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Shear and kink angles at the lüders band front
Scripta METALLURGICA
Vol. 23, pp. 1075-1078, 1989 Printed in the U.S.A.
Pergamon Press plc All rights reserved
S H E A R A N D K I N K A N G L E S AT T H E L U D E R S B A N D F R O N T V.S. Ananthan* and E.O.Hall Department of Chemical and Materials Engineering The University of Newcastle N.S.W. 2308, Australia (Received March 31, 1989) (Revised April 17, 1989)
Introduction Lfiders bands in mild steel are made visible to the naked eye by a macroscopic kink at the band front. Earlier estimates of the kink angle had been made by Hall (1) as the order of 1°, with the angle of the band front lying close to the plane of maximum shear. Such small values of the kink angles are difficult to measure accurately, and further reports of the kink angles are scarce. The present work was aimed at studying the effects of varying shapes and grain sizes on the magnitude of the kink angle, and thus obtaining quantitative information on the magnitude of the shear at the band front. Recent microscopic evidence had confirmed the presence of shear at the band front (2,3). Furthermore, at coarse grain sizes it was shown that the band front became diffuse, and the kink disappeared. This research was done as a part of a formal analysis of the shear and flow theory of Lfiders bands, proposed by Hall et al. (4).
Experimental Mild steel rods of differing cross-section were used to study the Liiders bands, these sections being hexagonal, square, circular, octagonal, and rectangular. Hard drawn hexagonal rods, 6.3 mm across the opposite faces and square rods of 5 mm side were kindly supplied by Australian Wire Industries Pty. Ltd. Specimens of circular and octagonal cross-sections were machined from hexagonal and square samples respectively. Marbrite steel (R1008) was used for the rectangular samples. This gave average side lengths of 5.9, 4.3, and 2.1 mm for the hexagonal, square, and octagonal samples respectively; the circular samples had an average diameter of 5.5 mm, while the rectangular specimens were 15.2 mm X 2.7 mm. The chemical compositions (in wt.%) of the steels used are given in the following Table. TABLE
% % % % % %
carbon nitrogen phosphorus manganese silicon sulphur
hexagonal circular 0.09-0.10 0.006 0.029-0.031 0.39-0.41 O.O34-0.036 0.019-0.200
rectangular 0.10 0.009 0.040 0.20-0.50 0.050
square octagonal 0.12 0.005 0.027 0.37 0.030 0.020
"Present address: Metallurgy Dept., Ris¢ National Laboratory, Postboks 49, DK-4000 Roskilde, Denmark.
1075 0036-9748/89 $3.00 + .00 Copyright (c) 1989 Pergamon Press plc
1076
LUDERS
BANDS
Vol.
23, No.
7
The test specimens of different cross-sections were cut to a nominal length of 200 mm and then straightened and ground. The surfaces of the specimens were cleaned by polishing with wet emery papers to a 600 grade finish. This enabled the removal of most of the grinding marks on the surface of the specimens. Further cleaning was done with alcohol and ether. The specimens were annealed in a tube furnace under vacuum to obtain different grain sizes, with every possible care being taken to prevent oxidation, and then electrolytically polished. The electrolytic solution was made from 1596 cc of glacial acetic acid, 300 grams of chromium trioxide, and 84 cc of distilled water. The polished specimens were then tensile tested in a Shimadzu Autograph OSS-5000 tensile machine with self aligning grips at a test speed of 0.1 mm/min. A single L/iders band was initiated in most of the specimens at this crosshead speed. Specimens which developed multiple bands were rejected. L/iders bands were clearly visible in smaller grain-sized specimens, but had diffuse fronts in larger grain-sized specimens. The bands were examined on the screen of a Vickers projection microscope at a magnification of 3X, and the angles between the band front and the applied load (tensile) direction were measured to an accuaxcy of 4-0.5 °. This enabled the calculation of the orientation angles as described by Hall et al. (4). A Talycontour machine was used for the studies relating to the kink angles. This machine was kindly made available by the CSIRO National Measurement Laboratory, Lindfield, N.S.W.. The machine had a sharp titanium carbide profiler that can trace and record the surface of specimen, and the slope of the surface can be magnified to a maximum of 20X. This method was adopted, as optical methods did not yield satisfactory results. The surface profiler was made to traverse along the specimen axis across the band on all the faces of the specimens. The kink angles of circular cross-sectional specimens were measured at 60 ° intervals around the circumference. This method enabled a measurement of the kink angle to an accuracy of 4-0.05 °, which was considered reasonable for the study. Two to three measurements were taken on each face of the specimens and the average kink angles were calculated from the magnified surface profile. Results
The kink angles of opposite sides were opposite in nature; i.e., a convex kink in side 1 of a hexagonal specimen had a concave kink in side 4 (opposite to side 1). The values of the kink angles on opposite sides of the specimens were also equal in many cases, but there were a small number of exceptions. The kink angles of the sides (K) of different cross-sectional specimens enable a calculation of the kink angle (k) between the axes of the strained and the unstrained regions. With the assumption that k lies in the plane of shear, the measured kink angles of the sides A,B,C,~tc., are the projection of the dihedral angle on those planes. Hence
k = KA/cos(90 -- a) = Ks/cos(90 - ~) = go~ cos(90 - 7) where the orientation angles a, ~ , 7 are defined in the sterogram in ref. (4). In specimens where the orientation angles of the Liiders band front approached 0 or 180 degrees, and the value of the measured kink angle (K) approached a low value, the k value calculated from this equation could be very large. In calculating the average value of k for the specimen, such values were rejected because of inherent error. The k(av) values were then plotted against the mean grain size (d) and illustrated in Fig. 1. Although errors in calculating the kink angles are large, a clear decreasing trend of the kink angle with the increasing grain sizes of the specimens of all cross-sections is evident. The regression equation is given with the diagram covering all cross- section types.
The kink angle between the strained and unstrained regions of the specimens ranged between approximately 1.3 ° for finer grained specimens to approximately 0.5 ° in coarse grained specimens. These angles
Vol.
23, No.
7
LUDERS BANDS
2.0
1
1077
I
I
•
~
o o •
1.5
*
£
Circular Octagonal Square Rectangular Hexagonal
1.o
0.5
0.0
I
0.1
I
I
0.2 0.3 GRAIN SIZE (d)(ram)
0.4
Regression line k(av) = 1.13 - 2.84d. Fig. 1. Variation of the average kink angle with grain size for specimens of different cross-sections.
~,Tensileaxis
\
~ Fig. 2. A simple representation of the kink angle in major and minor axis of the ellipse.
/4 Trace of the Luders band
Slope
Fig. 3. Stereogram showing the maximum kink directions for the hexagonal specimens.
1078
LUDERS
BANDS
Vol.
23, No.
compare well with the earlier reported work of Hall (1), Hall et al. (4) and Iricibar et al. (5). The kink angles are also consistent with the analysis of the Lfiders deformation (4) as arising from the combination of a shear deformation, leading to the macroscopic shape changes, followed by a flow deformation, and will be the subject of a later paper. Iricibar et al.'s (5) complete rejection of a shear mechanism in favour of a "ledge" mechanism would seem to be unwarranted. However, one discrepancy between the shear-flow mechanism of Hall et al. (4) and the present results is worthy of note, since that theory was developed on the basis that the direction of shear would lie along the line of greatest slope in the shear surface, i.e., the Liiders band front. The kink angles in the directions of major (kl) and minor (k2) axis of the elliptical cross-section of the specimen (Fig. 2) may be resolved using a similar expression for the strain calculation as given by Hall et al. (4) and yielding k12 k2
cos~/3 sin 2/3 sin 2/3 cos 23, sin 27 sin 2 ~'
KB
Ko
where KA, Ks, Ko refer to the measured values of the kink angles of the sides of the (hexagonal) specimens. Solving for the values of kl and k2, the direction of maximum kink (¢) and the magnitude of the maximum kink (k(max)) may be determined as ¢ = tan-~(k2/k,),
k(rnax)
= k l cos ¢.
The values of k(max) compare well with the values of k(av) calculated earlier for all cross-sections of the specimens. However, the values of ¢ are not scattered around the line of greatest slope, but are spread along the great circle which is the trace of the Liiders band front. Fig. 3, for example, shows the result for a "standard" sterogram for the hexagonal specimens where ~he tensile axis is vertical and the line of greatest slope is marked. This broad scatter of results is repeated with other cross-sections. This in turn suggests that there are other factors which may be involved in determining the direction of shear, such as misalignments in the specimen caused by the grips, or different bending moments associated with the specimen shape, machine design, or even the kink itself and its position along the specimen. It has not proved possible to elucidate this further, but does not invalidate the general thrust of the "shear plus flow" concept of the Liiders deformation. The values of the kink angles (k(av)) decreased with increasing grain size, irrespective of the specimen shape. The regression line drawn in Fig. 1 indicates that grain sizes of the order of 0.4-0.5 mm have to be reached for a zero kink angle. The normal scatter of grain sizes thus lead to a diffuse band front, and experimental difficulties associated with the generation of single bands and the measurement of the angles become extremely difficult in coarse grained material. References
1. 2. 3. 4. 5.
E.O.Hall, Proc. Phys. Soc., B64, 742, (1951). V.S.Ananthan and E.O.Hall, Scrip. Met., 21,519, (1987). V.S.Ananthan and E.O.Hall, Scrip. Met., 21, 1699, (1987). E.O.Hall, R.J.Carter and G.Vitullo, ICSMA 6, (ed. by R.C.Gifkins), p393, Oxford, (1983). R.Iricibar, G.Panizza and J.Mazza, Acta Met., 25, 1163, (1977).
7
Vol. 23, pp. 1075-1078, 1989 Printed in the U.S.A.
Pergamon Press plc All rights reserved
S H E A R A N D K I N K A N G L E S AT T H E L U D E R S B A N D F R O N T V.S. Ananthan* and E.O.Hall Department of Chemical and Materials Engineering The University of Newcastle N.S.W. 2308, Australia (Received March 31, 1989) (Revised April 17, 1989)
Introduction Lfiders bands in mild steel are made visible to the naked eye by a macroscopic kink at the band front. Earlier estimates of the kink angle had been made by Hall (1) as the order of 1°, with the angle of the band front lying close to the plane of maximum shear. Such small values of the kink angles are difficult to measure accurately, and further reports of the kink angles are scarce. The present work was aimed at studying the effects of varying shapes and grain sizes on the magnitude of the kink angle, and thus obtaining quantitative information on the magnitude of the shear at the band front. Recent microscopic evidence had confirmed the presence of shear at the band front (2,3). Furthermore, at coarse grain sizes it was shown that the band front became diffuse, and the kink disappeared. This research was done as a part of a formal analysis of the shear and flow theory of Lfiders bands, proposed by Hall et al. (4).
Experimental Mild steel rods of differing cross-section were used to study the Liiders bands, these sections being hexagonal, square, circular, octagonal, and rectangular. Hard drawn hexagonal rods, 6.3 mm across the opposite faces and square rods of 5 mm side were kindly supplied by Australian Wire Industries Pty. Ltd. Specimens of circular and octagonal cross-sections were machined from hexagonal and square samples respectively. Marbrite steel (R1008) was used for the rectangular samples. This gave average side lengths of 5.9, 4.3, and 2.1 mm for the hexagonal, square, and octagonal samples respectively; the circular samples had an average diameter of 5.5 mm, while the rectangular specimens were 15.2 mm X 2.7 mm. The chemical compositions (in wt.%) of the steels used are given in the following Table. TABLE
% % % % % %
carbon nitrogen phosphorus manganese silicon sulphur
hexagonal circular 0.09-0.10 0.006 0.029-0.031 0.39-0.41 O.O34-0.036 0.019-0.200
rectangular 0.10 0.009 0.040 0.20-0.50 0.050
square octagonal 0.12 0.005 0.027 0.37 0.030 0.020
"Present address: Metallurgy Dept., Ris¢ National Laboratory, Postboks 49, DK-4000 Roskilde, Denmark.
1075 0036-9748/89 $3.00 + .00 Copyright (c) 1989 Pergamon Press plc
1076
LUDERS
BANDS
Vol.
23, No.
7
The test specimens of different cross-sections were cut to a nominal length of 200 mm and then straightened and ground. The surfaces of the specimens were cleaned by polishing with wet emery papers to a 600 grade finish. This enabled the removal of most of the grinding marks on the surface of the specimens. Further cleaning was done with alcohol and ether. The specimens were annealed in a tube furnace under vacuum to obtain different grain sizes, with every possible care being taken to prevent oxidation, and then electrolytically polished. The electrolytic solution was made from 1596 cc of glacial acetic acid, 300 grams of chromium trioxide, and 84 cc of distilled water. The polished specimens were then tensile tested in a Shimadzu Autograph OSS-5000 tensile machine with self aligning grips at a test speed of 0.1 mm/min. A single L/iders band was initiated in most of the specimens at this crosshead speed. Specimens which developed multiple bands were rejected. L/iders bands were clearly visible in smaller grain-sized specimens, but had diffuse fronts in larger grain-sized specimens. The bands were examined on the screen of a Vickers projection microscope at a magnification of 3X, and the angles between the band front and the applied load (tensile) direction were measured to an accuaxcy of 4-0.5 °. This enabled the calculation of the orientation angles as described by Hall et al. (4). A Talycontour machine was used for the studies relating to the kink angles. This machine was kindly made available by the CSIRO National Measurement Laboratory, Lindfield, N.S.W.. The machine had a sharp titanium carbide profiler that can trace and record the surface of specimen, and the slope of the surface can be magnified to a maximum of 20X. This method was adopted, as optical methods did not yield satisfactory results. The surface profiler was made to traverse along the specimen axis across the band on all the faces of the specimens. The kink angles of circular cross-sectional specimens were measured at 60 ° intervals around the circumference. This method enabled a measurement of the kink angle to an accuracy of 4-0.05 °, which was considered reasonable for the study. Two to three measurements were taken on each face of the specimens and the average kink angles were calculated from the magnified surface profile. Results
The kink angles of opposite sides were opposite in nature; i.e., a convex kink in side 1 of a hexagonal specimen had a concave kink in side 4 (opposite to side 1). The values of the kink angles on opposite sides of the specimens were also equal in many cases, but there were a small number of exceptions. The kink angles of the sides (K) of different cross-sectional specimens enable a calculation of the kink angle (k) between the axes of the strained and the unstrained regions. With the assumption that k lies in the plane of shear, the measured kink angles of the sides A,B,C,~tc., are the projection of the dihedral angle on those planes. Hence
k = KA/cos(90 -- a) = Ks/cos(90 - ~) = go~ cos(90 - 7) where the orientation angles a, ~ , 7 are defined in the sterogram in ref. (4). In specimens where the orientation angles of the Liiders band front approached 0 or 180 degrees, and the value of the measured kink angle (K) approached a low value, the k value calculated from this equation could be very large. In calculating the average value of k for the specimen, such values were rejected because of inherent error. The k(av) values were then plotted against the mean grain size (d) and illustrated in Fig. 1. Although errors in calculating the kink angles are large, a clear decreasing trend of the kink angle with the increasing grain sizes of the specimens of all cross-sections is evident. The regression equation is given with the diagram covering all cross- section types.
The kink angle between the strained and unstrained regions of the specimens ranged between approximately 1.3 ° for finer grained specimens to approximately 0.5 ° in coarse grained specimens. These angles
Vol.
23, No.
7
LUDERS BANDS
2.0
1
1077
I
I
•
~
o o •
1.5
*
£
Circular Octagonal Square Rectangular Hexagonal
1.o
0.5
0.0
I
0.1
I
I
0.2 0.3 GRAIN SIZE (d)(ram)
0.4
Regression line k(av) = 1.13 - 2.84d. Fig. 1. Variation of the average kink angle with grain size for specimens of different cross-sections.
~,Tensileaxis
\
~ Fig. 2. A simple representation of the kink angle in major and minor axis of the ellipse.
/4 Trace of the Luders band
Slope
Fig. 3. Stereogram showing the maximum kink directions for the hexagonal specimens.
1078
LUDERS
BANDS
Vol.
23, No.
compare well with the earlier reported work of Hall (1), Hall et al. (4) and Iricibar et al. (5). The kink angles are also consistent with the analysis of the Lfiders deformation (4) as arising from the combination of a shear deformation, leading to the macroscopic shape changes, followed by a flow deformation, and will be the subject of a later paper. Iricibar et al.'s (5) complete rejection of a shear mechanism in favour of a "ledge" mechanism would seem to be unwarranted. However, one discrepancy between the shear-flow mechanism of Hall et al. (4) and the present results is worthy of note, since that theory was developed on the basis that the direction of shear would lie along the line of greatest slope in the shear surface, i.e., the Liiders band front. The kink angles in the directions of major (kl) and minor (k2) axis of the elliptical cross-section of the specimen (Fig. 2) may be resolved using a similar expression for the strain calculation as given by Hall et al. (4) and yielding k12 k2
cos~/3 sin 2/3 sin 2/3 cos 23, sin 27 sin 2 ~'
KB
Ko
where KA, Ks, Ko refer to the measured values of the kink angles of the sides of the (hexagonal) specimens. Solving for the values of kl and k2, the direction of maximum kink (¢) and the magnitude of the maximum kink (k(max)) may be determined as ¢ = tan-~(k2/k,),
k(rnax)
= k l cos ¢.
The values of k(max) compare well with the values of k(av) calculated earlier for all cross-sections of the specimens. However, the values of ¢ are not scattered around the line of greatest slope, but are spread along the great circle which is the trace of the Liiders band front. Fig. 3, for example, shows the result for a "standard" sterogram for the hexagonal specimens where ~he tensile axis is vertical and the line of greatest slope is marked. This broad scatter of results is repeated with other cross-sections. This in turn suggests that there are other factors which may be involved in determining the direction of shear, such as misalignments in the specimen caused by the grips, or different bending moments associated with the specimen shape, machine design, or even the kink itself and its position along the specimen. It has not proved possible to elucidate this further, but does not invalidate the general thrust of the "shear plus flow" concept of the Liiders deformation. The values of the kink angles (k(av)) decreased with increasing grain size, irrespective of the specimen shape. The regression line drawn in Fig. 1 indicates that grain sizes of the order of 0.4-0.5 mm have to be reached for a zero kink angle. The normal scatter of grain sizes thus lead to a diffuse band front, and experimental difficulties associated with the generation of single bands and the measurement of the angles become extremely difficult in coarse grained material. References
1. 2. 3. 4. 5.
E.O.Hall, Proc. Phys. Soc., B64, 742, (1951). V.S.Ananthan and E.O.Hall, Scrip. Met., 21,519, (1987). V.S.Ananthan and E.O.Hall, Scrip. Met., 21, 1699, (1987). E.O.Hall, R.J.Carter and G.Vitullo, ICSMA 6, (ed. by R.C.Gifkins), p393, Oxford, (1983). R.Iricibar, G.Panizza and J.Mazza, Acta Met., 25, 1163, (1977).
7