Crystal-Plasticity FundamentalsHenry R. Piehler, Carnegie Mellon University

THE PURPOSE of this article is to enablethe reader to understand through examples thefundamentals of crystal plasticity. The richhistorical development of crystal plasticity istraced in some detail to enable the reader toappreciate and critically analyze more sophisti-cated recent approaches to crystal-plasticitymodeling. While most of the examples involvecubic metals deforming by rate-insensitive plasticflow, the concepts outlined here can begeneralized to other crystal structures and loadingconditions as well.

Schmid’s Law

Crystal plasticity has as its origin Schmid’slaw, which states that crystallographic slip isinitiated when a critical resolved shear stresson a slip plane in a slip direction is reached.As shown in Fig. 1, crystallographic slip initiatesin uniaxial tension when the resolved shear stress

on the slip plane in the slip direction reaches acritical value k (Ref 1). This critical resolvedshear stress criterion can be expressed as:

tns ¼ s cos l cosf ¼ k (Eq 1)

where tns is the shear stress on the slip plane,with normal n in the slip direction s; s is theapplied uniaxial stress; cos l is the cosine ofthe angle between the tensile axis and the slipdirection; cosj is the cosine of the angle betweenthe tensile axis and the slip plane normal; and kis the critical resolved shear stress. If l, s, and nare unit vectors along the tensile axis l, the slipdirection s, and the slip plane normal n, cos land cos f can be found from the dot productsl � s and l � n.For cubic metals, it is convenient to express

the unit vectors l, s, and n in terms of the unitvectors i1, i2, and i3 along the [100], [010], and[001] crystallographic axes, respectively. Forexample, if a uniaxial stress is applied in the[001] direction of a face-centered cubic (fcc)single crystal, slip will occur on the (111) planein the ½0�11� direction when the normal stress sreaches k/cos l cos f (Eq 1). For slip to occuron this (111)½0�11� slip system in response to auniaxial normal stress applied along the [001]direction:

l ¼ i3; n ¼ 1ffiffiffi3

p ½11 þ i2 þ 13�; and s ¼ 1ffiffiffi2

p ½�i2 þ i3�

Performing the dot products l � s and l � nyields:

s½001� ¼ffiffiffi6

pk ¼ 2:45 k

It should be noted that the yield stress must beat least 2 times the critical resolved shear stress,the minimum value achievable when both theslip plane normal and the slip direction are ori-ented at 45� to the uniaxial stress axis. As seenfrom Eq 1 and indicated in Table 1, Schmid’slaw is identical for fcc metals deforming on{111}<110> systems and body-centered cubicmetals deforming on {110}<111> systems;only the identity of f and l are interchanged.One of the earliest crystal-plasticity calcula-

tions was the determination by Sachs (Ref 2)of the uniaxial yield stress of an isotropic fcc

metal in terms of the critical resolved shearstress k. Sachs’ calculation was an average overthe stereographic triangle of the uniaxial stressesnecessary to initiate slip on the most highlystressed {111}<110> slip system(s). This isos-tress model predicted that the uniaxial yield stressof an isotropic fcc metal should equal 2.22k. It isalso noteworthy that the single-crystal yield stressin the [111] direction is equal to:

1:5ffiffiffi6

pk

50% higher than the isostress yield stress forthe [001] orientation calculated earlier and65% above the Sachs average. It is this depen-dence of the single-crystal yield stress on crys-tallographic orientation that provides the basisfor texture hardening (Ref 3).

Generalized Schmid’s Law

Schmid’s law for the response of crystals to auniaxial normal stress can be generalized to threedimensions to include both normal and shearstresses. A critical resolved shear stress criterioncan again be used, this time in response to all sixcomponents of stress, chosen initially for conve-nience along the cubic axes of an fcc crystal.These stresses along the cubic axes include thethree normal stresses,s11,s22,s33, and the threeindependent shear stresses, s12 = s21, s23 = s32,and s31 = s31. These stresses can be referred toas sij, where i and j = 1, 2, 3.Expressions for the resolved shear stresses on

the twelve crystallographic slip systems (andtheir negatives) are written in terms of the sixindependent stresses referred to the cubic axesof the fcc crystal. The notation used by Bishopand Hill (Ref 4, 5) and Bishop (Ref 6) forthe twelve slip systems is shown in Fig. 2 andTable 1, where the individual slip planes andtheir corresponding slip directions are identified,along with the shear strains associated with thesesystems. The generalized Schmid’s law isobtained using a tensor transformation to sumthe contributions to each of the twelve shearstresses from the stressessij along the cubic axesof the fcc crystal. This generalized Schmid’s lawfor the critical resolved shear stress tns on each ofthe twelve fcc slip systems is:

Fig. 1 Representation of Schmid’s critical resolvedshear stress criterion: t

ns= s cos l cos j = k

tns ¼ ln1ls1s11 þ ln2ls2s22 þ ln3ls3s33 þ ln1ls2s12

þ ln2ls3s23 þ ln3ls1s31 þ ln2ls1s21 þ ln3ls2s32

þ ln1ls3s13

(Eq 2a)

or

tns ¼X3

j¼1

X3

i¼1

lnilsjsij (Eq 2b)

where lni and lsj are the direction cosines of theangle between the particular slip plane normal nand the ith cubic axis and the direction cosineof the angle between the particular slip direc-tion s and the jth cubic axis. Equation 2(a) canalso be written using the summation conventionthat the repeated indices i and j are summedfrom 1 to 3, or:

tns ¼ lnilsjsij (Eq 2c)

These direction cosines can again be foundfrom the dot products lni = n � ii and lsj = s � ij.Calculating the generalized Schmid’s law forthe ð111Þ½0�11� system considered previouslygives:

ln1 ¼ ln2 ¼ ln3 ¼ 1ffiffiffi3

p and

ls1 ¼ 0; ls2 ¼ � 1ffiffiffi2

p ; ls3 ¼ 1ffiffiffi2

p

and the generalized Schmid’s law for this systembecomes:

tns ¼ 1ffiffiffi6

p ð�s22 þ s33 � s12 þ s23 � s32 þ s13Þ

Since s23 = s32, this reduces to:

tns ¼ 1ffiffiffi6

p ðs22 þ s33 � s12 þ s13Þ

This ð111Þ½0�11� system is designated as �a1 byBishop and Hill (Ref 4, Table 1), whose proce-dure will subsequently make use of all 12 ofthese generalized Schmid’s law expressions.Notice that the critical resolved shear stressreduces to the same result calculated previouslyfor uniaxial tension along the [001] direction ifonly s33 is applied.

Taylor Model

G.I. Taylor (Ref 7) noted that the isostressmodel used by Sachs to calculate the uniaxialyield stress of an isotropic fcc aggregate interms of the critical resolved shear stress kfailed to satisfy the strain compatibility require-ments among grains for plastic deformation ofan isotropic material deformed in uniaxial ten-sion. These isostrain uniaxial tension require-ments are that, for a given plastic strain dealong the length l of a single crystal, the normalstrains along two axes perpendicular to l areequal to �de/2 (from constancy of volumerequirements), and all shear strains are equalto zero. Taylor then proceeded to calculate theuniaxial yield stress of an isotropic fcc aggre-gate in terms of the critical resolved shear stressusing his isostrain model, which is describednext.Taylor’s approach sought to determine the

crystallographic shear strains resulting fromthe operation of a particular set of five indepen-dent slip systems that will accommodate theuniaxial tension isostrain requirements imposedalong the primed axes associated with a partic-ular fcc crystal orientation. The first step is tofind the plastic strain increments referred tothe cubic axes of an fcc single crystal resultingfrom the imposition of the plastic strainsimposed along primed specimen axes. If theimposed strain increments along the primedspecimen axes are designated as dek0l0 usingthe summation convention, the strain incre-ments along cubic axes can be written in termsof these imposed strains as:

deij ¼ lik00 ljl0dek0l0 (Eq 3)

The results of this isostrain requirement for uni-axial tension along various orientations canthen be used to relate the plastic strains alongthe cubic axes and the contributions of each ofthe twelve possible slip systems (four <111>planes each containing three {110} slip direc-tions), that is:

deij ¼X4

n¼1

X3

s¼1

linljsdens (Eq 4a)

where dens is 1 of 12 true plastic strains asso-ciated with the operation of particular slip systemwith slip plane normal n and a slip directions.The true shear strain dens is equal to ½ the simpleshear strain gns, which contains a rotation equalto (½) dgns. This relationship between crystallo-graphic or simple shear, pure or tensorial shear,and rotation is shown in Fig. 3. Using the nomen-clature of Bishop and Hill contained in Table 1,the relationships between the five independentcomponents of strain referred to the cubic axesand the12 crystallographic simple shear strainsai (Eq 4a) become:

de11 ¼ 1ffiffiffi6

p ð�2a2 þ 2a3 � 2a5 þ 2a6 � 2a8 þ 2a9

� 2a11 þ 2a12Þ

de22 ¼ 1ffiffiffi6

p ð�2a1 � 2a3 þ 2a4 � 2a6 þ 2a7 � 2a9

þ 2a10 � 2a12Þ

de33 ¼ 1ffiffiffi6

p ðþ2a1 þ 2a2 � 2a4 � 2a5 � 2a7 � 2a8

� 2a10 þ 2a11Þ

de12 ¼ 1ffiffiffi6

p ðþa1 � a2 þ a4 � a5 � a7 þ a8 � a10 þ a11Þ

de23 ¼ 1ffiffiffi6

p ðþa2 � a3 � a5 þ a6 þ a8 � a9 � a11 þ a12Þ

de31 ¼ 1ffiffiffi6

p ðþa1 þ a3 þ a4 � a6 þ a7 � a9 � a10 þ a12Þ(Eq 4b)

Since there are five independent components ofstrain eij along the cubic axes (reduced from six bythe constancy of volume requirement de11 + de22+ de33 = 0), at least five slip systems (a’s) mustoperate for Eq 4(b) to be inverted to find theamounts of shear on particular slip systems thatmust operate to accommodate the strains imposedalong the cubic axes. This requirement for thesimultaneous operation of five independent slipsystems to accommodate an arbitrary strain statein an fcc crystal was first pointed out by vonMises(Ref 8). Since one can find more than one set offive independent shear strains to accommodate agiven imposed strain state, the solution for eij interms of ai (Eq 4b) is not unique.Taylor selected the particular set of five inde-

pendent crystallographic simple shear strains aithat accommodated the imposed macroscopicstrain state ek0 0l0 with a minimum of the totalshear strain (or shear strain energy):

X5

i�1

ai

Therefore, to calculate the value of the uni-axial yield stress for an isotropic fcc crystallineaggregate using Taylor’s isostrain minimum

Fig. 2 Schematic representation of Bishop-Hill slipsystem notation

Table 1 Slip system notation used by Bishop and Hill for cubic crystals

Slip plane (or direction) (111) ð�1�11Þ ð�111Þ ð1�11Þ

Slip direction (or plane) ½01�1�½�101�½1�10� ½0�1�1�½101�½�110� ½01�1�½101�½�1�10� ½0�1�1�½�101�½110�Shear strain a1, a2, a3 b1, b2, b3 c1, c2, c3 d1, d2, d3Slip system designation (a1), (a2), (a3) (b1), (b2), (b3) (c1), (c2), (c3) (d1), (d2), (d3)

2 / Fundamentals of the Modeling of Microstructure and Texture Evolution

shear criterion, one starts by imposing the plasticstrain increment dell = de along the sample ten-sile axis and�de/2 along two axes perpendicularto this axis. This strain state referred to the sam-ple axes is then transformed to the cubic axes ofthe fcc crystal using Eq 3. The particular set offive independent crystallographic shear strainsai that accommodate the imposed strains is foundnext by looking for a set of five ai’s that satisfyEq 4(b) and, at the same time, minimize the totalcrystallographic shear:

X5

i�1

ai

Knowing the minimum total crystallographicshear for a particular crystal, one can find thevalue of the uniaxial yield stress in terms ofthe critical resolved shear stress k by equatingthe strain energy from slip to the external workdone:

kX5

i�1

ai ¼ sde

or

s ¼ kX5

i�1

ai=de ¼ Mk

where M, the ratio of the sum of the crystallo-graphic shears to the uniaxial axisymmetricextension resulting from these shears, is theTaylor factor. More generally, M is defined asthe sum of the crystallographic shears per uniteffective strain.Averaging the results for M over the stereo-

graphic triangle using this isostrain procedure,Taylor predicted that the uniaxial yield stressof an isotropic fcc aggregate is equal to 3.06k.Taylor’s minimum shear criterion was initi-

ally based on intuition but was later shown byBishop and Hill (Ref 4, 5) to be rigorously cor-rect and equivalent to their procedure describedsubsequently. Taylor also did not address thequestion as to whether or not one could findstress states that will simultaneously reach thecritical resolved shear stress k on at least fiveindependent slip systems while remaining below

k on the remaining systems, a question that againwas addressed subsequently by Bishop and Hill(Ref 4, 5). Even if such stress states could befound, however, the issue of stress compatibilityacross grain boundaries remained. Nevertheless,Taylor’s isostrain model remains a significantadvance in crystal plasticity and is also currentlyused as the basis for several numerical codescapable of calculating Taylor factors for arbitraryimposed strain states rather than just axisymmet-ric deformation.

Bishop-Hill Procedure

Like that of Taylor (Ref 7), the procedure ofBishop and Hill (Ref 4, 5) and Bishop (Ref 6) isan isostrain model. However, their stress-basedapproach sought to directly find stress states thatcould simultaneously operate at least five inde-pendent slip systems. Bishop and Hill began byexamining the generalized Schmid’s lawrequirements for operation of the 12 {111}<110> fcc slip systems. They chose to describethis yielding in terms of the new stresses:

A ¼ s22 � s33;B ¼ s33 � s11;C ¼ s11 � s22;

F ¼ s23;G ¼ s31; and H ¼ s12

(Eq 5)

which leads to the 12 (24 with negatives) criti-cal resolved shear stress expressions given inTable 2.These 12 yield expressions (24 with nega-

tives) can be plotted in three separate three-dimensional stress spaces having coordinatesA, G, and H; B, H, and F; and C, F, and G.The yield condition for a particular slip systemis represented by a plane in one of the threethree-dimensional stress spaces. These planesmake equal angles to the three coordinate axes;their intersection forms three separate octahe-dral, as shown in Fig. 4. The intersection ofthese planes along lines and at corners repre-sents polyslip stress states. One notes that thereare two types of polyslip stress states: one thatis situated at the vertex of an octahedron (pointsx1 and x2 in Fig. 4) and the other that lies on aline oriented at 45� to two coordinate axes(points y1, y2, y3, and y4). The former will

simultaneously operate four slip systems, thelatter two. Hence, any polyslip stress state thatsimultaneously operates at least five slip sys-tems will actually operate six or eight. Slip willbe activated on eight slip systems if the stressstate is located at two vertices (e.g., points x1and x2) or at one vertex and two 45� lines(e.g., points x1, y1, y2). Six systems will be acti-vated if the stress state is located along three45� lines (e.g., points y1, y3, and y4). (One ver-tex and one 45� line would violate the conditionthat A + B + C = 0.) A summary of all permis-sible polyslip stress states is given in Table 2.The three cases considered previously corre-spond to stress states �2, �12, and �16. Ascan be shown with reference to the octahedralin Fig. 4, stress state �2 will operate the eightslip systems �(a2), (a3), �(b2), (b3), �(c2),(c3), �(d2), and (d3); stress state �12 will oper-ate the eight systems �(a2), (a3), �(b2), (b3),�(c2), (c3), �(d1), and (d3); and stress state�16 will operate the six systems �(b2), (b3),�(c1), (c2), �(d1), and (d3). A similar examina-tion of the remaining polyslip stress states inTable 3 will show that the stress states 1 to 12simultaneously operate eight slip systems,while stress states 13 to 28 operate six.The particular state of stress that acts to

accommodate an imposed state of strain canbe found by selecting from the 28 (56 withnegatives) permissible fcc polyslip stress statesshown in Table 3, which shows the particularstress state that maximizes the external workdone. This principle of maximum work wasfirst derived rigorously by Bishop and Hill(Ref 4, 5), who also showed it to be equivalentto the minimum shear approach taken earlier byTaylor (Ref 7). The increment of work done,with reference to the cubic axes, is:

dW ¼ s11de11 þ s22de22 þ s33de33 þ 2s12de12þ 2s23de23 þ 2s31de31

(Eq 6a)

or, in Bishop-Hill notation:

dW ¼ �Bde11 þ Ade22 þ 2Fde23 þ 2Gde31 þ 2Hde12(Eq 6b)

In the actual calculation procedure, the strainincrements deij along the cubic axes are first

Fig. 3 Decomposition of a simple crystallographic shear into a pure or tensorial shear plus a rotation

Table 2 Bishop-Hill shear stressexpressions

Slip system Yield expression

þ�(a1) A� Gþ H ¼ � ffiffiffi6

pk

þ�(a2) B� Hþ F ¼ � ffiffiffi6

pk

þ�(a3) C� Fþ G ¼ � ffiffiffi6

pk

þ�(b1) Aþ Gþ H ¼ � ffiffiffi6

pk

þ�(b2) B� H� F ¼ � ffiffiffi6

pk

þ�(b3) Cþ F� G ¼ � ffiffiffi6

pk

þ�(c1) Aþ G� H ¼ � ffiffiffi6

pk

þ�(c2) Bþ Hþ F ¼ � ffiffiffi6

pk

þ�(c3) C� F� G ¼ � ffiffiffi6

pk

þ�(d1) A� G� H ¼ � ffiffiffi6

pk

þ�(d2) Bþ H� F ¼ � ffiffiffi6

pk

þ�(d3) Cþ Fþ G ¼ � ffiffiffi6

pk

Crystal-Plasticity Fundamentals / 3

written in terms of the imposed macroscopicstrain state dek0l0 and substituted into Eq 6(b).The values of the 28 (56) permissible stressstates contained in Table 2 are then substitutedsequentially into Eq 6(b). That particular per-missible stress state that maximizes dW forthe given imposed strain state will operatethe six or eight fcc slip systems that can accom-modate this imposed strain.Like Taylor before them, Bishop and Hill

used an isostrain assumption to calculate theyield stress of an isotropic fcc crystalline aggre-gate in terms of the critical resolved shearstress k. Since, as shown by Bishop and Hill,their maximum work and Taylor’s minimumshear criteria are equivalent, not surprisinglytheir value for the uniaxial yield stress of an iso-tropic fcc crystalline aggregate was also 3.06k.

Bounds for Yield Loci fromTwo-Dimensional Sachs andBishop-Hill Averages

The isostress and isostrain approaches to cal-culating the uniaxial yield stress of an isotropicfcc aggregate in terms of the critical resolvedshear stress k can be generalized to two dimen-sions to calculate yield loci for single crystalsor ideal textural components (Ref 9). Individualsingle-crystal yield loci can then be averaged tocalculate yield loci for crystalline aggregates.These rather simple two-dimensional boundsare often sufficient to answer engineering ques-tions regarding texture hardening, for example.The basic approach is again illustrated for fccmetals deforming by rate-insensitive {111}<110> slip, but this general approach is appli-cable to other crystal structures with other slipbehaviors as well.Isostress or lower-bound yield loci for sheets

having single-crystal orientation are found by:

� Applying a biaxial principal stress state alongthe rolling and transverse directions of asingle-crystal or ideal texture component

� Calculating the resolved shear stresses on the12 {111}<110> slip systems resulting fromthis biaxial principal stress state applied alongthe rolling and transverse directions

� Setting the 12 resolved shear stresses equalto k, the critical resolved shear stress

� Plotting the 12 lines representing yielding inthe biaxial principal stress space

� Forming the lower-bound isostress yieldlocus from the intersection of these 12 lines

In general, if the rolling, transverse, andsheet normal direction are designated as r, t,and p, the six independent macroscopic stresscomponents are srr, stt,, spp, srt, stp, and spr.The stresses tns acting on the {111}<110>slip systems can then be written in termsof these macroscopic stresses using thetransformation:

Fig. 4 Generalized Schmid’s law plotted as octahedra using three separate coordinate systems

Table 3 Bishop-Hill stress states that simultaneously operate six or eight slip systems

Stress state number

Stress terms

A=ffiffiffi6

pk B=

ffiffiffi6

pk C=

ffiffiffi6

pk F=

ffiffiffi6

pk G=

ffiffiffi6

pk H=

ffiffiffi6

pk

1 1 �1 0 0 0 02 0 1 �1 0 0 03 �1 0 1 0 0 0

4 0 0 0 1 0 05 0 0 0 0 1 06 0 0 0 0 0 1

7 ½ �1 ½ 0 ½ 08 ½ �1 ½ 0 �½ 09 �1 ½ ½ ½ 0 0

10 �1 ½ ½ �½ 0 011 ½ ½ �1 0 0 ½12 ½ ½ �1 0 0 �½

13 ½ 0 �½ ½ 0 ½14 ½ 0 �½ �½ 0 ½15 ½ 0 �½ ½ 0 �½16 ½ 0 �½ �½ 0 �½17 0 �½ ½ 0 ½ ½18 0 �½ ½ 0 �½ ½19 0 �½ ½ 0 ½ �½20 0 �½ ½ 0 �½ �½21 �½ ½ 0 ½ ½ 022 �½ ½ 0 �½ ½ 023 �½ ½ 0 ½ �½ 0

24 �½ ½ 0 �½ �½ 025 0 0 0 ½ ½ �½26 0 0 0 ½ �½ ½27 0 0 0 �½ ½ ½28 0 0 0 ½ ½ ½

4 / Fundamentals of the Modeling of Microstructure and Texture Evolution

tns ¼ lnrlsrsrr þ lntlststt þ lnplspspp þ 2lnrlstsrt

þ 2lntlspstp þ 2lnplsrspr

(Eq 7)

For example, for the ideal cube-on-corner tex-ture, t is taken along ½1�10�, r is along ½11�2�,and p is along [111]; the corresponding unitvectors are:

t ¼ 1ffiffiffi2

p ði1 � i2Þ; r ¼ 1ffiffiffi6

p ði1 þ i2 � 2i3Þ; and

p ¼ 1ffiffiffi3

p ði1 þ i2 þ i3Þ

Calculating the direction cosines using theappropriate dot products and substitution intoEq 7 give the expressions in Table 4.A lower bound to the yield locus for this

ideal cube-on-corner crystal can be founddirectly from the equations in Table 4 bysetting spp = srt = stp = str = 0. The operationof each slip system will be governed by a linein srr�stt space; the intersections of these linesform the lower bound to the yield locus for theideal cube-on-corner crystal (Fig. 5).A similar procedure is used to relate the strain

incrementsalong the rolling, transverse, andsheetnormal directions of the ideal cube-on-cornerorientation and the summed contributions fromthe12 crystallographicshears,yielding:

derr ¼ 1

3ffiffiffi6

p ð�2b1 þ 2b2 � 3c1 þ c2 þ 2c3 � d1

þ 3d2 � 2d3Þ

dett ¼ 1ffiffiffi6

p ðc1 � c2 þ d1 � d2Þ

depp ¼ 2

3ffiffiffi6

p ðb1 � b2 þ c2 � c3 � d1 þ d3Þ

dert ¼ 1

3ffiffiffi2

p ð�b1 � b2 þ 2b3 � c1 þ c3 � d2 þ d3Þ

detp ¼ 1=12ð�3a1 � 3a2 þ 6a3 � b1 � b2

þ 2b3 � c1 � 3c2 þ 4c3 � 3d1 � d2 þ 4d3Þdepr ¼ 1

12ffiffiffi3

p ð9a1 � 9a2 þ 7b1 � 7b2 þ 3c1 � 5c2

þ 2c3 þ 5d1 � 3d2 � 2d3Þ(Eq 8)

These strain-slip equations are usedsubsequently.An upper bound to the yield locus for this

same crystal is be found by:

� Imposing different ratios of the principalstrains err and ett

� Transforming these principal strains to thecubic axes

� Using the maximum work principle to selectthe Bishop-Hill stress state that operates theslip systems necessary to accommodate thisstrain imposed along the cubic axes

� Transforming these Bishop-Hill stresses tothe rolling (r), transverse (t), and sheet normal

(p) coordinate system and supplementingthese stresses with the condition that spp = 0

� Plotting the results in srr�stt space

Using the maximum work principle, theresulting Bishop-Hill stress states for this crys-tal to accommodate biaxial principal strains inthe ideal cube-on-corner crystal are No. 6, 24,25, and 28, the specific stress state dependingon the imposed strain ratio. After performingthe transformation outlined previously, thereresults the upper-bound yield locus shown inFig. 6.Figures 5 and 6, together with the normality

condition (Ref 4, 5), are used to determinebounds for the biaxial stresses needed toenforce a specific strain state (in this case planestrain) as well as to illustrate the physical dif-ferences between these two bounds for theplane-strain yield stress.Applying the normality condition as shown

in Fig. 5, plane strain in the transverse direction(de horizontal), one sees that the biaxial stressstate needed to initiate this deformation is:

srr ¼ stt ¼ 3/2ffiffiffi6

pk

This stress state simultaneously operates twobranches of the yield locus; the first (horizontal)branch of the yield locus initiates yielding onsystems �(b1), +(b2), +(c3), and �(d3); the sec-ond on systems �(c2) and +(d1).Before proceeding further, it should first be

noted that the strains derr and dett that resultfrom the operation of each particular systemare normal to their respective branch of theyield locus. Focusing first on the horizontalbranch and letting slip occur by a unit amountin any one of the systems �(b1), +(b2), +(c3),and �(d3), the strain state from Eq 8 for eachof these four systems is:

derr ¼ �depp ¼ 2

3ffiffiffi6

p de and dett ¼ 0

which is normal to the horizontal branch. Simi-larly, if slip occurs by a unit amount on eithersystem �(c2) or (d1), the resulting strain is:

derr ¼ �ffiffiffi6

p

3; dett ¼

ffiffiffi6

p; and depp ¼ � 2

ffiffiffi6

p

3

which again is perpendicular to its branch of theyield locus. This two-dimensional normalityholds true even if the biaxial principal stressesactivate systems that result in shear strains aswell as normal strains; that is, the principaldirections of stress and strain do not coincide.Plane strain in the transverse direction of the

cube-on-corner crystal is achieved by combiningthe strain vectors normal to the two branchesof the yield locus to achieve the strain statederr = 0, dett = 1, and depp =�1. This is achievedby adjusting the amounts of shear on the opera-tive systems to be:

b1 ¼ d3 ¼ �ffiffiffi6

p

8; b2 ¼ c3 ¼

ffiffiffi6

p

8; c2 ¼ �

ffiffiffi6

p

2; and

d1 ¼ffiffiffi6

p

2

Substituting these values into Eq 8 results inplane strain in the transverse direction, sincederr = 0, dett = 1, and depp = �1. However, onlytwo of the shear strains, dert and detp, vanish;the remaining shear strain (depr) is equal to:

15ffiffiffi2

p

8

In order to impose a state of transverse planestrain in the absence of shear strain on thiscube-on-corner crystal, it is necessary both toalter the levels of srr and stt and to apply anadditional shear stress spr. After transformingBishop-Hill stress state No. 6:

s12 ¼ffiffiffi6

pk

to the r, t, and p coordinate system and imposingthe condition that spp = 0, one finds that:

srr ¼ffiffiffi6

p

3k;stt ¼ 5

ffiffiffi6

p

3k;spr ¼ � 2

ffiffiffi3

p

3k; and

spp ¼ srt ¼ stp ¼ 0

By substituting these stresses into Eq 8, one canverify that the critical resolved shear stress is

Table 4 Stress transformation in Bishop-Hill notation

Slip system Shear stress expression

þ�(a1) �1/ 2stp þ 1/2ffiffiffi3

pspr ¼ � ffiffiffi

6p

k

þ�(a2) �1/2stp � 1/2ffiffiffi3

pspr ¼ � ffiffiffi

6p

k

þ�(a3) stp ¼ � ffiffiffi6

pk

þ�(b1) � 2

3ffiffi6

p srr þ 2

3ffiffi6

p spp � 2

3ffiffi2

p srt � 1/6stp þ 7

6ffiffi3

p spr ¼ � ffiffiffi6

pk

þ�(b2)2

3ffiffi6

p srr � 2

3ffiffi6

p spp � 2

3ffiffi2

p srt � 1/6stp � 7

6ffiffi3

p spr ¼ � ffiffiffi6

pk

þ�(b3)4

3ffiffi2

p srt þ 1/3stp ¼ � ffiffiffi6

pk

þ�(c1) � 1ffiffi6

p srr þ 1ffiffi6

p stt � 2

3ffiffi2

p srt � 1/6stp þ 1

2ffiffi3

p spr ¼ � ffiffiffi6

pk

þ�(c2)1

3ffiffi6

p srr � 1ffiffi6

p stt þ 2

3ffiffi6

p spp � 1/2stp � 5

6ffiffi3

p spr ¼ � ffiffiffi6

pk

þ�(c3)2

3ffiffi6

p srr � 2

3ffiffi6

p stt þ 2

3ffiffi2

p srt þ 2/3stp þ 1

3ffiffi3

p spr ¼ � ffiffiffi6

pk

þ�(d1) � 1

3ffiffi6

p srr þ 1ffiffi6

p stt � 2

3ffiffi6

p spp � 1/2stp þ 5

6ffiffi3

p spr ¼ � ffiffiffi6

pk

þ�(d2)1ffiffi6

p srr � 1ffiffi6

p stt � 2

3ffiffi2

p srt � 1/6stp � 1

2ffiffi3

p spr ¼ � ffiffiffi6

pk

þ�(d3) � 2

3ffiffi6

p srr þ 2

3ffiffi6

p spp þ 2

3ffiffi2

p srt þ 2/3stp � 1

3ffiffi3

p spr ¼ � ffiffiffi6

pk

Crystal-Plasticity Fundamentals / 5

reached simultaneously on systems �(a1), (a2),�(b1), (b2), (c1), �(c2), (d1), and �(d2). Planestrain in the absence of shear strain can beachieved if the amounts of shear on these oper-ative systems are:

a1 ¼ffiffiffi6

p

6; a2 ¼ �

ffiffiffi6

p

6; b1 ¼ �

ffiffiffi6

p

2; b2 ¼

ffiffiffi6

p

2; c1 ¼ d1

¼ffiffiffi6

p

4; and c2 ¼ d2 ¼ �

ffiffiffi6

p

4

Thus, for the ideal cube-on-corner orientation,transverse plane strain in the absence of shearstress is accompanied by the additional shearstrain depr; transverse plane strain in theabsence of shear strain, on the other hand,requires the additional shear stress spr. Hence,in neither case do the principal directions ofstress and strain coincide. This lack of coinci-dence of the principal directions is notrestricted to the cube-on-corner orientation. Itshould also be pointed out that the aforemen-tioned solution for the amounts of shear neededto accommodate transverse plane strain in thisideal cube-on-corner orientation is not unique.Hence, the accompanying rotations leading totextural changes are not unique as well.There are various ways in which the slip

behaviors of multiple-crystal orientations canbe combined to find yield loci for crystallineaggregates. One such average is the originalyield locus for an isotropic fcc polycrystallineaggregate calculated by Bishop and Hill (Ref5). Their averaging process rounded the edgesthat resulted from the intersections of the linesgoverning the operation of specific slip systemsand resulted in a yield locus that is midwaybetween the Tresca and the von Mises locus.

Since the Bishop-Hill procedure is based onan isostrain model (violating stress continuityamong grains), this yield locus is an upperbound to the actual locus.

Recent Developments

The lack of a unique prediction of operativeslip systems resulting from the rate-insensitiveBishop-Hill procedure was remedied by usinga rate-sensitive model, first introduced byHutchinson (Ref 10). An example of a com-puter code that incorporates rate sensitivity isthe widely used Los Alamos polycrystal plastic-ity (LApp) code (Ref 11, 12). The LApp is amodified Taylor isostrain model that incorpo-rates not only rate sensitivity but also grainshape and relaxed constraints (Ref 12).Several slip behaviors other than <111>

{110} slip in fcc metals have been studied insome detail. It should again be pointed out that<111>{110} slip in fcc metals is equivalent to<110>{111} slip in body-centered cubic (bcc)metals (often referred to as restricted glide),since this only involves switching the identitiesof the angles l and f and hence leavesSchmid’s law unchanged. Another approach tomodeling bcc slip behavior is pencil glide,where slip is assumed to occur with equal easeon the most highly stressed planes containingthe <111> slip directions (Ref 13). This slipplane relaxation has resulted in a change ofthe Taylor factor for an isotropic crystallineaggregate from 3.06 for bcc restricted glide to2.73 for bcc pencil glide (Ref 14). The plasticdeformation of hexagonal close-packed (hcp)crystals has also been studied (Ref 15, 16).

Modeling the plastic behavior of hcp metals isinherently more complicated, because deforma-tion typically occurs by twinning in addition toslip (Ref 12).While most current computer modeling

efforts are isostrain based (Taylor or Bishopand Hill), many problems, especially thoseassociated with surface behavior, are betterdescribed using simpler crystallographicapproaches. A case in point is the characteriza-tion of surface roughening during plane-strainplastic deformation of aluminum alloy sheet(Ref 17). Orientation imaging and scanningelectron microscopy of individual surfacegrains were used to characterize surface rough-ening resulting from the presence of both slip-banded valley-forming grains and nonslip-banded hill-forming grains. What was foundwas that slip-banded valley-forming grains cor-related with Schmid factor-based unconstrainedmeasures rather than constrained measuresbased on the Taylor factor. This is but oneexample of the caution needed to select theplasticity model appropriate to the problembeing addressed.

REFERENCES

1. E. Schmid, Proc. Intern. Congr. Appl.Mech., Delft, 1924, p 342–352, J. Waitman,Jr., Delft, 1925

2. G. Sachs, Zur Ableitung einer Fliessbedin-gung, Z. Ver. deut.Eng., Vol 72, p 734–736, 1928

3. W.A. Backofen, W.F. Hosford, Jr., and J.J.Burke, Texture Hardening, Trans. ASM,Vol 55, p 264, 1962

Fig. 5 Lower-bound yield locus for ideal cube-on-corner texture. Sheetnormal is [111].

Fig. 6 Upper-bound yield locus for ideal cube-on-corner texture. Sheetnormal is [111].

6 / Fundamentals of the Modeling of Microstructure and Texture Evolution

4. J.F.W. Bishop and R. Hill, A Theory of thePlastic Distortion of a PolycrystallineAggregate under Combined Stresses, Phil.Mag., Vol 42, p 414, 1951

5. J.F.W. Bishop and R. Hill, A TheoreticalDerivation of the Plastic Properties of aPolycrystalline Face-Centered Metal, Phil.Mag., Vol 42, p 1298, 1951

6. J.F.W. Bishop, A Theoretical Examinationof the Plastic Deformation of Crystals byGlide, Phil. Mag., Vol 44, p 51, 1953

7. R. von Mises, Mechanik der plastischenFormanderung der Kristallen, Z. Angew.Math. Mech., Vol 8, p 161, 1928

8. G.I. Taylor, Plastic Strain in Metals, J. Inst.Metals, Vol 62, p 307, 1938

9. H.R. Piehler and W.A. Backofen, ThePrediction of Anisotropic Yield Loci forTextured Sheets, Textures in Research

and Practice, G. Wassermann andG. Grewen, Ed., Springer Verlag, Berlin,1969, p 436–443

10. J.W. Hutchinson, Bounds and Self-Consis-tent Estimates for Creep of PolycrystallineMaterials, Proc. Roy. Soc. London A, Vol348, No. 1652, (Feb. 10, 1976), p 101–127

11. U.F. Kocks, G.R.Canova, C.N. Tome, A.D.Rollett, and S.I. Wright, Computer CodeLA-CC-88-6 (Los Alamos, NM: LosAlamos National Laboratory)

12. U.F. Kocks, C.N. Tome, and H.-R. Wenk,Texture and Anisotropy, CambridgeUniversity Press, 1998

13. H.R. Piehler and W.A. Backofen, ATheoretical Examination of the PlasticProperties of Body-Centered Cubic Crys-tals Deforming by <111> Pencil Glide,Met. Trans., Vol 2, p 249–255, 1971

14. J.M. Rosenberg and H.R. Piehler, Calcula-tion of the Taylor Factor and Lattice Rota-tions for Bcc Metals Deforming by PencilGlide, Met. Trans., Vol 2, p 257–259, 1971

15. D.R. Thornburg and H.R. Piehler, AnAnalysis of Constrained Deformation bySlip and Twinning in Hexagonal Close-Packed Metals and Alloys, Met. Trans. A,Vol 6A, p 1511–1523, 1975

16. C. Tome and U.F. Kocks, The Yield Sur-face of HCP Single Crystals, Acta Metall,Vol 33, p 603–621, 1985

17. Y.S. Choi, H.R. Piehler, and A.D. Rollett,Formation of Mesoscale Roughening in6022-T4 Al Sheets Deformed in Plane-Strain Tension, Met. and Mat. Trans. A,Vol 35A, p 513–524, 2004

Crystal-Plasticity Fundamentals / 7