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Acta Materialia 60 (2012) 376–386

Mapping subgrain sizes resulting from severe simple shear deformation

S. Abolghasem, S. Basu, S. Shekhar, J. Cai, M.R. Shankar ⇑

Department of Industrial Engineering, 3700 O’Hara St., University of Pittsburgh, Pittsburgh, PA 15261, USA

Received 2 August 2011; received in revised form 28 September 2011; accepted 29 September 2011Available online 4 November 2011

Abstract

Grain refinement under interactive effects of severe shear strains, strain rates and temperatures often follows complex trajectories.Encapsulating the process–structure linkages under these conditions is central to controlling product outcomes from processes involvingsevere plastic deformation. This paper uses in situ characterization of deformation in large-strain machining using high-speed digitalimage correlation and IR thermography to examine the microstructural consequences across a swathe of directly quantified thermome-chanical conditions. Using electron microscopy, it is shown that the average subgrain sizes resulting from this deformation correlate withthe strain and the theoretical limit of the subgrain-size that is achievable. From this, a suitably parameterized map-space is proposed forcapturing average subgrain sizes resulting from severe shear deformation. The parameterization is characterized by the y axis as the strainand the x axis as a parameter R, which is a function of the strain-rate, temperature and material constants. Noting that the surfaces frommachining processes are essentially derived from severe thermomechanical conditions akin to those explored here, the implications ofsubgrain size-maps for controlling surface microstructures on components manufactured by machining processes are discussed.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Ultrafine grained microstructure; High-speed deformation; Severe plastic deformation (SPD)

1. Introduction

An array of emerging materials processing and manu-facturing practices use severe plastic deformation (SPD)to favorably modulate mechanical and functional proper-ties by achieving novel microstructural designs. Much ofthe SPD literature has focused on the achievement of thegreatest possible levels of grain refinement to maximizethe weight-specific strength [1]. However, it has becomeincreasingly apparent that the spectrum of realizablemicrostructures is not just a function of the severe strains,but can be further broadened by probing a wide range ofstrain-rates and deformation temperatures. For example,dynamic plastic deformation involving the imposition oflarge strains at high strain-rates in compression can achievea switch-over from typical ultrafine-grained microstruc-tures to nanotwinned structures [2]. Similar behavior hasbeen observed in high-rate SPD in simple shear, which

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.09.055

⇑ Corresponding author.E-mail address: ravishm@pitt.edu (M.R. Shankar).

can result in microstructures that are nanotwinned [3], havemultimodal grain size distributions [4] or are with tunablegrain boundary structure/energies as one-to-one functionsof severe plastic strain, strain-rate and temperature [5].However, despite such empirical anecdotes, underlyingmappings between the thermomechanics of severe defor-mation and the resulting microstructures often remainobscured.

More traditionally, Hopkinson bar tests and hot torsiontests have been used to probe material response under arange of thermomechanical conditions. While Hopkinsonbar tests can allow for probing of large strain rates often�102 s�1, they usually involve the imposition of somewhatsmall values of strain (typically < 1) [6]. In contrast, hottorsion experiments have been used to probe larger strains,even in excess of 10, but at smaller strain-rates, usually lim-ited to <102 s�1 [7,8]. These studies have often sought tomodel the constitutive response of metals as well as to cap-ture the microstructural evolution as a function of thethermomechanics of deformation across a swathe ofZener–Hollomon (Z) parameter values [8,9].

rights reserved.

S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386 377

In contrast, the present authors used plane-strain largestrain machining (LSM) to examine strains in the range1–10, strain-rates in the range 10–103 s�1 and temperaturesfrom near ambient to �400 K, in a simple-shear geometrywhich remains essentially isomorphic across the entirerange (Fig. 1) [5]. By examining microstructures close tothe center of the “chip” where homogeneous severe simpleshear is imposed, it was shown that the complex interactiveeffects of SPD strains with Z engendered an array of ultra-fine grain characteristics. However, it is unclear yet how thestrain, strain-rate and temperatures in the deformation zonemap to resulting grain characteristics across the wide rangeof deformation conditions typically achieved in LSM. If thisinterrelationship could be captured, it could enable flexibleframeworks for either customizing microstructures on-demand by suggesting appropriate severe deformation con-ditions or, inversely, help to predict deformed microstruc-tures, given the thermomechanical deformation state. Onepossible approach for achieving this could revolve aroundthe delineation of microstructure maps wherein characteris-tics of deformed grain structures are projected as a one-to-one function on a suitable map-space, which is parameter-ized in terms of the strain, strain-rate and temperature ofdeformation.

This characterization would be directly relevant tomanipulating microstructures on the surfaces of compo-nents created using ubiquitous machining processes suchas turning, milling and shaping. Microstructures onmachined surfaces are inherited from the same deformationzone that leads to the ultrafine-grained “chip” in LSM andare subjected to comparable levels of simple-shear SPD[5]. The surface microstructures would probably determinethe product life cycles in an array of engineered componentsas well as their surface mechanical and functional proper-ties. Also, this elucidation can be relevant to other shear-based deformation processes, including high-pressure tor-sion (HPT), equal channel angular pressing (ECAP)

Fig. 1. Deformation geometry during chip formation allowing for plane-strainzone for DIC along with IR thermography of the deformation zone are illustrfrom both the free surface and the tool–chip interface, where predominantly u

[10,11], large strain extrusion machining [12] and friction stirwelding processing [13].

2. Experimental methods

The empirical data was gathered in a characteristic face-centered cubic material, microcrystalline OFHC Cuannealed at 700 �C for 2 h, using a simple-shear SPDframework, LSM, which is schematically illustrated inFig. 1 and is characterized by a sheet-like sample whichis advanced against a wedge-shaped tool where its velocityvector is orthogonal to the cutting edge. Such a configura-tion can lead to a plane-strain deformation field where SPDis imposed over a domain corresponding to undeformedthickness a0. This SPD involves simple shear and leads toa deformed “chip” thickness ac. The choice of this configu-ration was motivated by the observation that LSMinvolves severe shear in a deformation zone ahead of atool-tip in an unobstructed configuration, which allowsfor in situ observation of the metal flow using high-speedvisible light imaging and IR thermography in a configura-tion akin to that schematized in Fig. 1. In plane-straindeformation of LSM, it is found that the effective deforma-tion strain (e) associated with the formation of the “chip” isgiven by [14]:

e ¼ cos a=ffiffiffi3p

sin / cosð/� aÞ ð1Þwhere a is the rake angle of the cutting tool, and / is theinclination of the shear plane as illustrated in Fig. 1. Thisshear plane angle is given by Eq. (2) in terms of the ratioof the undeformed material (a0) to that of the deformedchip (ac):

tan / ¼ a0=ac cos a1� a0=ac sin a

ð2Þ

By varying a from 0� to +40�, it is possible to achieve shearstrains ranging from �1 to 10 in a single deformation pass

LSM. Typical focal locations for high-speed imaging of the deformationated. Microstructures were characterized near the center of the chip, awayniform SPD is imposed.

378 S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386

in Cu. a is considered positive when measured with respectto the normal, as illustrated in Fig. 1. Table 1 lists the con-ditions corresponding to 16 configurations with four rakeangles (a = 0�, 20�, 30� and 40�), which were advanced atfour different speeds: V (i.e., low (L) = 50 mm s�1; medium(M) = 550 mm s�1; medium–high (MH) = 750 mm s�1;and high (H) = 1250 mm s�1). Resulting chip samples aredenoted 0L, 0M, 0MH, 20H, etc., where the numbers rep-resent the rake angle and the letters L, M, MH and H referto the machining speeds. In all the experiments, a0 was cho-sen to be �0.2 mm and the resulting ac values were mea-sured for the various conditions to estimate the strainvalue using Eqs. (1) and (2).

Effective strain-rate (_e) in LSM is a function of thegeometry and is proportional to V with [15]:

_e ¼ CV cos a sin /cosð/� aÞa0

ð3Þ

The proportionality constant C is a characteristic of thechosen material. The direct proportionality of _e with V of-fers the ability to scan over two orders of magnitude ofstrain-rates for the aforementioned deformation condi-tions. To determine C, the strain-rate in the deformationzone was measured in situ using digital image correlation(DIC) for a range of LSM conditions with known valuesof a, /, a0 and V. Then, C can be back-calculated usingEq. (3), and the C value thus determined can be used toestimate strain rate for any LSM condition. DIC was per-formed on a time-series of high-speed images of the defor-mation zone, which were acquired from the side of thedeformation zone, as illustrated in Fig. 1, using aPCO1200HS camera system. Prior to imaging and LSM,the plate-like samples in Fig. 1 were loosely sprayed onthe side with a black spray paint. This provided a field ofasperities or speckle patterns which could be tracked dur-

Table 1Thermomechanical deformation conditions accessed using the variousLSM parameters in this study; for the various samples effective strain (e),effective strain rate (_e), temperature measured using an IR camera (Texp),temperature values calculated at deformation zone using moving heatsource model (Tcalc) are listed as a function of the deformation speed (V)and tool angle (a).

Samples V (mm s�1) a e _e (1 s�1) Texp (K) Tcalc (K)

0L 50 0� 8.7 60 322 3630M 550 0� 5.9 940 – 4540MH 750 0� 5.6 1240 – 46420L 50 20� 5.9 80 342 34620M 550 20� 3.9 1290 378 41220MH 750 20� 3.6 1740 – 41620H 1250 20� 3.4 3130 – 43930L 50 30� 4 100 319 33230M 550 30� 2.6 1740 – 37930MH 750 30� 2.5 2290 – 38530H 1250 30� 2.3 4030 – 40240L 50 40� 2.6 140 324 32140M 550 40� 2.1 1930 336 36740MH 750 40� 2 2520 339 37240H 1250 40� 1.8 4680 – 381

ing the imaging of LSM. Using cross-correlation to com-pare speckle patterns of temporally adjacent images inthe recorded sequence, a displacement vector field for thedeformation zone was obtained. From this, the velocityvector field was calculated by division of displacementcomponents by the interframing time [16–18]. Spatial dif-ferentiation of the velocity field yields the strain-rate field,which was then used to calculate C. DIC was found to re-sult in strain-rate accuracies better than 5% in calibrationexperiments [17].

Analogously to the high-speed imaging of the deforma-tion zone for DIC, an IR camera (FLIR 325A) was used toperform calibrated thermography of the deformation zonefor several conditions in Table 1 (also see Ref. [5]). To con-vert IR images to temperature field data, the IR camerawas first calibrated. Cu sheets were first coated with a blacknon-reflective stove paint to achieve a uniform emissivity.Then, the samples were homogeneously heated to temper-atures between 298 K and 473 K and observed with thecamera to record the corresponding number of “counts”

as a function of temperature. This provided the calibrationcurve to convert the IR image into the temperature fielddata of the interrogated object. Calibrated accuracy �2 Kwas found to be possible in this temperature range. Then,the samples subjected to LSM were coated with the sameblack paint, and the IR camera was used to measure thetemperatures in the deformation zone. During this imag-ing, the camera was focused on the middle of the deforma-tion zone, where DIC confirmed the imposition of uniformshear, ahead of the tool tip.

Complementing the characterization of the thermome-chanics, the chips were examined in a scanning electronmicroscope (Phillips XL30) using electron back-scattereddiffraction (EBSD) analysis. The samples for this study wereprepared by excising suitable sections of the LSM chip sam-ples and then subjecting them to a series of metallographicpolishing steps, followed by vibratory polishing. Theregions close to the center of the chip, away from boththe tool–chip interface and the free surface, were focusedon, where the subsequently delineated results from DICand IR experiments showed uniform deformation condi-tions. The resulting EBSD micrographs were analyzed,and the average subgrain size (dm), defined as domains withmisorientation > 2�, along with the standard deviation (SN)was measured for the various conditions. The grain sizedetermination was performed by calculating the numberweighted average of the area of the subgrains over thescanned area using the OIM data-processing software.From the average subgrain area thus measured, the equiva-lent circle diameter was measured as the average subgrainsize, listed in Table 2. Several scans for the various condi-tions resulted in sampling of several hundred subgrains toyield reliable estimates for these samples. Transmission elec-tron microscopy (TEM) was also performed on electrolyti-cally thinned samples using a JEOL 200-CX microscope.Vickers microhardness tests were also performed on themetallographically polished samples. Typically, samples

Table 2Microstructural consequences of deformation conditions listed in Table 1: measured average subgrain size (dm), standard deviation for measured subgrainsize (SN), calculated subgrain size at the initiation of stage IV (dIV), exact solution for saturated value of subgrain size ðdexact

s Þ; approximate solution forsaturated value of subgrain size (dappr

s ), subgrain size calculated using Eq. (10) (dr) and the measured Vickers microhardness values for the various samples.

Samples dm (lm) SN dIV (lm) dexacts (lm) dappr

s (lm) dr (lm) Vickers hardness (kgf mm�2)

0L 0.24 0.01 0.303 0.217 0.214 0.236 154 ± 4.50M 0.25 0.02 0.316 0.236 0.221 0.327 147 ± 4.80MH – 0.314 0.238 0.222 0.334 –20L 0.33 0.07 0.295 0.209 0.210 0.311 163 ± 3.220M 0.4 0.11 0.298 0.217 0.214 0.376 161 ± 2.520MH – 0.297 0.217 0.214 0.383 –20H 0.46 0.09 0.299 0.221 0.215 0.392 159 ± 3.130L – 0.289 0.202 0.207 0.362 –30M 0.4 – 0.286 0.205 0.208 0.406 154 ± 5.930MH – 0.286 0.205 0.208 0.410 –30H 0.4 – 0.287 0.207 0.209 0.414 152 ± 4.740L 0.4 0.02 0.284 0.197 0.204 0.401 158 ± 3.540M – 0.282 0.200 0.206 0.418 157 ± 5.740MH – 0.282 0.200 0.206 0.420 –40H – 0.281 0.200 0.206 0.427 155 ± 5.3

Fig. 2. DIC of LSM performed with a 40� rake angle (a) tool withV = 25 mm s�1 to back-calculate C for Cu in Eq. (3). (a) Grid ofdisplacement field vectors calculated between two consecutive framesseparated by a time interval of 0.7 ms. (b) Corresponding strain-rate fieldcalculated by differentiating the displacement field spatially andtemporally.

S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386 379

were in an epoxy mount, and 50-g loads were used in theseindentation experiments.

3. Results

3.1. Strain and strain-rate of SPD in LSM

The deformation strains in LSM, measured using Eqs.(1) and (2) for the various conditions, are listed in Table1. Note that the strain values increase for decreasing valuesof a to cover a swathe of values ranging from 2 to 9. Com-plementing the strain estimates, the strain-rates were calcu-lated using Eq. (3). The C value in Eq. (3) was obtainedfrom DIC experiment results, illustrated in Fig. 2. Fig. 2ashows the displacement field calculated between two con-secutive frames separated by a time interval of 0.7 ms.The corresponding strain-rate field obtained by differenti-ating this field spatially and temporally is illustrated inFig. 2b. By directly measuring a, /, a0 and V from theseimages, C was back-calculated as 2.77 (V in mm s�1 anda0 in mm). Backed by this in situ confirmation, thestrain-rate was estimated using Eq. (3) for the various con-ditions in Table 1. Also noted is the uniform strain ratenear the center of the chip, away from the edges in thisimage, from where the subsequent microstructure charac-terizations were performed.

3.2. Deformation zone temperatures

Fig. 3a and b illustrate the typical temperature field inthe deformation zone, which were measured using cali-brated IR thermography for two very different thermome-chanical conditions corresponding to 30L (e = 4,_e = 100 s�1) and 40 M (e = 2.1, _e = 1930 s�1), respectively.Both images show the uniformity of the temperature in themiddle of the deformation zone (or shear plane in Fig. 1)where the camera is focused. The average temperatures in

the deformation zone (Texp) were measured for a range ofLSM conditions and are listed in Table 1. Note that itwould be misleading to draw any conclusions about thetemperature in the regions of the image other than at the

Fig. 3. IR thermographs showing the temperature in the deformationzone for (a) 30L (e = 4, _e = 100 s�1) and (b) 40 M (e = 2, _e = 1930 s�1).The camera was focused on the center of the deformation zone, i.e., themiddle of the shear plane as illustrated in Fig. 1.

380 S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386

middle of the deformation zone, considering that elsewherethe camera may be out of focus.

During such SPD, significant dissipation of plastic workoccurs by heat generation in a characteristic “moving heatsource” configuration. The heat source being the localizedshear plane across which mass transport occurs can be con-veniently calculated in plane-strain LSM using approachesakin to that in Ref. [19]. This approach follows the follow-ing analytical route; more details of the derivation can befound in Ref. [19]. The thermomechanically coupled tem-perature rise at the nominal shear plane in the deformationzone occurs in response to the plastic work associated withprogressive accumulation of the large shear strains to con-vert the undeformed bulk into the “chip” material by LSM.For incremental increase in the strain de, the temperaturerise dT is given by: qCpdT ¼ ð1� bÞ � rðe; _e; T Þde. b is thepartition parameter, which determines the fraction of heattransported by the bulk workpiece away from the chip andthe deformation zone in the moving heat source configura-tion that typifies LSM in Fig. 1. r is the shear-flow stress,and qCp is the heat capacity of Cu (=3.63 MJ m�3 K�1

[20]. b ¼ 14a erf

ffiffiffiapþ ð1þ aÞerfc

ffiffiffiap� e�affiffi

pp ð 1

2ffiffiap þ

ffiffiffiffiffiaÞ

p, where

a ¼ ðV � a0 � tan /Þ=4 � j, and j is the thermal diffusivityof Cu (=117 mm2 s�1) [20]. Assuming the Johnson–Cookmodel [21] for describing rðe; _e; T Þ; it can be shown thatthe total temperature rise can be obtained by integrating

qCpdT ¼ ð1� bÞ � rðe; _e; T Þde and rearranging terms toget [19]:Z T calc

T 0

qCpðT Þ1� T�T r

T m�T r

� �m dT ¼ ð1� bÞ Aeþ Bnþ 1

enþ1

� �

� 1þ C ln_e_e0

� �ð4Þ

A, B, C, m, _e0, Tr and n are parameters for the Johnson–Cook model for Cu which were obtained from Ref. [21].e and _e are calculated from Eqs. (1)–(3). T0 is the initialtemperature of the workpiece, which is taken to be theambient temperature 293 K. Solving this integral for mate-rial constants for Cu yields estimates of the temperaturerise associated with the LSM process, which compare wellwith the IR measurements.

The calculated values for deformation temperature(Tcalc) as a function of LSM parameters are listed inTable 1. Across the spectrum of the strains and strain ratesconsidered here, deformation temperatures ranging fromclose to ambient to �400 K were estimated. These Tcalc val-ues that show good corroboration with the IR measure-ments will be used in the subsequent modeling of themicrostructure refinement from this SPD. It is noted paren-thetically that the values of V chosen here offer strain/strain-rate combinations beyond those realizable with con-ventional SPD approaches. However, “high-speed machin-ing” conditions, which are often an order of magnitudelarger than the highest speed considered here, were notconsidered. Under the high speeds in Cu, it has beenobserved that dramatic temperature rises occur which leadto complete recrystallization of the deformed samples [5].Such transformations are not considered in the presentstudy with the choice of the smaller values of V.

3.3. Microstructure characterization

It is known that, in general, imposition of large shearstrains entails progressive refinement of the microstructure,which is qualitatively evident from the measured averagesubgrain sizes in Table 2. Fig. 4 illustrates the microstruc-tures for some characteristic conditions that illustrates ahighly refined, sub-micrometer-scale structure. The sam-ples shown in Fig. 4 correspond to three widely spacedthermomechanical conditions studied here to present asnapshot of the variety of microstructures that emergedfrom LSM. The 0L condition (e = 8.7, _e = 60 s�1) involv-ing the highest levels of strain at the smallest strain-rate(Table 1) is characterized by the finest subgrain size. The20 M sample, in contrast, is characterized by a higherstrain-rate and smaller levels of strain (e = 3.9,_e = 1290 s�1). At this higher deformation rate, the coupledtemperature rise is also higher. Intuitively, it is reasonableto expect a much coarser microstructure in this conditionthan in the 0L case, and this is indeed found in Fig. 4and Table 2. The third sample illustrated in Fig. 4 is the

Fig. 4. Microstructures for three widely spaced LSM conditions elucidated using EBSD analysis. Inverse pole-figure maps are shown for: (a) 0L, e = 8.7,_e = 60 s�1; (b) 20 M, e = 3.9, _e = 1290 s�1; (c) 30H, e = 2.3, _e = 4030 s�1. Black lines indicate the high-angle boundaries.

S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386 381

30H case, which was generated at much smaller levels ofstrain, but high strain-rates (e = 2.3, _e = 4030 s�1) and tem-peratures. This corresponds to a predominantly subgrain-dominated microstructure, which is not as refined as the20M or 0L samples. The inverse pole figure maps fromEBSD illustrate a subgrain structure which is found to clo-sely resemble the TEM images for each of the conditions inFig. 5. The black lines in the EBSD micrographs demarcatehigh-angle grain boundaries characterized by misorienta-tions >15�. In addition, the TEM images illustrate varyingdislocation contents across the various conditions. Asexpected, the highly strained 0L case shows a structure thatis remarkably free of dislocations in the interiors of therefined subgrains. The 20M case shows a greater dispersionof dislocation tangles, and the 30H case shows significantdislocation content in the interiors of the subgrains. Forthe various microstructures, the average subgrain size andits standard deviation were measured from the EBSD scansand are listed in Table 2. Fig. 6 also shows the misorienta-tion distribution, illustrating a typical mixture of low-angleand high-angle boundaries. Note the declining contribution

from the high-angle boundaries with the transition fromthe highly deformed 0L case to the moderately deformed20M and to the least deformed 30H case. For all thesemicrostructures, the hardness values appear to be saturatedat �155 kgf mm�2, as shown in Table 2. While the hard-ness value may not offer the most sensitive framework fordistinguishing mechanical properties, this observed “stresssaturation” is generally consistent with what is observedin SPD of Cu at large strains > 2, including by ECAPand HPT [1].

4. Discussion

This section presents a framework for relating themicrostructure from SPD in machining to the thermome-chanics of the SPD. At this stage, for a range of conditionslisted in Table 1, one has the strain, strain-rate and temper-ature data, as well as a characterization of the resultingmicrostructure in Table 2. The focus is now on the subgrainsize resulting across the broad spectrum of conditionsexamined here. The underlying premise is that this would

Fig. 5. Bright-field TEM images for: (a) 0L: e = 8.7, _e = 60 s�1; (b) 20 M, e = 3.9, _e = 1290 s�1; (c) 30H, e = 2.3, _e = 4030 s�1.

382 S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386

offer a step towards microstructure control in SPD config-urations involving the superposition of large strain-rates,such as those encountered by machined surfaces, whichcharacterize the most prevalent engineering components.

4.1. Mapping subgrain sizes from SPD in LSM

In LSM, the accumulation of strain occurs progressivelyin a single deformation pass in a deformation zone charac-terized by the strain-rates and temperatures listed inTable 1. Here, SPD is imposed in simple-shear to variousfinal values, starting from the undeformed state ahead ofthe cutting tool’s edge, as illustrated by the DIC micro-graph in Fig. 2. As the material is being progressively sub-jected to the various final strains listed in Table 1 (�2–9), ittransitions through the different work-hardening stages inthe narrow deformation zone for each sample. For finalstrains >2 considered here, all samples can be reasonablyassumed to have transitioned through to stage IV ofwork-hardening, albeit with microstructural characteristicsunique to the individual thermomechanical conditions[22,23]. Note that much of the current understanding ofSPD microstructures is based on low-strain-rate studies,which may be confounded by the superposition of thehigher strain-rates considered in this study. And the pres-ent authors are not aware of a framework for encapsulat-

ing the microstructural characteristics resulting from thebroader array of strains, strain-rate and temperature com-binations. But, in several low-strain-rate studies reviewedin Ref. [22], by extracting work-hardening coefficients (H)in deformation experiments which measured the flow stress(s) for strains up to �9, onset of stage IV was detected as acharacteristic “kink” in the H–s curves, typically in thevicinity of �2 for Cu. While, it is not analogously possibleto resolve the transitions through the various stages ofwork hardening in the narrow deformation zone of LSM,such earlier observations further the expectation that thesubgrain-dominated microstructures observed here are aconsequence of progressive deformation to stage IV.

As a starting point, in this paper the focus is on relatingthe subgrain size to the deformation thermomechanics,and this is first examined within the context of establishedmodels of stage IV subgrain microstructures developed fromlow-strain-rate SPD studies akin to that in Ref. [23]. Thesemodels predict a progressive refinement of subgrain struc-tures with strain that is dynamically limited by recovery pro-cesses, and often culminates in the achievement of saturationof the microstructure refinement to limit the smallest achiev-able subgrain sizes from SPD. It is reasonable to expect asimilar interplay of recovery and refinement mechanismshere. An assumption is made here that the subgrain sizesin the regime of large strains, strain-rates and temperatures

Fig. 6. Crystallographic misorientation distributions in degrees extracted from EBSD scans for: (a) 0L, e = 8.7, _e = 60 s�1; (b) 20 M, e = 3.9, _e = 1290 s�1;(c) 30H, e = 2.3, _e = 4030 s�1. The data from several independent sample scans are superposed to show the repeatability of the observed misorientationdistributions.

S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386 383

can be captured as a “semi-empirical analytic continuity” ofthe traditional models that explain the behavior at the smal-ler strain-rates. To accomplish this, the way in which theexperimental observations across the broad strain/strain-rate/temperature regimes correlate with predictions of theconventional models of stage IV microstructures is firstexamined. From this, suitable parameters are extracted asfunctions of strain, strain-rate and temperature to encapsu-late the observed subgrain sizes across the spectrum of con-ditions studied here. Naturally, this leads to theaccomplishment of the other critical aim of this study: tomap the thermomechanics of deformation to the resultingsubgrain size, thus offering a microstructure prediction anddesign tool relevant to shear deformation processing acrossa broad range of strain, strain-rates and temperatures. Whilesemi-empirical in nature, such analysis can offer insights onmicrostructures from SPD configurations such as LSM,where materials are subjected to large strains, progressivelyin a single deformation pass and in a narrow deformationzone.

The current understanding is that microstructure refine-ment is not merely a monotonic function of strain, but isoften dynamically limited by two competing mechanisms.In the athermal limit (corresponding to the 0 K limitingcase) and at large strains >2, which is nominally in stageIV, the refinement of subgrain size (d) is known to follow[23]:

dd�=de ¼ �ffiffiffi3p

b1=2

u3=2IV d2

IV KIV

d5=2 ð5Þ

where b is the Burgers vector (0.256 nm for Cu) [23], uIV isstage IV average subgrain boundary misorientation, forwhich the reasonable value uIV ffi 3

is used, and the con-

stant KIV for copper was calculated to be 30.87 [23]. dIV

is the subgrain size at the initiation of stage IV, albeit cal-culated using models validated with low-strain SPD [23]that are used as a starting point in the present analysis.The calculated values are listed in Table 2. Eq. (5) has itsorigins in detailed considerations of “principle of scaling”

or similitude, which posits that microstructure refinementvia the development of a substructure, in the absence of dy-namic recovery, would essentially scale in a self-similarmanner as a function of the deformation strain [23].

But, in reality, this refinement is dynamically counter-acted by thermally induced coarsening, according to [23]:

ddþ=dt ¼ _eddþ=de

¼ mDb2Bdffiffiffiqp

exp�USD

kT

� �2 sinh

PV a

kT

mDb2Bdffiffiffiqp

exp� U SD � PV a

kT

� �� �ð6Þ

where mD is the Debye frequency, Bd is a pre-exponentialconstant associated with thermal activation of subgrain

384 S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386

growth (=2 � 104) [23], and q is the density of dislocations.USD is the activation energy for self-diffusion in Cu(=3.271 � 10�19 J atom�1) [24], k is Boltzmann’s constant,and T is the deformation temperature. V a ffi b3

u is the activa-tion volume (u is the sub-boundary misorientation), and P

is the driving pressure, given by P ¼ 4csbd (csb is the sub-

boundary energy) [23]. Note that the hyperbolic term inthe Eq. (6) is often simplified to an exponential functionfor the deformation conditions and the resulting micro-structures observed here. This expression is generally appli-cable to well-defined subgrain structures observed in stageIV deformation, such as those observed here in Fig. 4, todescribe the coarsening response [23].

The microstructure evolution is a superposition of Eqs.(5) and (6), where dd=de ¼ ddþ=deþ dd�=de. It has beenargued that, at very large strains, typically in stage IV ofwork hardening, the subgrain size eventually reaches a“saturation value”, ds such that the subgrain size (d) is nolonger sensitive to progressive levels of deformation strain(e), i.e., dd/de = 0 at d = ds.

This criterion can be written as [23]:

2 sinh4ndGb4

dskT¼

ffiffiffi3p d3

s

d2IV bKs

_emD

� �exp

USD

kTds 6 dIV ð7Þ

where nd is considered to be 50 for Cu, G is the shear mod-ulus (=47 GPa) [23] and the constant Ks ¼ u2

IV Bdj1=2KIV .The solution for the implicit Eq. (7) can be evaluatednumerically for the various deformation conditions consid-ered. These exact solutions ðdexact

s Þ are listed in Table 2. Sur-prisingly, when dm/ds vs e was plotted, a correlation inFig. 7 was noticed, illustrating a gradual convergence to-wards the “saturated grain size” with increasing levels ofstrain, across a range of deformation conditions. Recallthat the strain-invariant, saturated subgrain size is essen-tially a function of the strain-rate and temperature, andis a limiting case. The 0L condition corresponding to thelargest strain and the smallest strain-rate (Table 1) appearsto converge to the saturated subgrain size. Also note that

Fig. 7. Variation of the ratio average subgrain size to saturated grain size(dm/ds) with deformation strain (e) for various samples.

this convergence does not appear to be a simple functionof the strain, but appears to follow a more complex trendover a swathe of the map-space in Fig. 7. This is probablya result of the interactive effects of the large strains with thestrain-rate and temperature, which complicate the trajecto-ries of microstructure refinement.

Nonetheless, from the distribution of the data points inFig. 7, it is evident that the strain and the limiting grain sizefor a given strain-rate and temperature may offer the ele-ments for the parameterizations aimed at capturing theresulting subgrain sizes. From here a map-space washypothesized which is parameterized in terms of the defor-mation parameters onto which the various severelydeformed microstructures map, one-to-one. A scheme pre-sented here uses the y axis as the effective deformationstrain which essentially encapsulates the “athermal” refine-ment with progressive deformation (i.e., Eq. (5)). That is,moving along the y axis, one will be scanning the limitingcase where refinement is not modified by dynamic coarsen-ing. The x axis, which would be orthogonal to the y axis,should then encapsulate the “strain-invariant” characteris-tics of the microstructure. That is, moving along the x axis,one should be scanning along the limiting case, where themicrostructure is essentially independent of the strain(i.e., y axis). Orthogonality between the x and y axes isstrictly accomplished if and only if, along the x axis, dd/de = 0, which is essentially that corresponding to the satu-ration grain size (ds). All real samples created at finite tem-peratures and which are not at the grain size saturation canthen be expected to be interspersed on a map-space boundby these two limiting cases as their axes. Unfortunately, theexpression for ds is an implicit Eq. (7), which does notallow a viable parameterization in terms of the strain-rateand the temperature.

But, Eq. (7) does undergo a very useful simplification,wherein by taking logarithms on both sides and droppingthe d ln d term, Eq. (7) can be approximated by Eq. (8),which still provides comparably accurate approximationsof the saturation grain size dappr

s . The d ln d term is a weakfunction which remains nearly a constant across the vari-ous conditions considered here and adding a constant cor-rection term g compensates for it, while allowing for adesirable separation of variables in Eq. (8). Table 2 illus-trates the accuracy of this approximation.

dapprs ¼ C0

Gb3

kTþ g

� �1

ln C1 þ ln Z

� �ð8Þ

where g = 190.43 is the correction factor, Z ¼ _e expðUSDkT Þ,

C0 = 4ndb and C1 ¼ffiffi3p

d2IV bKsmD

, for which the values are ob-tained as C0 = 51.2 nm and ln C1 = 14.77. Multiplying C1

and dividing the dropped term by suitable unit measures,the product C1Z is rendered unitless and dimensionallyconsistent. Note that, in approximating C1, the value fordIV has been considered to be a constant �0.284 lm, givenits insensitivity to the deformation conditions in the regimestudied here (see Table 2).

Fig. 8. Map of the subgrain size dr on the RSM space as a function ofeffective strain (e) and R. Values of average subgrain sizes and its standarddeviation from experimental conditions are marked on the plot. Also, ateach experimental point, the mean + standard deviation and mean � stan-dard deviation are shown using the same color-coding scheme as thecontour map to illustrate the fidelity of Eq. (10) in capturing the meansubgrain size across the various thermomechanical conditions.

S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386 385

Given the correlation already observed in Fig. 7, thisapproximation for the “saturated grain size” reveals a via-ble parameterization for the x axis as:

R ¼ Gb3

kTþ g

� �1

lnðC1Þ þ ln Z

� �ð9Þ

which can be considered a temporally dependent “rate”

function. It is noted parenthetically that the parameteriza-tion for R is roughly analogous to the empirical correla-tions which have been observed between subgrain size (d)and ln Z in hot working of Al alloys with: d / 1/(a + b ln Z) [25,26], where a and b are empirically fitted val-ues. This coincidence in the functional form is a furthermotivation to pursue this parameterization to define therate–strain–microstructure (RSM) space for projectingthe subgrain sizes, where the x axis is the R parameter inEq. (9), and the y axis is the effective strain.

On this space, Eq. (10) captures the variation in sub-grain sizes across the swathe of thermomechanical condi-tions on the RSM space:

dr ¼ 0:25� 0:030eþ 0:058Rþ 0:0003eR ð10ÞTable 2 lists the subgrain sizes, dr, calculated with Eq. (10)for the various conditions. Note the close correspondenceof this equation to the measured values dm. via a contourmap, Fig. 8 illustrates the variation in the subgrain sizeson the parameterized RSM space across a wide range ofstrains, strain-rates and temperatures with the experimentalmeasurements overlaid on it for a better perspective.

The choice of the form of Eq. (10) is of course guided bythe fact that it can be considered as a Taylor series expansionfunction for the subgrain size (d), i.e., an analytic continuitywritten as a function of two variables R and e. That is:

dr ¼ d0 þ@dr

@eeþ @dr

@RRþ @2dr

@e@ReR ð11Þ

with @2dr@e2 and @2dr

@R2 taken to be zero by ignoring second-ordereffects in e and R.

Comparing Eq. (10) with Eq. (11) also indicates the nat-ure of the interactions of e with R in determining the trajec-tories of refinement. Owing to the role of the large strainsin refining the grain size, for a constant R one shouldexpect in Eq. (11):

ddr

dejR¼const

@dr

@eþ @2dr

@e@RR < 0 ð12Þ

Substituting the coefficients in Eq. (12), one obtainsddrde jR¼constant ¼ �0:030þ 0:0003R, which for the range of R

is consistently negative, implying the expected monotonicrefinement with increasing strain. In Eq. (9), with increas-ing temperature, R usually increases across the conditionsconsidered here, and this is found to correlate with anincreasing saturation subgrain size ds (Eq. (8)). Therefore,for a constant strain, one should expect:

ddr

dR

����e¼const

@dr

@Rþ @2dr

@e@Re > 0 ð13Þ

Substituting from Eq. (10) reveals thatddrdR je¼const ¼ 0:058þ 0:0003e, which is positive for all val-ues of strain. Complementing these effects is the role ofinteractive effects involving the effect of finite tempera-ture, strain and strain-rate, which is manifested in the fi-nal second-order cross-term on the right-hand side in Eq.(10). This term encapsulates the often recognized effectthat rate of grain refinement as a function of strain be-comes more sluggish (or a less negative derivative withrespect to strain) with increasing deformation tempera-ture i.e., ddr

de jT¼T 1> ddr

de jT¼T 2if T1 > T2. Given that increas-

ing T implies increasing R over the range of conditionsexamined here, one can conclude that d

dR ðddrde Þ > 0 or

@2dr@e@R > 0. Indeed, in Eq. (11), one finds that

@2dr@e@R ¼ 0:0003 > 0:

It is anticipated that analogous parameterizations canbe accomplished for other microstructural characteristics,including that for dislocation densities to delineate twoorthogonal axes, from a mechanism-based analysis akinto that illustrated here. Here, the “athermal” y axis is stillanticipated to be the effective strain, although the parame-terization for R for dislocation densities would probablydiffer from that for the subgrain size. Such elucidation,which is currently in progress, could help to examine thecongruence and deviation from expectations of correlatedbehaviors between dislocation densities (q) and subgrainsizes (d) of the form d

ffiffiffiqp

=constant that have been reported[27]. In addition, modeling of mechanical strength as asuperposition of contributions from the grain size, sub-grain size and dislocation density could lead to delineationof property mappings which are essentially functions of themicrostructure maps. This can also be performed for stored

386 S. Abolghasem et al. / Acta Materialia 60 (2012) 376–386

energies, ultimately to use such mappings to encapsulatethe process–structure–performance triad for a broad spec-trum of SPD conditions.

4.2. Implications of the mappings

Such analysis could enable process design tools by relat-ing microstructural characteristics to subregions of aparameterized RSM space. This could be useful for con-trolling deformed microstructures in an array of severeshear-based manufacturing processes, including the ubiqui-tous machining processes that engender a severely shear-deformed surface on manufactured components. Aspointed out earlier, the deformation zone that producesthe chip also bequeaths a severely deformed surface micro-structure to the machined surface. Controlling the micro-structure on machined surfaces to achieve tunable levelsof refinement may be useful, considering recent demonstra-tions of modification of surface phenomena as a functionof the underlying grain size. These include the observationof enhanced proliferation of osteoblasts at the surfaces ofnanostructured metals [28] and the modification of corro-sion properties in the grain refined states [29]. To achievesuch enhanced functionalities inherited from the fine-grained state, mappings akin to that in Fig. 8 could be usedto identify the thermomechanical parameters of SPD toendow the desired microstructural characteristics usingprocesses, including machining. Accomplishment of suchbroader aims, however, would require a recursive enhance-ment of the fidelity of the mappings through the accumula-tion of more empirical data. This is also necessary foranalyzing the coefficients in Eq. (10) to offer a better under-standing of the interplay of phenomena leading to micro-structure refinement under such SPD conditions.

5. Conclusions

LSM was used to study SPD across a broad range ofstrains (�2–9), strain-rates (�10–103 s�1) and temperatures(�300–450 K) within an experimental framework whichallowed for in situ elucidation of the underlying thermom-echanics. The microstructures across these deformationconditions are typically ultrafine grained, but are charac-terized by a range of subgrain sizes. However, across theentire spectrum of the thermomechanics accessed, averagesubgrain sizes were found to correlate with strain and thetheoretical saturation size limit, which is essentially a func-tion of strain-rate and temperature. Drawing from this, aRSM mapping was developed, with the y axis as the strainand the x axis as a “rate” parameter R, which is a functionof strain-rate, temperature and material constants. On thisspace, a Taylor series-like expression was identified forencapsulating the average subgrain size, which offers ananalytic continuity between the more characterized realm

of low-strain-rate SPD and the conditions examined here.The utility of these mappings for suggesting appropriatedeformation conditions for creating customizable grain-refined surfaces using metal cutting processes wasidentified.

Acknowledgement

The authors gratefully acknowledge support from theNational Science Foundation (Awards 0826010 and0927410).

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