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Microscopic observation of elastic distortions caused by orowan loops
Acta metal/. Vol. 34, No. 9, pp. 1751-1758,1986 Printed in Great Britain. All rights reserved
oool-6160/86 $3.00+ 0.00 Copyright 0 1986Pergamon Journals Ltd
MICROSCOPIC OBSERVATION OF ELASTIC DISTORTIONS CAUSED BY OROWAN LOOPS Department
A. SATO, M. MORI, P. D. FLJNKENBUSCH and T. MORI of Materials Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku Yokohama 227, Japan (Received
19 September
1985; in revised form 6 January 1986)
Abstract-The elastic distortion induced by Orowan loops surrounding a-Fe particles embedded in a Cu matrix has been detected by observing the rotation of Moirt fringes formed by {11I},, and {1lo}, planes. On the basis of detailed electron microscopic observations, the loss of Orowan loops upon annealing has been examined and correlated with the softening behavior observed following macroscopic stress-strain tests. Quantitative analysis has, further, revealed that consideration of the elastic field produced by discrete Orowan loops gives the correct magnitude for the elastic distortion as well as explaining the detailed structure of the Moir& fringes. R&un~Nous avons mis en tvidence la distorsion tlastique provoqute par des boucles d’orowan entourant des particules de fer-a dans une matrice de cuivre en observant les franges de Moir& form&es par les plans {11I},, et { 1lo},. A partir d’observations d&ill&s de microscopic tlectronique, nous avons 6tudie la disparition des boucles d’orowan au tours d’un revenu et nous l’avons reliQ g l’adoucissement que l’on observe B la suite d’essais contrainte/dtformation macroscopiques. Une analyse quantitative a de plus montrk que la prise en considtration deu champ tlastique produit par des boucles d’orowan d&r&es foumit la grandeur correcte pour la distorsion tlastique et explique la structure d&ail& des franges de Moirb. Zusannnenfassung-Die elastiche Verzerrung, die von Orowanschleifen an a-Fe-Teilchen in einer CuMatrix ausgeht, wurde aus der Rotation von Moirke-Streifen, gebildet aus den (11 l}c,- und {1IO},-Gitterebenen, ermittelt. Ahhand ausfiihrlicher elektronenmikroskopischer Reobachtungen wurde die Verringerung der Orowanschliefen durch Ausheilen untersucht und mit dem Entfestigungsverhalten, das nach makroskopischen Spannungs-Dehnungsversuchen beobachtet wurde, korreliert. Die quantitative Analyse zeigt AuDerdem, da0 das von einer Orowanschliefe erzeugte elastisches Feld die richtige Gr6Benordnung fiir die elastische Verzerrung liefert und da13damit die Einzelheiten der Moirkestruktur erklart werden kBnnen.
1. INTRODUCTION
A Cu-Fe
In a previous study [1], we demonstrated that the elastic distortion caused by Orowan loops surrounding second phase particles may be observed by examining the rotation of MoirC fringes formed by nearly parallel matrix and particle lattice planes. We were also able to conclude, on the basis of these measurements, that such elastic distortion was responsible for the macroscopic stress-strain behavior of the Cu-Fe alloy used in the study. However, the elastic distortion observed during these experiments was considerably smaller than that predicted by an elasticity calculation. In the present study we seek to eliminate this discrepancy by taking tie discrete nature of Orowan loops into consideration. That is, we take into account the fact that the deformation is localized in the vicinity of the loops, rather than assuming a continuous “smeared-out” distribution. This appreach also allows us to assess the extent to which assumptions of uniform distortion are applicable in spite of the fact that Orowan loops actually produce inhomogeneous distortions within particles.
alloy similar to that used in [1] but with
a different crystallographic orientation was employed in this study. In this alloy a dispersion of a-Fe particles satisfying a K-S relationship with the matrix is produced by a simple thermomechanical treatment, and Moir& fringes, resulting from rotation of a {llO},, plane with respect to a (lll}c, plane, may then be imaged. The particle orientation examined here (Fig. 1) is convenient from the standpoint of detecting the local elastic distortion induced by discrete Orowan loops. By means of stereoscopic electron microscopy, the influence of the fiol surface is also clarified in the current work.
2. EXPERIMENTAL
Single crystal sheets of 2 x 20 x 200 mm were prepared under vacuum from a Cu-1.52 wt% Fe alloy by the Bridgman method using a seed. 100 mm long specimens with tensile direction T.D.1 (Fig. 2) were then cut using a spark cutter, polished mechanically, and cleaned in a mixture of HNO, and H,O. These
1751
1752
SAT0 et al.: ELASTIC DISTORTIONS CAUSED BY OROWAN LOOPS
Before Deformation (After Anneolino)
After Deformation
T.O. I 7.0.2
Fig. 1. Schematic illustration of a spherical a-Fe particle deformed in a Cu matrix. Distortion /?,*was measured using Moire fringes formed by (TIT),, and (TTO), planes. Fig. 2. Stereographic projection showing shear system and crystallographic orientation.
specimens were solution treated at 1223 K for 6 h, quenched into ice water, and aged at 973 K for either 6 or 24 h to introduce y-Fe particles of 36 or 47 nm average diameter, respectively. Subsequent deformation by 20% elongation at 77 K transformed the y-Fe particles into a-Fe and the samples were then annealed at 973 K for 2 h to make the a-Fe particles essentially twin free. Most of the particles satisfied the K-S relationship with the matrix described by (TlT)c,//(TTO),;
~~W,,//[~111,.
(1)
These specimens were further elongated by 2% at 77 K along T.D.2, crystallographically the same tensile direction as T.D.1, in order to introduce Orowan loops with Burgers vector a/2[011]. Thin foils, parallel to (ZTl),, and containing both the shear direotion [Ol l] and the shear plane normal [TIT], were sliced from the strained sample and examined in a high voltage electron microscope, H-1250s equipped with a 55” double tilt stage, at an operating voltage of 1000 kV. 0.5 h isochronal and 450 K isothermal softening treatments (to anneal out the Orowan loops) were conducted outside the electron microscope in a hydrogen atmosphere to prevent both evaporation and oxidation. For all observations, including stereographic microscopy, g = [TIT], i.e. g perpendicular to the Cu K-S plane, was used with specimen orientation adjusted to provide identical diffraction conditions by observing the location of the Kikuchi lines. As the Moire fringes had a somewhat wavy appearance, their orientation was determined on enlarged TEM micrographs by measuring the angle between the fringes in the center of the particles and the [TIT] direction. tin order to ensure that the Moire fringe rotation is caused only by annealing-out of deformation induced elastic strains, Orowan loop strain, we have conducted an experiment on a nondeformed sample. A thin foil was made out of the sample with twin-free Fe particles. Moire fringes were observed before and after annealing at 573 K for 0.5 h. No rotation of the fringes occurred on the annealing.
3. EXPERIMENTAL
RESULTS
A description of the general features observed in this material has been given in (1) and will not be repeated here. Instead we wish to focus on more detailed aspects of the behavior of the Moire fringes. Figure 3 is a representative micrograph of a deformed specimen. Particle E (with circular Moire fringes) does not satisfy equation (1) C has a complicated structure which is actually due to overlapping images formed by two particles, and D is a coherent y-Fe particle. Thus, of the particles in this figure, only A, a relatively large a-Fe particle located near the center of the foil, was suitable for analysis. Figure 4(a)-(d) show the behavior of particle A upon l/2 h isochronal annealing at various temperatures, and are representative of the results for large a-Fe particles. It is clear that both the steps and the curvature of the Moire fringes gradually disappear as the temperature is raised. Concurrently the fringes rotate in a clockwise direction, consistent with the expected relaxation of elastic distortion.?
Fig. 3. Representative view of as-deformed crystal containing twin-free a-Fe and y-Fe particles. B is a pit formed on a foil surface by particle dissolution.
SAT0 et al.: ELASTIC DISTORTIONS CAUSED BY OROWAN LOOPS
(a)
(b)
1753
bottom of the particle, where the loop radius is small, as anticipated from first principle considerations [5]. The appearance of small particles, such as particle B in Fig. 5, is somewhat different. The number of steps on the Moire fringes is smaller and, consistent with the particle size dependence of Orowan loop climb, the rotation is completed at a lower temperature. Finally, although there is a great deal of scatter in the data, the largest rotation angles, A& are observed for small rather than large particles, as shown in our previous study [I]. For example, particle B gives Afl = 0.14, 0.43 and 0.45” in Fig. 5(b), (c) and (d), respectively. This should be compared with the maximum value of 0.34” found for the large particle, A, in Fig. 4. This observation will be dis-
(a)
Cd)
Fig. 4. Rotation of Moirk fringes observed in a large particle during 0.5 h isochronal annealing. (a) Before annealing. (b) Annealed at 473 K. (c) 523 K. (d) 573 K. The actual rotation of the (TiO), planes with respect to the (11 f),, plane, fi, is given by [ 1,4]
where o[is the rotation angle of the Moire fringes with respect to the g vector, and d,,( = 0.2083 nm) and dFe(=0.2024 nm) are the lattice spacings of Cu and Fe, respectively. The difference A#?= j3(before anneaIing) - jI (after ann~ling)
(d)
(3)
determined from equation (2) is found to be O.I8, 0.30 and 0.34” for Fig. 4(b), (c) and (d), respectively. In addition it should be noted that the Moire fringes [Fig. 4(b) and (c)J start to rotate near the top and
Fig. 5. Rotation of Moire fringes observed in a small particle during 0.5 h isochronal annealing. (a) Before annealing. (b) Annealed at 473 K. (c) 523 K. (d) 573 K.
1754
SAT0 et al.:
ELASTIC DISTORTIONS
cussed later with reference to the inhomogeneous nature of slip. An example of the annealing behavior of Orowan loops during isothermal annealing is given in Fig. 6. The wavy structure of the Moire fringes becomes simpler with increasing annealing time, particularly at the beginning of annealing. This behavior is, again, attributable to the variation of loop climb rate as a function of position [S]. It is also apparent that some curvature of the fringes remains even after annealing for 90 min at 450 K. This is consistent with the fact that macroscopic annealing curves show that loss of work-hardening occurs over a rather wide temperature range [q.
(a)
(b)
CAUSED BY OROWAN LOOPS
i
600
-200 ,
t
IOOOnm
,
Fig. 7. Particle location determined by stereoscopic electron microscopy.
The height of the top and bottom surfaces of all thin foils was determined by stereographic electron microscopy, using the edges of pits formed by particle dissolution to locate the foil surfaces. All particles examined were then located with respect to the foil surfaces with an accuracy of better than a few nanometers. Figure 7 shows the distribution of particles in a typical foil cross-section obtained by this technique. In Fig. 8, the rotation after “full annealing” (623 K for 0.5 h) is plotted against the particle depth from the foil surface. Three important observations may be made from this data. First, there is no apparent tendency for the rotation angle to decrease as the foil surface is approached. In other words, there is little or no “surface relaxation effect”, contrary to our previous hypothesis [7]. Second, there is a large scatter in the data. The predominant reason for this must be a nonuniform distribution of slip. Particles intersected by a larger than average number of dislocations during the deformation will have correspondingly higher fringes rotation angles, while other particles, virtually unaffected by the slip, will exhibit rotation angles near zero. This effect should be particularly important for small particles. Although, from fundamental dispersion hardening theory [8], the aoerage distortion should not depend on particle
_____________ ________ _.J-_\ Cd)
Fig. 6. Rotation of Moirb fringes caused by isothermal annealing at 450 K. (a) Before annealing. (b) Annealed for 10min. fcl 30min. Id1 Wmin.
Depthfrom
surface
(nml
Fig. 8. A/?, after full annealing at 623 K for 0.5 h, plotted against particle depth from the foil surface.
SAT0 et al.:
ELASTIC DISTORTIONS
size, statistically it is much more likely that individual small particles will have very large (or near zero) distortions, simply because the number of active slip planes intersecting them is small. This is probably the reason why the very largest rotation angles are observed only in smaller particles. Finally, as noted previously [l], there is a large discrepancy between the calculated (theoretical) rotation angle and the experimental data. This problem is treated more fully in the following section.
1755
CAUSED BY OROWAN LOOPS
Table I. Transformation matrices of the coordinate systems used for solving equation (4) and determining rotation angle of the (TTO), plane (a) Transformation from the Fe cube axes to those of Cu for ;h; K-S variant satisfying (TtT)c,//(ITO),,;[ttul,.//[Ittl~~ Fe CU
WI
WI IW [OlO]
-l/6 (-2$+
WI
(2$ I)/6
(+ + V/6
(b) Transformation
+ I)/6 -l/6
(3 - 2)/b
WI ($5 - 2)/6 (4 + 2)/6
213
from the shear system to the cube axes 11001
[OlOl
lo011
4. DISCUSSION 4.1. Calculation
of A/l
The elastic state, produced by a homogeneous shear deformation (/I:: = y,,) occurring in a matrix containing nondeformable particles, can be obtained by introducing a hypothetical and homogeneous shear deformation of opposite sign (fir: = -yP) throughout the matrix and particles, thus producing a shear of /I:** = - yP in the particles while retaining the matrix plastically underformed [l, 81. The elastic state of an infinite body containing such a particle can be calculated by introducing an equivalent inclusion [9] possessing an eigenstrain (~2.) given by C,,,(&&”
- G/) = C$k,&,&n
- (/It*)*
(4)
Here, C,,, and C,$, are the elastic constants of the Cu matrix and the a-Fe particles, respectively. S,,,,,,, is given by [lo]
where c is a unit vector. NkP and D(f) are, respectively, the cofactor and the determinant of the matrix, the elements of which are &&&. The elastic distortion /Ilk, contributing to the rotation A/I in the present study is, then, given by Blk= %n&L - (-PI:*).
(6)
For the most numerous variant of the twin-free a-Fe particles, i.e. that satisfying the K-S orientation relationship given in equation (1) flu, is calculated from equations (4) (5) and (6) as
& =
-0.04128 0.68124 ( -0.00126
-0.31876
-0.00126
0.03064 -0.01797
-0.01797 0.01617 1
(7)
for a unit shear p r’* = 1 with all other components of /IX* = 0. The elastic constants used here are C ,,,, = 16.84 x 10”N/m2, C1,22= 12.14 x 1010N/m2 and CmZ = 7.54 x 10” N/m* for the Cu matrix, and Gill = 24.20 x 10” N/m’, Cr,,, = 14.65 x lOi N/m’ and C:,,, = 11.20 x 10” N/m2 for the a-Fe particle [1 11.The coordinate transformation matrices used in the above calculation are listed in Table 1. In the present study, the observation coordinates are the same as those of the shear system. For the shear
strain, flr2* = 0.04, given prior to annealing present experiment, j,2 is given as
in the
/I,* = 0.31876 /Iy2*= 0.01275 rad (0.73”).
(8)
This is the value shown by the broken line in Fig. 8 and, clearly, it represents large overestimation of the experimental values. Using equation (8) to convert the experimental fl,z values to /IF:, the number of Orowan loops removed during annealing (N,,) can be calculated under an assumption of homogeneous shear deformation. For example, N,, is 4.5 for particle A (a = 34 nm) in Fig. 4, 2.6 for particle B (a = 15 nm) in Fig. 5, and 3.3 for particle C (a = 81 nm) in Fig. 5. On the other hand, the theoretical numbers of loops (Nth) expected around the particles from a uniformly distributed shear deformation of 0.04 are 9.5, 4.2, and 11.3, respectively. The average value of AjI in Fig. 8 is 0.22, which yields N, = 1.6 for particles of 18 nm, while the theoretically expected value, Nth, is 5.2. Thus using an assumption of homogeneous shear deformation to compare the experimental data with theory clearly leads to a large discrepancy. In addition, if the deformation were truly homogeneous, the Moire fringes should also be straight and homogeneous, in contrast to the wavy and stepped fringes actually observed. Thus, to explain the experimental results, it is necessary to take into account the discrete nature of the Orowan loops. 4.2. Distortions
caused by discrete Orowan loops
Moire fringes are formed by the interference of electron waves passing through a thin foil specimen and scattered by the a-Fe particles and the Cu matrix [4]. Judging from the relatively small particle size compared with the foil thickness (Fig. 7) it might be expected that the amplitude of the electron waves scattered by the Cu matrix would be strong everywhere, giving a sharp peak at the Bragg angle on a diffraction spot. However, either the incident waves, or ones scattered once by the Cu matrix, will be scattered towards slightly different directions in the vicinity of the a-Fe particles, depending on the
SAT0
1756
et al.:
ELASTIC
DISTORTIONS
location of the scattering planes (since /3,zinduced by Orowan loops is not uniform). For simplicity, if the material is assumed to be elastically homogeneous and isotropic, /I,,,, produced by a dislocation loop (Orowan loop) is given by a line integral [12] as B”fn=
‘%jh C~~Gkrn,/(x
-
X’M’i
dxit
CAUSED
BY OROWAN
LOOPS
I 5-
i
4ON
a \ @if
32-
(9)
with
Y I .o
OOi Distance
from
origin
( 6’)
Fig. 10. Variation of /J2 along the X, y and z axes. & is normalized with respect to the value at the origin, /3e.
where x and x’ are the point observation and the location of a small dislocation segment, respectively, and %= x - x’. The distortion component of present interest, /In, at x = [0, y, 0] may be given in simple form as p,2
=
-
--Lb 8(1 -v)
r
x~l-2v)+2(1-v)($z}{l+($z~5’z
(11)
for a shear loop of radius r (see Fig. 9), lying on a (z, x) plane and having the Burgers vector b (= [b, 0,01). In the present experiments, the electron beam is taken along the z-axis in Fig. 9. The variation of /?,z with (y/r) is shown by curve Y in Fig. 10, normalized against the value of #II2 at y = 0(/I&). filz//Iyz is seen to increase to a maximum value of about 1.1 at y = r/2 and then decrease to 0.8 at y = r. Thus the variation of b,z with y is rather gradual. Note from equation (11) that as the loop radius, r, is decreased, p,z becomes larger. Thus Orowan loops located near the top and bottom of a particle should cause larger rotations. However, as r is decreased, these distortions become increasingly localized. Thus, such Orowan loops will have relatively little influence on the rotation of Moire fringes in the center of the particle. The maximum value of /II2 produced in the center of a particle by an Orowan loop may, there-
fore, be estimated by assuming that the loop lies near the center of the particle. This fringe rotation in the center of a particle is of particular interest here because the experimentally determined rotation angles were measured near the center of the particles, where there was a large area of relatively straight, continuous Moire fringes. Thus, the experimental data actually represents the fringe rotation near the center of the particles. In contrast with the above analysis, if the shear deformation is assumed to be spread out on a spherical particle of radius a, the distortion becomes constant inside the particle and is given as
y = 0, r = a into (1 l), and b$* = 0, /I Fz*= b/2a into equation
Using v = 0.3 and substituting equation
(12), we find that, when the loop is assumed to be smeared out, /?,z is overestimated by a factor of about 1.7 in comparison with the discrete loop calculation. We feel that this overestimation is a major cause of the discrepancy between the theoretical and calculated rotation values in Fig. 8. The difference noted above apparently comes from the inhomogeneous distribution of /?,z along the x and z directions. According to equation (9), the variation of B,z along the x and z directions is written as B = _ (t-2v)b 12 8n(l -v) (x/r)cos 0 - co.9 f3
n/2
I [
X
_n,2
\
4 \
z
\
-___.
/
/
/
/
Fig. 9. Coordinate system used to calculate the distortion caused by a dislocation loop. Outline of particle (dashed line) represents the case where loop radius = particle radius.
1 + (x/r)2 - 2(x/r)cos 0)“’
(x/r)cos 0 +
-
\
{
cos2
e
{ 1 + (x/r)* + 2(x/r)cos}“*
1 de
(13)
and (1 - 2v)b
812= - 4n(l - v)r x/2 X
s
_rr,2
~082
e
{ 1 + (z/r)2 - 2(z/r)sin 0}3/2de
(14)
-/ --+ -/ -c-3 SAT0 et al.:
--/ -/ -
--_
---
(a)
(b)
ELASTIC DISTORTIONS
tili~trm
_____
e
tiIiJcu trace
__---
Figures 11 and 12, drawn to incorporate the most important points from the above analysis, are schematic representations of the Moiri fringe structures expected in large and small particle specimens, respectively. The main features of Fig. 11 are; (1) a large number of steps on the fringes as a result of the presence of discrete dislocations along the particle interface, (2) a strong curvature of the fringes near the particle interface as a result of the inhomogeneous distortion in the vicinity of individual dislocations, and (3) discontinuities along the Moir6 fringes. This last feature results from the fact that /I,2 varies along small particle
__-._*e after
\
_-_-~-__-_0,x--
‘\,A ,
0.2/
/‘.
I/’
ox-;
’
- 0.6
‘a n b
5 ‘0 f L
nm
,d
/’
Ol-.
- 0.4
, -‘\
I’ _ _&--‘--’
49
nm
.$ ::
- 0.2;
0
1
500
1
700
600 temperature
0
(K)
Fig. 13. Comparison of microscopic Moire fringe rotation data (points) with macroscopic softening curves from Ref. (7). Average particle diameter of the samples used in the softening curves is indicated.
#4
4.3. Schematic representation of Moire fringe struc-
anneallng
AB
-
E 0”
35
Annrahng
tures
,“_
- 1.0
0.3-
:
400
at (x, 0,O) and (0, 0, z), respectively. Curves X and Z in Fig. 10 show /?,*obtained by numerical integration of the above equations. As would be anticipated, the distortion /?,2 increases drastically as the dislocation loop is approached. However, this distortion is, again, likely to be very localized and have relatively little influence on the rotation of Moirk fringes near the particle center.
(b)
p
1757
BY OROWAN LOOPS
I’
i
L e f <
Fig. 11. Schematic picture of Moirt fringes in a large particle. (a) Before and (b) after annealing.
(a 1 as-deformed
CAUSED
_
/--
Fig. 12. Schematic picture of Moirt fringes in a small particle. Note that fringes are simpler and the rotation angle is greater than in Fig. 11.
the z-axis, and the particle thickness (through which the electron beam must travel along the z-axis) also varies depending on the x and y coordinates. Discontinuities may, therefore, appear as a result of interference when electron waves are passed through such a crystal. For a small particle, Fig. 12, the number of steps should be smaller because fewer dislocations will encircle the particle. Further, although we would expect the Moirb fringes to curve near the particle interface in a manner similar to that observed in large particles, such curvature may not be observable from a practical standpoint because of the small crosssectional area involved. Finally, there is a fairly high probability that the fringe rotation will be considerably larger than that observed in a large particle. When Figs 11 and 12 are compared with actual micrographs (e.g. Figs 4 and 5, respectively), they are seen to well represent the observed MoirC fringe patterns. 4.4. Comparison annealing curves
of microscopic
and macroscopic
The average value of AD obtained by analyzing N 500 particles during 0.5 h isochronal annealing is plotted against annealing temperature in Fig. 13, in order to compare the results of microscopic analysis with the macroscopic softening behavior. The blocken lines are recovery of work-hardening curves obtained in the previous study [7] for two different particle sizes. (The vertical scales on the plot have been chosen so that the “saturation” values roughly coincide.) Note that the agreement between the lines and the rotation data is excellent in general form as well as in the temperature range of the recovery. As seen in Figs 4 and 5, rotation of Moir6 fringes certainly occurs at different temperatures depending on the particle size. This is in good accordance with the strong size dependence of the climb rate of Orowan loops, described previously [5]. A wide distribution of particle sizes, thus, causes the gradual
1758
SAT0 et al.:
ELASTIC DISTORTIONS
recovery of work hardening over a wide temperature range, as observed in Fig. 13.
CAUSED BY OROWAN LOOPS
comparison with that calculated for discrete Orowan loops. REFERENCES
5. CONCLUSIONS
By examining the detailed microstructure of Moire fringes formed by cr-Fe particles embeded in a Cu matrix, the lattice distortion induced by Orowan loops was detected and compared with the results of elasticity calculations. The conclusions of the present study are summarized as follows. 1. The distortion is seemingly independent of the particle depth in a thin foil specimen, indicating that any surface relaxation effect is not of detectable magnitude. 2. Data for the isochronal annealing of Orowan loops, observed by the rotation of Moire fringes, agrees well with an isochronal annealing curve obtained in macroscopic experiments [2]. 3. Assumption of homogeneous deformation leads to an overestimation of the magnitude of the Moire fringe rotation by a factor of approximately two in
1.
A. Sato, S. Onaka, R. Monzen, K. Kitagawa and T.
Mori, Acta metall. 32, 655 (1984). 2. M. Kato, R. Monzen and T. Mori, Acra metall. 26, 605 (1978). 3. R. Monzen, A. Sato and T. Mori, Acta metall. 31, 1267 (1983). 4. P. B. Hirsch, A. Howie, P. B. Nicholson, D. W. Pashley and M. J. Whelan, Electron Microscopy of Thin Crystals. Butterworths, London (1971). 5. T. Mori and H. Tokushige, Acta metall. 25,635 (1977). 6. A. Sato, M. Mori and-T. Mori, Trans. Jap& Ins;. Metals 25. 863 11984).
7. R. Mom&,
Y. i(awaguchi
and T. Mori, Acta metall.
30, 965 (1982).
8. K. Tanaka and T. Mori, Acta merall. 18, 931 (1970). 9. J. D. Eshelby, Proc. R. Sot. Land. A241, 376 (1957); A252, 561 (1959).
10. N. Kinoshita and T. Mura, Physica Status solidi (a) 5, 759 (1971).
11. J. P. Hirth and J. Lothe, Theory of Dislocations, p. 762. McGraw-Hill, New York (1968). 12. T. Mura, Phil. Mag. 8, 843 (1963).
oool-6160/86 $3.00+ 0.00 Copyright 0 1986Pergamon Journals Ltd
MICROSCOPIC OBSERVATION OF ELASTIC DISTORTIONS CAUSED BY OROWAN LOOPS Department
A. SATO, M. MORI, P. D. FLJNKENBUSCH and T. MORI of Materials Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku Yokohama 227, Japan (Received
19 September
1985; in revised form 6 January 1986)
Abstract-The elastic distortion induced by Orowan loops surrounding a-Fe particles embedded in a Cu matrix has been detected by observing the rotation of Moirt fringes formed by {11I},, and {1lo}, planes. On the basis of detailed electron microscopic observations, the loss of Orowan loops upon annealing has been examined and correlated with the softening behavior observed following macroscopic stress-strain tests. Quantitative analysis has, further, revealed that consideration of the elastic field produced by discrete Orowan loops gives the correct magnitude for the elastic distortion as well as explaining the detailed structure of the Moir& fringes. R&un~Nous avons mis en tvidence la distorsion tlastique provoqute par des boucles d’orowan entourant des particules de fer-a dans une matrice de cuivre en observant les franges de Moir& form&es par les plans {11I},, et { 1lo},. A partir d’observations d&ill&s de microscopic tlectronique, nous avons 6tudie la disparition des boucles d’orowan au tours d’un revenu et nous l’avons reliQ g l’adoucissement que l’on observe B la suite d’essais contrainte/dtformation macroscopiques. Une analyse quantitative a de plus montrk que la prise en considtration deu champ tlastique produit par des boucles d’orowan d&r&es foumit la grandeur correcte pour la distorsion tlastique et explique la structure d&ail& des franges de Moirb. Zusannnenfassung-Die elastiche Verzerrung, die von Orowanschleifen an a-Fe-Teilchen in einer CuMatrix ausgeht, wurde aus der Rotation von Moirke-Streifen, gebildet aus den (11 l}c,- und {1IO},-Gitterebenen, ermittelt. Ahhand ausfiihrlicher elektronenmikroskopischer Reobachtungen wurde die Verringerung der Orowanschliefen durch Ausheilen untersucht und mit dem Entfestigungsverhalten, das nach makroskopischen Spannungs-Dehnungsversuchen beobachtet wurde, korreliert. Die quantitative Analyse zeigt AuDerdem, da0 das von einer Orowanschliefe erzeugte elastisches Feld die richtige Gr6Benordnung fiir die elastische Verzerrung liefert und da13damit die Einzelheiten der Moirkestruktur erklart werden kBnnen.
1. INTRODUCTION
A Cu-Fe
In a previous study [1], we demonstrated that the elastic distortion caused by Orowan loops surrounding second phase particles may be observed by examining the rotation of MoirC fringes formed by nearly parallel matrix and particle lattice planes. We were also able to conclude, on the basis of these measurements, that such elastic distortion was responsible for the macroscopic stress-strain behavior of the Cu-Fe alloy used in the study. However, the elastic distortion observed during these experiments was considerably smaller than that predicted by an elasticity calculation. In the present study we seek to eliminate this discrepancy by taking tie discrete nature of Orowan loops into consideration. That is, we take into account the fact that the deformation is localized in the vicinity of the loops, rather than assuming a continuous “smeared-out” distribution. This appreach also allows us to assess the extent to which assumptions of uniform distortion are applicable in spite of the fact that Orowan loops actually produce inhomogeneous distortions within particles.
alloy similar to that used in [1] but with
a different crystallographic orientation was employed in this study. In this alloy a dispersion of a-Fe particles satisfying a K-S relationship with the matrix is produced by a simple thermomechanical treatment, and Moir& fringes, resulting from rotation of a {llO},, plane with respect to a (lll}c, plane, may then be imaged. The particle orientation examined here (Fig. 1) is convenient from the standpoint of detecting the local elastic distortion induced by discrete Orowan loops. By means of stereoscopic electron microscopy, the influence of the fiol surface is also clarified in the current work.
2. EXPERIMENTAL
Single crystal sheets of 2 x 20 x 200 mm were prepared under vacuum from a Cu-1.52 wt% Fe alloy by the Bridgman method using a seed. 100 mm long specimens with tensile direction T.D.1 (Fig. 2) were then cut using a spark cutter, polished mechanically, and cleaned in a mixture of HNO, and H,O. These
1751
1752
SAT0 et al.: ELASTIC DISTORTIONS CAUSED BY OROWAN LOOPS
Before Deformation (After Anneolino)
After Deformation
T.O. I 7.0.2
Fig. 1. Schematic illustration of a spherical a-Fe particle deformed in a Cu matrix. Distortion /?,*was measured using Moire fringes formed by (TIT),, and (TTO), planes. Fig. 2. Stereographic projection showing shear system and crystallographic orientation.
specimens were solution treated at 1223 K for 6 h, quenched into ice water, and aged at 973 K for either 6 or 24 h to introduce y-Fe particles of 36 or 47 nm average diameter, respectively. Subsequent deformation by 20% elongation at 77 K transformed the y-Fe particles into a-Fe and the samples were then annealed at 973 K for 2 h to make the a-Fe particles essentially twin free. Most of the particles satisfied the K-S relationship with the matrix described by (TlT)c,//(TTO),;
~~W,,//[~111,.
(1)
These specimens were further elongated by 2% at 77 K along T.D.2, crystallographically the same tensile direction as T.D.1, in order to introduce Orowan loops with Burgers vector a/2[011]. Thin foils, parallel to (ZTl),, and containing both the shear direotion [Ol l] and the shear plane normal [TIT], were sliced from the strained sample and examined in a high voltage electron microscope, H-1250s equipped with a 55” double tilt stage, at an operating voltage of 1000 kV. 0.5 h isochronal and 450 K isothermal softening treatments (to anneal out the Orowan loops) were conducted outside the electron microscope in a hydrogen atmosphere to prevent both evaporation and oxidation. For all observations, including stereographic microscopy, g = [TIT], i.e. g perpendicular to the Cu K-S plane, was used with specimen orientation adjusted to provide identical diffraction conditions by observing the location of the Kikuchi lines. As the Moire fringes had a somewhat wavy appearance, their orientation was determined on enlarged TEM micrographs by measuring the angle between the fringes in the center of the particles and the [TIT] direction. tin order to ensure that the Moire fringe rotation is caused only by annealing-out of deformation induced elastic strains, Orowan loop strain, we have conducted an experiment on a nondeformed sample. A thin foil was made out of the sample with twin-free Fe particles. Moire fringes were observed before and after annealing at 573 K for 0.5 h. No rotation of the fringes occurred on the annealing.
3. EXPERIMENTAL
RESULTS
A description of the general features observed in this material has been given in (1) and will not be repeated here. Instead we wish to focus on more detailed aspects of the behavior of the Moire fringes. Figure 3 is a representative micrograph of a deformed specimen. Particle E (with circular Moire fringes) does not satisfy equation (1) C has a complicated structure which is actually due to overlapping images formed by two particles, and D is a coherent y-Fe particle. Thus, of the particles in this figure, only A, a relatively large a-Fe particle located near the center of the foil, was suitable for analysis. Figure 4(a)-(d) show the behavior of particle A upon l/2 h isochronal annealing at various temperatures, and are representative of the results for large a-Fe particles. It is clear that both the steps and the curvature of the Moire fringes gradually disappear as the temperature is raised. Concurrently the fringes rotate in a clockwise direction, consistent with the expected relaxation of elastic distortion.?
Fig. 3. Representative view of as-deformed crystal containing twin-free a-Fe and y-Fe particles. B is a pit formed on a foil surface by particle dissolution.
SAT0 et al.: ELASTIC DISTORTIONS CAUSED BY OROWAN LOOPS
(a)
(b)
1753
bottom of the particle, where the loop radius is small, as anticipated from first principle considerations [5]. The appearance of small particles, such as particle B in Fig. 5, is somewhat different. The number of steps on the Moire fringes is smaller and, consistent with the particle size dependence of Orowan loop climb, the rotation is completed at a lower temperature. Finally, although there is a great deal of scatter in the data, the largest rotation angles, A& are observed for small rather than large particles, as shown in our previous study [I]. For example, particle B gives Afl = 0.14, 0.43 and 0.45” in Fig. 5(b), (c) and (d), respectively. This should be compared with the maximum value of 0.34” found for the large particle, A, in Fig. 4. This observation will be dis-
(a)
Cd)
Fig. 4. Rotation of Moirk fringes observed in a large particle during 0.5 h isochronal annealing. (a) Before annealing. (b) Annealed at 473 K. (c) 523 K. (d) 573 K. The actual rotation of the (TiO), planes with respect to the (11 f),, plane, fi, is given by [ 1,4]
where o[is the rotation angle of the Moire fringes with respect to the g vector, and d,,( = 0.2083 nm) and dFe(=0.2024 nm) are the lattice spacings of Cu and Fe, respectively. The difference A#?= j3(before anneaIing) - jI (after ann~ling)
(d)
(3)
determined from equation (2) is found to be O.I8, 0.30 and 0.34” for Fig. 4(b), (c) and (d), respectively. In addition it should be noted that the Moire fringes [Fig. 4(b) and (c)J start to rotate near the top and
Fig. 5. Rotation of Moire fringes observed in a small particle during 0.5 h isochronal annealing. (a) Before annealing. (b) Annealed at 473 K. (c) 523 K. (d) 573 K.
1754
SAT0 et al.:
ELASTIC DISTORTIONS
cussed later with reference to the inhomogeneous nature of slip. An example of the annealing behavior of Orowan loops during isothermal annealing is given in Fig. 6. The wavy structure of the Moire fringes becomes simpler with increasing annealing time, particularly at the beginning of annealing. This behavior is, again, attributable to the variation of loop climb rate as a function of position [S]. It is also apparent that some curvature of the fringes remains even after annealing for 90 min at 450 K. This is consistent with the fact that macroscopic annealing curves show that loss of work-hardening occurs over a rather wide temperature range [q.
(a)
(b)
CAUSED BY OROWAN LOOPS
i
600
-200 ,
t
IOOOnm
,
Fig. 7. Particle location determined by stereoscopic electron microscopy.
The height of the top and bottom surfaces of all thin foils was determined by stereographic electron microscopy, using the edges of pits formed by particle dissolution to locate the foil surfaces. All particles examined were then located with respect to the foil surfaces with an accuracy of better than a few nanometers. Figure 7 shows the distribution of particles in a typical foil cross-section obtained by this technique. In Fig. 8, the rotation after “full annealing” (623 K for 0.5 h) is plotted against the particle depth from the foil surface. Three important observations may be made from this data. First, there is no apparent tendency for the rotation angle to decrease as the foil surface is approached. In other words, there is little or no “surface relaxation effect”, contrary to our previous hypothesis [7]. Second, there is a large scatter in the data. The predominant reason for this must be a nonuniform distribution of slip. Particles intersected by a larger than average number of dislocations during the deformation will have correspondingly higher fringes rotation angles, while other particles, virtually unaffected by the slip, will exhibit rotation angles near zero. This effect should be particularly important for small particles. Although, from fundamental dispersion hardening theory [8], the aoerage distortion should not depend on particle
_____________ ________ _.J-_\ Cd)
Fig. 6. Rotation of Moirb fringes caused by isothermal annealing at 450 K. (a) Before annealing. (b) Annealed for 10min. fcl 30min. Id1 Wmin.
Depthfrom
surface
(nml
Fig. 8. A/?, after full annealing at 623 K for 0.5 h, plotted against particle depth from the foil surface.
SAT0 et al.:
ELASTIC DISTORTIONS
size, statistically it is much more likely that individual small particles will have very large (or near zero) distortions, simply because the number of active slip planes intersecting them is small. This is probably the reason why the very largest rotation angles are observed only in smaller particles. Finally, as noted previously [l], there is a large discrepancy between the calculated (theoretical) rotation angle and the experimental data. This problem is treated more fully in the following section.
1755
CAUSED BY OROWAN LOOPS
Table I. Transformation matrices of the coordinate systems used for solving equation (4) and determining rotation angle of the (TTO), plane (a) Transformation from the Fe cube axes to those of Cu for ;h; K-S variant satisfying (TtT)c,//(ITO),,;[ttul,.//[Ittl~~ Fe CU
WI
WI IW [OlO]
-l/6 (-2$+
WI
(2$ I)/6
(+ + V/6
(b) Transformation
+ I)/6 -l/6
(3 - 2)/b
WI ($5 - 2)/6 (4 + 2)/6
213
from the shear system to the cube axes 11001
[OlOl
lo011
4. DISCUSSION 4.1. Calculation
of A/l
The elastic state, produced by a homogeneous shear deformation (/I:: = y,,) occurring in a matrix containing nondeformable particles, can be obtained by introducing a hypothetical and homogeneous shear deformation of opposite sign (fir: = -yP) throughout the matrix and particles, thus producing a shear of /I:** = - yP in the particles while retaining the matrix plastically underformed [l, 81. The elastic state of an infinite body containing such a particle can be calculated by introducing an equivalent inclusion [9] possessing an eigenstrain (~2.) given by C,,,(&&”
- G/) = C$k,&,&n
- (/It*)*
(4)
Here, C,,, and C,$, are the elastic constants of the Cu matrix and the a-Fe particles, respectively. S,,,,,,, is given by [lo]
where c is a unit vector. NkP and D(f) are, respectively, the cofactor and the determinant of the matrix, the elements of which are &&&. The elastic distortion /Ilk, contributing to the rotation A/I in the present study is, then, given by Blk= %n&L - (-PI:*).
(6)
For the most numerous variant of the twin-free a-Fe particles, i.e. that satisfying the K-S orientation relationship given in equation (1) flu, is calculated from equations (4) (5) and (6) as
& =
-0.04128 0.68124 ( -0.00126
-0.31876
-0.00126
0.03064 -0.01797
-0.01797 0.01617 1
(7)
for a unit shear p r’* = 1 with all other components of /IX* = 0. The elastic constants used here are C ,,,, = 16.84 x 10”N/m2, C1,22= 12.14 x 1010N/m2 and CmZ = 7.54 x 10” N/m* for the Cu matrix, and Gill = 24.20 x 10” N/m’, Cr,,, = 14.65 x lOi N/m’ and C:,,, = 11.20 x 10” N/m2 for the a-Fe particle [1 11.The coordinate transformation matrices used in the above calculation are listed in Table 1. In the present study, the observation coordinates are the same as those of the shear system. For the shear
strain, flr2* = 0.04, given prior to annealing present experiment, j,2 is given as
in the
/I,* = 0.31876 /Iy2*= 0.01275 rad (0.73”).
(8)
This is the value shown by the broken line in Fig. 8 and, clearly, it represents large overestimation of the experimental values. Using equation (8) to convert the experimental fl,z values to /IF:, the number of Orowan loops removed during annealing (N,,) can be calculated under an assumption of homogeneous shear deformation. For example, N,, is 4.5 for particle A (a = 34 nm) in Fig. 4, 2.6 for particle B (a = 15 nm) in Fig. 5, and 3.3 for particle C (a = 81 nm) in Fig. 5. On the other hand, the theoretical numbers of loops (Nth) expected around the particles from a uniformly distributed shear deformation of 0.04 are 9.5, 4.2, and 11.3, respectively. The average value of AjI in Fig. 8 is 0.22, which yields N, = 1.6 for particles of 18 nm, while the theoretically expected value, Nth, is 5.2. Thus using an assumption of homogeneous shear deformation to compare the experimental data with theory clearly leads to a large discrepancy. In addition, if the deformation were truly homogeneous, the Moire fringes should also be straight and homogeneous, in contrast to the wavy and stepped fringes actually observed. Thus, to explain the experimental results, it is necessary to take into account the discrete nature of the Orowan loops. 4.2. Distortions
caused by discrete Orowan loops
Moire fringes are formed by the interference of electron waves passing through a thin foil specimen and scattered by the a-Fe particles and the Cu matrix [4]. Judging from the relatively small particle size compared with the foil thickness (Fig. 7) it might be expected that the amplitude of the electron waves scattered by the Cu matrix would be strong everywhere, giving a sharp peak at the Bragg angle on a diffraction spot. However, either the incident waves, or ones scattered once by the Cu matrix, will be scattered towards slightly different directions in the vicinity of the a-Fe particles, depending on the
SAT0
1756
et al.:
ELASTIC
DISTORTIONS
location of the scattering planes (since /3,zinduced by Orowan loops is not uniform). For simplicity, if the material is assumed to be elastically homogeneous and isotropic, /I,,,, produced by a dislocation loop (Orowan loop) is given by a line integral [12] as B”fn=
‘%jh C~~Gkrn,/(x
-
X’M’i
dxit
CAUSED
BY OROWAN
LOOPS
I 5-
i
4ON
a \ @if
32-
(9)
with
Y I .o
OOi Distance
from
origin
( 6’)
Fig. 10. Variation of /J2 along the X, y and z axes. & is normalized with respect to the value at the origin, /3e.
where x and x’ are the point observation and the location of a small dislocation segment, respectively, and %= x - x’. The distortion component of present interest, /In, at x = [0, y, 0] may be given in simple form as p,2
=
-
--Lb 8(1 -v)
r
x~l-2v)+2(1-v)($z}{l+($z~5’z
(11)
for a shear loop of radius r (see Fig. 9), lying on a (z, x) plane and having the Burgers vector b (= [b, 0,01). In the present experiments, the electron beam is taken along the z-axis in Fig. 9. The variation of /?,z with (y/r) is shown by curve Y in Fig. 10, normalized against the value of #II2 at y = 0(/I&). filz//Iyz is seen to increase to a maximum value of about 1.1 at y = r/2 and then decrease to 0.8 at y = r. Thus the variation of b,z with y is rather gradual. Note from equation (11) that as the loop radius, r, is decreased, p,z becomes larger. Thus Orowan loops located near the top and bottom of a particle should cause larger rotations. However, as r is decreased, these distortions become increasingly localized. Thus, such Orowan loops will have relatively little influence on the rotation of Moire fringes in the center of the particle. The maximum value of /II2 produced in the center of a particle by an Orowan loop may, there-
fore, be estimated by assuming that the loop lies near the center of the particle. This fringe rotation in the center of a particle is of particular interest here because the experimentally determined rotation angles were measured near the center of the particles, where there was a large area of relatively straight, continuous Moire fringes. Thus, the experimental data actually represents the fringe rotation near the center of the particles. In contrast with the above analysis, if the shear deformation is assumed to be spread out on a spherical particle of radius a, the distortion becomes constant inside the particle and is given as
y = 0, r = a into (1 l), and b$* = 0, /I Fz*= b/2a into equation
Using v = 0.3 and substituting equation
(12), we find that, when the loop is assumed to be smeared out, /?,z is overestimated by a factor of about 1.7 in comparison with the discrete loop calculation. We feel that this overestimation is a major cause of the discrepancy between the theoretical and calculated rotation values in Fig. 8. The difference noted above apparently comes from the inhomogeneous distribution of /?,z along the x and z directions. According to equation (9), the variation of B,z along the x and z directions is written as B = _ (t-2v)b 12 8n(l -v) (x/r)cos 0 - co.9 f3
n/2
I [
X
_n,2
\
4 \
z
\
-___.
/
/
/
/
Fig. 9. Coordinate system used to calculate the distortion caused by a dislocation loop. Outline of particle (dashed line) represents the case where loop radius = particle radius.
1 + (x/r)2 - 2(x/r)cos 0)“’
(x/r)cos 0 +
-
\
{
cos2
e
{ 1 + (x/r)* + 2(x/r)cos}“*
1 de
(13)
and (1 - 2v)b
812= - 4n(l - v)r x/2 X
s
_rr,2
~082
e
{ 1 + (z/r)2 - 2(z/r)sin 0}3/2de
(14)
-/ --+ -/ -c-3 SAT0 et al.:
--/ -/ -
--_
---
(a)
(b)
ELASTIC DISTORTIONS
tili~trm
_____
e
tiIiJcu trace
__---
Figures 11 and 12, drawn to incorporate the most important points from the above analysis, are schematic representations of the Moiri fringe structures expected in large and small particle specimens, respectively. The main features of Fig. 11 are; (1) a large number of steps on the fringes as a result of the presence of discrete dislocations along the particle interface, (2) a strong curvature of the fringes near the particle interface as a result of the inhomogeneous distortion in the vicinity of individual dislocations, and (3) discontinuities along the Moir6 fringes. This last feature results from the fact that /I,2 varies along small particle
__-._*e after
\
_-_-~-__-_0,x--
‘\,A ,
0.2/
/‘.
I/’
ox-;
’
- 0.6
‘a n b
5 ‘0 f L
nm
,d
/’
Ol-.
- 0.4
, -‘\
I’ _ _&--‘--’
49
nm
.$ ::
- 0.2;
0
1
500
1
700
600 temperature
0
(K)
Fig. 13. Comparison of microscopic Moire fringe rotation data (points) with macroscopic softening curves from Ref. (7). Average particle diameter of the samples used in the softening curves is indicated.
#4
4.3. Schematic representation of Moire fringe struc-
anneallng
AB
-
E 0”
35
Annrahng
tures
,“_
- 1.0
0.3-
:
400
at (x, 0,O) and (0, 0, z), respectively. Curves X and Z in Fig. 10 show /?,*obtained by numerical integration of the above equations. As would be anticipated, the distortion /?,2 increases drastically as the dislocation loop is approached. However, this distortion is, again, likely to be very localized and have relatively little influence on the rotation of Moirk fringes near the particle center.
(b)
p
1757
BY OROWAN LOOPS
I’
i
L e f <
Fig. 11. Schematic picture of Moirt fringes in a large particle. (a) Before and (b) after annealing.
(a 1 as-deformed
CAUSED
_
/--
Fig. 12. Schematic picture of Moirt fringes in a small particle. Note that fringes are simpler and the rotation angle is greater than in Fig. 11.
the z-axis, and the particle thickness (through which the electron beam must travel along the z-axis) also varies depending on the x and y coordinates. Discontinuities may, therefore, appear as a result of interference when electron waves are passed through such a crystal. For a small particle, Fig. 12, the number of steps should be smaller because fewer dislocations will encircle the particle. Further, although we would expect the Moirb fringes to curve near the particle interface in a manner similar to that observed in large particles, such curvature may not be observable from a practical standpoint because of the small crosssectional area involved. Finally, there is a fairly high probability that the fringe rotation will be considerably larger than that observed in a large particle. When Figs 11 and 12 are compared with actual micrographs (e.g. Figs 4 and 5, respectively), they are seen to well represent the observed MoirC fringe patterns. 4.4. Comparison annealing curves
of microscopic
and macroscopic
The average value of AD obtained by analyzing N 500 particles during 0.5 h isochronal annealing is plotted against annealing temperature in Fig. 13, in order to compare the results of microscopic analysis with the macroscopic softening behavior. The blocken lines are recovery of work-hardening curves obtained in the previous study [7] for two different particle sizes. (The vertical scales on the plot have been chosen so that the “saturation” values roughly coincide.) Note that the agreement between the lines and the rotation data is excellent in general form as well as in the temperature range of the recovery. As seen in Figs 4 and 5, rotation of Moir6 fringes certainly occurs at different temperatures depending on the particle size. This is in good accordance with the strong size dependence of the climb rate of Orowan loops, described previously [5]. A wide distribution of particle sizes, thus, causes the gradual
1758
SAT0 et al.:
ELASTIC DISTORTIONS
recovery of work hardening over a wide temperature range, as observed in Fig. 13.
CAUSED BY OROWAN LOOPS
comparison with that calculated for discrete Orowan loops. REFERENCES
5. CONCLUSIONS
By examining the detailed microstructure of Moire fringes formed by cr-Fe particles embeded in a Cu matrix, the lattice distortion induced by Orowan loops was detected and compared with the results of elasticity calculations. The conclusions of the present study are summarized as follows. 1. The distortion is seemingly independent of the particle depth in a thin foil specimen, indicating that any surface relaxation effect is not of detectable magnitude. 2. Data for the isochronal annealing of Orowan loops, observed by the rotation of Moire fringes, agrees well with an isochronal annealing curve obtained in macroscopic experiments [2]. 3. Assumption of homogeneous deformation leads to an overestimation of the magnitude of the Moire fringe rotation by a factor of approximately two in
1.
A. Sato, S. Onaka, R. Monzen, K. Kitagawa and T.
Mori, Acta metall. 32, 655 (1984). 2. M. Kato, R. Monzen and T. Mori, Acra metall. 26, 605 (1978). 3. R. Monzen, A. Sato and T. Mori, Acta metall. 31, 1267 (1983). 4. P. B. Hirsch, A. Howie, P. B. Nicholson, D. W. Pashley and M. J. Whelan, Electron Microscopy of Thin Crystals. Butterworths, London (1971). 5. T. Mori and H. Tokushige, Acta metall. 25,635 (1977). 6. A. Sato, M. Mori and-T. Mori, Trans. Jap& Ins;. Metals 25. 863 11984).
7. R. Mom&,
Y. i(awaguchi
and T. Mori, Acta metall.
30, 965 (1982).
8. K. Tanaka and T. Mori, Acta merall. 18, 931 (1970). 9. J. D. Eshelby, Proc. R. Sot. Land. A241, 376 (1957); A252, 561 (1959).
10. N. Kinoshita and T. Mura, Physica Status solidi (a) 5, 759 (1971).
11. J. P. Hirth and J. Lothe, Theory of Dislocations, p. 762. McGraw-Hill, New York (1968). 12. T. Mura, Phil. Mag. 8, 843 (1963).