Texture and elastic strains in hcp-iron plastically deformed up to 17.5 GPa and 600 K:

experiment and model

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2012 Modelling Simul. Mater. Sci. Eng. 20 024005

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 (17pp) doi:10.1088/0965-0393/20/2/024005

Texture and elastic strains in hcp-iron plasticallydeformed up to 17.5 GPa and 600 K: experiment andmodel

Sebastien Merkel1, Marion Gruson1, Yanbin Wang2,Norimasa Nishiyama3 and Carlos N Tome4

1 Unite Materiaux et Transformations (UMET), Universite Lille 1, CNRS, 59655 Villeneuved’Ascq, France2 Center for Advanced Radiation Sources, The University of Chicago, Chicago, IL, USA3 Geodynamics Research Center, Ehime University, Matsuyama, Japan4 MST Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

E-mail: sebastien.merkel@univ-lille1.fr

Received 1 June 2011, in final form 22 November 2011Published 24 February 2012Online at stacks.iop.org/MSMSE/20/024005

AbstractInternal elastic strains and textures are measured using monochromatic x-raydiffraction in ε-Fe plastically deformed up to 17.5 GPa and 600 K in thedeformation-DIA. We observe the development of a strong 0 0 0 1 compressiontexture along with a strongly non-linear behavior of 0 0 0 2 lattice strains.We then use an elastoplastic self-consistent polycrystal model to simulate themacroscopic flow curves, internal strain and texture development within thesample. Input parameters are single-crystal elastic moduli, critical resolvedshear stresses, and hardening behavior of the slip and twinning mechanisms.The model is found to reproduce the experiment data with basal slip as theeasiest and most active deformation mechanism. Other active mechanisms aretensile twinning, prismatic and pyramidal 〈c +a〉 slip. Tensile twinning is mostactive at lower temperatures (e.g. 400 K) and a higher strain rate. In most cases,the twinning activity occurs early in the deformation and later saturates. It isalso responsible for the non-linear behavior of 0 0 0 2 lattice strains. Later inthe deformation, the plastic activity of ε-Fe is controlled by basal, prismaticand pyramidal 〈c + a〉 slip.

(Some figures may appear in colour only in the online journal)

1. Introduction

Hexagonal closed packed ε-iron is stable at pressures above 15 GPa and it is believed to bethe main constituent of the Earth’s inner core [1]. Fe in the core may also contain 4 to 5 wt%

0965-0393/12/024005+17$33.00 © 2012 IOP Publishing Ltd Printed in the UK & the USA 1

Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

of Ni and 1 to 2 wt% of Si [2] but those alloys remain hcp [3], although other structures suchas bcc [4] or f cc [5] have also been proposed. The inner core shows significant seismicanisotropy with a complex substructure [6–8] and this anisotropy is directly related to theelastic properties and lattice preferred orientations in iron [9–12]. As such, understandingthe plastic properties of ε-iron is fundamental for constraining observations in this portion ofthe Earth.

Moreover, the plastic properties of metals with the hcp structure are of great interestbecause of their numerous technological applications. To that aim, the plastic deformationof hcp metals is often investigated using a combination of in situ x-ray or neutron diffractionmeasurements of strain and texture and numerical polycrystal models [13–18] and those modelshave been extended to be used under extreme conditions of pressures on Co [19]. Recently,pressure was also shown to be a useful tool for investigating the significance of twinning in hcp

metals as materials such as Zn and Cd show strong variations of c/a ratios with pressure [20].Understanding and modeling the plastic properties of ε-iron under pressure is therefore relevantfor the study of the plastic behavior of hcp metals in general.

In this paper, we present a new analysis and interpretation of the experimental results ofNishiyama et al [21]. In those experiments, ε-Fe was plastically deformed and studied in situusing a combination of the deformation-DIA (D-DIA) and monochromatic synchrotron x-rays[22] in which stress–strain curves were reported up to pressures of 19 GPa and temperaturesof 600 K. However, the interpretation of the experimental data is not straightforward andwas limited by the lack of a proper micro-mechanical model. Here, we use the Los Alamoselastoplastic self-consistent (EPSC) model, version 4, in order to understand and model theevolution of elastic strains and textures in the sample. The results are used to obtain importantinformation about the plastic behavior of ε-Fe under those conditions.

2. Experiment and data processing

2.1. Experimental details

Details of the experiment have been published before [21]. Here, we will only present a shortoverview of their most relevant features. The deformation experiment was performed at theGSECARS bending magnet beamline 13-BM-D at the Advanced Photon Source (Argonne,IL, USA) using a D-DIA with monochromatic x-ray diffraction of wavelength 0.191 Å and aradiographic imaging system.

The D-DIA is a multi-anvil apparatus consisting of six anvils squeezing a cubic pressuremedium. The top and bottom anvils are mounted onto symmetrical upper and lower guideblocks, while the four lateral anvils are mounted on the side faces of four wedge-shaped thrustblocks. Moving the guide blocks forward (toward each other) promotes the forward motionof the lateral anvils, hence symmetrical compression of all six faces of the cube at the centerof the apparatus. Once the desired pressure is reached, the deformation of the cubic mediumis promoted by moving top and bottom anvils forward, while constantly maintaining the oilpressure in the main ram of the apparatus. This allows the lateral anvils to slowly retract incompensation of the forward motion of the top and bottom anvils, hence maintaining a fairlyconstant medium volume and pressure.

During the experiment, stress and pressure are monitored using in situ x-ray diffractionwhile the resulting specimen strain and strain rates are quantified by time-resolvedx-ray radiography. Other details regarding strain paths, deformation geometry, sampleassembly, effects of friction, or x-ray diffraction in D-DIA experiments are describedin [22–26].

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

0 100 200 300 400 500 600Time (min.)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Axi

al s

trai

n ε 33

Cycle 2Cycle 3Cycle 4

Figure 1. Axial strain versus time in cycle. Black open circles, red open squares and green filledcircles are data for cycles 2 (600 K, 17.0 GPa, 0.5×10−5 s−1), 3 (400 K, 17.0 GPa, 1.1×10−5 s−1)and 4 (400 K, 17.5 GPa, 0.3 × 10−5 s−1), respectively. For each, solid lines with symbols areexperimental data and thin dashed lines are linear fits to extract strain rates at the beginning andthe end of the experiment.

The data were collected in a single run. Starting sample was a pure α-Fe rod, 0.5 mm indiameter and 0.5 mm in length. It was pressurized to ∼15 GPa and heated to 700 K. The α- to ε-Fe phase transition was monitored using x-ray diffraction until, after several minutes, α-Fe hadcompletely disappeared. After the transition, the pressure calculated using the equation of stateof ε-Fe [27] was 14.5 GPa at 700 K. After this synthesis, five shortening cycles were performedby advancing and retracting the differential ram pistons at various speeds. Data collected duringthe retraction of the rams are not considered in the analysis. Indeed, complicated processes,such as detwinning, take place during unloading and those cannot be accounted for in thepresent version of the numerical model. Moreover, during the experiment, unloading phaseswere performed at fairly high strain rates with limited data collection. The first and secondshortening cycles were carried out at 600 K, the third and fourth at 400 K and the fifth at roomtemperature. However, data were lost for the first and fifth cycles and, in this paper, we willonly analyze the results of cycles 2, 3 and 4. Note that, following each cycle, the distributionof grain orientations changes, and so does the hardening. The texture is measured after eachcycle, but no information about the hardening is gathered, although it is likely that at the highertemperature recovery takes place.

2.2. Strain rates

From the x-ray radiographs, sample strain ε can be deduced from sample length l usingε = ln(l0/l) (here in compression ε > 0), where l0 is the initial length of the specimen atgiven conditions. For each cycle, the initial length l0 is arbitrarily chosen when the apparentdifferential stress (as seen from x-ray diffraction) crosses 0. Strain rates ε are then deducedfrom ε versus time plots (figure 1).

Strain rates are not constant during the experiment. Most D-DIA experiments are used toextract stress levels under steady state deformation. Therefore, in the literature, most reportedD-DIA strain rates are fitted at the end of the deformation. In this study, however, we are

3

Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

Table 1. Temperatures, pressures, starting and final strain rates for cycles 2, 3 and 4. Pressuresare average pressures throughout entire deformation segments. Numbers in parentheses areuncertainties on the last digit.

Starting Final AverageTemperature Pressure strain rate strain rate strain rate

Cycle K (GPa) 10−5 s−1 10−5 s−1 10−5 s−1

2 600 17.0 0.2 0.8 0.5(3)3 400 17.0 0.6 1.5 1.1(5)4 400 17.5 0.2 0.4 0.3(1)

interested in the whole deformation cycle, including the transient domain. Therefore, wereport strain rates fitted at the start and at the end of each cycle (table 1). The differencesbetween both are used to estimate an uncertainty.

2.3. State of stress

Because of the geometry of the experiment, the stress applied to the sample is mostly axial with

σ =σ11 0 0

0 σ11 00 0 σ33

=

P 0 0

0 P 00 0 P

+

−t/3 0 0

0 −t/3 00 0 2t/3

, (1)

where P is the hydrostatic pressure and t = σ33−σ11 is the differential stress. By convention, athigh pressure, P and t are positive for compressive stresses. Therefore, during each shorteningcycle of the sample, t increases from 0 to a positive value. Unlike more conventionaldeformation experiments, pressure and stress are not measured directly and have to be extractedfrom the diffraction data.

2.4. Diffraction data processing

Representative diffraction patterns for cycle 3 are shown in figure 2. On those images, variationsof d-spacings and intensities with orientation are related to elastic strains and texture withinthe sample, respectively. Each of the 180 two-dimensional diffraction images was convertedinto one-dimensional diffraction data by binning the data in 5◦ intervals of the azimuthal angleδ. Each 1D pattern is then fitted individually using pseudovoigt peak profiles and a locallylinear background in order to extract the peak position and intensity for each reflection usingthe Multifit software [28].

In most high pressure x-ray diffraction experiments performed on plastically deformingsamples, it is observed that, to the first order, the fitted d-spacings are nearly linear whenplotted versus (1 − 3 cos2 ψ), where ψ is the angle between the diffracting plane normal andthe compression direction. Therefore, our dataset was reduced using the following definitionof lattice strain with respect to the pressure state

ε33(hkil, ψ) = dm(hkil, ψ) − dP (hkil)

dP (hkil)= Q(hkil)

(1 − 3 cos2 ψ

), (2)

where dm(hkil, ψ) are the measured d-spacing for the hkil line at angle ψ . dP (hkil) is thed-spacing for hkil line at ψ = 54.7◦ and Q(hkil) is the lattice strain parameter. Both areadjusted to the experimental data for each diffraction line at each deformation step using thePolydefix software [28]. Also note that, experimentally, d-spacings measured at ψ = 54.7◦

are found to correspond to those measured under hydrostatic equivalent pressure [19].

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

Figure 2. x-ray diffraction patterns for cycle 3 (17.0 GPa, 400 K, 1.1 × 10−5 s−1) (a) at ε33 = 0and (b) after 200 min at ε33 = 0.115. The direction of sample shortening is at δ = 0◦, δ = 180◦and δ = 360◦. Variation of d-spacings and intensities with azimuth are related to elastic strainsand texture within the sample, respectively. Miller indices of the ε-Fe diffraction lines are labeledin the figure.

0.00 0.02 0.04 0.06 0.08 0.10 0.12Axial strain ε33

0.0

0.5

1.0

1.5

2.0

∆P (

GP

a)

Cycle 2Cycle 3Cycle 4

Figure 3. Measured and simulated pressure increase versus strain. Black open circles, red opensquares and green filled circles are data for cycles 2 (600 K, 17.0 GPa, 0.5 × 10−5 s−1), 3 (400 K,17.0 GPa, 1.1 × 10−5 s−1) and 4 (400 K, 17.5 GPa, 0.3 × 10−5 s−1), respectively. Thin solid linesand thick solid lines are experimental measurements and results of EPSC simulations, respectively.

For each deformation step, values of dP (hkil) are used to estimate the average latticeparameters a and c of the hexagonal crystal and the hydrostatic pressure P using a knownequation of state [27] (figure 3). Extracted lattice strain parameters Q for cycles 2, 3 and 4 arepresented in figure 4.

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

0.00 0.05 0.10Axial strain ε33

0.0

1.0

2.0

3.0Q

(hkl

)×10

3

0.00 0.05 0.10Axial strain ε33

0.0

1.0

2.0

3.0

Q(h

kl)×

103

0.00 0.05 0.10Axial strain ε33

0.0

1.0

2.0

3.0

Q(h

kl)×

103

0.00 0.05 0.10Axial strain ε33

0.0

1.0

2.0

3.0

Q(h

kl)×

103

0.00 0.05 0.10Axial strain ε33

0.0

1.0

2.0

3.0

Q(h

kl)×

103

Cycle 2Cycle 3Cycle 4

1010 0002 1011

11201012

Figure 4. Measured and simulated lattice strain parameters versus strain for the 1 0 1 0, 0 0 0 2,1 0 1 1, 1 0 1 2 and 1 1 2 0 diffraction lines of ε-Fe. Black open circles, red open squares andgreen filled circles are data for cycles 2 (600 K, 17.0 GPa, 0.5 × 10−5 s−1), 3 (400 K, 17.0 GPa,1.1 × 10−5 s−1) and 4 (400 K, 17.5 GPa, 0.3 × 10−5 s−1), respectively. Thin solid lines and thicksolid lines are experimental measurements and results of EPSC simulations, respectively.

The variation of intensity along the diffraction rings is a measure of crystallographicpreferred orientation (figure 2). For each cycle, three diffraction images were also processedusing the Rietveld refinement method, as implement in the MAUD software [29], and followingthe procedure described in [30] in order to refine the orientation distributions functions (ODFs).Other parameters such as lattice parameters, grain sizes, and lattice strain parameters were alsorefined but are not used here. Sample texture was computed using the EWIMV algorithm [31] toproduce the ODFs assuming axial symmetry around the compression direction. The ODF fromMAUD was then exported and smoothed with a 7.5◦ filter to minimize artifacts from the ODFcell structures. Inverse pole figures (IPFs), which describe the orientation of the compressionaxis relative to the crystal coordinates, were then calculated and used for representation in thispaper (figure 5).

3. Experimental results

3.1. Pressure

Pressure is not constant during each deformation cycle (figure 3). This is due to difficulties incontrolling the relative movement of the horizontal and vertical rams in the D-DIA. At most,pressure is found to increase by 2 GPa over the entire deformation range, which is not criticalregarding the range of pressure considered here.

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

Figure 5. Experimental IPFs of the compression direction at ε33 = 0, ε33 = 0.05 and ε33 = 0.11illustrating the development of LPO in ε-Fe for cycles 2 (600 K, 17.0 GPa, 0.5×10−5 s−1), 3 (400 K,17.0 GPa, 1.1 × 10−5 s−1) and 4 (400 K, 17.5 GPa, 0.3 × 10−5 s−1). Equal area projection, linearscale in m.r.d. White color near 0 0 0 1 indicates regions for which intensities are higher than themaximum in the color scale.

On the other hand, the variation of pressure with strain implies a change of sample volumeduring the experiment. It will be accounted for during the modeling process by adjusting thelateral strains imposed to the sample.

3.2. Elastic lattice strains

For all cycles, lattice strains measured on 1 0 1 0 and 1 1 2 0 are of maximum amplitude,followed by, in decreasing amplitude order, lattice strains on 1 0 1 1 and 1 0 1 2 (figure 4).For those four reflections, lattice strains continuously increase and later saturate. Saturationoccurs at ∼3% strains in cycles 3 and 4, both performed at 400 K, and ∼6% strains for cycle 2,performed at 600 K.

Lattice strains measured on 0 0 0 2 show a more complex behavior. In cycle 2, lattice strainbehavior can be separated into three regions: ε33 < 1% and 1% < ε33 < 7%, where latticestrains linearly increase with applied strains, and ε33 > 7%, where a saturation is observed.For cycles 3 and 4, we observe three regions in which 0 0 0 2 lattice strains linearly evolve withapplied strains but with distinct slopes: ε33 < 2%, 2% < ε33 < 7% and ε33 > 7%.

In a previous analysis, the measured lattice strains were converted into stresses using apurely elastic model [21]. This resulted in different differential stresses measured for differentlattice planes and could not be fully interpreted. However, it is known that elastic models [32] donot account for stress variations induced by plastic relaxation and interpretation of experimentalresults based on those models can be biased [19, 33].

On the other hand, this strong non-linear behavior of strains for 0 0 0 2 peaks has beenobserved on other hcp metals, such as Mg, in ambient pressure experiments and have beeninterpreted as an effect of twinning [13, 15, 16]. This will be discussed later in the paper.

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

Table 2. Experimental and simulated integrated volume fraction of grains with 0 0 0 2 axes within11◦ of the compression direction.

Experimental Simulation

Strain ε33 Vol. fraction Strain ε33 Vol. fraction

Cycle 2 0% 7.9% 0% 7.1%5% 14.7% 5% 10.9%

12% 14.2% 12% 12.0%

Cycle 3 0% 3.6% 0% 3.3%5% 8.7% 5% 6.7%

11% 13.1% 12% 9.3%

Cycle 4 0% 4.7% 0% 4.2%5% 9.2% 5% 8.1%

11% 13.4% 12% 9.3%

3.3. Lattice preferred orientations

ε-Fe is known to develop strong 0 0 0 1 textures upon compression [34, 35] although, at highertemperatures this maximum tends to shift away from 0 0 0 1 [36, 37]. The likely reason forthis shift is pyramidal slip activity in those grains where the c-axis is close to the compressiondirection. Here, we observe the development of strong 0 0 0 1 textures upon compression inall the tested conditions (figure 5).

As the deformation rams are retracted, the sample undergoes latteral compression andlengthening. In this process, we observe a strong decrease of the 0 0 0 1 maximum. This iscorrelated with an increase in intensity at 1 0 1 0. As compression proceeds again, the 0 0 0 1maximum is formed again, indicating that the mechanism generating the 0 0 0 1 texture canbe reverted. In previous studies on Mg, this exchange of intensity between 0 0 0 1 and 1 0 1 0was correlated with twinning and detwinning of {1 0 1 2}〈1 0 1 1〉 tensile twins [38]. In hcp,tensile twins reorient the c-axes by nearly 90◦ from their original position, aligning themwith the compression direction. When the stress is reversed those twinned domains have thec-axis aligned with the tensile direction, and the same twinning variant is activated, leading todetwinning. Considering the c/a ratio of Fe in the range pressure of considered here, this willalso be true for ε-Fe [27, 39].

Texture strength can be analyzed in terms of intensity values at 0 0 0 1 in the IPF. However,this criterion is prone to statistical errors as the volume fraction of orientation space contributingto this intensity is very small. As such, the evolution of such texture is best followed by trackingthe integrated volume fraction of orientations with the c-axes within a given interval of thecompression direction. As shown in table 2, we find that in the experiment, the volume fractionof the c-axes within 11◦ of the compression direction increases quickly in cycle 2 while theincrease is more progressive in cycles 3 and 4, highlighting the effect of temperature on textureevolution.

4. Numerical model

4.1. EPSC model

The EPSC model we use here is an extension of the model of Turner and Tome [40] thatallows for grain rotation induced by slip and stress relaxation due to twinning [16, 41].The model represents the aggregate by a discrete number of orientations with associated

8

Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

Table 3. Elastic moduli used to model cycles 2, 3 and 4 (in GPa).

C11 C12 C13 C33 C44

Cycle 2 355 116 59 518 143Cycle 3 370 118 70 527 145Cycle 4 375 119 78 536 147

volume fractions chosen such as to reproduce the initial texture. Each grain is treated asan ellipsoidal elastoplastic inclusion embedded within a homogeneous elastoplastic effectivemedium with anisotropic properties characteristic of the textured aggregate. The externalboundary conditions (stress and strain) are fulfilled on average by the elastic and plasticdeformations at the grain level. The self-consistent approach explicitly captures the fact thatsoft-oriented grains tend to yield at lower stresses and transfer load to plastically hard-orientedgrains, which remain elastic up to rather large stress. Please note that our model is elastoplasticand therefore does not account for rate sensitivity directly.

Details of the extended EPSC model have been published before [16, 19, 41]. In summary,the model was extended to account for: (i) discrete crystal reorientation associated withtwinning, the formation of new child orientations, and stress relaxation taking place in theparent grain due to twinning [16]. The children are allowed to grow as further twin activitytakes place and, in the process, volume fraction is transferred from the parent to the child.After it has been created, the twin is treated as a new independent grain and the only couplingbetween parent and child is through the volume transfer; (ii) variation of the elastic constantswith hydrostatic pressure [19]; (iii) crystal rotations induced by slip, large strain deformation,and the associated texture evolution [41]. While our previous simulations for cobalt didnot benefit from the twinning and slip reorientation improvements [19], for the simulationspresented here we use all of the improvements listed above.

4.2. Parameters of the model

In this modeling, we assume that the sample is submitted to an axisymmetric axial compressionup to ε33 = 0.12. The lateral strains are not controlled during the experiment and are notaccessible to measurements. On the other hand, pressure is known. As a consequence, weenforce a strain path that gives hydrostatic stresses consistent with the experiment adjustingε11 = ε22 accordingly (figure 3). Since, in all 3 cycles, pressure versus strain curves show aprogressive change of slope, compression cycles were modeled with two subsequent continuousdeformation steps, between ε33 = 0 and ε33 = 0.05, and between ε33 = 0.05 and ε33 = 0.12,adjusting paths in ε11 and ε22 deformation to roughly match experimentally observed pressurechanges.

The model uses values of elastic moduli. Here, we used elastic moduli for ε-Fe calculatedusing first-principles methods [42], extrapolated to the experimentally measured unit cellvolumes (table 3). Maximum pressure change within a cycle is on the order of 2 GPa.According to the extrapolation of the first-principles results, this implies a change of 15%for C13 and less than 5% for other elastic moduli. However, this change in elastic moduli isnot very well constrained and would have little effect on the results presented here. Therefore,within each cycle, the effect of pressure on the elastic moduli was neglected.

For each cycle, a distribution of 5400 orientations corresponding to the experimentaltexture at ε33 = 0 is used as starting texture for the simulation. We chose this procedurebecause the simulation of the cycling process, involving detwinning, dislocation recovery anda temperature dependence of the hardening parameters, would be extremely complicated. The

9

Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

Table 4. CRSS and hardening parameters optimized for cycles 2, 3 and 4.

Deformationsystem τ0 (GPa) τ1 (GPa) θ0 (GPa) θ1 (GPa)

Cycle 2 Basal 0.08 0 0 0.05Prismatic 0.3 0.4 4 0.01Pyramidal 0.2 1.0 20 0.01Tensile twin 0.2 1 3 0.01

Cycle 3 Basal 0.05 0 0 0.02Prismatic 1.0 0 0 0.01Pyramidal 0.9 1.2 0.5 40.0Tensile twin 0.6 0 0 1.0

Cycle 4 Basal 0.01 0 0 0.02Prismatic 1.0 0 0 0.01Pyramidal 0.9 1.0 1.0 10.0Tensile twin 0.2 0 0 2.5

simplified procedure has the advantage that it guarantees starting from the right initial texturefor the cycle considered. Any accumulated hardening that took place in previous cycles, orchanges in hardening parameters with temperature, is accounted for empirically by selectingdifferent hardening parameters to simulate each cycle (table 4).

4.3. Choice of deformation mechanisms

The model uses combinations of deformation mechanisms typically found in hexagonal metals:slip of 1

3 〈1 1 2 0〉, or 〈a〉 dislocations, on basal (0 0 0 1) and prismatic {1 0 1 0} planes; slip of13 〈1 1 2 3〉, or 〈c + a〉 dislocations, on pyramidal {1 0 1 1} planes; and, finally, tensile twinningon {1 0 1 2} planes (table 4).

This choice of deformation mechanisms was guided by the fact basal and prismatic slipsare the most commonly observed slip systems in hexagonal metals. They are also inferredfrom modeling of textures produced in ε-iron in compression at ambient temperature [34, 35].Mechanisms such as pyramidal 〈c+a〉 and tensile twinning allow for reorientation of the c-axesand are necessary to properly model the texture evolution observed in the experiments. Theiractivation has also been successfully correlated with the strongly non-linear behavior of 0 0 0 2strains in Mg [15, 16].

For each slip and twin mode we describe the hardening of CRSS by means of an empiricalVoce hardening rule

τ = τ0 + (τ1 + θ1�)

{1 − exp

(−θ0�

τ1

)}, (3)

where τ is the instantaneous CRSS of the mechanism, τ0 and τ0 + τ1 are the initial and finalback-extrapolated CRSS, respectively, θ0 and θ1 are the initial and asymptotic hardening rates,and � is the accumulated plastic shear strain in the grain. For some systems, we reduce thenumber of adjustable parameters by assuming that τ1 = 0 and, in this case, the hardening lawbecomes linear according to

τ = τ0 + θ1�, (4)

and only two adjustable parameters remain.

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

4.4. Output of the model

Output of the simulation includes the relative activity of the various deformation mechanisms,the average stress in the polycrystal, stress and strain within each grain of the sample, predictedlattice strains, and the orientation of each grain.

As in a previous study [19], the simulated elastic lattice strains are compared withexperimental data by identifying the model grains whose crystallographic planes are orientedsuch as to contribute to the experimental signal. The lattice strain (peak shift) is calculatedas a weighted average over all grains that contribute to the peak. Specifically, we considered1 0 1 0, 0 0 0 2, 1 0 1 1, 1 0 1 2 and 1 1 2 0 diffraction lines at ψ = 0, 15, 30, 45, 60, 75 and 90◦.The region of orientation space which contributes to the signal was assumed to be within aninterval of ±7.5◦ with respect to the diffraction vector.

EPSC keeps track of each orientation and their respective volume fractions, including newchildren orientations created during twinning. From those orientation statistics, pole and IPFscan be calculated and compared with the experimental textures.

5. Results and discussion

5.1. Optimized models

The only ‘fitting parameters’ of the model are the parameters of the Voce hardening rule(equations (3) and (4)). For each cycle, parameters are first adjusted manually assuming alinear model for each plastic system. If the linear model is not sufficient, supplementaryparameters are included and a full Voce law is used. In all cases, parameters are manuallyoptimized for matching the experimental and simulated lattice strain parameters Q. Simulatedtextures were only checked at the end of the optimization and were not used in the optimization.

In the optimization, lattice strains for 1 0 1 0 and 1 1 2 0 are most sensitive to the CRSS ofprismatic slip, lattice strains for lines such as 1 0 1 1 and 1 0 1 2 are most sensitive to the CRSSof basal slip, and lattice strains for 0 0 0 2 are controlled by the CRSS of pyramidal 〈c + a〉 slipand tensile twinning

Table 4 and figure 6 present the optimized parameters and the corresponding functionaldependence, respectively, for each cycle. Relative activities for each mode are plotted infigure 7. For all cycles, basal slip is by far the easiest and most active slip system witha CRSS below 0.1 GPa, even after hardening. Tensile twinning is found to be the secondeasiest deformation mode early in the deformation. As deformation progresses, twinningbecomes harder and its activity decreases. Prismatic slip is also important as, later in thedeformation, its activity represents about 40% of the total plastic activity in the material.Finally, pyramidal 〈c+a〉 slip is the hardest system. It becomes active later in the deformation,after twinning saturates, to accommodate deformation along the c-axis. It leads to the rotationof the c-axes.

5.2. Lattice strains

Overall, we find parameters which provide very good agreement between the experimentaland simulated lattice strains (figure 4); the model qualitatively predicts the measured internalstrain data. Unexpected changes in slope in the 0 0 0 2 strain curves, with a flatteningof Q versus ε33 curve at ε33 ∼ 3% are correlated with the maximum activity for tensiletwinning. Once twinning saturates, Q factors for 0 0 0 2 increase again, until steady state isreached.

11

Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

0 0.1 0.2 0.3 0.4 0.5Accumulated strain Γ

0.0

0.5

1.0

1.5

2.0

2.5

3.0C

RS

S (

GP

a)

Prism.BasalPyr.Tens. twin

0 0.1 0.2 0.3 0.4 0.5Accumulated strain Γ

0.0

0.5

1.0

1.5

2.0

2.5

3.0

CR

SS

(G

Pa)

0 0.1 0.2 0.3 0.4 0.5Accumulated strain Γ

0.0

0.5

1.0

1.5

2.0

2.5

3.0

CR

SS

(G

Pa)

Cycle 2 Cycle 3 Cycle 4

Figure 6. CRSS versus accumulated shear strain in grain optimized using EPSC for cycles 2(600 K, 17.0 GPa, 0.5 × 10−5 s−1), 3 (400 K, 17.0 GPa, 1.1 × 10−5 s−1) and 4 (400 K, 17.5 GPa,0.3 × 10−5 s−1). Open black circles, open red squares, filled green circles and filled blue squaresare results for prismatic slip, basal slip, pyramidal 〈c+a〉 slip, and tensile twinning, respectively.

0 0.05 0.1Axial strain ε33

0

20

40

60

80

100

Rel

ativ

e ac

tivity

(%

)

0 0.05 0.1Axial strain ε33

0

20

40

60

80

100

Rel

ativ

e ac

tivity

(%

)

0 0.05 0.1Axial strain ε33

0

20

40

60

80

100R

elat

ive

activ

ity (

%)

Prims.BasalPyr <c+a>Tens. Twin.

Cycle 2 Cycle 3 Cycle 4

Figure 7. Predicted relative system activities versus strain for cycles 2 (600 K, 17.0 GPa,0.5 × 10−5 s−1), 3 ((400 K, 17.0 GPa, 1.1 × 10−5 s−1) and 4 (400 K, 17.5 GPa, 0.3 × 10−5 s−1).Open black circles, open red squares, filled green circles and filled blue squares are results forprismatic slip, basal slip, pyramidal 〈c+a〉 slip, and tensile twinning, respectively.

For cycle 2, performed at 600 K, steady state is reached at ε33 ∼ 9%. For cycles 3 and4, performed at 400 K, steady state is never reached in the current experiments as, both inexperiment and simulations, lattice strains for 0 0 0 2 are still increasing with applied strain atthe end of the cycle.

5.3. Textures

Simulated textures are presented in figure 8 for all cycles at 0%, 5% and 12% deformationand can be compared with the experimental textures of figure 5. After 5% deformation, the

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

Figure 8. Simulated IPFs of the compression direction at ε33 = 0, ε33 = 0.05 and ε33 = 0.12illustrating the development of LPO in ε-Fe for cycles 2 (600 K, 17.0 GPa, 0.5 × 10−5 s−1), 3(400 K, 17.0 GPa, 1.1×10−5 s−1) and 4 (400 K, 17.5 GPa, 0.3×10−5 s−1). Equal area projection,linear scale in m.r.d. A white area near 0 0 0 1 indicates regions for which intensities are higherthan the maximum in the color scale.

density in the IPF increases around 0 0 0 1 and decreases around 1 0 1 0. This is a signature ofthe activation of tensile twinning. At 10% deformation, the trend has continued.

Predicted textures are in overall good agreement with the experimental results. Thepredicted density near 0 0 0 1 is lower than in the experiment but, as mentioned earlier, thiscriterion is prone to statistical errors. Experimental and simulated integrated volume fractionsof orientations with the c-axes within 11◦ of the compression direction are shown in table 2. Incycles 2 and 4, this volume fraction increases quickly between 0% and 5% axial strain. In cycle3, performed at higher strain rates, the increase is more continuous. This can be correlatedwith the activity of tensile twinning that completely stops at a given strain in cycles 2 and 4,unlike cycle 3 where it is active until the end of the deformation (figure 7). However, thisshould be confirmed through further study. Strain rate changes are small in our experimentand, in other materials such as Be, twinning activity increases when orders of magnitude ofstrain rate increases are involved [43].

In calculations performed for Mg [16], the activation of pyramidal 〈c + a〉 slip induced adecrease in intensity at 0 0 0 1, with a reorientation of basal planes away from the load axis.This is not observed in both our simulation and experiment, probably because the activity ofpyramidal 〈c + a〉 slip remains fairly low at all stages.

5.4. Differential stress

Figure 9 shows the predicted evolution of the average differential stress versus strain for allcycles. This information is critical at high pressures: unlike ambient condition experiments,where stress markers can be used, the differential stress is not known in high pressureapparatuses. In a previous analysis of the current data [21], a plain elastic model was used toevaluate stress from experimental measurements of lattice strains. This was not satisfactoryas deduced stresses were inconsistent.

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

0 0.05 0.1Axial strain ε33

0.0

0.5

1.0

1.5

2.0

2.5

t (G

Pa)

Cycle 2Cycle 3Cycle 4

Figure 9. Predicted evolution of differential stress versus strain. Black open circles, red opensquares, and green filled circles are results for cycles 2 (600 K, 17.0 GPa, 0.5×10−5 s−1), 3 (400 K,17.0 GPa, 1.1 × 10−5 s−1) and 4 (400 K, 17.5 GPa, 0.3 × 10−5 s−1), respectively.

In the present calculation, we find that for cycles 3 and 4 performed at 400 K, stressesdo not saturate and linearly increase with applied strain above ε33 ∼ 0.03. At ε33 = 0.12,the stresses calculated for cycles 3 and 4 are 2.0 GPa and 1.9 GPa, respectively. For cycle 2,performed at 600 K, steady state is reached at the end of the simulation. In this case, we finda flow stress of 1.6 GPa.

Therefore, we find that under the present conditions, stress in ε-Fe is sensitive totemperature, but fairly insensitive to strain rate. However, please note that the range of strainrates investigated here is under an order of magnitude and therefore the effects of strain ratecould probably be observed over a wide range of measurements.

5.5. Effect of temperature and strain rate on ε-Fe plastic behavior

Overall, we find that for all conditions studied here, basal slip is by far the easiest and mostactive deformation mechanism. Other mechanisms involved are tensile twinning, prismaticand pyramidal 〈c + a〉 slip. In order to evaluate the accumulated activity of tensile twinning,figure 10 presents the predicted twin volume fraction induced by deformation for each cycle.For cycles 2 and 4, performed at lower strain rates, the twin volume fraction saturates andreaches values of 9% and 12%, respectively. For cycle 3, performed at higher strain rates, thetwin volume fraction does not saturate and reaches 15% at ε33 = 0.12. As usually assumed,we find that twinning is a low temperature/high strain rate deformation mechanism. At 600 K,twinning activity decreases but it is still a significant mechanism for the plastic behaviorof ε-Fe.

In all cases, once the transient state is over, deformation of ε-Fe is dominated by theactivation of basal, prismatic and pyramidal slip (figure 7). The ratio between the activityof these three systems does not show any significant strain rate dependence. It is, however,temperature dependent as the contribution of pyramidal 〈c+a〉 slip is more significant at 600 K.

The differences between observations in cycles 3 and 4 are surprising. Indeed, strain ratedifferences are of less than an order of magnitude and effects of the complex deformationhistory cannot be rule out. This will have to be investigated in the future.

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

0 0.05 0.1Axial strain ε33

0.0

5.0

10.0

15.0

Tw

in fr

actio

n (%

)

Cycle 2Cycle 3Cycle 4

Figure 10. Predicted twin volume fraction induced by deformation in cycle. Black open circles,red open squares and green filled circles are results for cycles 2 (600 K, 17.0 GPa, 0.5 × 10−5 s−1),3 (400 K, 17.0 GPa, 1.1 × 10−5 s−1) and 4 (400 K, 17.5 GPa, 0.3 × 10−5 s−1), respectively.

6. Conclusion

In this paper, we showed how self-consistent elastoplastic models can be used successfullyto interpret x-ray diffraction experiments performed on ε-Fe plastically deforming at highpressure. The model allows for the creation and growth of new twin grains as well as for therotation of grains induced by slip. It can be used to predict the evolution of internal strains,average stress, texture and twin volume fraction with strain.

In the past, strains measured during deformation experiments on ε-Fe have been analyzedin terms of elastic states, while textures were studied independently in terms of slip systems[21, 34, 35, 44]. Here, we demonstrate how a comparison between the experiment and EPSCmodel allows for a precise characterization of plastic activity in ε-Fe. In particular, the presentmethod is particularly powerful to distinguish effects of twinning and slip, which can be difficultbased on texture only [20]. Also, the obtained CRSS values are absolute, unlike results basedon texture comparison for which CRSS values are relative. Overall, we find that, under theconditions of the experiment, basal slip is by far the easiest and most active deformationmechanism. Other active mechanisms are tensile twinning, prismatic and pyramidal 〈c + a〉slip.

Tensile twinning is most active at lower temperatures (e.g. 400 K) and higher strain rate.In most cases, the twinning activity occurs early in the deformation (below ε33 ∼ 0.05) andlater saturates. Later in the deformation, the plastic activity of ε-Fe is controlled by basal,prismatic and pyramidal 〈c + a〉 slip.

Textures in ε-Fe deformed in axial compression at pressures up to 17.5 GPa andtemperatures up to 600 K show a distinct maximum exactly at 0 0 0 1, implying that basal planesare strongly aligned perpendicular to the compression direction. At higher temperatures, whenthe twinning activity decreases, we expect a shift of this maximum away from 0 0 0 1 because ofthe increased activity of pyramidal 〈c+a〉 slip. This has been observed experimentally [36, 37]but will have to be investigated in detail in the future.

The parameters deduced from this study can now be used for any arbitrary paths and startingtextures. However, one should be careful with extrapolations in pressure, temperature and strainrate. For applications to the Earth inner core, where strain rates are low and temperatures are

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Modelling Simul. Mater. Sci. Eng. 20 (2012) 024005 S Merkel et al

high, it is probable that the plastic activity of ε-Fe is mostly dominated by basal, prismatic andpyramidal 〈c + a〉 slip [12].

Future experiments addressing the plastic properties of ε-Fe for geophysical applicationsshould ensure that temperature and strain rates are appropriate as to limit the contribution oftensile twinning. As highlighted in this paper, and recent high pressure studies on other hcpmetals [20], twinning has profound implications on the plastic behavior of hcp metals. Forapplications to planetary cores, this mechanism should be avoided.

Acknowledgments

The D-DIA experiments were performed at the GeoSoilEnviroCARS (GSECARS) atthe Advanced Photon Source (APS). GSECARS is supported by the National ScienceFoundation Earth Sciences (EAR-0622171) and Department of Energy-Geosciences (DE-FG02-94ER14466). APS is supported by DOE-BES, under Contract No DE-AC02-06CH11357. YW thanks NSF supports EAR-0652574 and 0968456. SM was supportedby ANR program DiUP, N. ANR-07-JCJC-0136.

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