The plasticity of particle-containing polycrystals

Acta metall, mater. Vol. 38, No. 6, pp. 917-930, 1990

0956-7151/90 $3.00 + 0.00 Copyright © 1990 Pergamon Press plc

Printed in Great Britain. All rights reserved

THE PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS F. J. H U M P H R E Y S and P. N. KALU Materials Department, Imperial College, London SW7 2BP, England (Received 25 July 1989; in revised form 1 December 1989)

Abstract--The lattice misorientations adjacent to second-phase particles of silicon in a polycrystalline aluminium matrix deformed in compression have been measured by a TEM microtexture technique. The results have been analyzed in terms of a simple model which is based on a modification of Taylor's polycrystalline plasticity theory. A model in which independent deformation zones are formed in the vicinity of each particle for each active slip system, gives reasonable agreement with the experiments if overlap of the deformation zones is taken into account. The number of deformation zones formed at a particle and the size of the misorientation are found to be functions of particle size as well as strain. R~sum6--43n mesure, par une technique de microtexture en MET, les d~sorientations du r~seau adjacent fi des particules de silicium dans une matrice polycristalline d'aluminium d~form~e en compression. Les r~sultats sont analys~s d'apr~s un module simple bas~ sur une modification de la th~orie de la plasticit~ polycristalline de Taylor. Un module dans lequel des cones de d&ormation ind~pendantes se forment au voisinage de chaque particule pour chaque syst~me de glissement actif conduit/t un accord raisonnable avec l'exp~rience si le recouvrement des zones de d&ormation est pris en compte. Le nombre de zones de d~formation fortunes sur une particule et la valeur de la d~sorientation sont des fonctions de la taille des particules et de la d~formation. Zusammenfasseng--Die Gitterfehlorientierungen in der N/ihe yon Silizium-Teilchen in einer polykristallinen, im Druck verformten Aluminiummatrix wurde mit einem TEM-Mikrotexturverfahren gemessen. Die Ergebnisse werden nit einem einfachen Modell analysiert, welches auf einer Modifikation der Theorie der Polykristallplastizit~it von Taylor aufbaut. Ein Modell, bei dem unabh/ingige Verformungszonen in jedem aktiven Gleitsystem in der N/ihe eines jeden Teilchens gebildet werden, stimmt hinreichend mit den Experimenten fiberein fiir den Fall, dab fJberlapp der Verformungszonen berficksichtigt wird. Die Anzahl der bei jedem Teilchen gebildeten Verformungszonen und die Gr6Be der Fehlorientierung h~ingen nach den Ergebnissen von Teilchengr6Be und Verzerrung ab.

1. INTRODUCTION During the plastic deformation of a metal containing second-phase particles which do not deform, the deformation process is inevitably inhomogeneous (see for example [1-3]). The dislocation structures resulting from this inhomogeneous deformation have been studied in some detail in single crystals deformed to small strains, e.g. [3], and are found to be dependent on strain and particle size. However, for the practically important case of polycrystalline materials, or for materials deformed to large strains, the situation is less clear, The deformation processes in the vicinity of particles of diameter greater than about 0.1/1 m, result in the formation of locally rotated regions of material (deformation zones), adjacent to the particle. There have been some attempts to characterise these rotations [4-7] in polycrystals, although experimental difficulties have prevented detailed or systematic measurements. Most of these investigators have suggested that the deformation zones in the vicinity of the particles are randomly oriented, although the quality and quantity of the results is not really sufficient to

justify such a conclusion. The orientation of the deformation zones in particle-containing single crystals of aluminium and copper deformed in single slip were investigated in detail by Humphreys [8], using a dark field imaging technique in the transmission electron microscope. The zones were found to be misoriented with respect to the matrix about an axis perpendicular to the slip direction and the slip plane normal. Although the rotated deformation zones may have an effect on the mechanical properties of the material (see e.g. [9]), of equal significance in heavily deformed polycrystals is the fact that by changing the orientation of regions of the matrix, the presence of the deformation zones will affect the deformation texture, particularly if the volume fraction of particles is large. Experimentally, it has sometimes been found that high volume fractions of large second phase particle may weaken the deformation texture cornpared to that of a similar matrix without particles, although this is not always the case [10-12]. A proper understanding of the nature of t h e d e f o r m a t i o n zones developed during the deformation of polycrystals would thus provide a foundation for the development 917

918

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS

of a theory of deformation textures in particle-containing polycrystals. The deformation zones may also play an important role in recrystallisation. On annealing a cold worked metal containing particles of diameter larger than about 1 #m, recrystallisation may nucleate within the deformation zone [5]. The orientation of the new grains, and hence the final recrystallisation texture will thus be affected by the orientations within the deformation zones. An understanding of the deformation zones is therefore a prerequisite to formulating a theory of recrystallisation textures in particle-containing polycrystals.

"~ ~ -~ ~ ~ ~'@i'

~. [ - ~

~~ (b)

(o) ..

,," : ~ * ~ ~ ",

ImpenetrabLe

"",,,,/1 ~. 4 v ,, ,,

zone

."

", ~

THE PLASTICITY OF

TWO-PHASE ALLOYS 2.1. Single crystals

Ill/N/'I//IJll .b

-UnreLaxedorowan

PLastic relaxation

Loops to)

(d) :

* ~I ~:

The deformation zones are formed as a result of the plastic relaxation of the high energy dislocation arrays, such as Orowan loops, which build up at non-deformable particles during straining, A simple model of such a process is shown in Fig. 1. A particle within a crystal is surrounded by an Orowan loop [Fig. l(b)]. The stress can be relaxed and the loop absorbed into the particle-matrix interface, by the particle rotating [Fig. l(c)], and by the generation of secondary dislocations (not shown), This type of model, in which the dislocations form a low angle grain boundary at the particle interface, was first proposed by Ashby [13]. Other relaxation mechanisms have been discussed by Brown and Stobbs [9] and Humphreys and Stewart [14], and further details may be found in reviews [2, 3]. For a shear strain of % the number of dislocations of Burgers vector b trapped at a particle of diameter d is given [13] as ~d/b. If we assume that these dislocations relax to form a low angle grain boundary at the interface, then the spacing of the dislocations in the boundary is b/y and the misorientation of this boundary (0), i.e. the rotation of the particle, is then approximately equal to ),. We can now extend this model so as to predict the size of the deformation zone and the distribution of orientations within the zone. 2.1.1. A model for the deformation zone. We assume that a soft matrix containing a hard equiaxed particle is deformed on a single slip system,

Ondeformed

, ~ -r ,'

(c) 2. M O D E L L I N G

",

....

-,,,,,, ,,," ~ ~ * ~ ', ',

(b)

re)

Fig. 1. Schematic diagram of plastic relaxation by local lattice rotation,

Distance

(e) Fig. 2. Schematic diagram of the development of a deformation zone. Initially, dislocations interact with the particle and form unrelaxed Orowan loops as shown in Fig. 2(a). The high interface stresses due to the loops [15, 16] result in plastic relaxation, the generation of secondary dislocations, and rotation of the particle [Fig. 2(b)]. The nature of the secondary dislocation structures and the question of whether these are associated with lattice rotations has been extensively discussed [9, 14]. We assume, in agreement with Brown and Stobbs [9], that such rotations are very small ( < 1°), and that significant rotation is that associated with the particle. It should be noted that for the smaller particles, alternative mechanisms of relaxation involving prismatic punching of dislocations may lead to a similar or zero rotation of the particle [8, 9, 13, 14, 17, 18]. As straining proceeds, it will become increasingly difficult for further gliding dislocations to approach the particle, because of both the long range stresses associated with the unrelaxed shear loops, and also the forest effect of secondary dislocations. Thus the effective particle size increases as shown in Fig. 2(c). Further relaxation events then take place not at the particle surface, but at the interface between the matrix and the impenetrable structure near the particle, as shown in Fig. 2(d). Therefore as deformation proceeds, the impenetrable zoneasspreads outwards from the particle surface, and the deformation mechanism of Fig. 1 results in a rotation of the "undeformable" material, more and more matrix material will be rotated. The matrix very close to the particle will have been rotated throughout the straining, and material further away from the particle will have been rotated for a smaller strain and will therefore have been rotated through a smaller angle. There is thus an orientation gradient from the

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS particle outwards, and the spatial distribution of rotations appears somewhat as shown schematically in Fig. 2(e). If we can determine the size of the impenetrable zone, and its development with strain, then we will be in a position to determine the distribution of misorientations within the crystal. Such a model, albeit a very crude approximation, is developed in Appendix 1. 2.1.2. Justification of the model. The model of the development of the deformation zones presented above is essentially an extension of ideas originally proposed by Ashby [1], Brown and Stobbs [9] and Humphre~s [19]. The adoption of this model needs to be justified and compared with other models of deformation zones, There are three alternative models of the deformation zone to consider: • A Brown and Stobbs [9], considered a hard but not impenetrable forest of dislocations at the particle and predicted misorientations of the correct sign and of magnitude equal to the square of the strain. • B Argon et al. [20], considered a zone produced by prismatic punching of dislocations from the particle interface. This model does not predict any lattice misorientations within the zone. • C Humphreys [5], considered the zone to be formed directly from the secondary shear loops generated at the particle, which does not rotate. This model predicts the correct sign of the misorientations, and the magnitude of the misorientation, which is proportional to the strain, depends on the size of the plastic zone. Detailed measurement of the size and orientation distribution of deformation zones in single crystals was made by Humphreys [8]. It was found that the maximum misorientation within a zone was independent of particle diameter for particles larger than about 2 pro, and was equal to the shear strain 7. The region of maximum misorientation, which was close to the particle, was very small, and there was a steep orientation gradient. Although rotations of small ( < 0 . 1 / l m particles have been measured [17, 18]), there appears to be no experimental evidence as to the rotation of larger particles. The key experimental observation would seem to be that the maximum misorientation is found to be equal to the shear strain. In none of the three models discussed above does this occur, although in model C, this may occur if certain assumptions are made about the size of the deformation zone. The model presented in Section 2.1.1 in which the maximum misorientation is determined by the geometry of the crystal plasticity, regardless of the details of the dislocation model, appears to have the merit of simplicity as well as being in reasonable accord with the available experimental evidence,

2.1.3. Orientation

changes during deformation.

In Section 2.2, we will be discussing the changes

919

of orientation of matrix and deformation zones during uniaxial deformation of a polycrystal. It is perhaps useful to first clarify the situation for single crystals containing particles with deformation zones developed according to the model of Section 2.1. Figure 3 is an inverse polefigure showing the initial orientation (I) of the compression axis of a face centred cubic crystal oriented for single glide, before deformation. On compressive deformation by a shear strain 7, the planes of the crystal rotate relative to the compression axis through an angle (cos 2 sin ~b)7, where 2 and $ are the angles between the compression axis and the slip direction and slip plane normal, respectively [21]. The direction of rotation is such that the normal to the slip plane moves towards the compression axis. After a certain strain, the compression axis therefore has moved to the orientation M in Fig. 3. The maximum misorientation in the deformation zone is that of the small region which has rotated with the particle, as discussed in Section 2.1. Consideration of Fig. 1, shows that the particle rotates in the opposite sense, and is misoriented with respect to the matrix far from the particle by 7, or more exactly [21], by 2(cos 2 sin ~b)7. The maximum misorientation within the deformation zone is thus the point P shown in Fig. 3. The orientation of material within the deformation zone varies continuously between M and P, and the spread of orientations within the crystal is along the solid line in Fig. 3. The volume of material of each orientation may be calculated (approximately) from equation (A8) in Appendix 1.

2.2. Polycrystals Having formulated a model which will predict the misorientations which develop at particles in a deformed single crystal, we now examine the extension of this model to a polycrystal. The essential difference in the polycrystal, is that several slip systems will operate. 111

loo 1~o Fig. 3. Orientation changes during the compression of a single crystal.

920

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS

Consider a particle within a grain in the polycrystal, as in Fig. 4(a). Strain occurs on a slip system {hlkll~}(u~v]wj> and a rotated region RI adjacent to the particle is formed as shown in Fig. 4(b). Now a second slip system {h2k212}(u2t)2w2) operates within the grain, and this creates a second rotated zone R2. The position of the zones will depend on the slip system, and we assume, on the basis of previous work [8], that this is above and below the slip plane, In the case of a f.c.c, material, there is a possibility of forming 12 such zones at a particle. From the single slip model of Section 2.1, if we know the strain on each of the slip systems, then we can determine the maximum rotation associated with each of the deformation zones. However, there is an important problem which remains unresolved, and this concerns the interaction between the deformation zones. We can consider three possibilities, (I) Each deformation zone is independent. The rotation within the zone is that due to the cumulative strain on the appropriate slip system, (II) Zones are independent as in (I), but may physically overlap as in Fig. 4(d). In the region of overlap, the rotations will be the sum of the simple rotations. This is shown schematically in Fig. 4(el. If the slip on systems 1 and 2 is such as to produce rotations as shown by the vectors R1 and R2, then the rotation of the overlapping zone is R12. (III) A zone once formed behaves as if rigidly attached to the particle. When subsequent slip systems operate, the zones formed earlier will be reoriented. The rotations will to some extent be dependent on the sequence in which the various slip systems operate. The independent zone model (I), is considered below. The basis of the calculations to be considered in this section is as follows: - - F o r a grain of a particular starting orientation, a small increment of strain is applied and the strain on the various slip systems and the reorientation of the grain, is calculated from the theory of polycrystalline plasticity, R1

R1 , t-,J~' ~

...

1

t t

""

a2 R1 (b)

(Q) R12,~ R1 R

(d)

~-..

(el

i\Ri2"~ ! !

\ I R2

(e)

Fig. 4. Formation of more than one deformation zone at a particle,

- - T h e maximum misorientation in the deformation zone, which is essentially an equal and opposite rotation, according to the considerations of Section 2.1, is calculated for each of the 12 possible deformation zones. - - F u r t h e r increments of strain are applied, the matrix grain continues to be reoriented, and the cumulative misorientation for each deformation zone is calculated. --Thus, after a given amount of strain, the orienration of the matrix grain is known, and the orientation of the most highly misoriented part of the deformation zone is also known for each of the 12 possible deformation zones. 2.2.1. Matrix plasticity. As the rotations within the deformation zones are directly determined by the slip systems operative in the matrix grain it is clear that we are critically dependent on theories of polycrystalline plasticity and texture development for a cornplete solution to our problem. However, despite considerable activity in this field, the problems of polycrystalline plasticity have not yet been completely solved, and for detailed discussions of the state of this subject, the reader is referred to the recent reviews of Gil Sevillano et al. [22], van Houtte [23] and Hirsch and Lucke [24]. Although the most complete description of the plasticity of a polycrystal may, in principle, be obtained from self consistent elastic-plastic calculations [25], such models are very complex, and at present their use is restricted. The simpler model of Taylor [26], or modified versions of it are still widely regarded as giving a reasonable account of the deformation behaviour of polycrystals. In the Taylor, or full constraints model, each grain undergoes the same plastic strain as the macroscopic aggregate. This means that in general, 5 of the 12 available slip systems in an FCC material are simultaneously activated. The choice of the active slip systems and the strain on each of them is made on the principle of least work. Unfortunately, for many grain orientations, more than one set of 5 slip systems satisfies this requirement and such ambiguities must then be resolved on some other criterion. For example, it might be assumed that the slip is averaged among the contending sets of slip systems, that a random selection is made, that selection is governed by strain rate sensitivity or the favouring of either coplanar or cross slip. Discussion of these various approaches may be found in the reviews [22-24]. If the grains are severely flattened, as is expected after large strains, then it can be shown [27, 28] that less than five slip systems are required. Such relaxed constraints theories are now widely used to predict deformation textures after large strains. The basic Taylor theory predicts the deformation texture of aluminium reasonably well, and the various modifications to the theory result in relatively small changes [23, 29]. In addition, we are only going to consider relatively low strains and thus relaxed

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS constraints models are hardly justified. Therefore we adopt the simplest form of the Taylor theory, in which ambiguities in the choices of slip system are averaged. Whilst recognising the limitations of this approach, we feel that at this stage, a more complex theory is not justified, The models discussed in Section 2.2 and Fig. 4 were discussed in terms of slip occurring consecutively on different slip systems. The Taylor theory formally assumes simultaneous slip on 5 slip systems everywhere. There would therefore seem to be some discrepancy between the models and the Taylor theory, However, the orientation results of the Taylor theory are not significantly altered if slip is allowed to occur in bursts on successive slip systems, provided that over a small strain interval, the total strain on each system is that given by the theory. Moreover, it is well known that in real materials, slip is not homogeneous and does actually occur by groups of dislocations emanating from sources. There is therefore no reasons why our model, coupled with the Taylor

approach should not be used as a basis for predicting orientation changes. 2.2.2. Calculation o f misorientations at particles. Using the full constraints Taylor model as discussed above, the active slip systems and orientation paths of 50 grains (whose orientations were uniformly distributed throughout the stereographic triangle), subjected to uniaxial compression, were calculated [29, 30] for strains of up to 1.2 (70%). The reorientation of a typical grain towards [110] for a strain of 0.4 is shown in Fig. 5(a). The cumulative strain on each of the 12 slip systems during the deformation was calculated, and used to determine the maximum misorientation in each of the 12 possible deformation zones as discussed above. For the grain of Fig. 5(a) these zones are shown in Fig. 5(b). The matrix orientation has moved during the straining from I to M. The regions of maximum misorientation within the deformation zones are the orientations P,, and, as discussed in Section 2.1, orientations within the deformation zones will lie between M and P,. There are

111

100

921

111

110

100

,/

P3

(a)

(b)

111

111

110

(c)

Fig. 5. (a) Reorientation of a grain during compression. (b) Deformation zones predicted by independent zone model in grain after E = 0.3. (c) Deformation zone predicted by rigid zone model for same grain and strain.

110

922

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS

only three deformation zones which are associated with substantial rotations. The orientations of the other zones are less than 1° and lie too close to M to be shown. Such a result is typical. Although there are 12 possible deformation zones, the strain on most of them is almost always very small, and thus a maxim u m of perhaps three significant deformation zones will be formed at a particle. The activity predicted for the various deformation zones in the 50 grains of the model is shown in Table 1 for a strain of 0.4. In order to compare the experimental measurements with the predictions of the models discussed above, it is convenient to specify the direction of the rotation in a deformation zone by the angle (~t), which is defined in Fig. 6. It should be noted that because of the geometry of the stereographic projection this is not a precise measure, and rotations about a certain crystallographic axis for matrix orientations at different points in the unit triangle will give values of ~ varying by up to 3 °. However, this is smaller than the experimental error, and will give rise to no serious errors in the analysis, Figure 7 shows the angular distribution of deformation zones whose misorientation is greater than 2 °, for a compressive strain of 50% (E = 0.7). It can be seen that three angles predominate and that there is little activity on some of the zones (this histogram does not change significantly as a function of strain), This representation of the deformation zones does not in fact clearly distinguished between all of them, as some zones give rise to similar values of ct. The various slip systems are identified by number in Fig. 7. Although Table 1 shows that peak A is composed of similar amounts of zones 5 and 9, it can be seen that peak B is dominated by zone 4 and peak C by zone 2. 2.2.3. Rigidzone model. The model discussed in the previous section was based on the premise that the zones could act independently. The other extreme case is that when the zones are formed, they behave as if rigidly attached to the particle. Thus, for example, zone 1 is formed by slip on system 1, with a rotation R1. Then there is slip on system 2. This forms a zone of rotation R2, and also rotates the previously formed zone through the same angle. Thus the rotation of zone 1 is now R1 + R2 etc. The

111

M 1oo

11o Fig. 6. Definition of the angle cc

detailed results of such a model will depend on the temporal sequence of the operation of the slip systerns. However, if we neglect this factor, we can see that the result will be a single zone at the particle, whose rotation is given by R1 + R2 + R 3 . . . + RN. Figure 5(c) is an inverse polefigure on which the zone predicted by this model for a sample grain is shown. The similarity between this zone and that of the single crystal case in Fig. 3 is of course expected. Figure 8 shows the angular distribution of the zones for e = 0.7, and it should be compared with that for the independent zone model of Fig. 7. 3. EXPERIMENTAL

3.1. Materials An alloy of A1--0.8 wt% Si was chill cast into a copper mould, extruded to 12 m m diameter, and heat treated as described below. The grain size of the specimens was ~ 4 0 0 #m. The texture, measured by standard X-ray techniques was rather weak, and is shown in Fig. 9. The advantage of the alloy system is that it produces particles of silicon, most of which are equiaxed,

4/8 2/11

Table 1. Predicted activity of the deformation zones Total rotation Number of in zone Number Slip system zones > 2° (number x degrees) 1 (11 I) [01T] 9 36 2 (111) [101] 43 498

5/9

~ ~"

F

]

3

(11l) [IT0I

l0

58

45 6 7 8 9 10

(lIT)J101] (1 IT) [011] (11 T) lli'0] (ITI) [I 10] (l'f I) [10"i'] (ITI) [0111 (Tll) [01TI (TII) [101] (TI l)l 110]

50 34 2 14 13 47 9

1005 273 7 86 66 371 32

21

122

Fig, 7. The distribution of zone orientations according to

4

21

the independent zone model.

11 12

~°,160,

c

1i ~0

:5 2 12 . . . . . . . . . . . . . . . . . . . . . . . 0 50 100 200 250 300 AngLe alpha (degrees)

350

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS in a matrix of almost pure aluminium. These alloys have previously been found to be good model twophase materials [8, 31, 32], as the particles do not deform, and decohesion at the particle-matrix interface does not occur. Specimens were homogenised at 853 K and water quenched. This is followed by a heat treatment designed to produce a uniform precipitate of the required size. Details of the heat treatments are given in Table 2. The diameter of particles as measured in the SEM, and volume fractions as calculated from the phase diagram are also included in Table 2.

111

1.~

~ / //~.!~,~.o

100

3.2. Mechanical testing Cylindrical compression specimens of diameter 10 m m and length 20 ram, were tested in an Instron Screw testing machine, at room temperature, at a nominal strain rate of 10 -4 s-1 to true strains of 0.2, 0.7 and 1.2. 3.3. Determination ofmicrostructure and microtexture Thin foils of the deformed specimens were prepared by conventional electropolishing techniques from sections cut either parallel or perpendicular to the compression axis. The lattice misorientations adjacent to the particles and in the matrix areas were measured using the T E M microtexture technique [33, 34] in a JEOL 2000FX microscope operating at 200 keV. With this technique, a diffraction pattern is obtained from an area, typically of diameter 5-50 #m, of a thin foil, using the microscope in standard diffraction mode. A selected Debye ring, usually 111, is then scanned at 1.5 ° intervals over a small transmission electron detector, by microcomputer control of the beam deflector coils. The process is then repeated as the specimen is tilted over a range of + 5 0 °. The data are processed to produce a semi-quantitative partial polefigure. Regions contributing to a particular region of the pole figure may be selectively imaged in dark field. Some particles did not show local lattice rotations. As it is likely that this results from the removal of the deformation zone by electropolishing, or. of it being obscured by an

923

~10

Fig. 9. The texture of the specimens before compression.

electron opaque particle, such particles were ignored. The polefigures from a region containing a particle were analysed to determine the matrix orientation adjacent to the particle, and the maximum extent of the misorientations. In this way, the n u m b e r of deformation zones associated with a particle and their maximum misorientation could be ascertained. The angular resolution of the technique limits the m i n i m u m detectable misorientation to ~ 2 °, and deformation zones of less than this rotation are thus ignored. The orientations of matrix and deformation zones were then plotted on inverse polefigures so as to enable comparison with the theoretical predictions such as Fig. 5(b). Note that only the maximum misorientation in each deformation zone is measured, and that in the following sections this parameter will be taken to represent the orientation of the deformation zone. It should however be noted that, as discussed in Appendix A, there is a continuous range of orientations within the deformation zone, and that this factor should be taken into account if a complete description of the miscrostructure is required. 4. EXPERIMENTAL RESULTS The microtextures of over 200 particles of diameters between 0.5 and 8/tin were determined. In this section we present some typical results, and in Section 5, we analyse the full range of data. Table 2. Heat treatments and dispersion parameters of the alloy

,

Particle

o>, © tY__

0

F ~

,

.......................... 50 100 150 200 250 Angte alpha (degrees)

300

350

Fig. 8. The distribution of zone orientations according to the rigid zone model.

Heat treatment 853K for 6h, quench. 573 K for 24 h, quench. 763 K for 120 h, cool at 2~'/h to 573 K, quench. 853K for 6h, quench. 523 K for 5 h, quench. 693 K for 48 h, cool at 2°/h to 523 K, quench. 853K for 6h, quench. 523 K for 2 h, quench. 653 K for 48 h, cool at 2°/h to 523 K, quench.

size range (,am) 2 to 9

Volume fraction 7.7 × 10 3

0.5 to 3

8.3 × 10 3

0.l to 1

8.3 × 10 3

924

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS

/

Fig. 10. 111 microtexture polefigure of 5/~m diameter region containing a 2.5 #m particle. Compression axis (C), matrix (M) and deformation zone (P) are marked. * = 0.2. 4.1. Specimens deformed to E = 0.2

Figure 10 is a 111 polefigure from a region containing a particle of diameter 2.5 #m. The matrix oftentation is marked as M, and the single deformation zone as P. Figure 11 shows results from several particles, plotted on an inverse polefigure. The matrix orientations are indicated by open circles, and the deformation zones by solid circles. It can be seen that only one deformation zone per particle is detectable, and that the rotations between matrix and particle region are always in a similar direction.

....p.....

......

Fig. 12. !11 microtexture polefigure of5/zmregion containing a 3/am particle. E = 0.7. angles of rotation as was done for the theory in Fig. 7. Such a plot for all the experimentally measured deformation zones, regardless of strain or particle size is shown in Fig. 14. It can be seen that there are three distinct peaks in the data. However, comparison with the theoretical peaks (A, B, C) from Fig. 7 (independent zone model), which are superimposed onto Fig. 14, show that the correspondence between the experimental and theoretical peaks is rather poor. The experimental data may be analyzed in more detail by taking into account the effects of strain and particle size. Deformation zones at particles of

4.2. Specimens deformed to larger strains

111

Polefigures from a specimen deformed to E = 0.7 is shown in Fig. 12. It can be seen that in this case more than one deformation zone is formed. Inverse polefigures giving some of the results are shown in Fig. 13. In both Fig. 13(a) and (b), some preferential alignment of the rotations in directions other than that of Fig. 11 can be detected. 5. ANALYSIS OF THE RESULTS 5.1. The direction o f the rotation in the deformation zones

loo

In order to compare the experimental results with theory, it is convenient to plot the distribution of the

-

1fo

-

(o) 111

111

100

lO0

110

Fig. 11. Inverse polefigure showing orientations of matrix (open circles) and deformation zones (solid circles). ¢ = 0.2.

110

(b)

Fig. 13. Inverse polefigures showing data for a strain of 0.7. (a) Particles with one deformation zone. (b) Particles with more than one deformation zone.

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS

c

[,[,I,~ I ~ L ~

50

loo 15o ago aso 300 350 AngLealpha (degrees) Fig. 14. The experimentally determined distribution of zone orientations for all particle sizes and strains.

A

diameter between 0.5 and 8 g m were measured. The data were divided into two arbitrary size ranges-small particles of diameter up to 1.5 #m and large particles of diameter greater than 1.5/~m (there are insufficient data for division into more than two size groups). Three strains have been used---0.2, 0.7 and 1.2. The angular distribution of the zones for some of these variants is shown in Fig. 15. Comparison with the theory now reveals some interesting trends. For the small particles at a low strain (E = 0.2). The deformation zones are always close to the theoretical zone B (cf. Fig. 7). At higher strains there is some evidence of zones A and C, but many of the zones lie between A and B or between B and C.

C

100

200

925

A

300

B

0

100

Angle alpha (degrees)

C

200

300

Angle alpha (degrees)

bF6)

(e) A

C

A

u_

B

C

,,'-

i

O

:c)

1OO 200 Angle alpha (degrees)

A

300

i

i

i

i

i

i

i

i

1OO 200 Angle alpha (degrees)

i

i

i

300

c

o-

O

1OO 200 Angle olpha (degrees)

300

O

1OO 200 Angle alpha (degrees)

300

Fig. 15. The experimentally determined distribution of zone orientations as a ('unction of strain and particle size. (a) E =0.2, d < 1.5/~m; (b) E =0.2, d > ].5/~m; (c) E =0.7, d < 1.5#m; (d) E =0.7,

d>l.5#m;(e) E=l.2, d < l . 5 # m ; ( f ) E=l.2, d>1.5#m.

i

i

926

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS

For the larger particles, at low strain, there is evidence of the zones A, B and C, but again at higher strains the zones are usually oriented between A and B or more commonly between B and C.

5.2. Amount of rotation in the deformation zones The extent to which plastic relaxation by the rotation mechanism of Section 2 is occurring can be assessed by determining the amount by which the deformation zones are rotated. In Fig. 16 we show the amount of rotation per particle (number of zones multiplied by rotation of zone) as a function of strain for both experiment and theory. The following points

5

Theory

~ d, 1.s ~m o d
4

_~_--------+-"'-~

~ 3 ~, 2 ~ ~ ~ 1

o

-

~

o.'z

-

o14

~

~

o16

ola

Strain

, 1.o

, 1.a

1.4

should be noted: - - T h e experimentally measured rotation increases linearly with strain as expected from theory.

Fig. 17. The number of deformation zones with misorientation > 2° per particle.

- - T h e amount of rotation is greater for large particles than for small particles. --Even for the large particles, the measured rotations are lower than the predicted ones by more than

Taylor theory, and thus this result is to be expected. The trends in the orientations of the deformation

a factor of 2. 5.3. Number of zones per particle The number of zones per particle as calculated from the model, and as measured experimentally is shown in Fig. 17. It may be seen that the number of zones is least for low strains and small particlesl For the larger particles, the number of zones detected at the highest strain rises to 2.6, compared to the theoretical prediction of 4.2.

5.4. The end orientations of matrix and particles The experimentally measured orientations of the matrix grains and of the deformation zones, for strains of 0.2 and 1.2 are shown in Fig. 18, and these may be compared with the predictions of the independent zone model, which are shown in Fig. 19. For the matrix grains, it can be seen that although the grains move towards [110] during compression, as is expected, the rate of movement is much less than that predicted by theory. It is well known [23, 24] that in single phase polycrystals the rate of lattice reorientation is much slower than that predicted by the 14o leo

/ + Theory [] d>1.5bern

oa

<

/ /

t

s

~

,

m

~

loo so .g_ 6o ~o 4o ~ "~ 2o

/

o.Bf...--

D o.z

0.4

o.s o.a 1.o 1.2 1.4 strain Fig. 16. The amount of lattice rotation per particle,

zones at the particles [Fig. 18(c,d)] are less obvious. For the larger strain there are few orientations near [110], [100] and [111], and this concentration of orientations in the middle of the stereographic triangle is broadly in agreement with the theoretical predictions shown in Fig. 19(d), although more experimental data is really needed before firm conclusions can be drawn. 6. DISCUSSION

6. I. The model The experimental results presented above show very clearly that the deformation zones are misoriented with respect to the matrix in preferred crystallographic directions. This observation, which is consistent with previous work on single crystals [8] shows that it is incorrect to regard the deformation zones as being randomly oriented with respect to the matrix. Comparison of the experimental results with the rigid zone model discussed in Section 2.2.3 suggests that this model cannot completely account for the results. In particular, the number of zones per particle is generally larger than the single zone predicted by this model, and the experimentally measured directions of the rotations, particularly at the larger particles are often well outside the range of angles shown in Fig. 8. Nevertheless, particularly for the smaller particles at higher strains, the rotation directions do fall predominantly within the band predicted by the rigid zone model. The independent zone model does appear to be in better agreement with the results, particularly for low strains. However, at higher strains it is clear that a significant number of the deformation zones are rotated to angles intermediate between those predieted to be most active. Thus in Fig. 15(c-f), we find a predominance of zones rotated to angles in between those of the theoretical peaks B and C. This strongly

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS I'/1

•

eo

927 111

•

1oo

I1o

1oo

~Io

(o)

(c) 111

111

e ll • •

/

//.'. 100

110

(b)

eo

•

ee

•

,.- r... •

:..

eII

•

".'4.

/ oeO

o

-"

...-.. o

•

•

100

110

(d)

Fig. 18. The experimentally determined orientations of matrix and deformation zones. (a) Matrix, ~ = 0.2. (b) Matrix, E = 1.2. (c) Deformation zones, ~ = 0.2. (d) Deformation zones, E = 1.2. suggests that overlap of the zones contributing to peaks B and C is occurring, as shown schematically in Fig. 4(d. e), in which case we expect zones rotated somewhere between B and C, the exact positions depending on the relative activity of the various slip systems. Such a modification of the independent zone model is really a compromise between the two extremes of the independent and rigid zone models.

6.2. Effects of strain and particle size An important difference between small and large particles is that for small particles, stress relaxation by mechanisms other than local lattice rotations is possible (see [2, 3, 8]). In AI-Si single crystals it was found [8] that rotations occurred only at particles of diameter greater than 0.1 # m and that relaxation by non-rotation mechanisms such as prismatic punching, was significant for particles of diameter less than 2-3/~m. Therefore it is to be expected that small particles will show smaller rotations than do larger particles, as is shown in Fig. 16. It is interesting that, particularly at low strains [Fig. 15(a)], only one deformation zone (B) is found at small particles. This

may be accounted for if plastic relaxation on the most active slip system (lli)[101] occurs by local lattice rotation, and on the less active systems by other relaxation mechanisms. An alternative explanation of the result would be that at low strains, the 5 matrix slip systems required by the full constraints theory are not required, and that slip is as predicted by a Sachs [35] type model. There is experimental evidence that at low strains, f.c.c, polycrystals deform on fewer than 5 slip systems [36, 37]. However, the fact that the larger particles have more deformation zones at similar strains, makes it unlikely that this is the sole cause. This does raise the interesting possibility for future work in using particles as markers to determine the activity of matrix slip systems. In all cases, the amounts of rotation are significantly less than those predicted by theory (Fig. 16). It is difficult to know whether this is due to deficiencies in the model discussed or to matrix effects. As discussed in Section 5.4, it is established that in single phase polycrystals the rate of lattice reorientation is much slower than that predicted by the Taylor theory. This will therefore affect the particle rota-

928

HUMPHREYS and KALU: PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS

tions. A more a ~ u r a t e model for plasticity of single phase polycrystals would undoubtedly allow a better test of the two-phase model, 6.3. Texture

A detailed discussion of textures in deformed twophase polycrystals is not included in the present paper. However, there are certain points which should be noted. Although thedeformationzonesare rotated in crystallographic directions with respect to the matrix, for each matrix region with a well defined orientation, the effect of the particles is to spread the texture about the matrix orientation, e.g. Fig. 5(b). Consequently the particles are predicted to weaken the deformation texture. However, because the highly misoriented regions of the deformation zones are small, this will be a small effect unless the volume fraction of particles is large, The deformation zone model discussed in Appendix 1 allows us to calculate the volume of a deformation zone and the distribution of orientation within it. We can in principle therefore determine not

only the maximum misorientations associated with particles (e.g. Fig. 19), but also the total orientation distribution in the material. A simple use of this approach is the calculation of the amount of texture weakening in a particle-containing alloy [38]. Figure 19(d) suggests that particles may introduce some weak new components into the deformation texture. In the case of f.c.c, alloys deformed in compression, this appears to be a band of orientations running between [110] and [112]. As the rotations of the deformation zones are dependent on the operating slip systems, the rotations will be strongly dependent on the strain path and hence the initial orientations of the grains. In the present work, the starting texture of the experimental material was weak (Fig. 9) and the theoretical material had a random texture. However, different results would be obtained for strongly textured starting material. The experimental results clearly show that the types of deformation zone formed are dependent on particle size. As small particles tend to produce fewer

111

111

•

•

•

•

f

1oo

11o

•

•

•

•

•

go

go

8

•

•

•

11o

(c) 111

111

io •

o•

/" •

•

/ 110

/

• °o

eo

•

-

oo

" I .'- s ¢,,.. -.'2"~ ~o

/

ee •

•

/

(b)

•

lOO

(a)

100

el

/

•

• ..

•

~-

eo • o o

:. ", •

•

"

•

.

-

.

e

o

.'-...,-o.,r,,-,, -

•"u, oo

-~

100

110

(d)

Fig. 19. The orientations of matrix and deformation zones predicted on the independent zone model, (a) Matrix, E = 0.2. (b) Matrix, E = 1.2. (c) Deformation zones, E = 0.2. (d) Deformation zones, ~ = 1.2.

HUMPHREYS and KALU: deformation

zones

and

also

PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS

smaller

rotations,

we expect that, for a given volume fraction, small particles will result in less of a texture spread t h a n will large particles. 7. CONCLUSIONS 1. Models of the f o r m a t i o n o f local lattice rotations at second-phase particles in d e f o r m e d polycrystals are discussed a n d developed. Experimental m e a s u r e m e n t s of these rotations have been m a d e using the T E M microtexture technique. 2. The results clearly show t h a t the r o t a t i o n s are not r a n d o m , but are determined by the active slip systems in the matrix. However, the particles will result in a weakening of the d e f o r m a t i o n texture c o m p a r e d to a single phase m a t e r i a l . . 3. A model in ~ h i c h i n d e p e n d e n t d e f o r m a t i o n zones are formed for each active slip system gives reasonable agreement with the experiments, if overlap o f the d e f o r m a t i o n zones is taken into account. 4. Both theory and experiment show t h a t a l t h o u g h 5 slip systems may operate within a grain, the n u m b e r o f d e f o r m a t i o n zones with significant rotations is usually m u c h less than 5. 5. The a m o u n t of r o t a t i o n in the zones, a n d the n u m b e r of zones per particle are f o u n d to be a function of particle size as well as of strain. 6. The model discussed should be viewed very m u c h as a first a t t e m p t at the p r o b l e m a n d it is capable of further refinement. Ultimately it should be possible to extend it in order to develop a theory o f d e f o r m a t i o n textures in particle-containing polycrystals. 7. The work described in this p a p e r has i m p o r t a n t implications for the u n d e r s t a n d i n g of recrystallisation textures in polycrystalline two-phase alloys. This will paper.

be discussed

further

in a subsequent

929

12. D. J. Jensen, N. Hansen and F. J. Humphreys, Proc. 8th Int. Conf. on Textures, Sante Fe, U.S.A., p. 431 (1987). 13. M. F. Ashby, Phil. Mag. 14, 1157 (1966). 14. F. J. Humphreys and A. T. Stewart, Surf Sei. 31, 389 (1972). 15. L. M. Brown and W. M. Stobbs, Phil. Mag. 23, 1185 (1971). 16. P. M. Hazzledine and P. B. Hirsch, Phil. Mag. 30, 1331 (1974). 17. L. M. Brown and W. M. Stobbs, Phil. Mag. 34, 351 (1974). 18. M. Cahoreau and F. J. Humphreys, Acta metall. 32, 1365 (1984). 19. F. J. Humphreys, Ann. Chim. Fr. 5, 25 (1980). 20. A. S. Argon, J. Im and R. Safoglu, Metall. Trans. 6A, 825 (1975). 21. C. N. Reid, Deformation Geometry' for Materials Scientists. Pergamon Press, Oxford (1973). 22. J. Gil Sevillano, P. Van Houtte and E. Aernoudt, Prog. Mater. sci. 25, 69 (1980). 23. P. Van Houtte, Proc. 7th Int. Conf. on Strength of Metals and Alloys, Montreal, Canada, Vol. 3, p. 1701 (1985). 24. J. Hirsch and K. Lucke, Acta metall. 36, 2883 (1988). 25. M. Berveiller, A. Hihi and A. Zaoui, Proc. 2ndInt. Riso Symp., Riso, Denmark, p. 145 (1981). 26. G. I. Taylor, J. Inst. Metals 62, 307 (1938). 27. H. Honneff and H. Mecking, Proc. 5th Int. Conf. on Textures of Materials, Vol. 1, p. 265. Springer, Berlin (1978). 28. U. F. Kocks and G. R. Canova, Proc. 2nd Int. Riso Symp., Risg, Denmark, p. 35 (1981). 29. I. L. Dillamore and H. Katoh, in Proc. Conj. Quantitative Texture Analysis, Kracow, p. 315 (1971). 30. G. Y. Chin and E. L. Mammel, Trans. Am. Inst. Min. Engrs 239, 140 (1967). 31. A. T. Stewart and J. M. Martin, Aeta metall. 23, I (1975). 32. F.J. HumphreysandP. N. Kalu, Actametall. 35,2815 (1987). 33. F. J. Humphreys, Text. Microst. 6, 45 (1983). 34. F. J. Humphreys, Proc. 7th Int, Conf. on Textures, Holland, p. 771 (1984). 35. G. Sachs, Z. Verein Deut. lng. 72, 734 (1928). 36. T. Leffers, Proc. 2nd Int. Ris~ Symp., Riso, Denmark, p. 55 (1981). 37. R. L. Fleischer, Acta metall. 35, 2129 (1987). 38. P. L. Kalu and F. J. Humphreys, Proc. lOth Int. Riso Syrup., Riso, Denmark (1989).

Acknowledgement--This work was supported by the Science and Engineering Research Council.

APPENDIX REFERENCES

Lattice Rotations at a Particle

1. M. F. Ashby, Phil..~lag. 21, 399 (1970). 2. L. M. Brown, Proc. 5th Int. Conf. on Strength of Metals and Alloys, 3, 1551 (1980). 3. F. J. Humphreys, in Dislocations & Properties of Real Materials. p. 175. Inst. of Metals, London (1985). 4. D. T. Gawne and G. T. Higgins, in Textures in Research and Practice (edited by Grewen and Wasserman), p. 319. Springer, Berlin (1969). 5. F. J. Humphreys, Acta metall. 25, 1323 (1977). 6. P. Herbst and J. Huber, Proc. 5th Int. Conf. on Textures, Aachen, p. 453 (1978). 7. M. D. Ball and P. L. Morris, 9th Int. Congr. on Electron Miseroscopy, Toronto, Canada, Vol. 1, p. 622 (1978). 8. F. J. Humphreys, Acta metall. 27, 1801 (1979). 9. L. M. Brown and W. M. Stobbs, Phil. Mag. 23, 1201 (1971). 10. F. J. Humphreys, Metall. Forum 12, 123 (1978). 11. F. J. Humphreys and D. J. Jensen, Proc. 7th Int. Ris~ Symp., Riso, Denmark, p. 93 (1986).

In this section we develop a simple model of the size of the deformation zone at the particle, and of the distribution of orientations within this zone. We assume the model outlined in Section 2.1 and calculate the conditions for the deformation zone to be impenetrable to glide dislocations. The strength of the deformation zone is due to both unrelaxed long range stresses and to forest hardening. In this very much oversimplified approach, we only consider the latter contribution. This calculation is merely a first attempt at calculating the parameters of the deformation zone, several dubious assumptions are made, and should this model be found to be physically realistic, then more serious calculation of the zones would be warranted. The deformation zoneat aparticle ofradius R0is assumed to be a disc of thickness equal to the particle diameter as shown in Fig. A1. This is similar geometry to that assumed by Brown and Stobbs [9], and is in reasonable accord with observation [8].

AM 38/6~D

930

H U M P H R E Y S and KALU:

R

J

~

PLASTICITY OF PARTICLE-CONTAINING POLYCRYSTALS

Re I

The strength of this zone (trz) is given by

I

R i

(C%) gbx/P 4

o

Gb(R 0 d? ,]1/2 4 \ b dRRJ

(A4)

Fig. AI. Geometry of the deformation zone used in the

For the zone to be impenetrable, crz must be at least equal to the stress from the encircling Orowan loops, QGb/Ro, where Gb/Ro is the stress from a single loop, and Q, a small constant, is the stress concentration factor due to the total Orowan loop population at the particle (see e.g. [3]). Thus we find

model,

dR = R 3 d7 / 16bQ 2R.

For an increment of shear strain dT, the number (n) of secondary prismatic loops generated is given [13] by

On integration, with the limit that R = R0 when 7 = 0, then R 2 = R~ + R37/8bQ". (A6)

n = 2R 0 d ? / b .

Each value of R is reached at a particular strain (?c) given

(A1)

We assume that the loops are circles of diameter R0, and thus have a length of 2nR 0. The total dislocation length (L) is thus L = 47tR 2 d7/b.

(A2)

Let these dislocations occupy a shell of thickness dR around the disc. The volume o f this shell (V) is thus 4nR 0 R dR, and the dislocation density in the shell (p) is given by

p = RoT/bR dR.

(A3)

(A5)

by 7c =

8bQ2(R2 - R~)/R3o•

(A7)

The rotation (0) at any point within the zone is (7 - 7c), i.e.

0 = 7 -SbQ2( R z - R2o)/Rg•

(A8)

Comparison of the misorientation gradients and deformation zone sizes measured experimentally by Humphreys [8], with those predicted by equation (A8), give reasonable agreement with Q = 10.