The elastoplastic transition in channel die compression of fcc crystals

International Journal of Plasticity 35 (2012) 31–43

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The elastoplastic transition in channel die compression of fcc crystals Kerry S. Havner ⇑ Department of Civil Engineering, North Carolina State University, Raleigh, NC 27695-7908, USA Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7907, USA

a r t i c l e

i n f o

Article history: Received 14 September 2011 Received in final revised form 19 January 2012 Available online 23 February 2012 Keywords: Channel die compression Fcc crystals Elastoplastic analysis

a b s t r a c t A comprehensive analysis of the elastoplastic transition in fcc crystals in (110) channel die compression is presented. This range of very small crystal deformations and lattice rotations, of order 103 or less (typically with only two equally-stressed slip systems significantly active), precedes the 4-fold (or higher) large multiple-slip deformations that have been studied extensively, both experimentally and theoretically. General equations are derived for the strains, rotations, and crystal shearing in the elastoplastic transition, including a predictive equation for the compressive strain at the onset of finite deformation. From crystal elastic properties and experimental stress data for aluminum and copper in various lattice orientations, these very small elastoplastic strains, rotations, and shears are calculated over the range of possible lateral constraint directions. The results are contrasted with experimental and theoretical values of finite shears and lattice rotations in the same initial orientations, and a justification for the disregard of lattice straining in the finite deformation realm is given. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction and scope of study In the extensive experimental literature on channel die compression of single crystals (from Chin et al., 1966a, to Darrieulat et al., 2007), stress–strain plots typically are displayed with a strain scale of order 1, and a strain of 0.001 cannot be discerned as it is lost within the vertical stress-axis. At the onset of finite-deformation multiple-slip (commonly at least 4-fold in fcc metals), where experimentally determined plots intersect the stress axis at ‘zero’ finite strain, the actual compressive strain likely is of order 103. There will have been an ‘infinitesimal’ strain range of purely elastic response, followed by an elastoplastic transition with significant slip on only the highest-stressed system or systems. The orientation-dependent ratio between the lateral constraint stress and the load stress evolves from its initial value (theoretically determined by the elastic moduli) to the value required for there to be a sufficient combination of critical slip-systems for finite deformation to begin. The latter value is determined from multiple-slip kinematics, slip-system orientations, and the channel die constraints. From that point forward elastic straining typically is disregarded in finite-deformation analyses. (See, for example, Havner (2007a,b, 2008a,b, 2010, 2011), in which crystal hardening inequalities from Havner (2005), are applied to the analysis of (110) compression of fcc crystals in various orientation ranges.) This paper is concerned with a comprehensive analysis of the elastoplastic transition in fcc crystals in (110) channel die compression, an important family of load orientations to investigate, both experimentally and analytically, because (110) is a stable normal direction in sheet rolling of polycrystalline metals. In addition to derivation and solution of the general equations governing the elastoplastic transition, a particular objective is to give a rational basis for the preceding statement that finite deformation begins at a compressive strain of order 103 by developing a predictive equation for that strain in terms of

⇑ Address: Department of Civil Engineering, North Carolina State University, Raleigh, NC 27695-7908, USA. Tel.: +1 919 515 7632; fax: +1 919 515 7908. E-mail address: [email protected] 0749-6419/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2012.02.002

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K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

other experimental information. Also, the very small, orientation-dependent lattice rotations and crystal shears at the end of the elastoplastic transition are determined. The paper is organized as follows. The general stress and kinematic equations in (110) channel die compression, including lattice strain-rates on channel axes and the lattice rotation-rate, are given in Section 2. In Section 3 simplifications of lattice rotation-rate and channel-constraint equations are made, based on elastic properties and data from experiments on aluminum and copper crystals (presented in Appendix A) that cover the range of geometrically-distinct lattice orientations in (110) compression; and final solutions for slip-rates, rate-of-change of lateral constraint stress, and lattice rotation-rate during the elastoplastic transition are obtained. Section 4 presents the development of a general equation for the strain at the beginning of finite deformation (end of the transition), including evaluations in orientations spanning the range. In Section 5 solutions for the very small lattice rotations and crystal shears at the end of the elastoplastic transition are derived and evaluated for those same orientations. Contrasts with experimental and theoretical values at finite strains and a final justification for neglecting lattice straining in the finite deformation range are presented in Section 6, with closing remarks given in Section 7. Details of the basis in experiment for the simplification of general equations are given in Appendix A. The determination of the principal active slip-systems during the transition to finite deformations is presented in Appendix B, followed by the references.

2. General stress and kinematic equations in (110) compression  c1;  c2; The labeling of the 12 crystallographic slip systems in fcc crystals is shown in Table 1. Systems a1; b2;  b3; d1;  d2; a2;  b1; c3; &d3 (a bar above indicating the opposite-sense system from that defined in the table) are designated a3; 1–12 in order. The slip-rate in the jth slip system is denoted c_ j . We label (i) the loading, lateral constraint, and longitudinal channel axes by X, Y, Z, respectively, (ii) unit vectors normal to the positive X and Y crystal faces by i, j, and (iii) a unit vector coincident with the positive channel axis Z by l. Thus, ijl constitutes a right-handed orthogonal triad subsequently to be ^ denote the lattice co-rotational derivatives of i, j. In analdefined relative to the lattice axes [100], [010], [001]. Also, let ^i; j yses in Havner (2007b, 2008a,b, 2010, 2011) of finite deformation of fcc crystals in different orientation ranges and specific (‘singular’) orientations in (110) compression, various mathematical arguments for load-axis stability ð^i ¼ 0Þ are made. Here we shall simply adopt that stability as experimentally well-established and, in what follows, reduce the possible number of ^ cannot be taken as zero because lattice rotation about the load-axis is common different slip-rates accordingly. However, j in a range of orientations in channel die compression (e.g. Skalli et al., 1983; Skalli, 1984; Butler and Hu, 1989; Wrobel et al., 1996). 2.1. Lattice stress–strain relations, strain-rates, and resolved shear stresses The unit vectors in directions X, Y, Z in (110) compression may be expressed (in Miller index notation on lattice axes)

pffiffiffi

  2;  i ¼ ð110Þ; j ¼ ð11 bÞ; l ¼ ½bb b ¼ 2 cot /;

ð1Þ

 (or the anti-clockwise orientation of ½001  relative to Y), Fig. 1. where / is the clockwise orientation of Y with respect to ½001 The transformation matrix from lattice to channel axes then may be written

0 1=j pffiffiffi B pffiffiffi1 Q c ¼ ð1= 2Þj1 @ 2

1 1=j1 0 pffiffiffi pffiffiffi C  2  2b A; b

b

j1 ¼ 1=ðb2 þ 2Þ1=2 :

ð2Þ

2

Let f, g denote the ‘true’ compressive stresses in the loading and constraint directions respectively. From tensor transformations, the lattice strain components on channel axes for cubic lattices are



1 2







1 2

exx ¼  s11  sA f  s12 þ sA j21 g; exy ¼ 0; exz ¼ 0; sA ¼ s11  s12  s44 ; 





n

pffiffiffi



o

eyy ¼  s12 þ sA j21 f  s11  2sA j21 ð2  3j21 Þ g; eyz ¼ ð1= 2ÞsA bj21 f  2ðb2  1Þj21 g ; ezz

 

1 2 2 ¼  s12 þ sA b j21 f  s12 þ 3sA b j41 g; 2

ð3Þ

j21 ¼ 1=ðb2 þ 2Þ;

Table 1 Designation of slip systems in fcc crystals. Plane

(111)

Direction System

 ½011 a1

 11Þ  ð1  ½101 a2

 ½110 a3

[011] b1

 ð111Þ  1  ½10 b2

 ½110 b3

 ½011 c1

 ð111Þ  1  ½10 c2

[110] c3

[011] d1

 ½101 d2

 10  ½1 d3

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K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

(001) 90

(0 11)

b2

a1

(1 12)

b3

c1

d3

b2

c2

(1 10)

90

d1

d2

a2

(1 1 1)

II

(11 1)

(10 1)

Y b1

a2

90

(010)

a1

c3

b3

(1 12) (0 1 1)

b1

(100) X (110)

III

a3

c3

(111)

a3 (0 10)

(011)

(101)

-Z

(1 11)

c2

d1

c1

d3

d2

(01 1)

I 90

(00 1) Fig. 1. (100) stereographic projection identifying the slip systems in uniaxial compression of fcc crystals and showing the three ranges of constraint direction Y for (110) loading (axis X) in channel die compression (Z is the channel axis).

where s11, s12, s44 are the cubic elastic compliances in standard notation (with sA zero if isotropic), and j1 is the direction cosine of the Y-axis with the [100] lattice direction. (As is easily confirmed, exx + eyy + ezz = (s11 + 2s12)(f + g), the isotropic ‘linear compressibility’ of cubic lattice systems with inverse bulk modulus s11 + 2s12, (Nye, 1957).) The lattice strain-rates are then





pffiffiffi   1 _ e_ xx ¼  s11  sA f_  s12 þ sA j21 g_  2sA gbj21 /; e_ xy ¼ 0; e_ xz ¼ 0;

2 pffiffiffi         _ _eyy ¼  s12 þ sA j21 f_  s11  2sA j21 2  3j21 g_  2sA f  4g 1  3j21 bj21 /; n o n o pffiffiffi e_ yz ¼ ð1= 2ÞsA bj21 f_  2ðb2  1Þj21 g_ þ ðsA =2Þj21 ðb2  2Þf þ 2j21 ð9b2  b4  2Þg /_

ð4Þ

pffiffiffi pffiffiffi _ with /_ the anti-clockwise lattice rotation-rate about the load-axis X and the (from 2j21 b_ ¼ /_ and 2j1 j_ 1 ¼ 2bj21 /Þ, underlined terms the contributions of that rotation-rate to the strain-rate components on channel axes. (We shall not need an equation for e_ zz as it is a ‘free’ extension determined from e_ xx ; e_ yy by the isotropic linear compressibility relation above.) The stress tensor may be expressed r = fi  i  gj  j, from which the resolved shear stress in the kth slip system is

sk ¼ mk f þ rk g; mk ¼ iNk i; rk ¼ jNk j; Nk ¼ symðb  nÞk ;

ð5Þ

where bk, nk denote unit vectors in the slip and slip-plane normal directions, respectively, of the kth system. Upon substituting the unit vectors from Table 1, we have (Havner, 2008b, Eq. (3.8))

pffiffiffi 

pffiffiffi



s1 ¼ s2 ¼ ð1= 6Þ f  gbðb  1Þj21 ; s3 ¼ s4 ¼ ð1= 6Þgðb þ 2Þðb  1Þj21 ; pffiffiffi pffiffiffi s5 ¼ s6 ¼ ð2= 6Þgbj21 ; s7 ¼ s8 ¼ ð1= 6Þgðb þ 1Þðb  2Þj21 ; pffiffiffi   s9 ¼ s10 ¼ ð1= 6Þ f  gbðb þ 1Þj21 ; s11 ¼ s12 ¼ 0; j21 ¼ 1=ðb2 þ 2Þ:

ð6Þ

2.2. Lattice rotation-rate and channel constraint equations The lattice co-rotational derivative of a unit vector n normal to a material plane fixed in the channel frame (either the X or Y face) is

^  nx ¼ dnn n  n

X ðn  bj Þnj c_ j  ne_ ;

dnn ¼ e_ nn þ

X ðn  bj Þðn  nj Þc_ j ;

ð7Þ

where the possible lattice straining contribution has been added to the rigid-plastic multiple-slip Eq. (4.4) in Sue and Havner (1984) (also see Eq. (1) in Havner (2007a)). Here e_ is the lattice strain-rate tensor, x is the lattice spin tensor relative to the channel frame, and dnn ; e_ nn are the total Eulerian strain-rate and lattice strain-rate components in direction n. From Eqs. (1) and (4) we have e_ xx i  ie_ ¼ 0 and e_ yy j  je_ ¼ e_ yz l, whence

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K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

X fði  bj Þði  nj Þi  ði  bj Þnj gc_ j ; X ^ j ¼ fðj  bj Þðj  nj Þj  ðj  bj Þnj gc_ j  e_ yz l

^i ¼

ð8Þ

^ may be expressed ^ z ¼ l  j from Eq. (7). The lattice rotation rate /_  xx ¼ j

/_ ¼

X ðj  bj Þðl  nj Þc_ j þ e_ yz

ð9Þ

(from the orthogonality of j and l), and /_ ¼ e_ yz (the correct component of the lattice velocity gradient) in purely elastic response. Upon substitution of the bj, nj vectors (from Table 1) into Eq. (8)1, we find there must be pair-wise equality between sliprates in equally-stressed slip systems in order for there to be load-axis stability, ^i ¼ 0 (see Eq. (8)1 in Havner (2007a)):

c_ 1 ¼ c_ 2 ; c_ 3 ¼ c_ 4 ; c_ 5 ¼ c_ 6 ; c_ 7 ¼ c_ 8 ; c_ 9 ¼ c_ 10

ð10Þ

(with c_ 11 ¼ c_ 12 ¼ 0 as those systems are unstressed from Eq. (6)). Substituting these relations and the bj, nj vectors from Table 1 into Eq. (9), we have

o pffiffiffi n /_ ¼ ð2= 3Þ ðb  1Þc_ 1 þ ðb  1Þ2 c_ 3  2c_ 5  ðb þ 1Þ2 c_ 7 þ ðb þ 1Þc_ 9 j21 þ e_ yz :

ð11Þ

Lastly, the channel die constraints are

dxx ¼ iDi ¼ e_ L ; X D¼ N j c_ j þ e_ ;

dxy ¼ iDj ¼ 0;

dyy ¼ jDj ¼ 0;

ð12Þ

Nj ¼ symðb  nÞj ;

where eL = lnk is the logarithmic compressive strain, and k (<1) is the ‘spacing stretch’ in the X direction. Upon substituting the bj, nj vectors from Table 1 and the lattice strain-rates from Eq. (4), the constraint equations are

  pffiffiffi pffiffiffi 1 ð2= 6Þðc_ 1 þ c_ 9 Þ þ s11  sA f_ þ ðs12 þ sA j21 Þg_ þ 2sA gbj21 /_ ¼ e_ L ; 2 pffiffiffi 2 ¼ 0 : ð2= 6Þj1 fbðb  1Þc_ 1  ðb  1Þðb þ 2Þc_ 3  2bc_ 5  ðb þ 1Þðb  2Þc_ 7 þ bðb þ 1Þc_ 9 g pffiffiffi        ðs12 þ sA j2 Þf_  s11  2sA j2 2  3j2 g_  2sA f  4g 1  3j2 bj2 /_ ¼ 0; dxy  0:

dxx ¼ e_ L : dyy

1

1

1

1

1

ð13Þ

(dxy  0 from the pair-wise equality of the slip rates, Eq. (10), and from e_ xy ¼ 0, Eq. (4).) There remains the elimination of e_ yz _ and the slip-rates. between Eqs. (4) and (11) to obtain an explicit equation for /_ in terms of f_ ; g,

3. Simplifications of equations and solutions for rates-of-change 3.1. Lattice rotation rate _ will be neHenceforth the underlined terms in the lattice strain-rate Eq. (4), corresponding to the contributions of /, glected. This simplification is fully justified in Appendix A for aluminum and copper (and briefly argued for nickel), where it is found that these terms are of order 104 (or less) times the remaining terms in the elastoplastic transition, based on experimental data from initial orientations b = 100, 2, 1, & 0.05 (/ = 0.81°, 35.26°, 54.74°, & 88.0°, the last 2.0° from the ‘Goss’   These span the range of geometrically distinct lattice positions in (110) compression. orientation b = 0, or ð110Þð110Þ½00 1Þ. (Data for other fcc crystals in the channel die test is not readily available, but the terms likely also would be negligible for other metals.) The (simplified) final equation for /_ is then

pffiffiffi pffiffiffi 2 _ /_ ¼ ð2= 3Þj21 fðb  1Þc_ 1 þ ðb  1Þ2 c_ 3  2c_ 5  ðb þ 1Þ2 c_ 7 þ ðb þ 1Þc_ 9 g þ ð1= 2ÞsA bj21 ff_  2j21 ðb  1Þgg;

j21 ¼ 1=ðb2 þ 2Þ;

ð14Þ

throughout the elastoplastic transition. Here we shall only use this equation within rate-independent theory (with a critical resolved shear stress), corresponding to which only systems 1&2 ðc_ 1 ¼ c_ 2 Þ are active in the elastoplastic transition. However, the equation also is appropriate to use in a rate-dependent power-law with all stressed slip-systems active, although ‘elas rather than 7,8 would be active for  8 toplastic transition’ has a less clear meaning then. (Note: opposite-sense systems 7;  4  rather than 3,4 would be active for b < 1, as explained in Havner (2008b), Appendix b < 2, and opposite-sense systems 3; A(b).) 3.2. Constraint equations and rate-independent solutions for slip-rates and constraint stress-rate Let ()0 signify differentiation with respect to compressive strain eL. Upon substituting Eq. (14) into Eq. (13), the final (simplified) constraint equations may be written

K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

35

pffiffiffi   dxx ¼ e_ L : ð2= 6Þ c01 þ c09 þ C xx f 0 þ C xy g 0 ¼ 1; pffiffiffi     dyy ¼ 0 : ð2= 6Þj21 b ðb  1Þc01  2c05 þ ðb þ 1Þc09  ðb  1Þðb þ 2Þc03  ðb þ 1Þðb  2Þc07  C xy f 0  C yy g 0 ¼ 0; 1 C xx ¼ s11  sA ; 2

C xy ¼ s12 þ sA j21 ;

  C yy ¼ s11  2sA j21 2  3j21 ;

ð15Þ

j21 ¼ 1=ðb2 þ 2Þ;

where Cxx, Cxy, Cyy are the elastic compliances on channel axes in (110) compression. (These kinematic constraints also are appropriate to use with a rate-dependent power-law.)  are the most highly stressed in the elastic range for all orientations As shown in Appendix B, systems 1&2 ða1; b2Þ 1 > b > 0 (0 < / < 90°) in both Al and Cu. The initial stress ratio in the elastic range bE  (g/f)E is of course determined from eyy = 0, and is

bE  ðg=f ÞE ¼ C xy =C yy ¼

ðs12 þ sA j21 Þ s11  2sA j21 ð2  3j21 Þ

ð16Þ

(see Sue and Havner (1984), Eq. (3.6) or Havner (1992), Eq. (5.10)). Henceforth we shall consider only systems 1&2 to be active during the elastoplastic transition. However, without pre-compression, this reduction in active systems does not apply  to copper for b < 0.556 in range III (constraint directions within approximately 21.5° of the Goss orientation Y ¼ ð110ÞÞ, for  b1Þ are equally stressed as systems 1&2, Eq. (6). (Analysis of this range of orientations in which g = 0 and systems 9&10 (a2; 4-fold slip for finite deformations, disregarding lattice straining, may be found in Havner (2008a).) The systems that subsequently become critical, when stress ratio g/f has evolved from bE to a ratio bo marking the beginning of finite deformation, depend upon orientation and are fully identified in Skalli et al. (1983, Fig. 4) for the three orientation ranges I, II, &III (Fig. 1)  ð11  2Þ;  ð11  1Þ;  and ð110Þ.  and each singular orientation Y ¼ ð001Þ; Explicit equations for bo will be given in Section 4.2. With all slip-rates taken to be zero except c_ 1 ¼ c_ 2 , Eq. (14) for /_ reduces to

n o pffiffiffi pffiffiffi _ 2  1Þj21 /_ ¼ ð2= 3Þðb  1Þj21 c_ 1 þ ð1= 2ÞsA bj21 f_  2gðb

ð17Þ

and the constraint Eq. (15) during the elastoplastic transition becomes

dxx ¼ e_ L : dyy ¼ 0 :

  pffiffiffi   1 ð2= 6Þc01 þ s12 þ j21 sA g 0 ¼ 1  s11  sA h; h  df =deL ; 2 pffiffiffi   0 2 0 2 ð2= 6Þbðb  1Þj1 c1 þ ðs12 þ j1 sA Þ=bE g ¼ ðs12 þ j21 sA Þh;

ð18Þ

with solution

pffiffiffi ð 6=2Þð1  h=Ex Þ c ¼ ; 1  B1 0 1

1=Ex ¼ C xx þ bE C xy ;

g0 ¼

ðB1 =C xy Þ þ ðbE þ B1 C xx =C xy Þh ; 1  B1 2 1;

B1 ¼ bE bðb  1Þj

2 1

h 6 Ex ;

ð19Þ

2

j ¼ 1=ðb þ 2Þ;

where Ex is the modulus of elastic response rxx/exx (with eyy = 0) in (110) channel die compression (whence c01 ¼ 0 if h = Ex) and Cxx, Cxy, Cyy are the relevant elastic compliances from Eq. (15). (If bE < 0 from Eq. (16), as in Cu for b < 0.556, the crystal is laterally contracting elastically and gE = 0.) It may be confirmed algebraically that g0 = bEEx for h = Ex, corresponding to purely elastic response (as required).

3.3. Final lattice rotation rate _ e_ L is Upon substituting c_ 1  c01 e_ L ; g_  g 0 e_ L , and c01 ; g 0 from Eq. (19) into Eq. (17), the final equation for /0  /=

0

/ ¼ A1 þ A2 h;

A2 ¼

A1 ¼

n o pffiffiffi 2 2 2ðb  1Þj21 1  ðsA =C yy Þb ðb  1Þj41 ð1  B1 Þ

;

i o pffiffiffi h n 2 ð1= 2Þj21 sA b  ð3b þ 2ÞB1  2bðb  1Þj21 B1 C xx =C xy  2ðb  1ÞðC xx þ bE C xy Þ

ð20Þ

ð1  B1 Þ

(A1 dimensionless and A2 in units 105/MPa, corresponding to the same units for sA, Cxx, Cxy, and Cyy). Henceforward all analysis and evaluations for the elastoplastic transition will be based on Eqs. (19) and (20).

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K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

4. Elastic rotations and the strain eo at the end of the elastoplastic transition 4.1. Initial elastic rotations First consider the extremely small lattice rotation in the elastic range. Upon substituting h = Ex and the various definitions of terms from Eqs. (15), (16) and (19) into Eq. (20), one obtains

pffiffiffi /0E ¼ e0yz ¼ ð1= 2ÞsA Ex j21 fb  2ðb þ 1ÞB1 g:

ð21Þ

(This reduced equation for e0yz of course corresponds to elimination of the sAf and sAg terms multiplying /_ in the original e0yz Eq. (4), as noted at the beginning of Section 3.1.) The maximum elastic angle change, at a strain eE  fE/Ex, is then

pffiffiffi D/E ¼ ð1= 2ÞsA fE j21 fb  2ðb þ 1ÞB1 g;

fE ¼

pffiffiffi   6sE = 1  bE bðb  1Þj21 ;

ð22Þ

fE from Eq. (6), with a shear stress at the yield point sE = 1 MPa in both aluminum and copper from Appendix A.1 (based on experiments of Franciosi et al. (1980)). For the four b values (Section 3.1) spanning the range of distinct orientations in (110) compression, and the basic elastic compliances in Appendix A (from Nye (1957), Table 10), we have:

b ¼ 100

4 eAl E ¼ 0:450  10 ;

7 D/Al rad; E ¼ 0:301  10

4 bCu fECu ¼ 4:21 MPa; eCu E ¼ 0:423; E ¼ 0:209  10 ;

7 D/Cu rad; E ¼ 0:669  10

4 eAl E ¼ 0:334  10 ;

5 D/Al rad; E ¼ 0:170  10

: bAl E ¼ 0:365;

fEAl ¼ 3:83 MPa;

b¼2

: bAl E ¼ 0:369;

b¼1

4 5 bCu fECu ¼ 2:95 MPa; eCu D/Cu rad; E ¼ 0:508; E ¼ 0:497  10 E ¼ 0:167  10 ; pffiffiffi 4 5 Al Al : bE ¼ 0:337; fEAl ¼ 6 MPa; eAl D/E ¼ 0:237  10 rad; E ¼ 0:303  10 ; p ffiffiffi 4 5 fECu ¼ 6 MPa; eCu D/Cu rad; bCu E ¼ 0:279; E ¼ 0:840  10 E ¼ 0:177  10 ;

b ¼ 0:05 : bAl E ¼ 0:271; bCu E ¼ 0;

fEAl ¼ 2:79 MPa;

fEAl ¼ 2:43 MPa; pffiffiffi fECu ¼ 6 MPa;

4 eAl E ¼ 0:312  10 ;

ð23Þ

6 D/Al rad; E ¼ 0:224  10

4 6 eCu D/Cu rad: E ¼ 0:629  10 E ¼ 0:187  10 ;

(Note from Eq. (6) that s1, s2 are unaffected by constraint stress g in the Brass pffiffiffi orientation b = 1.) The largest D/E, for Cu at b = 1, is 4.8  104 degrees, hence neglect of the miniscule change in b ¼ 2 cot / is justified to more than three significant figures, greater than the accuracy of the elastic properties used to calculate bE and eE in that or any orientation. 4.2. Derivation of strain eo We begin with the orientation-dependent ratio bo between g and f that must be attained in each range (Fig. 1) for there to be sufficient critical slip-systems for finite deformation to begin (see Havner, 1992, Section 5.4): 2

2

Range I: 1 > b > 2 (0 < /o < 35.26°), systems 1&2 and 3&4 critical, bo ¼ 12 ðb þ 2Þ=ðb  1Þ. b = 2: /o = 35.26°, systems 5&6 additionally critical, bo = 1. Range II: 2 > b > 1 (35.26° < /o < 54.74°), systems 1&2 and 5&6 critical, bo = (b2 + 2)/(b2 + b).     b = 1: /o = 54.74°, only systems 1&2 critical, bo ¼ bE   s12 þ 13 sA = s11  23 sA (g elastically determined). Range III: 1 > b > 0 (54.74° < /o < 90°), systems 1&2 and 9&10 critical, which requires bo  0 if the slip-systems hardening during the elastoplastic transition is essentially isotropic. (Finite deformation analyses and evaluations of all these cases may be found in Havner (2007a,b, 2008a,b, 2010).) Consider the evolution of b  g/f from bE at the beginning of the elastoplastic transition to bo at its end. From Eq. (19),

g 0 ¼ a1 þ a2 h;

a1 ¼ ðB1 =C xy Þ=ð1  B1 Þ;

a2 ¼ ðbE þ B1 C xx =C xy Þ=ð1  B1 Þ

ð24Þ

(a1 in units 105 MPa and a2 dimensionless). Then, again assuming the change in b to be negligible (as will be confirmed in Section 5.1), we have

g o  bo fo ¼ g E þ a1 ðeo  eE Þ þ a2

Z

eo

hðeL ÞdeL ; eE

f o ¼ fE þ

Z

eo

hðeL ÞdeL ;

g E  bE fE ;

ð25Þ

eE

where eo is the compressive strain at the onset of finite deformation, from which

eo ¼ eE þ ð1=a1 Þfðbo  a2 Þfo  ðbE  a2 ÞfE g;

ð26Þ

a simple equation for this strain without the necessity of knowing the detailed shape of the stress–strain curve during the elastoplastic transition (a curve not shown in experimental papers).

K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

37

In addition to the total compressive strain eo in the different orientations, we also give values in Section 4.3 of the elastic part eo = fo/Ex and the ‘mean’ modulus hM = (fo  fE)/(eo  eE) during this transition, with Ex > hM > ho, where ho is the initial slope of the finite-deformation stress–strain curve (typically only several hundred MPa for Al&Cu). 4.3. Calculations of eo, eo, and hM For orientations b = 100, 2, &0.05 we obtain (from Eq. (26), experiment-based values for fo in Appendix A.2, fE values from Eq. (23), and the equations above for eo, hM):



EAl x ¼ 0:852 ;

Cu 3 3 eCu hM ¼ 1:753; ECu eCu o ¼ 0:265  10 ; o ¼ 0:233  10 ; x ¼ 2:016 ;

Al 2 3 b¼2: eAl eAl hM ¼ 0:277; EAl o ¼ 0:202  10 ; o ¼ 0:691  10 ; x ¼ 0:836 ;

Cu 3 4 eCu eCu hM ¼ 0:590; ECu x ¼ 1:768 ; o ¼ 0:255  10 ; o ¼ 0:962  10 ;

Al 2 3 eAl hM ¼ 0:0521; EAl b ¼ 0:05 : eAl o ¼ 0:379  10 ; o ¼ 0:282  10 ; x ¼ 0:779

b ¼ 100 :

3 eAl o ¼ 0:879  10 ;

Al 3 eAl hM ¼ 0:757; o ¼ 0:786  10 ;

ð27Þ

(with values of moduli hM and Ex in units 105 MPa, and Ex shown for comparison). In the Brass orientation b = 1 (for which Ex = 0.809  105, 1.385  105 MPa in Al and Cu, respectively), g is elastically determined throughout, Eq. (26) reduces to the identity go  gE = bE(fo  fE), and a specific eo, hence hM, cannot readily be calculated. (As noted in Section 3.2, the foregoing analysis does not apply to copper for b < 0.556 because slip systems 9&10 also would be active in that range.) From the results in Eq. (27) we see that eo is of order 103 or less, as anticipated. (The largest strain, 3.8  103 in Al at b = 0.05, is based upon reaching a theoretical bo  go/fo of zero. From the a1, a2 values in Eq. (24) for b = 0.05, g0 becomes negative after h has decreased to 0.580  104 MPa from its initial elastic value.) Note that: (i) the elastic part eo remains a significant portion of the total compressive strain throughout the elastoplastic transition (although this fraction is highly orientation-dependent, from less than 10 pct. in Al at b = 0.05 to nearly 90 pct. at b = 100); and (ii) the mean elastoplastic modulus hM is 30 pct. or more of the corresponding elastic modulus (save for Al at b = 0.05, where it is less than 4 pct.). 5. Lattice rotations and crystal shears at the end of the elastoplastic transition 5.1. Final lattice rotations Upon integrating Eq. (20) from the beginning to the end of the elastoplastic transition, again assuming the effect of very small change in orientation to be negligible, one obtains

D/o ¼ D/E þ A1 ðeo  eE Þ þ A2 ðfo  fE Þ;

pffiffiffi D/E ¼ ð1= 2ÞsA fE j21 fb  2ðb þ 1ÞB1 g:

ð28Þ

Thus, for the same orientations, experiment-based stresses fE, fo, and calculated strains eE, eo as before, we have

b ¼ 100 :

7 D/Al E ¼ 0:301  10 ;

5 D/Al o ¼ 0:204  10 ;

7 6 D/Cu D/Cu E ¼ 0:669  10 ; o ¼ 0:765  10 ;

b¼2:

5 D/Al E ¼ 0:170  10 ;

3 D/Al o ¼ 0:357  10 ;

5 4 D/Cu D/Cu E ¼ 0:497  10 ; o ¼ 0:451  10 ;

b¼1:

5 D/Al E ¼ 0:237  10 ;

ð29Þ

4 D/Al o ¼ 0:242  10 ;

5 4 D/Cu D/Cu E ¼ 0:840  10 ; o ¼ 0:288  10 ; 6 b ¼ 0:05 : D/Al E ¼ 0:224  10 ;

2 D/Al o ¼ 0:233  10

(all angles in radians). The D/E values from Eq. (23), included here for comparison, are an order-of-magnitude or more smaller than the corresponding total rotations D/o, except for Cu in the Brass orientation b = 1. There A1, B1 are zero in Eqs. (20) and (28), and the ratio reduces to the very simple result D/o/D/E = fo/fE for any metal, which gives 3.43 for copper using the experiment-based values. The largest lattice rotation, for Al at b = 0.05 (in the opposite direction from rotations in the other orientations), is a little more than a tenth of a degree. However, all other rotations are two-hundredths of a degree or less, justifying our use of the initial orientations in all calculations. 5.2. Horizontal shearing Let vx, vy denote the angles of shear in the horizontal (YZ) and vertical (XZ) channel planes (respectively the counterclockwise rotation about the X-axis and clockwise rotation about the Y-axis of material lines initially coincident with the Y and X directions). The general equations for their evolution are (Havner and Chidambarrao, 1987, Eq. (5.15))

38

K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

ðtan vx Þ_ ¼ 2dyz þ e_ L tan vx ;

ðtan vy Þ_ ¼ 2dxz þ 2e_ L tan vy :

ð30Þ

From Eq. (4) here and Eq. (13) in Havner (2010), dxz  0 in (110) compression with load-axis stability, whence vy = 0 and the only crystal shearing is vx in the horizontal plane. Consider first the general case of load-axis stability, in which all possible (pairwise equal) slip-rates are included. From dyz = jDl and Eq. (12) for D, one obtains

pffiffiffi 2 2 2 2 2 2dyz ¼ ð1= 3Þj21 fðb þ 4b  2Þc_ 1 þ ðb  8b  2Þc_ 3 þ 2ðb  2Þc_ 5  ðb þ 8b  2Þc_ 7 þ ðb þ 4b þ 2Þc_ 9 g þ 2e_ yz ;

ð31Þ

with e_ yz now given by Eq. (21) multiplied by e˙L. (Eq. (31), without lattice strain-rate e_ yz , is the special case of Eq. (55) in Havner (2010), corresponding to ^i ¼ 0.) A reduced equation for dyz in terms of slip rates may be derived with the aid of the channel die constraints dxx = e˙L and dyy = 0, Eq. (13). The result is

pffiffiffi pffiffiffi 2 2 2 2dyz ¼ 2e_ yz þ ð1= 2Þð1=bÞfb ðe_ L þ e_ xx Þ þ ðb  2Þe_ yy g  ð1= 3Þð1=bÞðb þ 2Þðc_ 3 þ c_ 7 Þ:

ð32Þ

With the same simplifications in e_ xx ; e_ xy ; e_ yy in Eq. (4) as discussed at the beginning of Section 3.1 (that is, neglecting the /_ terms), the final equation for dyz may be written

pffiffiffi pffiffiffi pffiffiffi 1 2 2 2 2 2dyz ¼ ð1= 2Þbe_ L  ð1= 3Þð1=bÞðb þ 2Þðc_ 3 þ c_ 7 Þ  ð1= 2Þð1=bÞ½fs11 b þ s12 ðb  2Þ  sA ðb þ 2Þgf_ 2 2 2 _ þ fs11 ðb  2Þ þ s12 b þ sA gg:

ð33Þ

The first two terms (i.e., without elastic compliances) are identical to Eq. (57) in Havner (2010), wherein lattice straining is neglected. We now may write an explicit equation for (tanvx)0  d(tanvx)/deL:

pffiffiffi pffiffiffi pffiffiffi   2 ðtan vx Þ0 ¼ tan vx þ ð1= 2Þb  ð1= 3Þð1=bÞðb þ 2Þ c03 þ c07  ð1= 2Þð1=bÞðS1 g 0 þ S2 hÞ; 1 2 2 2 2 2 S1 ¼ s11 ðb  2Þ þ s12 b þ sA ; S2 ¼ s11 b þ s12 ðb  2Þ  sA ðb þ 2Þ: 2

ð34Þ

As noted about Eqs. (14) and (15), this equation also is appropriate to use with a viscoplastic power-law in which all positively-stressed slip systems are active. With only systems 1&2 active during the elastoplastic transition (corresponding to rate-independent theory), upon substituting g0 from Eq. (24) we have

pffiffiffi 2 ðtan vx Þ0 ¼ tan vx þ ð1= 2Þð1=bÞfðb  S1 a1 Þ  ðS1 þ S2 a2 Þhg

ð35Þ

(with a1, a2 as in Eq. (24)). Evaluations for crystal shears will be based on Eq. (35). 5.3. Calculations of

vEx ; vox

With dyz ¼ e_ yz ¼ /_ E constant in the elastic range, the solution for the extremely small elastic shears vEx at the onset of slip in systems 1&2 is

tan vEx ¼ 2/0E ðexp eE  1Þ; /0E

eE  fE =Ex ;

with given by Eq. (21). The v For evaluation of the shears v (30) and (35))

2dyz ¼ ðc1 þ c2 hÞe_ L ;

E x values o x at the

ð36Þ

are presented below (in Eq. (39)). end of the elastoplastic transition, the equation for 2dyz may be written (from Eqs.

pffiffiffi 2 c1 ¼ ð1= 2Þð1=bÞðb  S1 a1 Þ;

pffiffiffi c2 ¼ ð1= 2Þð1=bÞðS2 þ S1 a2 Þ

ð37Þ

from which (tanvx)0 = tanvx + (c1 + c2h). The solution is (see Tuma, 1970, p. 164, for example)

tan vx ðeL Þ ¼ tan vEx þ c1 fexpðeL  eE Þ  1g þ c2 exp eL

Z

eL

hðnÞ expðnÞdn

eE

throughout the elastoplastic transition. As there is no information on the corresponding stress–strain curve readily available for any orientation, to simplify the integral we shall consider the values of exp(n) between eE and eo. For eE, from the values in Eq. (23), exp(eE) = 1.0000 (to 5 significant figures) in every orientation. For eo, from the values in Eq. (27), exp(eo) varies between 0.9997 (Cu at b = 2) and 0.9962 (Al at b = 0.05). Consequently, we shall take it as an acceptable approximation to factor-out exp(n)  1.00 from the integral. Thus the final equation for tan vox may be written (c1 dimensionless and c2 in units 105/MPa)

tan vox ¼ tan vEx þ c1 fexpðeo  eE Þ  1g þ c2 ðfo  fE Þ expðeo Þ:

ð38Þ

From Eqs. (36) and (38), the shears at the beginning and end of the elastoplastic transition are (all angles in radians):

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K.S. Havner / International Journal of Plasticity 35 (2012) 31–43





vEx Al ¼ 0:601  107 ;  E vx Cu ¼ 0:1339  106 ;  E vx Al ¼ 0:340  105 ; b¼2:  E vx Cu ¼ 0:995  105 ;  E vx Al ¼ 0:473  105 ; b¼1:  E vx Cu ¼ 0:1681  104 ;  E b ¼ 0:05 : vx Al ¼ 0:448  106 ;

b ¼ 100 :





vox Al ¼ 0:1074  103 ;  o vx Cu ¼ 0:402  104 ;  o vx Al ¼ 0:228  102 ;  o vx Cu ¼ 0:469  103 ;  o vx Al ¼ 0:537  103 ;  o vx Cu ¼ 0:722  103 ;  o vx Al ¼ 0:225  102 ;

5 D/Al o ¼ 0:204  10 ; 6 D/Cu o ¼ 0:765  10 ; 3 D/Al o ¼ 0:357  10 ; 4 D/Cu o ¼ 0:451  10 ;

D/Al o

ð39Þ

4

¼ 0:242  10 ;

4 D/Cu o ¼ 0:288  10 ; 2 D/Al o ¼ 0:233  10

(the D/o values included for comparison). As tan vEx differs only very slightly from vEx and eE is very small, vEx  2D/E , as may be confirmed by comparing with the D/E values in Eq. (29). (Because eo is not determined by Eq. (26) in the Brass orientation, the values of vox at b = 1 are based on e0 = 103, hence they are only order-of-magnitude estimates.) It is seen in Eq. (39) that the shears vox at the end of the elastoplastic transition, though still very small (the largest approximately a tenth of a degree in Al at each of b = 2&0.05), are at least one to two orders of magnitude greater than the elastic shears vEx (and much larger, of order 104, in Al at b = 100&0.05). Also, note that the crystal shears are an order-of-magnitude or more larger than the lattice rotations (with the exception of the near-Goss orientation b = 0.05 in Al, where they are roughly equal and both negative). 6. Finite deformation comparisons Beyond the elastoplastic transition, with four active slip-systems in (110) compression save for certain singular orienta 2Þ,  with a potential six active systems, and Y ¼ ð11  1Þ,  with only the original two), there is finite tions (in particular Y ¼ ð11 shearing vx in all orientations and finite lattice rotation D/ in range I (0 < / < 35.26°). The range I systems active during finite deformation, first identified in Skalli et al. (1983, Fig. 4) and analytically deter c1;  c2 (systems 1–4 here). As discussed in the latter work mined in Havner and Chidambarrao (1987, Section 4), are a1; b2; (p. 250), ‘‘a combination of slip rates on plane c and at least one of planes a and b’’ are required to satisfy the lateral constraint dyy = 0 at finite strain. It is that combination, not required in the other ranges to satisfy the constraints, that produces large lattice rotations. From Eq. (59) in Havner (2010) (lattice straining disregarded), the new orientation / in range I after finite logarithmic compressive strain eL (with load–axis stability) is determined as

pffiffiffi cos / ¼ ð1=3Þf 2g þ ð3  g2 Þ1=2 g;

n

g ¼ exp ln

pffiffiffi

o 2 cos /o  sin /o  eL ;

ð40Þ

where /o is the lattice orientation at the onset of finite deformation (end of the elastoplastic transition). Consider the initial orientation b = 100 calculated here, for which the change in / at the end of the elastoplastic transition is completely negligible (a ten-thousandth of a degree or less in both Al and Cu, Eq. (29) or (39)), as also is the corresponding strain eo (less than 103, Eq. (27)) compared with finite deformations. Thus we have /o = 0.81°, as before. (Note: with finite deformation the sA f_ ; sA g_ terms in Eq. (14) for /_ become negligible in comparison with the slip-rate terms because average slopes of crystal stress–strain curves reduce to 100 MPa or less for strains beyond 0.1. See Wonsiewicz and Chin (1970), Fig. 1; Skalli (1984), Fig. 45; Driver et al. (1994), Fig. 2; Darrieulat et al. (2007), Fig. 4. In rigid-plastic analyses, sA is of course effectively set to zero.)  Lattice rotations D/ were measured in initial (nominal) orientations Y ¼ ð001Þð/ o ¼ 0Þ in experiments of Butler and Hu (1989) on Al and Wrobel et al. (1996) on Cu. The latter state (p. 420) that a deformation ‘‘by channel die compression with 65% reduction of thickness brings about rotation of the initial orientation around the ND [normal direction] by approxi 2Þ  (b = 2). From eL = lnk, with k the ‘spacing stretch’ (ratio of current to original mately 35° toward the orientation’’ Y ¼ ð11 thickness), a 65pct. reduction (k = 0.35) gives eL = 1.05. Then, with that strain and /o = 0.81° in Eq. (40), one obtains / = 38.3° and a rotation of 37.5°, a rather good theoretical prediction of only 7pct. above the measured value.  110  Butler and Hu (1989) carried-out even larger deformations in a nominal ð001Þð001Þ½ orientation. They state (p. L31, with respect to their Fig. 2): ‘‘At 90% reduction, the crystal had rotated 50° about the normal direction.’’ With k = 0.1, the corresponding ‘true’ compressive strain is eL = 2.30 (as they state), for which (again using /o = 0.81°) Eq. (40) gives /  gives = 50.1° and a rotation of 49.3°, very close to the measured value. (Choosing an initial orientation 0.1° from Y ¼ ð001Þ a rotation of 50.0°! The authors do not specify the actual precise initial orientation.). The above rotations, measured in experiments and well-predicted by Eq. (40), obviously are thousands of times larger than those at the end of the elastoplastic transition, Eq. (29). This strongly supports the disregard of those transition deformations in finite-strain analysis of crystals. In the other ranges (with negligible lattice rotations) the comparisons must be between the crystal shears. The equation for finite shear in range I is

tan vx ¼

pffiffiffi 2ðsin /  sin /o Þ  ðcos /o  cos /Þ pffiffiffi 2 cos /  sin /

ð41Þ

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K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

(Havner, 2007a, Eq. (42), lattice straining disregarded). In both ranges II&III the finite shear is given by

pffiffiffi tan vx ¼ ðb= 2Þðexp eL  1Þ

ð42Þ

(Havner, 2008b, Eq. (7.14), range II, and Havner, 2008a, Eq. (27), range III, again disregarding lattice straining). Although the  (1&2): sysadditional active systems are different in ranges II&III, the slip planes are the same as the primary systems a1; b2  b3 (5&6) in range II (away from b = 2, Skalli et al., 1983); and a2;  b1 (9&10) in range III, all as noted in Section 4.2. tems a3; Hence there is no finite lattice rotation in ranges II& III (as previously remarked), and the equation for finite shear is of course very different from that in range I.  1ÞÞ,  the Brass orientation, in which only systems a1; b2  are Consider Eq. (42) for the intermediate position b ¼ 1ðY ¼ ð11 active. As noted in Havner (2008b, p. 1966), ‘‘the equation reduces to an equivalent one first derived by Chin et al. (1966b, Eq. (36)), which gave excellent agreement (see their Fig. 6) with experimental measurements of Chin et al. (1966a) for a Permalloy crystal.’’ Driver and Skalli (1982) carried out experiments on an Al single crystal to eL = 1 in this orientation. From their Fig. 5, the shear angle is determined as 46°, whereas Eq. (42) (with b = 1) gives vx = 50.5°. Perhaps the real crystal, even with standard Teflon coating, may have experienced some longitudinal frictional resistance, thus causing the 9pct. smaller shear than the theoretical prediction.  active. As those remain Recall Eq. (38) for shear vox at the end of the elastoplastic transition with only systems 1&2 (a1; b2Þ the only active systems in the Brass orientation throughout finite deformation, and as eo is not well-defined for b = 1 (as explained in Section 4.3), Eq. (38) may be extended as an elastic–plastic solution for vx to any strain in this orientation by replacing eo by eL and fo by f. The result is (from Eqs. (15), (16), (19), (24), (33), and (37))

pffiffiffi pffiffiffi tan vx ¼ tan vEx þ ð1= 2ÞfexpðeL  eE Þ  1g þ c2 ðf  fE Þ expðeL Þ; c2 ¼ ð1= 2ÞðS2 þ bE S1 Þ;     3 1 2 S1 ¼ s11 þ s12 þ sA ; S2 ¼ s11  s12  sA ; bE ¼  s12 þ sA s11  sA : 2 3 3

ð43Þ

5 E From the values in Eqs. (23) and (39), pffiffiffi eE and tan vx are of order 10 in both Al & Cu. Then, from the elastic compliances (all of 5 order 10 /MPa in Cu& Al), fE ¼ 6 MPa (Eq. (23)), and average experimental stress values of Wonsiewicz and Chin (1970, Fig. 1, curve ‘4’) for Cu (about 200 MPa) and of Driver et al. (1994, Fig. 2, curve ‘B’), for Al (about 60 MPa) over the finite defor3 3 mation range in this orientation, c2(f  fE) is less than 103 in both metals pffiffiffi (specifically, 0.39  10 in Al and 0.35  10 in Cu on average). Thus, the equation can be simplified to tan vx ¼ ð1= 2Þðexp eL  1Þ, which is Eq. (42) at b = 1. This comparison again supports the disregard of elastic strains in finite-deformation analysis of crystals and adoption of a stress–strain curve that begins with a stress fo at ‘zero’ finite strain.

7. Summary and closing remarks Herein we have presented a fully-developed elastoplastic transition analysis for (110) channel die compression of fcc crystals. An analytical equation for the orientation-dependent, elastoplastic strain eo at which finite deformation begins has been derived and evaluated for aluminum and copper over the full range of geometrically-distinct lattice orientations. The equation gives values of order 103 or less, as anticipated, based upon experimental data from the literature and the elastic properties of aluminum and copper, the two fcc metals most studied in channel die experiments. Moreover, we have shown that values of lattice rotation and crystal shearing in these orientations, at the end of the elastoplastic transition, also are very small (the former generally an order of magnitude smaller than the eo values). In Section 6 we have gone beyond the elastoplastic transition and demonstrated that disregard of lattice straining in finite deformation analyses is justified within rate-independent theory, shown in my previous papers (and briefly reviewed here) to give very good predictions of lattice rotation or stability (dependent upon orientation range) and finite crystal shearing in fcc crystals. In addition, we have derived lattice-straining/multiple-slip kinematics incorporating all possible slip-rates in (110) compression: for lattice rotation-rate (Eq. (14)), channel die constraints (Eq. (15)), and horizontal crystal shearing (Eq. (34)). These general equations are needed for any elastic-viscoplastic analysis of fcc crystal deformations. (As subsequently will be reported from a work in progress, it is critical in several crystal orientations, when applying a rate-dependent powerlaw, to include lattice straining in order to achieve realistic results.) Appendix A. Experiment-based simplifications of kinematic equations A.1. Simplification of equation for lattice rotation-rate _ one obtains Eq. (14), but with the entire rightWhen the original Eq. (4) for e_ yz is substituted into Eq. (11) and solved for /, hand side divided by

n o 2 2 4 denom ¼ 1  ðsA =2Þj21 ðb  2Þf þ 2j21 ð9b  b  2Þg :

ðA1Þ

We shall show, by evaluating the sA term for aluminum and copper in each of the four lattice orientations considered (and in _ b = 1 for nickel), that this term is negligible compared with 1, thus giving a justification for the simplified Eq. (14) for /.

K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

41

Independently of a particular metal, one obtains

b ¼ 100 :

denom: ¼ 1  sA ð0:4998f  0:9987gÞ   1 1 fþ g 6 2

b¼2:

denom: ¼ 1  sA

b¼1:

  1 2 f g denom: ¼ 1 þ sA 6 3

b ¼ 0:05 :

ðA2Þ

denom: ¼ 1 þ sA ð0:4988f þ 0:4931gÞ:

From s11, s12, s14 values in Nye (1957, Table 10), respectively (1.59, 0.58, 3.52)  105/MPa for aluminum (Al) and (1.49, 0.63, 1.33)  105/MPa for copper (Cu), the anisotropic shear compliance sA ¼ s11  s12  12 s44 equals 0.41  105/ MPa in Al and 1.455  105/MPa in Cu. (From other cubic crystal compliances in Nye’s table, sA is 0.689  105/MPa in nickel and 0.00 in bcc tungsten.) As reported experimental stresses up to the onset of finite deformation in Al and Cu crystals in (110) compression never reach 100 MPa (see the values below), it is evident that the sA terms in Eq. (A2) are negligible throughout the elastoplastic transition. (Neglecting them is identical with neglect of the underlined term in Eq. (4) for e_ yz .) Even for the Permalloy crystal in Chin et al. (1966a), which is 79% Ni, the fo stress in the Brass orientation b = 1 is only 110 MPa (given as 16 ksi in their Fig. 4). We consider specific experimental values for Al and Cu in the following, which values also are necessary input for the numeric results in Eqs. (23), (27), (29), and (39).  are the most highly stressed for all orientations 1 > b > 0 (0 < / < 90°) in As shown in Appendix B, systems 1&2 ða1; b2Þ both Al and Cu. Thus, whether or not other slip systems may be active during the elastoplastic transition (as with a viscoplastic power law, for example), the critical shear stress sE at the yield point will be defined by systems 1, 2. From Franciosi et al. (1980, Table 2), the shear stress sE for Al at the yield point in various single-slip orientations in axial tension ranges from 79 to 112 g/mm2 (as reported), with an average of 95.7 g/mm2, or 0.94 MPa. Fully half of the 14 tests on Al gave values between 0.98–1.10 MPa. In their 8 tests on Cu crystals in a comparable range of orientations, sE ranged from 75 to 120 g/mm2, with an average of 92.8 g/mm2, or 0.91 MPa. (5 of 8 tests were in the range 0.88–1.18 MPa.) In Schmid and Boas (1950, Table IX), they give a ‘critical shear stress at the yield point’ in Cu (purity > 99.9%) of 0.10 kg/mm2, or 0.98 MPa. (They also reported values for nickel, gold, and silver, but not aluminum.) Consequently, as no direct information on sE is available from channel die experiments (to my knowledge), it seems reasonable to consider the critical shear stress as essentially independent of orientation (the classic Taylor–Schmid law) and equal to 1 MPa in both Al and Cu (for simplicity, based on the above). Correspondingly, from Eqs. (16) and (22), using the elastic compliances above, the fE stress is between 2.4&4.25 MPa for Al and Cu throughout the range, and the gE = bEfE values are half those or less in each orientation. Therefore, at the beginning of the elastoplastic transition (end of the purely elastic range), the sA terms in Eq. (A2) are of order 105 or less for Al and Cu. Consequently, they are completely negligible compared with 1. (For nickel, Schmid and Boas, 1950, Table IX, give a critical shear stress of 0.58 kg/mm2, or 5.7 MPa. With bE, Eq. (16), calculated from the nickel elastic compliances, the fE stresses, Eq. (22), will be approximately 14–23 MPa over the range of orientations, with gE values always less than half those. Consequently, the sA terms in Eq. (A2) all will be less than 0.5–104.) Consider the following experiment-based values of fo (used in calculations throughout the paper). In Al crystals: (i) 67 MPa and 57.8 MPa for b = 100&2, from experimental curves of Skalli (1984, Fig. 45), in nominal orientations   2Þ;  (ii) 25 MPa for the Brass orientation b = 1, from Driver et al. (1994, Fig. 2, curve ‘B’); and (iii) 22 MPa for Y ¼ ð001Þ&ð1 1 b = 0.05, from an analytical curve in Havner (2011, Eq. (11)), which fits data of Darrieulat et al. (2007) very well in the nearby  (2° away) for 12 reported strain points from 0.05 to 1.0 (see Fig. 5 in Havner, 2011). In Cu crystals, Goss orientation Y ¼ ð110Þ for b = 100,2,1,&0.05, we have 47 MPa, 17 MPa (estimated), 8.4 MPa, and 17 MPa, respectively, from experimental stress– strain curves of Wonsiewicz and Chin (1970, Fig. 1). Consequently, at the end of the elastoplastic transition, the sA terms in Eq. (A2) are of order 104 (or less) in Al and Cu, and remain negligible compared with 1, from which the simplified lattice rotation-rate Eq. (14) follows. A.2. Simplification of equations for the constraints Although it now may seem logical to immediately neglect the underlined terms in the other lattice strain-rates in Eq. (4) (a simplification that leads directly to constraint Eq. (15)), because inclusion of the corresponding term in the e_ yz equation has a negligible effect on /_ in Al and Cu (and also an Ni alloy), we shall show independently that those terms are negligible. Upon substituting Eq. (14) into the general constraint Eq. (13), one obtains

n oi pffiffiffi h dxx ¼ e_ L : ð2= 6Þ f1 þ ðb  1Þd1 gc01 þ f1 þ ðb þ 1Þd1 gc09 þ d1 ðb  1Þ2 c03  2c05  ðb þ 1Þ2 c07 2

þ sA fC xx =sA þ ðb=2Þd1 gf 0  sA fðC xy =sA Þ þ bðb  1Þj21 d1 gg 0 ¼ 1;

42

K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

pffiffiffi    dyy ¼ 0 : ð2= 6Þj21 bð1  d2 Þ ðb  1Þc01  2c05 þ ðb þ 1Þc09  ðb  1Þfðb þ 2Þ þ bðb  1Þd2 gc03  ðb þ 1Þfðb  2Þ   n o 1 2 2 2 þ bðb þ 1Þd2 gc07 þ sA ðC xy =sA Þ  b j21 d2 f 0  sA ðC yy =sA Þ  b ðb  1Þj41 d2 g 0 ¼ 0; 2    2 4 2 d1 ¼ 2sA gbj1 ; d2 ¼ 2sA j1 f  4g 1  3j21 ; j21 ¼ 1=ðb þ 2Þ:

ðA3Þ

Consider the first of these equations and the ratio of the d1 term to the first term within each of the multipliers of c01 ; c09 ; f 0 , and g 0 . As is evident, the c01 , g0 ratios are identically zero in the Brass orientation b = 1, and all four would be zero in the Goss orientation b = 0. Otherwise, the largest ratio will be at the end of the elastoplastic transition (largest g); and for all three multipliers it is found to be at b = 2, where go = fo (Havner, 1992, Eq. (5.37)). Using the values 57.8 MPa in Al and 17 MPa in Cu for fo (Section A.1), the corresponding ratios are 2.63  105, 7.90  105, 0.779  105, &2.11  104 in Al, and 2.75  105, 8.25  105, 5.2  105, &1.0  104 in Cu. Thus the d1 term is negligible in comparison with the first term in each multiplier. Furthermore, the multipliers of slip-rates c03 ; c05 ; c07 all are of order 104 or less in comparison with the multipliers of c01 &c09 ; and c03 ; c05 ; c07 are much smaller than c01 during the elastoplastic transition (whether calculated from a power-law or simply taken as zero in rate-independent theory, as here). Therefore those terms also are negligible. In the second of Eq. (A3), d2 is less than 104 in magnitude in all orientations (and essentially zero at b = 100), hence it can be neglected in comparison with 1 in the common multiplier of c01 ; c05 ; c09 . In the multipliers of c03 &c07 , the d2 terms correspondingly are negligible in comparison with the first terms. In the multipliers of f0 , g0 , the ratios of the d2 terms to the respective first terms at b = 2 are 2.11  104&0.779  105 in Al and 1.0  104&5.2  105 in Cu (each pair the reverse of that in the first equation). At b = 1 the ratio is identically zero for g0 and 1.05  105&1.36  104 for f0 in Al&Cu. The ratios approach zero as b ? 0. Consequently, all d2 terms are negligible. From the above, we have seen that all d1 and d2 terms in Eq. (A3) are of order 104 or less in comparison with the other terms in the multipliers of slip-rates and stress-rates for Al and Cu crystals in (110) compression. Upon setting them all to zero, Eq. (A3) identically reduces to the simplified constraint Eq. (15) obtained by neglecting all the /_ terms in the lattice strain-rates. Appendix B. Determination of highest-stressed slip systems in the elastic range B.1. Determination of initial active slip-systems in aluminum Evaluating Eq. (16) for aluminum (compliances in Appendix A.1), we have (Havner, 1992, Eq. (5.38))

bE ¼

4

2

4

2

0:58b þ 1:91b þ 1:50 1:59b þ 4:72b þ 5:54

ðA4Þ

:

Upon substituting into the resolved shear stress Eq. (6) (with g = bEf), we obtain for the shear stress differences with systems 1&2 in the elastic range:

pffiffiffi

s1  s3 ¼ ð1= 6Þf ð0:43b6 þ 5:24b4 þ 15:80b2 þ 14:08Þ=DðbÞ; pffiffiffi s1  s5 ¼ ð1= 6Þf ð1:01b6  0:58b5 þ 5:99b4  1:91b3 þ 13:48b2  1:50b þ 11:08Þ=DðbÞ; pffiffiffi s1  s7 ¼ ð1= 6Þf ð0:43b6 þ 1:16b5 þ 5:24b4 þ 3:82b3 þ 15:80b2 þ 3b þ 14:08Þ=DðbÞ; pffiffiffi s1  s9 ¼ ð1= 6Þf ð1:16b5 þ 3:82b3 þ 3bÞ=DðbÞ; 6

4

ðA5Þ

2

DðbÞ ¼ 1:59b þ 7:90b þ 14:98b þ 11:08: pffiffiffi All differences are positive and finite, independent of b ¼ 2 cot / (with the exception that s1  s9 ? 0 as b ? 1, / ? 0). However, for any fcc crystal, we see from Eq. (6) that systems 7&8 are positively stressed in the opposite sense of slip for b < 2, and systems 3&4 are positively stressed in the opposite sense for b < 1. Therefore, we must evaluate the sign of s1 + s7 for b < 2 and the sign of s1 + s3 for b < 1 to confirm that systems 1&2 always are the most highly stressed. We have

pffiffiffi

s1 þ s7 ¼ ð1= 6Þf ð1:59b6 þ 6:74b4 þ 11:16b2 þ 8:08Þ=DðbÞ > 0; pffiffiffi s1 þ s3 ¼ ð1= 6Þf ð1:59b6 þ 1:16b5 þ 6:74b4 þ 3:82b3 þ 11:16b2 þ 3b þ 8:08Þ=DðbÞ > 0:

ðA6Þ

Thus, from these and the preceding equations, systems 1&2 are the most highly stressed. These systems will be the first to become active, at a critical resolved shear stress sE, in all orientations in (110) channel die compression. B.2. Determination of initial active slip- systems in copper Evaluating Eq. (16) for copper (compliances again in Appendix A.1), we have (Havner, 1992, Eq. (5.39)) 4

bE ¼

2

0:63b þ 1:065b  0:39 4

2

1:49b þ 0:14b þ 3:05

:

ðA7Þ

K.S. Havner / International Journal of Plasticity 35 (2012) 31–43

43

 b ¼ 0; / ¼ 90 Þ the crystal initially contracts laterally This is negative for b < 0.556 (/ > 68.5°). In this sub-range (to Y ¼ ð110Þ; (because of the very strong elastic anisotropy of copper crystals), and g must be set to zero in the elastic resolved shear stress equations. Also, the lateral constraint becomes dyy 6 0. Again substituting g = bEf into Eq. (6), we find for the shear stress differences with systems 1&2 in Cu, for b P 0.556,

pffiffiffi

s1  s3 ¼ ð1= 6Þf ð0:23b6 þ 2:25b4 þ 6:24b2 þ 5:32Þ=DðbÞ; pffiffiffi s1  s5 ¼ ð1= 6Þf ð0:86b6  0:63b5 þ 2:055b4  1:065b3 þ 3:71b2 þ 0:39b þ 6:10Þ=DðbÞ; pffiffiffi s1  s7 ¼ ð1= 6Þf ð0:23b6 þ 1:26b5 þ 2:25b4 þ 2:13b3 þ 6:24b2  0:78b þ 5:32Þ=DðbÞ; pffiffiffi s1  s9 ¼ ð1= 6Þf ð1:26b5 þ 2:13b3  0:78bÞ=DðbÞ; 6

4

ðA8Þ

2

DðbÞ ¼ 1:49b þ 3:12b þ 3:33b þ 6:10; all of which are positive. As before, we must evaluate systems 7&8 in the opposite sense of slip for b < 2, and systems 3&4 in the opposite sense for b < 1. We obtain

pffiffiffi

s1 þ s7 ¼ ð1= 6Þf ð1:49b6 þ 1:86b4 þ 1:20b2 þ 6:88Þ=DðbÞ; pffiffiffi s1 þ s3 ¼ ð1= 6Þf ð1:49b6 þ 1:26b5 þ 1:86b4 þ 2:13b3 þ 1:20b2  0:78b þ 6:88Þ=DðbÞ:

ðA9Þ

Each sum is positive for all b. Thus, systems 1&2 are the most highly stressed in Cu for b P 0.556. For b < 0.556 (/ > 68.5°), s1 = s9 and s3 = s5 = s7 = 0 in Eq. (6) (from g = 0), whence systems 1&2 and 9&10 are the only stressed slip systems in Cu within this range. References Butler Jr., J.F., Hu, H., 1989. Channel die compression of aluminum single crystals. Mater. Sci. Eng. A 114, L29–L33. Chin, G.Y., Nesbitt, E.A., Williams, A.J., 1966a. Anisotropy of strength in single crystals under plane strain compression. Acta Metal. 14, 467–476. Chin, G.Y., Thurston, R.N., Nesbitt, E.A., 1966b. Finite plastic deformation due to crystallographic slip. Trans. Metal. Soc. AIME 236, 69–76. Darrieulat, M., Poussardin, J.-Y., Fillit, R.-Y., Desrayaud, Ch., 2007. Homogeneity and heterogeneity in channel-die compressed Al-1%Mn single crystals: Considerations on the activity of the slip systems. Mater. Sci. Eng. A, 641–651. Driver, J.H., Juul Jensen, D., Hansen, N., 1994. Large strain deformation structures in aluminium crystals with rolling textures. Acta Metal. Mater. 42, 3105– 3114. Driver, J.H., Skalli, A., 1982. L’essai de compression plane de monocristaux encastres: methode d’etude du comportement d’un crystal soumis a une deformation plastique imposee. Revue Phys. Appl. 17, 447–451. Franciosi, P., Berveiller, M., Zaoui, A., 1980. Latent hardening in copper and aluminium single crystals. Acta Metall. 28, 273–283. Havner, K.S., 1992. Finite Plastic Deformation of Crystalline Solids. Cambridge University Press (Paperback Re-issue 2008). Havner, K.S., 2005. On lattice and material-frame rotations and crystal hardening in high-symmetry axial loading. Philos. Mag. 85 (25), 2861–2894. Havner, K.S., 2007a. Channel die compression revisited: application of a set of basic crystal hardening inequalities to (110) loading. Mech. Mater. 39, 610– 622. Havner, K.S., 2007b. Corrigendum to Channel die compression revisited: Application of a set of basic crystal hardening inequalities to (110) loading. Mech. Mater. 39, 893–895. Havner, K.S., 2008a. Investigation of basic crystal hardening inequalities in a range of stable lattice orientations in (110) channel die compression. Int. J. Plast. 24, 74–88. Havner, K.S., 2008b. Further investigation of crystal hardening inequalities in (110) channel die compression. Proc. R. Soc. A 464, 1955–1982. Havner, K.S., 2010. Analysis of fcc crystals in two singular orientation in (110) channel die compression. Mech. Mater. 42, 657–672. Havner, K.S., 2011. Perspectives on (110) channel die compression and analysis of the Goss orientation. Int. J. Plast. 27, 1512–1526. Havner, K.S., Chidambarrao, D., 1987. Analysis of a family of unstable lattice orientations in (110) channel die compression. Acta Mech. 69, 243–269. Nye, J.F., 1957. Physical Properties of Crystals. Oxford University Press. Schmid, E., Boas, W., 1950. Plasticity of Crystals. Hughes, 1935, Springer (reissue 1958, Chapman & Hall, Translation of: Kristallplastizität). Skalli, A., 1984. Etude theorique et experimentale de la deformation plastique en compression plane de cristaux d’aluminium. These d’Etat, Ecole des Mines de Saint-Etienne, France. Skalli, A., Driver, J.H., Wintenberger, M., 1983. Etude theorique et experimentale de la deformation plastique de monocristaux d’aluminium. Deuxieme partie: compression plane partiellement imposée. Mem. Et. Sci. Rev. Metall. 80, 293–302. Sue, P.L., Havner, K.S., 1984. Theoretical analysis of the channel die compression test – I. General considerations and finite deformation of f.c.c. crystals in stable lattice orientations. J. Mech. Phys. Solids 32, 417–442. Tuma, J.J., 1970. Engineering Mathematics Handbook. McGraw-Hill. Wonsiewicz, B.C., Chin, G.Y., 1970. Plane strain compression of copper, Cu 6Wt Pct Al, and Ag 4Wt Pct Sn crystals. Metal. Trans. 1, 2715–2722. Wrobel, M., Dymek, S., Blicharski, M., 1996. The effect of strain path on microstructure and texture development in copper single crystals with (110)[001]  initial orientations. Scr. Mater. 35, 417–422. and (110)½110