Delineation of first-order closures for plastic properties requiring explicit consideration of strain hardening and crystallographic texture evolution

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International Journal of Plasticity 24 (2008) 327–342

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Delineation of first-order closures for plasticproperties requiring explicit consideration of strainhardening and crystallographic texture evolution

Marko Knezevic a, Surya R. Kalidindi a,*, Raja K. Mishra b

a Department of Materials Science and Engineering, Drexel University, Philadelphia, PA 19104, USAb General Motors Research and Development Center, Warren, MI 48090, USA

Received 19 January 2007; received in final revised form 4 May 2007Available online 25 May 2007

Abstract

Microstructure Sensitive Design (MSD) is a novel mathematical framework that facilitates devel-opment of invertible linkages between statistical description of the material’s microstructure and itseffective properties. Property closures are an important outcome of the MSD methodology, anddelineate the complete set of theoretically feasible effective (homogenized) anisotropic property com-binations in a given material system for a selected homogenization theory. In recent publications, wehave reported first-order closures for the elastic and yield properties of both cubic and hexagonalpolycrystalline materials. In this paper, we present major extensions to the previously reportedframework to enable rigorous consideration of strain hardening and the concomitant evolution ofthe crystallographic texture with imposed plastic strain. These new extensions facilitate delineationof first-order closures for properties associated with finite plastic strains (e.g. ultimate tensilestrength, uniform ductility). The proposed approach has been successfully applied to an aluminumalloy and a copper alloy, and the results are presented and discussed in this paper.� 2007 Elsevier Ltd. All rights reserved.

Keywords: A. Ductility; A. Microstructures; B. Crystal plasticity; C. Numerical algorithms; Property closures

0749-6419/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2007.05.002

* Corresponding author. Tel.: +1 215 895 1311; fax: +1 215 895 6760.E-mail address: [email protected] (S.R. Kalidindi).

mailto:[email protected]

328 M. Knezevic et al. / International Journal of Plasticity 24 (2008) 327–342

1. Introduction

Property closures delineate the complete set of theoretically feasible macroscale(homogenized) anisotropic property combinations in a given material system, and are oftremendous interest in optimizing the performance of engineering components. Histori-cally, this problem has been referred to as the G-closure problem by the applied mathe-matics community (Cherkaev, 2000; Cherkaev and Gibiansky, 1996; Lurie, 2004; Muratand Tartar, 1985). To date, G-closures have been obtained only for a limited set oftwo-dimensional microstructures comprised of isotropic phases and have been largelyfocused on properties such as effective conductivity and elastic stiffness. In recent years,a novel spectral framework called Microstructure Sensitive Design (MSD) (Adamset al., 2001; Adams et al., 2004b; Kalidindi et al., 2004; Lyon and Adams, 2004) was pro-posed and demonstrated to provide approximations to the G-closures for a number ofcombinations of the elastic properties and yield properties of polycrystalline materials(Knezevic and Kalidindi, 2007; Proust and Kalidindi, 2006; Wu et al., 2007) and two-phase composites (Adams et al., 2004a; Kalidindi and Houskamp, in press).

In MSD, we start with a statistical description of the material microstructure using localstate distribution functions. The simplest of these, called 1-point distributions, reflect theprobability densities associated with realizing specified distinct local states (may be definedas a combination of several microstructural variables that are measurable at the lengthscale of interest) in the immediate neighborhood of a point thrown randomly into themicrostructure. Higher order descriptions, called n-point spatial correlation functions,are also possible (Brown, 1955; Torquato, 2002). These distributions are then used toestablish quantitative linkages between microstructure and macroscale properties (Adamset al., 2005; Beran, 1968; Kalidindi et al., 2006a; Kroner, 1977; Lyon and Adams, 2004;Proust and Kalidindi, 2006; Torquato, 2002). A salient aspect of MSD is that these link-ages are transformed into an efficient Fourier space, resulting in two main constructs: (i) amicrostructure hull that includes the complete set of theoretically feasible statistical distri-butions describing the important details of the microstructure, and (ii) delineation of prop-erty closures for a selected homogenization theory. The primary advantages of the MSDapproach lie in its (a) consideration of anisotropy of the properties at the local lengthscales, (b) exploration of the complete set of relevant microstructures leading to globaloptima, and (c) invertibility of the microstructure–property relationships.

All of the previous reports in the literature on delineation of property closures havefocused on a class of properties that treat the microstructure as being static. In consider-ation of a broader class of plastic properties of metals, we immediately encounter twoimportant features: (i) strain hardening and (ii) concurrent evolution of microstructuredue to plastic strain. Prime examples of such properties include the uniform ductilityand the ultimate tensile strength. Because of their influence on the toughness exhibitedby the material, these properties play an important role in materials selection for criticalstructural components. Since these properties directly influence the formability (andthereby the success of certain deformation processing operations), they are also of tremen-dous interest for deformation processing of metals. In order to successfully delineate abroader class of plastic property closures, the framework we presented in previous work(Knezevic and Kalidindi, 2007; Proust and Kalidindi, 2006; Wu et al., 2007) needed tobe extended to allow for the evolution of the associated local state variables. In recentwork, we demonstrated that the predictive capabilities of the Taylor-type crystal plasticity

M. Knezevic et al. / International Journal of Plasticity 24 (2008) 327–342 329

model for the evolution of the crystallographic texture and the anisotropic stress–strainresponse of polycrystalline fcc metals (with strain hardening) can be efficiently capturedin the spectral framework of MSD (Kalidindi et al., 2006b). The main goal of this workis to combine these recent advances in spectral representation of microstructure–prop-erty–processing relationships and produce for the first time a new class of property clo-sures that address plastic properties at finite strains (requiring explicit consideration ofstrain hardening and texture evolution).

In this paper, our focus is on property closures involving combinations of yieldstrength, ultimate tensile strength, uniform ductility, and the Lankford parameter R forfcc metals. The main microstructural detail considered in obtaining these closures is thecrystallographic texture (1-point distribution of lattice orientation in the microstructure;also referred to as orientation distribution function or simply ODF) and its evolution.Strain hardening plays a dominant role in some of the properties listed above. Highly non-linear material behavior, in conjunction with multidimensional Fourier space representingthe properties and textures, demands new mathematical approaches for delineation ofproperty closures. A novel mathematical approach has been established and has beenapplied successfully to several examples of property closures for two single-phase fcc met-als: oxygen-free-high-conductivity (OFHC) copper and Al 5754-O aluminum alloy sheets.These results will be described in detail in the rest of this paper.

2. Brief review of Taylor-type crystal plasticity model

Since the crystal plasticity modelling framework is at the core of the current paper,some of the main details are summarized below using a notation that is now standardin modern continuum mechanics textbooks. For finite deformations, the total deformationgradient tensor (F) can be decomposed into elastic and plastic components as (Asaro,1983)

F ¼ F�Fp; ð1Þwhere F* contains deformation gradients due to both elastic stretching as well as the latticerotation, while Fp denotes the deformation gradient due to plastic deformation alone. Theconstitutive equation for stress in the crystal can be expressed as (Kalidindi et al., 1992)

T� ¼ C½E��; T� ¼ F��1fðdet F�ÞTgF��T ; E� ¼ 1

2fF�T F� � 1g; ð2Þ

where C is the fourth-order elasticity tensor, T* and E* are a pair of work conjugate stressand strain measures, and T is the Cauchy stress in the crystal. The evolution of Fp can beexpressed in a rate form as

_Fp ¼ LpFp; Lp ¼X

a

_caSao; Sa

o ¼ mao � na

o; ð3Þ

where _ca is the shearing rate on the slip system a, and denote the slip direction and the slipplane normal of the slip system, respectively, in the initial unloaded configuration. For thefcc metals studied here, the family of f111g 1�10

� �slip systems were considered as poten-

tial slip systems. The shearing rate on the slip system is dependent on the resolved shearstress (sa) on the slip system and the slip resistance (sa) of that slip system, and can be ex-pressed in a power–law relationship (Asaro and Needleman, 1985) as


_ca ¼ _co

sa

sa

��1=m

sgnðsaÞ; sa � T� � Sao: ð4Þ

In Eq. (4), _co denotes a reference value of the slip rate (taken here as 0.001 s�1) and m rep-resents the strain rate sensitivity parameter (taken here as 0.01 to capture the behavior ofmost metals at low homologous temperatures). In the present study, for simplicity, wehave adopted a saturation type hardening law to describe the evolution of the slipresistances:

_sa ¼ ho 1� sa

ss

� �aXb

j _cbj: ð5Þ

The slip hardening parameters, ho, ss, and a, in Eq. (5) for annealed Al 5754-O and OFHCcopper have been established by calibrating the Taylor model predictions to experimentalmeasurements using procedures described in our earlier paper (Kalidindi et al., 1992). Thevalues for OFHC copper were obtained as ho = 180 MPa, ss = 148 MPa, a = 2.25, andso = 16 MPa. The values for Al 5754-O were obtained as ho = 745 MPa, ss = 130 MPa,a = 1.81, and so = 17 MPa. Note that these hardening parameters are expected to bestrongly influenced by composition (e.g. purity levels) and the grain size distribution inthe metal. Note also that the hardening law used here does not reflect any latent hardening(i.e. it assumes equal hardening of all slip systems). The extension of the spectral frame-work presented here to include latent hardening is relatively straightforward, and has beenavoided here for improved clarity of the presentation of the main new concepts. The latticespin W* in the crystal is given by

W� ¼Wapp �Wp; Wp ¼ 1

2ðLp � LpTÞ; ð6Þ

where Wapp is the applied overall spin on the polycrystal and Wp is the plastic spin (skew-symmetric component of Lp defined in Eq. (3)). The numerical procedures for the integra-tion of this elastic–viscoplastic constitutive model have been described in detail in the priorliterature (Kalidindi et al., 1992).

The most widely used approach to obtain the response of a polycrystal from theresponse of the individual grains is to use the extended Taylor’s assumption of iso-defor-mation gradient in all of the crystals comprising the polycrystal. This model has enjoyedremarkable successes in predicting both the anisotropic stress–strain response and the evo-lution of the underlying texture in single-phase medium to high stacking fault energy cubicmetals subjected to finite plastic strains in a broad range of deformation paths (e.g. Beaud-oin et al., 1995; Beaudoin et al., 1993; Bronkhorst et al., 1992; Delannay et al., 2002;Mathur and Dawson, 1989; Van Houtte et al., 2002).

In evaluating tensile yield strength ry1 using the Taylor-type model, the following mac-roscopic isochoric velocity gradient is imposed on each crystal:

L ¼ L ¼_�e 0 0

0 �q_�e 0

0 0 �ð1� qÞ_�e

0B@

1CA; ð7Þ

where q can take any value between 0 and 1. The local stresses, r0ijðqÞ, computed by theTaylor model are largely deviatoric. The hydrostatic component is computed by establish-


ing the value of q (denoted as q*) for which the averaged lateral stresses over the polycrys-tal are equal to each other, i.e. �r033ðq�Þ ¼ �r022ðq�Þ. The tensile yield strength of the polycrys-tal in the e1-direction is then computed as

ry1 ¼ �r011ðq�Þ � �r022ðq�Þ: ð8ÞIn this study, the yield strength is defined to correspond to a plastic strain of 0.2 % at animposed strain rate of 0.001 s�1.

The R1-ratio represents the ratio of the true width strain to the true thickness strain in atensile test and is a prime example of a macroscale plastic property of the metal that istypically of interest in sheet metal forming operations. The R1 value (corresponding to ten-sile loading in the e1-axis) can then be defined as

R1 ¼q�

1� q�: ð9Þ

Uniform ductility (eu) and the ultimate tensile strength (rUTS) are defined using the Con-sidere’s criterion (Considere, 1885). To facilitate the computation of these variables fromTaylor-type models, the evolution of the true stress–true strain curve in tension is analyzedto establish the point of necking as

drde

��e¼en

¼ rn; ð10Þ

where rn and en denote the true stress and true strain at the point of necking. The uniformductility, eu, and the ultimate tensile strength, rUTS, are then easily computed as

eu ¼ expðenÞ � 1; rUTS ¼rn

1þ en

: ð11Þ

3. Microstructure sensitive design framework

As mentioned earlier, our primary focus in this paper is on the crystal lattice orientationas the primary descriptor of the local state in single-phase polycrystalline metallic systems.The crystal orientation, g, is defined here using a set of three Bunge–Euler angles (Bunge,1993), i.e. g = (u1, U, u2). The local state space describing the complete set of physicallydistinct orientations relevant to a selected class of textures is referred to as the fundamen-tal zone (FZ). In this paper, we further restrict our attention to cubic–orthorhombic1 tex-tures. The 1-point statistics of the distribution of lattice orientations in the polycrystallinemetal (also referred to as the crystallographic texture or the orientation distribution func-tion or simply the ODF) is denoted as f(g), and reflects the normalized probability densityassociated with the occurrence of the crystallographic orientation g in the sample. ODF isformally defined as

dV g

V¼ f ðgÞdg;

ZFZ

f ðgÞdg ¼ 1:0; ð12Þ

1 The first symmetry in this notation refers to symmetry at the crystal level (resulting from the atomicarrangements in the crystal lattice) while the second refers to symmetry at the sample scale (resulting fromprocessing history).


where V denotes the total sample volume and dVg is the sum of all sub-volume elements inthe sample that are associated with a lattice orientation that lies within an incrementalinvariant measure, dg, of the orientation of interest, g. Cubic–orthorhombic texturescan be expressed efficiently in a Fourier series using generalized spherical harmonic(GSH) functions (Bunge, 1993) as

f ðgÞ ¼X1l¼0

XMðlÞl¼1

XNðlÞm¼1

Clml T

::lml ðgÞ; NðlÞ ¼ l

2þ 1 ð13Þ

where T::

lml ðgÞ denote the symmetrized cubic–orthorhombic GSH functions and Clm

l coeffi-cients represent uniquely the ODF. Note also that Eq. (13) allows the visualization ofODF as a single point in an infinite dimensional Fourier space (coordinates given byClm

l ). The set of all such points, corresponding to the complete set of all physically realiz-able2 ODFs, is called the texture hull in the MSD framework and has been depicted in sev-eral prior publications (Adams et al., 2001; Kalidindi et al., 2004; Lyon and Adams, 2004).It should be noted that any physically realizable texture has to have a representation insidethe texture hull.

The MSD approach brings together the microstructure description with the corre-sponding homogenization theory and establishes mathematical procedures to delineateapproximations to the closures for selected combinations of macroscale properties (Adamset al., 2005; Kalidindi et al., 2006a; Lyon and Adams, 2004). As mentioned earlier, ourfocus here is on the Taylor-type crystal plasticity model. In order to facilitate the delinea-tion of plastic property closures using this model, it is essential to formulate spectral link-ages that capture efficiently the main results of the Taylor-type crystal plasticity model. Inrecent work (Kalidindi et al., 2006b), we have demonstrated that the local stresses, the lat-tice spins, and the accumulated slip rates in individual crystals can be expressed as a func-tion of their lattice orientation for a specified monotonic deformation path using the GSHfunctions. For the deformation path of interest here (described by Eq. (7)), these relation-ships can be expressed as

W �ijðg; qÞ ¼

X1l¼0

XMðlÞl¼1

XN 0ðlÞm¼1

ijAlml ðqÞ

__T lml ðgÞ; N 0ðlÞ ¼ 2lþ 1 ð14Þ

r0iiðg; qÞ ¼ sX1l¼0

XMðlÞl¼1

XNðlÞm¼1

y1iiSlm

l ðqÞT::

lml ðgÞ; ð15Þ

Xa

_caj jðg; qÞ ¼X1l¼0

XMðlÞl¼1

XNðlÞm¼1

Glml ðqÞT

::lml ðgÞ: ð16Þ

Note that s in Eq. (15) represents the slip resistance (see Eq. (5)). In Eqs. (14)–(16), we haveformulated Fourier representations for the three independent components of the skew-symmetric W �

ijðg; qÞ, the three diagonal components of the deviatoric stress r0ijðg; qÞ, andfor the sum of the absolute values of the slip rates on the different slip systems in the crys-talP

aj _cajðg; qÞ. The variables in Eqs. (14)–(16) have been selected because they constitute

2 The term physically realizable texture is used here to refer to a texture that can be physically described orimagined. It is anticipated that a large number of these are not yet achievable in practice by currently knownmanufacturing options.


the essential information needed to predict the texture evolution and the anisotropicstress–strain behavior of cubic–orthorhombic polycrystals in tensile deformation, consis-tent with the Taylor-type crystal plasticity theory.

It should be noted that cubic–triclinic GSH functions, __T lml ðgÞ (Bunge, 1993), have been

used for the lattice spin components in Eq. (14), whereas the cubic–orthorhombic GSHfunctions have been used for the other variables in Eqs. (15) and (16). This is becausethe lattice spin components for individual crystals do not exhibit orthorhombic symmetryat the sample level, while the diagonal components of the deviatoric stress tensor and thesum of the absolute slip rates in the individual crystals do exhibit the orthorhombic sym-metry at the sample level for the selected deformation path. It should be noted that therepresentation using cubic–orthorhombic GSH functions requires substantially lowernumber of terms compared to the representation using cubic–triclinic GSH functions.For example, for representations to l = 12 used in this study, the cubic–orthorhombicexpansion contains 37 real Fourier coefficients, whereas the cubic–triclinic expansion tothe same level contains 130 complex Fourier coefficients. Needless to say, cubic–ortho-rhombic expansions are preferred whenever they are applicable.

The numerical procedures used to compute the Fourier coefficients in Eqs. (14)–(16)have been described in our prior papers (Kalidindi et al., 2006b). As an example,

ijAlml ðqÞ can be computed as

ijAlml ðqÞ ¼ ð2lþ 1Þ

ZFZ

W �ijðg; qÞ

__T lml ðgÞdg; ð17Þ

where the bar on top indicates a complex conjugate. In order to numerically evaluate theintegral in Eq. (17) (say, using the Simpson method (e.g. Press et al., 2002)), one needs toselect a distribution of grain orientations in the cubic–triclinic fundamental zone of the ori-entation space (Bunge, 1993) and compute the values of W �

ijðg; qÞ for each of the selectedorientation using the Taylor-type model for a small increment of the selected deformationmode (for a specified q value in Eq. (7)). Applying similar procedures for computing theFourier coefficients in Eqs. (15) and (16), one can obtain a highly efficient spectral calibra-tion of the Taylor-type model. These spectral linkages have been extensively validated inour prior work (Kalidindi et al., 2006b).

In this work, the values of the Fourier coefficients ijAlml ðqÞ, y1

iiSlml ðqÞ, and Glm

l ðqÞ wereestablished and utilized to l = 12 (note that in prior work (Proust and Kalidindi, 2006)we have only used coefficients up to l=8). At this level of truncation, we found that theaverage error between the spectral representation (due to truncation of the series) andthe Taylor-type model predictions for a broad set of single crystals and polycrystals wasless than 3 %. Furthermore, ijA

lml ðqÞ, y1


l ðqÞ have been computed for discretevalues of q, in increments of 0.1 in the range 0.0–1.0, and linearly interpolated for values inbetween. Note that the ijA

lml ðqÞ, y1


l ðqÞ coefficients are the same for all fccmetals in which slip occurs on the f111gh1�1 0i slip systems and the slip hardening canbe described with the simple isotropic slip hardening law (see Eq. (5)). Furthermore, theseFourier coefficients are also independent of the texture in the polycrystalline sample. Themain advantage of the spectral linkages described in Eqs. (14)–(16) is that once the Fouriercoefficients are computed (a one time computational cost which typically takes severalhours on a regular desktop PC), all subsequent simulations are extremely fast because theyonly involve evaluation of the series in Eqs. (14)–(16). Eqs. (14)–(16) can be used in arecursive manner to simulate large plastic strains on the polycrystals. Eqs. (8), (15), and


(11) can then been used to compute the various macroscale properties of interest for anygiven texture.

4. Property closures

In prior work (Kalidindi et al., 2006a; Knezevic and Kalidindi, 2007; Proust and Kalid-indi, 2006; Wu et al., 2007), we have successfully explored various approaches to delineat-ing the property closures using the spectral representations described above. Thesemethods have typically entailed searching the texture hull (representing the complete setof feasible ODFs) for optimized combinations of selected properties that lie on the bound-ary of the desired closures. In other words, the search for each desired boundary point onthe closure was formulated as a nonlinear constrained optimization problem, and solu-tions were successfully obtained using techniques such as sequential quadratic program-ming (e.g. Das and Dennis, 1998; Kim and de Weck, 2005; Lyon and Adams, 2004;Proust and Kalidindi, 2006) and Pareto-front solutions (e.g. Fullwood et al., 2007). Notethat all of the properties addressed in prior work were properties associated with a staticmicrostructure (i.e. the evolution of texture with imposed deformation and concomitantstrain hardening were not considered). In the present work, we found that these earlierapproaches could not be extended in simple ways for properties that need to explicitlyaccount for microstructure evolution. We have therefore explored a novel scheme to delin-eate the property closures of interest here.

The first-order3 closures presented in this paper differ substantially from those pub-lished previously by showing the set of theoretically feasible plastic property combinationswhile taking into account strain hardening and concurrent texture evolution due to plasticstrain. We present here results for two fcc metals: oxygen-free high-conductivity (OFHC)copper, and Al 5754-O. Below we provide three specific examples of closures and discussthem. It should, however, be noted that the mathematical framework presented here isquite general and can be applied to other desired combinations of properties of interest.

The methodology explored here for building first-order closures starts with a consider-ation of a set of points in the texture hull that correspond to ‘‘eigen textures” (Adamset al., 2005). Because of the orthorhombic sample symmetry used in this study, each eigentexture corresponds to the texture produced by a set of four equi-volume single crystalsthat are selected to satisfy the orthorhombic symmetry at the sample scale. The set of eigentextures is selected while ensuring an adequate coverage of the fundamental zone (FZ).The property combinations for these eigen textures are first evaluated using the spectrallinkages described in the previous section. Fig. 1 shows an example of a closure for thefeasible combinations of ry1 and rUTS (both these properties are defined along the samplee1 direction), based solely on the consideration of the eigen textures. A finite number oftextures corresponding to the boundary of this closure were then selected. The macroscaleproperty combinations for the weighted combinations of these textures, taking one pair oftextures at a time, were evaluated systematically (considering all possible combinations ofpairs of the selected textures). As expected, these computations revealed that some of theproperty combinations outside the eigen-texture closure were indeed possible, i.e. the

3 The closures produced here are referred to as first-order closures to remind us that they are based on the first-order Taylor-type model.

Fig. 1. First-order closures for the ultimate tensile strength (rUTS) and the yield strength (ry1) in OFHC Cu and5754-O Al based on Taylor-type models and only a consideration of the eigen textures (covering the cubic–orthorhombic fundamental zone).


results expanded the closure. Once again a new set of textures corresponding to the newboundary of the expanded closure were selected (this time these were a mixture of eigentextures and non-eigen textures) and the property combinations corresponding to theweighted combinations of these (for all possible pairs of selected textures) were evaluatedto see if they further expanded the closure. This process was repeated until the closure didnot expand in any discernable way. Fig. 2 shows such an expanded closure for ry1 andrUTS1. Compared to the eigen-texture closure shown in Fig. 1, the expanded closure inFig. 2 is larger and more convex. The method described above to produce a closure essen-tially follows the main ideas underlying genetic algorithms, where good solutions are pre-selected (as we have done here by selecting the textures producing property combinationson the boundary of the closure) and ‘‘mutations” or ‘‘cross-overs” (weighted combinationsof textures in our approach) are explored. A similar approach was also recently appliedsuccessfully to delineate elastic–plastic closures in hcp metals (Wu et al., 2007), where eventhe consideration of static microstructures required the use of a very large number of Fou-rier dimensions and precluded the use of the simpler optimization techniques mentionedearlier.

The first-order closure for the ultimate tensile strength (rUTS) and the yield strength(ry1) in Fig. 2 is expected to be of interest to mechanical designers as they seek superiorcombinations of these two properties in their designs. It is observed that the closuresfor the two metals selected in this study are strongly influenced by the strain hardeningparameters. The slight vertical translation of the 5754-O Al closure with respect to theOFHC Copper closure can be easily explained based on the slightly higher value of theinitial slip resistance used for 5754-O Al (17 MPa) compared to the value used for OFHCCopper (16 MPa). The significant horizontal translation of the 5754-O Al closure com-pared to the closure for OFHC Copper in Fig. 2 has to be attributed to the differencesin the slip hardening parameters for these two metals. It is especially noteworthy that5754-O Al exhibits higher values of rUTS in spite of a lower saturation value of slip resis-tance (130 MPa for 5754-O Al versus 148 MPa for OFHC Copper). These higher valuesrUTS are attributed to the significantly higher strain hardening rates in 5754-O Al

Fig. 2. First-order closures for the ultimate tensile strength (rUTS) and the yield strength (ry1) in OFHC Cu and5754-O Al based on Taylor-type models and a consideration of all theoretically possible textures (i.e. the elementsof the texture hull). The textures that are theoretically predicted to correspond to salient points of interest on theboundary are depicted. Textures A, C, and E are for OFHC Copper, while textures B, D, and F are for 5754-OAl.


compared to OFHC Copper (ho is 180 MPa for OFHC Copper while it is 745 MPa for5754-O Al).

Textures corresponding to corners A and B are theoretically identified to possess thecombination of the highest values of ry1 and rUTS feasible in the materials studied here.It should, however, be noted that these predictions are based on the selected homogeniza-tion theory (Taylor-type model) and the prescribed slip hardening parameters. Examplesof textures corresponding to corners A and B are presented in Fig. 2. It is seen that themain texture component providing this optimal combination of properties appears to bethe ð111Þ½1�10� in both materials, and is not a major component in any of the widely useddeformation processing operations by the metal working industry. On the other hand,


materials processing specialists relate the ratio of the yield strength to the ultimate tensilestrength of the metal to its strain hardening response. Generally, a lower value of ry1 /rUTS

is correlated to higher capacity for plastic deformation. For example, textures correspond-ing to points C and D in Fig. 2, for OFHC copper and 5754-O Al, respectively, exhibit thelowest value of ry1/rUTS in these metals. The dominant texture component in both tex-tures C and D is close to the ð110Þ½1�10� orientation. The slight differences between thesetextures and their influence on the properties of interest will be discussed shortly. Texturescorresponding to points E and F in Fig. 2 provide much lower values of both ry1 andrUTS, and may be of interest to processing specialists because of the lower load require-ments in metal shaping operations (provided they exhibit adequate ductility). The domi-nant texture component in both textures E and F is close to the ð121Þ½0�12� orientation,while there are again subtle differences between them.

The subtle differences in the optimal textures for the two different metals studied herereflect the important role of strain hardening in influencing strength. To illustrate this, wehave calculated the properties of 5754-O Al corresponding to salient textures A, C, and E(these were initially identified on the closure for OFHC Copper). Likewise, we have alsocomputed the properties of OFHC Copper corresponding to salient textures B, D, and F(initially identified on the closure for 5754-O Al). The property combinations for all tex-tures A through F are shown in Fig. 3 on closures for both metals. It is seen that the dif-ferences between textures A and B have very little effect on the ry1 and rUTS of these twometals, while the differences between textures C and D and between textures E and F resultin significant differences in the ry1 and rUTS exhibited by these two metals. These obser-vations confirm our expectation that the strain hardening parameters do play a significantrole on the class of plastic closures presented in this paper.

Fig. 4 depicts property closures for uniform ductility (eu) and ultimate tensile strength(rUTS), both defined along the e1 direction of the sample. These specific property combi-nations are of interest in developing high strength high-toughness alloys for structuralapplications. Also, in metal forming operations, uniform ductility (eu) is given specialimportance because it reflects workability of the metal. The textures predicted to exhibit

Fig. 3. Influence of the subtle differences in the textures on the ultimate tensile strength (rUTS) and the yieldstrength (ry1) exhibited by the metal. The properties corresponding to pairs of slightly different textures (A, B),(C, D), and (E, F) are shown on closures for 5754-O Al and OFHC Copper.

Fig. 4. Closure for the uniform ductility (eu) and ultimate tensile strength (rUTS) in two cubic metals based onTaylor-type models, and a consideration of all theoretically possible textures (i.e. the elements of the texture hull).The textures that are theoretically predicted to correspond to salient points of interest on the boundary aredepicted.


superior combinations of ultimate tensile strength and uniform ductility lie on the B–Hand A–G boundaries of the closures shown in Fig. 4. Different points on these boundariesprovide different trade-offs in the achievable combinations of ultimate tensile strength anduniform ductility in the two alloys studies. The main component in textures G and H isclose to ð1 10Þ½1�1 0� orientation, whereas the textures A and B have already been discussed(see Fig. 2). Although textures G and H appear to be similar to the textures C and D dis-cussed earlier (corresponding to the lowest values of ry1/rUTS in Fig. 2), there are indeedsignificant differences between these textures. In order to illustrate these differences, theproperty combinations corresponding to texture C are also shown on the closure forOFHC Copper in Fig. 4. It is clearly observed that the property combinations exhibitedby textures C and G are substantially different from each other.

As a final illustration of the potential benefits of the mathematical framework presentedin this paper, we present a different kind of example. Here we consider a closure for prop-erties that are defined at yield, and therefore there is no need to consider the evolution of

Fig. 5. Closure for the R1-ratio and the yield strength (ry1) in 5754-O Al based on Taylor-type model for alltheoretically possible textures. The desired texture is I. The evolution of textures and properties in typicaldeformation processes are also shown for three initial textures. These processes do not transform the properties inthe desired direction.


texture or strain hardening in delineating the closure. In other words, the closure itselfcould have been obtained using the methods we described in earlier papers for any mate-rial for which the strain hardening characteristics are known. Fig. 5, shows a closure forthe R1-ratio and the yield strength for 5754-O Al that was presented in earlier work (Knez-evic and Kalidindi, 2007). This closure is of interest to the sheet metal forming industrywhere the goal is to maximize the workability of the material (high R1-ratios) while keep-ing the yield strength low. We now explore here a range of processing paths in this closureto examine if the desired combination of properties can be obtained using several differentstarting textures and a set of readily available deformation processing options. The math-ematical framework presented in this paper allows us to very quickly evaluate the evolu-tion of the properties of interest during any imposed deformation path. Here, we haveselected three different initial textures and subjected them to two different deformation


paths. The initial textures, chosen for this study, include a random texture, a (110) fibertexture, and a (100)[0 01] cube texture, shown as J, K, and L, respectively, in Fig. 5. Thesetextures were subjected to plane-strain rolling and simple compression deformation. Theevolution of the properties of interest during the selected deformation processes is depictedin Fig. 5. It is observed that none of the combinations of initial textures and deformationpaths selected produced a substantial increase in the R1 value. The texture that is theoret-ically predicted to produce a high value of R1 is shown as texture I in Fig. 5. This domi-nant component in this desired texture is ð221Þ½1�10�, and is not seen as a major texturecomponent in any of the deformation processing operations typically used by the metalworking industry.

All of the examples presented here have highlighted the need and potential for thedevelopment of novel processing routes resulting in superior combinations of properties.

5. Conclusions

In this study, we have presented extensions to the MSD framework that facilitate delin-eation of a new class of property closures. This new class of closures deals with propertiesassociated with finite plastic strains (e.g. ultimate tensile strength and uniform ductility)and requires an explicit consideration of strain hardening and the concomitant evolutionof the crystallographic texture. A new mathematical procedure for successful delineationof these closures has been described and demonstrated.

Several examples of closures for selected combinations of plastic properties wereobtained for two specific metal alloys. It was seen that the closures obtained were sensitiveto the slip hardening characteristics exhibited by the alloys. Furthermore, it was alsoobserved that the best textures that correspond to the optimized combinations of macro-scale plastic properties were somewhat different for the two metals studied. This impliesthat different processing routes would be needed in the different metals to achieve the bestpossible performance. All of the examples presented in this study highlight the clear needand potential for the development of novel processing routes resulting in superior combi-nations of plastic properties in metallic alloys.

Acknowledgements

Financial support for this work was provided by the Army Research Office (ProposalNo. 46886 MS, Dr. David Stepp, Program Director) and by General Motors Researchand Development, Warren, MI.

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Delineation of first-order closures for plastic properties requiring explicit consideration of strain hardening and crystallographic texture evolution