Investigating the limits of polycrystal plasticity modeling

International Journal of Plasticity 21 (2005) 221–249 www.elsevier.com/locate/ijplas

Investigating the limits of polycrystal plasticity modeling Thomas E. Buchheit *, Gerald W. Wellman, Corbett C. Battaile Department 1851 MS 0889, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185, USA Received in final revised form 2 October 2003 Available online 20 February 2004

Abstract A material model which describes the rate-dependent crystallographic slip of FCC metals has been implemented into a quasistatic, large deformation, nonlinear finite element code developed at Sandia National Laboratories. The resultant microstructure based elastic–plastic deformation model has successfully performed simulations of realistic looking 3-D polycrystalline microstructures generated using a Potts-model approach. These simulations have been as large as 50,000 elements composed of 200 randomly oriented grains. This type of model tracks grain orientation and predicts the evolution of sub-grains on an element by element basis during deformation of a polycrystal. Simulations using this model generate a large body of informative results, but they have shortcomings. This paper attempts to examine detailed results provided by large scale highly resolved polycrystal plasticity modeling through a series of analyses. The analyses are designed to isolate issues such as rate of texture evolution, the effect of mesh refinement and comparison with experimental data. Specific model limitations can be identified with lack of a characteristic length scale and oversimplified grain boundaries within the modeling framework. Published by Elsevier Ltd. Keywords: A. microstructures; B. crystal plasticity; B. polycrystalline material; C. finite elements

*

Corresponding author. Tel.: +1-505-845-0298; fax: +1-505-844-4816. E-mail address: [email protected] (T.E. Buchheit).

0749-6419/$ - see front matter. Published by Elsevier Ltd. doi:10.1016/j.ijplas.2003.10.009

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1. Introduction A standard crystal plasticity formulation treats material deformation via the wellknown slip mechanism in a crystalline material. In an FCC metal, a well-established rate-dependent phenomenology exists which partitions plastic deformation amongst the crystallographic h1 1 1i{1 1 0} slip systems defined within a single crystal (Hill and Rice, 1972; Asaro and Rice, 1977; Hutchinson, 1976; Canova et al., 1988; Kocks et al., 1998). Literature examples abound with FCC crystal plasticity formulations incorporated into finite element code, allowing researchers to mesh and simulate the deformation of grain assemblages that more closely approximate the microstructure of a polycrystalline material (e.g., Mika and Dawson, 1998, 1999; Barbe et al., 2001a,b; Sarma et al., 2002; Kowalczyk and Gambin, 2003). In some cases, the crystal plasticity framework has been extended to beyond characterizing plastic deformation of FCC metals to deformation of crystalline materials with more complex microstructures or different crystallographic shearing mechanisms (Myagchilov and Dawson, 1999; Han et al., 2003). More recently, formulations which capture the effect of strain gradients (e.g., Huang et al., 2003) have been incorporated into the crystal plasticity framework in attempts to capture grain boundary and other non-local influences on the deformation response of single crystals and polycrystals (Shu and Fleck, 1999; Arsenlis and Parks, 2002; Acharya and Beaudoin, 2000; Beaudoin et al., 2000; Bammann, 2001). Although pursuing these avenues is likely to broaden the capability of microstructure-based crystal plasticity models, a standard FCC formulation incorporated into a finite element code with sub-grain resolution offers insight into both the polycrystalline model and real material behavior which has not yet been fully realized. Without question, the basic model framework does not capture important underlying mesoscopic aspects of microstructure evolution during plastic deformation of a polycrystalline material, but in certain cases, simulations using the model phenomenology can make very accurate predictions (Mathur and Dawson, 1990; Beaudoin et al., 1994, 1996; Bachu and Kalidindi, 1998). Thus, an underlying goal in this study is to provide results that establish connections and isolate deviations caused by model oversights between predictions and experimental observations obtained by state-of-the-art techniques in polycrystalline materials. Polycrystalline finite element simulations in this study contain about 200 grains composed, on average, of 100Õs of elements per grain, designed to look like a realistic metal microstructures. Often, in the context of this paper, they are referred to as highly resolved polycrystal plasticity simulations. These kind of simulations have the great advantage of allowing the finite element method enforce compatibility and equilibrium between and within grains. Local intra- and intergranular response of grains within a simulated polycrystal are driven by geometric constraints imposed by neighboring grains with different orientations (Beaudoin et al., 1996). Lattice rotation and material strength within each element is tracked, thus a finite element polycrystal plasticity simulation composed of several hundred elements per grain provides access to a large amount, e.g., in papers by Mika, Dawson and co-authors (1998,1999,2002), of local simulated microstructural information. A challenge ad-

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dressed in this paper is interpreting those results within the context of experimental microstructure deformation evolution observations. A simulation-experiment comparison section of this paper utilizes Electron Backscatter Pattern Imaging (EBSD), a commonly employed experimental technique for characterizing deformation substructure that maps microstructure by crystal lattice orientation (Randle, 1992; a textbook on the EBSD technique). Unlike stereologicalbased transmission electron microscopy (TEM) where substructure boundaries may be both observed and characterized, EBSD is a scanning electron microscopy based blind mapping technique but it can provide data directly comparable to simulated polycrystal plasticity results. The current authors are aware of two previous studies that directly compare simulated polycrystal plasticity data with EBSD results (Weiland and Becker, 1999; Bhattacharyya et al., 2001). The general conclusion from both studies was a polycrystal plasticity model cannot capture the influence of deformation evolved microstructural features, therefore it cannot accurately simulate the subgrain deformation evolution of polcrystalline materials. Bhattacharyya et al. (2001), do state that their polycrystal plasticity finite element model resolved to several hundred elements per grain at least captures the range of subgrain misorientations relatively well when directly compared with experimental EBSD data. Amongst other results and observations, this paper follows up these previous comparisons with experimental data by applying a concept recently put forth by Raabe et al. (2002,). These researchers outline a theory, supported by both simulated and experimental evidence, that the propensity for a grain to develop orientation gradients, or ‘‘break apart’’ during deformation in a polycrystalline material is governed primarily by a grainÕs initial orientation relative to the applied deformation direction. In this paper, grains with orientations unfavorable for breaking apart during deformation are referred to as intrinsically stable grains, grains with orientations favorable for breaking apart during deformation are referred to as intrinsically unstable grains. 2. Constitutive model The kinematic development of a single crystal plasticity model is well documented by several authors and has been recently published as a textbook chapter (Kocks et al., 1998; Chapter 8). Different researchers have implemented slightly different variations using the same constitutive framework into finite element codes. The model, as it is implemented into Sandia National Laboratories quasistatic finite element code JAS-3D (Biffle, 1987; Buchheit et al., 1997), is given in this section. It includes anisotropic elastic deformation and assumes all plastic deformation occurs via dislocation slip on {1 1 1}h1 1 0i crystallographic systems, the observed mode of low temperature plastic deformation in FCC metals. 2.1. Kinematic framework A brief discussion of the model may begin with describing distortion of a continuum using the velocity gradient which can be additively decomposed into symmetric and skew-symmetric parts:

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L ¼ X þ D;

ð1Þ

where D is the symmetric deformation rate tensor and X is the skew-symmetric spin rate tensor. Distortion of single crystals has been described as a combination of plastic flow due to crystallographic slip and lattice distortion. Lattice distortion includes elastic distortion and rigid body rotation of the crystal lattice. Thus, for single crystals the deformation rate, D, and spin rate, X, can be further decomposed into lattice and plastic parts as follows: D ¼ Dl þ Dp ;

ð2aÞ

X ¼ Xl þ Xp ;

ð2bÞ

where Dl represents the lattice deformation rate and Dp represents the plastic deformation rate due to crystallographic slip. Xl represents the crystal lattice spin rate and Xp represents the plastic spin rate. The plastic deformation rate and spin rate depend on the slip rates, c_ s , for the active slip systems, 12 X Dp ¼ ð3Þ c_ s Ps ; s¼1

Xp ¼

12 X

ð4Þ

c_ s Ws ;

s¼1

where Ps and Ws are the symmetric and skew-symmetric parts of the dyad which is formed from the lattice vectors for each slip system, defined as: Ps ¼ 12ðds
ns þ ns
ds Þ;

ð5Þ

Ws ¼ 12ðds
ns  ns
ds Þ;

ð6Þ

where ds represents a unit vector oriented in the slip direction for slip system ‘‘s’’ and ns represents a unit vector normal to the slip plane for slip system ‘‘s’’. This model uses a constitutive relation originally given by Asaro and Rice (1977), based on the assumption that the elastic lattice properties are unaffected by slip: r

rl þ r trðDl Þ ¼ E : Dl ; ð7Þ where r is the Cauchy stress, trðDl Þ represents the trace of Dl and E is the fourth r order elasticity tensor. rl is the co-rotational stress rate formed on axes which spin with the crystal lattice, defined as r

rl ¼ r_  Xl  r  r  Xl :

ð8Þ

Asaro and Rice also express the constitutive relation in terms of the co-rotational r stress rate formed on axes which spin with a material element, r: ! 12 X r 0 r þ r trðDl Þ ¼ E : D  ð9Þ c_ s Ps ; s¼1

where

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P0s ¼ Ps þ E1 : ðWs  r  r  Ws Þ:

225

ð10Þ

Lattice vectors which characterize the slip systems are affected by lattice distortion. A variety of assumptions can be made concerning the evolution of the crystal lattice vectors. In this version of the kinetic formulation, the crystal lattice vectors are assumed to remain orthogonal unit vectors that simply rotate at the lattice spin rate: d_ s ¼ Xl  ds ;

ð11Þ

n_ s ¼ Xl  ns :

ð12Þ

To complete the constitutive model, equations for the slip rates, c_ s , are needed. For the current implementation, a slip rate with a power-law dependence on the resolved shear stress as originally suggested by Hutchinson (1976) was used

ð1=m1Þ c_ s ss

ss

¼

; ð13Þ c0 js js where c_ is the reference slip rate, m is the rate sensitivity factor – set to 20 for all of the simulations in this study, ss is the resolved shear stress on slip system ‘‘s’’ and is an internal state variable which accounts for the hardening on slip system ‘‘s’’. The initial value of js on each slip system corresponds to the critical resolved shear stress on that system, i.e., the stress necessary on slip system ‘‘s’’ to activate slip in the rate insensitive limit. In the rate-dependent model, it is sometimes referred to as the reference shear stress. The resolved shear stress, ss , for slip system ‘‘s’’ generated by an applied Cauchy stress, r, can be obtained using SchmidÕs Law: ss ¼ Ps : r:

ð14Þ

2.2. Hardening model Material hardening is captured at the slip-system level by evolving the js term in Eq. (10). A preliminary part of this study focused on developing easy, reliable methods for capturing work-hardening in a polycrystal plasticity model. Slip-system activity is tracked within each element in a finite element based polycrystal plasticity model. Due to variable constraints caused by element and grain neighbors, multiple slip systems simultaneously activate at the onset of plastic deformation within an element in these simulations. This study assumes multiple slip-system activity occurring at the onset of plastic deformation overwhelms latent hardening effects observed in classical single crystal tension and compression test experiments (a topic covered in most mechanical metallurgy texts, e.g., Chapter 4 in Dieter (1986)), where it is possible for a single slip system to accommodate the plastic deformation. Thus, an assumption commonly employed in polycrystal plasticity modeling and used in all simulations in this study is that all slip systems harden equally. To evolve js , two straightforward hardening laws were considered, the first based on an integral form the hardening saturation law originally proposed by Voce (1948, 1955) and Kocks (1976) is given as

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 js ¼ j0 þ ðjsat  j0 Þ 1  exp js ¼ j0 þ Aenpl :



h  epl jsat  j0

 ;

ð15Þ ð16Þ

Both models are written as slip-system hardening laws, although they have often been used to model the tensile stress–strain response in a ductile polycrystalline metal. As they are expressed in Eqs. (15) and (16), h, jsat and j0 are material parameters in the modified Voce law, and A and n are material parameters in the power law. In both models, epl is the equivalent plastic strain within a material element, defined by Z t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=3Dp : Dp dt: ð17Þ epl ¼ 0

Constitutive model formulations that assume all slip systems harden equally invariably evolve material hardening using total shear accumulated on all of the slip systems, usually expressed as the sum of the absolute value of accumulated shear on each slip system. An assumption with substantial validity since the sum of crystallographic shear has significance beyond a relative measure of accumulated strain through its relationship to the Taylor factor (Mecking et al., 1996). Evolving hardening through equivalent plastic strain within a material element rather than using the algebraic sum of shears was chosen for this model because determining slip-system hardening parameters from a single polycrystalline stress–strain curve becomes very straightforward. Curve fitting a tensile test result on a randomly textured polycrystalline FCC metal using either Voce law or power law models, then dividing the fitted result by the Taylor factor provides a slip-system hardening model that, when used within a polycrystal plasticity simulation, reasonably captures the original experimentally observed result. Comparison results using this method for determining slip-system hardening parameters for both models are illustrated in Fig. 1, discussed in more detail in the next subsection.

3. Simulation procedure The JAS-3D finite element implementation of the model provided the framework to simulate the elastic–plastic deformation of a polycrystal (Biffle, 1987). In this study, eight node (hexahedral) 3-D isoparametric finite elements were used. The elements use a single integration point at the center to give an efficient and fast numerical formulation. Hourglass control, necessary for single-point integration of the finite elements, uses the method developed by Flanagan and Belytschko (1981). Numerical integration of the constitutive model was performed using a forward Euler algorithm with adaptive subincrementation developed by Zinkiewicz and Cormeau (1974). Restrictions are placed on the allowable time step size for the integration algorithm to remain stable and accurate. If the step size requested by the user is larger than the allowable size, the step is divided into subincrements of

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Fig. 1. (a) Simulated tensile response of a 100 grain-one element per grain polycrystal compared with an experimental copper tensile stress–strain curve. (b) Simulated tensile stress–strain response of 2-D and 3-D Copper polycrystal compared with experimental data. The simulated polycrystals are illustrated in (c) with superimposed predicted von Mises stress distribution after 10% strain.

allowable size within the constitutive subroutine. The microstructure for these simulations is usually evolved using a 2-D or 3-D Potts model (Anderson et al., 1984; Holm and Battaile, 2001). When appropriate, periodic boundary conditions were applied to all sides of the simulations. In the case of one element thick ‘‘2.5D’’ simulations, periodic boundary conditions applied to the front and back faces,

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equivalent to simulating a microstructure infinite in extent. The periodic boundary conditions used in the simulations are a generalization of classical periodic boundary conditions. In classical periodic boundary conditions, the displacement of a node on one face is identical to the displacement of the corresponding node on the opposite face. In this case, that concept is generalized to allow an overall average displacement of an entire face to be imposed on the periodic displacement. This average displacement may be kinematic, i.e., user imposed displacement, or may be defined iteratively such as the average displacement to achieve an imposed overall force. Fig. 1 illustrates comparisons between experiments and simulations fit to tensile test data on a copper polycrystal. To define the tensile test boundary conditions in the simulations, average displacement boundary conditions were applied to the top and bottom faces (or edges in 2.5-D simulations). Generalized periodic boundary conditions were applied to all faces and edges including those which displacement boundary conditions were applied to define uniaxial tension. Fig. 1(a) illustrates an experiment–simulation comparison using the method for hardening law parameter extraction defined in the previous section on a 100 grain simulation (10  10  1) composed of one element per grain. By fitting the experimental stress–strain curve, values determined for the parameters were, j0 ¼ 16:1 MPa, A ¼ 925 MPa and n ¼ 0:759 for power law and j0 ¼ 16:1 MPa, jsat ¼ 259:1 MPa and h ¼ 2490 MPa. Slip-system hardening was evolved by using these parameters in the evolution Eqs. (13) and (14), and dividing the work hardening rate by the Taylor factor at each time step. Mathematically, this may be expressed as given in Eq. (15): djs f ðep Þ ; ¼ M dt

ð18Þ

where f ðep Þ represents the slip-system work hardening evolution equation, M is the Taylor factor and dt is the time increment. Simulations using either power law or Voce law to evolve the slip-system hardening provided an excellent fit to the experimental data up to 10% strain. These same parameters were used for the paved 2.5-D and 40  40  40 3-D simulations. Each simulation is composed of 200 randomly oriented grains as part of a repeating microstructure. The simulated microstructures are illustrated with superimosed von Mises stress distributions after 10% strain in Fig. 1(c). The simulated stress–strain response, shown in Fig. 1(b), softens slightly in these multiple element per grain simulations, a consequence of increasing the number of degrees of freedom within each grain in the simulations. This result is also a manifestation of no length scale in these models, a critical missing length scale is grain size, known to play an important role in hardening in polcrystalline materials. Independent of this observation, Fig. 1(c) also illustrates wide stress variations within the simulated polycrystals. These variations provide initiation sites for fracture, account for the variable and statistically dependent deformation response of polycrystalline materials and can contribute to localized strain-enhanced coarsening of microstructure, leading to thermomechanical fatigue and failure of a polycrystalline material. The choice of hardening model, either Voce law or power law, has little influence on the distribution of stress within the simulated polycrystals. More

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importantly, these stress variations cannot be predicted by a standard continuum model and have not been validated by quantitative experimental analysis. Using this simulation tool, this paper presents results from a study that investigates three cases intended to address these mesh sensitive and stress-variation issues, ultimately providing a better understanding of results provided by this type of modeling. Each case (i), (ii) and (iii) is outlined below: (i) Texture evolution during isochoric deformation of a 3-D polycrystal composed of 200 grains with randomly generated crystallographic orientations and 64,000 elements. Isochoric deformation, also sometimes referred to as axisymmetric tension, is defined in Eq. (19) and chosen because of its simplicity in illustrating texture evolution results, is applied as a boundary condition to the entire polycrystal: 0 1 1 0 0 A dt: L ¼ @ 0 1=2 0 ð19Þ 0 0 1=2 Often, it has been indicated that polycrystal plasticity models overpredict the evolution of texture, suggesting that the problem is overconstrained (Butler and McDowell, 1998). Those statements are primarily based on results from simulations composed of one or more grains per finite element. Results from this simulation are compared with Taylor model results and a simulated polycrystal composed of one element per grain to illustrate the influence of mesh refinement to several hundred elements per grain on the evolution of texture in a polycrystal plasticity simulation. (ii) A series of three one element thick simulations where the elements are paved into the grains. As with the simulation illustrated in Fig. 1, these simulations were subjected to periodic boundary conditions, eliminating surface effects. Through element paving, the average element size to grain size ratio may be varied by factors of two and four within a simulated polycrystal composed of the same microstructure. The influence of element size to grain size can be investigated in these simulations, clearly illustrating a convergent solution and the influence of a missing length scale in polycrystal plasticity simulations. (iii) A direct comparison between a polycrystal plasticity simulation with experimental EBSD data collected from a copper polycrystal subjected to tensile deformation. EBSD data were collected from the surface of an OFHC (oxygen-free high conductivity) copper tension specimen, then used for comparison with a finite element polycrystal plasticity simulation using the implementation discussed in this paper. The OFHC copper used for the study was cast into 1 in. diameter by 12 in. high ingots then swaged to a 9/16 in. diameter bar. A standard sized flat tension specimen was machined from the swaged bar, annealed for 1 h at 800
C to achieve a large grain size, then polished for EBSD analysis. The data was collected at the same location on the tension specimen before deformation and after 0% and 10% tensile strain, and compared with a corresponding polycrystal plasticity finite element analysis simulating tensile deformation on the same section of microstructure. EBSD spacing was 2 lm, one finite element corresponded to one EBSD data point.

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4. Results 4.1. Evolution of texture in a polycrystal Isochoric deformation is best represented experimentally by wire drawing. Previous researchers have simulated wire drawing using the Taylor assumption, fully prescribing the equivalent deformation imposed on a polycrystal in each grain (Chin and Mammel, 1967; Zhou et al., 1993) or an aggregate of grains which underlie a material point in a Finite Element model (Mathur and Dawson, 1990). Applying the Taylor assumption to a series of grains with a periodic orientation distribution reveals how their rotation paths correspond to the development of wire drawing texture in a polycrystalline FCC material (Kocks et al., 1998, Chapter 9; Gambin, 2001, Chapter 6). Fig. 2 illustrates this concept by superimposing the grain rotation paths, predicted using the Taylor assumption, on to experimentally determined texture of polycrystalline copper subjected to wire drawing using an inverse pole figure representation, i.e., the rotation paths of the global wire drawing axes plotted in crystal orientation coordinates using an equal area projection. Since the fully prescribed deformation state of the polycrystal is imposed on each grain, the Taylor assumption predicts an overconstrained response, the rate of texture evolution is more rapid than experimentally observed (Butler and McDowell, 1998; Kocks et al., 1998, Chapter 12). In a finite element simulation, where energy is minimized across the entire model, grain rotation paths meander in response to the changing constraints imposed by neighboring grains with different orientations in the polycrystal, as illustrated in the 6  6  6 one element per grain result in Fig. 3.

Fig. 2. Superimposed grain rotation paths as predicted by the Taylor model on to experimentally determined texture by X-ray goniometer measurement (Stout and OÕRourke, 1988) on polycrystalline copper subjected to wire drawing. Contour levels indicate times random texture.

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Fig. 3. (a) Finite element model result and (b) grain rotation paths predicted by a 6  6  6 one element per grain polycrystal plasticity simulation of isochoric through 100% plastic strain. Each grain is randomly assigned an orientation, randomly assigned initial grain orientations are specified by black dots in the inverse pole figure.

A similar texture evolution is predicted but meandering grain rotation changes during deformation track longer paths toward their equilibrium points, slowing the predicted evolution of the wire drawing texture. Divergence within the resultant rotation fields given by Figs. 2 and 3 indicate regions of ‘‘intrinsic’’ orientation instability (Raabe, 2002). Grains with orientations near these regions have a propensity to build up significant orientation gradients, or break apart, during isochoric deformation simply due to their crystallographic orientation relative to the deformation condition. To investigate the influence of mesh refinement beyond one element per grain, a series of simulations on 6  6  6 polycrystals composed of cubic grains with different levels of mesh refinement were subjected to isochoric deformation using the Voce isotropic slip-system hardening model to a true strain of 1.5. Fig. 4 illustrates the stress–strain response along the major axis of deformation for each of the simulations. As the mesh is refined, the simulations show a slightly softer response due to an increased number of degrees of freedom, a result consistent with that in Fig. 1. Detailed orientation evolution results from these simulations are given in Fig. 5. Fig. 5(a) illustrates the initial orientation of the 216 grains within the polycrystal using an inverse pole figure representation. Figs. 5(b)–(d) illustrate the simulated orientations of each finite element in respective one element per grain, 8 elements per grain and 64 elements per grain simulations. The ability of the simulations to break-up crystal lattice orientations and form orientation gradients within an original grain is realized on an element by element basis. The results presented in Fig. 5 suggest that an increased level of finite element discretization give a more precise prediction of the final texture. That is, although the one element per grain and 8 element per grain results correctly predict intensities at the [0 0 1] and [1 1 1] poles, the 64 element per grain simulation more accurately captures a more subtle texture component, the band of intensity between the [0 0 1] and [1 1 1] poles.

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700

1 element per grain

True Stress (MPa)

600 512 elements per grain

500 400

Increasing degree of mesh refinement

300 200 100 0 0

0.2

0.4

0.6

0.8

1

True Strain Fig. 4. Predicted stress–strain response along the major axis of deformation in a polycrystal plasticity model with Voce slip-system hardening. The polycrystal material response softens slightly with an increasing degree of mesh refinement.

Results from a more detailed examination of how rotation paths evolve in three selected grains (labeled 1, 2, 3 in Fig. 5(a)) are illustrated in Fig. 6. Figs. 2 and 3 indicate the general rotation paths of grains within a polycrystal subjected to isochoric deformation. Based on those indicators, grain 1 is expected to rotate toward the [1 1 1] pole, grain 3 is expected to rotated toward the [0 0 1] pole and grain 2 has an orientation near the location of rotation divergence. Grain 2 has an intrinsically unstable orientation. Figs. 6(a) and (b) compare the grain rotation paths from the one element per grain (black lines), and 8 and 64 elements per grain (gray lines) simulations. Rotations toward the poles are as expected for grain 1 and grain 3, although increasing refinement the finite element mesh permits an increasing degree of spreading of final orientations not possible in the one element per grain simulations. Also, mesh refinement slightly alters constraints within and between these grains, slightly shifting the average rotation path in the 8 and 64 element per grain results. As might be expected, grain 2 behaves differently, the one element per grain rotation path of grain 2 begins to track toward the [1 1 1] pole, but then changes direction toward the [0 0 1] pole. Refining the mesh within this grain illustrates extremely divergent predicted rotation paths within the elements that compose the original grain. Further mesh refinement, as illustrated in Fig. 6(c) and (d) contributes to spreading from the original grain orientations during deformation, akin to a refined data resolution more accurately picking up the tail of a distribution. This result suggest that grains with initial orientations located near a rotation divergence region break apart readily and are responsible for generating the minor texture component between the [0 0 1] and [1 1 1] poles caused by isochoric deformation of an FCC polycrystal. A polycrystal plasticity model can only capture this component when it is highly resolved, i.e., containing perhaps 100Õs of elements per grain, such that it can capture these rapidly evolving rotation gradients within the intrinsically unstable grains.

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Fig. 5. Inverse pole figures illustrating evolved orientation distribution after isochoric deformation to 150% strain in a 6  6  6 (216 grains) polycrystal plasticity simulation with varying degrees of mesh refinement.

For the purpose of providing a result more directly comparable to the experimental result presented in Fig. 2, the discrete data from the isochoric deformation simulations was binned and contoured using an algorithm provided by HKL Inc. (www.hkltechnology.com) as part of their EBSD mapping package and illustrated in Fig. 7. Prior to contouring the data, the algorithm applies a Gaussian filter to each data point, the amount of Gaussian smoothing defined by a half width angle parameter. As expected, the intensity band between the [1 1 1] and [0 0 1] poles is more accurately captured in the contoured results from simulations with the highest mesh resolution. Fig. 7 also shows that further smoothing by increasing the Gaussian filter half-width lowers the intensities at the poles, at the expense of accurately capturing the lesser intensity band between the poles. No combination of mesh refinement or Gaussian smoothing leads to a contoured result quantitatively identical to the result

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Fig. 6. Detailed examination of how rotation paths evolve in three selected grains from the 6  6  6 polycrystal plasticity simulations of isochoric deformation.

given by X-ray goniometer measurements given in Fig. 2. The contours in Fig. 6(d), mesh resolution of 512 elements per grain and Gaussian half-width parameter of 5
, most closely resemble the experimental measurement in Fig. 2. However, the peak intensities in this simulated result remain high even at this degree of mesh resolution. Smoothing associated with the X-ray measurement is not the same as smoothing discrete data using a Gaussian filter, thus reasons behind the differences in an

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53.22 Gaussian Half-Width = 5º

1

20.63 Gaussian 3 Half-Width 1 0.5 = 10º

4 23

0.5

3

235

4 2

0.5 1

0.5 1

3

5.33

10.35

4 2

42

1 element per grain

(a)

49.73 Gaussian Half-Width = 5º

1

20.31 Gaussian 3 Half-Width 1 = 10º 0.5

4 23

4 2

0.5

0.5 31

5.70

10.79 4 2

(b)

4

0.5 21 3

8 elements per grain 19.22

43.81

Gaussian Half-Width = 5º

3 1

4 2

Gaussian Half-Width = 10º

0.5 0.5

1

3

4 2

0.5

0.5

1 3

3 5.79 4

10.39

2

1

4 2

(c)

64 elements per grain 36.41 Gaussian Half-Width = 5º

3 1

4 2 0.5

17.41

Gaussian Half-Width = 10º

3 1

4 2

0.5 0.5

1 3

1 4

10.12

32

4 2 0.5

(d)

512 elements per grain

Fig. 7. Contoured crystallographic texture results from isochoric deformation simulations with varying degrees of mesh refinement to 150% strain. A Gaussian filter is applied to the discrete orientation data prior to binning and contouring the results. The Gaussian half-width parameter defines the amount of filtering applied.

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experiment-simulation comparison of the contoured pole figures remain unclear. Fig. 7 results do suggest that further mesh refinement beyond 512 element per grain may continue to reduce the (over)predicted [1 1 1] peak intensity. 4.2. Refining the mesh in a polycrystal plasticity simulation The goal of this series of simulations is to generate results which characterize the influence of mesh refinement across a range of 10Õs–100Õs of elements per grain within a realistic looking microstructure. Paved one element thick meshes that can be refined at a 2:1 and 4:1 ratio along grain boundaries were placed on the same Potts model evolved microstructure composed of 200 grains. The resultant number of elements in the three simulations are 8792 elements, 33,771 elements and 118,507 elements, the degree of mesh refinement, expressed as mesh density, for each case is illustrated in Fig. 9. Each mesh was subjected to the same ‘‘tensile test’’ boundary conditions defined in the simulations whose results were given in Fig. 1. A periodic microstructure was used and periodic boundary conditions were applied to the ‘‘edges’’ and ‘‘faces’’ of the one element thick simulations. The polycrystalline stress– strain response of each mesh is illustrated up to 10% strain in Fig. 8(a) and the two coarser meshes up to 40% strain in Fig. 8(b). The saturated work-hardening response at high tensile strains correctly captures the influence of using the Voce law as the slip-system hardening model in these simulations. Fig. 8(a) reveals no discernible difference in the polycrystalline stress–strain response between simulations of different mesh densities. Similar to the slight softening due to mesh refinement from the isochoric deformation given in Figs. 4, 8(b) shows that at larger tensile strains a slight softening in stress–strain response occurs as the mesh is refined and the number of degrees of freedom is increased in the simulations. As stated in the introduction section, a polycrystal plasticity model tracks stress distribution and crystallographic orientation changes on an element by element basis during deformation. Fig. 9 plots local predicted von Mises stress distributions and local crystallographic rotation change as lattice rotation relative to the original crystallographic orientation within an element prior to deformation. Lattice rotation, in this case, is specifically defined as the minimum misorientation angle taken from the axis-angle pair description of the crystallographic rotation change. Prior to deformation, every element within a grain is initialized with the same orientation, thus the spatially defined lattice rotation plots in Fig. 9 show local misorientation relative to original grain orientation and give a visual indication of how a grain is predicted to evolve rotation gradients or break apart during deformation. Fig. 9 shows that predicted subgrain deformation evolution and von Mises stress distribution within and between grains are virtually mesh insensitive across this wide range of mesh refinement. A key result that clearly illustrates the influence of no length scale in these finite element simulations. Specifically, no mechanism exists within the basic model framework defining a relative grain size to element size ratio. Fig. 9 does show some subtle sharpening of local misorientation variation as the mesh is refined. Specific sharpened regions are highlighted with arrows in Fig. 9(c). A simple increase in degrees of freedom argument offers an explanation for this

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observation and indirectly supports observations of better texture predictions in models with increased mesh refinement. 4.3. Direct comparison between experiment and simulation This subsection presents results from a direct comparison of EBSD measurements with a polycrystal plasticity simulation of a microstructure section that has experienced deformation on the surface of a tensile specimen to 10% strain. To perform the comparison, EBSD measurements must be taken on the same microstructure section before and after deformation. Data taken prior to deformation must be conditioned and used to initialize the finite element model. Fig. 10 illustrates a part of the conditioning procedure. Figs. 10(a) and (b) illustrate a 200 lm  200 lm (101  101 measurements) section of EBSD data taken on a 2 lm grid for input into a polycrystal plasticity simulation. Fig. 10(a) shows a clean looking EBSD map of polycrystalline microstructure with a slight h1 1 0i texture relative to the imaging direction (this direction is normal to the tensile direction, it gave better contrast between grains) by using a inverse pole figure colorizing scheme given in the lower right corner of the figure. This representation can be misleading with regard to the state of the data for initializing a finite element polycrystal plasticity model. Fig. 10(b) illustrates this point by plotting a single value of the 3  3 crystallographic orientation matrix, as determined by the EBSD analysis, revealing non-unique measurements within several grains. The non-uniqueness is caused by crystal structure symmetry, which is factored out in the inverse pole figure representation. EBSD data collection software typically does not automatically assign orientation measurements to identical global reference axes. In addition to reassigning crystallographic coordinates to the same global reference frame, local measurement quality must be judged. In this study, a small number of ‘‘low’’ quality measurements were discarded. Once reassignment and data quality conditioning is completed, all measurements within a

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Fig. 9. Predicted von Mises stress distribution and lattice rotation relative to original grain orientation in a 200 grain polycrystal subjected to tensile test boundary conditions using a polycrystal plasticity model.

single grain, which varied by as much as 1 degree, were averaged. The effect of the averaging procedure is illustrated on the h100i micropole plot in Fig. 10(d). Each red cloud represents a cluster of measured locations within an individual grain and the corresponding black dot within each cloud represents the average of those measurements. Finally, discarded data points were assigned orientations based on their nearest neighbors, filling any holes within the 101  101 grid. The resultant grid is

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Fig. 10. A section of EBSD data captured from the surface of a flat, coarse grained copper tensile specimen for experiment-simulation comparison. (a) Data imaged by colorizing orientation relative to the ‘‘y’’ direction using inverse pole figure scheme, (b) data imaged by using a single value from the crystal orientation matrix, (c) data conditioned for input into polycrystal plasticity model and (d) Actual measurements (red) and averaged measurements (black) for each grain as a result of data conditioning for model input.

illustrated in Fig. 10(c), which is directly converted into finite elements for the simulations. Microstructure information beneath the surface of the tensile specimen is not known. Relatively complex boundary conditions were initially contrived to approximate real boundary conditions experienced by the surface of the specimen during tensile deformation. One example, illustrated in Fig. 10(c), extended the surface microstructure determined by EBSD to 5 elements deep, giving a total of 51,005 elements, and applied a symmetry boundary condition to the backside of the polycrystal plasticity model. The edges of the model were fixed with a displacement boundary condition applied parallel to the tensile axes. Comparison between 5 element thick and one element thick simulations with identical microstructure and the same boundary conditions gave virtually identical results. For this reason, simulated data from the one element thick simulations are presented and compared with experimentally measured local crystal rotation and misorientation data. As Fig. 10(a) indicates, EBSD revealed some texture in the tensile bar, a h1 1 1i component parallel to the tensile direction and a h1 1 0i component normal to the

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tensile direction. Stress–strain data from an identical polycrystalline Cu sample was fit to the power law slip-system hardening model to 10% strain, as outlined in the simulation procedure section. This power law fit was used for the comparison simulation. After performing the simulation and repeating EBSD experiments at 10% tensile strain, five grains near the center of the selected region were identified for detailed comparison. They are indicated in Fig. 11(a), which colorized grains relative to the tensile direction, using the same inverse pole figure colorizing scheme as given in Fig. 10. In Fig. 11, the simulated and experimental tensile orientations for each finite element within the five indicated grains are plotted on a inverse pole figure before deformation, and after 10% tensile strain. Each inverse pole figure illustrates a large black point, and a green cloud of data corresponding to simulated data before deformation and after 10% strain, and respective red and blue clouds of data representing experimental EBSD data before deformation and after 10% tensile strain. Comparison between simulated and experimental data after 10% strain, does not give a strong correlation. In a qualitative sense, this correlation is poorer than that obtained by Bhattacharyya et al. (2001) and Delaire et al. (2000) who, very recently, have performed similar comparisons. The 1
measurement error certainly dilutes the correlation and this kind of comparison is also sensitive to any difference in specimen alignment within the scanning electron microscope between pre- and postdeformation EBSD measurements. Fig. 11 results suggest that a degree of specimen alignment error exists in the experimental comparison because measured 10% rotation data is not tracking in expected directions based on simulated and experimental texture evolution results of an FCC metal experiencing tensile deformation (which are similar to isochoric deformation results). Still, certain aspects of the comparison give a favorable correlation such as grain 4 has an intrinsically unstable crystallographic orientation, favorable for subgrain break-up, both 10% experimental and simulated results show considerable spread in subgrain orientations. A different kind of graphical comparison of sub-grain evolution is illustrated in Fig. 12, where lattice rotation (local misorientation relative to the original grain orientation), using the axis-angle pair description of crystallographic rotation, is plotted as a function of position within each of the five identified grains. The same EBSD measurement differences in specimen alignment within the scanning electron microscope between pre- and post- deformation also influence this comparison. In this case, it appears to contribute to the general trend of measured maximum lattice rotations, which are consistently slightly larger than comparison simulated lattice rotations after 10% tensile deformation. Regardless, Fig. 12, in contrast to Fig. 11, shows a reasonable qualitative corroboration between experimentally measured and simulated sub-grain evolution during deformation. A noticeable feature in the simulated results given in Fig. 12 is the smooth deterministic trends in lattice rotation within grains after deformation. These trends are governed by the geometric influences of neighboring grains that control local constraints within a polycrystal plasticity model. They are not mesh sensitive, as illustrated by the mesh refinement results in Fig. 9, thus to maintain the same overall trend in crystallographic rotation across a grain, misorientation between neighboring elements decreases as the mesh is refined within a grain in a polycrystal plasticity model.

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Fig. 11. Comparison of simulated and experimental subgrain rotations after 10% tensile strain in 5 selected grains from a Cu tensile specimen. In each pole figure, simulated data is represented by a black point as the grain orientation at 0% strain and a green cloud of points as the crystal orientation of each finite element after 10% tensile strain. Experimental EBSD data in each pole figure is represented by a red cloud of points at 0% strain and a blue cloud of points at 10% strain.

5. Discussion The most important issue addressed in this study is mesh sensitivity within large scale polycrystal plasticity simulations. The results show, most clearly in Fig. 9, that stress distribution and orientation changes within grains are nearly mesh insensitive

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Fig. 12. Comparison of experimental and simulated subgrain evolution plotted as a function of lattice rotation relative to original grain orientation after 10% tensile deformation in the five grains identified in Fig. 11(a).

across a wide range of mesh densities, a consequence of no inherent length scale in finite element polycrystal plasticity modeling. Increasing mesh refinement within a polycrystal plasticity model does allow the model to better capture the evolution of minor texture components during deformation, as best demonstrated by the isochoric deformation simulations at mesh densities spanning from 1 to 512 elements per grain. The same series of simulations show that break-up of grains with intrinsically unstable orientations relative to the applied deformation direction, e.g., grain 2 in Fig. 6, accounts for the deformation induced development of the minor texture during isochoric deformation. Conversely, grains with intrinsically stable orienta-

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tions remain relatively intact even after significant strains, e.g., grains 1 and 3 in Fig. 6. Highly resolved polycrystal plasticity modeling also captures certain experimentally observed deformation substructure evolution aspects well. This is illustrated by the experiment-simulation comparison in Fig. 12 and the ability of the model to predict texture evolution well, illustrated in Figs. 5 and 7. These results are encouraging, but critical experimentally observed characteristics of subgrain evolution during polycrystalline deformation are not captured in these simulated results, thus the predictive capability of the model breaks down at the subgrain scale. The compendium of simulated and experimental results in this paper, discussed in the context of previously reported observations, offers insight for this breakdown. The evolution of substructure during deformation of polycrystalline materials looks statistical in nature. Three dimensional random walk simulations of crystallographic orientations diffusing in Euler space (Miodownik et al., 2001) illustrate results that fit a Gamma distribution (or a log-normal distribution, but previous experimental considerations, discussed below, did not consider this). It is given in Eqs. 14,20 where a and r are the statistical parameters and C is the Gamma function. Results also scale with the average misorientation angle, provided that the average misorientation between adjacent locations is less than 10
(Hughes et al., 1997; Hughes, 2002). f ðxÞ ¼

ar r1 x expðaxÞ: CðrÞ

ð20Þ

Physical meaning has been attached to the distributions when they are obtained from transmission electron microscope based stereological techniques, where subgrain boundaries are actually observed as well as measured (Hughes et al., 1997; Hughes, 2002; Liu et al., 1998). For example, the average normalized Gamma distribution (where r is set equal to a) of specific types of sub-grain boundaries, incidental dislocation boundaries (IDBs) and geometrically necessary boundaries (GNBs), appears to be strain independent and scale with the average misorientation angle. In turn, the average subgrain misorientation angle for both types of subgrain boundaries scales with effective strain. Subgrain break-up as predicted in a polycrystal plasticity model statistically evolves in a manner consistent with the random walk simulations, thus results from highly resolved polycrystal plasticity modeling should capture those statistics. Indeed they do, as best demonstrated from highly resolved simulations performed by Mika, Dawson et al. (1999) When inter-element misorientations within many grains are placed in the same distribution, the average misorientation angle scales with effective strain and seems to match experimental TEM observations. Specifically, Mika and Dawson (1999) performed a highly resolved polycrystal plasticity deformation simulation composed of 172 grains with 576 element per grain and placed every inter-element misorientation within each grain in a single statistical distribution. The results are astounding, they fit a Gamma distribution perfectly, nearly precisely scale with the average misorientation angle, and approach scaling behavior with effective plastic strain. However, on a grain by grain basis, a highly resolved polycrystal plasticity model with no length scale may generate Gamma distributions of

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intragranular misorientations, but these individual distributions do not necessarily match and will not scale with effective strain applied to the polycrystal. The interelement misorientation distribution of an intrinsically unstable grain will be broader than the inter-element misorientation distribution of an intrinsically stable grain after the same amount of applied strain to the polycrystal, given approximately the same mesh resolution within each grain. Consider Figs. 13(a) and (b), which illustrate the misorientation distributions after a strain of 1.5 within the 3 grains used to demonstrate the role of subgrain break-up and grain orientation stability in Section 4.1. The binning procedure for generating these distributions was performed exactly as outlined by Hughes et al. (1998). As expected based on inverse pole figure results given in Fig. 6, the misorientation distribution between elements is very diffuse in grain 2, so diffuse in this case, that the data, as binned, did not fit a Gamma distribution. The misorientation distributions of grains 1 and 3 are also somewhat different from each other, grain 3 giving a more diffuse distribution that, again, could be expected based on the results given in Fig. 6. Fig. 13(a) shows that a higher average misorientation angle corresponds to a broader distribution within a grain, suggesting that the misorientation distributions could scale with average inter-element misorientation. Fig. 13(b) suggests that this is not the case, especially given that the binned data for grain 2 does not fit a scaled Gamma Distribution well. The poor scaling behavior may be caused by an undefined relationship between grain size and element size in the simulation, i.e., if the mesh was refined within these grains, the inter-element distributions presented in Fig. 13(a) and (b) would change. Experimentally, TEM observations used to define the scaling behavior between subgrain misorientation and strain contain a length scale, subgrain size, since those observations take measurements across actual subgrain boundaries. Blind EBSD mapping, as it was performed in this study, does not measure orientation differences across observed subgrain boundaries, but across equidistant observation points, thus this experimental method contains no length scale. Consequently, the same consideration applied to the model results also applies to blind EBSD mapping. Furthermore, the measurement technique is statistical in itself, mimicking ‘‘random walk’’ about a single orientation. To demonstrate this, Fig. 13(c) illustrates the distribution of measurements within grains 1, 4 and 5 prior to deformation from the results on the annealed Cu polycrystal presented in Section 4.3. Assuming each grain has the same orientation at all locations, Fig. 13(c) depicts the statistics of the EBSD measurements, not surprisingly, they fit a Gamma distribution. Misorientation between EBSD measurements within a grain after a specified amount of deformation depends on both the distance between measurements, akin to element size to grain size ratio in highly resolved polycrystal plasticity modeling, and the propensity of a grain to break apart during deformation. Fig. 13(d) illustrates the misorientation distribution between EBSD measurements within grains 1, 4 and 5 from Section 4.3 after 10% tensile strain. As expected, orientation stability governs the distributions. Tensile deformation is similar to an isochoric deformation boundary condition, so the inverse pole figure representation in Fig. 2 may be used as a guide to determining intrinsic orientation stability of a given grain orientation. On a relative basis, grain 1

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Probability vs. Misorientation isochoric deformation - strain = 1.5

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Fig. 13. (a)- Probability vs. interelement misorientation angle from the 3 selected grains after isochoric strain to 1.5 in the 6  6  6 – 512 elements per grain simulations. (b) Scaled probability vs. interelement misorientation angle from the same 3 grains. (c) Probability vs. subgrain misorientation from adjacent EBSD measurements within selected grains 1, 4 and 5 before deformation. (d) Probability vs. subgrain misorientation from adjacent EBSD measurements within the same grains after 10% tensile deformation.

and grain 4 are expected to be intrinsically unstable orientations whereas grain 5 is expected to be an intrinsically stable orientation. Certainly, grains 1 and 4 exhibit wider misorientation distributions than grain 5, although grain 1 exhibits the widest distribution, not grain 4 as might be expected from noting the results given in Fig. 11. Presumably, the variable constraints within each grain imposed by its neighbors, exacerbated by the surface location of these grains, contributes to the variability of the distributions after a modest amount of imposed deformation. Referring back to Fig. 12, one can observe a relatively sharp orientation gradient in grain 1 likely caused by the variable constraints and manifested in the slightly wider than expected distribution.

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Finally, the most highly resolved simulation in this study, illustrated in Fig. 9(c), shows some evidence of orientation gradient banding in isolated grains, indicated by the arrows in the lattice rotation map in Fig. 9(c). Such banding could be associated with GNBs and must be completely driven by variable constraints imposed by neighboring grains. (Beaudoin et al., 1996) Increasing the effect of constraints imposed by grain boundaries and grain neighbors, which would certainly occur by incorporating a length scale, could encourage the formation of substructure associated with GNBs in a highly resolved polycrystal plasticity model. 6. Conclusions • Predicted stress-distributions within a highly resolved polycrystal plasticity simulation were not dependent on choice of slip-system hardening law, either powerlaw or Voce law hardening. If all slip systems are assumed to harden equally, the parameters required for these hardening model models can be extracted from a single stress–strain curve of a randomly textured FCC polycrystal. • Stress distribution and orientation changes within grains are mesh insensitive across a wide range of mesh densities. Accordingly, misorientation between adjacent elements within pre-defined grains after a prescribed amount of deformation is highly mesh sensitive. Also, polycrystal plasticity models contain no inherent length scale thus cannot implicitly capture any influence of grain size. • Subgrain mesh refinement within polycrystal plasticity models allows them to better capture deformation evolution of minor texture components primarily because grains with intrinsically unstable orientations are better able to break apart during deformation. • Inter-element misorientation distributions within a grain are highly dependent on grain orientation relative to direction of stress/strain applied to the polycrystal. Thus, on a grain by grain basis, misorientation distributions do not scale with strain in a highly revolved polycrystal plasticity model. • EBSD measurements, as they were performed in this study, contained no length scale. Misorientation between measurements depended on the distance between measurements, similar to predicted misorientations between elements depending on element size to grain size ratio in polycrystal plasticity modeling. Consequently, the characteristics of EBSD measured subgrain misorientation distributions in a deformed polycrystal were similar to inter-element misorientation distributions after deformation in a polycrystal plasticity simulation.

Acknowledgements The authors gratefully acknowledge Mike Nielsen and Brad Boyce for helpful discussion and comments concerning this manuscript and Matt Nowell at TSL Inc. (Draper, UT) who collected the EBSD data on the copper tensile sample. All other work was performed at Sandia National Laboratories in Albuquerque, NM. Sandia

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is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy, under Contract No. DEAC04-94AL85000.

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