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Texture softening and strain instability in F.C.C. metals: Plane-strain deformation
Acfa metal/. Vol. 35, No. 9, pp. 2307-2314, Printed in Great Britain. All rights reserved
TEXTURE
1987 Copyright
0
OOOI-6160/87 $3.00 +O.OO 1987 Pergamon Journals Ltd
SOFTENING AND STRAIN INSTABILITY IN F.C.C. METALS: PLANE-STRAIN DEFORMATION
T. 6ZTtJRK’ and G. J. DAVIES’ ‘Department of Metallurgical Engineering, Middle East Technical University, Ankara, Turkey and *Department of Metallurgy, University of Sheffield, England (Received
25 June 1984; in revised form
6 December
1986)
Abstract-An analysis was carried out into texture induced softening and the resulting strain instability in plane-strain deformation. A macroscopic model was employed in the analysis which viewed the deformation as the juxtaposition of suitably disposed positive and negative simple shears. With this model, the textural contribution to the rate of deformation hardening was computed in terms of a texture softening factor, S, this in a form that could readily be incorporated into the relevant conditions for strain instability. Maps were obtained which show the orientation dependence of texture softening factor and the geometry of the macroscopic shear. The former can be used to assess the susceptibility of a textured f.c.c. metal (or a single crystal) for strain instability, and the latter can be used to predict the geometry of shear bands resulting from the instability. RQsumP-Nous avons analysi l’adoucissement induit par la structure et l’instabilite de deformation qui en rtsulte dans le cas dune deformation plane. Nous avons utilise dans ce but un modele macroscopique considerant la deformation comme la juxtaposition de cisaillements simples positifs et negatifs convenablement disposes. A l’aide de ce modtle, nous avons calcult numeriquement la contribution de la texture a la vitesse de durcissement par deformation, en fonction d’un facteur d’adoucissement textural, S, exprime de faGon a pouvoir Ctre rapidement incorporb dans les conditions de l’instabilitt de la deformation. Nous avons obtenu des cartes qui montrent l’influence de l’orientation sur le facteur d’adoucissement textural, ainsi que la gtometrie du cisaillement macroscopique. On peut utiliser les premieres pour &valuer la predisposition dun metal cfc texture (ou dun monocristal) a l’instabilite de deformation; et les secondes pour predire la gtometrie des bandes de cisaillement resultant de l’instabilite. Zusammenfassung-Die Textur-induzierte Entfestigung und die daraus folgende Dehnungsinstabilitit bei Verformung im ebenen Spannungszustand wurden analysiert. Hierzu wurde ein Model1 benutzt, welches die Verformung als ein Nebeneinander von geeignet verteilten positiven und negativen Scherungen beschreibt. Mit diesem Model1 wurde der Texturbeitrag zur Verfestigung anhand eines TexturEntfestigungsfaktors S berechnet. Dieser Faktor war in einer Form, dal.3er sofort in die entsprechenden Bedingungen fur die Texturinstabilitlt eingefiihrt werden konnte. Es ergeben sich Karten, die die Orientierungsabhlngigkeit des Textur-Entfestigungsfaktors und die Geometrie der makroskopischen Scherung zeigen. Mit dem Textur-Entfestigungsfaktor kann die Ant%llligkeit eines kfz. Materials (oder eines Einkristalls) fur die Dehnungsinstabilitat abgeschatzt werden. Aus der Geometrie der Schenmg kann die Geometrie der aus dieser Instabilitlt herriihrenden Scherbander vorausgesagt werden.
1. INTRODUCTION
Plastic deformation in metals brings about various geometric and structural changes. Each of these changes is characteristically associated with an hardening or a softening effect. When the hardening effects are dominant strain distribution is uniform in the material and the deformation continues with a rising load. But in an on-going deformation, conditions often develop in which the softening effects tend to equalize or dominate over the hardening effects. Then the material is subject to strain instability and a load drop occurs. Conditions for strain instability once developed may persist throughout the deformation. Then localized deformation which ensues the instability is rapidly followed by fracture. However in some cases
the instability develops only temporarily, and after some localized strain the material reverts to homogeneous deformation. In these cases the effect of the instability is mostly confined to some structural features of localized deformation in the materials microstructure. Examples for the latter are Liider’s bands [l] and shear bands [2]. Even mechanical twinning can be considered as a strain inhomogeneity resulting from a temporary instability. Thus the occurrence of strain instability is quite a significant event not only for fracture but also for the development of microstructure in deformed metals (see Refs [3,4]). 2. CONDITIONS
FOR INSTABILITY
Since the localized deformation is associated with a load drop, the condition for its initiation can be
2307
2308
OZTuRK
and DAVIES:
TEXTURE
SOFTENING
written in the form
AND
STRAIN
INSTABILITY
stress-strain behaviour of the materials, and hence equation (4), e.g. strain rate sensitivity (see Ref. [I]).
Where C is a constant representing the geometric propensity of a particular working process for strain instability. In those working processes where there is a decrease in the cross-sectional area normal to the applied load, C is positive. Then the instability can occur relatively easily since a little decrease in the normalized rate of strain hardening which normally occurs with increasing strain, is sufficient to satisfy the condition. Examples of this case include planestrain tension as encountered in sheet forming, where C = 1. In most bulk deformation processes, however, of necessity, compressive loads are involved, and as a result the cross-sectional area increases. C is then negative indicating considerable geometric resistance for instability. In this wide range of working processes, the case of plane-strain compression is quite special. Since there is no area change in this process, C = 0 and the instability is neither aided nor restrained by the geometry of the deformation. Thus the instability in plane strain compression has purely material origin. For a given process, i.e. for a given C, material parameter that will influence the instability exercise their effect through the term (l/o) * (da/dt). Of such parameters the present work will concentrate on texture induced softening. To evaluate the effect of this softening on strain instability, it will be assumed that the stress-strain behaviour of the material is controlled solely by dislocational strain hardening and texture induced softening. It is further assumed that the former follow a hardening rule of the form r =k(y)
(2)
where k is a constant and z is the shear stress on an active slip system and y is the total dislocational shear strain on all active slip systems. Normal strain E is related to shear strain as c = y/M where M is the Taylor factor, and normal stress can be written as 0 = MT. Then the stress-strain behaviour of the material can be described as
from this relation the condition can he derived as
3. MODELLING OF PLANE-STRAIN DEFORMATION
To evaluate the texture softening factor, S, in single crystals (or in textured polycrystals) we adopt a macroscopic model for plane-strain deformation. The model views the deformation as the juxtaposition of a suitably disposed positive and negative simple shears. The shears are of equal magnitude and operate along planes in the block, that are macroscopic and inclined equally but oppositely to the compression plane, as shown in Fig. 1. Here fl is the angle between the shear planes and the compression plane and ~1= 45”-lb I. Simultaneous operation of these shears in the block produce uniform strain. With the geometry given, the required deformation can be produced by any shear pair irrespective of the value of the angle p. But at /7 = 45” the deformation requires minimum shear along the inclined planes. At other angles, to produce the same amount of macroscopic deformation it is necessary to increase the shear strains on the inclined planes by a factor of l/cos 2~. Thus plastic work considerations require that if isotropic the block will deform through shears at /I = 45”. Where the flow stress is directional, however, as in single crystals for example, the shears may operate at angles other than f3 = 45”. 4. PERMISSIBLE
SHEAR GEOMETRIES
To determine the operative shear pair in an anisotropic block, it is necessary to evaluate the plastic work for a prescribed macroscopic strain as a function of the shear angle J3.This calculation was carried out for single crystals. Crystals were specified in terms of three Euler angles $ 8 and 4 (see Appendix). Since the Taylor factor M is a relative measure of the crystals flow stress (see above), this amounts to determining M/cos 2~. For a given shear geometry, two calculations were made, one for positive and the other for the negative simple shears. Each of these shears was described by an incremental strain
for strain instability
Ida 6 --=;+(n+l)S
Fig. 1. Angles p and ac(=4S” - 1s I) depicting the geometry of a simple shear pair compatible with plane strain deformation. The pair is made up of positive and negative simple shears that operate on + /3and -/i planes respectively. The magnitude of shear strain is the same for each shear.
OZTuRK
and DAVIES:
TEXTURE
SOFTENING
AND STRAIN INSTABILITY
2309
0 =60
0
J.kLS%)
30(_+2.5)~~+j()(t2.5)
/ LOkz.s)rps50(‘2.5) lzil
q 35
(-,2S)S
lilll
l$S55(?2.5)
Fig. 2. Permissible shear angles in crystals under plane strain deformation. Note that the range of permissible shear angles is a sensitive function of crystal orientation. +, 0 and Q,are Euler angles which define crystal orientation.
displacement tensor and the M-value for the crystal was derived using Taylor-Bishop and Hill procedure [S, 61. The deformation was assumed to occur through slip on {111) ( 1lo} systems. In this way the plastic work requirement of the crystal was evaluated in terms of @/cos 2c( where I@ is the Taylor factor averaged for both shears. For most crystals, results based on piastic work considerations were not unequivocal in describing the shear geometry. It was true that for some specific crystals the plastic work was a minims at fl = 45” for which the shear geometry was clearly identified. But for most crystals the variation of &?/cos 2c( with 8 was associated with a plateau centered at /I = 45” and extended within a range as much as a = t 5-10”. As seen in Fig. 2, the limits of this range were a sensitive function of crystal orientation and could
be as wide as a = If: 25” in extreme cases, i.e. 20” < fl < 70”. (Figure 3 is a by-product of this calculation and shows the minimum values of ii;iJcos 21x.Since the permissible shear angles always include /3 = M”, i.e. CI= O”, these are also the conventional Taylor factors for plane-strain deformation.) Thus in general the geometry of the operative shear is not fixed by the plastic work considerations and can take any value within the observed limits. Then precisely which shear pair becomes active can be dete~ined with recourse to texture softening considerations. 5. TEXTURE SOFTENING Textural contribution to the rate of deformation hardening in the crystal was calculated in terms of the
2310
&TURK
and DAVIES: TEXTURE SOFTENING AND STRAIN INSTABILITY
1
Y
d-40
1 \\py 0 =
60
0 = 70
0 = 80
Fig. 3. Values of Taylor factor, M, for plane-strain deformation contours: from 2.2 to 4.6 by 0.3
previously defined texture softening factor
where M’ = M/cos 2a. Calculations were carried out at 2” intervals within the permissible angular range in a. For a given geometry, positive and negative simple shears were described by appropriate strain displacement tensors and the value of S was computed numerically for each shear by determining the rate of lattice rotation and the associated rate of change in the Taylor factor, i.e. dM -=-.-. dc
dR dM dE da
Lattice rotations were determined using a procedure described by Kallend and Davies [7]. The two svalues obtained in this way are then combined to give an average texture softening factor, s, for the particular shear angle for the crystal. The angle at which S was minimized not only defines the operative shear pair, but also gives the true value of texture softening factor for the crystal. The S-values determined for the crystals are given in Fig. 4. In regions of the orientation space where the
S-values are negative i.e. where the crystals texturally soften, contours are drawn as continuous lines. The values of the angle /l defining the geometry of the operative shear pair are given in Fig. 5. These results allow the following generalization to be made: 1. Based on texture softening considerations it can be concluded that the crystals always favour shears operating at /I < 45”, and moreover a correlation seems to exist between the Taylor factor M i.e. “hardness” and the shear angle /I, such that where the M-value is high, the crystals usually favour shears operating at /l<<45”. 2. Although there are some important exceptions, a similar correlation also holds with respect to the S-value, namely where the M-value is high, the S-value
t)ZTORK and DAVIES: TEXTURE SOFTENING AND STRAIN INSTABILITY found to be as much as 3 = -0.4. On the otherhand a range of other orientations exist at which the crystals are subject to the opposite effect of texture hardening. In these latter orientations the texture is a factor contributing to the stability of the deformation. Where there is pronounced texture softening, it is expected that limit strain L*, i.e. the strain level at which the instability initiates, will be reduced to a lower value. This strain level can be calculated with reference to the instability condition given in equation (4). This yields a limit strain of c: = n/{(n + 1)s) for plane-strain compression (C = 0), and CT = n/{(n + 1)s + I} for plane-strain tension (C = 1). Once these strain levels are reached the shears concentrate on macroscopic planes in the crystal, i.e. shear bands form. (This also corresponds to strain level at which necking initiates in planestrain tension.) The calculation of the limit strains as carried out above assumes that the S-value remains constant during the deformation. However, this is true only for some specific crystal orientations. For the majority,
lattice rotation which accompany the deformation alters the orientation and therefore the S-value. This continues until the crystal is rotated into an orientation stable under plane-strain. In f.c.c. metals orientations stable under plane strain constitutes a range between { llOj(112) and {112}(111). A range of orientations between {ll0}(112)and{110}(001)arealsonearlystable[8]. Figure 4 shows that all of these orientations are associated with slightly negative S-values. At (110}(001) $ =90”, e =45”, 4 =oc, s= -0.08 (/? = 40”), the softening increases slightly at {110}(112) $ = .54.7”, 0 = 45’, 4 = 0’ where s= -0.12 (p = 38”), it is further increased at {112) (11 l), I(/ = O”, 0 = 35.3”, 4 = 45”, for which s= -0.16 (p = 35”). This is in fact the order in which the M value increases in plane-strain. The numerical values given here are in good agreement with those reported by Dillamore et al. [I]. In order to test the validity of the predicted Svalues reference could be made to single crystal studies involving plane-strain deformation. However this is not a fruitful attempt. First of all shear bands
6 560 Fig. 4. Values of texture softening factor, s = (l/M) (dM/dc), (continuous):
-0.05,
-0.1,
-0.4;
2311
(dotted):
for plane-strain deformation. 0.0, 0.05, 0.1, 0.4.
Contours
2312
&TURK
and DAVIES:
TEXTURE
SOFTENING
AND STRAIN INSTABILITY
I9
QI- 60
cl
@=3!jZz.s
@ = fS22.5
= 40+
2.5
p=
3k2.5
Fig. 5. Values of the angle 8, defining the geometry of the operative shear pair in crystals deforming under plane-strain.
which would distinguished the basis that which cannot
form as a result of the instability is from other strain inhomogeneities on they cross grain boundaries, an aspect be observed in single crystals.?
tin plane-strain deformed single crystals of initial orientations such as (112) [l 1 l] there is an indirect evidence for the occurrence of strain inhomogeneity of the form under discussion here. According to Bauer et uf. [9] (112)[111] oriented crystal preserves its initial orientation until quite heavy reductions with some spread, but at heavier reductions it develops its complimentary {112}( 11l> orientation. The development of this complimentary orientation, which is frequently encountered in single crystal work [9, IO] is probably due to the formation of shear bands at heavy strains. While orientations such as (112) [I 1I] is stable under uniform strain i.e. simultaneous operation of the simple shears throughout the crystal, they become metastable when strain concentrates in shear bands. It was shown by Dillamore [ll] and &tiirk [12] that the manner of lattice rotations in shear bands are such that with little fluctuation, initially (112) [i 111oriented region rotate rapidly into its complimentary orientation.
In polycrystals subject to plane-strain deformation initially individual grains experience different degrees of texture softening or hardening depending on their orientation. But gradually the polycrystal develops the stable orientations mentioned above, and so it is subject to texture softening with an average S-value in the same order as those reported for the stable orientations. Dillamore et al. [l] considering an hypothetically sharp “pure metal” texture calculated an average softening factor of s = - 0.11. They further showed that shear banding as it occurs in heavily rolled f.c.c. metals were in good agreement with this prediction, moreover the angle of /3 = 35” at which the shear bands form in these metals were shown to correspond to the geometry at which the softening was maximized. With reasonably high strain hardening exponent n, the strain instability is postponed to relatively heavy strains, the level of which is expected to depend on whether the metal develops “pure metal” or “alloy” type texture [8]. Texture softening in metals that form
t)ZTURK
and DAVIES:
TEXTURE
SOFTENING
AND STRAIN INSTABILITY
2313
in thickness. Orientations developed are simple which can be described as { 110}(223) with some spread. The softening at { 110)(223) 1(1= 45”, 0 = 45”, 4 = O”, is about s N -0.1, even higher in parts of the spread close to {llO}(llO) $ = O”, 0 =45”, 4 = 0” where the softening is as high as s = -0.4. The structure of this metal was accordingly heavily banded as shown in Fig. 7. Shear bands in this structure occur on average at /3 = 33”. This is in good agreement with the predicted shear angle for {110}(223), see Fig. 5. 5.2. Texture softening
Fig. 6. 4 = 0” section of a.c.0.d.f. relating to a cross-rolled brass after 95% reduction in thickness. Contours: from 1.0 to 6.5 x random.
alloy type texture, i.e. low stacking fault energy metals is, however, outside the scope of the present paper. The S-values reported in Fig. 4 were calculated with the assumption of { 11 l}( 110) multiple slip alone and are not expected to be valid when mechanical twinning is an active deformation mode. Thus if discussion is confined to high-stacking fault energy metals, the texture softening may vary depending on the details of the pure metal texture developed. For instance if the texture consists of orientations near (112) (111) the metal is subject to more pronounced softening than would be the case for a more balanced case of uniform distribution of orientations between { 1 lO}( 112) and { 112)( 111). Thus the initial texture of the metal by favouring one stable orientation more than the others may influence the softening behaviour of the metal at heavy strains. In cross-rolling or in compression rolling, the deformation is still plane-strain compression, though the metals develop different textures. Figure 6 shows the texture of a cross-rolled brass after 95 reduction
in sheet forming
The role of the initial texture in influencing the softening behaviour of the metal is much more pronounced in sheet forming than it is for planestrain compression. This is partly due to the fact that plane-strain tension which is normally encountered in sheet forming mostly involve sheet metals that are strongly textured and, partly due to reduced strain levels at which the instability can initiate in tension. Cube textured metal sheets illustrate a good case of how the initial texture can influence the limit strains in plane-strain tension. The instability condition given in equation (4) shows that with no texture softening, i.e. S = 0, the instability occurs at ET = n. This approximately defines the limit strain when the cube textured sheet is subject to plane-strain tension along the original rolling direction (i.e. along {lOO}(OOl), $ = O’, 0 = O”, 4 = O”), see Fig. 4. However if instead the sheet is stretched 45” to the rolling direction (i.e. { lOO}( IlO), $ = 45”, 0 = O”, 4 = 0”), S = -0.4 and the limit strain is reduced to CT = n/{(n + 1)0.4 + 1). Thus with an hypothetically strong cube texture, even when the strain hardening exponent is as high as n = 0.5, the limit strain cannot be more than t: = 0.3. This is in agreement with the observations of Da Costa Viana [13] who found that cube textured copper sheets fail prematurely when
Fig. 7. The microstructure in a longitudinal section of a cross-rolled brass (rolled 90” to the rolling direction after each pass) after 95% reduction in thickness (the last rolling direction is horizontal). Shear bands occur on average at p z 33’.
2314
dZTURK and DAVIES: TEXTURE SOFTENING AND STRAIN INSTABILITY
tested in hydrolic bulging, the failure occurring in plane-strain, i.e. close to the clamped periphery of the sheet. 6. CONCLUDING REMARKS The analysis presented above indicates that the textural contribution to the rate of work wardening, as measured by texture softening factor S, is strongly orientation dependent. While certain textures promote instability through accompanying softening, the other result in hardening whereby contributing to the stability of deformation. Textures which normally develop in f.c.c. metals under plane-strain belong to the former, though the degree of softening associated with these textures is not very pronounced. The analysis further shows that the scope for instability control via texture is greater in plane-strain tension than it is for plane-strain compression. On this basis the extension of the present analysis to other strain states where tension is dominant, (e.g. states of biaxial strain encountered in sheet forming) would be highly beneficial. Acknowledgement-The authors would like to thank Professor I. L. Dillamore for invaluable discussion at various stages of this work.
REFERENCES 1. I. L. Dillamore, J. G. Robert and A. C. Bush, Me&Is sci. J. 12, 73 (1979). 2. B. J. Duggan, M. Hatherly, W. B. Hutchinson and P. T. Wakefield, Metals Sci. J. 12, 343 (1978). 3. T. Gztiirk, Text. Microstruct. 6, 39 (1983). 4. T. Haratani, W. B. Hutchinson, I. L. Dillamore and P. Bale, Metals Sci. J. 18, 51 (1984). 5. G. I. Taylor, J. Inst. Metals 62, 307 (1938). 6. J. F. W. Bishop and R. Hill, Phil. Mug. 42, 414, 1298 (1951). 7. J. S. Kallend and G. J. Davies, Phil. Mug. 7,361 (1972). 8. J. S. Kallend and G. J. Davies, Texture 1, (1972). 9. R. E. Batter, H. Mecking and K. Lticke, Muter. Sci. Engng. 27, 1631 (1977).
10. K. Brown and M. Hatherly, J. Inst. Metals 310, 98 (1970).
11. I. L. Dillamore and A. C. Bush, Texture of Materials (edited by C. Gottstein and K. Liicke), Vol. 1, p. 367. Springer, Berlin (1978). 12. T. Gztiirk, Ph.D. thesis, Univ. of Cambridge (1978). 13. C. S. Da Costa Viana, Ph.D. thesis, Univ. of Cambridge (1978). 14. G. J. Davies, D. J. Goodwill and J. S. Kallend, J. uppl Crystallogr. 4, 67 (1971).
APPENDIX Euler angles $ f3and I#Jrelate the crystallographic axes of the crystal to chosen axes in the sample material, e.g. normal direction, rolling direction, and transverse direction. Ideal orientation (hkl)[uvw] which is commonly used in describing the crystal orientation, correspond to Euler angles given by W
cm ti =
(II’+
(u2
+
y2 +
(h2
+
k2
w2)1/2
k* + 12)“*
(h* + k*)“*
1 ‘OS
e =
+
-h ~0s ti = (h2 + k2),,2 cos($ +d)=
/2)1,2
(h, k z 0)
(u2+u2u+w2)l,2
(h,k=O).
Here (hkl) refers to a crystallographic plane lying in the compression (rolling) plane of specimen and [uuw] is a crystallographic direction in this plane and parallel to the extension (rolling) direction. Alternatively, an ideal orientation can be found from known Euler angles using the relations h=-sinBcos4; u=cO~t,bOO~eOO~~
a=-cos+cos0sin#~ w = cos * sin e
k=sint?sin4; -sin$sin4 -sin$cos4
l=cos8
and rationalizing the resulting indices. In most cases, this analysis is unnecessarily time consuming and to facilitate the analysis a set of charts has been prepared for cubic materials [14]. These charts show the position in orientation space of all (hkl)[uvw] for cubic materials formed by permutating h, k, I, u, u, w through 3 to -3. The resulting Euler angles are taken within the ranges. o<$
~~12
o
0<4
which are adequate for a complete description of textures with cubic symmetry.
TEXTURE
1987 Copyright
0
OOOI-6160/87 $3.00 +O.OO 1987 Pergamon Journals Ltd
SOFTENING AND STRAIN INSTABILITY IN F.C.C. METALS: PLANE-STRAIN DEFORMATION
T. 6ZTtJRK’ and G. J. DAVIES’ ‘Department of Metallurgical Engineering, Middle East Technical University, Ankara, Turkey and *Department of Metallurgy, University of Sheffield, England (Received
25 June 1984; in revised form
6 December
1986)
Abstract-An analysis was carried out into texture induced softening and the resulting strain instability in plane-strain deformation. A macroscopic model was employed in the analysis which viewed the deformation as the juxtaposition of suitably disposed positive and negative simple shears. With this model, the textural contribution to the rate of deformation hardening was computed in terms of a texture softening factor, S, this in a form that could readily be incorporated into the relevant conditions for strain instability. Maps were obtained which show the orientation dependence of texture softening factor and the geometry of the macroscopic shear. The former can be used to assess the susceptibility of a textured f.c.c. metal (or a single crystal) for strain instability, and the latter can be used to predict the geometry of shear bands resulting from the instability. RQsumP-Nous avons analysi l’adoucissement induit par la structure et l’instabilite de deformation qui en rtsulte dans le cas dune deformation plane. Nous avons utilise dans ce but un modele macroscopique considerant la deformation comme la juxtaposition de cisaillements simples positifs et negatifs convenablement disposes. A l’aide de ce modtle, nous avons calcult numeriquement la contribution de la texture a la vitesse de durcissement par deformation, en fonction d’un facteur d’adoucissement textural, S, exprime de faGon a pouvoir Ctre rapidement incorporb dans les conditions de l’instabilitt de la deformation. Nous avons obtenu des cartes qui montrent l’influence de l’orientation sur le facteur d’adoucissement textural, ainsi que la gtometrie du cisaillement macroscopique. On peut utiliser les premieres pour &valuer la predisposition dun metal cfc texture (ou dun monocristal) a l’instabilite de deformation; et les secondes pour predire la gtometrie des bandes de cisaillement resultant de l’instabilite. Zusammenfassung-Die Textur-induzierte Entfestigung und die daraus folgende Dehnungsinstabilitit bei Verformung im ebenen Spannungszustand wurden analysiert. Hierzu wurde ein Model1 benutzt, welches die Verformung als ein Nebeneinander von geeignet verteilten positiven und negativen Scherungen beschreibt. Mit diesem Model1 wurde der Texturbeitrag zur Verfestigung anhand eines TexturEntfestigungsfaktors S berechnet. Dieser Faktor war in einer Form, dal.3er sofort in die entsprechenden Bedingungen fur die Texturinstabilitlt eingefiihrt werden konnte. Es ergeben sich Karten, die die Orientierungsabhlngigkeit des Textur-Entfestigungsfaktors und die Geometrie der makroskopischen Scherung zeigen. Mit dem Textur-Entfestigungsfaktor kann die Ant%llligkeit eines kfz. Materials (oder eines Einkristalls) fur die Dehnungsinstabilitat abgeschatzt werden. Aus der Geometrie der Schenmg kann die Geometrie der aus dieser Instabilitlt herriihrenden Scherbander vorausgesagt werden.
1. INTRODUCTION
Plastic deformation in metals brings about various geometric and structural changes. Each of these changes is characteristically associated with an hardening or a softening effect. When the hardening effects are dominant strain distribution is uniform in the material and the deformation continues with a rising load. But in an on-going deformation, conditions often develop in which the softening effects tend to equalize or dominate over the hardening effects. Then the material is subject to strain instability and a load drop occurs. Conditions for strain instability once developed may persist throughout the deformation. Then localized deformation which ensues the instability is rapidly followed by fracture. However in some cases
the instability develops only temporarily, and after some localized strain the material reverts to homogeneous deformation. In these cases the effect of the instability is mostly confined to some structural features of localized deformation in the materials microstructure. Examples for the latter are Liider’s bands [l] and shear bands [2]. Even mechanical twinning can be considered as a strain inhomogeneity resulting from a temporary instability. Thus the occurrence of strain instability is quite a significant event not only for fracture but also for the development of microstructure in deformed metals (see Refs [3,4]). 2. CONDITIONS
FOR INSTABILITY
Since the localized deformation is associated with a load drop, the condition for its initiation can be
2307
2308
OZTuRK
and DAVIES:
TEXTURE
SOFTENING
written in the form
AND
STRAIN
INSTABILITY
stress-strain behaviour of the materials, and hence equation (4), e.g. strain rate sensitivity (see Ref. [I]).
Where C is a constant representing the geometric propensity of a particular working process for strain instability. In those working processes where there is a decrease in the cross-sectional area normal to the applied load, C is positive. Then the instability can occur relatively easily since a little decrease in the normalized rate of strain hardening which normally occurs with increasing strain, is sufficient to satisfy the condition. Examples of this case include planestrain tension as encountered in sheet forming, where C = 1. In most bulk deformation processes, however, of necessity, compressive loads are involved, and as a result the cross-sectional area increases. C is then negative indicating considerable geometric resistance for instability. In this wide range of working processes, the case of plane-strain compression is quite special. Since there is no area change in this process, C = 0 and the instability is neither aided nor restrained by the geometry of the deformation. Thus the instability in plane strain compression has purely material origin. For a given process, i.e. for a given C, material parameter that will influence the instability exercise their effect through the term (l/o) * (da/dt). Of such parameters the present work will concentrate on texture induced softening. To evaluate the effect of this softening on strain instability, it will be assumed that the stress-strain behaviour of the material is controlled solely by dislocational strain hardening and texture induced softening. It is further assumed that the former follow a hardening rule of the form r =k(y)
(2)
where k is a constant and z is the shear stress on an active slip system and y is the total dislocational shear strain on all active slip systems. Normal strain E is related to shear strain as c = y/M where M is the Taylor factor, and normal stress can be written as 0 = MT. Then the stress-strain behaviour of the material can be described as
from this relation the condition can he derived as
3. MODELLING OF PLANE-STRAIN DEFORMATION
To evaluate the texture softening factor, S, in single crystals (or in textured polycrystals) we adopt a macroscopic model for plane-strain deformation. The model views the deformation as the juxtaposition of a suitably disposed positive and negative simple shears. The shears are of equal magnitude and operate along planes in the block, that are macroscopic and inclined equally but oppositely to the compression plane, as shown in Fig. 1. Here fl is the angle between the shear planes and the compression plane and ~1= 45”-lb I. Simultaneous operation of these shears in the block produce uniform strain. With the geometry given, the required deformation can be produced by any shear pair irrespective of the value of the angle p. But at /7 = 45” the deformation requires minimum shear along the inclined planes. At other angles, to produce the same amount of macroscopic deformation it is necessary to increase the shear strains on the inclined planes by a factor of l/cos 2~. Thus plastic work considerations require that if isotropic the block will deform through shears at /I = 45”. Where the flow stress is directional, however, as in single crystals for example, the shears may operate at angles other than f3 = 45”. 4. PERMISSIBLE
SHEAR GEOMETRIES
To determine the operative shear pair in an anisotropic block, it is necessary to evaluate the plastic work for a prescribed macroscopic strain as a function of the shear angle J3.This calculation was carried out for single crystals. Crystals were specified in terms of three Euler angles $ 8 and 4 (see Appendix). Since the Taylor factor M is a relative measure of the crystals flow stress (see above), this amounts to determining M/cos 2~. For a given shear geometry, two calculations were made, one for positive and the other for the negative simple shears. Each of these shears was described by an incremental strain
for strain instability
Ida 6 --=;+(n+l)S
Fig. 1. Angles p and ac(=4S” - 1s I) depicting the geometry of a simple shear pair compatible with plane strain deformation. The pair is made up of positive and negative simple shears that operate on + /3and -/i planes respectively. The magnitude of shear strain is the same for each shear.
OZTuRK
and DAVIES:
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SOFTENING
AND STRAIN INSTABILITY
2309
0 =60
0
J.kLS%)
30(_+2.5)~~+j()(t2.5)
/ LOkz.s)rps50(‘2.5) lzil
q 35
(-,2S)S
lilll
l$S55(?2.5)
Fig. 2. Permissible shear angles in crystals under plane strain deformation. Note that the range of permissible shear angles is a sensitive function of crystal orientation. +, 0 and Q,are Euler angles which define crystal orientation.
displacement tensor and the M-value for the crystal was derived using Taylor-Bishop and Hill procedure [S, 61. The deformation was assumed to occur through slip on {111) ( 1lo} systems. In this way the plastic work requirement of the crystal was evaluated in terms of @/cos 2c( where I@ is the Taylor factor averaged for both shears. For most crystals, results based on piastic work considerations were not unequivocal in describing the shear geometry. It was true that for some specific crystals the plastic work was a minims at fl = 45” for which the shear geometry was clearly identified. But for most crystals the variation of &?/cos 2c( with 8 was associated with a plateau centered at /I = 45” and extended within a range as much as a = t 5-10”. As seen in Fig. 2, the limits of this range were a sensitive function of crystal orientation and could
be as wide as a = If: 25” in extreme cases, i.e. 20” < fl < 70”. (Figure 3 is a by-product of this calculation and shows the minimum values of ii;iJcos 21x.Since the permissible shear angles always include /3 = M”, i.e. CI= O”, these are also the conventional Taylor factors for plane-strain deformation.) Thus in general the geometry of the operative shear is not fixed by the plastic work considerations and can take any value within the observed limits. Then precisely which shear pair becomes active can be dete~ined with recourse to texture softening considerations. 5. TEXTURE SOFTENING Textural contribution to the rate of deformation hardening in the crystal was calculated in terms of the
2310
&TURK
and DAVIES: TEXTURE SOFTENING AND STRAIN INSTABILITY
1
Y
d-40
1 \\py 0 =
60
0 = 70
0 = 80
Fig. 3. Values of Taylor factor, M, for plane-strain deformation contours: from 2.2 to 4.6 by 0.3
previously defined texture softening factor
where M’ = M/cos 2a. Calculations were carried out at 2” intervals within the permissible angular range in a. For a given geometry, positive and negative simple shears were described by appropriate strain displacement tensors and the value of S was computed numerically for each shear by determining the rate of lattice rotation and the associated rate of change in the Taylor factor, i.e. dM -=-.-. dc
dR dM dE da
Lattice rotations were determined using a procedure described by Kallend and Davies [7]. The two svalues obtained in this way are then combined to give an average texture softening factor, s, for the particular shear angle for the crystal. The angle at which S was minimized not only defines the operative shear pair, but also gives the true value of texture softening factor for the crystal. The S-values determined for the crystals are given in Fig. 4. In regions of the orientation space where the
S-values are negative i.e. where the crystals texturally soften, contours are drawn as continuous lines. The values of the angle /l defining the geometry of the operative shear pair are given in Fig. 5. These results allow the following generalization to be made: 1. Based on texture softening considerations it can be concluded that the crystals always favour shears operating at /I < 45”, and moreover a correlation seems to exist between the Taylor factor M i.e. “hardness” and the shear angle /I, such that where the M-value is high, the crystals usually favour shears operating at /l<<45”. 2. Although there are some important exceptions, a similar correlation also holds with respect to the S-value, namely where the M-value is high, the S-value
t)ZTORK and DAVIES: TEXTURE SOFTENING AND STRAIN INSTABILITY found to be as much as 3 = -0.4. On the otherhand a range of other orientations exist at which the crystals are subject to the opposite effect of texture hardening. In these latter orientations the texture is a factor contributing to the stability of the deformation. Where there is pronounced texture softening, it is expected that limit strain L*, i.e. the strain level at which the instability initiates, will be reduced to a lower value. This strain level can be calculated with reference to the instability condition given in equation (4). This yields a limit strain of c: = n/{(n + 1)s) for plane-strain compression (C = 0), and CT = n/{(n + 1)s + I} for plane-strain tension (C = 1). Once these strain levels are reached the shears concentrate on macroscopic planes in the crystal, i.e. shear bands form. (This also corresponds to strain level at which necking initiates in planestrain tension.) The calculation of the limit strains as carried out above assumes that the S-value remains constant during the deformation. However, this is true only for some specific crystal orientations. For the majority,
lattice rotation which accompany the deformation alters the orientation and therefore the S-value. This continues until the crystal is rotated into an orientation stable under plane-strain. In f.c.c. metals orientations stable under plane strain constitutes a range between { llOj(112) and {112}(111). A range of orientations between {ll0}(112)and{110}(001)arealsonearlystable[8]. Figure 4 shows that all of these orientations are associated with slightly negative S-values. At (110}(001) $ =90”, e =45”, 4 =oc, s= -0.08 (/? = 40”), the softening increases slightly at {110}(112) $ = .54.7”, 0 = 45’, 4 = 0’ where s= -0.12 (p = 38”), it is further increased at {112) (11 l), I(/ = O”, 0 = 35.3”, 4 = 45”, for which s= -0.16 (p = 35”). This is in fact the order in which the M value increases in plane-strain. The numerical values given here are in good agreement with those reported by Dillamore et al. [I]. In order to test the validity of the predicted Svalues reference could be made to single crystal studies involving plane-strain deformation. However this is not a fruitful attempt. First of all shear bands
6 560 Fig. 4. Values of texture softening factor, s = (l/M) (dM/dc), (continuous):
-0.05,
-0.1,
-0.4;
2311
(dotted):
for plane-strain deformation. 0.0, 0.05, 0.1, 0.4.
Contours
2312
&TURK
and DAVIES:
TEXTURE
SOFTENING
AND STRAIN INSTABILITY
I9
QI- 60
cl
@=3!jZz.s
@ = fS22.5
= 40+
2.5
p=
3k2.5
Fig. 5. Values of the angle 8, defining the geometry of the operative shear pair in crystals deforming under plane-strain.
which would distinguished the basis that which cannot
form as a result of the instability is from other strain inhomogeneities on they cross grain boundaries, an aspect be observed in single crystals.?
tin plane-strain deformed single crystals of initial orientations such as (112) [l 1 l] there is an indirect evidence for the occurrence of strain inhomogeneity of the form under discussion here. According to Bauer et uf. [9] (112)[111] oriented crystal preserves its initial orientation until quite heavy reductions with some spread, but at heavier reductions it develops its complimentary {112}( 11l> orientation. The development of this complimentary orientation, which is frequently encountered in single crystal work [9, IO] is probably due to the formation of shear bands at heavy strains. While orientations such as (112) [I 1I] is stable under uniform strain i.e. simultaneous operation of the simple shears throughout the crystal, they become metastable when strain concentrates in shear bands. It was shown by Dillamore [ll] and &tiirk [12] that the manner of lattice rotations in shear bands are such that with little fluctuation, initially (112) [i 111oriented region rotate rapidly into its complimentary orientation.
In polycrystals subject to plane-strain deformation initially individual grains experience different degrees of texture softening or hardening depending on their orientation. But gradually the polycrystal develops the stable orientations mentioned above, and so it is subject to texture softening with an average S-value in the same order as those reported for the stable orientations. Dillamore et al. [l] considering an hypothetically sharp “pure metal” texture calculated an average softening factor of s = - 0.11. They further showed that shear banding as it occurs in heavily rolled f.c.c. metals were in good agreement with this prediction, moreover the angle of /3 = 35” at which the shear bands form in these metals were shown to correspond to the geometry at which the softening was maximized. With reasonably high strain hardening exponent n, the strain instability is postponed to relatively heavy strains, the level of which is expected to depend on whether the metal develops “pure metal” or “alloy” type texture [8]. Texture softening in metals that form
t)ZTURK
and DAVIES:
TEXTURE
SOFTENING
AND STRAIN INSTABILITY
2313
in thickness. Orientations developed are simple which can be described as { 110}(223) with some spread. The softening at { 110)(223) 1(1= 45”, 0 = 45”, 4 = O”, is about s N -0.1, even higher in parts of the spread close to {llO}(llO) $ = O”, 0 =45”, 4 = 0” where the softening is as high as s = -0.4. The structure of this metal was accordingly heavily banded as shown in Fig. 7. Shear bands in this structure occur on average at /3 = 33”. This is in good agreement with the predicted shear angle for {110}(223), see Fig. 5. 5.2. Texture softening
Fig. 6. 4 = 0” section of a.c.0.d.f. relating to a cross-rolled brass after 95% reduction in thickness. Contours: from 1.0 to 6.5 x random.
alloy type texture, i.e. low stacking fault energy metals is, however, outside the scope of the present paper. The S-values reported in Fig. 4 were calculated with the assumption of { 11 l}( 110) multiple slip alone and are not expected to be valid when mechanical twinning is an active deformation mode. Thus if discussion is confined to high-stacking fault energy metals, the texture softening may vary depending on the details of the pure metal texture developed. For instance if the texture consists of orientations near (112) (111) the metal is subject to more pronounced softening than would be the case for a more balanced case of uniform distribution of orientations between { 1 lO}( 112) and { 112)( 111). Thus the initial texture of the metal by favouring one stable orientation more than the others may influence the softening behaviour of the metal at heavy strains. In cross-rolling or in compression rolling, the deformation is still plane-strain compression, though the metals develop different textures. Figure 6 shows the texture of a cross-rolled brass after 95 reduction
in sheet forming
The role of the initial texture in influencing the softening behaviour of the metal is much more pronounced in sheet forming than it is for planestrain compression. This is partly due to the fact that plane-strain tension which is normally encountered in sheet forming mostly involve sheet metals that are strongly textured and, partly due to reduced strain levels at which the instability can initiate in tension. Cube textured metal sheets illustrate a good case of how the initial texture can influence the limit strains in plane-strain tension. The instability condition given in equation (4) shows that with no texture softening, i.e. S = 0, the instability occurs at ET = n. This approximately defines the limit strain when the cube textured sheet is subject to plane-strain tension along the original rolling direction (i.e. along {lOO}(OOl), $ = O’, 0 = O”, 4 = O”), see Fig. 4. However if instead the sheet is stretched 45” to the rolling direction (i.e. { lOO}( IlO), $ = 45”, 0 = O”, 4 = 0”), S = -0.4 and the limit strain is reduced to CT = n/{(n + 1)0.4 + 1). Thus with an hypothetically strong cube texture, even when the strain hardening exponent is as high as n = 0.5, the limit strain cannot be more than t: = 0.3. This is in agreement with the observations of Da Costa Viana [13] who found that cube textured copper sheets fail prematurely when
Fig. 7. The microstructure in a longitudinal section of a cross-rolled brass (rolled 90” to the rolling direction after each pass) after 95% reduction in thickness (the last rolling direction is horizontal). Shear bands occur on average at p z 33’.
2314
dZTURK and DAVIES: TEXTURE SOFTENING AND STRAIN INSTABILITY
tested in hydrolic bulging, the failure occurring in plane-strain, i.e. close to the clamped periphery of the sheet. 6. CONCLUDING REMARKS The analysis presented above indicates that the textural contribution to the rate of work wardening, as measured by texture softening factor S, is strongly orientation dependent. While certain textures promote instability through accompanying softening, the other result in hardening whereby contributing to the stability of deformation. Textures which normally develop in f.c.c. metals under plane-strain belong to the former, though the degree of softening associated with these textures is not very pronounced. The analysis further shows that the scope for instability control via texture is greater in plane-strain tension than it is for plane-strain compression. On this basis the extension of the present analysis to other strain states where tension is dominant, (e.g. states of biaxial strain encountered in sheet forming) would be highly beneficial. Acknowledgement-The authors would like to thank Professor I. L. Dillamore for invaluable discussion at various stages of this work.
REFERENCES 1. I. L. Dillamore, J. G. Robert and A. C. Bush, Me&Is sci. J. 12, 73 (1979). 2. B. J. Duggan, M. Hatherly, W. B. Hutchinson and P. T. Wakefield, Metals Sci. J. 12, 343 (1978). 3. T. Gztiirk, Text. Microstruct. 6, 39 (1983). 4. T. Haratani, W. B. Hutchinson, I. L. Dillamore and P. Bale, Metals Sci. J. 18, 51 (1984). 5. G. I. Taylor, J. Inst. Metals 62, 307 (1938). 6. J. F. W. Bishop and R. Hill, Phil. Mug. 42, 414, 1298 (1951). 7. J. S. Kallend and G. J. Davies, Phil. Mug. 7,361 (1972). 8. J. S. Kallend and G. J. Davies, Texture 1, (1972). 9. R. E. Batter, H. Mecking and K. Lticke, Muter. Sci. Engng. 27, 1631 (1977).
10. K. Brown and M. Hatherly, J. Inst. Metals 310, 98 (1970).
11. I. L. Dillamore and A. C. Bush, Texture of Materials (edited by C. Gottstein and K. Liicke), Vol. 1, p. 367. Springer, Berlin (1978). 12. T. Gztiirk, Ph.D. thesis, Univ. of Cambridge (1978). 13. C. S. Da Costa Viana, Ph.D. thesis, Univ. of Cambridge (1978). 14. G. J. Davies, D. J. Goodwill and J. S. Kallend, J. uppl Crystallogr. 4, 67 (1971).
APPENDIX Euler angles $ f3and I#Jrelate the crystallographic axes of the crystal to chosen axes in the sample material, e.g. normal direction, rolling direction, and transverse direction. Ideal orientation (hkl)[uvw] which is commonly used in describing the crystal orientation, correspond to Euler angles given by W
cm ti =
(II’+
(u2
+
y2 +
(h2
+
k2
w2)1/2
k* + 12)“*
(h* + k*)“*
1 ‘OS
e =
+
-h ~0s ti = (h2 + k2),,2 cos($ +d)=
/2)1,2
(h, k z 0)
(u2+u2u+w2)l,2
(h,k=O).
Here (hkl) refers to a crystallographic plane lying in the compression (rolling) plane of specimen and [uuw] is a crystallographic direction in this plane and parallel to the extension (rolling) direction. Alternatively, an ideal orientation can be found from known Euler angles using the relations h=-sinBcos4; u=cO~t,bOO~eOO~~
a=-cos+cos0sin#~ w = cos * sin e
k=sint?sin4; -sin$sin4 -sin$cos4
l=cos8
and rationalizing the resulting indices. In most cases, this analysis is unnecessarily time consuming and to facilitate the analysis a set of charts has been prepared for cubic materials [14]. These charts show the position in orientation space of all (hkl)[uvw] for cubic materials formed by permutating h, k, I, u, u, w through 3 to -3. The resulting Euler angles are taken within the ranges. o<$
~~12
o
0<4
which are adequate for a complete description of textures with cubic symmetry.