Influence of in-grain mesh resolution on the prediction of deformation textures in fcc polycrystals by crystal plasticity FEM

Acta Materialia 55 (2007) 2361–2373 www.actamat-journals.com

Influence of in-grain mesh resolution on the prediction of deformation textures in fcc polycrystals by crystal plasticity FEM Z. Zhao a, S. Kuchnicki b, R. Radovitzky a

a,*

, A. Cuitin˜o

b

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, USA b Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ, USA Received 5 October 2006; received in revised form 9 November 2006; accepted 10 November 2006 Available online 5 February 2007

Abstract The ability of three different crystal plasticity finite element models to predict deformation textures in face-centered cubic metals observed in experiments is assessed. These methods are: (i) Taylor averaging, in which the interactions of the grains are considered in a homogenized manner; (ii) low-resolution simulation (LRS), in which grain interactions are considered explicitly albeit with low resolution; and (iii) direct numerical simulation (DNS), which provides high-resolution details of the deformation fields inside the grains and of the grain interactions. A quantitative comparison of the numerical results provided by these three methods against experimental plane-strain compression textures is performed via orientation distribution functions and fiber line analysis. It is found that some details of the texture which are inaccessible to either Taylor averaging and LRS approaches are captured by the DNS approach. This can be explained by the ability of the high-resolution DNS method to describe details of the grain interactions, including heterogeneous deformation under homogeneous macroscopic strain and smooth gradients of lattice rotations inside the grains which are missing in low-resolution models.  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Texture; Crystal plasticity; Finite element models; Taylor model

1. Introduction Deformation processes in polycrystalline metals are always accompanied by a change of crystallographic orientation of each grain, which is often referred to as texture evolution. The control of texture evolution in metal-forming processes is of significant interest in modern industry for the purpose of achieving specific anisotropic mechanical properties in the final products. With the advance of computational materials science in recent years, modeling and simulation has provided a powerful tool both for theoretical investigations and industrial applications of texture evolution in polycrystalline metals [1]. Early descriptions of the mechanical response of polycrystals were based on sweeping assumptions on the interactions among the polycrystal grains. For example, the full*

Corresponding author. E-mail address: [email protected] (R. Radovitzky).

constraints (FC) Taylor model [2] assumes that all the grains are subjected to the same macroscopic strain. As a consequence, the compatibility among the grains is satisfied but equilibrium of stresses is violated. By contrast, Sachs [3] assumed that each grain was subjected to the same stress state and only one slip system with the highest resolved shear stress can be activated due to the externally imposed stress. In this model, the deformation of individual grains is treated as if the grains were isolated single crystals undergoing single slip, which violates the compatibility of deformation among the grains. Predictions of texture evolution from these types of model are able to capture the global texture features, but discrepancies from experiments can be significant [4–7]. Numerous modifications to the original models of Taylor and Sachs have been proposed attempting to relax the strong assumptions of these models. For example, relaxed-constraints (RC) Taylor-type models [8–11] partially relax the strict compatibility hypothesis of the FC model. In the socalled modified Sachs model [12,13], multiple slip systems

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can be activated to reduce the shear incompatibility among neighboring grains. Self-consistent models [14–18] satisfy stress equilibrium and strain compatibility simultaneously by considering individual grains embedded in a matrix whose properties are taken as those of the average grain assembly. The improved models have provided more detailed information on the texture evolution compared with the original FC and Sachs’ models. However, further details on microstructural aspects of the deformation, such as local grain interaction and intra-grain inhomogeneities of deformation, are not accessible to these models. The combination of single crystal plasticity constitutive models [19–21] and the finite element method (CPFEM) has enabled the investigation of a variety of aspects of the heterogeneous plastic response of polycrystals under general boundary conditions, including the prospect for describing complex details of the interactions among the grains and their influence on the macroscopic texture. The uses of CPFEM that have been proposed can be categorized in the following three types of approaches: (i) those using a homogenized constitutive model at the integration point level; (ii) those considering each element of the mesh as a single crystal; and (iii) those in which each idealized grain is represented with a fine mesh in order to resolve intra-grain deformation gradients. Among the first category, Taylor averaging has perhaps been the most widely used approach in theoretical and industrial applications due to its simplicity of implementation in finite element codes. Other approaches based on self-consistent models have also been successfully used and discussed in detail elsewhere [22,23]. In the Taylor averaging method, each integration point represents the collective Taylor-averaged behavior of a large number of grain orientations. The properties at each integration point are calculated by averaging either the stress tensor [20] or stiffness matrix [24] of every crystal comprising the aggregate. Although this method has been successfully used in the simulation of texture evolution in face-centered cubic (fcc) [25] and body-centered cubic (bcc) [26] materials, it has been observed that it tends to overpredict peak texture intensities and to shift the position of texture components, especially at large strains [6,27] and in cases where experiments show evidence of inhomogeneous deformation inside grains [18]. Efficiency improvements of this method aiming to reduce the number of grains required for industrial applications have been proposed recently [28–30]. Among the second category involving low-resolution methods, a simple and effective way to represent polycrystal aggregates is achieved by assigning a different crystal orientation to each individual element of the finite element mesh, i.e. each element represents an individual grain. As a result, the interactions among neighboring grains can be accounted for explicitly with a reasonable computational cost. This method has been widely used for the computation of macroscopic texture [20,31–33,7]. However, the lack of resolution of the deformation fields due to the low-order interpolation inside each element precludes the description

of heterogeneous features of deformation such as in-grain subdivision. Methods in the third category address this limitation by attempting to compute the full-field solution of the polycrystal response. This is achieved by resolving the deformation fields inside each grain with a fine computational mesh. Harren and Asaro [6,34] adopted this approach to investigate the localization of plastic flow in plane strain compression using a two-dimensional finite element model with hexagonal grains and a limited number of elements per grain. Becker [35] used a similar approach to show orientation splitting inside each grain as well as the overall deformation texture. Advances in parallel computing in the last decade have enabled large-scale simulations and, thus, the investigation of polycrystal response in higher detail. The first three-dimensional study was published by Mika and Dawson [36,37], who used an idealized rhombic dodecahedral grain microstructure with 576 elements in each of the 316 grains considered to investigate in-grain subdivision and inter-grain misorientations leading to the so-called geometrically necessary boundaries (GNB). Additional simulations of this type have been presented by Barbe [38,39], who commented on the observed heterogeneity of the stress and strain fields, intergranular deformation structures and surface effects. In this reference, a Voronoi construction is used to represent the grains on top of a structured finite element mesh, which results in grain boundaries with a staircase shape. The main objective of this paper is to quantitatively compare the ability of these three polycrystal modeling approaches to predict details of deformation textures observed in fcc metals under plane-strain compression. Toward this objective, we conduct CPFEM simulations of the plane-strain compression of polycrystalline aluminum samples using the three modeling methodologies mentioned above. In our implementation of the high-resolution approach, we adopt idealized equiaxed polycrystal topologies respecting the ostensible flatness of grain boundaries. Each grain is discretized with a fine mesh and its constitutive response is described with a forest dislocation-based hardening model of crystal plasticity. The considerable computing effort is distributed among processors via a parallel implementation based on mesh partitioning and message passing. We refer to this approach in which the details of the mechanical deformation fields are explicitly and accurately resolved in the finite element approximation – versus modeled by recourse to an averaging or homogenization approach – as direct numerical simulation (DNS), by analogy to the use of this term in fluid mechanics [40]. A detailed analysis of the deformation textures predicted by the three approaches is conducted by extracting the orientation distribution functions (ODF) from the simulation results and by subsequently performing a skeleton line analysis using the ODF. Finally, the pole figures, the ODFs and the fiber line analyses are compared among the three methods and with the experimental rolling textures from the work of Hirsch [5].

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This paper is organized as follows. In Section 2 we summarize the formulation of the specific crystal plasticity model employed in calculations. This is followed by a description of the overall numerical approach and simulation setup in Section 3. The results of the numerical simulations are presented in Section 4 and discussed in Section 5. A summary of the paper with the main conclusions is given in Section 6. 2. Constitutive framework The fcc crystal plasticity constitutive model adopted in the calculations presented in this paper corresponds to the explicit formulation presented in Kuchnicki et al. [41]. This particular formulation provides significant performance improvements over the original implicit formulation of the forest dislocation hardening model presented in Cuitin˜o and Ortiz [21], thus rendering the model suitable for large-scale computations. We summarize the features of the model in this section for completeness. The total deformation of a crystal is the result of two main mechanisms: dislocation motion within the active slip systems and lattice distortion. Following Lee [42], this points to a multiplicative decomposition F ¼ Fe Fp

ð1Þ p

of the deformation gradient F into a plastic part F , which accounts for the cumulative effect of dislocation motion, and an elastic part Fe, which accounts for the remaining non-plastic deformation. Following Teodosiu [43] and others [44–48], we shall assume that Fp leaves the crystal lattice not only essentially undistorted, but also unrotated. Thus, the distortion and rotation of the lattice is contained in Fe. This choice of kinematics uniquely determines the decomposition, Eq. (1). By virtue of Eq. (1), the deformation power per unit undeformed volume takes the form P : F_ ¼ P : F_ e þ R : Lp

ð2Þ

where P ¼ PFpT R ¼ FeT PFpT Lp ¼ F_ p Fp1

ð3Þ

Here, P defines a first Piola–Kirchhoff stress tensor relative to the intermediate configuration Bt and R a stress measure conjugate to the plastic velocity gradients Lp on Bt . The work conjugacy relations expressed in Eq. (2) suggest plastic flow rules and elastic stress–strain relations of the general form Lp ¼ Lp ðR; QÞP ¼ PðFe ; QÞ

ð4Þ

Here, Q denotes some suitable set of internal variables defined on the intermediate configuration, for which equations of evolution, or hardening laws, are to be supplied. A standard exercise shows that the most general form of Eq. (2) consistent with the principle of material frame indifference is P ¼ Fe SðCe Þ;

Ce ¼ FeT Fe

ð5Þ

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where S ¼ Ce1 R is a symmetric second Piola–Kirchhoff stress tensor relative to the intermediate configuration Bt and Ce is the elastic right Cauchy–Green deformation tensor on Bt . For most applications involving metals, a linear – but anisotropic – relation between S and the elastic Lagrangian strain Ee ¼ ðCe  IÞ=2 can be assumed without much loss of generality. Higher-order moduli are given by Teodosiu [49]. From the kinematics of dislocation motion, it has been shown by Taylor [2] and Rice [48] that Eq. (4) is of the form X a Lp ¼ ð6Þ c_ a Sa  m a a

 a are where c_ is the shear rate on slip system a and Sa and m the corresponding slip direction and slip plane normal. At this point the assumption is commonly made that c_ a depends on stress only through the corresponding resolved shear stress sa, i.e. c_ a ¼ c_ a ðsa ; QÞ

ð7Þ

which is an extension of Schmid’s rule. If Eq. (7) is assumed to hold, then it was shown by Rice [48] and by Mandel [47] that the flow rule, Eq. (6), derives from a viscoplastic potential. In order to complete the constitutive description of the crystal, hardening relations governing the evolution of the internal variables Q need to be provided. In this work, we adopt the forest dislocation hardening model for fcc metals of Cuitin˜o and Ortiz [21]. A synopsis of the main assumptions of the model together with the key constitutive relations is provided below for completeness. The rate of shear deformation on slip system a is given by a power-law of the form:  8  1=m < c_ sa  1 ; if sa P 0 0 ga c_ a ¼ ð8Þ : 0; otherwise In this expression, m is the strain-rate sensitivity exponent, c_ 0 is a reference shear strain rate and ga is the current shear flow stress on slip system a. Implicit in the form in which Eq. (8) is written is the convention of differentiating between the positive and negative slip directions ±ma for each slip system, whereas the slip rates c_ a are constrained to be nonnegative. This rate-dependency law is slightly different from that of Cuitin˜o and Ortiz [21] in that its multiple root has been shifted. As it has been noted by several authors [50,51,41], this has the effect of mitigating the tendency of the original model to predict unrealistic values of slip for sa =ga much different from unity. For multiple slip, the evolution of the flow stresses is found from an analysis based on statistical mechanics to be governed by a diagonal hardening law: X g_ a ¼ haa c_ a ð9Þ a

where haa are the diagonal hardening moduli: "  # )  a  a 3 ( 2 sc g sac aa h ¼ a cosh  1 no sum in a cc sac ga

ð10Þ

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In this expression, pffiffiffiffiffiffiffi sac ¼ rlb pna and

a

bq cac ¼ pffiffiffiffiaffi no sum in a ð11Þ 2 n are a characteristic shear stress and strain for the slip system a, respectively. The values of sac and cac determine the location of the ‘bend’ in the resolved shear stress-slip strain curve associated with the observable yielding during experiments. Thus, sc correlates well with the value of the flow stress determined by back extrapolation. In expressions (11), l is the shear modulus, na is the density of obstacles in slip system a, qa is the dislocation density in slip system a, b is the Burgers vector and r is a numerical coefficient of the order of 0.3 that modulates the strength of the obstacle in slip plane a given by a pair of forest dislocations separated a distance l. This strength is estimated as lb s ¼r l a

ð12Þ

In order to complete the constitutive formulation, evolution equations for the obstacle density na and dislocation density qa are provided. Evidently, na is a function of the dislocation densities in all remaining systems. The experimental work of Franciosi and co-workers [52–56] is suggestive of a dependence of the form X na ¼ aab qb ð13Þ

Table 1 Constitutive model parameters for pure aluminum Parameter

Value

Parameter

Value

C11 C44 S c_ 0 q0 b

108 GPa 28.5 GPa 135 · 103 J m2 10 s1 1012 m2 2.56 · 1010 m

C22 g0 m csat qsat l

61 GPa 2 MPa 0.1 0.5% 1015 m2 26 GPa

1. In boundary value problems, the description of the constitutive response at the single crystal level is completed when an orientation of the crystal in space is assigned. 3. Computational framework for modeling polycrystals The continuum finite element formulation employed in this work is described in [57]. In calculations, 10-node quadratic tetrahedra are employed to avoid volumetric locking problems. The set up of the boundary value problem is dependent upon the polycrystal modeling approach, which we describe in the following. In all cases, the analysis is performed with 559 randomly chosen initial orientations. 3.1. Taylor averaging and low-resolution simulation set up

b

Experimentally determined values of the interaction matrix aab have been given by Franciosi and Zaoui [52] for the 12 slip systems belonging to the family of {1 1 1} planes and [1 1 0] directions in fcc crystals. They classify the interactions according to whether the dislocations belong to the same system (interaction coefficient a0), fail to form junctions (interaction coefficient a1), form Hirth locks (interaction coefficient a1), co-planar junctions (interaction coefficient a1), glissile junctions (interaction coefficient a2), or sessile Lomer–Cottrell locks (interaction coefficient a3), with a0 6 a1 6 a2 6 a3. Franciosi [55] has also found the interaction coefficients to be linearly dependent on the stacking fault energy of the crystal, the degree of anisotropy increasing with decreasing stacking fault energy. Finally, an analytical expression for the evolution of qa with the applied slip strain can be postulated by considering that the dislocation production is dominated by multiplication by cross glide and dislocation annihilation is proportional to the probability of having two dislocation segments of different sign in a small neighborhood of each other. The resulting expression is given by     q0 ca =csat a q ¼ qsat 1  1  e ð14Þ qsat where qsat and csat are the saturation dislocation density and saturation shear slip, which are determined by the multiplication and annihilation rates. The values of the model parameters used in all the simulations presented in Section 4 are collected in Table

In the case of the Taylor-averaging approach, all the 559 orientations are assigned to each integration point of the mesh. This requires as many evaluations of the constitutive model using the local value of the macroscopic deformation gradient at each loading step followed by an averaging of the stress tensor. As a result, the computational cost of the simulation can be considerable. For this reason, a coarse mesh consisting of 64 elements was used for the Taylor-averaging simulation (Fig. 1a). In the case of low-resolution simulation (LRS), the orientation of each element was selected randomly from the same initial pool of 559 orientations and a mesh consisting of 3072 tetrahedral elements was employed (Fig. 1b). 3.2. DNS A challenging aspect of the DNS approach lies in adopting a representation of the three-dimensional grain shapes which is adequate for the intended purpose of the model. While it is common practice to use EBSD analysis to obtain grain geometry and orientation in two dimensions and to subsequently use this information to create finite element meshes [58,59], three dimensional non-destructive microstructure digital reconstruction is still in its infancy [60]. A number of strategies have been proposed for modeling the geometry and shape distribution of the crystalline grains resulting from the solidification process or from microstructural transformations [61,62]. The simplest approach consists of assuming isotropic and uniform grain growth during solidification which

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Fig. 1. Mesh configuration for (a) Taylor-averaging and (b) LRS.

Fig. 2. (a) Polycrystal mesh configuration used in DNS simulation consisting of 559 grains with unique orientations. The shading indicates the location of grain boundaries. (b) Individual grain geometry and mesh. (c) Cut through the polycrystal model revealing internal grain structure.

results in a grain structure given by the Voronoi diagram of the randomly specified grain seeds. The current state of the art lacks ad hoc tools to automatically produce finite element meshes conforming to such polyhedra. Thus, a further simplification is usually adopted in which a – usually structured – computational mesh is created independently. In general, this mesh violates grain boundary conformity. The grain geometry is included in the finite element mesh by assigning material orientations according to spatial location [63]. As a consequence, grain boundaries lose their flatness, which subsequently changes the computed deformation and stress fields. Therefore, the computed interaction among grains is also adversely affected. Due to the emphasis of this paper on the role of grain interaction in determining details of the texture evolution of polycrystals, we require our computational meshes to satisfy the following two conditions: finite elements should

conform to the grain boundaries and the geometry of the grain boundary facets should be flat. To this end, we adopt the simplifying assumption that the geometry of the grains consists of the space-filling polyhedron corresponding to the Wigner–Seitz cell of a bcc lattice also known in the literature as tetrakaidecahedron. The advantage of this approach lies in the ease of automatically generating a structured mesh of the unit grain, which can then be replicated maintaining the boundary conformity, Fig. 2b. In spite of its inability to represent realistic microstructures, this approach provides a convenient way to define idealized geometries retaining the basic properties expected in a real polycrystal. A similar approach using rhombic dodecahedra corresponding to the Wigner–Seitz cell of an fcc lattice was proposed in [37]. In the DNS simulation presented in the next sections, we adopt the grain layout shown in Fig. 2, which consists of

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Fig. 3. Comparison of the deformed meshes among the three methods (a) Taylor-averaging (b) LRS and (c) DNS.

559 grains, including those partial grains that are necessary to conform to the flat boundaries of the macroscopic sample. The grain boundaries have been emphasized in the figure with a special shading. Each one of the grains is discretized into 192 tetrahedral elements and assigned a unique crystallographic orientation chosen from the pool of orientations used in the Taylor-averaging and LRS simulations. 4. Numerical simulation results Using the computational framework described above, we conduct simulations of plane-strain compression of aluminum specimens up to 50% height reduction by the three different simulation methods. Fig. 3 compares the deformed meshes resulting from the three methodologies. The Taylor averaging calculation, Fig. 3a, results in an ostensibly uniform deformation which

is a consequence of the homogenization. In the case of the LRS approach, Fig. 3b, the deformation is highly heterogeneous due to the plastic anisotropy of the randomly oriented single-element grains, which is manifested by the roughening of the free surface. The DNS approach, Fig. 3c, not only captures the heterogeneity of the deformation but also resolves the details of the gradients of the deformation owing to the large number of degrees of freedom inside each grain. This reveals the localization of deformation which nucleates in grains of soft behavior and is not accessible to the LRS approach. Next, we investigate the final texture predicted by the three methods by comparing {1 1 1} pole figures and orientation distribution functions (ODF), Fig. 4, and fiber line analysis, Fig. 5. It is clear from Fig. 4 that the overall textures resulting from the different methods have similar qualitative features typically observed in rolling (plane-strain compression)

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Fig. 4. {1 1 1} pole figures and ODF predicted by three methods: (a) Taylor averaging, (b) LRS and (c) DNS.

textures of fcc metals with high stacking fault energy such as pure aluminum or copper. In particular, in all cases the ODF analysis reveals the presence of characteristic texture

components, which include Goss, (u1, U, u2) = (0, 45, 0), Brass, (u1, U, u2) = (35, 45, 0), Copper, (u1, U, u2) = (90, 35, 45) and S, (u1, U, u2) = (59, 37, 63). At a qualitative

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Fig. 5. Fiber line analysis: (a) a-line, (b) b-line and (c) position line.

level, the only difference in the observed texture results is in the increased orientation scatter and overall smoothness with the increase of in-grain resolution provided, in turn, by LRS and DNS. However, a quantitative analysis exposes significant differences among the three methods. A first quantitative indication of the orientation scatter provided by in-grain resolution is a reduction in the maximum ODF density from 17.8 in the Taylor averaging method to 10.9 in DNS. A clearer quantitative assessment of the differences in the details of texture prediction is provided by the fiber-line analysis shown in Fig. 5 which gives a comparison with experiments on aluminum [64] and copper [5]. Unfortunately,

these references do not provide the full ODF, only the fiber line analysis. Three main differences may be observed: (1) The Goss orientation density increases with in-grain resolution (Fig. 5a). The densities predicted by Taylor averaging, LRS and DNS are 1.02, 1.86 and 5.25, respectively. The DNS prediction is in good agreement with experimental observations. (2) The density along b-line predicted by the Taylor averaging method is much higher than that for the other two methods. As in-grain mesh resolution is increased, a dramatic decrease in density and a flatter b-line profile is observed (Fig. 5b).

Fig. 6. Deformed mesh after 50% plane strain compression obtained with the DNS method (a) and cross-sectional view (b). Color indicates the orientation change between initial orientation and deformed configurations.

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(3) As we increase the in-grain resolution, the Brass texture component becomes stronger. Although the variation in magnitude is slight, the tendency is apparent (Fig. 5a and b). In order to better understand these texture predictions using the DNS method, a more detailed analysis of the simulation results is conducted which includes a qualitative description of the observed grain deformation and misorientation patterns, a quantitative comparison of the occurrence of shear strains in the deformed sample and an investigation of the occurrence of in-grain orientation splitting and subdivision. Fig. 6a shows the deformed mesh after 50% plane strain compression obtained using the DNS method. The color indicates the local orientation change between initial and deformed configurations. The interior deformed structure

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is revealed in Fig. 6b, which shows a view of a cross-section of the sample through the mid-plane perpendicular to the transverse direction (TD). The heterogeneity of the deformation due to the strong interactions among the anisotropic grains is evident both from the overall large distortion of the individual grains and from the inhomogeneous misorientation map. It is important to highlight the role of shear deformation in providing the deformation mechanism to accommodate inter-grain incompatibility. This mechanism is only realized to its full magnitude in the DNS approach. A more quantitative statement can be made by observing Fig. 7, which shows a frequency histogram of the occurrence of shear strain components in the three simulation methods. The shear strains in the Taylor simulation remain zero due to the homogenized response under the uniform plane-strain compression macroscopic boundary

Fig. 7. Quantitative comparison of the occurrence of local shear strain components among the three polycrystal modeling approaches. The subscripts 1, 2 and 3 refer, respectively, to the rolling, transverse and normal directions. The histogram shows the volume fraction of the specimen experiencing shear strains of a given value.

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condition. In the case of LRS, the development of shear is severely constrained by the stiff behavior of each single-element grain, thus resulting in small shear strains (maximum value 15%) in only a small fraction of the volume (30– 35%). In the case of DNS, the presence of shear strains is greatest (65% of the volume) with maximum values above 30–40%. Another important observation in Fig. 7 is that the shear component e12, i.e. in the TD–RD plane, is completely missing in both the Taylor averaging and LRS simulations, whereas in the DNS simulation its magnitude reaches 5% and occurs in almost 20% of the sample. It will be discussed later that the occurrence of this shear strain component plays an important role in the formation of the Brass texture component.

In addition to capturing local shear strains, the DNS method has the potential of resolving in-grain deformation-induced orientation gradients which may provide a basis to rationalize some detailed aspects of the texture such as the evolution of the Goss component in fcc crystals. This will be further discussed in Section 5. One issue that has been raised before is the in-grain mesh resolution required to capture all the physics available in – albeit within the limitations of – CPFEM models [37]. Toward the objective of answering this question, we repeated the DNS simulation with two additional levels of mesh resolution. The finer meshes were obtained by subdividing each tetrahedral element in the original 192-element mesh into eight elements using a recursive algorithm that does not

Fig. 8. Influence of mesh refinement in the resolution of intra-grain deformation and misorientation gradients. (a) 192 elements per grain, (b) 1536 elements per grain and (c) 12,288 elements per grain. Color scale indicates misorientation.

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deteriorate the element quality in a significant manner. The meshes thus obtained comprised 1536 and 12,288 elements per grain, respectively. The finer meshes required largescale computations on the ASC/DOE ALC supercomputer. Fig. 8 shows the mesh of a generic grain extracted from the polycrystal assembly for all three cases as well as the corresponding deformed configurations obtained in each simulation. Color contours of orientation change overlaid on the deformed meshes show that only the highest resolution simulation provides a smooth representation of the gradients in the misorientation field, which is indicative of convergence or sufficient resolution. For the particular extracted grain, this level of simulation detail reveals that the influence of the neighboring grains appears to be concentrated next to the grain boundaries, thus suggesting a soft response. 5. Discussion The preceding simulation results demonstrate the ability and limitations of the three models in capturing detailed aspects of the texture evolution in fcc polycrystals. Specifically, quantitative results have been provided on aspects of the polycrystal response which we now in turn discuss. As noted in Section 1, it has been a long-standing observation that deformation textures simulated via homogenized approaches tend to be sharper than experimental ones [1]. This has been interpreted as a consequence of the strong restrictions posed by the models on the interactions among the grains. In the case of the LRS and DNS approaches, the only restriction imposed on the neighbor interactions is the compatibility of deformation and equilibrium of stress at the grain boundary. Thus, the crystals have a much wider spectrum of responses which can deviate from the macroscopic boundary conditions, resulting in smoother textures (Fig. 4). For full-resolution DNS, in addition, the intra-grain deformation is spatially resolved, resulting in a continuous variation of the lattice rotations inside the grains and, thus, in smoother textures. These effects were quantified in the b-line analysis (Fig. 5b), which shows that the serious overprediction of Taylor averaging is brought to bear with the experiments by both the LRS and DNS methods. DNS gives a better match towards the two ends of the b-lines (Brass and Copper components) due to the resolved intra-grain gradients. In particular, the Brass texture component usually observed in cold-rolled fcc metals has been associated with the presence of shear strains in the TD–RD plane [8,5,31]. It was shown in Fig. 7 that DNS captures the development of shear strains of this type e12 of up to 10% in almost 20% of the sample, whereas in LRS and Taylor averaging they are virtually absent. This explains that the prediction of the Brass component of the texture by DNS is closer to experiments than the other methods and that its density increases with the in-grain mesh resolution (Fig. 5). It has been previously shown that the inability of the LRS and

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Taylor averaging methods to reproduce the Brass component is exacerbated for high reduction [7]. The prediction of the Goss texture component merits a separate discussion. It has been known experimentally that the density of the Goss component initially increases with the level of compression, reaches a peak value around 50% reduction and subsequently decreases until it finally disappears [5,64]. It is also known that, owing to its high symmetry with respect to the loading, the Goss orientation is stable in fcc metals provided there is no exterior interference from neighboring grains [7]. As a particular implication of this, Sachs model seriously overpredicts the Goss density. By contrast, the Taylor model is unable to capture the evolution of this component due to the severe restrictions imposed by the grain interaction assumptions. This external vulnerability of the Goss component can be explained by its relatively soft behavior [65]. The three polycrystal models presented reflect this response and, perhaps, provide some additional insight. It is clear from Fig. 5a that both Taylor averaging and LRS fail to describe the density peak of the Goss component at 50% reduction observed in experiments, whereas DNS succeeds. This can be rationalized by the ability of DNS to provide some shielding effect from the influence of the neighborhood inside the grain through the development of intra-grain plastic gradients (Fig. 8). As the deformation increases, it is expected that the volume of the grain influenced by its neighbors will grow towards the center at which point the Goss component will become unstable and disappear. 6. Summary and conclusions A comparison of three polycrystal modeling approaches based on combining a crystal plasticity constitutive model and the finite element method in various ways has been presented. Particular emphasis has been given to the assessment of the DNS approach, in which an attempt is made to resolve intra as well as inter-grain gradients in the solution fields. It is found that DNS enables the description of details of the texture evolution which are strongly dependent on the heterogeneous response and, thus, inaccessible to either homogenization or low-resolution approaches. In particular, the following main conclusions are drawn:  Increasing the in-grain mesh resolution has the overall effect of smoothing the predicted textures and improving the agreement with experimental results.  The increased description of heterogeneity provided by DNS method allows for the appearance of shear strain components in the simulations. In particular, the development of shear in the TD–RD plane results in increased density of the Brass component.  DNS successfully captures the peak density of the Goss component at 50% reduction observed in experiments which is virtually missing in the alternative methods

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