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On velocity discontinuities in elastoplastic bicrystals in channel die compression
Perg, amon
International Journal of Plasticity, Vol. 14, Nos. 1-3, pp. 61 74, 1998 © 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0749-6419/98 $19.00 + 0.00 P I I : S0749-6419(97)00040-5
ON VELOCITY DISCONTINUITIES IN ELASTOPLASTIC BICRYSTALS IN CHANNEL DIE COMPRESSION*
K. S. Havner Department of Civil Engineering, North Carolina State University, Raleigh, NC 27695-7908, U.S.A.
(Received in final revisedform 21 September 1996) Abstract--Possible consequences of including lattice straining in f.c.c, bicrystal models in channel die compression are investigated. It is determined that the tangential velocity discontinuities which emanate from the interface edges in rigid/plastic models are not eliminated by taking crystal elasticity into account. © 1998 Elsevier Science Ltd. All rights reserved
I. I N T R O D U C T I O N
In Wu and Havner (1995, 1997) and Havner and Wu (1995), an analytical rigid/plastic model of symmetric f.c.c, bicrystals in (110) channel die compression is rigorously demonstrated to exhibit tangential velocity discontinuities at the yield point. These velocity discontinuities, interpreted as the initiation of shear bands, essentially are a consequence of the geometric boundary conditions, slip-system kinematics, and f.c.c, crystal yield locus. They extend from the intersections of the interface plane with the rigid channel walls, and are in characteristic directions which depend on crystal orientation. The prediction of discontinuities holds for every constraint direction Y (relative to the lattice) in the (110) family of loading direction X. Moreover, each crystal subdivides into distinct regions among which both material strain-rates and lattice-rotation rates differ, consistent with the initiation of subgrains. The evolution of shear bands and continued formation of subgrains are well-established experimental phenomena in the plastic deformation of polycrystalline metals (see Bay et al., 1989, 1992; Hughes and Nix, 1989; Hughes, 1993), hence the theoretical results wottld appear encouraging. In addition to the presentation in Wu and Havner (1997) of complete analytical solutions for all orientation ranges in (110) compression, finite element results for several orientation,; at the end of the elastoplastic transition (from purely elastic to fully plastic response) a:re compared with the corresponding analytical solutions (also see Wu, 1995). The adopted finite element discretization does not admit tangential velocity discontinuities, but otherwise the numerical results for both stress states and velocity fields are in qualitative agreement with the analytical rigid/plastic solutions; and in some cases the results are quite close. *This paper is dedicated to the memory of James F. Bell, distinguished experimentist, scholar, and friend. 61
62
K.S. Havner
The question remains: is the unequivocal analytical requirement of tangential velocity discontinuities at the initiation of finite multiple-slip in the rigid/plastic bicrystal merely a consequence of that idealization, or does it still hold when lattice straining is taken into account in the crystal kinematics? This paper undertakes to answer that question. II. FORMULATIONAND GOVERNINGEQUATIONS Based upon the comprehensive review in Havner (1992), of both experimental and theoretical studies of finite deformation of single crystals in (110) channel die compression, and the formulation (and numerical solution) of the elastic bicrystal problem in Fuh (1989), Havner et al. (1994), draw the following conclusion. "The nonuniformity of deformation of adequately lubricated, symmetric fcc bicrystals in (110) channel die compression can be expected to take place entirely within Y Z planes .... The loading axis will remain normal to a (110) lattice plane throughout, while the lattice nonuniformly rotates about that axis, causing each crystal to break-up into a crystalline aggregate with a (110)texture." This is because, as stated in Wu and Havner (1997), "geometric incompatibility between differently oriented, freely shearing individual crystals in (110) loading exists only in Y Z planes, hence the nonuniform straining necessary to satisfy compatibility in bicrystals logically should take place entirely within those planes". This reduction to a spatially two-dimensional problem has been confirmed numerically for the elastoplastic transition by Wu (1995), who began with a fully three-dimensional finite element model. (For more detailed arguments and citations of experimental support for these conclusions, see Sections 1 and 2 of Havner et al., 1994 and Section 2 of Wu and Havner, 1997.) As a consequence of the foregoing, the general problem of bicrystals in (110) channel die compression is governed by the following equilibrium and kinematic relations (Wu and Havner, 1997, eqns (2.2)-(2.4)): trxy = trxz = O,
dxy = d~z = O,
< O,
-
dxx=-l,
+
Oy
Ov dyy=~y,
Wy = wz = O,
Wy = Wz = O,
(1)
--~-z = O,--@--y + ~ - z = O,
Ow Ov 2dyz=~-fy+~,
Ow dzz=~z,
Ow 2Wx-oy
(2) Ov Oz
(3)
with velocities u = - x , v(y, z), w(y, z), and lattice-rotation rate wx = ~(y, z) defined with respect to logarithmic compressive strain eL rather than with respect to time. The d u are components of Eulerian strain-rate D; to = (Wx, toy, ~oz)x and w = (wx, Wy, wz) T are axial vectors of lattice and material spin tensors to and W, respectively, relative to the channel axes; and q~is the (local) counterclockwise orientation from the Y-axis of lattice direction [00i] in the right-hand crystal (Fig. 1). The initial boundary conditions for that crystal (a unit cube), taken to apply up to the onset of finite multiple-slipping (and thus throughout the very small strain, elastoplastic transition stage), are: x=O:u=O; y=O,l
x=l:u=--l;
:ayz=O,
z=O:cryz=O,
(4)
v=O; w=O;
z=l
:O'y~=O'~=O.
On velocitydiscontinuities in elastoplastic bicrystals in channel die compression
63
Crystal A
Crystal B
,\
/
[oot-l
y[
Fig. 1. Initial configuration of symmetric bicrystal in (110) channel die compression as viewed from loading direction. (Rectangular grid representsunderlyinglattice.) The 12 crystallographic slip systems in f.c.c, crystals, defined in Table 1, are numbered as follows: al,b2=1,2;
c],c2=3,4;
a3, b 3 = 5 , 6 ;
dl,d2=7,8;
a2, b l = 9 , 1 0 ;
c3, d 3 = 1 1 , 1 2
(the bar above signifying the opposite sense system from that in the table). From eqn (1)1 and the standard tensor transformation between channel and lattice axes, one obtains (Havner et al., 1994, eqns (3.1)1, (3.1)3, and (5.2)1) Orll = 0"22,
0"13 = --0"23,
0"11 -I- 0"12= ~xx ~ O.
(5)
Then, because the resolved shear stresses rk on the twelve slip systems depend only upon trl2, tr13, and cr33 - trH, with rk = 0 in systems 11, 12, the initial yield locus is a decahedron in the three-dimensional lattice-stress space having these components as axes, as shown in Fig. 2. There r0 denotes the critical resolved shear stress ("critical strength") at the beginning of fully plastic response, assumed to be the same in all slip systems. (Possible effects of anisotropy in the small amount of hardening that develops during the elastoplastic transition stage are considered negligible. See the quantitative assessment of this hardening in single crystals in (110) compression in Havner, 1992. For further details regarding the yield locus, see Havner et al., 1994; or Wu and Havner, 1995). The Eulerian strain-rate D and plastic spin f2 = W - 09 may be expressed as D = symEO! ® n)k~k + ~,
(6)
, f2 = skw E(b ® n)k~'k
Table 1. Designation of slip systemsin f.c.c, crystals Plane Direction System
011 al
(111) 10i a2
il0 a3
0ii bl
(lli) 101 b2
il0 b3
01i cl
(lii) 101 c2
ii0 c3
011 dl
(lil) 10i d2
ii0 d3
64
K. S, Havner o'13 ,/'~----~o' Zd,3 (5,8),,~
/._
.
ut
,0.5
./ c~.,) / I '%,. / / __ \ ~
o_.~) _ \ o , -"~I~/0.~,-0.,,.,
* 1 / ~
(~)---~ ~ 0.1~ _ To'
\
(o~o) /
~ ~t2
/
~
.o ' - ~"
E I
Fig. 2. Initial yield locus in lattice stress space, with slip systems indicated in parentheses.
where Yk is the slip-rate in the kth system, bk, nk are unit vectors in the slip and normal directions of that system, and k is the lattice strain-rate. From eqns (1)2 and (3)~ and the tensor transformation which gave eqn (5), we have dll =d22,
d]] --{-d12 -~--dxx -- - 1 .
d13 ~-- -d23,
(7)
In addition, from transformation of the axial vector an = w - w of plastic spin ~2 there follows 1
a] -- ~32 = a2 -- ~13 -~- --~_ax,
a3 ~-- ~'221 = O.
(8)
Therefore, from eqns (6)-(8), there are five constraints on the 10 unknown slip rates. (yll, y12 are zero since the corresponding rk equal zero.) Because the lattice strain-rates satisfy the same basic tensor transformation as a and D, four of the constraints are the same as in the rigid/plastic model (Havner et al., 1994, eqn (4.6)): ~1+Y5=~2+~6,
Y3=Y4,
Y7=~8,
Yl+~9=~2+Y10;
(9)
and the fifth is (from eqn (7)3)
f/l --[- f/9 -~- --(~/r'a/2)(dxx-- Exx).
(lO)
The final equations for the Eulerian strain-rate components on lattice axes thus may be expressed (from eqns (6)1, (9), (10) and Table 1)
On velocity discontinuities in elastoplastic bicrystals in channel die compression
1
65
1
a l l = d22 = -~(dxx - ~xx) --I-~ ( ~ 3
- ~'7) -+- ~11,
2
d33 = - ( d x x
- Exx) - ---~ (J)3 - })7) -+" E33, vo
d13 = - d 2 3 = 1
l
(dxx - ixx) + 1
d12 --'=~ ( d x x -- ~xx) -- - - ~ ( ~
(ll)
.
~--~(2y1 + h + 2y5 + b ) + i13, -- ~'7) + ~12.
These generalize eqn (4.10) of the rigid/plastic model in Havner et al. (1994). Because I ~xx [<< -dxx = 1 at the onset of finite multiple-slip in the compressed bicrystal, at least one of systems 1 (al) and 9 (a2) necessarily is active in the positive sense from eqn (10). Consequently, every stress state in the right-hand crystal must lie either on face A B C D (systems 1,2) or face A D E F (systems 9,10) of the yield locus (Fig. 2), including their edges. For the rigid/plastic model it is concluded in Wu and Havner (1995, 1997) that al3 is non-negative for all 4'0, hence all stress states lie on face ABCD. Wu (1995) found this to be the case as well for the elastoplastic transition, from finite element calculations of a number of bicrystal orientations spanning the range 4'0 between 0 and 90 °. (See Wu and Havner (1997) for final stress states in two of these orientations.) Therefore, only face A B C D is investigated here. Moreover, we restrict consideration to the range of lattice orientations for which the stress state at each material point lies either on edge AB or within face A B C D (excluding the other edges). In the rigid/plastic model this is called range I and is given by 0 < 4'0 < 35.26 o , or channel constraint direction Y between [00i] and [1 i2], first analyzed in Wu and Havner (1995). (There it was assumed that all stress points lay on edge AB only.) With lattice straining included, the end points of this range cannot be so precisely defined analytically. However, they are expected to differ relatively little (no more than a degree or two) from [00i] and [1 i2.], and their exact positions need not concern us in a general analysis of range I. Along edge A B of the yield locus only slip systems 1,2 and 3,4 can be active, and within face A B C D only the first pair (see Fig. 2). Consequently, setting all other slip rates in eqns (9)-(11)) to be zero, we find from the requirement yk > 0, k = 1.... 4, that the normality constraints for range I orientations are -(d12 -- El2) = 2(d13 - ~13),
dll -- ~11 --> d12 -- El2
(12)
the latter of which may be expressed (from eqn (7)) -2d12 > -dxx +
Eli - - ~12,
-dxx = 1
(13)
(the equality satisfied only when systems 3,4 are inactive, as within face ABCD). These relations generalize those for edge A B and face A B C D of the rigid/plastic crystal in Wu and Havner (1995), eqn (8), and complete the set of governing equations. IH. GENERAL ANALYSIS OF VELOCITY FIELDS
We adopt the parameter b= ~
cot4~
(14)
66
K.S. Havner
introduced in Havner and Chidambarrao (1987), from which channel constraint direction Y = [1 lb] and the precise range I of the rigid/plastic model is defined by oo > b > 2. From the first normality constraint, eqn (12 h, and the standard strain-rate transformation between channel and lattice axes, one then can obtain (after some algebra) 2(b - l)2dxx + (b2 - 8b - 2)dyy + 2~/2(b 2 + b - 2)dyz
(15)
= b(b - 4)~v + 2(b2 + 2)(~12 + 2~13)
in which use has been made of the condition of zero plastic volumetric strain-rate (from eqn (6)1) (16)
dxx + dw + dzz = kv = 2klt "l- ,~33
to simplify eqn (15) (kv is the lattice volumetric strain-rate). Therefore, from eqn (3) and the above, the velocity field in the YZ plane is governed by the following pair of partial differential equations (also involving the unknown lattice strain-rates): 0u Al~+~z=C+e*,
A1 ,v/~(b2 + b 1
-
2)
1 0
u = ( v , w ) r,
i
,
c
=
2
-
1
(17)
]
c.={b(b_4)kv+2(b2+2)(k12+2k13) +-b
axx,
}r ,
(18)
(19)
These generalize the pair of hyperbolic equations in Wu and Havner (1995), eqns (4.4)(4.5), and differ from them only by the addition of vector e* in the lattice strain-rates. From Wu and Havner (1995), eqns (3.7)-(3.8), the characteristic directions of the homogeneous form of eqn (17) are rotated counterclockwise from the Y, Z directions by an angle 0 given as tan 0 -
b+2 q~(b - 1)
(20)
Whence, from eqn (14), cot(0 - q~) = ~
(21)
and the first characteristic direction is rotated counterclockwise from [001] by 35.26 ° for all lattice orientations in range I (Wu and Havner, 1995). Let oe,/3 denote these orthogonal characteristic directions, and u~, ue denote the corresponding velocity components: ot = y c o s 0 + z sin0, u,~ = vcos0 + wsin0,
/3 = - y sin 0 + z cos 0, ut~ = - v s i n 0 + wcos0.
(22)
On velocitydiscontinuities in elastoplastic bicrystalsin channel die compression
67
Then, from eqns (20) and (22), eqn (17) can be transformed into the following spectral pair:
bud 2 - - = -dxx + ~(ev + ~12 -+- 2el3), oa (23) 8u~
8¢3
--
1
3
2
6V -- ~"(El2 -F 2e13) .5
from which Oudl~ >> 0u~18/3, and 2. dud = (-dxx) dot + -~(ev + el2 + 2k13)dot
along ot-lines, (24)
1
du~ = ~ {k v - 2(e12 + 2k13)}dfl
along/3-lines,
generalizing the simple relations for the rigid/plastic bicrystal first given in Wu and Havner (1995), eqr~s (4.8). We introduce variables 0(ot,/3), ((or,/3), defined by 01/
2
= ,5"~(~V-'1-~12 "1"-2e13),
8( 1 3-"-~= 3 {ev -- 2(e12 + 2e13)}
(25)
(which gradients may be discontinuous across certain interior boundaries), noting that 8rl/Oot + 8(/8fl = kv. The general solution of eqns (23) or (24) now may be expressed as
u~=-otd~+g(/3)+O(ot,/3),
u~=f(a)+((ot,/3).
(26)
These reduce to the rigid/plastic solution in Wu and Havner (1995), eqn (4.9) (or Wu and Havner (1997), eqn (3.26)), with 0, ( of course zero. We also need the kinematic boundary conditions in the new variables. From eqns (4) and (22) these are: z=0
(w=0)"
cr+/3cot0=0,
ud+u, cot0=0,
y=0 y=l
(v=0)' (v=0)"
ot=13tan0, u~ -- u, tan 0, ot=/3tan0+sec0, u~=u~tan0.
(27)
Consider the subdivision of the right-hand crystal into the five regions shown in Fig. 3 (from Wu and Havner (1995), Fig. 4), and let J~, gi, Oi, (i denote the unknown functions in the ith region. Continuity of normal velocities across interior region boundaries gives:
f2(~)=A(a)+~12(~), g12(a)=gl(~, 0)-g2(~, 0), g3(/3)=gl(~)--I--O13(/3), Ol3(/3)=Ol(a, fl)--o3(a, fl), g4(~)=g2(~)-q-O24(fl), 024(/3)=02(a,/3)--o4(a,~), f4(ot)=f3(a)q-(34(a), ~ 3 4 ( a ) = ~3(~, 0 ) - - ~ 4 ( ~ , O)
(28)
68
K.S. Havner
o
Z
b
Z
d
~
OC
i1
b
"I °\ Y
(a)
(b)
Fig. 3. Grain subdivision into regions I-V for kinematic analysis of symmetric bicrystals
and, for 0 _> 45 ° (case (a), Fig. 3): fs(ot) =f4(ot) q- ~'45(0/), ~'45(0t) = ~'4(ot, c) - ~'5(og,c)
(29)
with a = cos 0 and c = a cot 0. (For 0 < 45 °, case (b) in Fig. 3, eqn (29) is replaced by similar relations in g and 0-functions.) Upon substituting eqns (28) and (29) into eqn (26) and the results into the first boundary condition, eqn (27)1, we have (henceforth defining fl = f a s in Wu and Havner (1995, 1997) gl(fl) = -{fldxx + f ( - / 3 c o t 0) + ( l ( - f l c o t 0, fl)} cot0 - 01(-flcotO, fl).
(30)
From the second boundary condition, eqn (27)2, applied to regions II and V (0 _> 45°),
gi(fl)
=
{fldxx +f,.(/3 tan 0) + (i(fl tan O, /3)} tan 0 - Oi(fl tan O, /3),
i = 2, 5
(31)
with f2 given by eqn (28)1 (the argument o~ of course replaced by fl tan 0) and f5 given in terms off3 through eqns (28)4 and (29). Function f3 is determined from the third boundary condition, eqn (27)3, as f3(ot) = {-otdxx + g3(-X) -k- 03(or, -X)} cot 0 - (3(or, -X),
X = csc 0 - a cot 0
(32)
g3(--X) = {Xdxx - f ( x c o t 0) - ~'l(XCOt 0, - X ) } cot0 - 0 1 ( x c o t 0, - X ) + 013(-X)
(33)
with
from eqns (28)2 and (30). All functions fi, gi now are expressed in terms of the single unknown function f and the lattice strain-rate-dependent functions Oj, ffj, J = 1..... 5, for
On velocity discontinuities in elastoplastic bicrystals in channel die compression
69
0 > 45 °. (Modified equations for f5 and g5 of course can be determined for 0 < 45 °, as in the rigid/plastic results given for that case by W u and Havner (1995), eqns (4.14)-(4.15).) Let us a,;sume there are no tangential velocity discontinuities along any o f the interior boundaries between the regions defined in Fig. 3, case (a). Along fl = 0, between regions I and II, we then would have, from eqns (26)1, (30) and (31), (Au~)12 = 0 = - i f ( 0 ) + ~1(0, 0)} sec 0 csc 0 - 0~2(0) + 01z(a), 012(o0 = rh (or, 0) - 02(or, 0)
(34)
from which r/12(ot) = const = 012(0). Consequently
f(o) + ~1(0, o) = 0
(35)
Similarly, applying our assumed condition o f no tangential velocity discontinuities to the other interior boundaries, we obtain ffl3(ot, 13) = const = ~'13(a, 0),
a = COS0,
f(a) + ~l (a, -- sin 0) = 0, ~24(a, ~) = const = ~24(a, 0),
(36)
r/s4(ot, 0) = const = r/34(a, 0), 045(ot, c) = const = r/45(a, C),
C = acot0.
Thus, from the two conditions on f u n c t i o n f ( e q n s (35) and (36)2), we must have flcos 0) - f(0) = ~l (0, 0) - ~l (cos 0, - sin 0)
(37)
for there to be no tangential velocity discontinuities. (Note that function ~'1(or, fl) is evaluated in this equation at the edges o f the interface plane.) A comparable analysis can be made for c,ase (b), 0 < 45 °. IV. APPLICATION OF THE SECOND NORMALITY CONSTRAINT
F r o m eqn (6)1 we may write
dyy ~- ~](K" bj)(K" nj)f/j --[- ~yy,
K = (1, - 1 , - b ) / v / ~
+ 2.
(38)
In range I,, with only four possible non-zero slip rates, we have
~'l = i/2 = -(~/6/2)(dxx - kxx) > 0,
)~3 = Y4
(39)
(from eqns (9) and (10)). Therefore, from Table 1 and the above:
dyy-
b(b-1) b2 ~ ~ (dxx - kxx)
2(b 2 + b + 2 ) . ~/-6(b 2 + 2) Y3 +
Eyy.
(40)
Thus, since the second normality constraint, eqn (13), is identically the condition y3 > 0, we must have
70
K.S. Havner
dyy <
b ( b - 1) --ff5~ ( d x x - ~ x ) + k y y .
(41)
T o proceed further, we need to express this inequality in terms o f functions f, g, r/, ~"within each region. F r o m eqn (26) the strain-rates on the or,/3 axes in the ith region are
d~ = -dxx +--~,
2d~e = f'i(ot) + g 'i(/3) +-~-~ +--O~t,
(42)
aee =
the prime signifying differentiation of a function with respect to its argument. Consequently, from the coplanar strain-rate transformation between Y, Z and or,/3 axes, eqn (20), k~,~ = ell + Sl2, and eqn (41), we find (after substantial algebra):
~{fti(ot)
-[- g ti(/3)} ~_>dxx
(~. 0¢;~ 1 - ~/~\ /3 + - ~ ) + ~ {El1
-
-
7k12-[- 4 (El3
-
-
833)}
(43)
which generalizes the inequality for the rigid/plastic bicrystal in Wu and H a v n e r (1995), eqn (4.21) (where, o f course, only dxx = - 1 appears on the right-hand side). Obviously from eqn (43), we require the derivatives o f functions J~, gi in each region. The results obtained from the corresponding differentiation o f eqns (28)-(33) m a y be summarized as follows:
region I: f](a)=f'(~), g](/3)=-dxxcotO+f'(-/3cotO)cot20+G~(y(/3))cscO, Go(y)=ol(-/3cotO,/3)+~l(--/3cotO,/3)cotO=~l(y(/3))+(l(y(/3))cotO,
(44)
y ( / 3 ) = --/3CSC0; region II:
f~(a) = f ' ( o t ) + ¢'12(~),
g'2(/3) = dxx tan 0 + {f'(/3 tan 0) + ¢'12(/3 tan 0)} tan 2 0 = F'o(Z(/3)) sec 0, Fo(z) = 02(/3 tan 0,/3) - ¢2(/3 tan 0,/3) tan 0 = 02(z(/3)) - (2(z(/3)) tan 0,
(45)
z(/3) = / 3 sec 0; region III" g~(/3) = -dxxcotO+f'(~(/3))cot20+ ~(/3) = - / 3 c o t 0 > 0, y(/3) = - / 3 c s c 0 > 0,
G'o(v(/3))cscO+ r1'13(/3),
f~ (et) = -dxx cot 0 csc 2 0 + f'(~(~(et)) cot 4 0 q- {G to(y(fl(ot)) csc 0 + r/13(~(a))} cot 2 0 + F ' 1(zl (or)) cot 0 csc 0,
(46)
F1 (zl) = rls(ot, ~(ot)) -- ~3(ot, ~(ot)) tan 0 = ~3(Zl(a)) -- ~3(zl (a0) tan 0, ~(a) = a c o t 0 -- csc0 < 0 region IV:
zl(o0 = a c s c 0 -- c o t 0 > 0;
f~(a) ----f~(ot) + (34(00, g~(/3) = g~(/3) + 0 ~4(/3);
(47)
On velocity discontinuities in elastoplastic bicrystals in channel die compression
71
f~(o0 = f~(a) + (34(0t) + (45(ff),
region V:
g'5(fl) = -dxx
cot 0 +f'(~) cot s 0 + {g"~4(~)
-'}-(45(~)} tan 2 0 0 '13(-X)} - F'o(Z) see 0,
+ {G'o(y) csc 0 + F ' l (zl) see 0 + =xcot0, Y=Xcsc0, zl=(sin0-x) X = csc 0 - ~ cot 0 = csc 0 -/3,
(48)
sec0,
~ =/3 tan 0,
z =/3 sec 0.
Note that l~anctions F0, Go, and Fl, wherever they appear, are evaluated along the crystal edges y = 0, z = 0, and y = 1, respectively. The points (or lines) of evaluation of functions in eqns (46)-(48) (regions III-V) are shown in Fig. 4. (Eqn (48) applies only for 0 _> 45°; similar equations can be worked out for 0 < 45 °, Fig. 3(b).) In Wu and Havner (1995) it is proved that for the rigid/plastic bicrystal the second normality constraint is critical in region II; whence if satisfied there, the rigid/plastic version of eqn (43) is a strict inequality in every other region (i.e. slip systems 3,4 are active). With the added complexity due to the unknown functions 0i, (~ in eqns (44)-(48), a direct proof that :region II still governs in the elastoplastic crystal is not apparent and may not be possible (although it seems likely that remains the case). However, it will be sufficient for our propose to evaluate the applicability of the condition of zero tangential velocity discontinuities, eqn (37), to the second normality constraint, eqn (43), only in region II, whether or not that is the critical one. For simplicity, the normality constraint is evaluated along y = 0 (/~ tan 0 = a), which encompasses the full range (0, cos 0) of ot in region II for 0 >_ 45 ° (Fig. 3(a)). One obtains (after substituting eqn (45) for f~, g ~)
f'(ot) sec2 0 >
- dxx( tan O--~2 ) - (',2(ot) sec2 0 + F'o(Z(a)) sec O 002 08
3(2 + ~(z(a)) O~ z
(49) / FO
Z
Go~
zl
Fl
1
Y Fig. 4. Points and lines of evaluation of functions needed for the determination of J~, gi in regions III-V (eqns (46)-(48)).
72
K.S. Havner
with
F~(z)=~(z(a))-~(z(~))tanO, 1
.
~=~-~{ell
--7e12+4(~13
z(~)=acsc0,
(50)
--~33)}y=0"
(Equation (49) through the dxx term is equivalent to the critical constraint condition for the rigid/plastic crystal in Wu and Havner (1995), eqn (4.26).) From the mean value theorem, there is a point or* within the range at which f'(ot*) = {f(cos 0) - f ( 0 ) } sec 0. Thu~, from eqn (37), for there to be no tangential velocity discontinuities we must have (upola substituting dxx = - 1) A~"1 see 0 + ~'12(ot*) - r'o(Z* ) cos 0 + \ ~
+ ~ - - ~ (z*) cos 2 0>_
tan0-
cos 20,
A~1 = ~1(0, 0) - ~l (cos 0, - sin 0).
(51) The minimum value of the right-hand side of the inequality is 0.1464 at 0 = 45°(4~ = 9.736°). At 0 = 60°(~b = 24.74°), the value is 0.2562, with a slightly smaller value (0.2357) at the nominal end of range II (0 = 70.53 °, 4~ = 35.26°). In contrast, from the basic definitions of functions 7, ~, and ~p (eqns (25) and (50)), each term on the left side is a direct function of lattice strain-rates k/j with respect to logarithmic compressive strain eL (corresponding to dxx = -1), or integrals of their gradients over distances less than unity. Thus, the left-hand side is of magnitude k, where k now represents a measure of the lattice strain-rates (say I k0.k0. ]1/2, or some comparable norm). From typical values of elastic moduli and hardening parameters in cubic metal crystals, we expect k to be 0(10 -2) at most, if not 0(l 0 -3) (see Appendix). Consequently, it is quite unlikely that eqn (51) can be satisfied since the right-hand side is greater than 0.1. Therefore, we conclude that the assumed condition of no tangential velocity discontinuities, eqn (37), violates the normality constraint in region II (to say nothing of the other regions) and so cannot be true. V. C L O S I N G
REMARKS
We have answered the question posed at the outset, albeit by calling upon standard order-of-magnitude estimates of lattice strain-rates in assessing the final inequality, eqn (51). The inclusion of lattice straining in an f.c.c, bicrystal model in channel die compression
does not eliminate the tangential velocity discontinuities rigorously predicted by rigid/plastic theory. This is because of the strength of those discontinuities. In range I, 0 > 45 °, the minimum possible strength of the largest discontinuity corresponds to (i) only systems 1,2 active in region II and (ii) equal discontinuities along lines ab and bd, Fig. 3(a) (or ab and oc for 0 = 45°). From eqns (4.26) and (5.1)-(5.2) of Wu and Havner (1995), the minimumstrength discontinuity in the rigid/plastic crystal (dxx = - 1 ) can be determined as
minlAual = ~
1-
cot0 cos0.
(52)
On velocity discontinuities in elastoplastic bicrystals in channel die compression
73
The values for 0 - - 4 5 °, 60 °, and 70 ° are 0.1036, 0.1479, and 0.1270, respectively. (However, Wu and Havner (1995), argue that a dominant discontinuity will emanate from the lower left corner a, Fig. 3(a), and continue, magnified by tan 0, along bd. If this is correct, at least near0 = 45 °, then the greatest of these discontinuities (IAu~l along ab for 0 = 45 °) will be nearly twice as large as the value above; see Wu and Havner (1997), Fig. 7.) Only for an order-of-magnitude discontinuity of 10 -2 or less in the rigid/plastic case might there be a :reasonable possibility that incorporation of lattice straining in the bicrystal model could eliminate these discontinuities (whereas from eqn (52) they are greater than 0.1 in range I). The initiation, as a tangential velocity discontinuity, of the physical feature of shear bands in metal polycrystals would appear from the present analysis (and that of Wu and Havner (1995, 1997)) to be a natural consequence of the mathematics and mechanics of crystalline slip, independent of the inclusion or exclusion of lattice straining. This result may have consequences for a physically meaningful finite-element modelling in crystal and polycrystal plasticity. Acknowledgements--Appreciation is expressed to the National Science Foundation, Mechanics and Materials Program, for support of this research through Grant CMS-941376.
REFERENCES
Bay, B., Hansen, N. and Kuhlmann-Wilsdorf, D. (1989) Deformation structures in lightly rolled pure aluminium. Mctter. Sci. Engng All3, 385. Bay, B., Hansen, N., Hughes, D. A. and Kuhlmann-Wilsdorf, D. (1992) Evolution of f.c.c, deformation structures in polyslip. Acta Metall. Mater. 40, 205. Fuh, S. (1989) Applications of a postulate of minimum plastic spin in crystal mechanics. Ph.D. dissertation, North Carolina State University. Havner, K. S. (1992), Finite Plastic Deformation of Crystalline Solids. Cambridge University Press, Cambridge. Havner, K. S. and Chidambarrao, D. (1987) Analysis of a family of unstable lattice orientations in (110) channel die comp:ression. Acta Mech. 69, 243. Havner, K. S. and Wu, S.-C. (1995) An analytical investigation of inhomogeneous straining, subgrain formation, and the initiation of microshear bands in bicrystals. In Proc. I U T A M Symp. on Anisotropy, Inhomogeneity, and Nonlinearity in Solid Mechanics, eds D. F. Parker and A. H. England, pp. 211-216. Kluwer Academic Publishers, Dordrecht. Havner, K. S., Wu, S.-C. and Fuh, S. (1994) On symmetric bicrystals at the yield point in (110) channel die compression. J. Mech. Phys. Solids 42, 361. Hughes, D. A. (1993) Microstructural evolution in a non-cell forming metal: AI-Mg. Acta Metall. Mater. 41, 1421. Hughes, D. A. and Nix, W. D. (1989) Strain hardening and substructural evolution in Ni-Co solid solutions at large strains. Mater. Sci. Engng A122, 153. Wu, S.-C. (1995) Analysis of symmetric bicrystals in (110) channel die compression. Ph.D. dissertation, North Carolina State University. Wu, S.-C. and Havner, K. S. (1995) Exact stress states and velocity fields in bicrystals at the yield point in channel die compression. Z. Angew. Math. Phys. 46, Special Issue, $446. Wu, S.-C. and Havner, K. S. (1997) Analytical and numerical investigation of nonuniform straining and subgrain initiation in bicrystals in channel die compression. Phil. Trans. R. Soc. Lond. A355, 1905.
APPENDIX A
The general hardening law in active systems for infinitesimal lattice straining may be written trNk~ = EHkjf/j,
Nk = sym(b ® n)k
(A1)
74
K.S. Havner
in which ~ denotes the lattice-corotational derivative of Cauchy stress tr, and the Hkj are slip-system hardening moduli (see the review in Havner (1992) for example). From eqn (10), for dxx = - 1 the slip rates are O(1) and so ~ is of order H, an appropriate mean of the Hkj. Consequently, since t~,l = ~,jktekl
(A2)
(again for infinitesimal lattice straining), we find ~ (the norm of the kkt) to be of order H/~, where is a suitable norm of the elastic moduli Yiikt. Typically, for cubic crystals such as aluminium and copper, H is 0(102 MPa) whereas ~" is 0(102 GPa). Thus, k is of likely order 10 -3, or at most 10 -2 if H is on the high side and ~ on the low for a given crystal.
International Journal of Plasticity, Vol. 14, Nos. 1-3, pp. 61 74, 1998 © 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0749-6419/98 $19.00 + 0.00 P I I : S0749-6419(97)00040-5
ON VELOCITY DISCONTINUITIES IN ELASTOPLASTIC BICRYSTALS IN CHANNEL DIE COMPRESSION*
K. S. Havner Department of Civil Engineering, North Carolina State University, Raleigh, NC 27695-7908, U.S.A.
(Received in final revisedform 21 September 1996) Abstract--Possible consequences of including lattice straining in f.c.c, bicrystal models in channel die compression are investigated. It is determined that the tangential velocity discontinuities which emanate from the interface edges in rigid/plastic models are not eliminated by taking crystal elasticity into account. © 1998 Elsevier Science Ltd. All rights reserved
I. I N T R O D U C T I O N
In Wu and Havner (1995, 1997) and Havner and Wu (1995), an analytical rigid/plastic model of symmetric f.c.c, bicrystals in (110) channel die compression is rigorously demonstrated to exhibit tangential velocity discontinuities at the yield point. These velocity discontinuities, interpreted as the initiation of shear bands, essentially are a consequence of the geometric boundary conditions, slip-system kinematics, and f.c.c, crystal yield locus. They extend from the intersections of the interface plane with the rigid channel walls, and are in characteristic directions which depend on crystal orientation. The prediction of discontinuities holds for every constraint direction Y (relative to the lattice) in the (110) family of loading direction X. Moreover, each crystal subdivides into distinct regions among which both material strain-rates and lattice-rotation rates differ, consistent with the initiation of subgrains. The evolution of shear bands and continued formation of subgrains are well-established experimental phenomena in the plastic deformation of polycrystalline metals (see Bay et al., 1989, 1992; Hughes and Nix, 1989; Hughes, 1993), hence the theoretical results wottld appear encouraging. In addition to the presentation in Wu and Havner (1997) of complete analytical solutions for all orientation ranges in (110) compression, finite element results for several orientation,; at the end of the elastoplastic transition (from purely elastic to fully plastic response) a:re compared with the corresponding analytical solutions (also see Wu, 1995). The adopted finite element discretization does not admit tangential velocity discontinuities, but otherwise the numerical results for both stress states and velocity fields are in qualitative agreement with the analytical rigid/plastic solutions; and in some cases the results are quite close. *This paper is dedicated to the memory of James F. Bell, distinguished experimentist, scholar, and friend. 61
62
K.S. Havner
The question remains: is the unequivocal analytical requirement of tangential velocity discontinuities at the initiation of finite multiple-slip in the rigid/plastic bicrystal merely a consequence of that idealization, or does it still hold when lattice straining is taken into account in the crystal kinematics? This paper undertakes to answer that question. II. FORMULATIONAND GOVERNINGEQUATIONS Based upon the comprehensive review in Havner (1992), of both experimental and theoretical studies of finite deformation of single crystals in (110) channel die compression, and the formulation (and numerical solution) of the elastic bicrystal problem in Fuh (1989), Havner et al. (1994), draw the following conclusion. "The nonuniformity of deformation of adequately lubricated, symmetric fcc bicrystals in (110) channel die compression can be expected to take place entirely within Y Z planes .... The loading axis will remain normal to a (110) lattice plane throughout, while the lattice nonuniformly rotates about that axis, causing each crystal to break-up into a crystalline aggregate with a (110)texture." This is because, as stated in Wu and Havner (1997), "geometric incompatibility between differently oriented, freely shearing individual crystals in (110) loading exists only in Y Z planes, hence the nonuniform straining necessary to satisfy compatibility in bicrystals logically should take place entirely within those planes". This reduction to a spatially two-dimensional problem has been confirmed numerically for the elastoplastic transition by Wu (1995), who began with a fully three-dimensional finite element model. (For more detailed arguments and citations of experimental support for these conclusions, see Sections 1 and 2 of Havner et al., 1994 and Section 2 of Wu and Havner, 1997.) As a consequence of the foregoing, the general problem of bicrystals in (110) channel die compression is governed by the following equilibrium and kinematic relations (Wu and Havner, 1997, eqns (2.2)-(2.4)): trxy = trxz = O,
dxy = d~z = O,
< O,
-
dxx=-l,
+
Oy
Ov dyy=~y,
Wy = wz = O,
Wy = Wz = O,
(1)
--~-z = O,--@--y + ~ - z = O,
Ow Ov 2dyz=~-fy+~,
Ow dzz=~z,
Ow 2Wx-oy
(2) Ov Oz
(3)
with velocities u = - x , v(y, z), w(y, z), and lattice-rotation rate wx = ~(y, z) defined with respect to logarithmic compressive strain eL rather than with respect to time. The d u are components of Eulerian strain-rate D; to = (Wx, toy, ~oz)x and w = (wx, Wy, wz) T are axial vectors of lattice and material spin tensors to and W, respectively, relative to the channel axes; and q~is the (local) counterclockwise orientation from the Y-axis of lattice direction [00i] in the right-hand crystal (Fig. 1). The initial boundary conditions for that crystal (a unit cube), taken to apply up to the onset of finite multiple-slipping (and thus throughout the very small strain, elastoplastic transition stage), are: x=O:u=O; y=O,l
x=l:u=--l;
:ayz=O,
z=O:cryz=O,
(4)
v=O; w=O;
z=l
:O'y~=O'~=O.
On velocitydiscontinuities in elastoplastic bicrystals in channel die compression
63
Crystal A
Crystal B
,\
/
[oot-l
y[
Fig. 1. Initial configuration of symmetric bicrystal in (110) channel die compression as viewed from loading direction. (Rectangular grid representsunderlyinglattice.) The 12 crystallographic slip systems in f.c.c, crystals, defined in Table 1, are numbered as follows: al,b2=1,2;
c],c2=3,4;
a3, b 3 = 5 , 6 ;
dl,d2=7,8;
a2, b l = 9 , 1 0 ;
c3, d 3 = 1 1 , 1 2
(the bar above signifying the opposite sense system from that in the table). From eqn (1)1 and the standard tensor transformation between channel and lattice axes, one obtains (Havner et al., 1994, eqns (3.1)1, (3.1)3, and (5.2)1) Orll = 0"22,
0"13 = --0"23,
0"11 -I- 0"12= ~xx ~ O.
(5)
Then, because the resolved shear stresses rk on the twelve slip systems depend only upon trl2, tr13, and cr33 - trH, with rk = 0 in systems 11, 12, the initial yield locus is a decahedron in the three-dimensional lattice-stress space having these components as axes, as shown in Fig. 2. There r0 denotes the critical resolved shear stress ("critical strength") at the beginning of fully plastic response, assumed to be the same in all slip systems. (Possible effects of anisotropy in the small amount of hardening that develops during the elastoplastic transition stage are considered negligible. See the quantitative assessment of this hardening in single crystals in (110) compression in Havner, 1992. For further details regarding the yield locus, see Havner et al., 1994; or Wu and Havner, 1995). The Eulerian strain-rate D and plastic spin f2 = W - 09 may be expressed as D = symEO! ® n)k~k + ~,
(6)
, f2 = skw E(b ® n)k~'k
Table 1. Designation of slip systemsin f.c.c, crystals Plane Direction System
011 al
(111) 10i a2
il0 a3
0ii bl
(lli) 101 b2
il0 b3
01i cl
(lii) 101 c2
ii0 c3
011 dl
(lil) 10i d2
ii0 d3
64
K. S, Havner o'13 ,/'~----~o' Zd,3 (5,8),,~
/._
.
ut
,0.5
./ c~.,) / I '%,. / / __ \ ~
o_.~) _ \ o , -"~I~/0.~,-0.,,.,
* 1 / ~
(~)---~ ~ 0.1~ _ To'
\
(o~o) /
~ ~t2
/
~
.o ' - ~"
E I
Fig. 2. Initial yield locus in lattice stress space, with slip systems indicated in parentheses.
where Yk is the slip-rate in the kth system, bk, nk are unit vectors in the slip and normal directions of that system, and k is the lattice strain-rate. From eqns (1)2 and (3)~ and the tensor transformation which gave eqn (5), we have dll =d22,
d]] --{-d12 -~--dxx -- - 1 .
d13 ~-- -d23,
(7)
In addition, from transformation of the axial vector an = w - w of plastic spin ~2 there follows 1
a] -- ~32 = a2 -- ~13 -~- --~_ax,
a3 ~-- ~'221 = O.
(8)
Therefore, from eqns (6)-(8), there are five constraints on the 10 unknown slip rates. (yll, y12 are zero since the corresponding rk equal zero.) Because the lattice strain-rates satisfy the same basic tensor transformation as a and D, four of the constraints are the same as in the rigid/plastic model (Havner et al., 1994, eqn (4.6)): ~1+Y5=~2+~6,
Y3=Y4,
Y7=~8,
Yl+~9=~2+Y10;
(9)
and the fifth is (from eqn (7)3)
f/l --[- f/9 -~- --(~/r'a/2)(dxx-- Exx).
(lO)
The final equations for the Eulerian strain-rate components on lattice axes thus may be expressed (from eqns (6)1, (9), (10) and Table 1)
On velocity discontinuities in elastoplastic bicrystals in channel die compression
1
65
1
a l l = d22 = -~(dxx - ~xx) --I-~ ( ~ 3
- ~'7) -+- ~11,
2
d33 = - ( d x x
- Exx) - ---~ (J)3 - })7) -+" E33, vo
d13 = - d 2 3 = 1
l
(dxx - ixx) + 1
d12 --'=~ ( d x x -- ~xx) -- - - ~ ( ~
(ll)
.
~--~(2y1 + h + 2y5 + b ) + i13, -- ~'7) + ~12.
These generalize eqn (4.10) of the rigid/plastic model in Havner et al. (1994). Because I ~xx [<< -dxx = 1 at the onset of finite multiple-slip in the compressed bicrystal, at least one of systems 1 (al) and 9 (a2) necessarily is active in the positive sense from eqn (10). Consequently, every stress state in the right-hand crystal must lie either on face A B C D (systems 1,2) or face A D E F (systems 9,10) of the yield locus (Fig. 2), including their edges. For the rigid/plastic model it is concluded in Wu and Havner (1995, 1997) that al3 is non-negative for all 4'0, hence all stress states lie on face ABCD. Wu (1995) found this to be the case as well for the elastoplastic transition, from finite element calculations of a number of bicrystal orientations spanning the range 4'0 between 0 and 90 °. (See Wu and Havner (1997) for final stress states in two of these orientations.) Therefore, only face A B C D is investigated here. Moreover, we restrict consideration to the range of lattice orientations for which the stress state at each material point lies either on edge AB or within face A B C D (excluding the other edges). In the rigid/plastic model this is called range I and is given by 0 < 4'0 < 35.26 o , or channel constraint direction Y between [00i] and [1 i2], first analyzed in Wu and Havner (1995). (There it was assumed that all stress points lay on edge AB only.) With lattice straining included, the end points of this range cannot be so precisely defined analytically. However, they are expected to differ relatively little (no more than a degree or two) from [00i] and [1 i2.], and their exact positions need not concern us in a general analysis of range I. Along edge A B of the yield locus only slip systems 1,2 and 3,4 can be active, and within face A B C D only the first pair (see Fig. 2). Consequently, setting all other slip rates in eqns (9)-(11)) to be zero, we find from the requirement yk > 0, k = 1.... 4, that the normality constraints for range I orientations are -(d12 -- El2) = 2(d13 - ~13),
dll -- ~11 --> d12 -- El2
(12)
the latter of which may be expressed (from eqn (7)) -2d12 > -dxx +
Eli - - ~12,
-dxx = 1
(13)
(the equality satisfied only when systems 3,4 are inactive, as within face ABCD). These relations generalize those for edge A B and face A B C D of the rigid/plastic crystal in Wu and Havner (1995), eqn (8), and complete the set of governing equations. IH. GENERAL ANALYSIS OF VELOCITY FIELDS
We adopt the parameter b= ~
cot4~
(14)
66
K.S. Havner
introduced in Havner and Chidambarrao (1987), from which channel constraint direction Y = [1 lb] and the precise range I of the rigid/plastic model is defined by oo > b > 2. From the first normality constraint, eqn (12 h, and the standard strain-rate transformation between channel and lattice axes, one then can obtain (after some algebra) 2(b - l)2dxx + (b2 - 8b - 2)dyy + 2~/2(b 2 + b - 2)dyz
(15)
= b(b - 4)~v + 2(b2 + 2)(~12 + 2~13)
in which use has been made of the condition of zero plastic volumetric strain-rate (from eqn (6)1) (16)
dxx + dw + dzz = kv = 2klt "l- ,~33
to simplify eqn (15) (kv is the lattice volumetric strain-rate). Therefore, from eqn (3) and the above, the velocity field in the YZ plane is governed by the following pair of partial differential equations (also involving the unknown lattice strain-rates): 0u Al~+~z=C+e*,
A1 ,v/~(b2 + b 1
-
2)
1 0
u = ( v , w ) r,
i
,
c
=
2
-
1
(17)
]
c.={b(b_4)kv+2(b2+2)(k12+2k13) +-b
axx,
}r ,
(18)
(19)
These generalize the pair of hyperbolic equations in Wu and Havner (1995), eqns (4.4)(4.5), and differ from them only by the addition of vector e* in the lattice strain-rates. From Wu and Havner (1995), eqns (3.7)-(3.8), the characteristic directions of the homogeneous form of eqn (17) are rotated counterclockwise from the Y, Z directions by an angle 0 given as tan 0 -
b+2 q~(b - 1)
(20)
Whence, from eqn (14), cot(0 - q~) = ~
(21)
and the first characteristic direction is rotated counterclockwise from [001] by 35.26 ° for all lattice orientations in range I (Wu and Havner, 1995). Let oe,/3 denote these orthogonal characteristic directions, and u~, ue denote the corresponding velocity components: ot = y c o s 0 + z sin0, u,~ = vcos0 + wsin0,
/3 = - y sin 0 + z cos 0, ut~ = - v s i n 0 + wcos0.
(22)
On velocitydiscontinuities in elastoplastic bicrystalsin channel die compression
67
Then, from eqns (20) and (22), eqn (17) can be transformed into the following spectral pair:
bud 2 - - = -dxx + ~(ev + ~12 -+- 2el3), oa (23) 8u~
8¢3
--
1
3
2
6V -- ~"(El2 -F 2e13) .5
from which Oudl~ >> 0u~18/3, and 2. dud = (-dxx) dot + -~(ev + el2 + 2k13)dot
along ot-lines, (24)
1
du~ = ~ {k v - 2(e12 + 2k13)}dfl
along/3-lines,
generalizing the simple relations for the rigid/plastic bicrystal first given in Wu and Havner (1995), eqr~s (4.8). We introduce variables 0(ot,/3), ((or,/3), defined by 01/
2
= ,5"~(~V-'1-~12 "1"-2e13),
8( 1 3-"-~= 3 {ev -- 2(e12 + 2e13)}
(25)
(which gradients may be discontinuous across certain interior boundaries), noting that 8rl/Oot + 8(/8fl = kv. The general solution of eqns (23) or (24) now may be expressed as
u~=-otd~+g(/3)+O(ot,/3),
u~=f(a)+((ot,/3).
(26)
These reduce to the rigid/plastic solution in Wu and Havner (1995), eqn (4.9) (or Wu and Havner (1997), eqn (3.26)), with 0, ( of course zero. We also need the kinematic boundary conditions in the new variables. From eqns (4) and (22) these are: z=0
(w=0)"
cr+/3cot0=0,
ud+u, cot0=0,
y=0 y=l
(v=0)' (v=0)"
ot=13tan0, u~ -- u, tan 0, ot=/3tan0+sec0, u~=u~tan0.
(27)
Consider the subdivision of the right-hand crystal into the five regions shown in Fig. 3 (from Wu and Havner (1995), Fig. 4), and let J~, gi, Oi, (i denote the unknown functions in the ith region. Continuity of normal velocities across interior region boundaries gives:
f2(~)=A(a)+~12(~), g12(a)=gl(~, 0)-g2(~, 0), g3(/3)=gl(~)--I--O13(/3), Ol3(/3)=Ol(a, fl)--o3(a, fl), g4(~)=g2(~)-q-O24(fl), 024(/3)=02(a,/3)--o4(a,~), f4(ot)=f3(a)q-(34(a), ~ 3 4 ( a ) = ~3(~, 0 ) - - ~ 4 ( ~ , O)
(28)
68
K.S. Havner
o
Z
b
Z
d
~
OC
i1
b
"I °\ Y
(a)
(b)
Fig. 3. Grain subdivision into regions I-V for kinematic analysis of symmetric bicrystals
and, for 0 _> 45 ° (case (a), Fig. 3): fs(ot) =f4(ot) q- ~'45(0/), ~'45(0t) = ~'4(ot, c) - ~'5(og,c)
(29)
with a = cos 0 and c = a cot 0. (For 0 < 45 °, case (b) in Fig. 3, eqn (29) is replaced by similar relations in g and 0-functions.) Upon substituting eqns (28) and (29) into eqn (26) and the results into the first boundary condition, eqn (27)1, we have (henceforth defining fl = f a s in Wu and Havner (1995, 1997) gl(fl) = -{fldxx + f ( - / 3 c o t 0) + ( l ( - f l c o t 0, fl)} cot0 - 01(-flcotO, fl).
(30)
From the second boundary condition, eqn (27)2, applied to regions II and V (0 _> 45°),
gi(fl)
=
{fldxx +f,.(/3 tan 0) + (i(fl tan O, /3)} tan 0 - Oi(fl tan O, /3),
i = 2, 5
(31)
with f2 given by eqn (28)1 (the argument o~ of course replaced by fl tan 0) and f5 given in terms off3 through eqns (28)4 and (29). Function f3 is determined from the third boundary condition, eqn (27)3, as f3(ot) = {-otdxx + g3(-X) -k- 03(or, -X)} cot 0 - (3(or, -X),
X = csc 0 - a cot 0
(32)
g3(--X) = {Xdxx - f ( x c o t 0) - ~'l(XCOt 0, - X ) } cot0 - 0 1 ( x c o t 0, - X ) + 013(-X)
(33)
with
from eqns (28)2 and (30). All functions fi, gi now are expressed in terms of the single unknown function f and the lattice strain-rate-dependent functions Oj, ffj, J = 1..... 5, for
On velocity discontinuities in elastoplastic bicrystals in channel die compression
69
0 > 45 °. (Modified equations for f5 and g5 of course can be determined for 0 < 45 °, as in the rigid/plastic results given for that case by W u and Havner (1995), eqns (4.14)-(4.15).) Let us a,;sume there are no tangential velocity discontinuities along any o f the interior boundaries between the regions defined in Fig. 3, case (a). Along fl = 0, between regions I and II, we then would have, from eqns (26)1, (30) and (31), (Au~)12 = 0 = - i f ( 0 ) + ~1(0, 0)} sec 0 csc 0 - 0~2(0) + 01z(a), 012(o0 = rh (or, 0) - 02(or, 0)
(34)
from which r/12(ot) = const = 012(0). Consequently
f(o) + ~1(0, o) = 0
(35)
Similarly, applying our assumed condition o f no tangential velocity discontinuities to the other interior boundaries, we obtain ffl3(ot, 13) = const = ~'13(a, 0),
a = COS0,
f(a) + ~l (a, -- sin 0) = 0, ~24(a, ~) = const = ~24(a, 0),
(36)
r/s4(ot, 0) = const = r/34(a, 0), 045(ot, c) = const = r/45(a, C),
C = acot0.
Thus, from the two conditions on f u n c t i o n f ( e q n s (35) and (36)2), we must have flcos 0) - f(0) = ~l (0, 0) - ~l (cos 0, - sin 0)
(37)
for there to be no tangential velocity discontinuities. (Note that function ~'1(or, fl) is evaluated in this equation at the edges o f the interface plane.) A comparable analysis can be made for c,ase (b), 0 < 45 °. IV. APPLICATION OF THE SECOND NORMALITY CONSTRAINT
F r o m eqn (6)1 we may write
dyy ~- ~](K" bj)(K" nj)f/j --[- ~yy,
K = (1, - 1 , - b ) / v / ~
+ 2.
(38)
In range I,, with only four possible non-zero slip rates, we have
~'l = i/2 = -(~/6/2)(dxx - kxx) > 0,
)~3 = Y4
(39)
(from eqns (9) and (10)). Therefore, from Table 1 and the above:
dyy-
b(b-1) b2 ~ ~ (dxx - kxx)
2(b 2 + b + 2 ) . ~/-6(b 2 + 2) Y3 +
Eyy.
(40)
Thus, since the second normality constraint, eqn (13), is identically the condition y3 > 0, we must have
70
K.S. Havner
dyy <
b ( b - 1) --ff5~ ( d x x - ~ x ) + k y y .
(41)
T o proceed further, we need to express this inequality in terms o f functions f, g, r/, ~"within each region. F r o m eqn (26) the strain-rates on the or,/3 axes in the ith region are
d~ = -dxx +--~,
2d~e = f'i(ot) + g 'i(/3) +-~-~ +--O~t,
(42)
aee =
the prime signifying differentiation of a function with respect to its argument. Consequently, from the coplanar strain-rate transformation between Y, Z and or,/3 axes, eqn (20), k~,~ = ell + Sl2, and eqn (41), we find (after substantial algebra):
~{fti(ot)
-[- g ti(/3)} ~_>dxx
(~. 0¢;~ 1 - ~/~\ /3 + - ~ ) + ~ {El1
-
-
7k12-[- 4 (El3
-
-
833)}
(43)
which generalizes the inequality for the rigid/plastic bicrystal in Wu and H a v n e r (1995), eqn (4.21) (where, o f course, only dxx = - 1 appears on the right-hand side). Obviously from eqn (43), we require the derivatives o f functions J~, gi in each region. The results obtained from the corresponding differentiation o f eqns (28)-(33) m a y be summarized as follows:
region I: f](a)=f'(~), g](/3)=-dxxcotO+f'(-/3cotO)cot20+G~(y(/3))cscO, Go(y)=ol(-/3cotO,/3)+~l(--/3cotO,/3)cotO=~l(y(/3))+(l(y(/3))cotO,
(44)
y ( / 3 ) = --/3CSC0; region II:
f~(a) = f ' ( o t ) + ¢'12(~),
g'2(/3) = dxx tan 0 + {f'(/3 tan 0) + ¢'12(/3 tan 0)} tan 2 0 = F'o(Z(/3)) sec 0, Fo(z) = 02(/3 tan 0,/3) - ¢2(/3 tan 0,/3) tan 0 = 02(z(/3)) - (2(z(/3)) tan 0,
(45)
z(/3) = / 3 sec 0; region III" g~(/3) = -dxxcotO+f'(~(/3))cot20+ ~(/3) = - / 3 c o t 0 > 0, y(/3) = - / 3 c s c 0 > 0,
G'o(v(/3))cscO+ r1'13(/3),
f~ (et) = -dxx cot 0 csc 2 0 + f'(~(~(et)) cot 4 0 q- {G to(y(fl(ot)) csc 0 + r/13(~(a))} cot 2 0 + F ' 1(zl (or)) cot 0 csc 0,
(46)
F1 (zl) = rls(ot, ~(ot)) -- ~3(ot, ~(ot)) tan 0 = ~3(Zl(a)) -- ~3(zl (a0) tan 0, ~(a) = a c o t 0 -- csc0 < 0 region IV:
zl(o0 = a c s c 0 -- c o t 0 > 0;
f~(a) ----f~(ot) + (34(00, g~(/3) = g~(/3) + 0 ~4(/3);
(47)
On velocity discontinuities in elastoplastic bicrystals in channel die compression
71
f~(o0 = f~(a) + (34(0t) + (45(ff),
region V:
g'5(fl) = -dxx
cot 0 +f'(~) cot s 0 + {g"~4(~)
-'}-(45(~)} tan 2 0 0 '13(-X)} - F'o(Z) see 0,
+ {G'o(y) csc 0 + F ' l (zl) see 0 + =xcot0, Y=Xcsc0, zl=(sin0-x) X = csc 0 - ~ cot 0 = csc 0 -/3,
(48)
sec0,
~ =/3 tan 0,
z =/3 sec 0.
Note that l~anctions F0, Go, and Fl, wherever they appear, are evaluated along the crystal edges y = 0, z = 0, and y = 1, respectively. The points (or lines) of evaluation of functions in eqns (46)-(48) (regions III-V) are shown in Fig. 4. (Eqn (48) applies only for 0 _> 45°; similar equations can be worked out for 0 < 45 °, Fig. 3(b).) In Wu and Havner (1995) it is proved that for the rigid/plastic bicrystal the second normality constraint is critical in region II; whence if satisfied there, the rigid/plastic version of eqn (43) is a strict inequality in every other region (i.e. slip systems 3,4 are active). With the added complexity due to the unknown functions 0i, (~ in eqns (44)-(48), a direct proof that :region II still governs in the elastoplastic crystal is not apparent and may not be possible (although it seems likely that remains the case). However, it will be sufficient for our propose to evaluate the applicability of the condition of zero tangential velocity discontinuities, eqn (37), to the second normality constraint, eqn (43), only in region II, whether or not that is the critical one. For simplicity, the normality constraint is evaluated along y = 0 (/~ tan 0 = a), which encompasses the full range (0, cos 0) of ot in region II for 0 >_ 45 ° (Fig. 3(a)). One obtains (after substituting eqn (45) for f~, g ~)
f'(ot) sec2 0 >
- dxx( tan O--~2 ) - (',2(ot) sec2 0 + F'o(Z(a)) sec O 002 08
3(2 + ~(z(a)) O~ z
(49) / FO
Z
Go~
zl
Fl
1
Y Fig. 4. Points and lines of evaluation of functions needed for the determination of J~, gi in regions III-V (eqns (46)-(48)).
72
K.S. Havner
with
F~(z)=~(z(a))-~(z(~))tanO, 1
.
~=~-~{ell
--7e12+4(~13
z(~)=acsc0,
(50)
--~33)}y=0"
(Equation (49) through the dxx term is equivalent to the critical constraint condition for the rigid/plastic crystal in Wu and Havner (1995), eqn (4.26).) From the mean value theorem, there is a point or* within the range at which f'(ot*) = {f(cos 0) - f ( 0 ) } sec 0. Thu~, from eqn (37), for there to be no tangential velocity discontinuities we must have (upola substituting dxx = - 1) A~"1 see 0 + ~'12(ot*) - r'o(Z* ) cos 0 + \ ~
+ ~ - - ~ (z*) cos 2 0>_
tan0-
cos 20,
A~1 = ~1(0, 0) - ~l (cos 0, - sin 0).
(51) The minimum value of the right-hand side of the inequality is 0.1464 at 0 = 45°(4~ = 9.736°). At 0 = 60°(~b = 24.74°), the value is 0.2562, with a slightly smaller value (0.2357) at the nominal end of range II (0 = 70.53 °, 4~ = 35.26°). In contrast, from the basic definitions of functions 7, ~, and ~p (eqns (25) and (50)), each term on the left side is a direct function of lattice strain-rates k/j with respect to logarithmic compressive strain eL (corresponding to dxx = -1), or integrals of their gradients over distances less than unity. Thus, the left-hand side is of magnitude k, where k now represents a measure of the lattice strain-rates (say I k0.k0. ]1/2, or some comparable norm). From typical values of elastic moduli and hardening parameters in cubic metal crystals, we expect k to be 0(10 -2) at most, if not 0(l 0 -3) (see Appendix). Consequently, it is quite unlikely that eqn (51) can be satisfied since the right-hand side is greater than 0.1. Therefore, we conclude that the assumed condition of no tangential velocity discontinuities, eqn (37), violates the normality constraint in region II (to say nothing of the other regions) and so cannot be true. V. C L O S I N G
REMARKS
We have answered the question posed at the outset, albeit by calling upon standard order-of-magnitude estimates of lattice strain-rates in assessing the final inequality, eqn (51). The inclusion of lattice straining in an f.c.c, bicrystal model in channel die compression
does not eliminate the tangential velocity discontinuities rigorously predicted by rigid/plastic theory. This is because of the strength of those discontinuities. In range I, 0 > 45 °, the minimum possible strength of the largest discontinuity corresponds to (i) only systems 1,2 active in region II and (ii) equal discontinuities along lines ab and bd, Fig. 3(a) (or ab and oc for 0 = 45°). From eqns (4.26) and (5.1)-(5.2) of Wu and Havner (1995), the minimumstrength discontinuity in the rigid/plastic crystal (dxx = - 1 ) can be determined as
minlAual = ~
1-
cot0 cos0.
(52)
On velocity discontinuities in elastoplastic bicrystals in channel die compression
73
The values for 0 - - 4 5 °, 60 °, and 70 ° are 0.1036, 0.1479, and 0.1270, respectively. (However, Wu and Havner (1995), argue that a dominant discontinuity will emanate from the lower left corner a, Fig. 3(a), and continue, magnified by tan 0, along bd. If this is correct, at least near0 = 45 °, then the greatest of these discontinuities (IAu~l along ab for 0 = 45 °) will be nearly twice as large as the value above; see Wu and Havner (1997), Fig. 7.) Only for an order-of-magnitude discontinuity of 10 -2 or less in the rigid/plastic case might there be a :reasonable possibility that incorporation of lattice straining in the bicrystal model could eliminate these discontinuities (whereas from eqn (52) they are greater than 0.1 in range I). The initiation, as a tangential velocity discontinuity, of the physical feature of shear bands in metal polycrystals would appear from the present analysis (and that of Wu and Havner (1995, 1997)) to be a natural consequence of the mathematics and mechanics of crystalline slip, independent of the inclusion or exclusion of lattice straining. This result may have consequences for a physically meaningful finite-element modelling in crystal and polycrystal plasticity. Acknowledgements--Appreciation is expressed to the National Science Foundation, Mechanics and Materials Program, for support of this research through Grant CMS-941376.
REFERENCES
Bay, B., Hansen, N. and Kuhlmann-Wilsdorf, D. (1989) Deformation structures in lightly rolled pure aluminium. Mctter. Sci. Engng All3, 385. Bay, B., Hansen, N., Hughes, D. A. and Kuhlmann-Wilsdorf, D. (1992) Evolution of f.c.c, deformation structures in polyslip. Acta Metall. Mater. 40, 205. Fuh, S. (1989) Applications of a postulate of minimum plastic spin in crystal mechanics. Ph.D. dissertation, North Carolina State University. Havner, K. S. (1992), Finite Plastic Deformation of Crystalline Solids. Cambridge University Press, Cambridge. Havner, K. S. and Chidambarrao, D. (1987) Analysis of a family of unstable lattice orientations in (110) channel die comp:ression. Acta Mech. 69, 243. Havner, K. S. and Wu, S.-C. (1995) An analytical investigation of inhomogeneous straining, subgrain formation, and the initiation of microshear bands in bicrystals. In Proc. I U T A M Symp. on Anisotropy, Inhomogeneity, and Nonlinearity in Solid Mechanics, eds D. F. Parker and A. H. England, pp. 211-216. Kluwer Academic Publishers, Dordrecht. Havner, K. S., Wu, S.-C. and Fuh, S. (1994) On symmetric bicrystals at the yield point in (110) channel die compression. J. Mech. Phys. Solids 42, 361. Hughes, D. A. (1993) Microstructural evolution in a non-cell forming metal: AI-Mg. Acta Metall. Mater. 41, 1421. Hughes, D. A. and Nix, W. D. (1989) Strain hardening and substructural evolution in Ni-Co solid solutions at large strains. Mater. Sci. Engng A122, 153. Wu, S.-C. (1995) Analysis of symmetric bicrystals in (110) channel die compression. Ph.D. dissertation, North Carolina State University. Wu, S.-C. and Havner, K. S. (1995) Exact stress states and velocity fields in bicrystals at the yield point in channel die compression. Z. Angew. Math. Phys. 46, Special Issue, $446. Wu, S.-C. and Havner, K. S. (1997) Analytical and numerical investigation of nonuniform straining and subgrain initiation in bicrystals in channel die compression. Phil. Trans. R. Soc. Lond. A355, 1905.
APPENDIX A
The general hardening law in active systems for infinitesimal lattice straining may be written trNk~ = EHkjf/j,
Nk = sym(b ® n)k
(A1)
74
K.S. Havner
in which ~ denotes the lattice-corotational derivative of Cauchy stress tr, and the Hkj are slip-system hardening moduli (see the review in Havner (1992) for example). From eqn (10), for dxx = - 1 the slip rates are O(1) and so ~ is of order H, an appropriate mean of the Hkj. Consequently, since t~,l = ~,jktekl
(A2)
(again for infinitesimal lattice straining), we find ~ (the norm of the kkt) to be of order H/~, where is a suitable norm of the elastic moduli Yiikt. Typically, for cubic crystals such as aluminium and copper, H is 0(102 MPa) whereas ~" is 0(102 GPa). Thus, k is of likely order 10 -3, or at most 10 -2 if H is on the high side and ~ on the low for a given crystal.