolov et al., 1995).

Earlier deformation by folding at different scale levels has been observed elsewhere (Lychagin, 2006; Lychagin et al., 2006;Lychagin et al., 2011) and related to a complex process of accommodating the subsurface layer to deformation in the bulk of amaterial. In this work we tried to reveal only basic fold structures and relationships but in view of many types of foldsobserved experimentally the results of such a study will be reported elsewhere in a separate paper. Analyzing the EBSD shearand reorientation map (Fig. 17), we can trace the general course of deformation in [111]-single crystals under compression.

3.6. Structural units localization and their relationship with the stress-strain curve stages

The development of shear in these single crystals starts from generation of rectilinear slip bands with further generation ofstrain macrobands (Fig. 17(aed)). The areas occupied by basic deformation domains are distributed non-uniformly over thesingle crystal's faces so that slip systems related to different planes occupy different surface areas. The deformation pattern on

Fig. 16. The SEM images of bending band between segments III and IV (a) and system of folds at the top of segment IV (b) on (112) face.

Fig. 17. Schematic layout of surface deformation pattern on the lateral faces of single crystal showing positions of basic deformation domains and segments forstrain degrees 4 / 19%.

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D.V. Lychagin et al. / International Journal of Plasticity 69 (2015) 36e53 49

the (112)-faces is represented by folds which start appearing from the very onset of deformation Fig. 17a, c. The (111)-planeslip band systems are separated from the corresponding macrobands by the constraint shear areas (Fig. 17bed).

The macrobands are developed under deformation ε ¼ 4e13% (Fig. 17eeh) from the slip band systems. Also new slip bandsystems relating to the basic planes appear in the process. The appearance of the reoriented areas leads to generation of(111)-plane slip band systems. The single crystal's face becomes divided into 5 segments. The number of intersecting slipband systems is low as compared to single crystals of other orientations to be found within the corners of the standardstereographic triangle (Lychagin, 2006; Lychagin et al., 2006; Harren et al., 1988).

Folding is a most intensely developed process on (112)-faces as compared to that of (110)-face. The bending bands appearin the subsurface of faces in contact with the test machine anvils as well as between the segments. The deformation domainsbecome divided into smaller parts in the course of reorientation since slip systems belonging to each forming subdomainhave their own Schmid factors.

No newmacroband systems tend to appear when increasing the strain degree up to ε¼ 19% (Fig.17, i, j). The deformation isdeveloped by (111) and (111) planes as well as by crystalline lattice rotation in the reoriented areas.

The deformation in the single crystal is developing in accordance to a course of the strain-hardening curve as follows:

1. Shear by slip planes at the onset of the strain hardening stage II (see A in Fig. 18a, b);2. The onset of macroband generation (В in Fig. 18a, c);3. Reorientation of crystalline lattice or formation of the reoriented segments during the crossover from stage II to stage III.

Development of reorientation and shear in the reoriented segments by new slip planes (see C in Fig. 18a, d).

So the specificity of the surface deformation pattern generation is related to the changes of the strain hardening coefficientvalue, i.e. to appearance of a new stage on the strain-hardening curve. Reduction of the strain-hardening coefficient isobserved during both the crystalline lattice reorientation and changing of the Schmid factor value for the slip systems. Let usnote here that a relationship between strain hardening curve stages and volume percentage of different dislocation

Fig. 18. The stress-strain (1) and strain hardening (2) curves of the samples studied (a), schematics of for slip bands (b), macrobands (c), and reorientationsegments (d).

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substructures has been shown earlier by Koneva and Lychagin et al. (1985a,b, 1988). Below we discuss the reasons behindmisorientation and softening in more details.

4. Discussion

Several reasons responsible for development of macroscale reorientation in [111]-single crystals may be identified such asstress distribution inhomogeneity or stress state difference between near-platen and central parts of the sample, the presenceof constraint shear zones where reverse sign stress are induced in the slip planes by the opposite testing machine's platen,and the symmetry misfit between equal-loaded slip planes and lateral faces in the crystal.

The differences between stress states that occurred under loading may be illustrated by Fig. 19a where three zones can bedelineated each of them having its own normal stress diagram. The central zone is subject to uniform uniaxial compressionwhereas two other ones located in the vicinity of sample's ends are under conditions of inhomogeneous hydrostaticcompression. The crystalline lattice rotation starts from the boundary between zones of different stress states. A structuralunit of a finite volume located on the boundary between the zones relating to different stress states ðsijss0

ijÞ is loaded by amoment of force, which contributes to the reorientation (Fig. 19b). We note here that stress distribution changes whenchanging the sample's aspect ratio and this is what occurs when testing samples of a height/width aspect ratio other thanused in our experiments, i.e. h/d ¼ 2 are tested.

The load applied to the sample is distributed inhomogeneously so that corners and near-platen edges become the stressconcentrators. Also the most intensive deformation occurs by octahedral planes, which pass through them. Therefore,orientation of all the sample faces will influence the development of shear in the single crystal. That such is true wasillustrated by Teplyakova et al. (2004) who qualitatively showed the role played by stress concentrators in the specificity ofshear pattern formed on aluminum [110]-single crystals.

It was indicated above that constraint shear zones for all the slip planes occupy substantial volume inside the singlecrystal. There is a zone inside the single crystal with its slip planes being confined by the test machine's platen (denoted as CSin Fig. 19, a). This volume is obtained by intersecting constraint shear zones shown in Fig. 1 and its boundaries are close tothose of primary segments. This zone is at angle with respect to compression axis while secondary segments a1 and a2 appearin the vicinity of both the constraint shear and inhomogeneous hydrostatic compression zones.

As a result, dislocations of one sign glide in the stress fields exerted by one platenwhile other reverse sign ones glide in thefield exerted by other platen. Such a configuration makes possible the hindering of shear on the constraint shear zoneboundary. The dipole excess dislocation density zones appear under these conditions and the whole situation ends up in themisorientation of the said volume (Lychagin, 2006).

The effect of hindered dislocation pile-up on the reorientation has been suggested by Jia et al. (2012a,b) where slip wasassumed starting from the near-end edge and when propagating further it initiates secondary shear from the vertical edge.The hindered slip forms a dipole configuration of excess same sign dislocation density which then causes the reorientation ofthe structural component in the polar structure zone, i.e conditions created there for development of local volume misori-entation according to Fig. 19c.

Meanwhile we have to stress some features important for further development of the above stated model considerations.It was noted above that shear is propagating from the highest stress zones such as near-end corners and edges, i.e linear and

Fig. 19. Schematics of uniaxial (1), hydrostatic compression (2) and constrained shear (CS) zones for all slip planes. Moment of force action developed on anallocated unit between the zones with different normal stress patterns (b) because of excess dislocation density accumulation in shear suppression in the CS (c).

Table 1Schmid factor for slip planes as changed in deformation.

Loading axis [111]

Slip plane (111) m Slip plane (111) m Slip plane (111) m Slip plane (111) m Slip plane (111) m

ε ¼ 0% ε ¼ 25% ε ¼ 0% ε ¼ 25% ε ¼ 0% ε ¼ 25% ε ¼ 0% ε ¼ 25%

[110] 0 [110] 0 [101] 0 0 [011] 0 0 [110] 0 0[011] 0.27 [011] 0.41 [011] 0.27 0 [101] 0.27 0 [101] 0 0.41[101] 0.27 [101] 0.41 [110] 0.27 0 [110] 0.27 0 [011] 0 0.41

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point-like stress concentrators; therefore its occurrence is greatly dependent on the lateral face orientations. Apparently 2DFEM simulations (Jia et al., 2012a) can not to the full extent take into account both crystallographic and geometrical aspects ofthe real 3D shear pattern as well as the stress inhomogeneities. Using both crystallographic and geometrical approaches tothe octahedral shear development allows distinguishing among free, constraint or partially constraint volumes as well asforecasting high-strain or rotation plasticity zone's localization (Lychagin, 2006). Studying the surface deformation patternspecific to compression loaded fcc metals or single-phase alloys makes it possible to reveal the orientations belonging to thestandard stereographic triangle's corners, which lead to deformation occurrence by slip plane packs basically and generationof both meso- and macrobands. Also the lateral faces' orientations serving to release accommodative shear bands have beenidentified. These shear bands have been observed on single crystals having the Goss orientations {011}<001>. At the sametime, the shear band localization zones corresponded to those obtained in modeling the Cooper orientations {112}<111> (Jiaet al., 2012a,b). The development of accommodative shear bands for Cooper orientation and reduced samples' height is ingood agreement with our observations made on copper, nickel and aluminum samples having the Goss orientation.

Let us dwell on the changes of slip system in reorientation and the effect they have on the surface deformation relief of the[111]- samples (see Table 1) for the three equally loaded slip systems with Schmid factor value 0.27. Slip systems by (111) stayunloaded with zero Schmid factor. The shear stresses in octahedral slip planes change in the course of deformation-inducedreorientation so that Schmid factors change too as shown in Table 1. The Schmid factor of only (111) slip plane is increasedwhile it is decreased to zero for other slip planes. At the same time slip systems belonging to the plane perpendicular to theloading direction become activated and their Schmid factors increase to that of the (111)-plane. Such changes are inaccordance to basic conceptions of texturing in metals and alloys. Experimental data obtained in this work allow to specifythis process which showed up in the process of segmentation.

5. Conclusion

The sequence of deformation processes in these single crystals in relation to the strain hardening stages may be defined asfollows. At the first stage several primary macrosegments are generated. The intensive shear by one of the equally-loadedplanes leads to bending the sample. Also the bending is stimulated by development of constraint shear and different pri-mary stress zones. The above-designated factors contribute to reorientation of the single crystal zones and formation ofreoriented segments. The Schmid factor is increased for one of the equally-loaded planes and decreased for other ones. Such asituation is resolved by a breakthrough of permanently created barriers, thus leading to softening, which is expressed mostclearly at the stage of reorientation and slipping by previously not loaded planes. The result is reducing the strain-hardeningcoefficient's value and initiating the deformation represented by the strain hardening curve stage III. The single crystals oforientations studied here demonstrate different surface deformation patterns which denote the differences in shear occur-rence at macro- and mesolevels. It was established that along with systems of parallel slip bands, which represent thepredominating shear development in strain domains, there were generated other deformation pattern components such asmacro- andmesobands as well as folding structures. Moreover, these folding structures were found to be the basic macrolevelcomponents. The specificity of folding structures is that there are slip bands between them, however they are arranged in amanner that allows them implementing an order of magnitude higher displacements as compared to those achieved by theneighboring area's slip bands. The mesobands serve as auxiliary surface deformation pattern components and form in thesegments that have already suffered reorientation from (111) to (110).

Acknowledgments

Investigations were carried out using the equipment center for collective use “Analytical Center Geochemistry of NaturalSystems” National Research Tomsk State University (NR TSU). This work has been supported by the Program to improve thecompetitiveness of NR TSU. Financial support has been provided also by Program for Basic Scientific Research of the StateAcademy of Science on 2013-2020 and RFBR grant 13-08-00324.

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