banding in single crystals

Banding in single crystals during plastic deformation

M.Arul Kumar a ,Sivasambu Mahesh a ,b ,?

a

Department of Mechanical Engineering,Indian Institute of Technology,Kanpur 208016,India b Department of Aerospace Engineering,Indian Institute of Technology,Kanpur 208016,India

a r t i c l e i n f o Article history:Received 29October 2011Received in ?nal revised form 26February 2012Available online 16March 2012Keywords:Crystal plasticity Single crystal Macroscopic shear band Regular deformation band Dislocation boundary

a b s t r a c t

A rigid-plastic rate-independent crystal plasticity model capable of capturing banding in

single crystals subjected to homogeneous macroscopic deformation is proposed.This

model treats the single crystal as a ‘stack of domains’.Individual domains deform homoge-

neously while maintaining velocity and traction continuity with their neighbors.All the

domains collectively accommodate the imposed deformation.The model predicts lattice

orientation evolution,slip distribution,strain localization and band orientation in copper

single crystals with imposed plane strain deformation.In quantitative agreement with

experimental observations reported in the literature,macroscopic shear banding and reg-

ular deformation banding are predicted in initially copper and rotated cube oriented single

crystals,respectively,while banding is not predicted in initially Goss oriented single crys-

tals.The model does not,however,predict the experimentally observed orientation of

smaller scale dislocation boundaries such as dense dislocation walls.

ó2012Elsevier Ltd.All rights reserved.1.Introduction

1.1.Banding

Polycrystal models that predict mechanical response and texture evolution during plastic deformation commonly regard each grain of the polycrystal as a homogeneously deforming entity (Taylor,1938;Lebensohn and Tomé,1993;Van Houtte et al.,1999;Mahesh,2009,2010).However,an enormous body of experimental metallurgical literature indicates that defor-mation of grains is inhomogeneous.The inhomogeneity of deformation leads to grain subdivision into misoriented domains called bands,which are demarcated by dislocation boundaries.Dislocation boundaries are of two kinds:incidental and geo-metrically necessary.Incidental dislocation boundaries are comprised of statistically trapped dislocations.Geometrically necessary dislocation boundaries,on the other hand,are necessary for accommodating incompatibility of plastic deforma-tion (Nye,1953).The volumes delineated by these boundaries are called bands.They are classi?ed based on their size-scale and structure,e.g.,Gil Sevillano et al.(1980)and Hansen and Juul Jensen (1999).The present work focuses on the formation and evolution of two types of bands,viz.,regular deformation bands (RDBs)and macroscopic shear bands (MSBs).The lon-gest dimension of RDBs and MSBs is comparable to the grain size.Smaller scale geometrically necessary dislocation bound-aries,which delineate regions whose longest dimension is of the order of tens of dislocation mean free paths,called dense dislocation walls/microbands (DDW-MBs)are also of interest presently.

The constraint experienced by an individual grain in a polycrystal is mediated by the grains surrounding it and generally differs from the macroscopic constraint imposed upon the polycrystal.However,single crystals subjected to plastic deformation experience the imposed macroscopic constraint.This makes single crystals better suited for the study of band-ing under imposed homogeneous deformation.

0749-6419/$-see front matter ó2012Elsevier Ltd.All rights reserved.https://www.wendangku.net/doc/df8083149.html,/10.1016/j.ijplas.2012.03.008

?Corresponding author at:Department of Mechanical Engineering,Indian Institute of Technology,Kanpur 208016,India.

E-mail address:smahesh@iitk.ac.in (S.Mahesh).

16M.Arul Kumar,S.Mahesh/International Journal of Plasticity36(2012)15–33

1.2.Models of banding

Van Houtte et al.(1979)and Dillamore et al.(1979)proposed that strain localization occurs within MSBs when the effec-tive hardening rate therein becomes negative on account of the local geometrical softening due to lattice rotation being in excess of the local hardening rate of the slip systems.A rigorous analytical methodology embodying this physical notion was given by Asaro(1979),formally based on the considerations of Biot(1965)and Hill and Hutchinson(1975).In this approach, banding was identi?ed with the existence of inhomogeneous solutions of the equations governing the stress-rate?elds in the crystal.As the governing equations are not directly solvable,Asaro(1979)considered a reduced problem wherein the inhomogeneity is con?ned within a rectangular strip,or band,that cuts across the crystal.The inhomogeneous?elds that solve the reduced problem are obtained either directly(Asaro,1979)or by phrasing the problem in variational terms.The latter approach,which also allows for the energetic contribution of band boundaries,has been taken by Kratochvíl et al. (2007).

The analytical approach(Asaro,1979;Kratochvíl et al.,2007)yields conditions for the onset of banding in closed form at practically no computational cost for speci?c lattice orientations.It is ideally suited to lattice orientations wherein only a few slip systems are activated and slip system hardening has a simple form,as in Asaro(1979)and Kratochvíl et al.(2007).This approach becomes highly laborious for lattice orientations undergoing general multi-slip and non-uniform hardening.

Studies based on the crystal plasticity?nite element method(CPFEM)have sought to capture the development of grain-level deformation inhomogeneity for grains of arbitrary crystallography undergoing general multi-slip with possibly non-uniform hardening(Pierce et al.,1982;Raabe et al.,2004;Rezvanian et al.,2006;Kuroda and Tvergaard,2007;Si et al., 2008;Groh et al.,2009;Zhang et al.,2009;Kanjarla et al.,2010).An initial perturbation in model single crystal sample geom-etry(Pierce et al.,1982;Zhang et al.,2009),local lattice orientation(Raabe et al.,2004),shear imbalance(Rezvanian et al., 2006)or slip system hardness(Kuroda and Tvergaard,2007)triggers the development of local slip and/or local lattice ori-entation inhomogeneities.Pierce et al.(1982)predicted a band of higher plastic strain than the matrix and associated it with shear banding observed experimentally.More recently,Rezvanian et al.(2006)and Si et al.(2008)have identi?ed RDBs with-out intense shear localization in their initially cube oriented simulated aluminum crystals.Under imposed plane strain deformation,they?nd good agreement between their predicted lattice orientation variation along the normal direction with the experimental observations of Liu and Hansen(1998).Kanjarla et al.(2010)predicted RDB formation in a nearly rotated cube oriented grain in a model columnar polycrystal.They found the RDB pattern in their model grain to be affected by the inter-granular interactions.

CPFEM simulations,however,come at a high computational cost.Spatial resolution of the material response in millimeter scale single crystals wherein much of the deformation is con?ned to shear bands,each only a few microns wide(Wróbel et al.,1994;Wagner et al.,1995),requires a?ne mesh(Pierce et al.,1983;Anand and Kalidindi,1993),higher order elements (Kuroda and Tvergaard,2007)or the explicit treatment of large strain gradients across shear band boundaries(Anand et al., 2012).Spatial resolution of the deformation?eld in single crystals undergoing homogeneous deformation or those forming RDBs without intense shear localization is more computationally tractable(Rezvanian et al.,2006;Si et al.,2008;Kanjarla et al.,2010),as a coarser mesh suf?ces to resolve the smaller strain gradients.

Approaches to predict banding in single crystals that lie in between the computationally light but laborious analytical approach and the computationally intensive but general CPFEM approach are provided by crystal plasticity based models. These are two to three orders of magnitude computationally less intensive than CPFEM and yet suf?ciently general to treat arbitrary slip activity and hardening laws.A class of such models capable of predicting the formation of arbitrarily oriented shear bands on the basis of existence of inhomogeneous solutions to the governing equations is due to Needleman and Rice (1978)and Kuroda and Tvergaard(2007).Another class of such models,due to Chin et al.(1969),Lee and Duggan(1993),Lee et al.(1993),Ortiz and Repetto(1999)and Ortiz et al.(2000)is based on experimental observations of patchy slip in grains, whereby deformation by single slip in spatially separate domains is observed in preference to homogeneous multi-slip.The model of Lee and Duggan(1993)and Lee et al.(1993)predicts regular deformation banding by seeking to reduce the work of plastic deformation.The variational theory of Ortiz and Repetto(1999)seeks to minimize a functional that,in effect,depends on both plastic power and the rate of plastic power.The latter term incorporates the latent hardening.The variational theory predicts banding because the functional can be reduced by lowering latent hardening,which in turn is accomplished by replacing multislip activity in a homogeneously deforming crystal with single slip activity within spatially isolated bands.

Another class of crystal plasticity based models capable of predicting regular deformation banding and macroscopic shear banding has been developed in foregoing works(Mahesh and Tomé,2004;Mahesh,2006).In these works,a banded single crystal is represented as a pair of regions that model a pair of compatibly deforming bands.The lattice orientation of both regions are initially identical apart from a small perturbation.The growth of this perturbation with deformation,called ori-entational instability(Mahesh,2006),is identi?ed with banding,while its diminution with further deformation,called ori-entational stability,points to the dominance of a homogeneous deformation mode.

While the aforementioned models and the present model for the inhomogeneous deformation of a single crystal have an energetic basis,a non-energetic reason for banding,viz.,Taylor ambiguity,was suggested by Chin and Wonsiewicz(1969) and termed Type1banding by Chin(1971).Taylor ambiguity refers to the possibility of accommodating the imposed defor-mation by activating different set of slip systems in different parts of the single crystal or grain.Leffers(2001a,b)suggested that Taylor ambiguity may underlie the experimentally observed subdivision of grains by DDW-MBs.For two assumed ori-entations of the band boundaries,he investigated the possible solutions for the slip rates in a pair of compatibly deforming

M.Arul Kumar,S.Mahesh/International Journal of Plasticity36(2012)15–3317 bands caused by Taylor ambiguity,subject to the constraint that a total of8slip systems must activate in both bands to accommodate the imposed plane strain deformation.

1.3.Present work

A model capable of predicting the formation and evolution of MSBs and RDBs during plastic deformation of copper single crystals is presently developed.This model is based on the recently proposed‘stack of domains’model(Arul Kumar et al., 2011).The single crystal is regarded as a one-dimensional stack of domains that collectively accommodate the imposed deformation.The domains deform homogeneously and compatibly with their neighbors.Notable differences with the‘stack of domains’model are:(1)sub-structure based hardening of domain slip systems,(2)introduction of rules to identify MS

B and RDB domains and(3)accounting for mobility of domain boundaries.

The present model is used to simulate plane strain compression of copper single crystals.In agreement with experimental observations reported in the literature,the model predicts the formation of MSBs and RDBs in the initially copper and rotated cube oriented single crystals,respectively,and the formation of neither in the Goss oriented single crystal.

2.Experimental evidence

Relevant facts from the experimental literature about three types of bands,viz.,dense dislocation walls-microbands (DDW-MBs),macroscopic shear bands(MSBs)and regular deformation bands(RDBs),are now presented.

2.1.Dense dislocation walls and microbands(DDW-MBs)

Dislocation structures called dense dislocation walls(DDW)and microbands(MBs)that bound regions a few dislocation mean free paths long have been reported in plastically deformed copper.Bay et al.(1989)noted that DDWs and MBs appear together as if forming one general feature,which they called DDW-MBs.These structures,across which the lattice misori-entation is of the order of a few degrees,are mobile relative to the crystal material(Albou et al.,2010).

Depending on the lattice orientation and the imposed deformation,DDW-MBs may or may not lie parallel to active slip planes(Hughes and Hansen,1993;Liu et al.,1998).Winther et al.(1997)have attempted to correlate the formation of crys-tallographic DDW-MBs with the coplanar slip fraction(CSF),which they de?ned as the ratio of the largest accumulated slip in a crystallographic slip plane to the total accumulated slip in a grain.They observed experimentally that CSF P0.45in about75%of rolled aluminum grains,which form crystallographic DDW-MBs parallel to the most active crystallographic slip plane and that CSF<0.45in about75%of the experimentally studied grains forming non-crystallographic DDW-MBs.The CSF thus provides a good indicator for the formation of crystallographic DDW-MBs.The CSF indicator,however,misclassi?es about25%of the grains.

2.2.Macroscopic shear bands

Crystallographic DDW-MBs aligned with the dominant crystallographic slip plane act as barriers to slip systems non-coplanar with the dominant crystallographic slip plane.Non-coplanar slip activity therefore concentrates in regions where the crystallographic DDW-MBs barriers are few or weak and results in severe localized deformation in these regions.The sub-structural elements formed during such localized deformation are the long and narrow microscopic shear bands(Nakay-ama and Morii,1982;Duggan et al.,1978).In f.c.c.copper single crystals deformed quasistatically at room temperature,the presence of a single set of pre-existing DDW-MBs parallel to the dominant crystallographic slip plane is a necessary condi-tion for microscopic shear banding(Nakayama and Morii,1982).

A macroscopic shear band(MSBs)in a single crystal is comprised of a cluster of microscopic shear bands.It typically ex-tends across the crystal and undergoes much more slip than the surrounding material,called the matrix(Duggan et al.,1978; Morii and Nakayama,1981;Nakayama and Morii,1982;Hatherly and Malin,1984;Wagner et al.,1995;Jasienski et al., 1996).The typical misorientation of an MS

B with the matrix is of the order of tens of degrees.

Microscopic shear bands,once nucleated,propagate to their?nal dimensions rapidly and remain inoperative thereafter (Duggan et al.,1978;Hatherly and Malin,1984;Wagner et al.,1995);nucleation and propagation of new microscopic shear bands elsewhere within the MSB is required to accommodate its subsequent deformation.The direction of propagation of microscopic shear bands evolves gradually during the deformation resulting in apparent mobility of the MSB boundaries (Jasienski et al.,1996).Both microscopic shear band and MSB boundaries do not have simple crystallographic alignments (Wagner et al.,1995;Jasienski et al.,1996).

Various types of dislocation structures have been reported within MSBs in copper single crystals deformed quasistatically at room temperature:microbands parallel to the MSB walls(Wróbel et al.,1994),thin elongated cell structure(Morii and Nakayama,1981),equiaxed cells(Paul et al.,2010),dynamically recovered cell structure(Wróbel et al.,1996)and no appar-ent cell structure(Korbel and Szczerba,1982).Korbel and Szczerba(1982)have suggested that enhanced dynamic recovery (D.R.-2)occurs in the MSB due to the activation of slip systems not coplanar with pre-existing dislocation structures.Blichar-ski et al.(1995)have suggested that dynamic recovery enhancement in MSBs may be aided by temperature rise,which may accompany rapid local strain accumulation even in thermally conductive copper.Finally,Huang et al.(2006)have reported

deformation twinning within shear bands formed in coarse-grained copper at room temperature deformed quasistatically.They have suggested that the high resolved shear stress required for deformation twinning under these conditions is realized within MSBs after substantial hardening of the MSB material.

2.3.Regular deformation bands

A single crystal may divide into lath-shaped regions called regular deformation bands,each several tens of dislocation mean free paths wide and extending across the entire crystal.The deformation and lattice orientation within each region is approximately uniform and distinct from that of its neighbors.Intense shear localization is not normally associated with such regular deformation banding.RDBs have been variously termed type-2bands (Chin,1971),deformation bands (Lee et al.,1993),matrix bands (Liu and Hansen,1998),primary regular deformation bands (Kulkarni et al.,1998;Kuhlmann-Wilsdorf,1999)and special bands of secondary slip (Wert et al.,2005).

In f.c.c.copper and aluminum,RDBs develop from geometrically necessary dislocation walls,which form along {110}planes perpendicular to the direction of predominant slip (Cahn,1951;Heye and Sattler,1971;Cizek et al.,1995).Misori-entation between RDBs progressively increases with deformation (Barrett,1939,1940)and may be of the order of tens of degrees (Akef and Driver,1991;Liu and Hansen,1998).RDB boundaries also deviate increasingly from the crystallographic {110}plane (Cahn,1951)with continuing deformation.Regular deformation banding of single crystals typically begins in the early stages of deformation and the number and volume of bands does not change with further deformation (Cahn,1951;Heye and Sattler,1971;Lee et al.,1993;Wert et al.,2005).Unlike DDW-MBs and MSBs,RDB boundaries are immobile rel-ative to the material (Wert et al.,2005).Regular deformation banding is observed only in the absence of pre-existing crys-tallographic DDW-MBs (Wróbel et al.,1994;Liu and Hansen,1998).However,once formed,slip is predominantly con?ned to a single slip plane and results in the formation of a set of crystallographic DDW-MBs (Wróbel et al.,1988;Liu and Hansen,1998).

3.Model

3.1.Standard rate-independent crystal plasticity

Every material point of a single crystal is assumed to follow the standard rigid-plastic rate-independent volume preserv-ing constitutive response (Taylor,1938;Bishop and Hill,1951;Kocks et al.,1998).If the slip-rate tensor at the material point is denoted by L ss ,

L ss ?P S s ?1

_c s b s n s ;e1T

where _c

s denotes the slip-rate of slip system s 2{1,2,...,S }and b s is the Burgers vector or unit slip directional vector of slip system s with slip plane normal n s .b s and n s depend upon the lattice orientation of the material point described completely by the orthonormal tensor X ,which transforms the vectors B s and N s that describe the unit slip directional vector and slip plane normal of slip system s in the crystal coordinate system into the b s and n s in the reference coordinate system following

b s ?X B s ;

and n s ?X N s :e2TThe symmetric part of the slip-rate tensor,L ss ,is the strain-rate tensor,_

,_ ?L ss tL T ss =2?P

S s ?1_c s m s ;e3T

where,m s denotes the Schmid tensor of slip system s

m s ?eb s n s tn s b s T=2:e4T

If L denotes the velocity gradient at the material point,the lattice spin tensor,_W

c of the material point,is given by (Kocks et al.,1998)

_W

c ?skew eL àL ss T:e5TThe deformation gradient of a material point,F ,describes the local shape an

d evolves following th

e ?ow law (Gurtin,1981),_F ?LF :e6TLet r denote the deviatoric part o

f the Cauchy stress at the material point.Schmid’s law (Kocks et al.,1998)states that slip system s may have non-zero slip-rate only if the resolved shear stress,r :m s tr (r m s T ),equals the critical resolved shear stress (CRSS)on slip system s ,s s .Thus,

_c s P 0;if r :m s ?s s ;

?0;if r :m s

18M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–33

Taylor’s principle (Taylor,1938)asserts that for given _

of all possible slip-rate combinations f _c s ;s 2f 1;...;S gg that respect the constraint given by Eq.(3),those which minimize the plastic power of deformation

P ?P

S s ?1s s _c s e8T

are preferred.Chin and Mammel (1969)have shown that the set of slip-rates,_c

s ,which minimize P subject to the constraint given by Eq.(3)automatically satis?es Eq.(7).f _c

s ;s 2f 1;...;S gg that minimizes P given by Eq.(8)subject to the constraint Eq.(3)may be non-unique.In this case,the consistency condition and algorithm for its implementation given by Anand and

Kothari (1996)is invoked to ensure that _c

s >0in the largest number of slip systems.Slip systems harden with deformation.The evolution of the critical resolved shear stress of the s th slip system is taken to follow (Hill,1966;Kocks et al.,1998)

_s s ?d s d C P S s 0?1H ss 0_c s 0;e9T

where [H ]is the latent hardening matrix,

C ?P

S s ?1c s ;e10T

is the total accumulated slip at the material point and c s is the accumulated slip in the s th slip system.In the present work,s is taken to follow the extended Voce law given by Toméet al.(1984),

s eC T?s 0tes 1th 1C T?1àexp eàC h 0=s 1T ;

e11T

where s 0,s 1,h 0and h 1are material hardening parameters.

3.2.The ‘stack of domains’model of a single crystal 3.2.1.Geometry

A model of a rigid-plastic rate-independent single crystal capable of representing banding is presently described.The sin-gle crystal,shown schematically in Fig.1,is represented as a stack of N parallelepiped-shaped domains each of which is en-dowed with a uniform lattice orientation and is assumed to deform homogeneously.A single domain or a group of neighboring domains may represent a band.The geometric shapes of the individual domains are assumed to be identical,so that the volume fraction q [l ]of the l th domain is simply q [l ]=1/N ,for l =1,2,...,N .Stacking is assumed to be repeated peri-odically so that neighbors of domain l in the stack are (l +1mod N )and (l à1mod N ).Domain boundaries between adjacent parallelepiped-shaped domains are assumed to be planar and identically oriented.For convenience,the domain boundary between domains (l mod N )and (l +1mod N )is numbered as the l th domain boundary.Quantities with superscripts en-closed in square brackets,e.g.,{á}[l ],are associated with domain l while,quantities with superscripts enclosed in parenthe-ses,e.g.,{á}(l ),are associated with domain boundary l

.

M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–3319

All the domain boundaries in the ‘stack of domains’model are assumed to be oriented identically with normal m ,as shown in Fig.1.This is reasonable,since only the boundaries enclosing domains representing MSBs or RDBs,which are experimentally observed (Section 2)to be delineated by approximately parallel boundaries,are physically signi?cant.This assumption implies that only one set of bands can be predicted by the present model even in crystals wherein deforma-tion bands at different length scales simultaneously form.Moreover,the predicted bands will necessarily be the largest bands in the single crystal under consideration.Two sets of cartesian coordinates:the sample coordinate system xyz and a domain boundary ?xed coordinate system XYZ with the Y -axis always aligned with the normal vector m are also shown.

The present geometric arrangement of domains within a single crystal is similar to the geometric arrangement of grains within a polycrystalline sub-aggregate studied by Arul Kumar et al.(2011).

https://www.wendangku.net/doc/df8083149.html,ttice orientation perturbation

If the lattice orientation in all the domains comprising the stack were identical,the ‘stack of domains’model would rep-resent a perfect single crystal.In this case,each domain would undergo the same deformation as the single crystal,i.e.,single crystal would deform homogeneously.However,even in an initially perfect single crystal,lattice orientation perturbations arise during plastic deformation.Across incidental dislocation boundaries,for instance,Hughes et al.(1997,1998)have observed that the average absolute misorientation angle hj x ?ji scales with von Mises strain vM as

hj x ?ji ?k 1=2vM :e12TThe misorientation rate follows from Eq.(12),and is given by,

hj _x ?ji ?k _ vM 2???????? vM

p ;e13T

where,the von Mises strain-rate,_ vM ,is given by _ vM ???????????????????23tr e_ 2Tr :e14T

Following Eq.(13),the lattice rotation rate in the l th domain is prescribed as

_x ?l ??p k _ vM 4???????? vM p sin 2p l ;e15T

as schematized in Fig.1.This lattice rotation is about a unit misorientation vector,^m

,uniformly distributed on the unit sphere.The tensorial lattice spin of the l th domain caused by the lattice orientation perturbation is thus

_W ?l ??_x ?l ?e^m ?T;e16Twhere,^m

?denotes the skew-symmetric tensor whose axial is the vector ^m .It turns out that the state evolution of the ‘stack of domains’model is not sensitive to the functional form assumed in Eq.

(15).Even if the _x

?l ?were drawn from a uniform distribution of mean hj _x ?ji given by Eq.(13),the predictions of the present model,given in Section 4,remain practically unchanged.The role of the lattice orientation perturbation is simply to trigger the initial slip inhomogeneity amongst domains in the numerical calculation.

3.2.3.Kinematics Let L denote the macroscopic velocity gradient of the ‘stack of domains’representing the model single crystal.L is taken to be the volume fraction weighted average of the velocity gradients,L [l ],of individual domains,l ,i.e.,

P

N l ?1q ?l L ?l ?L :e17T

The shape of the model single crystal represented by stack of domains is described by deformation gradient,F .The ?ow law,which determines the evolution of F parallels Eq.(6)and is _F ?L F :e18T

3.2.

4.Boundary and continuity conditions

Let L imp be the velocity gradient imposed upon the single crystal that must be collectively accommodated by its domains,i.e.,

L ?L imp :

e19TLet _ imp ?eL imp tL imp T T=2be the imposed strain-rate on the single crystal.It then follows from Eqs.(3),(5),(17)and (19)that

20M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–33

P

N l ?1q ?l _ ?l ?_ imp :e20T

Let p [l ]denote the scalar hydrostatic pressure in domain l so that the Cauchy stress therein is p [l ]I +r [l ].Traction continuity conditions between neighboring domains in the model in the XYZ coordinate system are then:

r ?

l mod N XY ?r ?l t1mod N XY ;

r ?

l mod N YZ ?r ?l t1mod N YZ ;

p ?l mod N tr ?l mod N YY ?p ?l t1mod N tr ?l t1mod N YY :e21T

It is not required to enforce the third equation of Eq.(21)explicitly because its satisfaction is automatic:It can be shown

(Arul Kumar et al.,2011)that the third equation of Eq.(21)can always be satis?ed for arbitrary f r ?l YY :l 2f 1;2;...;N gg by

suitable choice of the domain hydrostatic pressure components {p [l ]:l 2{1,2,...,N }}.Moreover,p [l ]does not affect the ri-gid-plastic volume-preserving deformation of the domains.

Traction continuity and compatibility conditions between neighboring domains that must be explicitly enforced in the model are thus (Arul Kumar et al.,2011):

r ?

l mod N XY ?r ?l t1mod N XY ;

r ?l mod N YZ ?r ?l t1mod N YZ ;

_ ?

l mod N XX ?_ ?l t1mod N XX ;

_ ?

l mod N ZZ ?_ ?l t1mod N ZZ ;and

_ ?

l mod N XZ ?_ ?l t1mod N XZ ;e22T

for l 2{1,2,...,N }.

Following Hill (1961),the velocity continuity condition between domains l and l +1can be written as

s L el Tt ?L ?l t1mod N àL ?l mod N ?k el T m ;

e23T

where k (l ),Hadamard’s characteristic segment of domain boundary l ,is given by (Mahesh,2006)k el T?2s _

el Tt m tes _ el Tt m ám Tm :e24TEq.(19)may be written using Eq.(5)as

P

N l ?1q ?l L ?l ss t_W ?l c ?L imp :e25TAlgebraic manipulation detailed in Arul Kumar et al.(2011)of Eqs.(23)and (25)using Eqs.(24)and (5)yields

_W ?l c ?skew L imp àL ?l ss tU ?l àU imp ;

e26T

where,U ?l ??2_

?l m àe_ ?l m ám Tm m and U imp ??2_ imp m àe_ imp m ám Tm m :e27T

By comparing Eqs.(26)and (5)it is seen that the lattice spin of domain l in the ‘stack of domains’model subjected to imposed velocity gradient L imp differs from that of a material point subjected to the same velocity gradient by skew(U [l ]àU imp ).It also follows from Eq.(27)that P N l ?1q ?l skew eU ?l àU

imp T?0.Therefore,skew(U [l ]àU imp )represents the local deviation of the lattice spin of domain l from that of the average (Kocks et al.,1998,§11.4.2).Parallel to Eq.(24),this deviation can be written as

skew eU ?l àU imp T?K ?l m ;e28T

where,K ?l ?2e_

?l à_ imp Tm tee_ ?l à_ imp Tm ám Tm denotes the Hadamard characteristic segment between two material points deforming with strain-rates _

?l and _ imp compatibly across an interface oriented normal to m .The net lattice spin in the l th domain is simply the sum of the lattice spins imposed by the lattice orientation perturbation and by the requirement for compatibility across domain interfaces.Thus,

_W ?l ?_W ?l ?t_W ?l c ;e29T

where,the individual terms on the right side are given by Eqs.(16)and (26).An algorithm to solve for the slip rates,_c

s ,in all the domains subject to the conditions imposed by Eqs.(7),(20)and (22),and consistency conditions (Havner,1992)has been developed previously (Arul Kumar et al.,2011).

M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–33

21

3.3.Orientational stability

The notion of orientational stability proposed by Mahesh(2006)is now extended to the‘stack of domains’model.Let X[l mod N]and X[l+1mod N]be the orthonormal lattice orientation tensor of the domains l mod N and l+1mod N separated by domain boundary l that transform vectors from the crystal coordinate system to the sample coordinate system following Eq.(2).Then the misorientation vector between these domains is given by(Kocks et al.,1998)

^melT?axialeskeweX?l mod N X?lt1mod N TTT:e30TThe axial vector of the rate of misorientation between these neighboring domains is

^welT?axial f_W?l mod N à_W?lt1mod N g;e31Twhere_W?l mod N and_W?lt1mod N denote the lattice spin rates of domains l mod N and l+1mod N given by Eq.(26).The do-main boundary l between domains l mod N and l+1mod N is said to be orientationally unstable if lelT?^melTá^welT>0;e32Tand orientationally stable otherwise.

Domain boundaries that remain orientationally stable throughout the deformation separate domains of negligible mis-orientation.They,therefore,do not correspond to physical geometrically necessary dislocation boundaries.On the other hand,domain boundaries that are orientationally unstable,during part or whole of the deformation process may separate misoriented domains and may represent geometrically necessary physical band boundaries.Also,because of the periodicity in the stacking of domains,assumed in Section3.2.1,exactly one domain boundary cannot be orientationally unstable;the number of orientationally unstable domain boundaries can only be zero,two or more.

3.4.Sub-structure

3.4.1.Slip system dominance

As noted in Sections2.2and2.3,the presence of crystallographic DDW-MBs is a pre-requisite for macroscopic shear band-ing,while the absence of crystallographic DDW-MBs is necessary for regular deformation banding.Sub-structure is not di-rectly represented in the present model;DDW-MBs are not explicitly evolved in the course of the simulated deformation of the‘stack of domains’model.The presence or absence of crystallographic DDW-MBs in a domain is inferred from the dom-inance or non-dominance of slip activity of a single crystallographic slip plane in that domain,respectively.

A quantitative criterion for the formation of crystallographic DDW-MBs based on slip activity in crystallographic slip planes,suggested by Winther et al.(1997),has been discussed in Section2.1.This criterion is unable to discriminate between crystallographic and non-crystallographic DDW-MBs for about25%of the grains studied by Winther et al.(1997).Another measure of slip concentration in a crystallographic slip plane,which accounts for simultaneously activated cross-slip sys-tems is,therefore,presently suggested.Two slip systems s and t are said to be simultaneously activated cross-slip systems if_c s>0,_c t>0,b s?b t and n s–±n t.Screw dislocations gliding in one of a pair of simultaneously activated cross-slip systems may readily cross-slip into the other.They are therefore unlikely to get trapped in their original glide plane.Let S p denote the set of slip systems in the crystallographic slip plane p with normal n p,whose cross-slip systems are not simultaneously activated.Thus,

S p?f s:n s??n p;_c s_c t?0if b s?b t and n s–?n t ge33Tdenotes the set of slip systems that may contribute dislocations to DDW-MBs aligned parallel to the plane p.Even though the mechanism of cross-slip only applies to the screw component of gliding dislocations,for simplicity,both screw and edge dis-location contributions from simultaneously activated cross-slip systems are neglected in Eq.(33).

The effective total slip activity,_C p,contributing to DDW-MB formation in the crystallographic slip plane p is then

_C p ?

P

s2S p

j_c s j:e34T

Let_C p?and_C p??be the total slip rates of the most active and second most active crystallographic slip planes,p?and p??, respectively.The criterion for dominance of the crystallographic slip plane p?,which is also taken to be the criterion for the formation of DDW-MBs parallel to p?in pure copper is(Mahesh,2006)

_C

p?

_C

p??

P1:2:e35T

3.4.2.Sub-structure based hardening

A sub-structure comprised of dislocation cells or non-crystallographic DDW-MBs is referred to as type(i)and a sub-struc-ture comprised of crystallographic DDW-MBs as type(ii).Non-satisfaction and satisfaction of Eq.(35)in a domain of a model copper single crystal are taken to imply the formation of type(i)and type(ii)sub-structures therein,respectively.It has been 22M.Arul Kumar,S.Mahesh/International Journal of Plasticity36(2012)15–33

noted in Section 2.2that crystallographic DDW-MBs interrupted by microscopic shear bands is one type of sub-structure observed within MSBs.This is termed the type (iii)sub-structure.Type (ii)and type (iii)sub-structures are schematically shown in Figs.2(a)and (b),respectively.Plastic anisotropy is accounted for by assigning different hardening matrices,

[H (i)],[H (ii)]and [H (iii)]in Eq.(9)to each of the three sub-structure types.

In a domain with type (i)sub-structure,plastic anisotropy is assumed negligible,i.e.,all slip systems are assumed to hard-en equally.Thus,

H ei Tss 0?1;s ;s 02f 1;2;...;S g :e36TIn a domain with type (ii)sub-structure,shown in Fig.2(a),DDW-MBs parallel to plane p ?act as directional barriers to slip and cause anisotropy in the domain’s plastic response.In such domains,the CRSS of slip systems intersecting p ?will exceed that of slip systems parallel to p ?.To re?ect this,the elements of the hardening matrix,[H (ii)]are taken as

H eii Tss 0?a P 1;if n s –?n p ?;n s 0??n p ?and s –s 0;

1;otherwise : e37T

As noted in Section 2.2,enhanced dynamic recovery and/or deformation twinning may occur within MSBs.This may reduce the hardening rate of slip systems and alter the plastic anisotropy in the MSB material.Neglecting the latter effect for sim-plicity,it is presently assumed that the hardening rate [H (iii)]in an MSB domain is proportional to [H (ii)]:

H eiii Tss 0?v H eii Tss 0;s ;s 02f 1;2;...;S g ;e38Twhere,06v 61indicates a reduced hardening rate in MSB domains.It is emphasized that Eq.(38)does not imply softening of the MSB domain;only a reduction in the hardening rate is proposed.Also,the reduced hardening rate given by Eq.(38)is applied only after MSB nucleation;it is not therefore responsible for MSB nucleation.Finally,it is assumed in Eq.(38)that the mechanisms responsible for reduced hardening of MSB material become active immediately upon MSB nucleation;in reality,these may only be activated after substantial MSB deformation.

3.5.Identi?cation of banding

A contiguous set of domains D ?f l ;l t1;...;m g is taken to represent a band if their slip pattern and lattice rotations are similar,but differ from that in other domains,denoted D c .A robust and reliable indication of divergence of the slip pattern and lattice rotations of D from that of D c is provided by orientational instability of the two domain boundaries separating D from D c ,i.e.,l (l à1)>0and l (m )>0.Orientationally unstable domain boundaries thus represent band boundaries and the set of domains contained between a pair of nearest orientationally unstable domain boundaries represent a band.

The decision tree followed to identify a set of contiguous domains D as a particular type of band is shown in Fig.3.Do-mains identi?ed as constituents of a band are assumed to remain so during subsequent deformation.

3.6.Domain boundary orientation

In order to capture the experimentally observed mobility of band boundaries noted in Sections 2.1and 2.2,domain boundaries are assumed mobile relative to the crystal material.Following Mahesh (2006),their orientation m is taken to be that which minimizes the plastic power of the model crystal,i.e.,

m ?argmin m ?P N l ?1q ?l P S s ?1s ?l s _c ?l s em ?T;e39T

subject to the constraints given by Eqs.(3),(20)and (22).Since the objective function in Eq.(39)is non-smooth,a section search method (Chang,2009)is employed for minimization over the unit hemispherical surface spanned by m ?.Predicted domain boundaries need not coincide with {111}planes.

It was noted in Section 2.3that RDB boundaries are not mobile relative to the crystal material and therefore,evolve in conformity with the shape of the crystal.Eq.(39)therefore,is not applied to model crystals containing RDB bands.Instead,if m 0is the domain boundary normal and 0the deformation gradient at the instant of RDB formation,the domain boundary orientation m when the crystal deformation gradient becomes F is (Gurtin,1981

)

M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–3323

m ?F T 0F

àT m 0k 0àT m 0k :e40T

In crystal orientations forming RDBs,Eq.(39)applies until the instant of banding and Eq.(40)thereafter.m 0is,therefore,gi-ven by Eq.(39).

4.Results

4.1.Initial lattice orientation

The banding response of copper single crystals subjected to plane strain compression in the plane containing the rolling direction (RD)and the normal direction (ND)is studied.The specimen dimension along the transverse direction (TD)is main-tained constant.Three experimentally well-studied crystal orientations are analyzed:copper:ND/RD ?e112T=?11 1 (Bunge angles:(/1,u ,/2)=(à90°,35.26°,45°)),rotated cube:ND/RD =(001)/[110]:(Bunge angles:(/1,u ,/2)=(90°,0°,à135°))and Goss ND/RD =(110)[001](Bunge angles:(/1,u ,/2)=(90°,90°,45°)).For brevity,these will henceforth be referred to as the C crystal,the RC crystal and the G crystal.

Figs.4(a 1),(b 1)and (c 1)show the initial orientation of these crystals together with the four {111}h 110i slip systems acti-vated during full constraints plane strain compression,following the Schmid-Boas notation (Ortiz and Repetto,1999).In the C crystal,potential slip activity is divided between two coplanar (CP:b 2?e111T?0 11 and b 4?e111T? 101 )and two co-direc-tional slip systems (CD:a 6?e 111T?110 and d 6?e1 11T?110 ).The RC crystal can potentially activate slip on two pairs of coplanar slip systems (CP 1:b 2?e111T?0 11 and b 4?e111T? 101 ;CP 2:c 1?e 1 11T?011 and c 3?e 1 11T?101 ).Likewise,in the G crystal two pairs of coplanar slip systems (CP 1:b 2?e111T?0 11 and b 4?e111T? 101 ;CP 2:c 1?e 1 11T?011 and c 3?e 1

11T?101 )are activated.It is evident from Fig.4that all three crystals considered are crystallographically symmetric about the RD–ND plane.In view of the symmetry of the imposed plane strain deformation also about the TD,two dimensional analyses of these orien-tations with two slip systems each,as shown in Figs.4(a 2),(b 2)and (c 2),will suf?ce.Crystal symmetry about the RD–ND plane also ensures that all lattice rotations occur about TD.Thus,a scalar x ,related to the second Bunge angle u through

x ?u t62:64 ;for the C crystal ;

u ;for the RC crystal and u à90 ;for the G crystal ;8><>:e41T

and representing the inclination of the angle bisector of the projection of the two slip directions onto the RD–ND plane,to-gether with the included angle 2/between these projections,suf?ces to represent the lattice orientations of the two-dimen-sional crystals (Figs.4(a 2),(b 2)and (c 2)).

A two-dimensional ‘stack of domains’model of a single crystal,which accommodates the plastic slip and lattice rotation within the RD-ND plane is shown in Fig.5.The sample coordinate system,xy ,coincides with the RD-ND system.The domain boundary coordinate system,XY ,with Y -axis always normal to the domain boundaries,is con?ned to the RD-ND plane.

A

24M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–33

scalar h ,also shown,speci?es the domain boundary orientation relative to the xy -system.Plane strain deformation under fully constrained imposed velocity gradient

?L RD àND ?100à1

e42T

in the [RD-ND]system is simulated following the procedure detailed in Section 3.The single crystal is discretized into a stack of N =16

domains.

M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–3325

4.2.Hardening parameters

Three sets of hardening parameters,s 0,h 0,s 1and h 1(Eq.(11))were obtained by Toméet al.(1984)by ?tting experimental stress–strain curves for uniaxial tension,compression and torsion in polycrystalline OHFC copper.The response predicted by the present simulations using all three parameter sets are qualitatively similar.Therefore,in the following,predictions cor-responding to only one of the parameter sets of Toméet al.(1984):s 0=12MPa,h 0=160MPa,s 1=98MPa and h 1=7MPa,

is

26M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–33

discussed.The hardening matrices corresponding to type (i),type (ii)and type (iii)sub-structures,discussed in Section 3.4.2,are taken to be

?H ei T ?11

11 ;?H eii T ?1a a 1

and ?H eiii T ?v 1a

a 1 ;e43T

respectively,where,a =3.1and v =0.5.These numbers represent limiting values for these parameters in that for a <3.1or v >0.5,macroscopic shear banding in the C crystal,reported below,is not predicted.a )1signi?es a substantial

anisotropic

M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–3327

hardening in domains with types (ii)and (iii)sub-structure,while v signi?cantly smaller than 1indicates substantial reduc-tion of the hardening rate in MSB domains.

The predicted lattice rotations,localization response,domain boundary orientation evolution and CRSS evolution in all three crystals are now described.

https://www.wendangku.net/doc/df8083149.html,ttice rotation

The evolution with strain of the lattice orientation,x [l ],in individual domains,l =1,2,...,N =16of the C,RC and G crystals is shown in Figs.6(a),(b)and (c),respectively.It is seen from Fig.6(a)that in the C crystal,for vM <0.29,the lattice orien-tations of all domains evolve similarly toward the D-orientation (Bunge angles:(/1,u ,/2)=(à90°,27.22°,45°)).For vM 60.29,Eq.(35)is satis?ed in all domains of the model crystal,as shown below.At vM =0.29the domain boundaries of a certain domain become orientationally unstable.Following Section 3.5,this domain is therefore identi?ed as an MSB.The lattice orientation of the MSB domain increasingly deviates from that of the other N à1=15domains with strain.

For vM 60.29,in all the domains of the C crystal,as noted in Section 4.1,two coplanar (b 2and b 4)and two co-directional slip systems (a 6and d 6)are activated.The co-directional slip systems a 6and d 6are simultaneously activated cross-slip sys-tems,de?ned in Section 3.4and therefore,their activities do not enter into the summation of Eq.(34).Identifying the crys-tallographic b-plane with p ?for which,_C p ?>0and any of the other crystallographic planes with p ??for which,_C p ???0,the ratio between two largest slip-rates,_C

p ?=_C p ???1,which implies the satisfaction of Eq.(35).Fig.6(b)shows the evolution of the lattice orientation in all domains of the model RC crystal.Starting already at vM =0,two orientationally unstable domain boundaries separate the model crystal into two sets of contiguous domains.One set of 7domains rotate clockwise and the other set of 9domains rotate counter-clockwise about TD.At vM =0,Eq.(35)is not sat-is?ed in any of the model domains,as shown below.Following Section 3.5,therefore,these two sets of domains are identi?ed as RDBs and denoted B 1and B 2

.

28M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–33

In the RC crystal,as noted in Section 4.1,two sets of coplanar slip systems (CP 1:b 2and b 4,and CP 2:c 1and c 3)are acti-vated.If the crystallographic b-plane is identi?ed with p ?and the crystallographic c-plane with p ??,_C

p ?=_C p ??%1,so that Eq.(35)for the dominance of a single crystallographic slip plane is not satis?ed at the instant of banding.

Finally,Fig.6(c)shows the lattice orientation evolution in all domains of the G crystal with deformation.It is clear that no domain is predicted to rotate signi?cantly away from the initial Goss orientation.Thus,the G crystal is predicted to form neither MSBs or

RDBs.

M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–3329

4.4.Slip accumulation

The total accumulated slip,C [l ],in each of the N =16domains of the C,RC and G crystals is shown in Fig.7.In the model C crystal it is seen from Fig.7(a)that one domain,which was identi?ed as an MSB in Section 4.3,accumulates one order of magnitude more slip compared to the other 15domains,which were identi?ed as the matrix band by vM =1.The latter set of domains are seen to undergo relatively little slip once macroscopic shear banding begins at vM =0.29,as evidenced by the nearly constant value of C [l ]in Fig.7(a)for vM >0.29.No such separation of domains on the basis of slip activity is observed in the case of the RC or G crystals in Fig.7(b)and (c).

To quantify the concentration of slip in a single domain,a localization parameter,L ,is de?ned as

L ?max 16l 6N C ?l P N

l ?1C ?l àmax 16l 6N C ?l .eN à1T:e44T

L is thus the ratio of the maximum accumulated slip in a domain to the average accumulated slip,where the average is cal-culated by excluding the contribution of the domain with the maximum accumulated slip.Fig.8shows that L in the mac-roscopic shear banding C crystal at vM =1is one order of magnitude greater than that in either the regular deformation banding RC or homogeneously deforming G crystals.Intense shear localization in the C crystal occurs even though macro-scopic shear banding was identi?ed only in terms of lattice orientation deviation and pre-existing type (ii)substructure.Slip localization is thus orientation dependent.

The distribution of slip activity between the coplanar and co-directional slip systems in the domains of the C crystal is shown in Fig.9.The slip accumulated in the coplanar and co-directional slip systems of the MSB domain are labeled c CP,MSB and c CD,MSB ,respectively.The coplanar and co-directional slip activity,averaged over the 15domains comprising the matrix region are also shown,labeled as h c CP,M i and h c CD,M i ,respectively.It is noteworthy that the bulk of the deforma-tion in the C crystal is accommodated by slip in the coplanar systems of the domain representing the MSB.

The total slip averaged separately over the domains of RDBs B 1and B 2in the RC crystal is shown in Fig.10.h c CP 1;B 1i ,h c CP 2;B 1i ,h c CP 1;B 2i and h c CP 2;B 2i represent the average accumulated slip of the two coplanar slip systems CP 1and CP 2,in bands B 1and B 2,respectively.In band B 1,CP 1dominates CP 2,while in band B 2,CP 2dominates CP 1.The quantitative condition for the dominance of one slip system in each of the two bands,Eq.(35),is satis?ed for vM P 0.06in all the domains of both bands.Macroscopic shear banding may thus occur within the RDBs,as indeed observed by Wróbel et al.(1988).This second-ary banding is not captured by the

model.

Table 1

Simulation time,in seconds,of the C,RC and G crystals with N =16domains (t stack )and N =1domain (Taylor model,t Taylor ).It is

seen that t stack $Nt Taylor .

Crystal

t stack (s)t Taylor (s)t domain /t Taylor C

68524 1.78RC

42321 1.26G 43522 1.24

30M.Arul Kumar,S.Mahesh /International Journal of Plasticity 36(2012)15–33

M.Arul Kumar,S.Mahesh/International Journal of Plasticity36(2012)15–3331 4.5.Domain boundary orientation

The domain boundary orientation,h,for the macroscopic shear banding C crystal evolves from about15°to45°following Eq.(39),as shown in Fig.11(a).According to Eq.(39),the initial orientation of the domain boundaries in the RC crystal is h=à3.75°and evolve with crystal shape(Section3.6)into alignment with the rolling plane.The predicted evolution of the very slightly misoriented domain boundaries in the G crystal is shown in Fig.11(b).As in the C crystal,these boundaries also align close to h=45°eventually.

4.6.Evolution of the critical resolved shear stress

The evolution of the normalized CRSS,^s?s=s0,of the CP and CD slip systems in all the domains of the model C crystal is shown in Fig.12.It is seen that the CRSS in all the domains evolve similarly,following[H(ii)]of Eq.(43),until vM=0.29.At

=0.29,as noted in Section4.3,the MSB forms.Accordingly,further hardening of the MSB domain is governed by the la-vM

tent hardening matrix,[H(iii)]of Eq.(43).This,together with the favorable orientation of the MSB,results in slip concentra-tion in the CP system of the MSB,as seen in Fig.9.This causes substantial latent hardening of the CD slip system in the MSB domain,as shown in Fig.12.It is found that if v>0.5in Eq.(43),deformation localization does not occur.

RDBs form in the RC crystal already at vM=0,as noted in Section4.3.At vM=0,Eq.(35)is not satis?ed in the domains of either RDB,so that the hardening follows[H(i)]of Eq.(43).For vM P0.06,however,Eq.(35)is satis?ed in all the domains of both RDBs and[H(ii)]represents the hardening matrix thereafter.In the G crystal,Eq.(35)is not satis?ed throughout the deformation so that hardening follows[H(i)]of Eq.(43)throughout.

https://www.wendangku.net/doc/df8083149.html,putational time

The wall clock computational time t stack,in seconds,required to simulate plane strain deformation to vM=1of the C,RC and G crystals comprised of N=16domains is given in Table1.The time,in seconds,required for the corresponding Taylor model(Taylor,1938),t Taylor,is also given.The simulations are carried out in a standard2.6GHz processor PC.It is seen from Table1that t stack=b Nt Taylor,where1

5.Discussion

5.1.C crystal

In agreement with experimental observations(Wagner et al.,1995;Jasienski et al.,1996)the model C crystal undergoes macroscopic shear banding.Model MSBs form only after considerable rolling reduction,as in the experiment of Wagner et al. (1995).Also in accord with experimental observations(Wagner et al.,1995;Jasienski et al.,1996),model domains rotate toward the D-orientation prior to macroscopic shear banding.After banding,while the MSB domain rotates progressively toward the Goss orientation,the matrix domains rotate back toward the C orientation.Quantitative comparison of the pres-ent predictions with experimentally measured orientations(Wagner et al.,1995;Jasienski et al.,1996)is shown in Fig.6(a) at three rolling reductions and agreement to within a few degrees is found.Finally,the predicted h=45°inclination of the macroscopic shear band boundary,shown in Fig.11(a),is close to the experimentally observed MSB inclination of42°after 27%reduction in channel die compression,reported by Jasienski et al.(1996).

5.2.RC crystal

For the model RC crystal,quantitative agreement of the predicted lattice orientations of RDBs,B1and B2,with the exper-imental measurements from the literature(Heye and Sattler,1971;Bauer et al.,1977;Butler and Hu,1989;Wróbel et al., 1994)at different strain-levels is shown in Fig.6(b).In agreement with experimental observations(Heye and Sattler, 1971),localization is not observed and RDB formation occurs already at vM=0.The experimentally observed alignment of RDB boundaries with the rolling plane(Heye and Sattler,1971)is also captured by the present model,as noted in Section 4.5.

5.3.G crystal

In the G crystal,in agreement with experimental observations(Bauer et al.,1977;Wróbel et al.,1994;Lee et al.,1999)to within a few degrees,no domain is predicted to rotate signi?cantly away from the initial Goss orientation,as shown in Fig.6(c).The predicted domain boundaries in the G-crystal,which separate only slightly misoriented domains,represent nei-ther MSB nor RDB boundaries.It may be supposed that they represent smaller scale dislocation boundaries such as DDW-MBs.However,the predicted boundary orientation is not in accord with this supposition.Because,while h%45°boundaries are predicted in Fig.11(b),it is experimentally observed(Ananthan et al.,1991)that DDW-MBs in Goss oriented grains in a

32M.Arul Kumar,S.Mahesh/International Journal of Plasticity36(2012)15–33

copper polycrystal align with the active slip planes,which are inclined approximately35°to RD.It is thus clear that the pres-ent approach is unsuccessful in predicting DDW-MB orientations.

6.Conclusions

A computationally ef?cient crystal plasticity model of a single crystal,based on the‘stack of domains’model of Arul Ku-mar et al.(2011)and capable of predicting experimentally observed regular deformation banding and macroscopic shear banding has been developed.Predicted macroscopic banding response,lattice orientation evolution,slip distribution and band boundary evolution during plane strain compression in three copper single crystals initially in the copper,rotated cube and Goss orientations compare well with experimental observations reported in the literature.The present model is unable to predict the alignment of smaller scale dislocation boundaries such as DDW-MBs,however.

Acknowledgments

We thank Prof.V.Parameswaran for encouragement and for comments that helped improve this manuscript.Funding was provided by the Indira Gandhi Center for Atomic Research,Kalpakkam.

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2003年度中国对外直接投资统计公报介绍

前言 为了科学、有效地组织全国的对外直接投资统计工作，客观、真实地反映我国对外直接投资的实际情况，保障统计资料的准确性、及时性和完整性，加强对我国企业开展境外投资活动的宏观动态监管，为各级政府管理部门掌握情况、制定政策、指导工作以及建立我国资本项目预警机制提供依据，2002年12月原外经贸部（现商务部）、国家统计局共同制定了《对外直接投资统计制度》（外经贸合发[2002]549号）。制度所涉及的对外直接投资的定义、统计原则及计算方法等是以经济合作与发展组织（OECD）《关于外国直接投资的基准定义》（第三版）及国际货币基金组织（IMF）国际收支手册（第五版）为基础建立的。制度规定，境内投资主体所属行业类别按中华人民共和国《国民经济行业分类》（GB/T4754-2002）执行，境外企业所属行业类别参照执行；中国的对外直接投资统计也包括境内投资主体对香港、澳门及台湾地区的投资。 国家统计局为《对外直接投资统计制度》有关指标及统计方法等设定方面做了大量的指导，铁道部、司法部、国家广电总局、国家烟草专卖局、国家质量监督检验检疫总局、中国科学院、中国国际贸易促进委员会等部门在2003年对外直接投资统计数据收集方面做了大量工作，为第一次对社会公众发布我国非金融类对外直接投资统计公报打下了基础，深表感谢。 本公报所发布有关中国对外直接投资数据均为非金融类对外直接投资，不包括香港、澳门、台湾地区对其他国家（地区）及中国内地的直接投资。

一、中国对外直接投资概况 2003年，中国对外直接投资总额29亿美元，扣除对外直接投资企业对境内投资主体的反向投资，投资净额为28.5亿美元， 比上年增长5.5%; 截至2003年,中国累计对外直接投资总额334亿美元, 扣除对外直接投资企业对境内投资主体的反向投资,累计对外直接投 资净额332亿美元。 据联合国贸发会议（UNCTAD）发布的2003年世界投资报告显示，2002年全球外国直接投资流出总额为6470亿美元，存量为 68660亿美元，以此为基期进行测算，2003年中国对外直接投 资分别相当于全球对外直接投资流量、存量的0.45%和0.48%。

中国对非洲直接投资的发展历程与未来趋势

中国对非洲直接投资的发展历程与未来趋势 朴英姬 一、中国对非洲直接投资的发展历程 1979年以前，中国对非洲的直接投资很少，仅限于企业执行特定的政府项目。1979-1990年底，中国对非洲的直接投资与贸易、援助相辅相成，在非洲共投资102个项目，投资总额5119万美元，每个项目平均投资额约50万美元，也有一些大中型项目，如在刚果（金）建立的金沙萨木材加工厂，投资额超过500万美元。 90年代初，中国开始将对非洲的援助转化为双边企业间的合资合作。1995年中国政府改革援外方式，将中非合作的主体从政府转向企业，实行援外方式和资金的多样化。中国积极帮助受援国建立生产项目以获得经济发展动力，将援外与直接投资、工程承包、劳务合作、外贸出口紧密结合起来。1995至1999年底，中国政府与23个非洲国家签订了政府间有关贷款框架协议，从资金方面帮助中国公司和企业到非洲投资。中国企业只要在这些非洲国家找到了合适项目，便可申请有关贷款。 此外，中国政府1995年-1997年在埃及、几内亚、马里、科特迪瓦、尼日利亚、喀麦隆、加蓬、坦桑尼亚、赞比亚和莫桑比克设立了11个“投资开发贸易中心”，专门为中国企业到非洲开展经贸业务提供具体服务及安全保障等。1998年，国家计划

委员会（现发展改革委）确定对非投资规划方案，第一次就对非投资领域、规模及投资目标进行量化分析，并提出相关指导意见。这标志着中国对非投资工作开始孕育面向新世纪的战略转变，即由贸易型投资逐渐向生产加工和资源开发投资转变。 中国政府从2000年起实施“走出去”战略，积极应对经济全球化挑战。中国企业在纺织、家电、建材、农业、食品加工等行业技术成熟，质高价廉产品给非洲人民带来实惠；投资非洲除享有当地优惠政策外，还享有欧美等发达国家对非洲国家的优惠政策。因此，非洲市场是中国实施“走出去”战略的重点地区之一。中国采取了一系列政策，鼓励企业到非洲投资建厂，如适当放宽了企业境外投资限制, 建厂投资的设备、零件、原材料享受出口退税。对于在国外投资带动国内相关产品出口的企业及新开拓出口市场的企业和产品，中国政府从简化手续、减免税费征收等方面实行政策倾斜，加大鼓励力度。政府还允许境外企业在开业的5年内所获外汇实行全额留成，以用于扩大再生产。2006年，中国大力推动有信誉、有实力、有比较优势的各类企业积极参与中非各个领域的经济技术合作。 总体来说，受国家鼓励政策和非洲经济复苏的积极影响，1996年以来中国对非洲的直接投资增长迅速。表现在： 1.2000年中国在非洲新设立投资企业57家，双方协议投资金额 2.51亿美元，中方实际投资额2.16亿美元，比1999年增长1倍多，约占中国当年对外投资总额的39.2%，达到最高点。

中国对外直接投资特征分析

中国对外直接投资分析 王文雷 摘要：改革开放以来，中国对外直接投资快速增长，特别是“走出去”战略的深入实施直接带动了我国对外直接投资的跨越式发展。本文从投资规模、地区分布、投资行业和资金来源等方面深入分析了中国对外直接投资的特点及原因，对中国对外直接投资存在的问题提出了一些基本的对策。 关键词：对外直接投资投资规模地区分布 一、引言 加入WTO以来，中国正在成为世界最大的海外投资国之一（Unctad，2004）。据商务部、国家统计局、国家外汇管理局发布的《2010年度中国对外直接投资统计公报》，2010年，中国对外直接投资净额（流量）为688.1亿美元，同比增长21.7%，连续九年保持增长势头，年均增速为49.9%。其中，非金融类601.8亿美元，同比增长25.9%；金融类86.3亿美元。根据联合国贸发会议《2011年世界投资报告》，2010年中国对外直接投资占全球当年流量的5.2%，位居全球第五，首次超过日本（562.6亿美元）、英国（110.2亿美元）等传统对外投资大国。 尽管我国对外直接投（ODI）增长较快，但这种较快增长势头是否可以持续？哪些因素制约着我国ODI的增长速度和质量？我国ODI有哪些特征和问题？这些问题都是值得思考的。因此有必要深入全面分析我国ODI特征及其成因，从中发现投资过程中的问题，并结合我国现状提出具有针对性的政策措施，这些是本文研究的出发点。 二、我国对外直接投资的发展阶段及现状 1. 我国对外直接投资的发展阶段 从1982—2010年间我国对外直接投资流量和存量的发展趋势来看，主要经历了三个阶段： 第一阶段（1982—1991年）。这一阶段对外直接投资主要是试验性的，并且在国家严格管制以及资金稀缺的条件下进行的。几乎所有的对外直接投资项目都是国有贸易公司及其他国有企业在政府主导下进行的。但投资额在整个时期很小，平均每年流量为5.3亿美元，到1991年底对外直接投资存量约为53亿美元。这一阶段，中国的跨国公司主要是垄断行业经营的大型国有企业，如金融服务业、航运业、国际贸易以及自然资源行业。最具代表性的企业有中信集团、中远集团、中国建筑工程总公司、中石油、中石化、中化集团、中海油、中国五矿和中粮集团等。 第二阶段（1992—2002年）。这一阶段中国对外直接投资发展迅速，每年平均流量为28.9亿美元，截止2002年底对外直接投资存量372亿美元。这一阶段中国对外直接投资流量波动幅度较大，这反映了国内、国际上政治经济环境的变化。 第三阶段（2003年至今）。2003年以来，中国对外直接投资的增长率和投资额急剧上升。近几年中国对外直接投资流量年均达208.64亿美元，存量也从2003年的330亿美元上升到

中国对日本直接投资.1

随着中国对外直接投资的不断发展，中国对于发达国家的逆向直接投资也呈现快速增加的趋势。 尽管2012年出现的钓鱼岛“购岛”事件，再次使中日关系处于紧张局面，两国政治外交关系达到了冰点，这有可能会使中日双方投资者望而却步，但2012至2013年中日之间的直接投资仍然出现了较快增长。统计显示，日本对华投资2012年达73.8亿美元，同比增长16.3%；而2012年中国对日本则实现了两位数增长，同比增长47.8%。 中国对日本直接投资流量和存量 资料来源：中国对外直接投资统计公报2013 2008～2013年中国对日本直接投资企业数量 年份2008 2009 2010 2011 2012 2013 合计65 90 114 109 131 112 制造业 4 18 14 16 30 24 比重(%) 6.2% 20% 12.3% 14.7% 22.9% 21.42% 非制造业61 72 100 93 101 88 比重(%) 93.8% 80% 87.7% 85.3% 77.1% 78.57% 资料来源：中华人民共和国商务部境外投资企业（机构）名录整理 据日本贸易振兴机构（JETRO）统计，2014年中国对日本的投资额约为600亿日元，另一方面，日本的对华投资额为7200亿日元。两者相差12倍，JETRO 大连事务所的市场开拓部部长高山博表示“中国对日投资扩大的余地还很大”。 投资日本的中国企业 Ctrip（携程旅行网）2014 年在日本东京成立株式会社CTRIP JAPAN。为

了满足急剧增加的中国赴日游客的需求，携程公司正在逐步扩大在日业务规模。 大连奥托股份有限公司（Dalian Automobile Technology CO., LTD,）2013年 3月在爱知县名古屋市设立了全资子公司“株式 会社Auto Tech Japan”。大连奥托股份有 限公司，自1990年设立以来，以中国国内 的大型汽车制造厂商为中心，提供各类产品 和服务，随着中国汽车市场的扩大该公司规 模也进一步扩大。该公司除中国以外，在德 国和南非设有据点，随着汽车行业的全球化 积极的向海外市场拓展，进一步扩大海外的 商业合作。今后，为了进一步拓展与强化已 投资中国市场的日本企业间的合作交流，同 时，也为了扩大日本市场的销路，该公司决 定在汽车产业活跃的名古屋市设立据点。 江苏三鑫特殊金属材料股份有限公司（Jiangsu Tri-M Special Metals Co. Ltd.）2012年10月在爱知县名古屋市设立了“三鑫金属日本株 式会社”，该公司主要从事手表，汽车零部 件，电子产品，医疗器械用品等使用的特殊 金属材料及加工制品的制造和销售，此次为 扩大在日本的销路，在爱知县设立了日本法 人。今后，随着销售业绩的上升，会进一步 考虑投资设立工厂。 上海吉祥航空有限公司2012年9月在冲绳县开设了日本首家分公司。该公司取 得了日本国土交通省颁发的外国人国际航 空运输业务的经营许可证，为了能在上海-- 那霸开通定期航线进行准备。 江苏威洋物流有限公司2012年9月在大阪设立了株式会社威洋物流。总公司设 立在苏州，在哈尔滨，天津，苏州，上海及 广州有仓库，已经在中国主要城市形成了配 送网络。除了运输以外，还在仓库内进行检 收，品质检查，商品贴标，分类，加工包装 等业务。 上海春秋国际旅行社有限公司2012年11月在东京设立了日本春秋旅行株式会社。上海 春秋国际旅行社有限公司是1981年创业的， 是总公司在中国上海的大型旅行代理店，主 要负责观光业务，酒店预订，安排，运营，

中国对非洲投资情况概览

中国对非洲投资情况概览 中国对非洲投资的比重相比其在亚洲及欧洲53.4%及14.8%投资的比重并不算高。中国企业在非洲大陆投资仅占其全球投资额度的12.1%。但以对外直接投资覆盖率来衡量，截至2010年根据中国商务部及国家外汇管理局提供的数据，在非洲59个国家之中中国境外企业投资覆盖的国家数达到50个，投资覆盖率达85%。紧随亚洲国家之后，居各大洲第二（见图1、表1） 图1: 2010年末中国境外企业在世界各地区覆盖比率 表1：2010年末中国对外直接投资企业在全球的地区分布

实际上，伴随着中国企业对非投资覆盖率的升高，中国企业对非洲直接投资额度也在迅速增长。据相关机构提供的数据，对非直接投资额由2003年的7481万美元上升至2010年的21.12亿美元（见图2） 投资行业扩大至农业、采矿业、制造业、机器设备、家电、轻工业、纺织业、其它制造业、服务业等。投资涉及非洲国家包括：阿尔及利亚、安哥拉、贝宁、博茨瓦纳、布隆迪、喀麦隆、佛得角、中非、乍得、科摩罗、刚果（金）、刚果（布）、科特迪瓦、吉布提、埃及、赤道几内亚、厄立特里亚、埃塞俄比亚、加蓬、冈比亚、加纳、几内亚、肯尼亚、莱索托、利比里亚、利比亚、马达加斯加、马拉维、马里、毛里塔尼亚、毛里求斯、摩洛哥、莫桑比克、纳米比亚、尼日尔、尼日利亚、卢旺达、塞内加尔、塞舌尔、塞拉利昂、南非、苏丹、坦桑尼亚、多哥、突尼斯、乌干达、赞比亚、津巴布韦等。中国企业在上述非洲国家投资流量在2003年到2010年之间的变化在表2及表3中将详细列举。

表2：中国对非洲各国投资流量（2003-2010） 表3：中国对非洲各国投资流量（2003-2010）续