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Influence of ‘floating’ parallel alignment on luders band configurations
Scripta
METALLUR(;ICA
Vol. 14, pp. 915 - 918, 1980 P r i n t e d i n the U.S.A.
INFLUENCE OF 'FLOATING'
PARALLEL ALIGNMENT
Pergamon Press Ltd. All rights reserved.
ON LUDERS BAND CONFIGURATIONS
A. W. Sleeswyk and H. Hiemstra Department of Applied Nijenborgh 18, 9747
Physics, University of Groningen, AG Groningen, The Netherlands.
(Received
June
5,
1980)
The jaws in which the specimen is gripped during a tensile test tend to be displaced sideways relative to each other if the plastic deformation response of the material contains an asymmetric component. The phenomenon may occur in tensile tests of single crystals, and of materials in which deformation bands or Luders bands are formed. Often the jaws are mounted between universal joints in order to allow a transverse relative movement to take place. If both jaws rotate over the same angle, the relative tensile displacement of the jaws takes place without any relative rotation being superimposed on it, which is a desirable characteristic. The two jaws rotate independently, however, and consequently the two rotations are not necessarily equal, and then the heads are no longer parallel: bending is superimposed on the lengthwise straining. An indication of lack of parallelism is that when there are two Luders band fronts formed in the specimen these are generally not parallel, which has the same overall effect as the presence of a single L~ders band, viz. a slight bending of the specimen. The overall deformation in the L~ders band front region is a shear parallel to the front, so the L{~ders band is a sheared region. The effect of a single L~ders band is schematically depicted in Fig. la, and in Fig. Ib the specimen with two non-parallel L~ders band fronts is depicted schematically, and in Fig. Ic the specimen with parallel Luders band fronts and parallel heads. In order to ensure parallelism of the specimen heads during plastic deformation we designed and built an alignment system incorporating a four bar linkage, as schematically depicted in Fig. Id. The lower jaw is fixed (at the end of a cage extension to the tensile tester which has not been drawn), between the upper jaw and the hinged pull rod the symmetrical four bar linkage is mounted; the latter is a device well-known in mechanical engineering and consists of four hinged bars, of which at least one has a length different from that of the others. The design of the system aims at: I. having the axis of the pull rod passing through the centre of the gauge section of the specimen; 2. having the deflection of the four bar linkage cause the lower bar (length 'a'), and the upper jaw attached to it, to counter-rotate relative to the pull rod, such that a = - B. The first of these aims is fulfilled if the specimen's gauge section undergoes a strain which has a homogeneous transverse component, as in Fig. Id. If the transverse shear is inhomogeneously distributed over the gauge length, however, the axis of the pull rod will pass only near the centre, but not exactly through it. However, if the distance from the centre of the specimen to the hinge in the pull rod is large compared to the gauge length this error is small. In the tester we built, the ratio of these two lengths is approximately 15:1, resulting in a maximum error of less than I% in the maximum sideways displacement of the uppcr head during Luders band formation. Because of the inhomogeneity of these bands this error is unavoidable, but it does not seem to be a serious one. Similarly, the second aim is approximated closely, but not fulfilled entirely. With finite sideways displacement of the upper specimen head the compensating eounterrotatiun wil I start to
0036-9748/80/080915-04502.00/0 Copyright (c) 1 9 8 0 P e r g a m o n Press
Ltd.
LUDERS B A N D
916
CONFIGURATIONS
Vol.
14,
No.
8
to deviate from its initial rate, resulting in a slight uncompensated rotation. In practice, we found that this uncompensated rotation can be kept well within normal machining accuracy. In our experiment, with: h=425, a=50, b=62.5 and c=179 mm, we calculated - using a formula given by Schuh [I] - that with the range the pull rod had to rotate, from 0 to 0.2 ° in our tests, the four bar linkage compensates at least some 991% of the rotation of the pull rod, as is evident from Table I, where the compensating rotation ~ is presented for different values of the length b and the angle ~; in addition, the value of dB/d~ is given for ~ = 0. TABLE
I
Values of the Compensating Counter-Rotation
b (u~n)
(d~)~ =0
62.0
-
62.5
-
63.0
-
B.
~ = 0"1000°
~ = 0"2000o
~ = 0"3000o
0.954
- 0.0954 °
- 0.1908
- 0.2862
0.994
- 0.0994 °
- 0.1987
- 0.2981
1.033
-
-
-
1.0033
°
0.2067
0.3101
It is evident from the table that a practical difficulty in optimizing the system resides in its sensitivity to dimensional inaccuracies, in particular of the distances between hinges in the linkage. A near-perfect compensation would result for: b=62.58 mm, but the accuracy implied in this dimension would not only be expensive, but also illusory as we used strips of phosphorus-bronze as hinges which fixed the position of the hinge at an estimated + 0.05 mm. In order to obtain 'floating' of the head in all transverse directions, a second fo]r bar linkage, oriented at 90 ° to the first one, was placed in series to it. Tensile tests were performed on specimens of Armco ingot iron, 0.2 mm thick, with a gauge section 17 rmm wide and I00 n~n long. One of the surfaces of each specimen was polished carefully to a mirror finish, after which the specimens were vacuum-annealed. The composition of the material is given in Table II; the average grain size varied from 22 to 31.I0 -° m. TABLE II Composition of Armco Ingot Iron (wt. perc.). C Mn P S Si
.015 .02 .01 .01 .003
A1 Cu Sn Cr Ni
<
.0l .045 .005 .02 .045
Mo O N Ti V
< <
.004 .087 .055 .002 .002
The tests were run at extension velocities ranging from 5.10 -6 to ].67.10 -8 m.s. -l, while photographs of the Luders band configurations were taken, using the grid reflection technique developed by Verel and Sleeswyk [2]. Four heats of specimens, each of ca. I0 specimens, were divided in two batches: one batch was tested in a conventionally aligned tester with independently rotable jaws, the other was tested with the 'floating' parallel alignment. From the photographs the Luders band front angles were determined, but as the Luders band front was often somewhat curved, the angular intervals were chosen sufficiently large -5 °- that the angles could be represented meaningfully. The results are shown in the histograms presented in Fig. 2. In a number of specimens in both batches only one Luders band with one growing front the other one extending into the jaw gripping the specimen - was formed. The distributions of angles of single Luders band fronts with a line perpendicular to the tensile axis turned out to be not significantly different for the two batches; in Fig. 2a the distribution obtained with two independently rotahle jaws is given. Results obtained on specimens containing
two moving Luders band fronts obtained with the same
Vol.
14,
No.
8
I.tJDERS
BAND C O N F I G U R A T I O N S
917
t e s t e r a r e p r e s e n t e d i n F i g . 2b. The a n g l e s p r e s e n t e d i n t h i s d i a g r a m a r e t h o s e b e t w e e n t i l e two L ~ d e r s b a n d s in t h e same s p e c i m e n f o r a number of s p e c i m e n s . C o n f i g u r a t i o n s s u c h a s sho~m s c h e matically [n F i g s . lb and Ic b o t h o c c u r .
h
pull
rod
a
linkage b
I.
c
d
/
;
specimen
7
".,'///7//4,/. FIG. Showing r o t a t i o n of specimen heads fronts. If the fronts are parallel f o u r b a r l i n k a g e (d) may c a u s e t h e t h e e f f e c t of t h e r o t a t i o n (~ of t h e
I
i f (a) L ~ d e r s band f r o n t i s p r e s e n t , or (b) two non p a r a l l e l ( c ) , t h e h e a d s a r e s h i f t e d s i d e w a y s but r e m a i n p a r a l t e l . A u p p e r head t o c o u n t e r - r o t a t e o v e r an a n g l e ~, d i m i n i s h i n g pull rod.
percentage
a
b
c
50 40
-L,
2oJ lo
I
ol
0
r
I0
20
30
40
0
10
20
30
40
50
6'o angles
FIG.
o
,o
2o
3'o ,'o
(degrees)
2
Histograms of: (a) the angle between a single Luders band front and a perpendicular to the tensile axis; (b) the angle between two Luders band fronts in specimens tested betwee, independently rotatable jaws; (c)the angle between two Luders band fronts in specimens tested between floating parallel jaws. The distribution between Luders band fronts obtained on specimens tested between the floating jaws is presented in Fig. Ic. Configurations such as depicted in Fig. Ib, with differently inclined fronts were not observed. The L~ders band fronts in this batch of specimens were always
918
LUDERS BAND CONFIGURATIONS
inclined to the same side and were more or less parallel, tribut~ons in Fig. Ib and Fig. Ic.
Vol.
which accounts
14,
No.
for the different dis-
It may be concluded that the 'floating' head arrangement approximates the ideal conditions of parallelism and minimum bending in the specimen sufficiently to suppress differently inclined pairs of L~ders band fronts, such as depicted in Fig. lb. On the other hand, single Luders band fronts, as in Fig. la, are not suppressed, which may be due to a lack of stiffness. Our study shows, however, that the construction of a 'floating' parallel head is entirely feasible and that the constraint exercised by this arrangement influences L~ders band configurations. ACKNOWLEDGMENTS The work was supported by the Netherlands programme on Dislocation Dynamics.
Science Foundation
F. O. M. as part of a research
REFERENCES I. 2.
F. Schuh, Leerboek der Theoretische Mechanica Pt II, Ch. 5, Amsterdam D. J. Verel and A. W. Sleeswyk, Acta Met. 2, 1087 (1973).
8
(1948).
METALLUR(;ICA
Vol. 14, pp. 915 - 918, 1980 P r i n t e d i n the U.S.A.
INFLUENCE OF 'FLOATING'
PARALLEL ALIGNMENT
Pergamon Press Ltd. All rights reserved.
ON LUDERS BAND CONFIGURATIONS
A. W. Sleeswyk and H. Hiemstra Department of Applied Nijenborgh 18, 9747
Physics, University of Groningen, AG Groningen, The Netherlands.
(Received
June
5,
1980)
The jaws in which the specimen is gripped during a tensile test tend to be displaced sideways relative to each other if the plastic deformation response of the material contains an asymmetric component. The phenomenon may occur in tensile tests of single crystals, and of materials in which deformation bands or Luders bands are formed. Often the jaws are mounted between universal joints in order to allow a transverse relative movement to take place. If both jaws rotate over the same angle, the relative tensile displacement of the jaws takes place without any relative rotation being superimposed on it, which is a desirable characteristic. The two jaws rotate independently, however, and consequently the two rotations are not necessarily equal, and then the heads are no longer parallel: bending is superimposed on the lengthwise straining. An indication of lack of parallelism is that when there are two Luders band fronts formed in the specimen these are generally not parallel, which has the same overall effect as the presence of a single L~ders band, viz. a slight bending of the specimen. The overall deformation in the L~ders band front region is a shear parallel to the front, so the L{~ders band is a sheared region. The effect of a single L~ders band is schematically depicted in Fig. la, and in Fig. Ib the specimen with two non-parallel L~ders band fronts is depicted schematically, and in Fig. Ic the specimen with parallel Luders band fronts and parallel heads. In order to ensure parallelism of the specimen heads during plastic deformation we designed and built an alignment system incorporating a four bar linkage, as schematically depicted in Fig. Id. The lower jaw is fixed (at the end of a cage extension to the tensile tester which has not been drawn), between the upper jaw and the hinged pull rod the symmetrical four bar linkage is mounted; the latter is a device well-known in mechanical engineering and consists of four hinged bars, of which at least one has a length different from that of the others. The design of the system aims at: I. having the axis of the pull rod passing through the centre of the gauge section of the specimen; 2. having the deflection of the four bar linkage cause the lower bar (length 'a'), and the upper jaw attached to it, to counter-rotate relative to the pull rod, such that a = - B. The first of these aims is fulfilled if the specimen's gauge section undergoes a strain which has a homogeneous transverse component, as in Fig. Id. If the transverse shear is inhomogeneously distributed over the gauge length, however, the axis of the pull rod will pass only near the centre, but not exactly through it. However, if the distance from the centre of the specimen to the hinge in the pull rod is large compared to the gauge length this error is small. In the tester we built, the ratio of these two lengths is approximately 15:1, resulting in a maximum error of less than I% in the maximum sideways displacement of the uppcr head during Luders band formation. Because of the inhomogeneity of these bands this error is unavoidable, but it does not seem to be a serious one. Similarly, the second aim is approximated closely, but not fulfilled entirely. With finite sideways displacement of the upper specimen head the compensating eounterrotatiun wil I start to
0036-9748/80/080915-04502.00/0 Copyright (c) 1 9 8 0 P e r g a m o n Press
Ltd.
LUDERS B A N D
916
CONFIGURATIONS
Vol.
14,
No.
8
to deviate from its initial rate, resulting in a slight uncompensated rotation. In practice, we found that this uncompensated rotation can be kept well within normal machining accuracy. In our experiment, with: h=425, a=50, b=62.5 and c=179 mm, we calculated - using a formula given by Schuh [I] - that with the range the pull rod had to rotate, from 0 to 0.2 ° in our tests, the four bar linkage compensates at least some 991% of the rotation of the pull rod, as is evident from Table I, where the compensating rotation ~ is presented for different values of the length b and the angle ~; in addition, the value of dB/d~ is given for ~ = 0. TABLE
I
Values of the Compensating Counter-Rotation
b (u~n)
(d~)~ =0
62.0
-
62.5
-
63.0
-
B.
~ = 0"1000°
~ = 0"2000o
~ = 0"3000o
0.954
- 0.0954 °
- 0.1908
- 0.2862
0.994
- 0.0994 °
- 0.1987
- 0.2981
1.033
-
-
-
1.0033
°
0.2067
0.3101
It is evident from the table that a practical difficulty in optimizing the system resides in its sensitivity to dimensional inaccuracies, in particular of the distances between hinges in the linkage. A near-perfect compensation would result for: b=62.58 mm, but the accuracy implied in this dimension would not only be expensive, but also illusory as we used strips of phosphorus-bronze as hinges which fixed the position of the hinge at an estimated + 0.05 mm. In order to obtain 'floating' of the head in all transverse directions, a second fo]r bar linkage, oriented at 90 ° to the first one, was placed in series to it. Tensile tests were performed on specimens of Armco ingot iron, 0.2 mm thick, with a gauge section 17 rmm wide and I00 n~n long. One of the surfaces of each specimen was polished carefully to a mirror finish, after which the specimens were vacuum-annealed. The composition of the material is given in Table II; the average grain size varied from 22 to 31.I0 -° m. TABLE II Composition of Armco Ingot Iron (wt. perc.). C Mn P S Si
.015 .02 .01 .01 .003
A1 Cu Sn Cr Ni
<
.0l .045 .005 .02 .045
Mo O N Ti V
< <
.004 .087 .055 .002 .002
The tests were run at extension velocities ranging from 5.10 -6 to ].67.10 -8 m.s. -l, while photographs of the Luders band configurations were taken, using the grid reflection technique developed by Verel and Sleeswyk [2]. Four heats of specimens, each of ca. I0 specimens, were divided in two batches: one batch was tested in a conventionally aligned tester with independently rotable jaws, the other was tested with the 'floating' parallel alignment. From the photographs the Luders band front angles were determined, but as the Luders band front was often somewhat curved, the angular intervals were chosen sufficiently large -5 °- that the angles could be represented meaningfully. The results are shown in the histograms presented in Fig. 2. In a number of specimens in both batches only one Luders band with one growing front the other one extending into the jaw gripping the specimen - was formed. The distributions of angles of single Luders band fronts with a line perpendicular to the tensile axis turned out to be not significantly different for the two batches; in Fig. 2a the distribution obtained with two independently rotahle jaws is given. Results obtained on specimens containing
two moving Luders band fronts obtained with the same
Vol.
14,
No.
8
I.tJDERS
BAND C O N F I G U R A T I O N S
917
t e s t e r a r e p r e s e n t e d i n F i g . 2b. The a n g l e s p r e s e n t e d i n t h i s d i a g r a m a r e t h o s e b e t w e e n t i l e two L ~ d e r s b a n d s in t h e same s p e c i m e n f o r a number of s p e c i m e n s . C o n f i g u r a t i o n s s u c h a s sho~m s c h e matically [n F i g s . lb and Ic b o t h o c c u r .
h
pull
rod
a
linkage b
I.
c
d
/
;
specimen
7
".,'///7//4,/. FIG. Showing r o t a t i o n of specimen heads fronts. If the fronts are parallel f o u r b a r l i n k a g e (d) may c a u s e t h e t h e e f f e c t of t h e r o t a t i o n (~ of t h e
I
i f (a) L ~ d e r s band f r o n t i s p r e s e n t , or (b) two non p a r a l l e l ( c ) , t h e h e a d s a r e s h i f t e d s i d e w a y s but r e m a i n p a r a l t e l . A u p p e r head t o c o u n t e r - r o t a t e o v e r an a n g l e ~, d i m i n i s h i n g pull rod.
percentage
a
b
c
50 40
-L,
2oJ lo
I
ol
0
r
I0
20
30
40
0
10
20
30
40
50
6'o angles
FIG.
o
,o
2o
3'o ,'o
(degrees)
2
Histograms of: (a) the angle between a single Luders band front and a perpendicular to the tensile axis; (b) the angle between two Luders band fronts in specimens tested betwee, independently rotatable jaws; (c)the angle between two Luders band fronts in specimens tested between floating parallel jaws. The distribution between Luders band fronts obtained on specimens tested between the floating jaws is presented in Fig. Ic. Configurations such as depicted in Fig. Ib, with differently inclined fronts were not observed. The L~ders band fronts in this batch of specimens were always
918
LUDERS BAND CONFIGURATIONS
inclined to the same side and were more or less parallel, tribut~ons in Fig. Ib and Fig. Ic.
Vol.
which accounts
14,
No.
for the different dis-
It may be concluded that the 'floating' head arrangement approximates the ideal conditions of parallelism and minimum bending in the specimen sufficiently to suppress differently inclined pairs of L~ders band fronts, such as depicted in Fig. lb. On the other hand, single Luders band fronts, as in Fig. la, are not suppressed, which may be due to a lack of stiffness. Our study shows, however, that the construction of a 'floating' parallel head is entirely feasible and that the constraint exercised by this arrangement influences L~ders band configurations. ACKNOWLEDGMENTS The work was supported by the Netherlands programme on Dislocation Dynamics.
Science Foundation
F. O. M. as part of a research
REFERENCES I. 2.
F. Schuh, Leerboek der Theoretische Mechanica Pt II, Ch. 5, Amsterdam D. J. Verel and A. W. Sleeswyk, Acta Met. 2, 1087 (1973).
8
(1948).