A study of surface roughening in fcc metals using direct numerical simulation

Acta Materialia 52 (2004) 5791–5804 www.actamat-journals.com

A study of surface roughening in fcc metals using direct numerical simulation Z. Zhao a, 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 19 July 2004; received in revised form 25 August 2004; accepted 26 August 2004 Available online 30 September 2004

Abstract Grain-scale surface roughening due to plastic straining in polycrystalline aluminum is studied with the aid of a three-dimensional finite-element crystal-plasticity model. An improved understanding of the origin of surface roughening profiles in plastically strained aluminum is sought. Large-scale, Direct Numerical Simulation enables the computation of full-field solutions and the explicit consideration of the deformation of individual crystals as well as of their crystalline texture evolution and interaction with neighboring grains. Simulations are conducted on idealized flat sheet polycrystalline aluminum samples under uniaxial and biaxial loading conditions. The results obtained show that the ensuing surface profiles are controlled by several factors: applied boundary conditions, Taylor factor and shear tendency of the individual grains and the spatial distribution of grain neighborhood orientations. The conditions leading to different ridging profiles, e.g., corrugated and ribbed surface profiles, are discussed.
2004 Published by Elsevier Ltd on behalf of Acta Materialia Inc. Keywords: Surface roughening; Crystal plasticity; Finite element simulation

1. Introduction Surface roughening is a common phenomenon observed in plastically deformed polycrystalline metals, as undesirable in industrial practice as seemingly unavoidable. For instance, the development of this type of defects can produce an unsmooth appearance of the surface of formed sheet metal and provide initiation sites for strain localization [1]. It is commonly accepted that this phenomenon originates in the heterogeneity of the polycrystalline material, whose grains have an anisotropic elastic and plastic behavior and different crystal orientations leading to – grain boundary impeded – incompatibilities of deformation arising from the interactions between neighboring grains. At the free surface, *

Corresponding author. Tel.: +1 617 252 1518; fax: +1 617 253 0361. E-mail address: [email protected] (R. Radovitzky).

these incompatibilities do not arise, which results in differential deformations in the direction normal to the surface (ND) and, therefore, in grain-scale roughening. Despite this basic understanding of the reasons causing grain-scale roughening, the details of how this phenomenon occurs and the factors determining the resulting roughening patterns are not yet properly understood or quantitatively characterized. In general, surface roughening that originates in the microstructure – as opposed to friction, contact or other sources – can be grouped into two phenomena: orange peel and ridging. Orange peel is characterized by outof-plane displacements which roughly map the surface grain shapes and is commonly rationalized as originating when the spatial distribution of orientations gives rise to grain interactions which are predominantly local. The ridging phenomenon, also known as roping, is a collective deformation of larger sets of grains typically resulting in a banded surface topology. Ridging often

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2004 Published by Elsevier Ltd on behalf of Acta Materialia Inc. doi:10.1016/j.actamat.2004.08.037

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occurs in the form of corrugated surfaces which typically extend along the rolling direction (RD) of the stretched metal sheet [2]. Early experimental observations showed that the overall roughness decreases linearly with grain size and increases linearly with plastic strain, at least for a limited range of strains [1]. However, the dependence of roughness on strain is expected to deviate from a linear relation at large plastic strains. Chao [3] and Wright [2] studied the dependence of roughness on the experimental boundary conditions. They found that the loading direction relative to the RD of the sample had a significant influence on the roughness and ridging profiles, which suggested a connection between roughness and crystallographic texture. In an effort to establish this connection, they conducted X-ray texture analysis and investigated the influence of macrotexture on roughness. Modern experimental techniques, e.g. EBSD, have enabled the characterization of microtexture and grain morphology and have, therefore, found widespread application in the analysis of grain-scale surface roughness [4–9]. These studies have suggested that the individual orientation (texture component) and the spatial arrangement of the grains play an important role on the deformation mechanisms leading to surface roughening. In fcc materials, the Cube texture component appears as soft compared to other components due to its high crystallographic symmetry and low Taylor factor (M). Therefore, surface grains with this orientation are expected to stretch and sink into valleys in the roughened surface. On the other hand, S or other b fiber textures are much harder due to their higher Taylor factor [5,6] and, therefore, expected to protrude from the surface. The Goss orientation is an extreme example of an orientation with strong anisotropy whose Taylor factor changes from MRD = 2.45 to MTD = 4.9 when the load changes from RD to transverse direction (TD). It has been reported to play an important role in the formation of ridging [7,8]. Numerical simulation based on crystal-plasticity finite element models has arisen as a useful tool to complement experimental studies of surface roughening [4,5,8,9]. Within the limitations of the modeling assumptions, the value of numerical simulation lies in enabling the investigation of the independent and combined influence of the different factors affecting roughening, which cannot be done easily experimentally. Becker [4] pioneered the application of finite-element crystal-plasticity models to the analysis of surface roughening. The simulations presented in that study suggest that strain localization at the surface and, thus, strain hardening, crystallographic texture and material homogeneity play an important role in surface roughening. The analysis was limited to two-dimensional generalized plain strain conditions due to the computing hardware and software limitations of the time. The simulation results provided

useful insights and gave reasonable predictions of experimentally measured roughness. However, the inadequacy of the two-dimensional assumption and the need for high-resolution three-dimensional models to improve the predictive quality of the simulations were also recognized. Beaudoin et al. [5] presented the first three-dimensional simulations of surface roughening. Their results show that grains of similar orientations can act collectively to define the roughening profile. In particular, they support the idea that textural banding is a source of ridging. Another important observation made in that study is that grain interactions play a significant role in the deformation of surface grains and, therefore, that conclusions assuming an average behavior may be unfounded or even misleading. In particular, some of the grains of Cube orientation which, on the basis of the low Taylor factor, are expected to behave softly and therefore sink to create the surface valleys, show deviations from this behavior. In a recent study on bcc steel, Shin et al. [9] showed that different combination of texture components can lead to different ridging shapes in bcc materials. Their work established a connection between ridging profiles in steel and the existence of certain texture components. For bcc materials, the ridging profile has a close relationship with three different texture components: {0 0 1} Æ1 1 0 æ, {1 1 1} Æ1 1 0æ and {1 1 2} Æ1 1 0æ. Their simulations show that shear deformation tendency as well as the arrangement of these texture components can influence the ridging shape considerably. In this work, the role played by different factors known to influence surface roughening profiles is analyzed with the aid of a crystal plasticity finite element model. Toward this end, we propose a computational strategy for modeling the polycrystalline mechanical response accounting for the deformation of the individual grains as well as for their interactions. We adopt a ‘‘fullfield’’ solution approach in which the details of the mechanical fields resulting from grain interactions are accurately resolved by recourse to large-scale simulation. The polycrystal is modeled as a continuum with discontinuities of lattice orientation at grain boundaries. The computational approach is based on a Lagrangian large-deformation finite-element formulation of the continuum problem. Idealized equiaxed polycrystal topologies respecting the ostensible flatness of grain boundaries are adopted in calculations. 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 approxima-

Z. Zhao et al. / Acta Materialia 52 (2004) 5791–5804

tion – as opposed to 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 [10]. The resulting computational framework is applied to the investigation of the influences of specific crystallographic texture components and their spatial arrangement, as well as of different macroscopic boundary conditions, in determining ridging profiles in plastically deformed pure aluminum sheets. This paper is organized as follows: in Section 2 we briefly review the continuum formulation. In Section 3, we review the general constitutive framework and the crystal plasticity hardening model. This is followed by a description of the numerical formulation and simulation setup in Section 4. Section 5 presents the results of the numerical simulations. A discussion of these results is presented in Section 6. We conclude this paper with a short summary of conclusions in Section 7.

2. Continuum field equations in the Lagrangian framework For completeness, we start by briefly describing the general Lagrangian form of the continuum field equations assumed to govern the dynamic deformation of polycrystals. The formulation accounts for finite kinematics, inertia and general constitutive behavior. We select the configuration B0
Rd of the body at time t0 as the reference configuration. The coordinates X of points in B0 are used to identify material particles throughout the motion. The motion of the body is described by the deformation mapping, x ¼ uðX; tÞ;

X 2 B0 :

ð1Þ

Thus, x is the position of material particle X at time t. We shall denote by Bt the deformed configuration of the body at time t. The material velocity and acceleration _ € ðX; tÞ, X2B0, fields follow from (1) as uðX; tÞ and u respectively, where a superposed (.) denotes partial differentiation with respect to time at fixed X. The local deformation of an infinitesimal material neighborhood is described by the deformation gradient, F ¼ r0 uðX; tÞ;

X 2 B0 ;

ð2Þ

where $0 denotes the material gradient of a function defined over B0. Thus, the components of $0f are the partial derivatives of f with respect to X. The scalar function, J ¼ det ðFðX; tÞÞ;

ð3Þ

is the Jacobian of the deformation, and measures the ratio of the deformed to undeformed volume of an infinitesimal material neighborhood. The local form of linear momentum balance is

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€  r0  P ¼ q0 B; q0 u

in B0 ;

ð4Þ

where q0 is the mass density in the reference configuration, B(X,t) are the body forces per unit mass, and P(X,t) is the first Piola–Kirchhoff stress tensor. The Cauchy stress tensor follows from P through the relation, r ¼ J 1 PFT :

ð5Þ

The formulation of the initial boundary-value problem requires the specification of appropriate boundary conditions,
; u¼u

ð6Þ

on oB01 ;


ðX; tÞ is the prescribed deformation mapping where u and oB01 is the portion of the boundary oB0 where the displacements are prescribed,
P  N ¼ T; on oB02 ; ð7Þ
where TðX; tÞ are the prescribed tractions, N is the unit outward normal and oB02 is the portion of the boundary oB0 where tractions are prescribed; and the initial conditions: uðX; 0Þ ¼ u0 ðXÞ;

ð8Þ

_ uðX; 0Þ ¼ u_ 0 ðXÞ:

ð9Þ

3. Constitutive framework 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 [11], this points to a multiplicative decomposition, F ¼ Fe Fp ;

ð10Þ

of the deformation gradient F into a plastic part Fp, which accounts for the cumulative effect of dislocation motion, and an elastic part Fe, which accounts for the remaining non-plastic deformation. Following Teodosiu [12] and others [13–17], 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 (10). By virtue of (10), the deformation power per unit undeformed volume takes the form e

p


: F_ þ R
;
:L P : F_ ¼ P

ð11Þ

where
p ¼ F_ p Fp1 :
¼ PFpT ; R
¼ FeT PFpT ; L ð12Þ P
defines a first Piola–Kirchhoff stress tensor relHere, P
t , and R
a stress ative to the intermediate configuration B
p on measure conjugate to the plastic velocity gradients L
t . The work conjugacy relations expressed in (11) B

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suggest plastic flow rules and elastic stress–strain relations of the general form,
P

p ðR;
¼ PðF
e ; QÞ:
p ¼ L
QÞ L

ð13Þ


denotes some suitable set of internal variables Here, Q 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 (11) consistent with the principle of material frame indifference is
C
e Þ;
¼ Fe Sð P


e ¼ FeT Fe ; C

ð14Þ


¼C
e1 R
is a symmetric second Piola–Kirchhoff where S stress tensor relative to the intermediate configuration
e is the elastic right Cauchy–Green deforma
t , and C B
t . For most applications involving mettion tensor on B
and als, a linear – but anisotropic – relation between S
e  IÞ=2 can be as
e ¼ ðC the elastic Lagrangian strain E sumed without much loss of generality. Higher-order moduli are given by Teodosiu [18]. From the kinematics of dislocation motion, it has been shown by Taylor [19] and Rice [17] that (13) is of the form, X
p ¼
a; L ð15Þ c_ a
sa  m a


a where c_ a is the shear rate on slip system a and
sa and m are 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; ð16Þ which is an extension of SchmidÕs rule. If (16) is assumed to hold, then it was shown by Rice [17] and by Mandel [16] that the flow rule (15) derives from a viscoplastic potential. In order to complete the constitutive description of the crystal, hardening relations governing the evolution
need to be provided. In this of the internal variables Q work, we adopt the forest dislocation hardening model for fcc metals of Cuitin˜o et al. [20]. 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 a < if sa P 0; c_ 0 gsa a c_ ¼ ð17Þ : 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 (17) 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. 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 haa c_ a ; ð18Þ g_ a ¼ a

where haa (no sum in a) 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 ð19Þ In this expression, pffiffiffiffiffiffiffi bqa sac ¼ rlb pna and cac ¼ pffiffiffiffiaffi 2 n

no sum in a

ð20Þ

are a characteristic shear stress and strain for the slip system a, respectively. The values of sac and cac determine the location of the ÔbendÕ 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 (20), 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 sa ¼ r

lb : l

ð21Þ

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 [21–25] is suggestive of a dependence of the form X na ¼ aab qb : ð22Þ b

Experimentally determined values of the interaction matrix aab have been given by Franciosi and Zaoui [21] 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), coplanar junctions (interaction coefficient a1), glissile junctions (interaction coefficient a2), or sessile LomerCottrell locks (interaction coefficient a3), with a0 6 a1 6 a2 6 a3. Franciosi [24] has also found the

Z. Zhao et al. / Acta Materialia 52 (2004) 5791–5804 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

€ n are where the subscript n refers to time tn, and u_ n and u the material velocity and acceleration fields. Details on the constitutive update algorithm which enable the computation of the stresses at time tn + 1 are given elsewhere [20]. The spatial finite-element discretization of the timediscretized linear momentum balance equation (26) is based on the weak form, Z Z € nþ1  v dX þ q0 u Pnþ1 : r0 v dX B0

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 dislocations 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 ð23Þ 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 5 are collected in Table 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.

4. Numerical formulation The preceding field equations may be rendered into a form suitable for computation by a combination of a time discretization of the momentum and constitutive equations and a finite-element discretization of the reference configuration of the solid. Some key aspects of the particular approach adopted here are summarized next for completeness and later reference. More detailed accounts may be found elsewhere [26]. Here and subsequently we envision an incremental solution procedure aimed at sampling the solution at discrete times t0, . . ., tn,tn + 1 = tn + Dt, . . . The linearmomentum balance equation (4) is discretized in time by recourse to the second-order accurate explicit central-difference time-stepping algorithm: unþ1 ¼ un þ Dt u_ n þ

Dt2 € ; u 2 n

ð24Þ

€ n; u_ nþ1 ¼ u_ n þ Dt u

ð25Þ

€ nþ1  r0  Pnþ1 ¼ q0 Bnþ1 ; q0 u

ð26Þ

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¼

Z

B0


 v dS þ T

oB02

Z

q0 Bnþ1  v dX;

8v 2 V ;

ð27Þ

B0

where V is the space of admissible displacements, i.e., such incremental displacements, or, alternatively, velocities, that satisfy the essential boundary conditions (6) in the sense of traces. This weak statement is also known as the principle of virtual work. We consider finite-element interpolations of the form, uh ðXÞ ¼

N X

xa N a ðXÞ;

ð28Þ

a¼1

where uh is the deformation mapping interpolant; Na are the displacement shape functions, respectively; the sum on a ranges over the N nodes in the mesh, whereas the sum on e ranges over the E elements in the mesh. The displacement shape functions Na must be conforming. In calculations we employ standard quadratic ten-node tetrahedral elements [27]. 4.1. Model setup One of the remaining challenges in the modeling of polycrystals lies in the realistic three-dimensional representation of the actual polycrystal grain structure. Whereas 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 [4,28], a three-dimensional counterpart of this technology is not readily available. In order to fulfill this necessity, a number of strategies have been proposed for modelling the geometry and shape distribution of the crystalline grains resulting from the solidification process or from microstructural transformations [29,30]. The simplest approach consists of assuming isotropic and uniform grain growth during solidification which results in a grain structure given by the Voronoi diagram of the randomly specified grain seeds. Due to the lack of ad hoc tools to automatically produce finite element meshes that conform to the convex Voronoi polyhedra, a further simplification is usually adopted in which a – usually structured – computational mesh is created independently, violating the conformity at grain boundaries. The grain geometry is induced in the finite element mesh by conveniently

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assigning material orientations according to spatial location [5,9,31]. As a consequence, grain boundaries lose their flatness, which subsequently affects the computed deformation and stress fields and, therefore, the computed interaction among grains. Due to the importance of grain interaction in determining surface roughening, we require of 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 spacefilling polyhedra corresponding to the Wigner–Seitz cell of a bcc lattice. 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. 1. In spite of its obvious 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 [32]. In the roughening simulations presented in the next sections, we adopt the grain layout shown in Fig. 1. The geometry of the sample is 6 mm wide, 6 mm long and 1 mm thick and consists of 134 grains, including those partial grains that are necessary to conform to the flat boundary. It is important to note that references to absolute lengths in this article are given for the sole purpose of defining the geometry of the calculations. It should not be interpreted that the simulations presented are size dependent, as the model proposed does not in-

clude such effects. In all cases studied in this article, we assume the effects of body forces such as gravity are negligible, and thus, we set B(X,t) to zero. Each one of the grains is discretized into 192 tetrahedral elements and assigned a crystallographic orientation. The grooves in these figures indicate the location of grain boundaries. The idealized polycrystal model introduced in Fig. 1 can be effectively used in the investigation of the orange peel roughening mechanism when combined with random initial textures. In the case of ridging, an important role is played by clusters of similarly oriented grains aligned with the sheet RD. This preferred direction normally results from the discontinuous and simultaneous nucleation of similarly oriented new grains in one original hot or cold band grain or by continuous subgrain coarsening [33]. Based on this observation, in our studies of ridging three different grain arrangements were considered to represent the grain cluster topology, Fig. 2. In all three arrangements, the samples comprise two types of grain columns aligned along RD, which are differentiated in the figure by light and dark gray colors. A different crystallographic orientation is then assigned to each type of grain column. The crosssectional view emphasizes the differences between these three types of spatial arrangement. The first arrangement (Type-I) approximates a bamboo arrangement in which the grain clusters penetrate through the thickness, Fig. 2(a). The second arrangement (Type-II) corresponds to a sheared bamboo shape, Fig. 2(b). In the third arrangement (Type-III), the graycolored grain columns are only loosely connected, approximating the case of a given orientation immersed in a matrix of a different orientation.

Fig. 1. Polycrystal mesh configuration. (a) A total of 134 grains are used to create a sample with an aspect ratio 6:6:1. The grooves indicate the location of grain boundaries. (b) Part of the top buffer layer is removed to reveal the interior arrangement of the grains. (c) Individual grain shape.

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Fig. 2. Three types of grain arrangement are used in the simulation of ridging profiles. A different orientation is assigned to the light and dark gray grains to define the grain clusters. (a) Type-I arrangement is characterized by a bamboo shape in the cross-section while (b) Type-II has a sheared bamboo shape. (c) Type-III arrangement corresponds to an arrangement of banded clusters (dark gray grains) immersed in a matrix (light gray grains).

5. Numerical simulations of surface roughening In this section, we apply the computational framework presented in the foregoing to the investigation of the influence of key factors suspected of affecting the development of surface roughening profiles in aluminum sheets. These factors include the loading direction (boundary conditions), the kinematic hardening (Taylor factor), the shear tendency and the spatial arrangement of specific texture components and their interaction. 5.1. Simulations of orange peel One hundred and thirty four orientations were selected from a set of random orientations and assigned to each of the grains in the polycrystal model shown in Fig. 1. Three boundary conditions were considered: uniaxial stretch along two different orthogonal directions and biaxial stretch. Twenty percent elongation was reached in each case of uniaxial stretch and ten percent in each direction in the case of biaxial stretch. The resulting surface topographies are shown in Fig. 3. In these figures, the color scale indicates final z-location relative to the original z-location of the top surface in millimeters. Typical orange peel profiles can be observed for all three boundary conditions. It can also be observed in these figures that the specific roughening profile obtained differs depending on the boundary condition ap-

plied. This behavior can be rationalized in terms of the different single crystal response (Taylor factor and shear tendency) of each grain orientation to different loading directions, leading to different neighborhood interactions. In order to ascertain the importance of the role played by the Taylor factor, in Fig. 4 we plot its value for each grain for the case of stretching along RD as calculated from full constraints Taylor model. It can be clearly seen in this figure that there is no direct relationship between the Taylor factor of the individual grains and the positions of the valleys and hills. This suggests that the Taylor factor alone cannot fully explain surface roughening and that other aspects of the anisotropic behavior of the individual crystals, as well as their interactions, play an important role in determining it. It may also be observed in these figures that, as a consequence of grain interactions, hills and valleys in the resulting surface topography always involve several grains. This observation is consistent with previous experimental observations [8], which show a collective behavior of surface grains in contributing to surface roughness. 5.2. Ridging: surface roughening in the presence of banded grain clusters In this section, we investigate the roughness profiles arising when the grains adopt a banded cluster arrangement. The Cube orientation (Euler angles: u1 = 0,

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Fig. 3. Simulation of orange peel. The initial texture is random. The profiles shown correspond to three different boundary conditions: (a) uniaxial stretch along RD, (b) uniaxial along TD and (c) biaxial stretch. The color scale indicates final z-location relative to the original z-location of the top surface in millimeters.

U = 0, u2 = 0) is a predominant texture component in hot-rolled and cold-rolled annealed aluminum sheets. As discussed in previous work [6], it always plays an important role in the formation of surface roughness in fcc materials. Ridging profiles arising from the combination of banded Cube clusters with four other common texture components: Goss (u1 = 0, U = 45, u2 = 0), S

(u1 = 60, U = 32, u2 = 65), Brass (u1 = 35, U = 45, u2 = 0) and Copper (u1 = 90, U = 35, u2 = 45) are simulated using the idealized Type-I polycrystal model described in Section 4.1 and shown in Fig. 2(a). The light gray grains are assigned the Cube orientation and the dark gray grains are assigned the second orientation under consideration. The samples were subjected to the three different boundary conditions described in Section 5.1. Fig. 5 shows the resulting ridging profiles. It can be observed in these figures that ridging is significantly dependent on the loading direction. The roughness generated by stretching along RD is relatively small compared to TD. Furthermore, ridging occurs when stretching in TD irrespective of the texture component combined with the Cube component. As shown in Fig. 5(a), the combination of Cube and Goss orientations results in a flat surface when stretched along the RD but exhibits strong ridging under TD elongation. By contrast, when stretching along RD, the combinations Cube–Brass, Cube–Copper and Cube–S, mild roughness develops and, again, the effect of stretching in TD is to exacerbate this ridging tendency, Fig. 5(b)–(d). Fig. 6 shows the cross-sections of the deformed samples, allowing to inspect the resulting roughness on both the top and bottom surfaces. It can be clearly observed in these figures that the roughness patterns corresponding to stretch along TD exhibit a synchronized ridging profile between the top and bottom surfaces. In the first three orientation combinations, Fig. 6(a)–(c), the profiles are symmetric, leading to the so-called ribbed shape. In the case of the Cube– S combination, Fig. 6(d), the sections along RD appear to displace rigidly but nonuniformly in ND, leading to a corrugated shape. The inhomogeneities in the roughening patterns at the boundaries are due to the boundary conditions

Fig. 4. Relationship between Taylor factor and surface roughening profiles. (a) Taylor factor mapping in the undeformed mesh. (b) Surface roughening profiles after 20% stretch along RD.

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Fig. 5. Ridging profiles resulting from different texture combinations and boundary conditions. Figures in the first column correspond to uniaxial stretch along RD; in the second column along TD and in the third column to biaxial stretch. The different rows correspond to the different orientation pairs considered: (a) Cube and Goss, (b) Cube and Brass, (c) Cube and Copper, (d) Cube and S. The color scale indicates final z-location relative to the original z-location of the top surface in millimeters.

which do not preclude the shearing tendency of some of the texture components. In particular, this out-ofplane shearing appears only in those cases in which the grain orientation exhibits a shear tendency, (Brass, Copper, S, Fig. 6(b)–(d)) and not in the case of the Cube–Goss combination, Fig. 6(a), both of whose components are stable to shape changes due to shear. Finally, the roughening patterns obtained from biaxial stretch exhibit qualitatively similar features to the cases of RD and TD stretching combined.

5.3. Influence of shear tendency on ridging profiles The S orientation is a common deformation texture component in fcc metals that has a strong tendency to shear in the ND-TD plane when plastically stretched in the RD–TD plane due to its plastic anisotropy [5]. As is well known, hot and cold rolled metal sheets adopt a 222 sample symmetry, which results in four crystallographically equivalent S orientations: S1 (u1 = 60, U = 32, u2 = 65), S2 (u1 = 300, U = 148, u2 = 245), S3 (u1 = 120, U = 148, u2 = 245) and S4 (u1 = 240,

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Fig. 6. Cross-sections of the ridging profiles in Fig. 5). The combination of Cube with either Goss (a), Brass (b) or Copper (c) exhibits a ribbed-shape profile. The combination of Cube with S exhibits a corrugated-shape profile.

U = 32, u2 = 65). In particular, these four orientations share the same Taylor factors MRD = 3.46 when stretched along RD and MTD = 3.27 when stretched along TD. Yet, the shear tendency is not the same in all of them and, for some pairs of orientations (S1–S2 and S3–S4), the shear tendency is, in fact, opposite. This provides an additional reason to suspect that the Taylor factor alone may not suffice for characterizing the contribution of specific texture components to roughening. In this section, we investigate the influence of shear tendency of texture components on surface roughening. Toward this end, we assign the S1 and S2 orientations to the grain arrangement in Fig. 2(a) and conduct uniaxial (both along RD and TD) and biaxial stretch simulations as before. Fig. 7 shows the ridging profiles obtained. It is clear from these figures that a corrugated shape is common to the three boundary conditions and that this tendency is more pronounced for uniaxial TD stretching. An important feature observed in these profiles is that the roughening protrusions and depressions take place at the S1–S2 grain boundaries on the surface, which indicates that the corrugated shape is caused by the opposite shear deformation of the two different S orientations. The boundary effects observed in Fig. 7(a) and (c) are due to the boundary-impeded shear tendency on the RD–ND plane of the S1 and S2 orientations under RD stretching. 5.4. Influence of grain spatial distribution on ridging profiles The in-plane and through-thickness spatial distribution or arrangement of grain orientations is expected to play an important role in determining surface rough-

ening profiles. We explore its influence by conducting simulations on the three different arrangements presented in Fig. 2. Two types of texture component combinations were used: (i) Cube and Goss, (ii) Cube and S (u1 = 60, U = 32, u2 = 65). The boundary conditions considered were restricted to TD stretching. Fig. 8 shows the simulated surface profiles obtained after 20% elongation. The following observations can be made: in the case of the combination of the Cube and Goss orientations, Fig. 8(a), the Type-I arrangement shows a clear ribbed-shape ridging profile, whereas in the Type-III arrangement the resulting profile is clearly corrugated. In the case of the Type-II arrangement the roughness developed is considerably lower and does not show a clear profile shape. In the case of the combination of the Cube and S orientation, a dominant corrugatedshape profile results regardless of the type of arrangement, Fig. 8(b).

6. Discussion Among the causes of roughness investigated, the Taylor factor, as a measure of the ability of a given grain orientation to accommodate plastic flow or grain anisotropic kinematic hardening, is shown to play a very important role. Thus, some of the ridging profiles obtained can be entirely rationalized on the basis of the Taylor factors of the orientations involved. The emblematic example of this behavior corresponds to the RD-banded clusters of Cube and Goss orientations stretched along either RD and TD presented in Section 5.2. These orientations were chosen because, the Goss

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Fig. 8. Influence of grain spatial distribution on ridging profiles. Figure (a) shows the three arrangements considered. The profiles shown are obtained after 20% elongation along TD for three different types of grain arrangement combining two different texture orientations: (b) Cube and Goss; (c) Cube and S. The light gray grains correspond to the Cube orientation.

Fig. 7. Ridging profiles resulting from the combination of two crystallographically equivalent – but mechanically different – orientations: S1 (u1 = 60, U = 32, u2 = 65), S2 (u1 = 300, U = 148, u2 = 245). The different figures correspond to the three different boundary conditions considered: (a) RD, (b) TD and (c) biaxial.

orientation presents a large plastic anisotropy, i.e. its Taylor factor as calculated from full constraints Taylor theory changes from MRD = 2.45 under RD stretching to MTD = 4.9 under TD stretching, whereas the Cube orientation preserves the same value of MTD = MRD = 2.45 under either loading direction. The matching Taylor factors in the combined sample when loading along RD results in no roughness, whereas their difference when loading along TD explains the significant ridging obtained. Furthermore, the deformation localizes in the softer Cube-oriented clusters, resulting in a ribbed-shape ridging profile. This behavior was previously pointed out by Raabe [8]. It bears emphasis that in this case neither orientation exhibits a tendency to shear out of the plane of the sheet.

It was also found that surface roughening may deviate from the behavior expected from Taylor-factor considerations alone. This was illustrated in the simulations of orange peel presented in Section 5.1, which showed a lack of correlation between the Taylor factors of the randomly chosen orientations and the locations of the protrusions and depressions in the roughened surface. This is an indication that other aspects of the grainsÕ anisotropy such as the shear tendency and the interactions among the grains may play an important role in determining surface roughening. Modified versions of the Taylor factor including local deformation information have been proposed [34,28] in order to take into account the influence of the grainsÕ neighborhood interaction. In order to investigate the influence of shear tendency, we considered the ridging profiles produced in the extreme case of combining two orientations with the same Taylor factor but opposite shear tendency, Section 5.3. In that case, the slip systems of the two equivalent S orientations considered have mirror symmetry and an opposite shear tendency when stretching along TD. The deformed samples exhibit significant ridging

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without any strain localization at the cluster boundaries, an indication that there is no hardening mismatch. The peaks and valleys coincide with the grain cluster boundaries, which supports the explanation of ridging in terms of shear incompatibility in this case. In order to provide additional support for the statements above, we analyzed the single and bi-crystal behavior of the Cube, Goss and S orientations, Figs. 9 and 10. Fig. 9(a) and (b) show that both the single crystal Cube and Goss orientations remain flat after TD stretching, whereas Fig. 9(c) shows that the symmetric (ribbed) ridging pattern originates in the localized deformation of the Cube clusters. Fig. 10(a) and (b) show that the two equivalent S orientations (S1 and S2) shear outside of the plane in opposite directions when stretched along TD as single crystals and, by contrast, lead to that corrugated profile shown in Fig. 10(c) when combined, without any localization. Pole figures showing the initial

and final bicrystal textures accompany these plots in Figs. 9(d) and 10(d). Boundary (loading) conditions are found to have a significant influence on the development of roughness. Most importantly, it is found that different roughening profiles, both qualitatively and in overall roughening magnitude, can be obtained under uniaxial tension along TD and RD in the presence of banded grain clusters along RD, Fig. 5. These results are in agreement with recently published experimental findings [7]. When loading along TD, soft and hard band clusters along RD will experience the same level of stress, leading to more elongation and thinning of the soft clusters. Therefore, a ridging profile with hard clusters protruding from the surface and soft clusters sinking on the surface to form valleys develops. By contrast, when loading along RD, soft and hard band clusters must elongate by the same amount, resulting in much weaker levels of roughness

Fig. 9. Single and bi-crystal behavior of Cube and Goss orientations. (a) Cube and (b) Goss oriented single crystals exhibit no shear tendency. (c) Ribbed-shape ridging profiles appearing in the combined periodic bi-crystal showing the localization of strain in the Cube grains. (d) Corresponding initial and final bi-crystal textures.

Fig. 10. Single and bi-crystal behavior of S1 and S2 orientations. (a) S1 and (b) S2 oriented single crystals exhibit opposite shear tendency. (c) Ridging profiles appearing in the combined periodic bi-crystal is characterized by corrugated upper and bottom surfaces. (d) Corresponding initial and final bi-crystal textures.

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and, presumably, in tensile (compressive) residual stresses in the hard (soft) clusters. Another factor investigated was the influence of the spatial arrangement of grain clusters. It was found that the ridging profile changes fundamentally (from ribbed to corrugated shape), Fig. 8(a), by re-arranging the spatial distribution of grain-clusters. It was also found that this change did not take place in the case of shear incompatibility, Fig. 8(b). This supports the idea that in real fcc materials, in which grain orientations deviate from the ideal texture components, the shape stability of the Cube and Goss components and their corresponding ribbed-shape ridging profiles is never found [7].

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shape. With randomly distribution of different oriented grains, roughening are dominated by the orange peel shape.

Acknowledgement The support of DOE through Caltech ASCI Center for the Simulation of the Dynamic Response of Materials is gratefully acknowledged. The authors are grateful to Stephen Kuchnicki for his useful contributions to improving the robustness of the crystal plasticity model. We also wish to thank Dierk Raabe for helpful comments.

7. Summary and conclusions We have presented a numerical study of grain-scale surface roughening due to plastic straining in polycrystalline aluminum. The numerical model is based on a crystal-plasticity finite-element formulation, which explicitly accounts for the deformation of individual grains as well as of their interaction. The simulations show that the surface profiles obtained are the synthetic result of the hardening difference and shear compatibility between neighboring grains, the spatial distribution of similarly oriented grains and the loading conditions applied on the samples. Some of the specific features affecting roughness observed in simulations include:  samples with grain clusters aligned along RD exhibit significantly less roughness when deformed along RD than when deformed algon TD. In the latter case, localized deformation in the lower Taylor factor regions develops.  the Goss orientation has a strong in-plane anisotropy when strained along RD and TD. Sheet combined with Cube oriented and Goss oriented grain-clusters shows pronounced ribbed ridging profiles when stretch along TD direction, which results from the hardness difference between these two texture components and also their same shape stability during uniaxial and biaxial straining. As a kinematic hardness index, the Taylor factor serves as a good explanation for the origin of this kind of roughening.  corrugated roughness profiles may occur through the combination of kinematically equivalent S orientations (i.e. same Taylor factors). In this case, the roughening mechanism originates in the opposite shear tendency of each texture components.  spatial distribution of grains or same oriented grainclusters plays an important role on the surface roughening shapes. Different arrangements of the same predominant texture components may lead to a change of roughening profile from a ribbed to a corrugated

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