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Shear band post-bifurcation in oriented copper single crystals
Acta metall, mater. Vol. 42, No. 8, pp. 2763-2774, 1994
Pergamon
0956-7151(94)E0037-H
Copyright © 1994ElsevierScienceLtd Printed in Great Britain.All rights reserved 0956-7151/94 $7.00+ 0.00
SHEAR BAND POST-BIFURCATION IN ORIENTED C O P P E R SINGLE CRYSTALS S. YANG and C. REY LPMTM-CNRS, Universit6 Paris Nord, Av. J.-B. Clement, 93430 Villetaneuse, France (Received 20 July 1993; in revised form II November 1993)
Abstract--By in situ SEM observation performed in oriented copper single crystals strained in tension, the formation and propagation processes of macroscopic shear bands (MSB) as well as the consequences of their interactions are analysed. We found that an MSB consisted of a collection of parallel shear bands on slip lines scale. The observed lengthening and thickening of MSB were related to saturation of the individual shear bands and to nucleation of new ones in the adjacent matrix. In order to understand this phenomenon, a post-bifurcation model is applied to an idealized planar double slip single crystal. We show that shear banding is effectively characterized by the three observed stages: bifurcation, large shear strain localization and saturation. The bifurcation is accompanied by unloading in the matrix. The saturation is due to the fact that the geometrical softening leading to the bifurcation decreases with localized shear.
1. INTRODUCTION In tension tests, ductile single crystals exhibit deformation modes, mostly homogeneous on a sample scale. However, after a finite amount of plastic strain, some localized deformation modes such as necking followed by shear banding, leading to a drop of load, appear in the samples. Failure usually follows either by shear banding down to a "chisel edge" or by rupture within intense shear bands [1-5]. These phenomena depend on both macroscopic conditions (loading conditions, sample geometry) and microstructural properties of materials. The localization of the plastic flow in the thin bands with severe shearing, also appears in polycrystals during some metal forming processes, in particular during cold rolling process, which leads to modifications of some mechanical properties of materials and then to failure. Such consequences explain the great deal of experimental and theoretical studies on this subject, in single crystals and in polycrystals, deformed by rolling and by tension tests. Early studies have been summarized in several overviews [6-12]. In order to clarify the complex mechanisms of shear banding in polycrystals, it is essential to analyse the shear band bifurcation in more simple situation such as in single crystals. Many investigations at different scales (microscopic as well as macroscopic) have been performed on single crystals of pure metals and alloys [13-20]. These investigations have pointed out some physical characteristics of localization such as correlations between shear band formation and microstructure of dislocations, the relative orientation of the shear plane with respect to the activated slip planes, the evolution of the lattice orientation or the micro-hardness in the vicinity of the shear bands.
In mechanical modelling, shear band formation is viewed as a material constitutive instability, and thus, it depends strongly on material constitutive laws. In order to understand the physical origin of shear band formation, bifurcation models taking into account the crystallography have been proposed by different authors for single crystals [21-25]. The bifurcation conditions have been analysed as a function of slip geometry, applied stress field and strain hardening behaviour of materials. According to these models, the bifurcation is led by a geometrical softening induced by the lattice rotation accompanied with shear banding. More recently, different approaches of bifurcation phenomena, using finite element simulations have been developed in order to validate the bifurcation analysis under different boundary conditions applied to the single crystals and to the multicrystals [7, 8, 19, 20, 26]. But post-bifurcation behaviour in single crystals, especially shear band saturation phenomenon, has not yet been analytically studied. Note that a relevant study of post-bifurcation behaviour has been carried out by Hutchinson and Tvergaard [27]. The authors used a phenomenological corner theory of plasticity to represent a polycrystal, and studied the shear band formation by growth from a preexisting material band which presents a softer behaviour than the surrounding material. The saturation of shear localization is predicted when in the band, the stiffening effect associated with the ceases of the total loading is emphasized by a blunt comer or by suppressing the total loading. In single crystals, the corner exists in the loading point of the yield surface and has a physical meaning. Shear band formation can be studied by bifurcation and post-bifurcation analysis which permits a better understanding of basic physical origin of shear band formation on a grain scale.
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YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
This work has two main purposes: • To get an accurate description of the mechanisms of propagation and saturation of shear bands in oriented copper single crystals strained in tension, as well as the effects of the interactions between shear bands on the post-bifurcation behaviour. • To analyse the observed phenomena and especially the conditions of saturation of shear band, through a post-bifurcation model performed in an idealized planar double slip single crystal, able to take into account the correlated effects of strain hardening behaviour, as well as the applied stresses and the lattice rotation, i.e. the geometrical softening. The first section of this paper is devoted to the analysis of the mechanisms of propagation and inter-
Samples A B c D
Tensileaxis []'121 [T23] [T22] [01H
Table I Largeface (1TI)or (110) OTl) (0l-D (0i-1)
Max. load 50% 60% 60% 80%
conditions on shear band pattern was examined on samples A. The other tested samples had the same geometry (13 x 3.5 x 2 mm3). 2.2.2. Slip pattern and shear band pattern. Samples A: these samples were oriented in double slip B4 and C1 (according to the Schmid and Boas notation [29]), respectively. The two slip directions are parallel to (ITI) faces and symmetric with respect to the tensile axis. The two slip planes were activated homogeneaction of shear bands in copper singlecrystalsstrained ously from the beginning of the deformation. On in tension under an SEM. In order to realizejudicious (IT1) faces the angle between the tensile axis and situations for this analysis, several crystallographic the traces of the two activated slip planes is of orientations (particularly the []'12]orientation cor- about 30 deg. With such slip geometry, the thickness responding to symmetric double slip) and different in direction [1"i'1] of the samples remained constant samples geometry were studied. In the second section, during straining. One or two families of shear bands the two-dimensional post-bifurcationmodel is recalled were activated within the neck according to the and applied to the singlecrystalpresenting the double sample geometry. Samples B: the orientation corresponded to a sliporientation,which permits to predict the observed shear band formation, propagation and saturation. single slip activation. The activation of the primary slip system (B4) led to a lattice rotation around the [ITI] axis. The activation of the secondary slip system 2. EXPERIMENTAL RESULTS ((21) was observed at elongation approximately equal 2.1. Experimental procedure to 35%. Two families of shear bands symmetric with Oriented copper single crystals were elaborated by respect to the tensile axis appeared within the neck. the Bridgmann technique. The samples were cut by Then the behaviour of the samples was identical to sparking machine, then submitted to a mechanical the A samples. Samples C: the crystallographic initial orientations and electrolytic polishing. The crystallographic orientations were checked with the back-reflection corresponded to a coplanar double slip (B4, B5) Laue method. Before the shear band formation, (weak interaction). Only one trace of slip plane was fiducial grids of 5/~m steps were deposited, using a observed before necking. In the necking area, some microlithographic method [28], on the sample surface traces of a new crystallographic plane (identification in order to visualize the deformation induced by was very difficult) were observed. The gradient of strains and that of material rotations induced by the shear banding. The samples were deformed at room temperature necking were so high in these samples that it was by uniaxial tension in a Schenk apparatus or in a impossible to specify if the localization within the neck micro-tensile machine built in a Scanning Electron corresponded to shear bands or to coarse slip bands. Samples D: the tensile axis was very close to [011]. Microscope (SEM) with a deformation rate of 10 -4 s -1. The surfaces of the samples were observed at The deformation of such samples was heterogeneous, with two families of slip lines corresponding to the different stages of deformation by SEM. planes A and B, localized in different areas of the 2.2. Experimental results samples respectively. Large gradient of strains and that In order to determine the role of the initial of material rotations were observed. Two families of orientation on the threshold of strain localization shear bands appeared within the neck. The traces and on shear band pattern, different samples were of shear planes were slightly different from the traces tested with various initial crystallographic orientations of the activated slip systems. 2.2.3. Effect of the sample geometry on shear band named A, B, C and D, respectively. By using different sample geometry we have obtained particular situ- pattern. Different geometry of the samples A was ations allowing to study the mechanisms of shear tested: band propagation and interaction. L--15mm, l=2mm, e = 6 . 5 m m , I/L=O.I 2.2.1. Threshold of localization. In all studied L=13mm, l=3.5mm, e=2mm, 1/L=0.3 samples, shear bands appeared within the neck. The elongation values corresponding to the maximum load L - - 1 5 m m , l = 6 . 5 m m , e = 6 . 5 m m , 1/L=0.4 of the tensile curve are given in Table 1. The effect of L=25mm, l=15mm, e = 2 . 5 m m , ILL=0.6 the sample geometry which is related to the boundary
YANG and REY:
SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
2765
where l denotes the initial width of the samples, L is the initial length and e is the initial thickness. Two symmetric families of shear bands were observed for l/L <~0.4, but only one for l/L = 0.6. This phenomenon was probably due to the structure effects associated with the clamping of the tensile machines. From observations carried out on these two kinds of samples (l/L = 0.6 and I/L <. 0.4), we analysed the mechanisms of formation and propagation of a macroscopic shear band (MSB) on a sample scale, as well as interactions between them.
2.2.4. Mechanisms of formation and propagation of an MSB. The following described mechanisms, analysed on the large samples ( l / L = 0.6) were identical for all tested single crystals. As in all samples, the localization of plastic flow in an MSB took place within the neck, where the distribution of the two activated slip systems B4 and C1 was homogeneous. As already observed by Dubois [30, 31], the MSB was composed, on slip lines scale, of a collection of short segments of shear bands (about 50/tm of length) parallel to each other, with a spacing of about 0.5 #m. The thickness of shear bands was not identifiable by SEM. The MSB was limited by two approximately parallel interfaces named S and U respectively (Fig. 1). The interface S was stable, independent of
Fig. 2. Nucleation of new shear bands at the instable interface of the MSB. increasing straining. On the contrary, the interface U was unstable, which lengthened and spread out, leading to the lengthening and the thickening of the MSB. The interface S corresponded to the first shear bands. The propagation of the interface U is due to activation of new short shear bands in the adjacent matrix. These new short shear bands constituted the interface U and so was parallel to the latter (Fig. 2). The in situ observation showed that the deformation was mostly localized in these new shear bands, whereas the old ones saturated. This explains that the interface S was macroscopically less well-defined than the interface U. When the MSB was confined in the sample, the directions of the shearing, of the thickening and of the lengthening were not independent. The thickening took place always beside the interface at which the shearing direction was opposite to the lengthening direction (Fig. 3). Figure 4 showed the large localized shear within the MSB in early stage, which had reached a value of about 1.2, whereas the matrix remained almost undeformed. With increasing straining, the localized shear within the MSB increased slowly, as can be remarked by contrasting the unstable interface in Fig. l(a) and (b). As concerns the matrix strain, even though it is small compared with the localized shear, it increased with the band widening, pointed out by the deformation of the fiducial grids (Fig. 2). Shear
Lea
enin
Shear Fig. 1. (a) MSB limited between the stable interface and the unstable interface; (b) thickening of the MSB and propagation of a crack along the unstable interface.
Fig. 3. Sketch of the propagation directions of the MSB.
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YANG and REY:
SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
strain and lattice rotation gradient induced by the necking. 2.2.5. Interaction between MSBs. The samples with a low IlL value presented two families of MSBs, each of them slightly misoriented vs the traces of the two activated slip planes. It is worth noting that, the orientation of the traces of the two activated slip planes before the shear band formation was nonuniform within the neck, the rotation can reach a maximum value of about 10 deg from the centre to the sides of the samples. The sequences of propagation of these shear bands are sketched in Fig. 5. The main points may be summarized up as follows: • Nucleation from the sides within the neck of four MSBs (named E, F, G and H respectively) at about 45 deg with respect to the tensile axis. The MSBs E and F, G and H, presented respectively a same shear plane and a same shear direction but opposite directions of propagation. • The four MSBs grew in thickness and in length by nucleation of new short shear bands, as described here above. The magnitude of the localized shear was not uniform along these MSBs. • The four bands never reached the centre of the sample but they induced in this spot a strain twice larger than in the remaining part of the matrix and sufficient to fulfil the conditions of activation of four new MSBs named E', F', G' and H' respectively. E' and F', G' and H', had the same shear plane and
Fig. 4. Shear localization in the MSB initiating from the side
within the neck. The formation and the propagation of the MSB can be described as follows: the first shear bands grew from the side of the sample within the neck and then constituted the MSB. The MSB propagated through consecutive activation of new shear bands in the adjacent matrix and saturation of old ones in such a way that the thickness of the MSB increased with increasing straining and can reach some millimetres before rupture. It is worth noting that the shear bands were slightly misoriented with respect to one family of the activated slip systems, and that the unstable interface U, constituted by the new short shear bands, presented a curvature, which is related to the
(a)
(b)
(e)
(d)
(c)
°
E
i i (f)
/ H'
F'
Fig. 5. (a-e) Sketch of propagation and interaction processes of the M SBs in the [TI 2] tensile tested single crystal with increasing straining. (f) Details of (d) at the intersections of the MSBs.
G
YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
2767
the same shear direction, but opposite directions of propagation, respectively. A large gradient of strains and material rotations was observed at the intersection of these MSBs. We observed that gradually, the former MSBs E, F, G and H became inactive. • The very pronounced neck was then divided into two regions, the new MSBs E', F', G' and H', and the matrix which corresponded to the no-sheared areas. With increasing straining, the extent of these no-sheared areas decreased progressively whereas the MSBs E', F', G' and H' grew in thickness as shown in Figs. 5(c,d). It can be noticed that close to the bands, the strains were much larger than elsewhere (Fig. 6). • The active shear bands E' and G', F' and H' intersected each other in M and N respectively [Fig. 5(f)]. At higher strain, the distance MN increased [Fig. 6(a--c)], and a triangular crack appeared in M from the centre of the samples, propagated by tearing along shear bands E' and G', and finally induced the rupture of the samples. 2.2.6. Fracture. The discussion will be brief since this is not the main focus of this paper. As already noted by Chang and Asaro [13] and by Drve et al. [19], two modes of fracture were observed. In the case where the MSB initiated from the side within the neck, a crack initiated and grew steadily along the unstable interface of the MSB (Fig. 1), by alternate shearing of the MSB and important slip concentrated at the tip of the crack. The final fracture occurred almost by complete shearing off of the localized band at the unstable interface. However, in the case where the four symmetric MSBs initiated from the centre and grew to the sides, any crack was not observed at the intersections of the MSBs and the sides. Instead, at final stage, a triangular cavity formed at the intersection of two MSBs (a "diamond" shaped cavity was observed by DOve et al. [19]). It should be noted that in the latter case, the alternate shearing of the MSBs by one crossing another was not observed. As a consequence, at the intersection of the MSBs, a boundary parallel to the tensile axis which separated the MSBs was clearly observed [Fig. 6(b,c)]. At this boundary, large internal stresses could be induced, as illustrated by Yang [32]. The large internal stresses may explain the cavity formation at the intersection. But why not in "diamond" shape? This may be probably due to the increased distance of MN in Fig. 5(f), thus the shear band pattern was dissymmetric at the intersections. This triangular cavity did not grow by sliding off along the instable interfaces of the MSBs. The explanation is as follows: according to the growth mechanism of the crack in Fig. 1, an alternate shearing is necessary for a crack to grow along the unstable interface of MSB. When the triangular cavity formed, at the tips of the cavity, an alternate shearing by slip across the
Fig. 6 (a-c).
Fig. 6. (a,b) Micrographs of the propagation and interaction of the MSBs in the [-1-12]tensile tested single crystal with increasing straining. (c) Formation of a crack at the intersection of the MSBs.
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YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
MSB would be difficult to operate, while there was saturation of old shear bands, thus the MSB could not propagate by thickening like that in Fig. 1. 2.3. Discussion on the mechanisms o f shear banding
Different points must be stressed upon the MSB formation and propagation. These mechanisms are clearly the consequence of coupled effects: boundary conditions (sample geometry, effects of the clamping), initial crystallographic orientations, and also strain hardening behaviour due to the activated slip systems before localization. In these conditions, the resulting stress field could not be assumed to be a uniaxial tension, nevertheless whatever was the stress field, the mechanisms of shear banding present many common features. It is worth noting that in all cases studied, the activated shear bands were roughly parallel to one of the activated slip planes, and the MSB propagation mechanisms across the samples were the same. Concerning the initial crystallographic orientations (all other conditions being identical), they affect only the threshold of localization, as already shown by Gasp6fini and Rey [33], the measured elongation corresponding to the neck formation, presented weak values when initial orientation of the tensile axis was close to a "hard" double slip position (formation of Lormer-Cottrell locks). The problem of the shear localization is more complex because both boundary conditions and initial crystallographic orientation ruled the lattice rotation gradient in the sample and consequently the conditions of shear band bifurcation. The mechanisms of propagation of an MSB could be summed up as follows: • Some collections of shear bands appeared in the most deformed areas of the necking region and constituted the MSBs with increasing straining. The MSB was always limited by two interfaces: one stable and the other propagating by activation of new shear bands in the adjacent matrix. This led to the thickening and lengthening of the macroscopic band. The direction of propagation by thickening and that of lengthening were strongly correlated. In the case where the MSBs initiated from the sides, new shear bands nucleated beside the unstable interface but not beside the two interfaces limiting the MSB. This may be attributed to the accompanied crack formation and growth, which could induce strain concentration near the tip of the crack, thus led to the alternate shearing by the MSB and by important slip concentrated at the tip, and therefore the MSB thickened at one side. In the case where four symmetric MSBs initiated at the centre of the sample, it has been shown [32] that the internal stresses induced by the four MSBs have effect to favour shear band nucleation in the observed directions, and to deactivate shear band nucleation at another side of the MSBs. • Activation conditions of the shear bands were very sensible to the lattice rotation and the resulting MSB was generally curved.
• Observation of the deformation of the fiducial grids pointed out the following phenomena: firstly, during the early stage of the MSB formation, the magnitude of the localized shear was not uniform along these bands, such inhomogeneities, according to Mussot et al. [34], induced internal stresses which would be locally accommodated by activation of some slip lines at the head of the MSB and so would lead to the lengthening of the MSB. A more or less uniform localized shear was observed only when the MSB was well-developed. Secondly, the MSB presented two different zones, one beside the stable interface, another beside the unstable interface. This latter zone, whose thickness (about 5 gm) can be considered as constant independent of increasing straining, was constituted by the new activated shear bands. On the contrary, the former zone was constituted by the deactivated shear bands, its thickness grew with increasing straining. Thirdly, four MSBs could induce a local but large strain field at their vicinity as observed in samples A. Finally, during the propagation of the MSBs, the matrix far from the bands had a rigid behaviour. Experiments on samples A showed that a change of the sample geometry could modify the number and the distribution of MSBs. It was pointed out that the MSBs induced a large range of internal stresses probably of the same order of the applied stresses, able to create new MSBs. Interactions between MSBs were strong and led to heterogeneous shear band patterns. The propagation mechanisms of an MSB is in fact, according to the aforementioned experimental observations, due to the activation and the saturation of the individual shear bands. The behaviour of the individual shear bands was characterized by their bifurcation, the large shear strain localization and the saturation. It depends essentially on the local behaviour of the crystal. In order to have a more accurate description of the parameters which rule the shear banding, a theoretical study is hereafter proposed. In the next section, we propose an analysis of the propagation mechanism, in particular of the saturation of a shear band, in terms of slip geometry, crystal strain hardening behaviour and loading conditions, by using a two-dimensional single crystal model which is sufficiently representative of the here above studied [1"12] single crystals.
3. POST-BIFURCATION MODEL 3. I. Slip geometry and notations
The geometry of the idealized planar double slip single crystal is sketched in Fig. 7. The crystal is assumed to be deformed according to two slip systems characterized by their slip directions s 1 and s 2 and their slip plane normals ml and m2, respectively, tp is the angle between s I (s2) and the 2-axis (~p will be
YANG and REY:
SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS AD=-~(g®n+n®g)=~
2769
p, i - P r o ) P ) ,
(2)
i
Ca)
x
where
~n 'l x.~" ~'a m2
p,=½(si®m,+mi®s,).
(3)
p~ and p~ may be calculated as a function of tp. Thus ~lb and ~ in the band can be deduced from (2) as
I',,
"
2~__
2 [-"
' 1\
'
~ =~
L.e,
1
[J. sin 2(0 + ~o - ~)
+ ~ sin 2(29 - ~t) + ~ sin 2~t]
(4)
, ',,
~ = ~ /V'/~.~
" ~Nj~
(b)
/J
[~. sin 2(0 - ~o - e) - )~ sin 2e
+ ~p sin 2(2¢p + e)].
~
At the interface, the discontinuity A ~ of the total spin is the sum of the jumps in the plastic and the lattice spins, Aff~p and A~*, respectively. Thus
(e)
Fig. 7. (a) The planar double slip single crystal model; (b) a shear band forms at plane normal to [] and in direction g and (c) its processing scheme.
Af2*= ~ ( g ® n - n ® g ) - ~
(w~/b- wm@m), (5)
where taken to be 30 ° to represent the slip geometry of the [-1-12]single crystal in tension). The interface between the band and the matrix is denoted by the shear direction g or the normal n to the shear plane in the current configuration (n and g are unit vectors). The deformation in the band and in the matrix during shear banding is assumed to be uniform respectively. For a given localized shear 2, 0 is the angle between the interface and the 2-axis (00 being the initial angle at the beginning of shear banding), ct is the lattice misorientation between the band and the matrix.
un rs an
va=s
ta os of o =ation
by shear banding in the single crystal, the crystallographic slips y~ and y: on the two activated slip systems are calculated inside the band and inside the matrix (indexed later by superscripts b and m, respectively) as a function of 2. These four values are analytically deduced from the following shear banding kinematic conditions, static conditions and strain hardening behaviour as well as loading conditions,
3.2. Constitutive law 3.2.1. Kinematics of shear banding. The crystal is assumed to be rigid-plastic (the elastic strains are neglected, but the lattice rigid-body rotation is taken into account). All the manipulations are carried out in the co-rotational (g, n)-coordinate system, Shear banding is described at every moment by the discontinuity AL in velocity gradient at the interface band/matrix AL = ).g®n,
(1)
where A denotes a jump, ~. is the localized shear rate. The symmetric part, i.e. the discontinuity AD of the strain rate is
w~ = ½(s~®m ~ - mi® s~). The lattice misofientation rate is given by -- g. Aft*. n.
(6) (7)
We obtain then from (7), (5), (6) and (4) ~F cos2(0 _ ~t)O ~ = 2 L1 c~ss2-~ l{I cosc-~s 2(~02~-- 0C)]'md + 1 ])/1
J
-
I
1
1}
coscos 2(~o2tp + e) ~p .
(8)
The rotation of the shear plane with respect to the (1, 2)-coordinate system depending only on matrix velocity gradient, one can easily deduce that 0 = s i n 2 ( 0 - q ~ ) ~ - s i n 2 ( 0 + ~0)~. (9)
3.2.2. Statics of shear banding. At the interface band/matrix, the equilibrium equation, for an incompressible material in two-dimensional case, can be written as A~ = 0 (10) where ~ , = g - a .n, representing the resolved shear stress on the shear band. The rates of the resolved shear stress on the two sides of the interface can be expressed as follows ~ . zb = g- ( V, b - ~rn~)(flg~) b ~ b a ) b • n - (~rgg
(11) v, m
[-z~ = g ( a )
m
m
* m
-n - (t~g - an~)(flg~) ,
where (~,)b and (v*) m are the rates of the Cauchy stress tensor in the co-rotational coordinate system bound to the lattice in the band and to that in the
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YANG and REY: SHEAR BAND POST-BIFORCATION IN COPPER CRYSTALS
matrix, respectively; (Q.)b, (~r'~n)m are the lattice rotation rates in the (g, n)-coordinate system. 3.2.3. Hardening matrix. We assume that the crystal obeys the Schmid's law. The elastic part of the strain rate D * being neglected, the rate of the resolved shear stress on the (i) slip system can be written as zi = s," v , . mj.
equation (10) can be written in terms of the resolved shear stress rate on the inactive slip system A ~ ? + B2"t~ = C.
(17)
For the case where the system (2) is active and the system (1) is inactive, similarly we obtain B~f~n + A:~)~ = C
(12)
(18)
where
{
The (i) slip system is active if and only if
{~,
A~ = A - hmq~B2 A 2 = B - hmqmBt
= ~
~, = ~
(13)
(19)
and
where z~ is the critical shear stress, z~ can be expressed as a function of a symmetric latent hardening matrix, whose components h 0 are positive, namely
"~ = h,j~j
(14)
fnl
=
sin sin 2(04(p + cp) (20)
sin 2(0 - ~0) 2= sin 4~
where we take h e to have elements h H = h22 = h and hi2 = h21 = hq, with h > 0 and q > 0. 3.3. Calculation o f slips inside the band and inside the matrix
The experimental observations showed that, during shear banding, the major amount of plastic deformation was localized inside the band, whereas the contribution of the matrix was weak. So, we assume, in the following, that two slip systems are active in the band. But for the matrix, all cases will be analysed: case 1--two active slip systems, case 2---only one active slip system, case 3--no active slip system, i.e. a rigid matrix. Three equations will be hereafter set up corresponding respectively to the above three cases. They are deduced from equilibrium equation (1 0) and expressions (l l) as well as all the aforementioned kinematic equations. Case 1: both slip systems (1) and (2) are active in the matrix. In this case, the increments of the resolved
shear stress on the two slip systems, in the band and in the matrix, can be expressed by (14). We obtain thus a linear equation in terms of ~? and ~ A ~ + B~n = C
(15)
Case 3: rigid matrix. In this case, the equilibrium equation (10) can be expressed in terms of the resolved shear stress rates on the two inactive slip systems in the matrix "~gn = Bt ~T + B2z~n = C.
(21)
In all cases, the parameter C is given by equation (16) which corresponds, when the matrix is rigid, to the resolved shear stress rate fg, on the interface inside the band (normalized by the localized shear rate 2). The sign of C depends on relative weight of strain hardening and geometrical softening given by a,~)ct, with & being the lattice spin in the band in rigid matrix case. 3.4. Bifurcation conditions
Let us assume that shear band bifurcation occurs with the two slip systems in the matrix being still active, and that at the very moment of the bifurcation, no discontinuity exists in the crystal, hence we have: ~t = 0, Aa = 0 and Ah0 = 0. Then in equation (15) we have: A = B = 0 and C = Co with Co = l h { ( 1 +q)~fi~-~)/sin 20°'X2
with C
=
g.(~,)b.n 1 [1 2L
-
( ~ =b -
. / c o s 200'~2~ +(1-q,~) ;
a . .b)
cos 2(0-~OS 2 ~ CQ]
(16)
where ~ , $~ and v, are normalized by ).. In the following, all incremental variables are normalized by the localized shear rate 2. The expressions of A, B and C are given in Appendix. These parameters depend on slip geometry, lattice misorientation ct and orientation 0 of the band as well as stress field and hardening moduli in the band and in the matrix. Case 2: one active slip system in the matrix. Consider the case in which the slip system (1) is active and the slip system (2) is inactive. For this case, equilibrium
1(1 -- (a~, - an,) "~-
c°s 20°'~ cos 2~o/
(22)
and equation (15) gives the bifurcation condition Co=0. The condition Co = 0 added to the condition dCo/dOo = 0 allows us to determine initial orientation 00 of critical shear band. In case of an uniaxial tension parallel to the 2-axis, equation (22) and the condition Co = 0 permit to determine the critical value (h/~r),et :for shear band bifurcation with initial orientation 00, where ~ represents the tensile stress. This is the same bifurcation conditions as Asaro's [22].
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SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
2771
F o r e < n/4, the lattice rotation ct presents, when 2 tends to be infinite, an asymptotic value equal to 8.0 (0 - e ) , the condition (24) i.e. ~t ~< (0 - e ) holding for ). ---,oo. The computed ratio of the crystallographic .i~-. 6 . 0 slip rates increases with shear banding. This means that the slip plane (1) in the shear band rotates with "~ 4.O 0=40* 1: ~=30* the lattice towards the shear plane. In this case, a 2: ¢--60" saturation of the shear band cannot be kinematically 2. o 2: " ~ , / ~ predicted. 0.0 I5 I0 15 3.5.2. Statics. To have a better understanding of O.O O. 1 . 1 . .O l shear band development in the case e < n / 4 , let Fig.8. The latticerotation~ and the ratio~/9~ of the slip us calculate under a rigid matrix assumption, the rates insidethe shear band 8 = 40° as a function of the variation 6z~ of the resolved shear stress on the interlocalizedshear )~for two differentcases: h ¢p = 30° and 2: face as a function of 2, for different strain hardening = 60°. behaviour. According to equation (21), we have 10.0
2: O.
3.5. Post-bifurcation analysis It will be shown in the following that shear band bifurcation can be accompanied by a rigid unloading of the matrix. This solution of the problem, compared with the homogeneous deformation mode and other post-bifurcation modes, is the most favourable deformation mode, and characterizes the observed shear band formation. 3.5.1. Kinematics. Assuming that the matrix is rigid during shear banding [ ~ = ~,~ = 0, and so 0 = 00 according to (9)], we have calculated the ratio of the shear rate 9tb/~b and the lattice rotation ~t in the band as a function of 2, for different values of e and 0, from equations (4) and (8) respectively. ~ and @2 b in equation (4) must be positive, thus we obtain
( 0 e [ 2 - e' q~l
for e > re/4
(23)
for e < n/4.
(24)
~ e,-~- e
e[o,0 - e l In rigid matrix case, the lattice rotation e in the band can be calculated by integration of (8), assumin.g that e = 0 at the beginning of shear banding (,~ = 0). The main results are shown in Fig. 8. The analysis of the computed curves shows that two situations may occur: For e > ~ / 4 , the condition (23) implies a ~<0 -(n/2-e), ct reaching its maximum value for = 2m~x, given by 1 , ~tg0+tge 2 ~ = ~ - ~ - ~ ,n ~ ~g ~ - t - ~ .
1-tg2e~ l - ~ ~gg2; j .
(25)
This result points out a limited localized shear in the hand for a given slip geometry, independent of applied stress field.
6Zgn = BI c~z~ + B25~ ~ =
S;
C d2
(26)
where 5 represents a variation computed from the bifurcation moment, B~ and B2 are positive according to (20), (23) and (24). After the shear band bifurcation, if the parameter C becomes negative, according to (26), the matrix will be unloaded because of negative value of 5zg. and positive values of B t and B 2. It has been noted previously, according to (16), that the parameter C depends on the relative importance of strain hardening, the lattice spin and stress level. So, one can expect that, with shear banding and under some conditions (which will be hereafter analysed), the sign of C would change so that the matrix would be reloaded and the assumption of rigid matrix might be wrong. From (26), one can deduce that when 6zg~ becomes positive, the rigid matrix assumption no longer holds, because 6zT and 6z~ must be negative if the matrix is rigid. The evolution 6zg, with the localized shear 2 have been studied in two cases: • firstly, the strain hardening moduli are supposed to be the same in the band and in the matrix and to stay constant with increasing straining, i.e. h ~ = h ~ = const; • secondly, the strain hardening moduli are assumed to decrease with increasing straining according to the law
dh = - k h dy
(27)
where k is a material constant (k/> 0), and 7 = 7~ + 72. This law describes a linear relation between the strain hardening moduli and the resolved shear stress, which corresponds to ductile single crystal strain hardening properties at stage III [18, 35-38]. We note that the first case corresponds to k = 0 in (27). In the second case, the hardening moduli in the band are different from that in the matrix after the bifurcation. Our results are summarized in Figs 9 and 10. In the first case, the rigid matrix assumption which corresponds to 6Zgn< 0 is fulfilled only for 0 > 36 °. We notice that 5zg, reaches positive values for 2 > 2r.
2772
YANG and REY:
SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
'/,:,,y
0.03
,.....
,
0.25
0,2
o°°', o. I " J ff
,
,
,
,
,
,
@=30*,O=400, k=O.5, q = l . 4
I
!
0.15
................................ ! 0.i
~' -0.01 -0.02
0.05
",
-0.03
,'"'" i 1
i 2
~= 30°,k=O,q = IA i 3
i 4
I
I
I
I
I
0.5
1
1.5
2
2.5
I, 3
3.5
4
t Fig. 9. The variation 6T_ • ip~ of the resolved shear •stress . as a function of the locahzed shear 2. The calculaaon is performed for different kinematically admissible shear band orientation 0, where the matrix is assumed to be rigid and k = 0, ~0 represents the tensile stress at the bifurcation moment. This means that, beyond this localized shear, the assumption of rigid matrix no longer holds. It can be noticed that 2r strongly depends on strain hardening behaviour, namely • for k = 0, 2, depends on 0; this is due to the fact that each angle 0 corresponds to a different (h/tr)crit value. • for k > 0, the computations point out that a large value of k, i.e. an important microstructural softening, induces an important value of hr" Let us now study the case when 6z~ becomes positive due to a positive value of C, which corresponds to that the plastic flow takes up again in the matrix. To determine ~ , ~ , ylb and ~2b, the equilibrium equation and the two kinematic equations are not sufficient, the loading conditions must be taken into account. Assuming that the matrix is submitted to an uniaxial stress parallel to the 2-axis (i.e. z'~ = zT, thus ~ ' = ~ ) , ~ and ~ may be computed as function of 2, for different value of ~p, 0, q and k. The results are given in Fig. 11. In this figure, one can see that ~,~ (or 7~') increases sharply and has an asymptotic axis which corresponds to 2 = ~ . The slip ~ ( o r 7~) grows faster 0,03
,
0.02
, i / ,
¢p: 300, O = 400, q = 1.4
,
/
o ~"
........... i 0
..............................
-0.01 k=l
-0.02
"" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-0.03
I 1
I 2
I 3
I 4
5
Fig. 10. The variation 6x~ of the resolved shear stress in the critical shear band (0 = 40°) as a function of the localized shear 2, for different values of k, where the matrix is assumed to be rigid, tr0 represents the tensile stress at the bifurcation moment.
Fig. 1I. The crystallographic slip ?T (or ~ ) in the matrix as a function of the localized shear 2 during shear banding.
than the localized shear 2 in the band, meaning that the shear band will rapidly become inactive when the matrix restarts to be deformed. 4. CONCLUSION We have investigated the mechanisms of deformation by shear banding in ductile copper single crystals subject to tensile straining• In all tested single crystals which presented different initial crystallographic orientations, the mechanisms of propagation of an MSB were identical: a collection of short shear bands, initiated in the most deformed area within the neck, then they constituted the MSB; the latter grew by thickening and by lengthening through consecutive activation of new short parallel shear bands and saturation of old ones. The propagation mechanisms of the MSBs were in fact a reflection of that of the individual shear bands, the latter having the same orientation as the unstable interface of the MSBs. Attention was thus focused to model one shear band locally, in particular, to model its bifurcation, the large shear strain localization and its saturation. In order to model these phenomena, postbifurcation analysis for a shear band was performed. We began for a sake of simplicity, from Asaro's idealized double slip single crystal strained in tension. Although two-dimensional, this crystal model is well illustrated by the []'12] single crystal subject to tensile straining. The latter presented at (1T1) faces, two symmetric slip systems simultaneously activated (~p = 30 °) with respect to the tensile axis, and two symmetric families of MSBs, at 0 ~ 40 ° near the centre of the samples. The post-bifurcation analysis deals with a material element, where the deformation is assumed uniform in the band and outside the band respectively. The calculations are performed under the hypothesis that the uniaxial tension is always applied on this material element. This model is able to predict the observed mechanisms of shear banding: shear band bifurcation (when the necessary and sufficient condition is fulfilled) then development with all strain concentrated in the band (rigid matrix) and then restart of plastic flow in the matrix and saturation of the shear band. Thus one
YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
2773
can deduce that, because of the restart of the plastic the influence of different parameters, such as slip flow in the matrix, some new shear bands will be geometry and strain hardening properties, were qualiactivated in the matrix when new bifurcation con- tatively valid. To have a good quantitative prediction, ditions are fulfilled there. Such sequence depends it is necessary to have a better but tractable strain on the slip geometry and the evolution of the strain hardening law for finite deformations, to take into hardening moduli with increasing straining. It is worth account the strain gradient within the neck and the recalling from the in situ observation that nucleation internal stresses induced by the MSBs, as well as rate of new shear bands appeared in the adjacent matrix sensitivity. The latter have significant retardation and near the saturated bands. The model, treating two stabilizing effects on shear band formation [8, 39, 40]. uniformly strained regions, the band and the matrix, can not predict that. But taking into account the REFERENCES nonuniform strain state induced by the necking and the internal stresses induced by the MSBs as well as I. W. F. Hosford, R. L. Fleisher and W. A. Backofen, the stress concentration induced by the crack formActa metall. 8, 187 (1960). 2. S. Saimoto, W. F. Hosford and W. A. Backofen, Phil. ation, one can understand that. Qualitatively, accordMag. 21, 319 (1965). ing to the model, the first shear bands tend to initiate 3. C.N. Reid, A. Gilbert and G. T. Hahn, Acta matall. 14, at some the most deformed regions within the neck, 975 (1966). e.g. preferentially at the sides. Because of the 4. B. Pegel, Scipta metall. 13, 47 (1980). 5. W. A. Spitzig, Acta metall. 29, 1359 (1981). nonuniform strain state, the internal stresses and 6. J. G. Sevillano, P. V. Houte and E. Aernoudt, Prog. the stress concentration, the restarted strain will be Metal Sci. 25, 69 (1981). larger in the adjacent matrix and in one side of MSB 7. D. Peirce, R. J. Asaro and A. Needleman, Acta metall. than else. Thus new shear bands tend to nucleate 30, 1087 (1982). there. 8. D. Peirce, R. J. Asaro and A. Needleman, Acta metall. 31, 1951 (1983). We show, by analysing the kinematics of shear 9. R. J. Asaro and A. Needleman, Scripta metall. 18, 429 banding, that one slip plane in the band progressively (1984). tends to parallel to the shear band, and thus the 10. M. Hatberly and A. S. Malin, Scripta metall. 18, 449 localized shearing is essentially produced by this slip (1984). system. In consequence, the "geometrical" softening l 1. L. Anand, Scripta metall. 18, 423 (1984). 12. G. R. Canova, U. F. Kocks and M. G. Strut, Scripta will become negligible at large localized shearing, and metall. 18, 437 (1984). thus the matrix will start to be reloaded, and then to 13. Y. W. Chang and R. J. Asaro, Acta metall. 29, 241 be redeformed leading to the shear band saturation. (1980). But if the k value in (27) is greater than some critical 14. K. Morii and Y. Nakayama, Trans. Japan. Inst. Metals 22, 851 (1981). value depending on crystallographic slip geometry, 15. Y. Nakayama and K. Morii, Trans. Japan. Inst. Metals the hardening moduli in the band will decrease 23, 422 (1982). rapidly and tend to be zero with the localized shear 16. K. Morii and Y. Nakayama, Scripta metall. 19, 185 strain, so that although the "geometrical" softening (1985). is negligible at large localized shearing, the matrix will 17. K. Morii, H. Mecking and Y. Nakayama, Acta metall. 33, 379 (1985). not be enough reloaded to be deformed plastically. In this case, the localized shearing is theoretically 18. Z. Jasienski, Mater. Tech., p. 91 (1985). 19. H. Dfve, S. Harren, C. Mccullough and R. J. Asaro, infinite. Acta metall. 36, 341 (1988). It is worth noting that the sign of the C parameter 20. S. Harren, H. Dfve and R. J. Asaro, Acta metall. 36, 2435 (1988). controls this analysis, which corresponds in rigid matrix case to the resolved shear stress rate on the 21. R. J. Asaro and J. R. Rice, J. Mech. Phys. Solids 25, 309 (1977). interface band/matrix, normalized by the localized 22. R. J. Asaro, Acta metall. 27, 445 (1979). shear rate. The sign of C depends on the relative con- 23. R. J. Asaro, J. appl. Mech. 50, 921 (1983). tribution of two terms given by (16). The first term 24. R. J. Asaro, Adv. appl. Mech. 23, 1 (1983). corresponds to the instantaneous response to the 25. D. Peirce, J. Mech. Phys. Solids 31, 133 (1983). localized shearing (strain hardening of the material), 26. S. V. Harren and R. J. Asaro, J. appl. Mech. 37, 191 (1989). the second term depends on the lattice spin in the 27. J. W. Hutchinson and V. Tvergaard, Int. J. Solids band (geometrical softening). A more detailed analysis Struct. 17, 451 (1981). 28. T. Bretheau and D. Caldemaison, Proc. 4th Ris~ Int. of this model is given by Yang [32]. Syrup. (1983). It must be indicated that, the magnitude of the 29.. E. Schmid and N. Boas, Kristal Plastititiit, Berlin, table localized shear predicted by this model is valid only VI (1962). when the neck is not pronounced before shear band 30. P. Dubois, Ph.D. thesis, Univ. of Paris XIII (1987). formation. In copper single crystals, because of the 31. P. Dubois, M. Gaspfrini, C. Rey and A. Zaoui, Arch. Mech. 40, 25 (1988). deep neck before shear band formation, it is not interesting to compare the localized shear predicted 32. S. Yang, Ph.D. thesis, Univ. of Paris XIII (1990). 33. M. Gaspfrini and C. Rey, MECAMAT'91, p. 229 by this post-bifurcation model with the experimental (1991). one. Nevertheless, we think that the predicted various 34. P. Mussot, C. Rey and A. Zaoui, Res. Mech. 14, 69 (1985). stages of deformation of shear banding as well as AM 4 2 / g ~
2774
YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
35. H. Meeking, Working Hardening in Tension and Fatigue (edited by A. W. Thompson), p. 67. Metals Soc. AIME (1977). 36. B. Nicolas and H. Mecking, ICSMA-5, p. 351. Pergamon Press, Oxford (1979). 37. P. N. B. Anongba, Ph. D. thesis, Eeole Polyteehnique F6d6rale de Lausanne (1990). 38. M. Zehetbauer and V. Seumer, Acta metall, mater. 41, 577 (1993). 39. P. Pan, Int. J. Solids Struct. 19, 153 (1983). 40. A. Molinari, C. R. Acad. Sci. Paris t.306, S~rie II, 841 (1988).
kbsin2(O--a)sin2(¢ +a)
1 (I-kin . 20
- [k~ cos 20 - k~ cos 2(0 -cosa)COS2¢ 2(¢ + ct) ]
+[Atr= - Ao-,,~]sin2(0+ ¢) C
1 (k b sin2 2(0 - =) cos" 2(0 - ~) =7'[' ~ +k~ c o ~ }
APPENDIX
b 1
The expressions of A, B and C. A f ~1 { [ k t m s i n 2 0 - - k b .l_r".
.
.
.
.
L~, c o s z " - ~ ,
+ [tr=-Crnnl
1
~
~
c-~-s2~
cos 2(0 - ~t) ] cos 2¢ J
where
-")]
bOOS2(0 -- a)cos 2(¢ --=)
- [A~rn - A~Jsin2(O - ¢)
[-
- o-on] 7l 1
]
sin+2¢q) f kI h(1 k =h(1 - q ) cos 2¢ The parameters kt and the matrix.
k 2 are
different in the band and in
Pergamon
0956-7151(94)E0037-H
Copyright © 1994ElsevierScienceLtd Printed in Great Britain.All rights reserved 0956-7151/94 $7.00+ 0.00
SHEAR BAND POST-BIFURCATION IN ORIENTED C O P P E R SINGLE CRYSTALS S. YANG and C. REY LPMTM-CNRS, Universit6 Paris Nord, Av. J.-B. Clement, 93430 Villetaneuse, France (Received 20 July 1993; in revised form II November 1993)
Abstract--By in situ SEM observation performed in oriented copper single crystals strained in tension, the formation and propagation processes of macroscopic shear bands (MSB) as well as the consequences of their interactions are analysed. We found that an MSB consisted of a collection of parallel shear bands on slip lines scale. The observed lengthening and thickening of MSB were related to saturation of the individual shear bands and to nucleation of new ones in the adjacent matrix. In order to understand this phenomenon, a post-bifurcation model is applied to an idealized planar double slip single crystal. We show that shear banding is effectively characterized by the three observed stages: bifurcation, large shear strain localization and saturation. The bifurcation is accompanied by unloading in the matrix. The saturation is due to the fact that the geometrical softening leading to the bifurcation decreases with localized shear.
1. INTRODUCTION In tension tests, ductile single crystals exhibit deformation modes, mostly homogeneous on a sample scale. However, after a finite amount of plastic strain, some localized deformation modes such as necking followed by shear banding, leading to a drop of load, appear in the samples. Failure usually follows either by shear banding down to a "chisel edge" or by rupture within intense shear bands [1-5]. These phenomena depend on both macroscopic conditions (loading conditions, sample geometry) and microstructural properties of materials. The localization of the plastic flow in the thin bands with severe shearing, also appears in polycrystals during some metal forming processes, in particular during cold rolling process, which leads to modifications of some mechanical properties of materials and then to failure. Such consequences explain the great deal of experimental and theoretical studies on this subject, in single crystals and in polycrystals, deformed by rolling and by tension tests. Early studies have been summarized in several overviews [6-12]. In order to clarify the complex mechanisms of shear banding in polycrystals, it is essential to analyse the shear band bifurcation in more simple situation such as in single crystals. Many investigations at different scales (microscopic as well as macroscopic) have been performed on single crystals of pure metals and alloys [13-20]. These investigations have pointed out some physical characteristics of localization such as correlations between shear band formation and microstructure of dislocations, the relative orientation of the shear plane with respect to the activated slip planes, the evolution of the lattice orientation or the micro-hardness in the vicinity of the shear bands.
In mechanical modelling, shear band formation is viewed as a material constitutive instability, and thus, it depends strongly on material constitutive laws. In order to understand the physical origin of shear band formation, bifurcation models taking into account the crystallography have been proposed by different authors for single crystals [21-25]. The bifurcation conditions have been analysed as a function of slip geometry, applied stress field and strain hardening behaviour of materials. According to these models, the bifurcation is led by a geometrical softening induced by the lattice rotation accompanied with shear banding. More recently, different approaches of bifurcation phenomena, using finite element simulations have been developed in order to validate the bifurcation analysis under different boundary conditions applied to the single crystals and to the multicrystals [7, 8, 19, 20, 26]. But post-bifurcation behaviour in single crystals, especially shear band saturation phenomenon, has not yet been analytically studied. Note that a relevant study of post-bifurcation behaviour has been carried out by Hutchinson and Tvergaard [27]. The authors used a phenomenological corner theory of plasticity to represent a polycrystal, and studied the shear band formation by growth from a preexisting material band which presents a softer behaviour than the surrounding material. The saturation of shear localization is predicted when in the band, the stiffening effect associated with the ceases of the total loading is emphasized by a blunt comer or by suppressing the total loading. In single crystals, the corner exists in the loading point of the yield surface and has a physical meaning. Shear band formation can be studied by bifurcation and post-bifurcation analysis which permits a better understanding of basic physical origin of shear band formation on a grain scale.
2763
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YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
This work has two main purposes: • To get an accurate description of the mechanisms of propagation and saturation of shear bands in oriented copper single crystals strained in tension, as well as the effects of the interactions between shear bands on the post-bifurcation behaviour. • To analyse the observed phenomena and especially the conditions of saturation of shear band, through a post-bifurcation model performed in an idealized planar double slip single crystal, able to take into account the correlated effects of strain hardening behaviour, as well as the applied stresses and the lattice rotation, i.e. the geometrical softening. The first section of this paper is devoted to the analysis of the mechanisms of propagation and inter-
Samples A B c D
Tensileaxis []'121 [T23] [T22] [01H
Table I Largeface (1TI)or (110) OTl) (0l-D (0i-1)
Max. load 50% 60% 60% 80%
conditions on shear band pattern was examined on samples A. The other tested samples had the same geometry (13 x 3.5 x 2 mm3). 2.2.2. Slip pattern and shear band pattern. Samples A: these samples were oriented in double slip B4 and C1 (according to the Schmid and Boas notation [29]), respectively. The two slip directions are parallel to (ITI) faces and symmetric with respect to the tensile axis. The two slip planes were activated homogeneaction of shear bands in copper singlecrystalsstrained ously from the beginning of the deformation. On in tension under an SEM. In order to realizejudicious (IT1) faces the angle between the tensile axis and situations for this analysis, several crystallographic the traces of the two activated slip planes is of orientations (particularly the []'12]orientation cor- about 30 deg. With such slip geometry, the thickness responding to symmetric double slip) and different in direction [1"i'1] of the samples remained constant samples geometry were studied. In the second section, during straining. One or two families of shear bands the two-dimensional post-bifurcationmodel is recalled were activated within the neck according to the and applied to the singlecrystalpresenting the double sample geometry. Samples B: the orientation corresponded to a sliporientation,which permits to predict the observed shear band formation, propagation and saturation. single slip activation. The activation of the primary slip system (B4) led to a lattice rotation around the [ITI] axis. The activation of the secondary slip system 2. EXPERIMENTAL RESULTS ((21) was observed at elongation approximately equal 2.1. Experimental procedure to 35%. Two families of shear bands symmetric with Oriented copper single crystals were elaborated by respect to the tensile axis appeared within the neck. the Bridgmann technique. The samples were cut by Then the behaviour of the samples was identical to sparking machine, then submitted to a mechanical the A samples. Samples C: the crystallographic initial orientations and electrolytic polishing. The crystallographic orientations were checked with the back-reflection corresponded to a coplanar double slip (B4, B5) Laue method. Before the shear band formation, (weak interaction). Only one trace of slip plane was fiducial grids of 5/~m steps were deposited, using a observed before necking. In the necking area, some microlithographic method [28], on the sample surface traces of a new crystallographic plane (identification in order to visualize the deformation induced by was very difficult) were observed. The gradient of strains and that of material rotations induced by the shear banding. The samples were deformed at room temperature necking were so high in these samples that it was by uniaxial tension in a Schenk apparatus or in a impossible to specify if the localization within the neck micro-tensile machine built in a Scanning Electron corresponded to shear bands or to coarse slip bands. Samples D: the tensile axis was very close to [011]. Microscope (SEM) with a deformation rate of 10 -4 s -1. The surfaces of the samples were observed at The deformation of such samples was heterogeneous, with two families of slip lines corresponding to the different stages of deformation by SEM. planes A and B, localized in different areas of the 2.2. Experimental results samples respectively. Large gradient of strains and that In order to determine the role of the initial of material rotations were observed. Two families of orientation on the threshold of strain localization shear bands appeared within the neck. The traces and on shear band pattern, different samples were of shear planes were slightly different from the traces tested with various initial crystallographic orientations of the activated slip systems. 2.2.3. Effect of the sample geometry on shear band named A, B, C and D, respectively. By using different sample geometry we have obtained particular situ- pattern. Different geometry of the samples A was ations allowing to study the mechanisms of shear tested: band propagation and interaction. L--15mm, l=2mm, e = 6 . 5 m m , I/L=O.I 2.2.1. Threshold of localization. In all studied L=13mm, l=3.5mm, e=2mm, 1/L=0.3 samples, shear bands appeared within the neck. The elongation values corresponding to the maximum load L - - 1 5 m m , l = 6 . 5 m m , e = 6 . 5 m m , 1/L=0.4 of the tensile curve are given in Table 1. The effect of L=25mm, l=15mm, e = 2 . 5 m m , ILL=0.6 the sample geometry which is related to the boundary
YANG and REY:
SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
2765
where l denotes the initial width of the samples, L is the initial length and e is the initial thickness. Two symmetric families of shear bands were observed for l/L <~0.4, but only one for l/L = 0.6. This phenomenon was probably due to the structure effects associated with the clamping of the tensile machines. From observations carried out on these two kinds of samples (l/L = 0.6 and I/L <. 0.4), we analysed the mechanisms of formation and propagation of a macroscopic shear band (MSB) on a sample scale, as well as interactions between them.
2.2.4. Mechanisms of formation and propagation of an MSB. The following described mechanisms, analysed on the large samples ( l / L = 0.6) were identical for all tested single crystals. As in all samples, the localization of plastic flow in an MSB took place within the neck, where the distribution of the two activated slip systems B4 and C1 was homogeneous. As already observed by Dubois [30, 31], the MSB was composed, on slip lines scale, of a collection of short segments of shear bands (about 50/tm of length) parallel to each other, with a spacing of about 0.5 #m. The thickness of shear bands was not identifiable by SEM. The MSB was limited by two approximately parallel interfaces named S and U respectively (Fig. 1). The interface S was stable, independent of
Fig. 2. Nucleation of new shear bands at the instable interface of the MSB. increasing straining. On the contrary, the interface U was unstable, which lengthened and spread out, leading to the lengthening and the thickening of the MSB. The interface S corresponded to the first shear bands. The propagation of the interface U is due to activation of new short shear bands in the adjacent matrix. These new short shear bands constituted the interface U and so was parallel to the latter (Fig. 2). The in situ observation showed that the deformation was mostly localized in these new shear bands, whereas the old ones saturated. This explains that the interface S was macroscopically less well-defined than the interface U. When the MSB was confined in the sample, the directions of the shearing, of the thickening and of the lengthening were not independent. The thickening took place always beside the interface at which the shearing direction was opposite to the lengthening direction (Fig. 3). Figure 4 showed the large localized shear within the MSB in early stage, which had reached a value of about 1.2, whereas the matrix remained almost undeformed. With increasing straining, the localized shear within the MSB increased slowly, as can be remarked by contrasting the unstable interface in Fig. l(a) and (b). As concerns the matrix strain, even though it is small compared with the localized shear, it increased with the band widening, pointed out by the deformation of the fiducial grids (Fig. 2). Shear
Lea
enin
Shear Fig. 1. (a) MSB limited between the stable interface and the unstable interface; (b) thickening of the MSB and propagation of a crack along the unstable interface.
Fig. 3. Sketch of the propagation directions of the MSB.
2766
YANG and REY:
SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
strain and lattice rotation gradient induced by the necking. 2.2.5. Interaction between MSBs. The samples with a low IlL value presented two families of MSBs, each of them slightly misoriented vs the traces of the two activated slip planes. It is worth noting that, the orientation of the traces of the two activated slip planes before the shear band formation was nonuniform within the neck, the rotation can reach a maximum value of about 10 deg from the centre to the sides of the samples. The sequences of propagation of these shear bands are sketched in Fig. 5. The main points may be summarized up as follows: • Nucleation from the sides within the neck of four MSBs (named E, F, G and H respectively) at about 45 deg with respect to the tensile axis. The MSBs E and F, G and H, presented respectively a same shear plane and a same shear direction but opposite directions of propagation. • The four MSBs grew in thickness and in length by nucleation of new short shear bands, as described here above. The magnitude of the localized shear was not uniform along these MSBs. • The four bands never reached the centre of the sample but they induced in this spot a strain twice larger than in the remaining part of the matrix and sufficient to fulfil the conditions of activation of four new MSBs named E', F', G' and H' respectively. E' and F', G' and H', had the same shear plane and
Fig. 4. Shear localization in the MSB initiating from the side
within the neck. The formation and the propagation of the MSB can be described as follows: the first shear bands grew from the side of the sample within the neck and then constituted the MSB. The MSB propagated through consecutive activation of new shear bands in the adjacent matrix and saturation of old ones in such a way that the thickness of the MSB increased with increasing straining and can reach some millimetres before rupture. It is worth noting that the shear bands were slightly misoriented with respect to one family of the activated slip systems, and that the unstable interface U, constituted by the new short shear bands, presented a curvature, which is related to the
(a)
(b)
(e)
(d)
(c)
°
E
i i (f)
/ H'
F'
Fig. 5. (a-e) Sketch of propagation and interaction processes of the M SBs in the [TI 2] tensile tested single crystal with increasing straining. (f) Details of (d) at the intersections of the MSBs.
G
YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
2767
the same shear direction, but opposite directions of propagation, respectively. A large gradient of strains and material rotations was observed at the intersection of these MSBs. We observed that gradually, the former MSBs E, F, G and H became inactive. • The very pronounced neck was then divided into two regions, the new MSBs E', F', G' and H', and the matrix which corresponded to the no-sheared areas. With increasing straining, the extent of these no-sheared areas decreased progressively whereas the MSBs E', F', G' and H' grew in thickness as shown in Figs. 5(c,d). It can be noticed that close to the bands, the strains were much larger than elsewhere (Fig. 6). • The active shear bands E' and G', F' and H' intersected each other in M and N respectively [Fig. 5(f)]. At higher strain, the distance MN increased [Fig. 6(a--c)], and a triangular crack appeared in M from the centre of the samples, propagated by tearing along shear bands E' and G', and finally induced the rupture of the samples. 2.2.6. Fracture. The discussion will be brief since this is not the main focus of this paper. As already noted by Chang and Asaro [13] and by Drve et al. [19], two modes of fracture were observed. In the case where the MSB initiated from the side within the neck, a crack initiated and grew steadily along the unstable interface of the MSB (Fig. 1), by alternate shearing of the MSB and important slip concentrated at the tip of the crack. The final fracture occurred almost by complete shearing off of the localized band at the unstable interface. However, in the case where the four symmetric MSBs initiated from the centre and grew to the sides, any crack was not observed at the intersections of the MSBs and the sides. Instead, at final stage, a triangular cavity formed at the intersection of two MSBs (a "diamond" shaped cavity was observed by DOve et al. [19]). It should be noted that in the latter case, the alternate shearing of the MSBs by one crossing another was not observed. As a consequence, at the intersection of the MSBs, a boundary parallel to the tensile axis which separated the MSBs was clearly observed [Fig. 6(b,c)]. At this boundary, large internal stresses could be induced, as illustrated by Yang [32]. The large internal stresses may explain the cavity formation at the intersection. But why not in "diamond" shape? This may be probably due to the increased distance of MN in Fig. 5(f), thus the shear band pattern was dissymmetric at the intersections. This triangular cavity did not grow by sliding off along the instable interfaces of the MSBs. The explanation is as follows: according to the growth mechanism of the crack in Fig. 1, an alternate shearing is necessary for a crack to grow along the unstable interface of MSB. When the triangular cavity formed, at the tips of the cavity, an alternate shearing by slip across the
Fig. 6 (a-c).
Fig. 6. (a,b) Micrographs of the propagation and interaction of the MSBs in the [-1-12]tensile tested single crystal with increasing straining. (c) Formation of a crack at the intersection of the MSBs.
2768
YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
MSB would be difficult to operate, while there was saturation of old shear bands, thus the MSB could not propagate by thickening like that in Fig. 1. 2.3. Discussion on the mechanisms o f shear banding
Different points must be stressed upon the MSB formation and propagation. These mechanisms are clearly the consequence of coupled effects: boundary conditions (sample geometry, effects of the clamping), initial crystallographic orientations, and also strain hardening behaviour due to the activated slip systems before localization. In these conditions, the resulting stress field could not be assumed to be a uniaxial tension, nevertheless whatever was the stress field, the mechanisms of shear banding present many common features. It is worth noting that in all cases studied, the activated shear bands were roughly parallel to one of the activated slip planes, and the MSB propagation mechanisms across the samples were the same. Concerning the initial crystallographic orientations (all other conditions being identical), they affect only the threshold of localization, as already shown by Gasp6fini and Rey [33], the measured elongation corresponding to the neck formation, presented weak values when initial orientation of the tensile axis was close to a "hard" double slip position (formation of Lormer-Cottrell locks). The problem of the shear localization is more complex because both boundary conditions and initial crystallographic orientation ruled the lattice rotation gradient in the sample and consequently the conditions of shear band bifurcation. The mechanisms of propagation of an MSB could be summed up as follows: • Some collections of shear bands appeared in the most deformed areas of the necking region and constituted the MSBs with increasing straining. The MSB was always limited by two interfaces: one stable and the other propagating by activation of new shear bands in the adjacent matrix. This led to the thickening and lengthening of the macroscopic band. The direction of propagation by thickening and that of lengthening were strongly correlated. In the case where the MSBs initiated from the sides, new shear bands nucleated beside the unstable interface but not beside the two interfaces limiting the MSB. This may be attributed to the accompanied crack formation and growth, which could induce strain concentration near the tip of the crack, thus led to the alternate shearing by the MSB and by important slip concentrated at the tip, and therefore the MSB thickened at one side. In the case where four symmetric MSBs initiated at the centre of the sample, it has been shown [32] that the internal stresses induced by the four MSBs have effect to favour shear band nucleation in the observed directions, and to deactivate shear band nucleation at another side of the MSBs. • Activation conditions of the shear bands were very sensible to the lattice rotation and the resulting MSB was generally curved.
• Observation of the deformation of the fiducial grids pointed out the following phenomena: firstly, during the early stage of the MSB formation, the magnitude of the localized shear was not uniform along these bands, such inhomogeneities, according to Mussot et al. [34], induced internal stresses which would be locally accommodated by activation of some slip lines at the head of the MSB and so would lead to the lengthening of the MSB. A more or less uniform localized shear was observed only when the MSB was well-developed. Secondly, the MSB presented two different zones, one beside the stable interface, another beside the unstable interface. This latter zone, whose thickness (about 5 gm) can be considered as constant independent of increasing straining, was constituted by the new activated shear bands. On the contrary, the former zone was constituted by the deactivated shear bands, its thickness grew with increasing straining. Thirdly, four MSBs could induce a local but large strain field at their vicinity as observed in samples A. Finally, during the propagation of the MSBs, the matrix far from the bands had a rigid behaviour. Experiments on samples A showed that a change of the sample geometry could modify the number and the distribution of MSBs. It was pointed out that the MSBs induced a large range of internal stresses probably of the same order of the applied stresses, able to create new MSBs. Interactions between MSBs were strong and led to heterogeneous shear band patterns. The propagation mechanisms of an MSB is in fact, according to the aforementioned experimental observations, due to the activation and the saturation of the individual shear bands. The behaviour of the individual shear bands was characterized by their bifurcation, the large shear strain localization and the saturation. It depends essentially on the local behaviour of the crystal. In order to have a more accurate description of the parameters which rule the shear banding, a theoretical study is hereafter proposed. In the next section, we propose an analysis of the propagation mechanism, in particular of the saturation of a shear band, in terms of slip geometry, crystal strain hardening behaviour and loading conditions, by using a two-dimensional single crystal model which is sufficiently representative of the here above studied [1"12] single crystals.
3. POST-BIFURCATION MODEL 3. I. Slip geometry and notations
The geometry of the idealized planar double slip single crystal is sketched in Fig. 7. The crystal is assumed to be deformed according to two slip systems characterized by their slip directions s 1 and s 2 and their slip plane normals ml and m2, respectively, tp is the angle between s I (s2) and the 2-axis (~p will be
YANG and REY:
SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS AD=-~(g®n+n®g)=~
2769
p, i - P r o ) P ) ,
(2)
i
Ca)
x
where
~n 'l x.~" ~'a m2
p,=½(si®m,+mi®s,).
(3)
p~ and p~ may be calculated as a function of tp. Thus ~lb and ~ in the band can be deduced from (2) as
I',,
"
2~__
2 [-"
' 1\
'
~ =~
L.e,
1
[J. sin 2(0 + ~o - ~)
+ ~ sin 2(29 - ~t) + ~ sin 2~t]
(4)
, ',,
~ = ~ /V'/~.~
" ~Nj~
(b)
/J
[~. sin 2(0 - ~o - e) - )~ sin 2e
+ ~p sin 2(2¢p + e)].
~
At the interface, the discontinuity A ~ of the total spin is the sum of the jumps in the plastic and the lattice spins, Aff~p and A~*, respectively. Thus
(e)
Fig. 7. (a) The planar double slip single crystal model; (b) a shear band forms at plane normal to [] and in direction g and (c) its processing scheme.
Af2*= ~ ( g ® n - n ® g ) - ~
(w~/b- wm@m), (5)
where taken to be 30 ° to represent the slip geometry of the [-1-12]single crystal in tension). The interface between the band and the matrix is denoted by the shear direction g or the normal n to the shear plane in the current configuration (n and g are unit vectors). The deformation in the band and in the matrix during shear banding is assumed to be uniform respectively. For a given localized shear 2, 0 is the angle between the interface and the 2-axis (00 being the initial angle at the beginning of shear banding), ct is the lattice misorientation between the band and the matrix.
un rs an
va=s
ta os of o =ation
by shear banding in the single crystal, the crystallographic slips y~ and y: on the two activated slip systems are calculated inside the band and inside the matrix (indexed later by superscripts b and m, respectively) as a function of 2. These four values are analytically deduced from the following shear banding kinematic conditions, static conditions and strain hardening behaviour as well as loading conditions,
3.2. Constitutive law 3.2.1. Kinematics of shear banding. The crystal is assumed to be rigid-plastic (the elastic strains are neglected, but the lattice rigid-body rotation is taken into account). All the manipulations are carried out in the co-rotational (g, n)-coordinate system, Shear banding is described at every moment by the discontinuity AL in velocity gradient at the interface band/matrix AL = ).g®n,
(1)
where A denotes a jump, ~. is the localized shear rate. The symmetric part, i.e. the discontinuity AD of the strain rate is
w~ = ½(s~®m ~ - mi® s~). The lattice misofientation rate is given by -- g. Aft*. n.
(6) (7)
We obtain then from (7), (5), (6) and (4) ~F cos2(0 _ ~t)O ~ = 2 L1 c~ss2-~ l{I cosc-~s 2(~02~-- 0C)]'md + 1 ])/1
J
-
I
1
1}
coscos 2(~o2tp + e) ~p .
(8)
The rotation of the shear plane with respect to the (1, 2)-coordinate system depending only on matrix velocity gradient, one can easily deduce that 0 = s i n 2 ( 0 - q ~ ) ~ - s i n 2 ( 0 + ~0)~. (9)
3.2.2. Statics of shear banding. At the interface band/matrix, the equilibrium equation, for an incompressible material in two-dimensional case, can be written as A~ = 0 (10) where ~ , = g - a .n, representing the resolved shear stress on the shear band. The rates of the resolved shear stress on the two sides of the interface can be expressed as follows ~ . zb = g- ( V, b - ~rn~)(flg~) b ~ b a ) b • n - (~rgg
(11) v, m
[-z~ = g ( a )
m
m
* m
-n - (t~g - an~)(flg~) ,
where (~,)b and (v*) m are the rates of the Cauchy stress tensor in the co-rotational coordinate system bound to the lattice in the band and to that in the
2770
YANG and REY: SHEAR BAND POST-BIFORCATION IN COPPER CRYSTALS
matrix, respectively; (Q.)b, (~r'~n)m are the lattice rotation rates in the (g, n)-coordinate system. 3.2.3. Hardening matrix. We assume that the crystal obeys the Schmid's law. The elastic part of the strain rate D * being neglected, the rate of the resolved shear stress on the (i) slip system can be written as zi = s," v , . mj.
equation (10) can be written in terms of the resolved shear stress rate on the inactive slip system A ~ ? + B2"t~ = C.
(17)
For the case where the system (2) is active and the system (1) is inactive, similarly we obtain B~f~n + A:~)~ = C
(12)
(18)
where
{
The (i) slip system is active if and only if
{~,
A~ = A - hmq~B2 A 2 = B - hmqmBt
= ~
~, = ~
(13)
(19)
and
where z~ is the critical shear stress, z~ can be expressed as a function of a symmetric latent hardening matrix, whose components h 0 are positive, namely
"~ = h,j~j
(14)
fnl
=
sin sin 2(04(p + cp) (20)
sin 2(0 - ~0) 2= sin 4~
where we take h e to have elements h H = h22 = h and hi2 = h21 = hq, with h > 0 and q > 0. 3.3. Calculation o f slips inside the band and inside the matrix
The experimental observations showed that, during shear banding, the major amount of plastic deformation was localized inside the band, whereas the contribution of the matrix was weak. So, we assume, in the following, that two slip systems are active in the band. But for the matrix, all cases will be analysed: case 1--two active slip systems, case 2---only one active slip system, case 3--no active slip system, i.e. a rigid matrix. Three equations will be hereafter set up corresponding respectively to the above three cases. They are deduced from equilibrium equation (1 0) and expressions (l l) as well as all the aforementioned kinematic equations. Case 1: both slip systems (1) and (2) are active in the matrix. In this case, the increments of the resolved
shear stress on the two slip systems, in the band and in the matrix, can be expressed by (14). We obtain thus a linear equation in terms of ~? and ~ A ~ + B~n = C
(15)
Case 3: rigid matrix. In this case, the equilibrium equation (10) can be expressed in terms of the resolved shear stress rates on the two inactive slip systems in the matrix "~gn = Bt ~T + B2z~n = C.
(21)
In all cases, the parameter C is given by equation (16) which corresponds, when the matrix is rigid, to the resolved shear stress rate fg, on the interface inside the band (normalized by the localized shear rate 2). The sign of C depends on relative weight of strain hardening and geometrical softening given by a,~)ct, with & being the lattice spin in the band in rigid matrix case. 3.4. Bifurcation conditions
Let us assume that shear band bifurcation occurs with the two slip systems in the matrix being still active, and that at the very moment of the bifurcation, no discontinuity exists in the crystal, hence we have: ~t = 0, Aa = 0 and Ah0 = 0. Then in equation (15) we have: A = B = 0 and C = Co with Co = l h { ( 1 +q)~fi~-~)/sin 20°'X2
with C
=
g.(~,)b.n 1 [1 2L
-
( ~ =b -
. / c o s 200'~2~ +(1-q,~) ;
a . .b)
cos 2(0-~OS 2 ~ CQ]
(16)
where ~ , $~ and v, are normalized by ).. In the following, all incremental variables are normalized by the localized shear rate 2. The expressions of A, B and C are given in Appendix. These parameters depend on slip geometry, lattice misorientation ct and orientation 0 of the band as well as stress field and hardening moduli in the band and in the matrix. Case 2: one active slip system in the matrix. Consider the case in which the slip system (1) is active and the slip system (2) is inactive. For this case, equilibrium
1(1 -- (a~, - an,) "~-
c°s 20°'~ cos 2~o/
(22)
and equation (15) gives the bifurcation condition Co=0. The condition Co = 0 added to the condition dCo/dOo = 0 allows us to determine initial orientation 00 of critical shear band. In case of an uniaxial tension parallel to the 2-axis, equation (22) and the condition Co = 0 permit to determine the critical value (h/~r),et :for shear band bifurcation with initial orientation 00, where ~ represents the tensile stress. This is the same bifurcation conditions as Asaro's [22].
YANG and REY:
SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
2771
F o r e < n/4, the lattice rotation ct presents, when 2 tends to be infinite, an asymptotic value equal to 8.0 (0 - e ) , the condition (24) i.e. ~t ~< (0 - e ) holding for ). ---,oo. The computed ratio of the crystallographic .i~-. 6 . 0 slip rates increases with shear banding. This means that the slip plane (1) in the shear band rotates with "~ 4.O 0=40* 1: ~=30* the lattice towards the shear plane. In this case, a 2: ¢--60" saturation of the shear band cannot be kinematically 2. o 2: " ~ , / ~ predicted. 0.0 I5 I0 15 3.5.2. Statics. To have a better understanding of O.O O. 1 . 1 . .O l shear band development in the case e < n / 4 , let Fig.8. The latticerotation~ and the ratio~/9~ of the slip us calculate under a rigid matrix assumption, the rates insidethe shear band 8 = 40° as a function of the variation 6z~ of the resolved shear stress on the interlocalizedshear )~for two differentcases: h ¢p = 30° and 2: face as a function of 2, for different strain hardening = 60°. behaviour. According to equation (21), we have 10.0
2: O.
3.5. Post-bifurcation analysis It will be shown in the following that shear band bifurcation can be accompanied by a rigid unloading of the matrix. This solution of the problem, compared with the homogeneous deformation mode and other post-bifurcation modes, is the most favourable deformation mode, and characterizes the observed shear band formation. 3.5.1. Kinematics. Assuming that the matrix is rigid during shear banding [ ~ = ~,~ = 0, and so 0 = 00 according to (9)], we have calculated the ratio of the shear rate 9tb/~b and the lattice rotation ~t in the band as a function of 2, for different values of e and 0, from equations (4) and (8) respectively. ~ and @2 b in equation (4) must be positive, thus we obtain
( 0 e [ 2 - e' q~l
for e > re/4
(23)
for e < n/4.
(24)
~ e,-~- e
e[o,0 - e l In rigid matrix case, the lattice rotation e in the band can be calculated by integration of (8), assumin.g that e = 0 at the beginning of shear banding (,~ = 0). The main results are shown in Fig. 8. The analysis of the computed curves shows that two situations may occur: For e > ~ / 4 , the condition (23) implies a ~<0 -(n/2-e), ct reaching its maximum value for = 2m~x, given by 1 , ~tg0+tge 2 ~ = ~ - ~ - ~ ,n ~ ~g ~ - t - ~ .
1-tg2e~ l - ~ ~gg2; j .
(25)
This result points out a limited localized shear in the hand for a given slip geometry, independent of applied stress field.
6Zgn = BI c~z~ + B25~ ~ =
S;
C d2
(26)
where 5 represents a variation computed from the bifurcation moment, B~ and B2 are positive according to (20), (23) and (24). After the shear band bifurcation, if the parameter C becomes negative, according to (26), the matrix will be unloaded because of negative value of 5zg. and positive values of B t and B 2. It has been noted previously, according to (16), that the parameter C depends on the relative importance of strain hardening, the lattice spin and stress level. So, one can expect that, with shear banding and under some conditions (which will be hereafter analysed), the sign of C would change so that the matrix would be reloaded and the assumption of rigid matrix might be wrong. From (26), one can deduce that when 6zg~ becomes positive, the rigid matrix assumption no longer holds, because 6zT and 6z~ must be negative if the matrix is rigid. The evolution 6zg, with the localized shear 2 have been studied in two cases: • firstly, the strain hardening moduli are supposed to be the same in the band and in the matrix and to stay constant with increasing straining, i.e. h ~ = h ~ = const; • secondly, the strain hardening moduli are assumed to decrease with increasing straining according to the law
dh = - k h dy
(27)
where k is a material constant (k/> 0), and 7 = 7~ + 72. This law describes a linear relation between the strain hardening moduli and the resolved shear stress, which corresponds to ductile single crystal strain hardening properties at stage III [18, 35-38]. We note that the first case corresponds to k = 0 in (27). In the second case, the hardening moduli in the band are different from that in the matrix after the bifurcation. Our results are summarized in Figs 9 and 10. In the first case, the rigid matrix assumption which corresponds to 6Zgn< 0 is fulfilled only for 0 > 36 °. We notice that 5zg, reaches positive values for 2 > 2r.
2772
YANG and REY:
SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
'/,:,,y
0.03
,.....
,
0.25
0,2
o°°', o. I " J ff
,
,
,
,
,
,
@=30*,O=400, k=O.5, q = l . 4
I
!
0.15
................................ ! 0.i
~' -0.01 -0.02
0.05
",
-0.03
,'"'" i 1
i 2
~= 30°,k=O,q = IA i 3
i 4
I
I
I
I
I
0.5
1
1.5
2
2.5
I, 3
3.5
4
t Fig. 9. The variation 6T_ • ip~ of the resolved shear •stress . as a function of the locahzed shear 2. The calculaaon is performed for different kinematically admissible shear band orientation 0, where the matrix is assumed to be rigid and k = 0, ~0 represents the tensile stress at the bifurcation moment. This means that, beyond this localized shear, the assumption of rigid matrix no longer holds. It can be noticed that 2r strongly depends on strain hardening behaviour, namely • for k = 0, 2, depends on 0; this is due to the fact that each angle 0 corresponds to a different (h/tr)crit value. • for k > 0, the computations point out that a large value of k, i.e. an important microstructural softening, induces an important value of hr" Let us now study the case when 6z~ becomes positive due to a positive value of C, which corresponds to that the plastic flow takes up again in the matrix. To determine ~ , ~ , ylb and ~2b, the equilibrium equation and the two kinematic equations are not sufficient, the loading conditions must be taken into account. Assuming that the matrix is submitted to an uniaxial stress parallel to the 2-axis (i.e. z'~ = zT, thus ~ ' = ~ ) , ~ and ~ may be computed as function of 2, for different value of ~p, 0, q and k. The results are given in Fig. 11. In this figure, one can see that ~,~ (or 7~') increases sharply and has an asymptotic axis which corresponds to 2 = ~ . The slip ~ ( o r 7~) grows faster 0,03
,
0.02
, i / ,
¢p: 300, O = 400, q = 1.4
,
/
o ~"
........... i 0
..............................
-0.01 k=l
-0.02
"" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-0.03
I 1
I 2
I 3
I 4
5
Fig. 10. The variation 6x~ of the resolved shear stress in the critical shear band (0 = 40°) as a function of the localized shear 2, for different values of k, where the matrix is assumed to be rigid, tr0 represents the tensile stress at the bifurcation moment.
Fig. 1I. The crystallographic slip ?T (or ~ ) in the matrix as a function of the localized shear 2 during shear banding.
than the localized shear 2 in the band, meaning that the shear band will rapidly become inactive when the matrix restarts to be deformed. 4. CONCLUSION We have investigated the mechanisms of deformation by shear banding in ductile copper single crystals subject to tensile straining• In all tested single crystals which presented different initial crystallographic orientations, the mechanisms of propagation of an MSB were identical: a collection of short shear bands, initiated in the most deformed area within the neck, then they constituted the MSB; the latter grew by thickening and by lengthening through consecutive activation of new short parallel shear bands and saturation of old ones. The propagation mechanisms of the MSBs were in fact a reflection of that of the individual shear bands, the latter having the same orientation as the unstable interface of the MSBs. Attention was thus focused to model one shear band locally, in particular, to model its bifurcation, the large shear strain localization and its saturation. In order to model these phenomena, postbifurcation analysis for a shear band was performed. We began for a sake of simplicity, from Asaro's idealized double slip single crystal strained in tension. Although two-dimensional, this crystal model is well illustrated by the []'12] single crystal subject to tensile straining. The latter presented at (1T1) faces, two symmetric slip systems simultaneously activated (~p = 30 °) with respect to the tensile axis, and two symmetric families of MSBs, at 0 ~ 40 ° near the centre of the samples. The post-bifurcation analysis deals with a material element, where the deformation is assumed uniform in the band and outside the band respectively. The calculations are performed under the hypothesis that the uniaxial tension is always applied on this material element. This model is able to predict the observed mechanisms of shear banding: shear band bifurcation (when the necessary and sufficient condition is fulfilled) then development with all strain concentrated in the band (rigid matrix) and then restart of plastic flow in the matrix and saturation of the shear band. Thus one
YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
2773
can deduce that, because of the restart of the plastic the influence of different parameters, such as slip flow in the matrix, some new shear bands will be geometry and strain hardening properties, were qualiactivated in the matrix when new bifurcation con- tatively valid. To have a good quantitative prediction, ditions are fulfilled there. Such sequence depends it is necessary to have a better but tractable strain on the slip geometry and the evolution of the strain hardening law for finite deformations, to take into hardening moduli with increasing straining. It is worth account the strain gradient within the neck and the recalling from the in situ observation that nucleation internal stresses induced by the MSBs, as well as rate of new shear bands appeared in the adjacent matrix sensitivity. The latter have significant retardation and near the saturated bands. The model, treating two stabilizing effects on shear band formation [8, 39, 40]. uniformly strained regions, the band and the matrix, can not predict that. But taking into account the REFERENCES nonuniform strain state induced by the necking and the internal stresses induced by the MSBs as well as I. W. F. Hosford, R. L. Fleisher and W. A. Backofen, the stress concentration induced by the crack formActa metall. 8, 187 (1960). 2. S. Saimoto, W. F. Hosford and W. A. Backofen, Phil. ation, one can understand that. Qualitatively, accordMag. 21, 319 (1965). ing to the model, the first shear bands tend to initiate 3. C.N. Reid, A. Gilbert and G. T. Hahn, Acta matall. 14, at some the most deformed regions within the neck, 975 (1966). e.g. preferentially at the sides. Because of the 4. B. Pegel, Scipta metall. 13, 47 (1980). 5. W. A. Spitzig, Acta metall. 29, 1359 (1981). nonuniform strain state, the internal stresses and 6. J. G. Sevillano, P. V. Houte and E. Aernoudt, Prog. the stress concentration, the restarted strain will be Metal Sci. 25, 69 (1981). larger in the adjacent matrix and in one side of MSB 7. D. Peirce, R. J. Asaro and A. Needleman, Acta metall. than else. Thus new shear bands tend to nucleate 30, 1087 (1982). there. 8. D. Peirce, R. J. Asaro and A. Needleman, Acta metall. 31, 1951 (1983). We show, by analysing the kinematics of shear 9. R. J. Asaro and A. Needleman, Scripta metall. 18, 429 banding, that one slip plane in the band progressively (1984). tends to parallel to the shear band, and thus the 10. M. Hatberly and A. S. Malin, Scripta metall. 18, 449 localized shearing is essentially produced by this slip (1984). system. In consequence, the "geometrical" softening l 1. L. Anand, Scripta metall. 18, 423 (1984). 12. G. R. Canova, U. F. Kocks and M. G. Strut, Scripta will become negligible at large localized shearing, and metall. 18, 437 (1984). thus the matrix will start to be reloaded, and then to 13. Y. W. Chang and R. J. Asaro, Acta metall. 29, 241 be redeformed leading to the shear band saturation. (1980). But if the k value in (27) is greater than some critical 14. K. Morii and Y. Nakayama, Trans. Japan. Inst. Metals 22, 851 (1981). value depending on crystallographic slip geometry, 15. Y. Nakayama and K. Morii, Trans. Japan. Inst. Metals the hardening moduli in the band will decrease 23, 422 (1982). rapidly and tend to be zero with the localized shear 16. K. Morii and Y. Nakayama, Scripta metall. 19, 185 strain, so that although the "geometrical" softening (1985). is negligible at large localized shearing, the matrix will 17. K. Morii, H. Mecking and Y. Nakayama, Acta metall. 33, 379 (1985). not be enough reloaded to be deformed plastically. In this case, the localized shearing is theoretically 18. Z. Jasienski, Mater. Tech., p. 91 (1985). 19. H. Dfve, S. Harren, C. Mccullough and R. J. Asaro, infinite. Acta metall. 36, 341 (1988). It is worth noting that the sign of the C parameter 20. S. Harren, H. Dfve and R. J. Asaro, Acta metall. 36, 2435 (1988). controls this analysis, which corresponds in rigid matrix case to the resolved shear stress rate on the 21. R. J. Asaro and J. R. Rice, J. Mech. Phys. Solids 25, 309 (1977). interface band/matrix, normalized by the localized 22. R. J. Asaro, Acta metall. 27, 445 (1979). shear rate. The sign of C depends on the relative con- 23. R. J. Asaro, J. appl. Mech. 50, 921 (1983). tribution of two terms given by (16). The first term 24. R. J. Asaro, Adv. appl. Mech. 23, 1 (1983). corresponds to the instantaneous response to the 25. D. Peirce, J. Mech. Phys. Solids 31, 133 (1983). localized shearing (strain hardening of the material), 26. S. V. Harren and R. J. Asaro, J. appl. Mech. 37, 191 (1989). the second term depends on the lattice spin in the 27. J. W. Hutchinson and V. Tvergaard, Int. J. Solids band (geometrical softening). A more detailed analysis Struct. 17, 451 (1981). 28. T. Bretheau and D. Caldemaison, Proc. 4th Ris~ Int. of this model is given by Yang [32]. Syrup. (1983). It must be indicated that, the magnitude of the 29.. E. Schmid and N. Boas, Kristal Plastititiit, Berlin, table localized shear predicted by this model is valid only VI (1962). when the neck is not pronounced before shear band 30. P. Dubois, Ph.D. thesis, Univ. of Paris XIII (1987). formation. In copper single crystals, because of the 31. P. Dubois, M. Gaspfrini, C. Rey and A. Zaoui, Arch. Mech. 40, 25 (1988). deep neck before shear band formation, it is not interesting to compare the localized shear predicted 32. S. Yang, Ph.D. thesis, Univ. of Paris XIII (1990). 33. M. Gaspfrini and C. Rey, MECAMAT'91, p. 229 by this post-bifurcation model with the experimental (1991). one. Nevertheless, we think that the predicted various 34. P. Mussot, C. Rey and A. Zaoui, Res. Mech. 14, 69 (1985). stages of deformation of shear banding as well as AM 4 2 / g ~
2774
YANG and REY: SHEAR BAND POST-BIFURCATION IN COPPER CRYSTALS
35. H. Meeking, Working Hardening in Tension and Fatigue (edited by A. W. Thompson), p. 67. Metals Soc. AIME (1977). 36. B. Nicolas and H. Mecking, ICSMA-5, p. 351. Pergamon Press, Oxford (1979). 37. P. N. B. Anongba, Ph. D. thesis, Eeole Polyteehnique F6d6rale de Lausanne (1990). 38. M. Zehetbauer and V. Seumer, Acta metall, mater. 41, 577 (1993). 39. P. Pan, Int. J. Solids Struct. 19, 153 (1983). 40. A. Molinari, C. R. Acad. Sci. Paris t.306, S~rie II, 841 (1988).
kbsin2(O--a)sin2(¢ +a)
1 (I-kin . 20
- [k~ cos 20 - k~ cos 2(0 -cosa)COS2¢ 2(¢ + ct) ]
+[Atr= - Ao-,,~]sin2(0+ ¢) C
1 (k b sin2 2(0 - =) cos" 2(0 - ~) =7'[' ~ +k~ c o ~ }
APPENDIX
b 1
The expressions of A, B and C. A f ~1 { [ k t m s i n 2 0 - - k b .l_r".
.
.
.
.
L~, c o s z " - ~ ,
+ [tr=-Crnnl
1
~
~
c-~-s2~
cos 2(0 - ~t) ] cos 2¢ J
where
-")]
bOOS2(0 -- a)cos 2(¢ --=)
- [A~rn - A~Jsin2(O - ¢)
[-
- o-on] 7l 1
]
sin+2¢q) f kI h(1 k =h(1 - q ) cos 2¢ The parameters kt and the matrix.
k 2 are
different in the band and in