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Plane strain compression of silicon-iron single crystals
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Acta metall. mater. Vol. 38, No.4, pp. 581-594, 1990 Printed in Great Britain. All rights reserved
Copyright
© 1990 Pergamon Press pic
PLANE STRAIN COMPRESSION OF SILICON-IRON SINGLE CRYSTALS B. ORLANS-JOLIET, J. H. DRIVER and F. MONTHEILLET Materials Department, Ecole Nationale Superieure des Mines de Saint-Etienne, 158 Cours Fauriel, 42023 Saint-Etienne, Cedex 2, France
(Received 12 January 1989; in revised form 27 September 1989) Abstract-Six different orientations of Fe-3 wt% Si single crystals have been deformed in plane strain compression (using a channel die) up to true strains of 0.5. The finite strain behaviours, i.e. the shape changes, lattice rotations and stress-strain curves, are compared with the predictions of the generalized Taylor analysis of partially constrained crystal deformation. The influence of the relative critical resolved shear stresses on the {11O} and {1I2}<11l) glide systems has been systematically examined. It is shown that for most crystals under multiple slip conditions the shape changes and lattice rotations are consistent with the hypothesis of glide on {l12} being somewhat easier than on {1I0}. Comparison with previous work on b.c.c. crystals undergoing large strains leads to the suggestion that: (i) under conditions of single or colinear slip, glide on {1I0} is easier than on {l12}; (ii) under conditions of intersecting
1. INTRODUCTION
The large strain plastic properties of polycrystalline aggregates depend essentially upon the behaviour of individual grains undergoing similar strains in constrained flow. Under such conditions single crystal plastic behaviour is often treated by using the classical Taylor [1] or Bishop and Hill [2] analyses-or appropriate modifications thereof. These methods are 581
used, for example, in polycrystal simulations of deformation texture development, plastic anisotropy and constitutive relations. It is clearly necessary to verify the models of constrained single crystal plasticity with the real experimental behaviour of crystals undergoing large, partially or completely, imposed strains. "Behaviour" here is taken to mean the operative slip systems, shape changes, lattice reorientations and stress-strain
582
ORLANS-JOLIET et at.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
relations. There have been several such studies of f.c.c. crystals deformed in plane strain compression, e.g. Chin et al. [3], Kocks and Chandra [4], Driver et al. [5]. However, relatively little work has been carried out on b.c.c. systems; the results published so far describe some tests on high symmetry orientations aimed at evaluating the resolved shear stress-shear strain curves. For example Crutchley et al. [6] have reported the stress-strain curves corresponding to 2 orientations, (110)[110] and (110)[001] of Cr, Mo, V and Nb crystals. A more detailed examination of different high symmetry orientation of Mo crystals has also been described by Carpay et al. [7,8]. But, so far there has been no complete finite strain study of arbitrarily oriented b.c.c. crystals deformed under the mixed boundary conditions of the plane strain compression test. The aim of the present paper is to compare the predictions of the generalized Taylor analysis [9], with the complete finite strain behaviour of Fe-Si crystals in plane strain compression. Silicon-iron is an appropriate b.c.c. material since previous studies, albeit mostly in single slip conditions, have established the type of active slip systems and their relative critical resolved shear stresses CRSS [10, 11]. The slip systems are considered to be in order of decreasing importance, {11O} (111), {112}(lII), {123}(lII) and pencil glide. The {112}(111) slip systems also exhibit stress asymmetry; slip in the twinning sense (TS) is generally easier than slip in the anti-twinning sense (ATS). Calculations will be presented here with different values of the CRSS for these systems (including asymmetrical effects) to determine the influence of the active slip systems on the crystal deformation behaviour. The latter is characterized experimentally by a detailed determination of the crystal shape changes, lattice reorientations and stress-strain relations during strains up to 0.5. In the course of this study on several orientations, it became apparent that, while most crystals deform homogeneously (in accordance with the assumption of the theory), a minority deform very heterogeneously-by separation into two or more different orientations. To clarify the presentation, only the behaviour of the homogeneously deforming crystals will be treated here; the heterogeneous deformation of particular (often high symmetry) orientations will be described in a separate paper.
2. CRYSTAL DEFORMATION SIMULATIONS
The general method used to simulate the rigidplastic deformation of single crystals in plane strain compression has been described in detail for example by Driver et al. [5] and Kocks and Chandra [4]. The method is essentially an application of the Renouard and Wintenberger [9] analysis of (rate-insensitive)
crystal deformation by crystallographic slip under mixed boundary conditions. Using the conventional notation, the strain rate tensor £ij is given in terms of the slip rates yk of the kth system by (1 )
mt
where the are generalized Schmid (orientation) factors of the slip systems of normal n and direction b
mt = !(M nj + bj nn
(no summation over k).
(2)
The stress state (fij is distributed on the slip systems as a shear stress 1: k (3)
and the slip systems are activated when the resolved shear stress attains a critical value 1:~, according to the Schmid law (4)
Under mixed boundary conditions the stress state is that which maximises the plastic work rate done by the non-imposed stresses on the imposed strain rate components (indexed !1.f3) (5)
In the case of plane strain compression when the crystal is compressed along the - X3 direction and constrained to extend along X2 , the imposed strain rate components are
The appropriate non-imposed stresses are, therefore
(the stresses on the free X 2 face normal are zero). There is a problem with the shear stress component since, as pointed out by Sue and Havner [12], lubrication of the crystal surfaces prevents tangential forces from being applied to create a shear stress (flJ; the above authors, therefore, assume (f lJ = O. However, as discussed by Fortunier [13], the condition alJ = 0 for an arbitrary crystal orientation leads to a non-zero shear strain ElJ which is incompatible with the boundary conditions of the test. The only reasonable solution, therefore, is that (flJ takes non-zero values in the volume of the sample but varies with distance such that (f13 tends to zero at the surface. Since we are concerned here with the bulk crystal behaviour we shall, therefore, take (f13 as nonzero and calculate its value in the volume from equation (5). The critically stressed systems are obtained from the stress state given by equation [5] and the application of the Schmid law [4] to the resolved shear stresses on the systems [equation (3)]. The slip rates of these systems are then determined by inversion of equation [1]. For the present case of partially imposed alJ
ORLANS-JOLIET et at.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
strains of non-symmetrical orientations there is virtually no 'l ambiguity (only 3 imposed independent strains and, as described below, mixed slip on {IIO} and {112}). When ambiguity did occur it was so small that, for practical purposes, all current methods of resolution (averaging, random choice or second order analysis) gave the same result. On the rare occasions that ambiguity occurred, an averaging procedure was used to determine the yk. Given the yk, the non-imposed strain rate components, namely £12 and £23 for the case of plane strain compression, were determined from equation (I). After a strain £ = In 12 /12 (0), the corresponding finite shear strains £12 and £23 can be described by the angles IX and p. Application of finite strain analysis, e.g. [12], leads to the following relations for the shears after the nth strain increment ~£ tan IX =2£12=2[£12(n tan
-1)+~£12(n -I)]exp(~£)
p= 2£23 =2[£23(n -1)+~£2/n
-1)]exp(2~£).
(6)
The factor 2 in the exponential term of £23 arises from the decreasing thickness of the sample during the test. Finally, the components of the current lattice are given [9] by rotation rate
'ij
'21 = btn~yk,
'32 =
-b~n~yk,
'13 = b~ntyk.
(7)
These values are used to obtain the finite rotation matrix for each strain increment and hence the new crystal orientation.
2.1. Choice of slip systems D.C.C. metal and alloys are known to slip on different planes along the
by Taoka et al. [10], and in shear by Roche [II], it is agreed that silicon-iron slips on {lID} but can also slip on {112} and occasionally on intermediate planes (wavy slip). Slip on {112} in the twinning sense (TS) is easier, by a factor of 5 or 10%, than slip on the same plane in the anti-twinning sense (ATS). However, the exact values of the CRSS on the different systems are not known precisely-particularly in multiple slip conditions. To analyse the experimental results in terms of the above models we have, therefore, made the following hypotheses (Table I) for the slip systems and the CRSS ratios ~ (where ~ = CRSS of {hkl} system/CRSS of {I 10} system, if {IIO} is allowed). Hypotheses 2 to 5 are consistent with the usual assumptions for slip in b.c.c. metals, i.e. mixed {IIO} and {112} slip with slip on {IIO} easier than on {112} or, alternatively, mixed {l1O} + {l12} + {123} slip. Note that the latter (hypothesis 5) corresponds very closely to pencil glide, i.e. slip on any plane about (111). The types of behaviour characterised by hypotheses 2 to 5 are often observed in single slip oriented crystals. Hypotheses 6 and 7 also assume mixed {IIO} and {112} slip but with slip on {l12} easier than on {liD}. Hypothesis 7 represents the conclusion of Carpay et al. [7] after plane strain compression tests on multiple slip oriented Mo crystals. As will be shown below, hypothesis 6 describes the best fit between predictions and experimental results for the present study on Fe-Si single crystals. Finally, hypotheses I and 8 correspond to restricted slip on only {IIO} and {l12} respectively. Calculations were performed for all orientations with all 8 hypotheses but, to simplify the presentation, detailed analyses will only be discussed in terms of the more common assumptions, hypotheses
Table I. Different slip system hypotheses Hypothesis No.
Possible slip planes
e
Remarks
I
110
I
Restricted {IIO} slip
2
110 112TS 1I2ATS
I 1.1 1.22
e values according to
110 112TS 112ATS
1 1.05 1.10
Mixed {l10} + {1I2} slip ~ values according to Taoka el al. [10]
110 112TS 1I2ATS
I I I
Mixed {l10} + {l12} slip
110 112 TS + 112 ATS 123
I 1 1
Mixed {IIO} + {l12} + {123} slip (Pseudo pencil glide)
4
583
Mixed {11O} + {112} slip
Roche [II]
6
110 112 TS 112 ATS
I 0.93 0.96
Mixed {1I0}+{112} slip with easy {l12} glide
7
110 112TS 112ATS
I 0.77 I
Mixed {I1O} + {112} slip with very easy {1I2}TS following Carpay el al. [7]
8
112 TS + 112 ATS
I'
Restricted {112} slip
'In this case where {l1O} slip is not allowed ~ is defined differently so that <,{l12} = 1 and <,{lIO} = 00.
584
ORLANS-JOLIET et af.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
1, 3, 5 and 6 (and particularly 3 and 6) which will be abbreviated as follows: Restricted {11O} slip -+ {l1O} 3 Mixed with easy {11 O} slip -+ ME 110 5
Pseudo pencil glide -+ Pencil
.s
6 Mixed with easy {112} slip -+ ME 112.
0
0
~ ~~
~ ~~
As is well known, the plastic work rate per unit volume W, and hence the Taylor factor M depend upon the choice of slip systems and ~. This means that the critical resolved shear stress-total shear strain curves obtained from the experimental 0'(£) plots also depend upon ~. This parameter determines the current Taylor factor defined as
-e: i
~ ~ r-'O I I
o
0
"'0
c2
o
0
o
0
a- "' ... 00'" ee
o
0
"'0
I I
I
M=~L1:~yk=~(LY·+~PLyb+ ... ). 1: £ £. P e
o
0
'" '"
~~
~~
~~
~N
cS
ct,
0
00 '"
c
0 '" oor-
(8)
where 1:~ is the CRSS on the reference system, usually taken here as the {l1O} <111 > system. From equation (8) one obtains
0
~~
o ~~
1:~f
o
'" a-aI I '"I '"I
'"6 o
W M=-
o 0 '" 00
a-a- ...
I I ~~
(9)
k
o 0 000
... '"
The b.racketed term is an equivalent total shear strain rate f for all the families (ct, P, ... ) of slip systeins [14]. It is used to define the current value of M (({JI ¢J ((J2) which varies during the deformation due to the lattice rotation. The CRSS-total shear strain curves 1:~(f) are then obtained from the experimental 0'33(£22) plots and M via the relations (10)
"'a-a-
~~
o 0 0'" c
~
So u
i
o 0 00 00
,
& ~~ ~ 0 Po
3. EXPERIMENTAL
The silicon iron single crystals were obtained by controlled horizontal solidification of high purity alloys of nominal composition Fe-3 wt% Si. In fact, crystal compositions varied from 2.7 to 3 wt% Si. To eliminate the possible influence of Si, almost all the crystal orientations have been tested at two different compositions: (1) 3% Si and (2) 2.7% Si. The 7 x 5 x 4 mm compression samples were machined from the large single crystals using a water-cooled cutting disc and then annealed 8 h at 1250°C under argon. The samples were then electropolished and the orientations determined by the back-reflexion Laue X-ray method. The compositions and orientations of the different crystals described here are given in Table 2, where the orientations are specified in terms of the approximate Miller indices and the Bunge Euler angles ({Jj ¢J ({J2' Before deformation, a grid of spacing - 1 mm was inscribed with a diamond microhardness indenter on
0
"'''' "'''' I I
' ' 0-
(II)
0
r- '" 00 00 o
.g
and
o
00 00
I I
o
'O'O
0
r- r-
o
"'''' o c 00 00
:=:=
"''''
I I
0';;) 00
\Co
I I
"'''' 00"
'" '" 00
o~o
-0
000
r-'O
00 00
...
'O'"
r- r'O'O
o '" '"
~ 66 ~
.S
0
r-a-
00
I I
I I
~~
'" '" 00 00
I I
...... 00 00
r-
00
0 ",a-
I I
r- 'O I I
-'" 00 -
... -
00 a- a00 I I
ee: e:e: -'" e:e:
e:e:
I:I:lI:l:l
0
~M oct=:' ...... ~~ 00'" 00 00 "'''' r-'O 00 00 'O'O a-a"'''' "'''' 00 0"0 00 rr'O'O "'''' '" '" r- -
~"~,,
-N
o 00
oo o
-N
-N
UU CC
ORLANS-JOLIET et at.:
585
COMPRESSION OF Si-Fe SINGLE CRYSTALS
a compression face and a tranverse face. The grid lines, initially parallel to Xl' X 2 , X) served as reference markers to measure the shear angles IX and P, and to check strain homogeneity. Plane strain compression tests were carried out at room temperature using a lubricated channel and die arrangement in an Instron 25 ton machine, operating at a cross-head speed of 0.2 mm/min. A thin (0.05 mm) Teflon film was placed around the crystal for lubrication. The deformation was imposed in strain increments of about 10% after which the load was removed, the sample dimensions measured and the teflon renewed. The true stress-strain curves were thus obtained using the sample dimensions measured after each increment to give (J = (F/S) and [= -In/)//) (0). After deformations of approximately 10, 30 and 50%, the crystal orientations were determined from {11O} pole figures obtained with a fully automated X-ray texture goniometer [IS]. The grid was also analysed to determine the average shears [12 and (2) from 1/2 tan IX and 1/2 tan p. As was done by Skalli [17], the sheared ends of the crystals were then mechanically polished off to return the crystal to a rectangular shape. This avoids creating inhomogeneous stresses, and strain rates in the sheared zones. Attempts at slip line trace analysis after small strain increments were made by optical microscopy of the specimen surface and in volume. Nomarski interference microscopy was used on the compression and transverse surfaces. Slip lines within the sample were examined by mechanically polishing off the surface layers, electropolishing in a solution of 16% HCI04 , 16% C6 H I6 0 2 and 68% C 2 H4 0 2 and immediate etching in 10% HNO). Slip traces were clearly visible but, as usual in b.c.c. metals, were very wavy and often varied significantly across the sample. The traces of the expected slip systems could only be positively identified when the crystal orientation was such that only 1 or 2 systems were predominant. For the more interesting orientations
where 3 or even 4 slip systems could be active, precise trace analysis turned out to be impracticable. 4. RESULTS
The following aspects of finite strain crystal deformation were examined in detail during the plane strain compression tests on the silicon-iron crystals: -finite shears -crystal lattice reorientations -stress-strain relations -active slip systems. It is important to note that for all orientations tested, variations of the silicon content in the range 2.7-3 wt% had no significant influence on the finite shears and the lattice reorientations. The same orientation with slightly different Si contents gave reproducible shears and lattice rotations, albeit with critical resolved shear stresses some 15% higher for the higher silicon content alloy.
4./. Finite shears
Accurate measurements of the crystal dimensions after each successive deformation showed that the crystals were effectively deformed in plane strain compression, i.e. Ell < 0.01 in all cases. The experimental shears [12 and (2) (=! tan IX and ! tan P) are shown as a function of [22 in Figs I to 6 for each crystal orientation. The error bars on the experimental results correspond to the standard deviation of the set of angular measurements of the reference lines on the grid. When the crystal deformation is perfectly homogeneous, tan IX is measured accurately but the error on tan p increases during deformation due to the decreasing thickness. When there is a tendency to inhomogeneous deformation (for example grain E) the error bars on both tan IX and tan p also indicate the degree of shear strain heterogeneity across the crystal. Figures 1 to 6 also show the theoretical values of [12 and (2) ([22) computed according to the ME 110
;0 Tan~
o ME 110 + MEl12
Tana
0.8
0.8 /+
0.6
0.6
/-+'
/+ /+ /-+'
0.4
0.4
,f'
0.2 0
/ /
0_0-0 ...0'"
/ ~~~oro0.2
0.4
0.2
*
0 E22
#
/
I
dJl
f
df
0.2
0.4
E22
Fig. 1. Experimental and theoretical [120 tan IX) and [2J(! tan P) shears of crystal A2. Theoretical values are computed for the ME 110 and ME 112 hypotheses.
586
ORLANS-JOLIET et at.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
Tan a
Tan~
0.4
0.4
0.2
0.2
0
. .....
........
....." 0.2
-0.2
'~
...
0.4
E22
o MEllO + MEl12
0
·£22
-0.2
~~ ~
-0.4
-0.4
Fig. 2. Experimental and theoretical E120 tan IX) and E23(~ tan fJ) shears of crystal 82. Theoretical values are computed for the ME 110 and ME 112 hypotheses.
Tan a
Tan~
0.6
0.6
./ / ' ~r
0.4
0.4
.....
0.2
0.2
",,:f 0
0.2
0
0.4
o ME 110 + ME 112
e::= £22
E22 -0.2
-0.2
Fig. 3. Experimental and theoretical EI2 0 tan IX) and E23(~ tan fJ) shears of crystal C2. Theoretical values are computed for the ME 110 and ME 112 hypotheses.
(a)
Tana
Tan~
0.6
0.6
0.4
0.4
0.2
0.2
0 -0.2
+
x ....0 .....0
0.2 .....0_
0.4
0 E22
0 -0-0_0_
0-0
-0.2
o ME 110 + ME 112
0\ 0.4 0\ 0
-0.4 -0.6
-0.4 -0.6 Fig. 4(a) Caption on facing page.
£22
\
0 ..... 0
ORLANS-JOLIET et at.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
(b) Tana
Tan~
0.6
0.6
0.4
0.4
587
+ 110 o MEllO x MEll2
• PENCIL 0.2
o
-+""'~,,""*
0.2* - t __
0.4
-.-.-.-·-tb -.-................
r-'"
~.-.-.-. .......
.......
-0.2
.
~
-0.4 -0.6
-0.6 Fig. 4(b)
Fig. 4. (a) Experimental and two theoretical £12(4 tan IX) and £23(4 tan f3) shear of crystal D2. (b) Computed theoretical shears according to the 110, ME 110, Pencil and ME I 12 hypotheses for crystal D2.
Tan a
Tanf3
0.4
0.4
0.2
0.2
0
o MEllO
+ MEI12
0 E22
+ E22
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
Fig. 5. Experimental and theoretical £12(4 tan IX) and £23(4 tan f3) shears of crystal E2. Theoretical values are computed for the ME IlO and ME Il2 hypotheses.
lana 0.4
lanf3 0.4
0.2 ... cp.T41,,~"41
0
0.2
..~.~_~"'ljl"i 0.4
o MEllO
+ ME 112
0.2 0 En
-0.2
-0.2
-0.4
-0.4
Fig. 6. Experimental and theoretical £12 <4 tan IX) and £23 <4 tan f3) shears of crystal F2. Theoretical values are computed for the ME IlO and ME Il2 hypotheses.
and ME 112 hypotheses. For a comparison of some of the different CRSS predictions, Fig. 4(b) also depicts the theoretical shears for the 4 models, namely llO, ME 110, ME 112 and Pencil. Comparison of the computed and experimental AMM 38/4-D
shears (Figs 1-6) shows that for several orientations there is good agreement for both the ME 110 and ME 112 hypotheses; this agreement is found for orientations B (Fig. 2), C (Fig. 3) and F (Fig. 6). In these cases, the models accurately predict the signs
588
ORLANS-JOLIET et at.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
and rates of shear of both (12 and (23' In particular for crystal B, both models accurately predict a monotonically increasing (negative) (12 shear and a (23 shear which goes through a maximum at ( - 0.3 as a consequence of the lattice rotation during the deformation. However, for crystals A, D and E there are some significant discrepancies between the experimental shears and some of the model predictions. Crystal A (Fig. I) exhibits (12 shears that tend to be lower than the computed values and, in particular, much lower than allowed by the ME 112 hypothesis. The (23 shears of this orientation are, however, in good agreement with the calculated values. Crystal D [Fig. 4(a) and (b)] undergoes a positive (23 shear and a practically negligible (12 shear. This behaviour is predicted quite well by the ME 112 hypothesis whereas the ME 110 predicts increasing negative (12 and (23 shears, particularly after strains of - 0.2. Figure 4(b) illustrates the general influence of the CRSS of the different systems on the expected shear strains of this orientation. In this case, only the ME 112 hypothesis appears to predict shears in reasonable agreement with the measured values. Finally, crystal E has a certain tendency to inhomogeneous deformation. It should be noted, however, that here again the ME 112 predictions are in reasonable agreement with the measured shears (slightly negative for (12 and -zero for (23)' while ME 110 predicts a small positive (12 shear, and a large negative (23 shear at ( > OJ. To summarize at this point, the measured and predicted unconstrained deformation components are often in reasonable agreement with both the ME 110 and ME 112 hypotheses. Where there is a significant difference between the two, then it is the ME 112 predictions that tend to be closer to the
.~
!------,il....,.---+------....---t+ X2
X1
Fig. 7. Experimental {1I0} pole figures of crystal 82 at strains of 0, 0.10 and 0.50. The lattice rotation occurs essentially about the XI axis.
experimental results, the single exception to this rule being the case of the (12 shear of crystal A. 4.2. Crystal lattice reorientations
The experimentally measured lattice orientations (CPI CP2) of the 12 crystals are given in Table 2 as a function of the nominal imposed strains. These orientations are determined from the type of {11O} pole figures shown in Fig. 7 for crystal B by determining the average Euler angles corresponding to the intensity peak maxima. The experimental lattice rotations are depicted by the positions of the X 3 and X 2 axes in standard 001-101-111 triangles in Figs 8-13 for crystals A to F. The latter also include the computed
111
o MEllO x
MEl12
111
110
101
010
001
Fig. 8. Experimental orientations of crystal A2 (axes X 3 and X 2 ) at strains of 0, 0.11, 0.30 and 0.50 (indicated by successively larger circles). Theoretical orientations for ME 110 and ME 112 are indicated at strain intervals of 0.05 up to 0.5 strain.
ORLANS-JOLIET et af.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
HI
589
o MEllO )( ME112
111
010
011
100
101 X3(0)
Fig. 9. Experimental orientations of crystal B2 at strains of 0, 0.12, 0.29, 0.47 and the computed ME 110 and ME 112 previsions.
o ME 110
)( ME 112
111
110
010
Fig. 10. Experimental orientations of crystal C2 at strains of 0, 0.13, 0.29,0.51 and the computed ME 110 and ME 112 previsions. orientations at 0.05 strain intervals according to the ME 110 and ME 112 hypotheses. From the set of 6 types of initial crystal orientation, two orientations, Band F, were such that both ME 110 and ME 112 (plus the 110 and Pencil models) predicted virtually identical rotations. Crystal B undergoes a very large rotation of about 30° about XI towards ~(211)[OII], (Fig. 9) and both the rotation path and rates are predicted perfectly by the computations. Crystal F (Fig. 13) rotates towards ~ (010)(201) and although the %2 orientation is predicted very well, the computations tend to make X 3 deviate somewhat from (010) at higher strains. The remaining crystal orientations exhibit different rotation paths according to the CRSS hypotheses.
Crystal A2 (Fig. 8) rotates moderately towards ~(458)[583]. This is predicted accurately by ME 110
but ME 112 makes the X 3 axis shift somewhat towards [III]. Note this descrepancy is consistent with the problem of the £12 shear of the same grain calculated with ME 112 (an error in the X 3 orientation leads to errors in the £12 shear about the X 3 axis). Crystal C2 (Fig. 10) undergoes a very large rotation from ~(101)[5'5] towards (447)[IlO]. The rotation towards ~(101)[13l] predicted according to ME 110 is in complete disagreement with the real behaviour. The rotation predicted according to ME 112 is certainly better, (i.e. X 3 to ~(212) and X 2 to [120]) but nevertheless is significantly different from the experiments and, in particular, is less than
ORLANS-JOLIET et at.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
590
(0)
111
o ME 110 x
ME 112
100
110 + 110
111
(b)
110
110
o MEllO x ME 112 • PENCIL
100
100
Fig. II. (a) Experimental orientations of crystal 02 at strains of 0, 0.14, 0.29, 0.47 and the computed ME 110 and ME 112 previsions. (b) Theoretical orientations of crystal 02 at strain intervals of 0.05, computed according to the 110, ME 110, Pencil and ME 112 hypotheses. the true rotation. In fact, the rotations for all 8 ~ values tabulated in Table I give rather poor prediction; the ~ values ~ I tend to favour the ME 110 path of Fig. 10, whereas ~ values ~ I, which facilitate {112} slip (including pure {112} slip), favour the ME 112 path in somewhat better agreement with the measurements. Figure II(a) shows that the ME 110 rotations are also in error for crystal D. The latter rotates towards ~(l1l)[IIO] as predicted by the ME 112 hypothesis, whereas ME 110 predicts a rotation towards (l11)[SS3]. The different rotation paths of the 4 models (ME 110, ME 112, 110 and Pencil) are illustrated in Fig. II(b) for this crystal. All predict that the compression plane rotates towards (111) but the rotation of the elongation direction X2 is very sensitive to the ~ value; ~ values ~ I, including Pencil,
ultimately predict a rotation towards the centre of the triangle, only ME 112 gives a rotation towards [IIO]. The behaviour of crystal E (Fig. 12) is somewhat similar, i.e. an experimental rotation of moderate amplitude towards ~(575)[IOI] which is predicted quite well by ME 112. Here again, if {1l0} slip is favoured, then the XJ rotation is reasonable, but X2 is expected to rotate towards the centre of the triangle. In summary, it is clear that measurements of the lattice rotations of certain crystals undergoing large imposed strains, constitute a sensitive technique for evaluating the relative ease of slip on different slip systems. For a limited number of crystals (here Band F), the rotations predicted by all reasonable hypotheses for the CRSS ratios ~ are in good agreement with the experimental measurements. For the others, the
ORLANS-JOLIET et af.:
591
COMPRESSION OF Si-Fe SINGLE CRYSTALS
111
o MEno x MEl12
111
fo 1
100
010
110
Fig. 12. Experimental orientations of crystal E2 at strains of 0,0.11,0.29, 0.47 and the computed ME 110 and ME 112 previsions.
rotation paths differ significantly according to ~. Crystals C, D and E change orientation according to the ME 112 model which favours {I12} slip. Only crystal A tends to follow a path which agrees with the conventional ME 110 hypothesis. 4.3. Stress-strain curves
The true stress-true strain, U33 (£22) curves of the silicon-iron crystals are depicted in Fig. 14(a) for the 3% Si alloy and (b) for the 2.7% Si alloy. These curves illustrate the strong orientation dependence of the flow stresses, a dependence which is in fact identical for both silicon contents. Thus, at a strain of 0.25, the flow stresses increase in the order B, F, C, A, E, D, which corresponds quite closely to the Taylor factors of the different orientations, respectively 2.31, 2.34,
2.44, 2.90, 3J2, 3.69 (hypothesis ME 110 at £ = 0.25). As described in Section 2.1, critical resolved shear stress L ~ 10 vs total equivalent shear strain f curves were derived from the u(E) curves using the appropriate Taylor factors. The latter were computed for the experimental orientations at strains of 0, 0.1, OJ and 0.5 and by interpolation at intermediate strains. The L~IO(f) curves are plotted for the Fe-2.7 wt% Si crystals in Fig. 15 (a) and (b) according to the ME 110 and ME 112 ~ hypotheses, respectively. The first point is that the Le(f) curves are now regrouped within a stress range of - ± 12%. This scatter is somewhat higher than the experimental reproducibility so that some orientation dependence of the Le(f) curves remains, but is markedly reduced by the Taylor factors. The second point is that the
o ME 110 x ME 112
111
To 1
100
110 Fig. 13. Experimental orientations of crystal F2 at strains of 0, 0.12, 0.32, 0.49 and the computed ME 110 and ME 112 previsions.
ORLANS-JOLIET et af.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
592 (0)
0"33
(b)
(Mfa)
0"33
1300
1300
1200
1200
1100
1100
1000
1000
900
900
800
800
700
700
600
600
500
500
(Mfa)
400 300
300
o
0.1
0.2
0.3
0.4
0.5
£
o
22
0.1
0.2
0.3
0.4
Fig. 14. True stress-strain curves of silicon-iron crystals; (a) Fe-3% Si, (b) Fe-2.7% Si.
ME 110 and ME 112 hypotheses produce very similar 'CeO") curves. These curves are consistent with previous 'Ce(Y) curves obtained on Fe-3% Si under conditions of single slip. Thus the shear experiments of Roche [11] on the {110} system of Fe-2.8% Si indicates a yield shear stress of - 150 MPa and a flow stress at Y = 0.2 of - 200 MPa. Zarubova and Sestak [16] found a similar yield shear stress in tension - 150 MPa, although with a slightly lower work hardening rate (at y - 0.4 'Ce - 180 MPa).
5. DISCUSSION AND CONCLUSIONS
The results clearly show that the overall finite strain behaviour of b.c.c. crystals undergoing partially imposed plane strain compression can be described in the framework of the generalized Taylor model of crystal plasticity. Of the six different crystal orientations described here, the lattice rotations and finite shears of at least five crystals can be quite accurately described by this model (incorporating the
(a) TC
(b)
(Mfa)
TC
(Mfa)
(110)
(110)
ME 110
ME112 400
400 ¢
02
300
200
100
100
o
r
o'---,---,----,r--,---,----,r--,---,----,r--,--
OJ Q8 1.2 16 2D 1.2 1.6 2.0 Fig. 15. Critical shear stresses on the {110}
r
0.8
r
ORLANS-JOLIET et al.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
appropriate boundary conditions of the channel die test). However, for certain orientations, relatively small variations of the critical strengths of the {l12} and {l1O}
e
(i) crystals Band F slip on the same systems according to both ME 110 and ME 112, giving rise to virtually identical rotations and non-imposed shears (in good agreement with experiment); (ii) crystal A slips essentially on a single {112} system according to ME 112, whereas the introduction of a second {IlO} system (ME 110) clearly gives better agreement with the experimental rotations and shears; (iii) in crystal C, the operation of two {112}
Crystal
the slip on two {112}
test, the evalues of the {Il2} systems in Fe-Si must be lower than 1. However, several tensile and shear experiments on single slip oriented Fe-Si crystals have found the converse, > 1. This can only be rationalized if the flow stresses of the {112} systems are lower than those of the {IIO} under conditions of intersecting slip characteristic of most multiple slip tests. There is some justification for this from the present results. The only crystal which seems to prefer easier {I 10} slip to {112} is A; in this case, the predominant systems predicted by ME 110 are colinear, i.e. without any significant intersecting slip directions. Crystals C, D and E which prefer easier {112} slip, all deform with intersecting slip directions. These observations on Fe-3% Si can also be extended to other b.c.c. metals. Similar orientations of Nb crystals have been recently tested in plane strain compression and identical results obtained [17], i.e. relatively easy {112} slip orientations characterized by intersecting slip directions. It should also be recalled that, according to Carpay et at. [7], the plane
e
£ = 0.25 according to the ME 110 and ME 112 hypotheses
Slip systems (in order of importance) ME 110 ME 112
A
(011)[11 I) (121) [II I) TS minor (101) [TTI]b
minor (211) [TTl) ATS
B
(110)[11 I) minor systemsb
(110) [III] minor systemsb
C
(110) [III] (011)[111] minor(121) [IlT] TS b
(l21)[III)TS
(011)[111) (l21)[III]TS (211)[111] ATS
D
(101) [lIT) (Ol1)[TTl) (211)[lIT] TS
(112) [TTl] TS (II2)[TIT) ATS (101) [ITT] or (Oil) [III]
E
(211)[lIT)TS (110) [lIT) (011)[11 I]
(1IO)[lIT] (121) [TIT] TS (011) [TTl]
F
(121)[lll]ATS (110)[1 II]
(121)[111] ATS (110)[1 II]
'Predominant taken to mean yk ~ 0.4 £. bMinor taken to mean 0.4 ~ yk ~ 0.2 £.
593
594
ORLANS-JOLIET et al.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
strain compression results on high symmetry Mo [7], Cr and V[6J crystals are all consistent with ~ {II2} TS values of - 0.77. The orientations used by these authors all involve intersecting slip directions. It is therefore suggested that as a general rule in b.c.c. metals undergoing large strains:
Acknowledgements-The authors wish to thank Miss H.
Bruyas and Mr R. Fillit for their indispensable help with the X-ray pole figure analyses.
(I) under conditions of single or colinear slip, glide on {1I0} is easier than on {1I2}; (2) under conditions where the predominantslip directions intersect, then glide on {112} is easier than on {1I0}. The present results have enabled us to specify the values appropriate to silicon iron under multiple slip conditions. We have carried out computations with a wide variety of ~ {l12} values. The choice of ~ makes a significant difference to the reorientations and finite shears of certain crystals, in particular A, C, D and E, but has less influence on the resolved shear stress-shear strain curves. As previously mentioned, computer simulations with ~ values greater than 1 did not predict the rotation of crystals C, D and E which deform in multiple slip. Furthermore, it was found that to obtain a reasonable prediction for the rotation of crystal E, it was necessary to use ~TS values of less than 0.94 and a difference between ~TS and ~ATS of about 3%. But, if ~TS values lower than 0.92 were used, then the shears of crystals D and E were the same as those predicted by the restricted 112 slip hypothesis and were in disagreement with the experimental results. We conclude therefore that best agreement can be obtained with the following values: ~
~IlO=I
~ 112 TS - 0.93 ~1I2ATS
-1.03 012 TS.
REFERENCES 1. G. I. Taylor, J. Inst. Metals 62, 307 (1938). 2. J. F. W. Bishop and R. Hill, P/lil Mag. 42, 414, 1298 (1951). 3. G. Y. Chin, E. A. Nesbitt and J. A. Williams, Acta metall. 14, 467 (1966). 4. U. F. Kocks and H. Chandra, Acta metall. 30, 695 (1982). 5. J. H. Driver, A. Skalli and M. Wintenberger, Phil. Mag. 49, 505 (1984). 6. D. E. Crutchley, C. N. Reid and T. H. Webster, Proc. 2nd Int. Con! on Strength of Metals and Alloys, Vol. I, p. 127 (1970). 7. F. M. A. Carpay, G. Y. Chin, S. Mahajan and J. J. Rubin, Acta metall. 23, 1473 (1975). 8. F. M. A. Carpay, S. Mahajan, G. Y. Chin and J. J. Rubin, Acta metall. 25, 149 (1977). 9. M. Renouard and M. Wintenberger, c.r. Acad. Sci., Paris 2838, 237 (1976). 10. T. Taoka, S. Takeuchi and J. Furubayashi, Phys. Soc. Japan 9, 701 (1964). 11. C. Roche, Doctoral thesis, Universite Paris VI (1976). 12. P. L. Sue and K. S. Havner, J. Mech. Phys. Solids 32, 417 (1984). 13. R. Fortunier, Doctoral thesis, INPG-ENSMSE (1987). 14. R. Fortunier, B. Orlans-Joliet, F. Montheillet and 1. H. Driver, Proc. 8th Rise Int. Symp., Constitutive Relations and their Physical Basis (edited by S. I. Andersen et al.), p. 317 (1987). 15. R. Y. FiIlit, H. Bruyas and F. Patay, European Patent: DOSOPHATEX No. 85-400 95405. 16. N. zarubova and B. Sestak, Physica status solidi 30 A, 479 (1975). 17. B. Orlans-Joliet, Doctoral thesis, ENSMP-ENSMSE (1989).
Acta metall. mater. Vol. 38, No.4, pp. 581-594, 1990 Printed in Great Britain. All rights reserved
Copyright
© 1990 Pergamon Press pic
PLANE STRAIN COMPRESSION OF SILICON-IRON SINGLE CRYSTALS B. ORLANS-JOLIET, J. H. DRIVER and F. MONTHEILLET Materials Department, Ecole Nationale Superieure des Mines de Saint-Etienne, 158 Cours Fauriel, 42023 Saint-Etienne, Cedex 2, France
(Received 12 January 1989; in revised form 27 September 1989) Abstract-Six different orientations of Fe-3 wt% Si single crystals have been deformed in plane strain compression (using a channel die) up to true strains of 0.5. The finite strain behaviours, i.e. the shape changes, lattice rotations and stress-strain curves, are compared with the predictions of the generalized Taylor analysis of partially constrained crystal deformation. The influence of the relative critical resolved shear stresses on the {11O} and {1I2}<11l) glide systems has been systematically examined. It is shown that for most crystals under multiple slip conditions the shape changes and lattice rotations are consistent with the hypothesis of glide on {l12} being somewhat easier than on {1I0}. Comparison with previous work on b.c.c. crystals undergoing large strains leads to the suggestion that: (i) under conditions of single or colinear slip, glide on {1I0} is easier than on {l12}; (ii) under conditions of intersecting
1. INTRODUCTION
The large strain plastic properties of polycrystalline aggregates depend essentially upon the behaviour of individual grains undergoing similar strains in constrained flow. Under such conditions single crystal plastic behaviour is often treated by using the classical Taylor [1] or Bishop and Hill [2] analyses-or appropriate modifications thereof. These methods are 581
used, for example, in polycrystal simulations of deformation texture development, plastic anisotropy and constitutive relations. It is clearly necessary to verify the models of constrained single crystal plasticity with the real experimental behaviour of crystals undergoing large, partially or completely, imposed strains. "Behaviour" here is taken to mean the operative slip systems, shape changes, lattice reorientations and stress-strain
582
ORLANS-JOLIET et at.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
relations. There have been several such studies of f.c.c. crystals deformed in plane strain compression, e.g. Chin et al. [3], Kocks and Chandra [4], Driver et al. [5]. However, relatively little work has been carried out on b.c.c. systems; the results published so far describe some tests on high symmetry orientations aimed at evaluating the resolved shear stress-shear strain curves. For example Crutchley et al. [6] have reported the stress-strain curves corresponding to 2 orientations, (110)[110] and (110)[001] of Cr, Mo, V and Nb crystals. A more detailed examination of different high symmetry orientation of Mo crystals has also been described by Carpay et al. [7,8]. But, so far there has been no complete finite strain study of arbitrarily oriented b.c.c. crystals deformed under the mixed boundary conditions of the plane strain compression test. The aim of the present paper is to compare the predictions of the generalized Taylor analysis [9], with the complete finite strain behaviour of Fe-Si crystals in plane strain compression. Silicon-iron is an appropriate b.c.c. material since previous studies, albeit mostly in single slip conditions, have established the type of active slip systems and their relative critical resolved shear stresses CRSS [10, 11]. The slip systems are considered to be in order of decreasing importance, {11O} (111), {112}(lII), {123}(lII) and pencil glide. The {112}(111) slip systems also exhibit stress asymmetry; slip in the twinning sense (TS) is generally easier than slip in the anti-twinning sense (ATS). Calculations will be presented here with different values of the CRSS for these systems (including asymmetrical effects) to determine the influence of the active slip systems on the crystal deformation behaviour. The latter is characterized experimentally by a detailed determination of the crystal shape changes, lattice reorientations and stress-strain relations during strains up to 0.5. In the course of this study on several orientations, it became apparent that, while most crystals deform homogeneously (in accordance with the assumption of the theory), a minority deform very heterogeneously-by separation into two or more different orientations. To clarify the presentation, only the behaviour of the homogeneously deforming crystals will be treated here; the heterogeneous deformation of particular (often high symmetry) orientations will be described in a separate paper.
2. CRYSTAL DEFORMATION SIMULATIONS
The general method used to simulate the rigidplastic deformation of single crystals in plane strain compression has been described in detail for example by Driver et al. [5] and Kocks and Chandra [4]. The method is essentially an application of the Renouard and Wintenberger [9] analysis of (rate-insensitive)
crystal deformation by crystallographic slip under mixed boundary conditions. Using the conventional notation, the strain rate tensor £ij is given in terms of the slip rates yk of the kth system by (1 )
mt
where the are generalized Schmid (orientation) factors of the slip systems of normal n and direction b
mt = !(M nj + bj nn
(no summation over k).
(2)
The stress state (fij is distributed on the slip systems as a shear stress 1: k (3)
and the slip systems are activated when the resolved shear stress attains a critical value 1:~, according to the Schmid law (4)
Under mixed boundary conditions the stress state is that which maximises the plastic work rate done by the non-imposed stresses on the imposed strain rate components (indexed !1.f3) (5)
In the case of plane strain compression when the crystal is compressed along the - X3 direction and constrained to extend along X2 , the imposed strain rate components are
The appropriate non-imposed stresses are, therefore
(the stresses on the free X 2 face normal are zero). There is a problem with the shear stress component since, as pointed out by Sue and Havner [12], lubrication of the crystal surfaces prevents tangential forces from being applied to create a shear stress (flJ; the above authors, therefore, assume (f lJ = O. However, as discussed by Fortunier [13], the condition alJ = 0 for an arbitrary crystal orientation leads to a non-zero shear strain ElJ which is incompatible with the boundary conditions of the test. The only reasonable solution, therefore, is that (flJ takes non-zero values in the volume of the sample but varies with distance such that (f13 tends to zero at the surface. Since we are concerned here with the bulk crystal behaviour we shall, therefore, take (f13 as nonzero and calculate its value in the volume from equation (5). The critically stressed systems are obtained from the stress state given by equation [5] and the application of the Schmid law [4] to the resolved shear stresses on the systems [equation (3)]. The slip rates of these systems are then determined by inversion of equation [1]. For the present case of partially imposed alJ
ORLANS-JOLIET et at.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
strains of non-symmetrical orientations there is virtually no 'l ambiguity (only 3 imposed independent strains and, as described below, mixed slip on {IIO} and {112}). When ambiguity did occur it was so small that, for practical purposes, all current methods of resolution (averaging, random choice or second order analysis) gave the same result. On the rare occasions that ambiguity occurred, an averaging procedure was used to determine the yk. Given the yk, the non-imposed strain rate components, namely £12 and £23 for the case of plane strain compression, were determined from equation (I). After a strain £ = In 12 /12 (0), the corresponding finite shear strains £12 and £23 can be described by the angles IX and p. Application of finite strain analysis, e.g. [12], leads to the following relations for the shears after the nth strain increment ~£ tan IX =2£12=2[£12(n tan
-1)+~£12(n -I)]exp(~£)
p= 2£23 =2[£23(n -1)+~£2/n
-1)]exp(2~£).
(6)
The factor 2 in the exponential term of £23 arises from the decreasing thickness of the sample during the test. Finally, the components of the current lattice are given [9] by rotation rate
'ij
'21 = btn~yk,
'32 =
-b~n~yk,
'13 = b~ntyk.
(7)
These values are used to obtain the finite rotation matrix for each strain increment and hence the new crystal orientation.
2.1. Choice of slip systems D.C.C. metal and alloys are known to slip on different planes along the
by Taoka et al. [10], and in shear by Roche [II], it is agreed that silicon-iron slips on {lID} but can also slip on {112} and occasionally on intermediate planes (wavy slip). Slip on {112} in the twinning sense (TS) is easier, by a factor of 5 or 10%, than slip on the same plane in the anti-twinning sense (ATS). However, the exact values of the CRSS on the different systems are not known precisely-particularly in multiple slip conditions. To analyse the experimental results in terms of the above models we have, therefore, made the following hypotheses (Table I) for the slip systems and the CRSS ratios ~ (where ~ = CRSS of {hkl} system/CRSS of {I 10} system, if {IIO} is allowed). Hypotheses 2 to 5 are consistent with the usual assumptions for slip in b.c.c. metals, i.e. mixed {IIO} and {112} slip with slip on {IIO} easier than on {112} or, alternatively, mixed {l1O} + {l12} + {123} slip. Note that the latter (hypothesis 5) corresponds very closely to pencil glide, i.e. slip on any plane about (111). The types of behaviour characterised by hypotheses 2 to 5 are often observed in single slip oriented crystals. Hypotheses 6 and 7 also assume mixed {IIO} and {112} slip but with slip on {l12} easier than on {liD}. Hypothesis 7 represents the conclusion of Carpay et al. [7] after plane strain compression tests on multiple slip oriented Mo crystals. As will be shown below, hypothesis 6 describes the best fit between predictions and experimental results for the present study on Fe-Si single crystals. Finally, hypotheses I and 8 correspond to restricted slip on only {IIO} and {l12} respectively. Calculations were performed for all orientations with all 8 hypotheses but, to simplify the presentation, detailed analyses will only be discussed in terms of the more common assumptions, hypotheses
Table I. Different slip system hypotheses Hypothesis No.
Possible slip planes
e
Remarks
I
110
I
Restricted {IIO} slip
2
110 112TS 1I2ATS
I 1.1 1.22
e values according to
110 112TS 112ATS
1 1.05 1.10
Mixed {l10} + {1I2} slip ~ values according to Taoka el al. [10]
110 112TS 1I2ATS
I I I
Mixed {l10} + {l12} slip
110 112 TS + 112 ATS 123
I 1 1
Mixed {IIO} + {l12} + {123} slip (Pseudo pencil glide)
4
583
Mixed {11O} + {112} slip
Roche [II]
6
110 112 TS 112 ATS
I 0.93 0.96
Mixed {1I0}+{112} slip with easy {l12} glide
7
110 112TS 112ATS
I 0.77 I
Mixed {I1O} + {112} slip with very easy {1I2}TS following Carpay el al. [7]
8
112 TS + 112 ATS
I'
Restricted {112} slip
'In this case where {l1O} slip is not allowed ~ is defined differently so that <,{l12} = 1 and <,{lIO} = 00.
584
ORLANS-JOLIET et af.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
1, 3, 5 and 6 (and particularly 3 and 6) which will be abbreviated as follows: Restricted {11O} slip -+ {l1O} 3 Mixed with easy {11 O} slip -+ ME 110 5
Pseudo pencil glide -+ Pencil
.s
6 Mixed with easy {112} slip -+ ME 112.
0
0
~ ~~
~ ~~
As is well known, the plastic work rate per unit volume W, and hence the Taylor factor M depend upon the choice of slip systems and ~. This means that the critical resolved shear stress-total shear strain curves obtained from the experimental 0'(£) plots also depend upon ~. This parameter determines the current Taylor factor defined as
-e: i
~ ~ r-'O I I
o
0
"'0
c2
o
0
o
0
a- "' ... 00'" ee
o
0
"'0
I I
I
M=~L1:~yk=~(LY·+~PLyb+ ... ). 1: £ £. P e
o
0
'" '"
~~
~~
~~
~N
cS
ct,
0
00 '"
c
0 '" oor-
(8)
where 1:~ is the CRSS on the reference system, usually taken here as the {l1O} <111 > system. From equation (8) one obtains
0
~~
o ~~
1:~f
o
'" a-aI I '"I '"I
'"6 o
W M=-
o 0 '" 00
a-a- ...
I I ~~
(9)
k
o 0 000
... '"
The b.racketed term is an equivalent total shear strain rate f for all the families (ct, P, ... ) of slip systeins [14]. It is used to define the current value of M (({JI ¢J ((J2) which varies during the deformation due to the lattice rotation. The CRSS-total shear strain curves 1:~(f) are then obtained from the experimental 0'33(£22) plots and M via the relations (10)
"'a-a-
~~
o 0 0'" c
~
So u
i
o 0 00 00
,
& ~~ ~ 0 Po
3. EXPERIMENTAL
The silicon iron single crystals were obtained by controlled horizontal solidification of high purity alloys of nominal composition Fe-3 wt% Si. In fact, crystal compositions varied from 2.7 to 3 wt% Si. To eliminate the possible influence of Si, almost all the crystal orientations have been tested at two different compositions: (1) 3% Si and (2) 2.7% Si. The 7 x 5 x 4 mm compression samples were machined from the large single crystals using a water-cooled cutting disc and then annealed 8 h at 1250°C under argon. The samples were then electropolished and the orientations determined by the back-reflexion Laue X-ray method. The compositions and orientations of the different crystals described here are given in Table 2, where the orientations are specified in terms of the approximate Miller indices and the Bunge Euler angles ({Jj ¢J ({J2' Before deformation, a grid of spacing - 1 mm was inscribed with a diamond microhardness indenter on
0
"'''' "'''' I I
' ' 0-
(II)
0
r- '" 00 00 o
.g
and
o
00 00
I I
o
'O'O
0
r- r-
o
"'''' o c 00 00
:=:=
"''''
I I
0';;) 00
\Co
I I
"'''' 00"
'" '" 00
o~o
-0
000
r-'O
00 00
...
'O'"
r- r'O'O
o '" '"
~ 66 ~
.S
0
r-a-
00
I I
I I
~~
'" '" 00 00
I I
...... 00 00
r-
00
0 ",a-
I I
r- 'O I I
-'" 00 -
... -
00 a- a00 I I
ee: e:e: -'" e:e:
e:e:
I:I:lI:l:l
0
~M oct=:' ...... ~~ 00'" 00 00 "'''' r-'O 00 00 'O'O a-a"'''' "'''' 00 0"0 00 rr'O'O "'''' '" '" r- -
~"~,,
-N
o 00
oo o
-N
-N
UU CC
ORLANS-JOLIET et at.:
585
COMPRESSION OF Si-Fe SINGLE CRYSTALS
a compression face and a tranverse face. The grid lines, initially parallel to Xl' X 2 , X) served as reference markers to measure the shear angles IX and P, and to check strain homogeneity. Plane strain compression tests were carried out at room temperature using a lubricated channel and die arrangement in an Instron 25 ton machine, operating at a cross-head speed of 0.2 mm/min. A thin (0.05 mm) Teflon film was placed around the crystal for lubrication. The deformation was imposed in strain increments of about 10% after which the load was removed, the sample dimensions measured and the teflon renewed. The true stress-strain curves were thus obtained using the sample dimensions measured after each increment to give (J = (F/S) and [= -In/)//) (0). After deformations of approximately 10, 30 and 50%, the crystal orientations were determined from {11O} pole figures obtained with a fully automated X-ray texture goniometer [IS]. The grid was also analysed to determine the average shears [12 and (2) from 1/2 tan IX and 1/2 tan p. As was done by Skalli [17], the sheared ends of the crystals were then mechanically polished off to return the crystal to a rectangular shape. This avoids creating inhomogeneous stresses, and strain rates in the sheared zones. Attempts at slip line trace analysis after small strain increments were made by optical microscopy of the specimen surface and in volume. Nomarski interference microscopy was used on the compression and transverse surfaces. Slip lines within the sample were examined by mechanically polishing off the surface layers, electropolishing in a solution of 16% HCI04 , 16% C6 H I6 0 2 and 68% C 2 H4 0 2 and immediate etching in 10% HNO). Slip traces were clearly visible but, as usual in b.c.c. metals, were very wavy and often varied significantly across the sample. The traces of the expected slip systems could only be positively identified when the crystal orientation was such that only 1 or 2 systems were predominant. For the more interesting orientations
where 3 or even 4 slip systems could be active, precise trace analysis turned out to be impracticable. 4. RESULTS
The following aspects of finite strain crystal deformation were examined in detail during the plane strain compression tests on the silicon-iron crystals: -finite shears -crystal lattice reorientations -stress-strain relations -active slip systems. It is important to note that for all orientations tested, variations of the silicon content in the range 2.7-3 wt% had no significant influence on the finite shears and the lattice reorientations. The same orientation with slightly different Si contents gave reproducible shears and lattice rotations, albeit with critical resolved shear stresses some 15% higher for the higher silicon content alloy.
4./. Finite shears
Accurate measurements of the crystal dimensions after each successive deformation showed that the crystals were effectively deformed in plane strain compression, i.e. Ell < 0.01 in all cases. The experimental shears [12 and (2) (=! tan IX and ! tan P) are shown as a function of [22 in Figs I to 6 for each crystal orientation. The error bars on the experimental results correspond to the standard deviation of the set of angular measurements of the reference lines on the grid. When the crystal deformation is perfectly homogeneous, tan IX is measured accurately but the error on tan p increases during deformation due to the decreasing thickness. When there is a tendency to inhomogeneous deformation (for example grain E) the error bars on both tan IX and tan p also indicate the degree of shear strain heterogeneity across the crystal. Figures 1 to 6 also show the theoretical values of [12 and (2) ([22) computed according to the ME 110
;0 Tan~
o ME 110 + MEl12
Tana
0.8
0.8 /+
0.6
0.6
/-+'
/+ /+ /-+'
0.4
0.4
,f'
0.2 0
/ /
0_0-0 ...0'"
/ ~~~oro0.2
0.4
0.2
*
0 E22
#
/
I
dJl
f
df
0.2
0.4
E22
Fig. 1. Experimental and theoretical [120 tan IX) and [2J(! tan P) shears of crystal A2. Theoretical values are computed for the ME 110 and ME 112 hypotheses.
586
ORLANS-JOLIET et at.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
Tan a
Tan~
0.4
0.4
0.2
0.2
0
. .....
........
....." 0.2
-0.2
'~
...
0.4
E22
o MEllO + MEl12
0
·£22
-0.2
~~ ~
-0.4
-0.4
Fig. 2. Experimental and theoretical E120 tan IX) and E23(~ tan fJ) shears of crystal 82. Theoretical values are computed for the ME 110 and ME 112 hypotheses.
Tan a
Tan~
0.6
0.6
./ / ' ~r
0.4
0.4
.....
0.2
0.2
",,:f 0
0.2
0
0.4
o ME 110 + ME 112
e::= £22
E22 -0.2
-0.2
Fig. 3. Experimental and theoretical EI2 0 tan IX) and E23(~ tan fJ) shears of crystal C2. Theoretical values are computed for the ME 110 and ME 112 hypotheses.
(a)
Tana
Tan~
0.6
0.6
0.4
0.4
0.2
0.2
0 -0.2
+
x ....0 .....0
0.2 .....0_
0.4
0 E22
0 -0-0_0_
0-0
-0.2
o ME 110 + ME 112
0\ 0.4 0\ 0
-0.4 -0.6
-0.4 -0.6 Fig. 4(a) Caption on facing page.
£22
\
0 ..... 0
ORLANS-JOLIET et at.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
(b) Tana
Tan~
0.6
0.6
0.4
0.4
587
+ 110 o MEllO x MEll2
• PENCIL 0.2
o
-+""'~,,""*
0.2* - t __
0.4
-.-.-.-·-tb -.-................
r-'"
~.-.-.-. .......
.......
-0.2
.
~
-0.4 -0.6
-0.6 Fig. 4(b)
Fig. 4. (a) Experimental and two theoretical £12(4 tan IX) and £23(4 tan f3) shear of crystal D2. (b) Computed theoretical shears according to the 110, ME 110, Pencil and ME I 12 hypotheses for crystal D2.
Tan a
Tanf3
0.4
0.4
0.2
0.2
0
o MEllO
+ MEI12
0 E22
+ E22
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
Fig. 5. Experimental and theoretical £12(4 tan IX) and £23(4 tan f3) shears of crystal E2. Theoretical values are computed for the ME IlO and ME Il2 hypotheses.
lana 0.4
lanf3 0.4
0.2 ... cp.T41,,~"41
0
0.2
..~.~_~"'ljl"i 0.4
o MEllO
+ ME 112
0.2 0 En
-0.2
-0.2
-0.4
-0.4
Fig. 6. Experimental and theoretical £12 <4 tan IX) and £23 <4 tan f3) shears of crystal F2. Theoretical values are computed for the ME IlO and ME Il2 hypotheses.
and ME 112 hypotheses. For a comparison of some of the different CRSS predictions, Fig. 4(b) also depicts the theoretical shears for the 4 models, namely llO, ME 110, ME 112 and Pencil. Comparison of the computed and experimental AMM 38/4-D
shears (Figs 1-6) shows that for several orientations there is good agreement for both the ME 110 and ME 112 hypotheses; this agreement is found for orientations B (Fig. 2), C (Fig. 3) and F (Fig. 6). In these cases, the models accurately predict the signs
588
ORLANS-JOLIET et at.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
and rates of shear of both (12 and (23' In particular for crystal B, both models accurately predict a monotonically increasing (negative) (12 shear and a (23 shear which goes through a maximum at ( - 0.3 as a consequence of the lattice rotation during the deformation. However, for crystals A, D and E there are some significant discrepancies between the experimental shears and some of the model predictions. Crystal A (Fig. I) exhibits (12 shears that tend to be lower than the computed values and, in particular, much lower than allowed by the ME 112 hypothesis. The (23 shears of this orientation are, however, in good agreement with the calculated values. Crystal D [Fig. 4(a) and (b)] undergoes a positive (23 shear and a practically negligible (12 shear. This behaviour is predicted quite well by the ME 112 hypothesis whereas the ME 110 predicts increasing negative (12 and (23 shears, particularly after strains of - 0.2. Figure 4(b) illustrates the general influence of the CRSS of the different systems on the expected shear strains of this orientation. In this case, only the ME 112 hypothesis appears to predict shears in reasonable agreement with the measured values. Finally, crystal E has a certain tendency to inhomogeneous deformation. It should be noted, however, that here again the ME 112 predictions are in reasonable agreement with the measured shears (slightly negative for (12 and -zero for (23)' while ME 110 predicts a small positive (12 shear, and a large negative (23 shear at ( > OJ. To summarize at this point, the measured and predicted unconstrained deformation components are often in reasonable agreement with both the ME 110 and ME 112 hypotheses. Where there is a significant difference between the two, then it is the ME 112 predictions that tend to be closer to the
.~
!------,il....,.---+------....---t+ X2
X1
Fig. 7. Experimental {1I0} pole figures of crystal 82 at strains of 0, 0.10 and 0.50. The lattice rotation occurs essentially about the XI axis.
experimental results, the single exception to this rule being the case of the (12 shear of crystal A. 4.2. Crystal lattice reorientations
The experimentally measured lattice orientations (CPI CP2) of the 12 crystals are given in Table 2 as a function of the nominal imposed strains. These orientations are determined from the type of {11O} pole figures shown in Fig. 7 for crystal B by determining the average Euler angles corresponding to the intensity peak maxima. The experimental lattice rotations are depicted by the positions of the X 3 and X 2 axes in standard 001-101-111 triangles in Figs 8-13 for crystals A to F. The latter also include the computed
111
o MEllO x
MEl12
111
110
101
010
001
Fig. 8. Experimental orientations of crystal A2 (axes X 3 and X 2 ) at strains of 0, 0.11, 0.30 and 0.50 (indicated by successively larger circles). Theoretical orientations for ME 110 and ME 112 are indicated at strain intervals of 0.05 up to 0.5 strain.
ORLANS-JOLIET et af.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
HI
589
o MEllO )( ME112
111
010
011
100
101 X3(0)
Fig. 9. Experimental orientations of crystal B2 at strains of 0, 0.12, 0.29, 0.47 and the computed ME 110 and ME 112 previsions.
o ME 110
)( ME 112
111
110
010
Fig. 10. Experimental orientations of crystal C2 at strains of 0, 0.13, 0.29,0.51 and the computed ME 110 and ME 112 previsions. orientations at 0.05 strain intervals according to the ME 110 and ME 112 hypotheses. From the set of 6 types of initial crystal orientation, two orientations, Band F, were such that both ME 110 and ME 112 (plus the 110 and Pencil models) predicted virtually identical rotations. Crystal B undergoes a very large rotation of about 30° about XI towards ~(211)[OII], (Fig. 9) and both the rotation path and rates are predicted perfectly by the computations. Crystal F (Fig. 13) rotates towards ~ (010)(201) and although the %2 orientation is predicted very well, the computations tend to make X 3 deviate somewhat from (010) at higher strains. The remaining crystal orientations exhibit different rotation paths according to the CRSS hypotheses.
Crystal A2 (Fig. 8) rotates moderately towards ~(458)[583]. This is predicted accurately by ME 110
but ME 112 makes the X 3 axis shift somewhat towards [III]. Note this descrepancy is consistent with the problem of the £12 shear of the same grain calculated with ME 112 (an error in the X 3 orientation leads to errors in the £12 shear about the X 3 axis). Crystal C2 (Fig. 10) undergoes a very large rotation from ~(101)[5'5] towards (447)[IlO]. The rotation towards ~(101)[13l] predicted according to ME 110 is in complete disagreement with the real behaviour. The rotation predicted according to ME 112 is certainly better, (i.e. X 3 to ~(212) and X 2 to [120]) but nevertheless is significantly different from the experiments and, in particular, is less than
ORLANS-JOLIET et at.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
590
(0)
111
o ME 110 x
ME 112
100
110 + 110
111
(b)
110
110
o MEllO x ME 112 • PENCIL
100
100
Fig. II. (a) Experimental orientations of crystal 02 at strains of 0, 0.14, 0.29, 0.47 and the computed ME 110 and ME 112 previsions. (b) Theoretical orientations of crystal 02 at strain intervals of 0.05, computed according to the 110, ME 110, Pencil and ME 112 hypotheses. the true rotation. In fact, the rotations for all 8 ~ values tabulated in Table I give rather poor prediction; the ~ values ~ I tend to favour the ME 110 path of Fig. 10, whereas ~ values ~ I, which facilitate {112} slip (including pure {112} slip), favour the ME 112 path in somewhat better agreement with the measurements. Figure II(a) shows that the ME 110 rotations are also in error for crystal D. The latter rotates towards ~(l1l)[IIO] as predicted by the ME 112 hypothesis, whereas ME 110 predicts a rotation towards (l11)[SS3]. The different rotation paths of the 4 models (ME 110, ME 112, 110 and Pencil) are illustrated in Fig. II(b) for this crystal. All predict that the compression plane rotates towards (111) but the rotation of the elongation direction X2 is very sensitive to the ~ value; ~ values ~ I, including Pencil,
ultimately predict a rotation towards the centre of the triangle, only ME 112 gives a rotation towards [IIO]. The behaviour of crystal E (Fig. 12) is somewhat similar, i.e. an experimental rotation of moderate amplitude towards ~(575)[IOI] which is predicted quite well by ME 112. Here again, if {1l0} slip is favoured, then the XJ rotation is reasonable, but X2 is expected to rotate towards the centre of the triangle. In summary, it is clear that measurements of the lattice rotations of certain crystals undergoing large imposed strains, constitute a sensitive technique for evaluating the relative ease of slip on different slip systems. For a limited number of crystals (here Band F), the rotations predicted by all reasonable hypotheses for the CRSS ratios ~ are in good agreement with the experimental measurements. For the others, the
ORLANS-JOLIET et af.:
591
COMPRESSION OF Si-Fe SINGLE CRYSTALS
111
o MEno x MEl12
111
fo 1
100
010
110
Fig. 12. Experimental orientations of crystal E2 at strains of 0,0.11,0.29, 0.47 and the computed ME 110 and ME 112 previsions.
rotation paths differ significantly according to ~. Crystals C, D and E change orientation according to the ME 112 model which favours {I12} slip. Only crystal A tends to follow a path which agrees with the conventional ME 110 hypothesis. 4.3. Stress-strain curves
The true stress-true strain, U33 (£22) curves of the silicon-iron crystals are depicted in Fig. 14(a) for the 3% Si alloy and (b) for the 2.7% Si alloy. These curves illustrate the strong orientation dependence of the flow stresses, a dependence which is in fact identical for both silicon contents. Thus, at a strain of 0.25, the flow stresses increase in the order B, F, C, A, E, D, which corresponds quite closely to the Taylor factors of the different orientations, respectively 2.31, 2.34,
2.44, 2.90, 3J2, 3.69 (hypothesis ME 110 at £ = 0.25). As described in Section 2.1, critical resolved shear stress L ~ 10 vs total equivalent shear strain f curves were derived from the u(E) curves using the appropriate Taylor factors. The latter were computed for the experimental orientations at strains of 0, 0.1, OJ and 0.5 and by interpolation at intermediate strains. The L~IO(f) curves are plotted for the Fe-2.7 wt% Si crystals in Fig. 15 (a) and (b) according to the ME 110 and ME 112 ~ hypotheses, respectively. The first point is that the Le(f) curves are now regrouped within a stress range of - ± 12%. This scatter is somewhat higher than the experimental reproducibility so that some orientation dependence of the Le(f) curves remains, but is markedly reduced by the Taylor factors. The second point is that the
o ME 110 x ME 112
111
To 1
100
110 Fig. 13. Experimental orientations of crystal F2 at strains of 0, 0.12, 0.32, 0.49 and the computed ME 110 and ME 112 previsions.
ORLANS-JOLIET et af.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
592 (0)
0"33
(b)
(Mfa)
0"33
1300
1300
1200
1200
1100
1100
1000
1000
900
900
800
800
700
700
600
600
500
500
(Mfa)
400 300
300
o
0.1
0.2
0.3
0.4
0.5
£
o
22
0.1
0.2
0.3
0.4
Fig. 14. True stress-strain curves of silicon-iron crystals; (a) Fe-3% Si, (b) Fe-2.7% Si.
ME 110 and ME 112 hypotheses produce very similar 'CeO") curves. These curves are consistent with previous 'Ce(Y) curves obtained on Fe-3% Si under conditions of single slip. Thus the shear experiments of Roche [11] on the {110} system of Fe-2.8% Si indicates a yield shear stress of - 150 MPa and a flow stress at Y = 0.2 of - 200 MPa. Zarubova and Sestak [16] found a similar yield shear stress in tension - 150 MPa, although with a slightly lower work hardening rate (at y - 0.4 'Ce - 180 MPa).
5. DISCUSSION AND CONCLUSIONS
The results clearly show that the overall finite strain behaviour of b.c.c. crystals undergoing partially imposed plane strain compression can be described in the framework of the generalized Taylor model of crystal plasticity. Of the six different crystal orientations described here, the lattice rotations and finite shears of at least five crystals can be quite accurately described by this model (incorporating the
(a) TC
(b)
(Mfa)
TC
(Mfa)
(110)
(110)
ME 110
ME112 400
400 ¢
02
300
200
100
100
o
r
o'---,---,----,r--,---,----,r--,---,----,r--,--
OJ Q8 1.2 16 2D 1.2 1.6 2.0 Fig. 15. Critical shear stresses on the {110}
r
0.8
r
ORLANS-JOLIET et al.:
COMPRESSION OF Si-Fe SINGLE CRYSTALS
appropriate boundary conditions of the channel die test). However, for certain orientations, relatively small variations of the critical strengths of the {l12} and {l1O}
e
(i) crystals Band F slip on the same systems according to both ME 110 and ME 112, giving rise to virtually identical rotations and non-imposed shears (in good agreement with experiment); (ii) crystal A slips essentially on a single {112} system according to ME 112, whereas the introduction of a second {IlO} system (ME 110) clearly gives better agreement with the experimental rotations and shears; (iii) in crystal C, the operation of two {112}
Crystal
the slip on two {112}
test, the evalues of the {Il2} systems in Fe-Si must be lower than 1. However, several tensile and shear experiments on single slip oriented Fe-Si crystals have found the converse, > 1. This can only be rationalized if the flow stresses of the {112} systems are lower than those of the {IIO} under conditions of intersecting slip characteristic of most multiple slip tests. There is some justification for this from the present results. The only crystal which seems to prefer easier {I 10} slip to {112} is A; in this case, the predominant systems predicted by ME 110 are colinear, i.e. without any significant intersecting slip directions. Crystals C, D and E which prefer easier {112} slip, all deform with intersecting slip directions. These observations on Fe-3% Si can also be extended to other b.c.c. metals. Similar orientations of Nb crystals have been recently tested in plane strain compression and identical results obtained [17], i.e. relatively easy {112} slip orientations characterized by intersecting slip directions. It should also be recalled that, according to Carpay et at. [7], the plane
e
£ = 0.25 according to the ME 110 and ME 112 hypotheses
Slip systems (in order of importance) ME 110 ME 112
A
(011)[11 I) (121) [II I) TS minor (101) [TTI]b
minor (211) [TTl) ATS
B
(110)[11 I) minor systemsb
(110) [III] minor systemsb
C
(110) [III] (011)[111] minor(121) [IlT] TS b
(l21)[III)TS
(011)[111) (l21)[III]TS (211)[111] ATS
D
(101) [lIT) (Ol1)[TTl) (211)[lIT] TS
(112) [TTl] TS (II2)[TIT) ATS (101) [ITT] or (Oil) [III]
E
(211)[lIT)TS (110) [lIT) (011)[11 I]
(1IO)[lIT] (121) [TIT] TS (011) [TTl]
F
(121)[lll]ATS (110)[1 II]
(121)[111] ATS (110)[1 II]
'Predominant taken to mean yk ~ 0.4 £. bMinor taken to mean 0.4 ~ yk ~ 0.2 £.
593
594
ORLANS-JOLIET et al.: COMPRESSION OF Si-Fe SINGLE CRYSTALS
strain compression results on high symmetry Mo [7], Cr and V[6J crystals are all consistent with ~ {II2} TS values of - 0.77. The orientations used by these authors all involve intersecting slip directions. It is therefore suggested that as a general rule in b.c.c. metals undergoing large strains:
Acknowledgements-The authors wish to thank Miss H.
Bruyas and Mr R. Fillit for their indispensable help with the X-ray pole figure analyses.
(I) under conditions of single or colinear slip, glide on {1I0} is easier than on {1I2}; (2) under conditions where the predominant
~IlO=I
~ 112 TS - 0.93 ~1I2ATS
-1.03 012 TS.
REFERENCES 1. G. I. Taylor, J. Inst. Metals 62, 307 (1938). 2. J. F. W. Bishop and R. Hill, P/lil Mag. 42, 414, 1298 (1951). 3. G. Y. Chin, E. A. Nesbitt and J. A. Williams, Acta metall. 14, 467 (1966). 4. U. F. Kocks and H. Chandra, Acta metall. 30, 695 (1982). 5. J. H. Driver, A. Skalli and M. Wintenberger, Phil. Mag. 49, 505 (1984). 6. D. E. Crutchley, C. N. Reid and T. H. Webster, Proc. 2nd Int. Con! on Strength of Metals and Alloys, Vol. I, p. 127 (1970). 7. F. M. A. Carpay, G. Y. Chin, S. Mahajan and J. J. Rubin, Acta metall. 23, 1473 (1975). 8. F. M. A. Carpay, S. Mahajan, G. Y. Chin and J. J. Rubin, Acta metall. 25, 149 (1977). 9. M. Renouard and M. Wintenberger, c.r. Acad. Sci., Paris 2838, 237 (1976). 10. T. Taoka, S. Takeuchi and J. Furubayashi, Phys. Soc. Japan 9, 701 (1964). 11. C. Roche, Doctoral thesis, Universite Paris VI (1976). 12. P. L. Sue and K. S. Havner, J. Mech. Phys. Solids 32, 417 (1984). 13. R. Fortunier, Doctoral thesis, INPG-ENSMSE (1987). 14. R. Fortunier, B. Orlans-Joliet, F. Montheillet and 1. H. Driver, Proc. 8th Rise Int. Symp., Constitutive Relations and their Physical Basis (edited by S. I. Andersen et al.), p. 317 (1987). 15. R. Y. FiIlit, H. Bruyas and F. Patay, European Patent: DOSOPHATEX No. 85-400 95405. 16. N. zarubova and B. Sestak, Physica status solidi 30 A, 479 (1975). 17. B. Orlans-Joliet, Doctoral thesis, ENSMP-ENSMSE (1989).