M. F. Ashby, Phil. Msg. 21, 399 (1970). 2. H. Fujita and ... N. Hansen, Acta metal!. 25, 863 (1977). 7. S. Miyazaki ... Baskes, Phil. Mag. 28,. 301 (1973). 23. Y. Ono ...
Acta Mutalhrqira. Vol. 27. pp. 855 lo 862 0 Pergamon Press Ltd 1979 Printed in Great Brifain
EFFECT OF SPECIMEN THICKNESS ON MECHANICAL PROPERTIES OF POLYCRYSTALLINE AGGREGATES WITH VARIOUS GRAIN SIZES SHUICHI
MIYAZAKI,
KENJI SHIBATAt
and HIROSHI FUJITA
Department of Materials Science and Engineering, Faculty of Engineering, Osaka University, Yamada-Kami, Suita, Osaka 565. Japan (Receiced 11 July 1978; in rrcised form 22 September 197X)
Abstract-The flow stress of polycrystalline AI, Cu. Cu-13 at:, Al and Fe have been investigated as a function of the grain size and the specimen thickness. The flow stress decreases with decreasing specimen thickness when the ratio of specimen thickness (t) to grain size (d) is smaller than a critical value. The critical value of t/d increases with decreasing both grain size and stacking fault energy. Based on the experimental results the radius of affected zone, in which individual grains strongly interact with each other caused by a deformed grain. is estimated using a simple model. The result shows that the long-range interaction among individual grains expands into a wide region across the first nearest-neighbor grains. Riswn-On a Ctudit la contrainte d’ecoulement de polycristaux tion de la taille des grains et de I’epaisseur des Cchantillons. lorsqu’on diminue l’tpaisseur de l’echantillon, a condition que grains soit inferieur a une valeur critique. Cette valeur critique des grains et I’tnergie de dtfaut d’empilement.
d’Al, Cu. Fe et Cu-13at~~~Aien foncLa contrainte d’ecoulement diminue Ic rapport de celle-ci a la taille des augmente lorsqu’on diminue la taille
Zusammenfassung- Die FlieDspannung von polykristallinem Al, Cu. Cu-13At.%AI und Fe wurde in Abhlngigkeit von KorngriiBe und Probendicke untersucht. Die FlieBspannung nimmt mit abnehmender Probendicke ab, wenn das Verh~ltnis von Probendicke (t) und KorngrGBe (d) einen kritischen Wert unterschreitet. Dieser kritische Wert t/d nimmt zu sowohl mit der KorngraDe als such mit der Stapelfehlerenergie.
1. INTRODUCTION The slip mode in individual grains of polycrystals is strongly affected by the interaction among adjacent grains [i-7]. Thus the mechanical properties of polycrystalline aggregates are closely related to the interaction among grains. This interaction is considered to expand into a wide region across the first nearestneighbor grains so that the flow stress markedly decreases with decreasing specimen thickness when the number of grains contained along the thickness direction becomes smaller than a critical value [S-11]. Many theories and experiments have been carried out to investigate the interaction [12-181, but most of them were concerned with the interaction among the first nearest-neighbor grains. Practically, however, the long-range interaction among individual grains is very important in the deformation of polycrystals 1191. The long-range interaction is a function of the slip mode in each individual grain, and thus it is very sensitive to both the grain size and the stacking fault energy of the specimens. From this point of view the present experiment has been carried out to investigate the flow stress of various polycrystals t Now at Research and Development Center, Tokyo Shibaura Electric Co., Ltd., Horikawa-cho 72, Kawasaki, Kanagawa 210, Japan. 855
as a function of the long-range interaction among grains by changing both grain size and stacking fault energy. Based on the experimental results, the long-range interaction among grains in polycrystals is estimated and the deformation mechanism of polycrystals is discussed. 2. SPECIMENS AND EXPERIMENTAL PROCEDURES Materials
used were poIycrystalline
99.99 wto/o Al,
99.99 wt.%Cu, Cu-13 at% Al and Fe. The chemical compositions are shown in Table 1. These materials were heavily cross-rolled at room temperature and followed by annealing in order to avoid the rolling texture. Grain size was controlled in the range from 16 to 1801.~m in all the specimens. After these treatments, the plate specimens were finished for tensile testing and the specimen thickness was changed in the range from 0.045 to 1.840mm for each grain size by chemical- and electro-polishing. The plate specimens used for tensile testing were 6mm in width and I2 mm in gauge length, and the shoulder of the part was 37.5 mm parallel in the radius of curvature. The specimens were stretched with a Shimadzu Autograph IS-5000 tensile machine of Instron type with a cross-
856
MIYAZAKI, SHIBATA
FUJITA:
AND
SPECIMEN
THICKNESS
EFFECT ON FLOW STRESS
Table 1. Chemical compositions (wt.%) Fe
Si
cu
S
99.994wt%Al
0.002
0.002
0.601
-
99,99lwt%Cu
0.0035
-
-
0.0032
0.006
-
0.002
Fe
_
head velocity of 0.5 mm/min. Slip mode and dislocation structures have been observed by optical and electron microscopies. The electron microscope used was of Hitachi HU-2000 type operated at 2 MV. 3. EXPERIME~AL
REZSUL’IS
3.1 Thickness &ecf on the flow stress Figures l(a), (b), (c) and (d) show relationships between the flow stresses at various strains and the ratio of the specimen thickness (t) to the grain size (6) in Al, Cu, Cu-13 at% Al and Fe, respectively. Grain sizes of these specimens are (a) 180, (b) 65, (c) 40 and (d) 25pm, respectively. It is noted in Fig. 1 that the fiow stresses at various strains of Al, Cu and Cu-13 at% Al decrease with decreasing specimen thickness when the value of t/d becomes smaller than a fixed value, i.e. the critical value, independent of the amount of strain. In Fig. 2 the flow stresses at various strains in Fig. 1 are normalized by each saturated value in the
Bi
Pt 0.0008 -
c _
Mn -
P
Al -
0.0005
-
-
-
-
-
0.009
0.001
0.003
0.001
thick ranges. It is noted in Fig. 2 that all curves taken at various strains almost coincide with each other in Al, Cu and Cu-13at%Al. Figure 3 shows the effect of grain size on the critical value of t/d in Al, Cu, Cu-13 at% AI and Fe at 20% strain. There are two ~mmon tendencies in all metals and alloy: (1) the critical value of t/d increases with decreasing grain size; (2) extrapolation of curves takes the same flow stress at zero value of t/d independent of the grain size. The extrapolated value of flow stress corresponds to the flow stress of two-dimensional polycrystal of these metals and alloy in which the interaction among grains does not occur along the thickness direction. 3.2 Thickness effect on the ~eformution of surface grains Figures 4(a) and (b) show the thickness effect on the deformation of surface grains in the Cu specimens of 0.25 and 1.50mm thick, respectively, at 20”/, strain. Grain size in both specimens is about 180pm, and
(b)
cu 6.65 w
lx).
Fig. 1. Thickness effect on the flow stresses at various strains in polycrystalline (a) Al, (b) Cu, (c) Cu-13 at% Al and (d) Fe.
MIYAZAKI,
01.
‘.
SHIBATA
“.
AND
FUJITA:
SPECIMEN
aJ
IO
5
Ttvcknesr/Gfom
THICKNESS
01
Size
5
IO size
ThiiknW./GfOm
I5
Fig. 2. Thickness effect on the normalized flow stresses at various strains in polycrystalline (b) Cu, (c) Cu-13 at% Al and (d) Fe.
thus the value of t/d in (a) and (b) is about 1.4 and 8.3, respectively. It is noted in Fig. 4 that crystal rotation of each individual grain is considerably larger in the thin specimen compared with that in the thick specimen. This means that the constraining force of each grain caused by the interaction among grains markedly decreases with decreasing specimen thickness. 3.3 Effect of constrainingforce ture
on the dislocation ,
struc-
lt is expected that the constraining force decreases near the specimen surface compared with that of the interior region of specimen. Figures 5(a) and (b) show the dislocation structures in the interior region and the surface layer, respectively, in a Cu specimen deformed to 1% strain. In the interior region, many dislocations are equally distributed not only near the grain boundaries but also inside the grain, as seen in Fig. 5(a). In the surface layer the dislocation tangles are found only near the three-fold node of grain boundaries and a few dislocations are heterogeneously distributed inside the grain, as seen in Fig. 5(b). Therefore, it is concluded that the constraining force acts only near the grain boundaries in the surface layer but homogeneously through the grain in the interior region.
857
EFFECT ON FLOW STRESS
(a) Al.
4. DISCUSSION 4.1 Estimation
of the interaction
force
among
grains
When a grain is deformed, an affected zone is formed around the deformed grain so that further deformation is constrained with surrounding grains. Figure 6 is a schematic illustration of the affected zone, in which t is the specimen thickness, x the distance between the center of the deformed grain and the top surface, and R0 and R radii of the grain and the affected zone, respectively. In Fig. 6 the constraining force of a grain at position G is given by H(x) =
fY
h(r)d K
(1)
where V is the volume of affected zone and h(r) is the constraining force at a distance of r from the center of the deformed grain. If a grain is deformed hydrostaticaliy, the elastic stress field around the grain decreases inversely as r*. So it is simply assumed here that h(r) is a/r2, where a is an arbitrary constant. When the constraining force h(r) is normalized in such a way that R a ri) 3 4nr’dr
JRo\’
/
=
1,
(2)
MIYAZAKI,
858
SHIBATA
AND
FUJITA:
SPECIMEN
THICKNESS
EFFECT
ON
FLOW
STRESS
CU 0 l
0.
ou. 5 Thickness/Groin
Fig. 3. Thickness
Fig. 4. Deformation
IO Size
5
IO Thickncss/Grom
r*2oX I40 uln 65wn :;$I
15
‘r
Size
effect on the flow stresses in polycrystalline (a) Al, (b) Cu, (c) Cu-13 at%Al (d) Fe with various grain sizes.
and
of the surface grains in the specimens of (a) 0.25 and (b) 1.50 mm thick, respectively. The grain size is 180pm.
MIYAZAKI. SHIBATA
AND
FUJITA:
SPECIMEN
THICKNESS
EFFECT ON FLOW STRESS
859
Fig. 5. Dislocation structures in (a) the interior region and (b) the surface layer of polycrystalline Cu deformed to 1% strain. 1
(3)
a = 4n( R - R,) * Then h(r) =
(4)
4n(R : R,,)r’
Near the specimen surface, part of the affected zone is cut off by the surface, as shown in Fig. 6. In such a case, the constraining force is reduced by the top and bottom surfaces as follows:
l-----+-----f Fig. 6. Schematic illustration of an affected zone around a deformed grain.
1
=I-
/
2(R - R,) i
1
R-a-xlogc
a1
1 (5)
-2(R
where tj is an azimuth angle, and a and p are given by (a) ar = R. (b) c1= x (c) cc = R
ifOSx
