Numerical simulations of necking during tensile deformation of … · 2008. 2. 19. · UNCORRECTED PROOF 1 2 Numerical simulations of necking during 3 tensile deformation of aluminum

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Available online at www.sciencedirect.com

INTPLA 1100 No. of Pages 21, Model 1+

17 January 2008; Disk Used ARTICLE IN PRESS

International Journal of Plasticity xxx (2008) xxx–xxx

www.elsevier.com/locate/ijplas

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FNumerical simulations of necking duringtensile deformation of aluminum single crystals

F. Zhang a,*, A.F. Bower a, R.K. Mishra b, K.P. Boyle c

a Division of Engineering, Brown University, Providence, RI 02912, USAb Materials and Processes Laboratory, GM R&D Center, Warren, MI 48090, USA

c CANMET Materials Technology Laboratory, Ottawa, ON, Canada K1A 0G1

Received 16 October 2007; received in final revised form 13 December 2007

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Finite element simulations are used to study strain localization during uniaxial tensile straining ofa single crystal with properties representative of pure Al. The crystal is modeled using a constitutiveequation incorporating self- and latent-hardening. The simulations are used to investigate the influ-ence of the initial orientation of the loading axis relative to the crystal, as well as the hardening andstrain rate sensitivity of the crystal on the strain to localization. We find that (i) the specimen fails bydiffuse necking for strain rate exponents m < 100, and a sharp neck for m > 100. (ii) The strain tolocalization is a decreasing function of m for m < 100, and is relatively insensitive to m form > 100. (iii) The strain to localization is a minimum when the tensile axis is close to (but not exactlyparallel to) a high symmetry direction such as [100] or [111] and the variation of the strain to local-ization with orientation is highly sensitive to the strain rate exponent and latent-hardening behaviorof the crystal. This behavior can be explained in terms of changes in the active slip systems as theinitial orientation of the crystal is varied.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Localization; B. Crystal plasticity; C. Finite element

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

doi:10.1016/j.ijplas.2007.12.006

* Corresponding author. Tel.: +1 401 863 2674; fax: +1 401 863 9009.E-mail address: [email protected] (F. Zhang).

Please cite this article in press as: Zhang, F. et al., Numerical simulations of necking during ...,Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2007.12.006

mailto:[email protected]

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2 F. Zhang et al. / International Journal of Plasticity xxx (2008) xxx–xxx



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1. Introduction

Strain localization limits the formability of many ductile materials in processes such aspunching, drawing, and hemming. This is a particular concern to the automotive industry,where efforts to replace steel with lightweight Al and Mg alloys are impeded by the lowerformability of the replacement materials (Taub et al., 2007). Complex parts such as dooror lift-gate inner panels either need to be redesigned to accommodate the lower formabilityof Al and Mg alloys at room temperature, or else must be manufactured using moreexpensive and slower processes such as elevated temperature Quick Plastic Forming (Taubet al., 2007). These demands are inspiring a renewed interest in the problem of strain local-ization during plastic flow. A principal goal of this effort is to determine the roles of alloycomposition and microstructure in controlling flow localization, with a view to improvingformability through alloy and process design.

Forming limits depend on the part geometry, loading conditions (especially deforma-tion rate, temperature, and the presence and type of lubrication), and the material prop-erties. Nevertheless, tensile tests are frequently used as a rapid means to estimate theability of a material to resist shear localization. The roles of material properties, specimengeometry, and loading conditions in controlling localization during a tensile test have beenextensively studied. Early work (Considere, 1888; Hill and Hutchinson, 1975; Hutchinsonand Neale, 1977) based on classical plasticity theory, focused on the influence of strainhardening and strain rate sensitivity of the specimen on the conditions necessary to initiatelocalization. Although these models oversimplify the constitutive behavior of the solid,their main conclusions remain broadly applicable even in more sophisticated analyses:namely, (i) materials with a high rate of strain hardening compared with their flowstrength will resist strain localization; (ii) plastic anisotropy can lead to the formationof vertices in the yield surface, which tend to make the material more prone to shear local-ization; and (iii) high strain rate sensitivity leads to the formation of diffuse necking andalso delays localization. More recent work has exploited the development of constitutivelaws for single crystals, which have been used to investigate the influence of plastic anisot-ropy resulting from texture formation in polycrystals on localization (Petryk, 1997; Shen,1993; Tvergaard and Needleman, 1993; Inal et al., 2002a,b), as well the influence of geo-metric softening resulting from crystallographic rotation during tensile deformation of sin-gle crystals (Asaro, 1979; Peirce et al., 1982, 1983; Balke and Estrin, 1994; Yang and Rey,1995; Stuwe and Toth, 2003).

While these studies have provided insight into the mechanics of strain localization, westill lack the detailed connection between localization and alloy composition or micro-structure that is required for alloy design. This is partly because models of localizationare based on phenomenological constitutive laws, and it is not easy to relate constitutiveparameters in these models to the underlying processes that control deformation. Currentresearch is attempting to address this issue: studies range from efforts to establish the roleof alloy chemistry at the atomic scale on plastic flow (Curtin and Olmsted, 2006), toimproving the description of defects and microstructural features in constitutive models(Ma et al., 2006). As part of this effort, Boyle and Mishra (2005) and Boyle (2005, submit-ted for publication) have recently developed a hardening law for FCC crystals, with a par-ticular focus on pure Al and Cu, which includes a more direct description of themicroscopic processes that control hardening. This model is based on the standarddescription of rate dependent slip in single crystals, but extends prior models by incorpo-


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rating a more detailed description of latent and self-hardening processes. In particular, itaccounts for the three stages of hardening, including the temperature dependence of StageIII; and also contains a particularly detailed model of the evolution of latent-hardeningwith plastic straining in the crystal. Although a large number of phenomenological param-eters remain in the constitutive equations, which are currently calibrated using experimen-tal data, many of these (such as the strengths of junctions between dislocations on thevarious active slip systems) can in principle be computed from atomic scale studies,thereby providing a more direct link between macroscopic flow and localization behaviorand microscopic deformation mechanisms.

In this paper, we briefly review the main features of the refined model and the hardeninglaws and show that an appropriately calibrated constitutive law captures the principaltrends observed in experimental measurements of the tensile behavior of pure Al singlecrystals, including the influence of crystal orientation on the hardening response and thestrain to localization. We use the model to explore the influence of changes in the proper-ties of the crystal, with a particular focus on its strain rate sensitivity, orientation, andlatent-hardening behavior.

2. Model

The problem to be solved is illustrated in Fig. 1. A tensile bar with length L and circularcross-section is subjected to prescribed displacements so as to extend the specimen in uni-axial tension at constant nominal strain rate. Specifically, we enforce u3 = 0 on the plane atx3 = 0 throughout straining and constrain u1 = u2 = 0 at the origin. Axial rotation of thesolid is constrained by enforcing u2 = 0 at x1 = R, x3 = 0. The end of the specimen atx3 = L in the initial configuration is subjected to a constant velocity parallel to the x3

direction. The loading is assumed to be quasi-static. Following Hutchinson and Neale(1977) we introduce a small geometric imperfection in the specimen, which serves to initi-ate localization in a controlled manner. This geometric imperfection takes the form of avariation in the cross-sectional area of the bar along its length given by

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Fig. 1. Schematic representation of a single-crystal tensile specimen, including the finite element mesh andgeometric parameters used in the simulation (symbols are defined in the text).


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A ¼ A0 1þ g cos2px3

L

� �� ð1Þ

where the A0 is the mean cross-sectional area of the bar and g is the amplitude of the var-iation in cross-sectional area.

The model uses the standard description of the kinematics of slip in a crystal. Let xi bethe position of a material particle in the undeformed crystal. The solid is subjected to adisplacement field ui(xk), such that the point at xi moves to yi = xi + ui after deformation,as illustrated in Fig. 2. The deformation gradient and its Jacobian are given by

F ij ¼ dij þoui

oxj; J ¼ detðFÞ ð2Þ

The velocity gradient L, stretch rate D and spin W are given by

Lij ¼o _ui

oyj

¼ _F ikF �1kj ; Dij ¼ ðLij þ LjiÞ=2; W ij ¼ ðLij � LjiÞ=2 ð3Þ

The total deformation gradient is decomposed into elastic and plastic parts by assumingthat deformation takes place in two stages. The plastic strain is assumed to shear the lat-tice, without stretching or rotating it. The elastic deformation rotates and stretches the lat-tice. We think of these two events occurring in sequence, as illustrated in Fig. 2, with theplastic deformation first, and the stretch and rotation second, giving

F ij ¼ F eikF p

kj ð4Þ

The velocity gradient may then be decomposed into elastic and plastic parts by noting that

Lij ¼ _F ikF �1kj ¼ _F e

ikF pkl þ F e

ik_F p

kl

� �F p�1

lm F e�1mj

� ¼ _F e

ikF e�1kj þ F e

ik_F p

klFp�1lm F e�1

mj ð5Þ

The velocity gradient contains two terms, one of which involves only measures of elasticdeformation, while the other contains measures of plastic deformation. We use this todecompose L into elastic and plastic parts as

Lij ¼ Leij þ Lp

ij Leij ¼ _F e

ikF e�1kj Lp

ij ¼ F eik

_F pklF

p�1lm F e�1

mj ð6Þ

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Fig. 2. Sketch showing the decomposition of plastic strain into elastic and plastic parts.


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Plastic flow in the crystal occurs by shearing a set of N slip systems. An FCC crystal hasN = 12 slip systems, which are listed in Table 1 and labeled according to the Schmidt andBoas convention. For computational purposes the slip systems are characterized by unitvectors parallel to slip directions sa

i and slip plane normals mai in the undeformed solid.

The rate of shear on the ath system is denoted by _ca. The velocity gradient due to thisshearing is

_F pikF p�1

kj ¼XN

a¼1

_casai ma

j ð7Þ

The lattice shearing does not alter the slip plane normal or the slip direction. Conse-quently, sa

i and mai are stretched and rotated only by the elastic part of the deformation

gradient, so that in the deformed configuration

s�ai ¼ F eiksa

km�ai ¼ makF e�1

ki ð8ÞThe plastic part of the velocity gradient can be expressed in terms of the shearing rates andthe deformed slip vectors as

Lpij ¼

XN

a¼1

_cas�ai m�aj ð9Þ

Finally, the elastic and plastic parts of the velocity gradient can be decomposed into sym-metric and skew symmetric parts, representing stretching and spin, respectively, as

Deij ¼ ðLe

ij þ LejiÞ=2; W e

ij ¼ ðLeij � Le

jiÞ=2

Dpij ¼ ðLp

ij þ LpjiÞ=2; W p

ij ¼ ðLpij � Lp

jiÞ=2ð10Þ

In particular, the plastic stretching and spin can be expressed in terms of the latticeshearing as

Dpij¼XN

a¼1

_caðs�ai m�aj þ s�aj m�ai Þ,

2 W pij ¼

XN

a¼1

_caðs�ai m�aj � s�aj m�ai Þ=2 ð11Þ

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Table 1Slip systems for FCC single crystals

Slip plane Slip direction

(111) ½0�11� a1

½10�1� a2

½�110� a3

ð�1�11Þ [011] b1

½�10�1� b2

½1�10� b3

ð�111Þ ½0�11� c1

½�10�1� c2

[110] c3

ð1�11Þ [011] d1

½10�1� d2

½�1�10� d3


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Internal forces in the crystal are characterized by the Cauchy (‘‘true”) stress rij, repre-senting the force per unit deformed area in the solid. The Kirchoff, Nominal, and Materialstress tensors as follows:

sij ¼ Jrij Sij ¼ JF �1ik rkj; Rij ¼ JF �1

ik rklF �1jl ð12Þ

respectively. Plastic shearing on the slip systems is driven by the resolved shear stress, de-fined as

sa ¼ Jm�ai rijs�aj ð13Þ

The constitutive equations for the crystal must relate the elastic and plastic parts of thedeformation gradient (or strain rates derived from them) to the stresses acting on the crys-tal. The elastic constitutive equation relates the elastic stretch rate to the Kirchhoff stressas

sijr ¼ Ce

ijklDekl ð14Þ

where sijr

is the Jaumann rate of Kirchhoff stress,

sijr ¼ dsij

dt� W e

ikskj þ sikW ekj ð15Þ

and

Ceijkl ¼ F e

inF ejmCnmpqF e

kpF elq ð16Þ

where Cijkl are the components of the elastic stiffness tensor for the material with orienta-tion in the undeformed configuration. For an FCC crystal the elastic modulus tensor canbe characterized by the tensile and shear moduli c11 � C1111, c12 � C1122 and c44 � C1212

with the remaining terms determined by symmetry. The moduli c11,c12,c44 are temperaturedependent, so we set

cij ¼ c0ij þ cT

ijT ð17Þ

where c0ij is the modulus at T = 0, cT

ij is the rate of change of modulus with temperature,and T is absolute temperature. The values used for all the material constants are listedin Table 2. Elastic constants were fit to the data collected by Simmons and Wang (1971).

The plastic constitutive equations specify the relationship between the stress on thecrystal and slip rates _ca on each slip system. We adopt a power-law rate-dependent flowfor this purpose, with the form

_ca ¼ _c0signðsaÞ sa

ga

� �m

ð18Þ

where sa ¼ Jm�ai rijs�aj is the resolved shear stress on the slip system, ga is its currentstrength (which evolves with plastic straining as defined below) and _c0, m are materialproperties.

The constitutive law for describing the slip system hardening behavior of the bar isbased on a model for fcc metals developed by Boyle and Mishra (2005) and Boyle(2005, submitted for publication). The details of the constitutive model and its calibrationare discussed in detail elsewhere (Boyle, submitted for publication). Here, we will merelysummarize the governing equations and give values for the relevant material parameters.


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Table 2List of values used for material parameters for Al

c011 180395.51 MPa

cT11 �46.71 MPa K�1

c012 63255.16 MPa

cT12 �9.52 MPa K�1

c044 32156.62 MPa

cT44 �13.85 MPa K�1

l0 29720.70 MPalT �14.492 MPa K�1

ho 60.0 MPahI 15.0 MPaso 0.8 MPasI 1.0 MPasvo 115.5 MPaA/l 0.2 � 10�19 m3

_c1 107s�1

co 0.005cII 0.0vN

m=l 1.0vC

m=l 0.0vG

m=l 1.0vH

m=l 0.75vS

m=l 2.15qN

m 2.0qC

m 1.1qG

m 2.0qH 0

m 1.6qS

m 2.0qs 1.1




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ECThe hardening rule specifies the relationship between the slip system strengths ga and

the plastic strain. At time t = 0 each slip system has the same initial strength g0. Thereaf-ter, the slip systems evolve as a result of the plastic shearing. The ath system may harden asa result of shearing on the ath system itself or a coplanar system (self-hardening). It mayalso harden as a result of shearing on a non-coplanar system (latent-hardening). The twomodes of hardening are distinguished by setting

_ga ¼XN

b¼1

~qabhbj _cbj ð19Þ

where hb is a self-hardening modulus, and ~qab are a set of latent-hardening moduli. Thelatent-hardening moduli satisfy ~qab ¼ 1 for a = b. For a 6¼b, ~qab evolve with plastic strain-ing, as discussed below. It is convenient to discuss the constitutive equations for self- andlatent-hardening in turn.

2.1. Self-hardening

Strain hardening in crystals is conventionally divided loosely into three (or more) sep-arate stages, which correspond to characteristic features of the uniaxial tensile stress–straincurve, and are associated with different microscopic processes that contribute to strain


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hardening (Kocks and Mecking, 2003). Stage I hardening (or ‘easy glide’) is observed incrystals that are oriented to activate a single slip system. It is associated with a low, tem-perature independent hardening rate. Stage II hardening initiates when a second slip sys-tem is activated. It is associated with a high rate of hardening, with a roughly linearrelationship between stress and strain. The hardening rate in Stage II is also insensitiveto temperature. Stage III hardening occurs for large strains, and corresponds to a progres-sively decreasing hardening rate. The rate of hardening in stage III is very sensitive to thetemperature and rate of deformation.

To describe this behavior, the self-hardening modulus hb for the three stages (I, II, andIII) of hardening is expressed as

hb ¼ hIIIb ðhI

b þ hIIb Þ ð20Þ

The stage I modulus is expressed as

hIb ¼ ðho � hIÞsech2½ðho � hIÞcb=ðsI � soÞ� þ hI ð21Þ

where ho and hI are the initial and final hardening rates during stage I, and

cb ¼Z t

0

j _cbjdt ð22Þ

is the accumulated plastic slip on the bth system.Stage II hardening is assumed to be activated by secondary slip with the stage II mod-

ulus given by

hIIb ¼

XN

j¼1

vbjf ðnacj=coÞ; f ðnac

j=coÞ ¼0 cj < cII

tanh2ðnacj=coÞ cj > cII

ð23Þ

Here, cII is a material constant that represents the amount of secondary slip necessary forthe initial activation of stage II, na is the number of active slip systems, and c0 is a materialconstant, defined so that co/na + cII is the amount of slip after which the interactionreaches �60% of peak strength. In this expression, a slip system is considered active ifthe fractional shearing rate is greater than a critical amount, i.e.

na ¼Xn

a¼1

nðaÞa ; nðaÞa ¼1 for _caj j=

P_cbj jP 0:1

0 for _caj j=P

_cbj j < 0:1

ð24Þ

The coefficients vbj in Eq. (23) are a set of material constants that control the magnitude ofthe dislocation interaction strengthening between slip systems b and j. The value of vbj fora pair of slip systems depends on the type of dislocation reaction that occurs between thetwo systems. We consider the following types of interaction:

(1) Systems with parallel slip directions, with jsb � sj j = 1, form no junctions and havevbj = vN;

(2) Coplanar systems with non-parallel slip directions (jsb � s jj < 1, jmb � mjj < 1) formcoplanar junctions and have vbj = vC;

(3) Systems with mutually perpendicular slip planes and slip directions (mb � sj = 0 ormj � sb = 0) form a glissile lock, and have vbj = vG;


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(4) The remaining systems, which must have sj � mb 6¼ 0 and sb � mj 6¼ 0 form a Hirthlock, with vbj = v H if the slip directions combine to produce a vector of [100] type(sj + sb � [001]), and form a sessile lock, with vbj = vS, otherwise.

The stage III modulus is sensitive to temperature and strain rate and is given by

hIIIb ¼ 1� tanh4 2

gb

sv

� �ð25Þ

where, following Kocks and Mecking (2003), we take the saturation stress sv to be given by

sv

l¼ svo

lo

_cb

_c1

� �kTA

ð26Þ

where l = l0 + lTT is a temperature dependent average shear modulus for the crystal, lo

and svo are the shear modulus and staturation stress at 0 K, _c1 is a representative strainrate, k is Boltzmann’s constant, T is the absolute temperature, and A is a material param-eter which depends on the stacking fault energy of the crystal. In our computations, A istaken to be a fixed fraction of the shear modulus, as listed in Table 2.

2.2. Latent-hardening

The latent-hardening ratios ~qab specify the rate of strain hardening due to shearing onnon-co-planar slip systems. Following Boyle (submitted for publication), the latent-hard-ening coefficient ~qab are expressed in terms of the slip system strengths as

~qab ¼ 2

sI � so

½sIðqabm � qsÞ þ soðqab

s � 1Þ�sech2 2

sI � so

ðgb � soÞ� �

þ qs ð27Þ

where b = 1, . . . ,N, and no summation over b is implied. Among the internal variables, so

and sI are the initial and back-extrapolated critical yield stresses in shear for stage I, whileqab

m and qs are maximum and saturation latent-hardening ratios. The values of qabm depend

on the types of dislocation reaction that occur between slip systems a and b, and are clas-sified using the same convention that was used to assign values to vab above. For example,systems which form no junction have qab

m ¼ qNm; systems which form coplanar junctions

have qabm ¼ qC

m; systems which form glissile locks have qabm ¼ qG

m; systems which form Hirthlocks have qab

m ¼ qHm ; and systems which form sessile locks have qab

m ¼ qSm.

Parameter values for material constants representing pure Al are listed in Table 2.The calibration of the model is described in more detail in Boyle (submitted forpublication).
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C3. Finite element implementation

The boundary value problem was solved using the commercial finite element codeABAQUS, augmented by a user-subroutine that implements the constitutive equationsdiscussed above. Fig. 1 illustrates a representative finite element mesh, which consists of8-noded reduced integration brick elements. Mesh sensitivity was tested by repeatingselected simulations with four times the number of elements: our results indicate that local-ization strains are changed by less than 2.5% with the refined mesh.


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4. Results and discussion

4.1. Orientation dependence of limit strain

Fig. 3 shows a set of representative nominal stress–nominal strain curves for uniaxialtensile deformation of model single crystals with properties intended to represent pureAl at a temperature of 273 K. Table 2 lists the values used for parameters in the constitu-tive model. The parameter g characterizing the variation of the cross-sectional area of thespecimen (Eq. (1)) was g = 0.005 and the solid was deformed at a constant nominal strainrate of 0.05 s�1. Results are shown for several initial orientations of the tensile axis withrespect to the crystal, including loading along [12 3], [11 2], [10 0], and [11 1] directions.The behavior of crystals loaded near a high symmetry orientation such as [10 0] and[111] is extremely sensitive to small misorientations. Consequently, results are shown bothfor perfect [100] and [111] orientated crystals, and also for specimens in which the loadingaxis was given a small (0.05�) random misorientation away from the high-symmetrydirection.

In our simulations, crystals that were loaded along or near the [100] and [111] direc-tions failed by strain localization. For these cases, Fig. 3 shows the necking strain, whichis defined as the strain at the instant that the strain rate near the ends of the specimendrops to zero.

The orientation dependence of the localization strain is simply a consequence of thehigh initial rate of hardening that occurs when the crystal is loaded along a symmetry axis,due to slip on multiple systems. This results in a higher flow stress at strains exceeding10%; at the same time the hardening rate decreases with continued straining which makes

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Fig. 3. Comparison of predicted nominal stress–strain curves for pure Al deformed in uniaxial tension with thetensile axis initially parallel to [100], [110], [111] and [123] orientations. Necking strains are marked by a dot oncurves. FEM results are shown as solid lines while experimental results are from Hosford et al. (1960). The FEMresults for perfect [100] and [111] orientations are shown as dotted lines, while solid lines for these orientationscorrespond to a deviation of 0.05� from the ideal direction.


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the material more prone to localization, as predicted by first approximation calculations(Considere, 1888) over a century ago.

For comparison, Fig. 3 also shows the experimental stress–strain curves for Al singlecrystals obtained by Hosford et al. (1960). Although our simulations do not give a perfectmatch to the experimental stress–strain curves, the model clearly reproduces the principalfeatures of experimental data, including (i) the qualitative dependence of the flow stress ofthe crystal on the direction of straining: the [111] orientation has the highest strength; fol-lowed by [100], [112], and [123] oriented crystals as well as the crossover of the stress–strain behavior at high strains for [10 0] and [112] orientations; (ii) the model correctlypredicts that crystals with [100], [111], and [112] orientations localize, whereas that with[123] orientation show no evidence of localization up to 50% nominal strain. In addition,the localization strains predicted by the simulations are close to those observed experimen-tally. The experiments and computations show detailed features that are not in perfectagreement, however. For example, the model predicts a strong increase in hardening rateat approximately 40% strain for the [123] oriented crystal: this is associated with the stageI–stage II transition at the onset of secondary slip in the crystal. Hosford et al. (1960)found no evidence of a stage I–II transition in their experiments.

Our main objective in the remainder of this paper is to study in more detail the influenceof geometric and material parameters in controlling the localization strain for these modelcrystals.

4.2. Effects of specimen geometry

It is important to note that critical localization strains predicted by the simulations areinfluenced by specimen geometry, especially geometric imperfections. For example, Fig. 4shows the critical localization strain as a function of the dimensionless defect parameter g,which characterizes the initial geometric imperfection in the specimen as defined in Eq. (1).The results are shown for [100] loading of an Al crystal at 273 K, with two values of the

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Fig. 4. Necking strains (defined to be the critical nominal strain where the strain rate outside the localized regionof the specimen drops to zero; see text for details) as a function of the dimensionless geometric defect g defined inEq. (1). Results are shown for [100] (with deviation) loading of Al at 273 K.


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slip rate exponent m = 20 and m = 150 in Eq. (18). With a low rate exponent m = 20, thelocalization strain is evidently proportional to 1/log(g). For the high rate exponentm = 150, the localization strain still decreases with the increasing value of g, but is notdescribed by a simple linear relationship with log(g). The sensitivity of localization strainto specimen geometry makes it difficult to compare quantitatively the predicted localiza-tion strains with experiment. However, the qualitative influences of crystal orientation,temperature and material properties on localization strain are independent of geometry.

In the remaining simulations reported in this paper, a small, fixed geometric defect isused to initiate localization in all simulations. This is not strictly necessary: the inhomoge-neity associated with the finite element discretization and rounding errors during the com-putation will cause even a geometrically perfect specimen to localize. In this case,localization strains would, however, be sensitive to mesh design and machine precision,so it is preferable to introduce a controlled defect. In actual experiments, localizationmay be triggered by a variety of processes, but the qualitative variation of localizationstrains with material and loading parameters is not sensitive to the details of the initiationprocess.

4.3. Effects of strain rate sensitivity

Fig. 5 shows the results of simulations in which the strain rate exponent m in Eq. (18)was systematically varied. Results are shown for [100] loading of Al at 273 K, with a geo-metric defect g = 0.005. We observe two regimes of behavior: for m < 100 the specimenfails by forming a relatively broad neck near the center of the specimen, and in this regimethe localization is highly sensitive to the strain rate exponent m. For m > 100, the specimenforms a diffuse neck during the early stages of deformation, but then localizes near theedges of this necked region. In this regime both the stress–strain curves and the criticalstrain to localization show a reduced sensitivity to m. Similar trends have been reportedby earlier studies, e.g. Hutchinson and Neale (1977), Peirce et al. (1983).

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Fig. 5. Necking strains as a function of strain rate exponent m as defined in Eq. (18). Results are shown for [100](with deviation) loading of Al at 273 K, with a geometric defect g = 0.005.


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The localization modes for perfectly oriented [100] crystals are different in these tworegimes. When m < 100, localization occurs by the formation of a single neck, as shownin Fig. 6a. In contrast, for m > 100, we observe two necking regions, as shown inFig. 6b. However, the double neck is observed only for a crystal with perfect [100] orien-tation: a very small deviation of the loading axis can change the deformation mode back tothe standard, single neck, as shown in Fig. 6c. In contrast, specimens with lower rate sen-sitivity (m < 100) always fail by formation of a single neck, regardless of orientation. Theneck becomes progressively more diffuse as m is reduced.

The results show that strain rate sensitivity clearly plays a critical role in governingmaterial behavior, and from the standpoint of alloy design, modifying alloy chemistryand microstructure to increase strain rate sensitivity may be a more effective way toimprove formability than attempting to modify hardening behavior. Assuming that theabove trend for a single crystal holds true for polycrystalline alloys where m valuesbetween 10 and 250 in Al–Mg alloys (Yao et al., 2000) is achievable, this result can betaken as a guide to designing high ductility Al alloys.

4.4. Effects of initial orientation of the crystal

In this section, we study in more detail the influence of the initial orientation of the sin-gle crystal on its strain to localization. Throughout this section, the orientation of the ten-sile axis with respect to the crystal will be displayed graphically on an inverse pole figure,such as that shown in Fig. 7. A detailed description of the construction of an inverse pole

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Fig. 6. Localization modes for Al loaded along [100] at 273 K: (a) m = 5; (b) m = 150; and (c) m = 150 but with asmall deviation from the perfect [100] orientation.


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Fig. 7. A contour map showing the strain to localization as a function of orientation of the initial loading axis inthe standard triangle. The results are for Al, deformed at 273 K with strain rate sensitivity m = 150 and aspecimen defect parameter g = 0.005. Lines marked ‘a’, ‘b’ and ‘c’ refer to deformation paths referred to in Figs. 8and 9. The simulations were terminated when the nominal strain reached 60%, so no localization occurred in thered regions of the map.




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TEfigure can be found in Hosford, 1993). Briefly, the figure displays stereographic projections

of characteristic crystallographic directions – in the convention adopted here, the projec-tions of the [01 0] and [001] directions form a Cartesian coordinate system in the plane ofthe figure, while the [100] direction points out of the plane. The orientation of the tensileaxis with respect to the crystal can then be specified by the projection of a unit vector par-allel to the tensile axis, which corresponds to a. The full inverse pole figure for an fcc crys-tal can be divided into 24 triangular regions, each of which (100), (011), and (11 1)directions at the three vertices. Fig. 7 shows two such regions. Each point within these tri-angles corresponds to a particular loading direction – for example, the vertices indicateloading parallel to [100], [11 0], or [111] directions. Due to the symmetry of the crystal,each of the 24 triangular regions in the pole figure are crystallographically identical,and it is only necessary to explore behavior with the tensile axis in a single representativetriangle. Accordingly, Fig. 7 shows a contour map of the strain to localization as a func-tion of the initial orientation of the tensile axis relative to the [100], [110], or [111] direc-tions. The results are for Al deformed at 273 K with strain rate sensitivity m = 150 and aspecimen defect parameter value of 0.005. The necking strain is smallest for initial orien-tations near the two high symmetry orientations [111] and [100].

The influence of initial orientation on necking strain is intimately connected to the evo-lution of the crystal’s orientation with straining. Fig. 8 shows the progressive orientationchange of the crystal as a function of straining for crystals with a range of initial orienta-tion plotted on an inverse-pole figure. The initial orientations correspond to points that liealong lines marked ‘a’, ‘b’ and ‘c’ in Fig. 7. Each line in Fig. 8 is color-coded to indicate thenominal strain variation with the orientation of the loading axis.

Fig. 8 shows that the crystal rotation may follow one of three general trends dependingon the initial orientation:


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Fig. 8. Inverse pole figures showing the evolution of the tensile axis for crystals with various initial orientations inthe standard triangle. The figure shows results for initial orientations along the lines marked ‘a’, ‘b’ and ‘c’ inFig. 7. The curves are color coded to indicate the variation of nominal strain along the orientation path. Mostinitial orientations deform initially in single slip, then transition to double slip after the loading axis leaves thestandard triangle, and approach the stable orientation with loading axis parallel to the [112] direction. If theloading axis is initially close to a high symmetry orientation, however, it may spiral around the high symmetrydirection.




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TED(1) If the crystal is loaded with the tensile axis perfectly parallel to the [100] or [111]

axis, it does not rotate at all during deformation.(2) If the initial loading axis is far from the [100] and [111] vertices, the crystal initially

deforms in single slip, and consequently the loading axis rotates along a great circletowards the [101] orientation. Shortly after the tensile axis leaves the standard trian-gle, a second slip system is activated: as a result, the crystal then begins to rotate soas to bring the tensile axis towards the stable [11 2] orientation. If the crystal couldcontinue to deform indefinitely without localization, the loading axis would eventu-ally converge to the [112] direction. However, our simulations show that in all caseslocalization sets in shortly after the onset of double slip. We note in passing that thecritical strain to activate double slip depends strongly on the latent-hardening behav-ior of the crystal. This behavior is explored in more detail in the next section.

(3) If the tensile axis lies very close to, but not exactly on, the [100] or [11 1] loadingdirection, our simulations predict extremely complex behavior, as illustrated inFig. 8. In this case, multiple slip systems are activated, and the loading axis showsa tendency to spiral around the high-symmetry orientation. In most cases, the crystaleventually settles to a state of double slip, but the choice of slip systems in this state ishighly sensitive to the initial misorientation of the crystal, as well as its latent-hard-ening behavior.

Fig. 9 shows the influence of initial orientation for crystals loaded near [100] and [111]orientations in more detail. Fig. 9a and b corresponds to crystal orientation variationsalong lines ‘a’ and ‘b’ in Fig. 7, respectively. Fig. 9a can be thought of as representingthe effects of misaligning the loading axis from the [100] direction by rotating the crystalabout the ½0�611� direction; similarly, Fig. 9b represents the influence of misaligningthe loading axis from the [11 1] direction by rotating the crystal about its ½�15�4� direction.


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Fig. 9. Variation of necking strain with misorientation of the tensile axis from a high symmetry direction: (a)results for tensile axis near [100], following the path marked ‘a’ in Fig. 7 and (b) results for the tensile axis near[111], along the path ‘b’ in Fig. 7.




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TEDSeveral features of these results are worth noting. Firstly, the strain to localization is a

monotonically increasing function of the misorientation angle for angles exceeding 4�.For smaller misorientations, however, the variation of localization strain with misorienta-tion is complex. A very small misorientation away from the perfect [100] or [111] orien-tation causes a rapid decrease in localization strain: the effect is particularly noticeablenear the [11 1] orientation in Fig. 9b, where the localization strain drops from 0.38 to0.22 as a result of a misorientation of only 0.04�. The drop is less pronounced for crystalsloaded near [100], but in this case, the necking strain is a fluctuating function of the mis-orientation. These fluctuations are not numerical noise: they are associated with changes inthe history of rotation of the crystal (see Fig. 8). This behavior occurs because the activeslip systems are highly sensitive to small changes in orientation when the crystal is loadednear a high symmetry orientation. The hardening behavior of the crystal and its rate sen-sitivity play a critical role in controlling the active slip systems, as discussed in more detailbelow.

4.5. Effects of rate sensitivity and latent-hardening on variation of localization strain

The behavior of crystals loaded near high symmetry orientations (e.g. with loading axesclose to [10 0] or [111] directions) is determined by the number of active slip systems andthe relative shear rates on each active system. These, in turn, are strongly sensitive to therate sensitivity and strain hardening behavior of the crystal.

The influence of strain rate sensitivity and latent-hardening on the orientation depen-dence of localization strain is illustrated in Fig. 10, which shows the variation of localiza-tion strain with the misorientation of the tensile axis for a crystal loaded near the [111]direction. (Fig. 10b shows the same data as (a), but re-plotted to show the behavior atsmall misorientations more clearly.) Results are shown for crystals with isotropic harden-ing and m = 150 as well as with latent-hardening with rate exponent m = 20 and m = 150.To understand the implications of this comparison, it is helpful to recall that Al has


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Fig. 10. (a) The variation of strain to localization with misorientation for crystals loaded with the tensile axis nearthe [111] direction with isotropic hardening and latent-hardening and (b) an enlarged view of the results for smallmisorientations.




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~qab > 1 (the value of q evolves with straining), which implies that slip on the ath systemcauses inactive systems (b 6¼ a) to harden more rapidly than the active system. In caseof isotropic hardening, with ~qab ¼ 1, all the systems harden at the same rate.

Reducing the strain rate exponent m retards localization for all orientations, asexpected from the results shown in Fig. 5. It also changes the sensitivity of the neckingstrain to misorientation. The underlying reason is straightforward: with high strain ratesensitivity (m = 150), the active slip systems change very rapidly as the loading axis isrotated away from the perfect [111] orientation. With a lower strain rate exponent(m = 20), the slip activity changes more gradually with misorientation. The influence ofthis behavior on the rotation of the crystal is shown in Fig. 11, which should be contrastedwith the results shown in Fig. 8 for a crystal with m = 150. In Fig. 8, the loading axis spi-rals around the high symmetry orientation only if the tensile axis is very close to the [111]

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Fig. 11. An inverse pole figure plot showing the rotation of the tensile axis for a crystal with various initialorientations near the [111] direction. The results are for the model including latent-hardening with strain ratesensitivity m = 20 and a specimen defect parameter g = 0.005.


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Fig. 12. An inverse pole figure plot showing the rotation of the tensile axis for a crystal with various initialorientations near the [111] direction. The results are for an isotropically hardening material, deformed at 273 Kwith strain rate sensitivity m = 150 and a specimen defect parameter g = 0.005.




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Pdirection. In contrast, with m = 20, this behavior is observed for larger misorientations.Consequently, with m = 20 we find that the localization strain is minimized for a misori-entation around 2�.

Latent-hardening has little influence on the strain to localization if the crystal is loadedalong a high symmetry orientation. In this case, latent-hardening merely increases theoverall strain hardening behavior of the crystal, and its effects can be understood usingthe classical Considere analysis of necking (Considere, 1888). For a misoriented crystal,however, latent-hardening has a much more significant effect. For small misorientations,the material with Boyle’s model of latent-hardening localizes earlier than an isotropicallyhardened material as shown in Fig. 12. This behavior is again a consequence of the influ-ence of hardening on the active slip systems: latent-hardening tends to prevent multipleslip in crystals that are loaded close to a high symmetry orientation. In contrast, whenthe initial loading is not close to a high symmetry orientation, latent-hardening increasesthe overshoot prior to the onset of secondary slip, and consequently increases the strain tolocalization.

While the above results are for aluminum single crystal, they provide important cluesabout designing aluminum alloys to increase the localization strain and enhance ductility.Reducing geometric imperfections has implications for control of second phase intermetal-lic distribution. Manipulating orientation is related to controlling texture. Changing strainrate exponent can be accomplished through solid solution strengthening and precipitationhardening, which also alter latent-hardening. Assuming that the above trends for singlecrystal also holds for aluminum alloys, a combination of these microstructural featurescan lead to a high ductility material.

5. Summary

Finite element simulations have been used to study localization in single crystals withproperties representing Al under uniaxial tensile straining. A refined constitutive lawwhich includes a detailed description of latent and self-hardening in the crystal (Boyleand Mishra, 2005; Boyle, 2005, submitted for publication) was used in the simulation.


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The simulations were used to investigate the influence of specimen geometry, the orienta-tion of the loading axis with respect to crystallographic directions, and the properties ofthe material, in particular, the slip rate sensitivity exponent m and the latent-hardeningratio on the strain-to-localization in uniaxial tension. Our simulations showed that:

(1) Two general forms of localization may occur, depending on the value of the strainrate exponent m. For m < 100, failure occurs by the gradual development of a diffuseneck at the center of the specimen; for m > 100, a sharp neck forms, and if the load-ing axis is exactly parallel to the [100] direction, two necks may form. In the diffusenecking regime (m < 100) the strain to localization decreases rapidly with increasingm, while for values of strain rate exponent m > 100 the strain to localization is onlyweakly sensitive to m.

(2) The strain to localization is sensitive to specimen geometry: for example, the strain tolocalization varies inversely with log(g), where g is a dimensionless measure of thelocal thinning at the center of the specimen, defined in Eq. (1).

(3) The influence of the orientation of the loading axis with respect to the crystal onstrain to localization was explored in detail. Our simulations showed that crystalslocalized most readily when the tensile axis is close to (but not exactly parallel to)a high symmetry orientation such as [100] or [111]. Broadly, this behavior occursbecause a larger number of slip systems are active when the crystal is loaded nearthe [100] or [111] direction, which causes a higher rate of hardening. For larger mis-orientations, the crystal initially deforms by single slip, and rotates until a secondaryslip system is activated, whereupon the crystal tends to rotate so as to align the [112]direction with the loading axis. In these cases we found that localization invariablyoccurred shortly after secondary slip was activated.

(4) Strain rate sensitivity has a strong influence on the orientation dependence of thelocalization strain. Small values of the strain rate exponent m (e.g. m = 20) tendto promote multiple slip, and the strain-to-localization has a minimum value whenthe loading axis is misoriented by approximately 2� from a high symmetry orienta-tion. For high values of rate sensitivity, the minimum value of strain-to-localizationoccurs when the loading axis is misaligned by approximately 0.05� to the loadingaxis.

(5) The influence of latent-hardening on the variation of strain-to-localization with mis-orientation was probed by contrasting the behavior of the model material withparameters representing Al with an isotropically hardening solid. In the model Almaterial, the hardening is such that slip on a single system tends to harden inactivesystems more rapidly than the active system; whereas in the isotropically hardeningmaterial, all slip systems harden at the same rate. We found that if the loading axis isperfectly aligned with a high symmetry orientation such as [10 0] or [111], latent-hardening has almost no effect on strain to localization. If the loading axis is misa-ligned with the high symmetry orientation by less than 2�, the isotropically hardeningmaterial has a higher strain to localization than the model material with propertiesrepresenting Al. In contrast, if the loading axis is misoriented by more then 2�, thetrend is reversed. This behavior is a consequence of the influence of latent-hardeningon the rotation of the crystal. When the loading axis is very near the high-symmetryorientation, the isotropically hardening solid deforms by slip on more than two sys-tems, while the material with properties representing Al tends to deform in double


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slip, and localizes more readily. When the initial loading axis is misoriented by morethan 2�, both materials initially deform in single slip, before eventually transitioningto double slip just prior to localization. Under these conditions higher latent-hard-ening tends to delay the onset of double slip, and consequently delays localization.

Acknowledgement

This work was supported by General Motors at Brown University as part of GM-Brown Collaborative Research Laboratory (CRL).
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Numerical simulations of necking during tensile deformation of … · 2008. 2. 19. · UNCORRECTED PROOF 1 2 Numerical simulations of necking during 3 tensile deformation of aluminum