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Relationship between loading process and Masing behavior in cyclic deformation
Materials Science and EngineeringA, 101 (1988) LI -L5
LI
Letter
Relationship between loading process and Masing behavior in cyclic deformation ZHIRUI WANG and CAMPBELL LAIRD l)epartment of Materials Science and Engineering, University q['Penno'h,ania, l'hiladelphia, f"4 19104-6272(U.S.A.) (ReceivedJanuary 12, 1988; in revised form February 18, 1988)
Abstract Masing behavior was investigated based upon the obsen,ed shapes of hysteresis loops in polycrystalline copper cycled with different loading methods. It was found that ramp-loaded specimens showed strong Masing behavior, whereas specimens tested in load control behaved much less in this aspect. To understand the above results, an elastic-perfect plastic muhiphase model was given and friction stress and back stress analyses were carried out in support. The mechanism and resuhs given in this letter are consistent with other results on cyclic deformation which contribute to understanding Masing behavior.
1. Introduction The idea of Masing behavior was first proposed in 1923 [1] to explain the stress-strain curve of two (or more) phase material when the two phases have distinct mechanical properties, e.g. yield stress, and to study deformation behavior upon stress reversal. As multiphase and composite materials have gained many more applications in industry, the study of Masing's theory has received increasing attention. Asaro discussed Masing's theory at length and connected it with the mechanical memory feature of the material [2, 3]. Many other attempts have been made to modify Masing's theory or to use the theory 1o rationalize deformation behavior. Interested readers can find stimulating ideas about the Masing model in the following references: Brown and Stobbs [4], Wilson [5], Wilson and Konnen [6], Wang [7], Lasalmonie and Martin [8], Wang and Margolin [9], and Mughrabi [10]. In fact, today we can say that Masing's theory is so important that, 0921-5093/88/$3.50
whenever deformation inhomogeneity exists, Masing's theory or a modified form of it demands application. Masing's model has received much attention in cyclic deformation, where Masing behavior is defined by comparing the shapes of hysteresis loops with the cyclic stress-strain curve. If the shape of the loop matches with the cyclic stress-strain curve (stress ordinate multiplied by two) then Masing behavior is considered to apply. If these two curves do not match then Masing behavior is considered not to apply (for pictorial representations, see Fig. 4 of ref. 11 ). Mughrabi et al. [11] recently studied changes of hysteresis loop shape in relation to testing mode and found that strong Masing behavior occurs in specimens incrementally step tested but it is not observed in specimens tested at constant strain. They have related this difference to the difference between the dislocation microstructural states for each type of specimen. The present authors have shown [12] that the form of the cyclic stress-strain curve can be controlled by the method of testing the specimen, with special reference to strain localization. Masing behavior is used here to explore the associated dislocation behavior and a model is offered to explain the Masing effect.
2. Experimental procedure Some 99.98% oxygen-free high-conductivity polycrystalline specimens, annealed and having a grain size of about 70 pm, were tested under three different loading conditions: Neumann-type ramp loading, ramp loading followed by cycling in strain control, and testing in constant load control. The main feature of the ramp-loading process in the present work is that 20 000 cycles were used in the ramp, i.e. starting from zero load, a stress increase of only about 0.05 MPa was applied in each cycle up to a varying maximum stress. For further details of the experimental procedure see ref. 12. In the present work, specimen 18 was ramp loaded to about 98 MPa in 20 000 cycles and then cycled under strain control at a plastic strain amplitude of 2.9 x 10 4. This amplitude was the same © Elsevier Sequoia/Printed in The Netherlands
L2 amplitude that the specimen reached naturally at the peak stress of the ramp. Specimen 19 was only ramp loaded to 120 MPa and then not tested further. Specimen 20 was tested in load control at the same stress as that attained by specimen 18 at saturation when it was cycled in strain control. Figure 1 schematically illustrates the above three tests. During all these tests, hysteresis loops were carefully recorded periodically. After each test the specimen surface was observed and the dislocation structures were investigated by transmission electron microscopy but these results are reported elsewhere [12]. 3.
Results
and discussion
Figures 2(a) and 2(b) show typical hysteresis loops of specimens 19 and 20. The loops which were picked out at different numbers of cycles during the test are arranged in such a way that all of the lower loop tips are joined together. Therefore the vertical axes and the horizontal axes in Figs. 2(a) and 2(b) represent o - ami, and e - tmi. respectively. Here o and ami, are the current applied stress and the minimum applied stress in each cycle respectively and e and tmin are, respectively, the strain and the minimum strain in each cycle. This method of comparing loops is the standard method of investigating whether or not Masing behavior applies and has been used widely, e.g. ref. 11 and references therein. When the stress-strain curves in the direction of tensile loading coincide then Masing behavior is considered to operate while the opposite holds if the curves do not overlap. As is obvious from Fig. 2(a) the ramploaded specimen 19 shows strong Masing behavior especially in the stress range of (o-amin) above about 190-200 MPa. On the other hand, for specimen 20 which was tested in load control (see Fig. 2(b)), the distinction between the loops is so marked that Masing behavior obviously does not apply.
a.
o L__ 98MPo ~
120MPo 98 M P ~
Note that for cycles higher than 250 the shape of the loop did not change -- saturation was attained. To explain the above results a schematic stress-strain curve of an elastic-perfect plastic multiphase material is given in Fig. 3. Here we assume that there are three components in the sample with different yield stresses OIy, Oily and Ollly (see Fig. 3). The volume fractions of the components are taken arbitrarily for illustrative purposes as 0.4, 0.4 and 0.2 respectively. Now consider the deformation behavior as stress is applied. As the applied stress r e a c h e s Oly, component I will start to deform plastically and the observed stress-strain curve will follow OYIY 2 since components II and III are still in the elastic region. Path YxY2 is obtained from the equation 3
o = Y F, o, ,=1
250
.
t"-
SPECIMEN2 0
Fig. 1. Schematic illustration of loading procedures in the present work.
/
/
"F///////
5o
~
5
I |0
(a)
I 1,5
I 20
I 25 X | 0
-4
E' - E'min.
250t o_ 200o
5
- 150c__ E
[:3 100
°, V V V"'°" ~CYCLES SPECIMEN19
150
I~ 100
50 5
(b)
SPECIMEN18
,38~_....~/ /
200
6
o
1680017340 1 5 1 ~
I 10
I 15
I 20
I 2 5 x ]0 - 4
E ' - E m in.
Fig. 2. Shape changes of hysteresis loops: (a) ramp loading test, specimen 19; (b) load control test, specimen 20. The numbers at each loop indicate the numbers of cycles from which the loops were taken.
L3 where F i and oi are, respectively, the volume fraction and the stress of component i. If the applied stress is removed at A, a half loop OY~AB will be obtained. Further increase of the applied stress will cause the observed curve (which reflects overall behavior) to develop along Y1YzY3, OY~Y2CD, OY~Y2Y3EF .... This is exactly the case during the ramp-loading test of specimen 19 in which all grains with different orientations can be considered as components with different "yield stresses" because of their different primary Schmidt factors. As the applied stress in ramping increases, more and more grains join the plastic deformation and so the observed overall flow curve will eventually have a rounded yielding form of stress-strain curve instead of a sharp yield point which each component would have if it deformed individually. At any point on the rounded yielding curve, i.e. at any stress level in this range, if the applied stress is reversed then Masing behavior will apply. Of course a cyclic test carried out in a ramp-loading mode reproduces this situation precisely and Masing behavior is observed. It is also worthwhile to mention that in the stress reversals the Bauschinger effect will also operate since residual stresses exist in the material due to the inhomogeneity of deformation. For a specimen simply tested in load control, e.g. specimen 20 (Fig. 2(b)), Masing behavior was observed not to occur. This case is unlike the ramploading test in which the flow process spreads from soft-oriented grains to hard-oriented grains as the stress increases cycle by cycle. Rather, in the load control test, most of the grains join the flow process in the early cycles so that the width of the loop narrows cycle by cycle as the material hardens and no Masing behavior appears because the structure is constantly changing.
In real copper, in a ramp-loading test, dislocation loop patches accumulate in the variously oriented grains and the structure is very homogeneous [12]. When persistent slip bands (PSBs) form in these loop patches they do so at a well-defined stress [12] and confer something of perfect plasticity on the specimen. Thus the connection with the thought experiment on elastic-plastic material described above is apt. Moreover, the structural type tends to be constant as the ramp develops. This situation does not apply in a load control test where multiplicity of slip (and structure) varies from grain to grain [13]. The interpretation of Mughrabi et al. [11] therefore applies. Kuhlmann-Wilsdorf and Laird [14] proposed a method to estimate friction stress and back stress from the shape of the hysteresis loop and, therefore, to correlate the loop shape with the microstructure of the specimen. The relation between friction stress oF,,back stress OB, peak applied stress az, and yield stress a~ in each cycle is given by (YE "q" (Ys
O F --
2 O"E -- O"s
2 Using the above formulae, measurements and calculations were carried out on the specimens of interest here and the results are shown in Fig. 4. For specimen 19, which was ramp loaded and which shows strong Masing behavior (Fig. 2), ov and a B are both observed to increase with increasing number of cycles of applied stress, formally just as in a single crystal [14] but with an important difference.
60L
p
O" 0"~) " oo
O'rry
1I
~:3
30
.
•
7. . . . . . .
o-• - BACK STRESS
0.4
t9
/
/
/
,o
0.4 0.2 I
I01
Fig. 3. Stress-strain curve of an elastic-perfect plastic multiphase material showing the basis of Masing behavior.
SPECIMEN
/ 5/ //
b" 2o
~y
i/ 5o,
lJ / ~
E
5- F2- F3-
~rl
tO 2
I
103 No OF CYCLES
I
104
Fig. 4. A n a l y s i s of f r i c t i o n s t r e s s o F a n d b a c k s t r e s s o B in the t h r e e d i f f e r e n t l y t e s t e d s p e c i m e n s . T h e results for s p e c i m e n s 18 a n d 19 are the s a m e d u r i n g t h e i r r a m p l o a d i n g a n d t h o s e i n d i c a t e d for s p e c i m e n 18 a p p l y to the test in s t r a i n c o n t r o l
after ramp loading.
L4
18
0
20
2OO
2-
1~160 120 O3
a: i.--
80
J
40
0 i-
! I
I
I
5 10 15 20 xlO - 4 ) TOTAL STRAIN E-E'mi n
Fig. 5. Comparison of saturation hysteresis loops between specimens 18 and 20.
In a single crystal the friction stress leads the back stress [14] but here the relative magnitudes of the stresses are reversed; OF< OB. This different behavior is attributed to the effect of grain structure which increases the back stress considerably but has little effect on the friction stress. The increase in a v and a B with increasing number of cycles continues until the stress is very high (over 100 MPa) where the volume fraction of plastically deforming material is considered close to 1, i.e. most (and probably all) of the grains have yielded and multislip has appeared in many grains [12]. This is because secondary stress has been excited by the requirements of compatibility between grains. The change in a F and oB shown in Fig. 4 is consistent with the Masing behavior shown in Fig. 2(a). On the other hand, the change in a F and o8 in specimen 18 after cycling in strain control is unremarkable and both stresses reached saturation in less than 20 cycles (see Fig. 4). Specimen 18 was of course ramp loaded first from 20 000 cycles and then tested in strain control; the o F and aB curves of the ramploading part are not shown here since they are the same as those of the ramp-loading part in specimen 19. However, the values of OF and o 8 in the subsequent strain control test are shown in Fig. 4 to be almost the same as those in specimen 20. This implies that the microstructures stabilized after ramp loading and quickly adjusted to their saturation state after the transfer of strain control. No Masing behavior was observed during the strain
control test because neither the stress nor the strain increased (slight cyclic softening occurred). To reveal the microstructural differences between specimens 18 and 20, in addition to the evidence considered from the behavior of o F and oB, Fig. 5 is provided for comparison of two hysteresis loops from these specimens presented in the same way as in Fig. 2. They represent the saturation loops of specimen 18 (in the strain control test) and of specimen 20 respectively. At the same stress level specimen 20 produced a larger plastic strain. This is due to its overall dislocation cell structure involving two or more slip systems, which is usually produced by a test in load control. In specimen 18, even if there is a large volume fraction of PSBs due to ramp loading [12], the overall plastic strain is still smaller. This is because any increase of plastic strain would create strong interactions between PSBs on different slip systems. Due to these differences of dislocation structures, no Masing behavior is observed in Fig. 5.
4. Conclusions In summary, copper specimens tested in ramp loading are found to obey Masing behavior because the dislocation structures are uniform in the polycrystalline mass and are constant in their roughly two-phase nature -- PSBs and matrix structures. Specimens tested in load control do not show Masing behavior because dislocation structures involving multislip vary greatly from grain to grain and vary differently with cycling. The observations reported here are consistent with the interpretation of Mughrabi et al. [11] on relating Masing behavior to structural type. They are also consistent with the observation of plateau behavior in the cyclic stress-strain response of ramp-loaded specimens [12].
Acknowledgments This work was supported by the National Science Foundation under Grant DMR 85-13259. The authors also wish to express their deep appreciation to the Laboratory for Research on the Structure of Matter, University of Pennsylvania, for its help in experimental work and the supply of equipment.
References 1 G. Masing, Wiss., Veroeff., Siemens-Werken, 3 (1927) 231. 2 R.J. Asaro, Acta. Metall., 23(1975) 1255.
L5 3 R. J. Asaro, in A. W. Thomson (ed.), Proc. Symp. on Work-hardening in Tension and Fatigue, Cincinnati, Ohio, 1975, Metallurgical Society of AIME, Warrendale, PA, 1975, p. 206. 4 k. M. Brown and W. M. Stobbs, Philos. Mag., 23 ( 1971 ) 1185.
5 D.V. Wilson, Acta Metall., 13 (1965) 807. 6 l). V. Wilson and Y. A. Konnen, Acta Metall. 12 (1964) 617. 7 Z. Wang, PhD. Thesis, Polytechnic Institute of New York, 1986. 8 A. Lasalmonie and J. W. Martin, Scr. Metall., 8 (1974) 377.
9 Z. Wang and H. Margolin, Res. Mech., 21 (3)(1987) 249. 10 H. Mughrabi, Acre Metall., 31 (1983) 1367. 11 H. Mughrabi, M. Bayerlein and H.-J. Christ, l'roc. 8th Riso Int. Symp. on Constitutive Equations and Their Physical Basis, Roskilde, Denmark, 198Z 12 Z. Wang and C. Laird, Cyclic stress-strain response of polycrystalline copper under fatigue conditions producing enhanced strain localization, Mater. Sci. Eng., 10,0 (1988) 57. 13 J. C. Figueroa, S. P. Bhat, R. de la Veaux, S. Murzenski and C. Laird, Acta Metall., 29(1981 ) 1667. 14 D. Kuhlmann-Wilsdoff and C. Laird, Mater. Sci. Eng., 37 (1979) 111.
LI
Letter
Relationship between loading process and Masing behavior in cyclic deformation ZHIRUI WANG and CAMPBELL LAIRD l)epartment of Materials Science and Engineering, University q['Penno'h,ania, l'hiladelphia, f"4 19104-6272(U.S.A.) (ReceivedJanuary 12, 1988; in revised form February 18, 1988)
Abstract Masing behavior was investigated based upon the obsen,ed shapes of hysteresis loops in polycrystalline copper cycled with different loading methods. It was found that ramp-loaded specimens showed strong Masing behavior, whereas specimens tested in load control behaved much less in this aspect. To understand the above results, an elastic-perfect plastic muhiphase model was given and friction stress and back stress analyses were carried out in support. The mechanism and resuhs given in this letter are consistent with other results on cyclic deformation which contribute to understanding Masing behavior.
1. Introduction The idea of Masing behavior was first proposed in 1923 [1] to explain the stress-strain curve of two (or more) phase material when the two phases have distinct mechanical properties, e.g. yield stress, and to study deformation behavior upon stress reversal. As multiphase and composite materials have gained many more applications in industry, the study of Masing's theory has received increasing attention. Asaro discussed Masing's theory at length and connected it with the mechanical memory feature of the material [2, 3]. Many other attempts have been made to modify Masing's theory or to use the theory 1o rationalize deformation behavior. Interested readers can find stimulating ideas about the Masing model in the following references: Brown and Stobbs [4], Wilson [5], Wilson and Konnen [6], Wang [7], Lasalmonie and Martin [8], Wang and Margolin [9], and Mughrabi [10]. In fact, today we can say that Masing's theory is so important that, 0921-5093/88/$3.50
whenever deformation inhomogeneity exists, Masing's theory or a modified form of it demands application. Masing's model has received much attention in cyclic deformation, where Masing behavior is defined by comparing the shapes of hysteresis loops with the cyclic stress-strain curve. If the shape of the loop matches with the cyclic stress-strain curve (stress ordinate multiplied by two) then Masing behavior is considered to apply. If these two curves do not match then Masing behavior is considered not to apply (for pictorial representations, see Fig. 4 of ref. 11 ). Mughrabi et al. [11] recently studied changes of hysteresis loop shape in relation to testing mode and found that strong Masing behavior occurs in specimens incrementally step tested but it is not observed in specimens tested at constant strain. They have related this difference to the difference between the dislocation microstructural states for each type of specimen. The present authors have shown [12] that the form of the cyclic stress-strain curve can be controlled by the method of testing the specimen, with special reference to strain localization. Masing behavior is used here to explore the associated dislocation behavior and a model is offered to explain the Masing effect.
2. Experimental procedure Some 99.98% oxygen-free high-conductivity polycrystalline specimens, annealed and having a grain size of about 70 pm, were tested under three different loading conditions: Neumann-type ramp loading, ramp loading followed by cycling in strain control, and testing in constant load control. The main feature of the ramp-loading process in the present work is that 20 000 cycles were used in the ramp, i.e. starting from zero load, a stress increase of only about 0.05 MPa was applied in each cycle up to a varying maximum stress. For further details of the experimental procedure see ref. 12. In the present work, specimen 18 was ramp loaded to about 98 MPa in 20 000 cycles and then cycled under strain control at a plastic strain amplitude of 2.9 x 10 4. This amplitude was the same © Elsevier Sequoia/Printed in The Netherlands
L2 amplitude that the specimen reached naturally at the peak stress of the ramp. Specimen 19 was only ramp loaded to 120 MPa and then not tested further. Specimen 20 was tested in load control at the same stress as that attained by specimen 18 at saturation when it was cycled in strain control. Figure 1 schematically illustrates the above three tests. During all these tests, hysteresis loops were carefully recorded periodically. After each test the specimen surface was observed and the dislocation structures were investigated by transmission electron microscopy but these results are reported elsewhere [12]. 3.
Results
and discussion
Figures 2(a) and 2(b) show typical hysteresis loops of specimens 19 and 20. The loops which were picked out at different numbers of cycles during the test are arranged in such a way that all of the lower loop tips are joined together. Therefore the vertical axes and the horizontal axes in Figs. 2(a) and 2(b) represent o - ami, and e - tmi. respectively. Here o and ami, are the current applied stress and the minimum applied stress in each cycle respectively and e and tmin are, respectively, the strain and the minimum strain in each cycle. This method of comparing loops is the standard method of investigating whether or not Masing behavior applies and has been used widely, e.g. ref. 11 and references therein. When the stress-strain curves in the direction of tensile loading coincide then Masing behavior is considered to operate while the opposite holds if the curves do not overlap. As is obvious from Fig. 2(a) the ramploaded specimen 19 shows strong Masing behavior especially in the stress range of (o-amin) above about 190-200 MPa. On the other hand, for specimen 20 which was tested in load control (see Fig. 2(b)), the distinction between the loops is so marked that Masing behavior obviously does not apply.
a.
o L__ 98MPo ~
120MPo 98 M P ~
Note that for cycles higher than 250 the shape of the loop did not change -- saturation was attained. To explain the above results a schematic stress-strain curve of an elastic-perfect plastic multiphase material is given in Fig. 3. Here we assume that there are three components in the sample with different yield stresses OIy, Oily and Ollly (see Fig. 3). The volume fractions of the components are taken arbitrarily for illustrative purposes as 0.4, 0.4 and 0.2 respectively. Now consider the deformation behavior as stress is applied. As the applied stress r e a c h e s Oly, component I will start to deform plastically and the observed stress-strain curve will follow OYIY 2 since components II and III are still in the elastic region. Path YxY2 is obtained from the equation 3
o = Y F, o, ,=1
250
.
t"-
SPECIMEN2 0
Fig. 1. Schematic illustration of loading procedures in the present work.
/
/
"F///////
5o
~
5
I |0
(a)
I 1,5
I 20
I 25 X | 0
-4
E' - E'min.
250t o_ 200o
5
- 150c__ E
[:3 100
°, V V V"'°" ~CYCLES SPECIMEN19
150
I~ 100
50 5
(b)
SPECIMEN18
,38~_....~/ /
200
6
o
1680017340 1 5 1 ~
I 10
I 15
I 20
I 2 5 x ]0 - 4
E ' - E m in.
Fig. 2. Shape changes of hysteresis loops: (a) ramp loading test, specimen 19; (b) load control test, specimen 20. The numbers at each loop indicate the numbers of cycles from which the loops were taken.
L3 where F i and oi are, respectively, the volume fraction and the stress of component i. If the applied stress is removed at A, a half loop OY~AB will be obtained. Further increase of the applied stress will cause the observed curve (which reflects overall behavior) to develop along Y1YzY3, OY~Y2CD, OY~Y2Y3EF .... This is exactly the case during the ramp-loading test of specimen 19 in which all grains with different orientations can be considered as components with different "yield stresses" because of their different primary Schmidt factors. As the applied stress in ramping increases, more and more grains join the plastic deformation and so the observed overall flow curve will eventually have a rounded yielding form of stress-strain curve instead of a sharp yield point which each component would have if it deformed individually. At any point on the rounded yielding curve, i.e. at any stress level in this range, if the applied stress is reversed then Masing behavior will apply. Of course a cyclic test carried out in a ramp-loading mode reproduces this situation precisely and Masing behavior is observed. It is also worthwhile to mention that in the stress reversals the Bauschinger effect will also operate since residual stresses exist in the material due to the inhomogeneity of deformation. For a specimen simply tested in load control, e.g. specimen 20 (Fig. 2(b)), Masing behavior was observed not to occur. This case is unlike the ramploading test in which the flow process spreads from soft-oriented grains to hard-oriented grains as the stress increases cycle by cycle. Rather, in the load control test, most of the grains join the flow process in the early cycles so that the width of the loop narrows cycle by cycle as the material hardens and no Masing behavior appears because the structure is constantly changing.
In real copper, in a ramp-loading test, dislocation loop patches accumulate in the variously oriented grains and the structure is very homogeneous [12]. When persistent slip bands (PSBs) form in these loop patches they do so at a well-defined stress [12] and confer something of perfect plasticity on the specimen. Thus the connection with the thought experiment on elastic-plastic material described above is apt. Moreover, the structural type tends to be constant as the ramp develops. This situation does not apply in a load control test where multiplicity of slip (and structure) varies from grain to grain [13]. The interpretation of Mughrabi et al. [11] therefore applies. Kuhlmann-Wilsdorf and Laird [14] proposed a method to estimate friction stress and back stress from the shape of the hysteresis loop and, therefore, to correlate the loop shape with the microstructure of the specimen. The relation between friction stress oF,,back stress OB, peak applied stress az, and yield stress a~ in each cycle is given by (YE "q" (Ys
O F --
2 O"E -- O"s
2 Using the above formulae, measurements and calculations were carried out on the specimens of interest here and the results are shown in Fig. 4. For specimen 19, which was ramp loaded and which shows strong Masing behavior (Fig. 2), ov and a B are both observed to increase with increasing number of cycles of applied stress, formally just as in a single crystal [14] but with an important difference.
60L
p
O" 0"~) " oo
O'rry
1I
~:3
30
.
•
7. . . . . . .
o-• - BACK STRESS
0.4
t9
/
/
/
,o
0.4 0.2 I
I01
Fig. 3. Stress-strain curve of an elastic-perfect plastic multiphase material showing the basis of Masing behavior.
SPECIMEN
/ 5/ //
b" 2o
~y
i/ 5o,
lJ / ~
E
5- F2- F3-
~rl
tO 2
I
103 No OF CYCLES
I
104
Fig. 4. A n a l y s i s of f r i c t i o n s t r e s s o F a n d b a c k s t r e s s o B in the t h r e e d i f f e r e n t l y t e s t e d s p e c i m e n s . T h e results for s p e c i m e n s 18 a n d 19 are the s a m e d u r i n g t h e i r r a m p l o a d i n g a n d t h o s e i n d i c a t e d for s p e c i m e n 18 a p p l y to the test in s t r a i n c o n t r o l
after ramp loading.
L4
18
0
20
2OO
2-
1~160 120 O3
a: i.--
80
J
40
0 i-
! I
I
I
5 10 15 20 xlO - 4 ) TOTAL STRAIN E-E'mi n
Fig. 5. Comparison of saturation hysteresis loops between specimens 18 and 20.
In a single crystal the friction stress leads the back stress [14] but here the relative magnitudes of the stresses are reversed; OF< OB. This different behavior is attributed to the effect of grain structure which increases the back stress considerably but has little effect on the friction stress. The increase in a v and a B with increasing number of cycles continues until the stress is very high (over 100 MPa) where the volume fraction of plastically deforming material is considered close to 1, i.e. most (and probably all) of the grains have yielded and multislip has appeared in many grains [12]. This is because secondary stress has been excited by the requirements of compatibility between grains. The change in a F and oB shown in Fig. 4 is consistent with the Masing behavior shown in Fig. 2(a). On the other hand, the change in a F and o8 in specimen 18 after cycling in strain control is unremarkable and both stresses reached saturation in less than 20 cycles (see Fig. 4). Specimen 18 was of course ramp loaded first from 20 000 cycles and then tested in strain control; the o F and aB curves of the ramploading part are not shown here since they are the same as those of the ramp-loading part in specimen 19. However, the values of OF and o 8 in the subsequent strain control test are shown in Fig. 4 to be almost the same as those in specimen 20. This implies that the microstructures stabilized after ramp loading and quickly adjusted to their saturation state after the transfer of strain control. No Masing behavior was observed during the strain
control test because neither the stress nor the strain increased (slight cyclic softening occurred). To reveal the microstructural differences between specimens 18 and 20, in addition to the evidence considered from the behavior of o F and oB, Fig. 5 is provided for comparison of two hysteresis loops from these specimens presented in the same way as in Fig. 2. They represent the saturation loops of specimen 18 (in the strain control test) and of specimen 20 respectively. At the same stress level specimen 20 produced a larger plastic strain. This is due to its overall dislocation cell structure involving two or more slip systems, which is usually produced by a test in load control. In specimen 18, even if there is a large volume fraction of PSBs due to ramp loading [12], the overall plastic strain is still smaller. This is because any increase of plastic strain would create strong interactions between PSBs on different slip systems. Due to these differences of dislocation structures, no Masing behavior is observed in Fig. 5.
4. Conclusions In summary, copper specimens tested in ramp loading are found to obey Masing behavior because the dislocation structures are uniform in the polycrystalline mass and are constant in their roughly two-phase nature -- PSBs and matrix structures. Specimens tested in load control do not show Masing behavior because dislocation structures involving multislip vary greatly from grain to grain and vary differently with cycling. The observations reported here are consistent with the interpretation of Mughrabi et al. [11] on relating Masing behavior to structural type. They are also consistent with the observation of plateau behavior in the cyclic stress-strain response of ramp-loaded specimens [12].
Acknowledgments This work was supported by the National Science Foundation under Grant DMR 85-13259. The authors also wish to express their deep appreciation to the Laboratory for Research on the Structure of Matter, University of Pennsylvania, for its help in experimental work and the supply of equipment.
References 1 G. Masing, Wiss., Veroeff., Siemens-Werken, 3 (1927) 231. 2 R.J. Asaro, Acta. Metall., 23(1975) 1255.
L5 3 R. J. Asaro, in A. W. Thomson (ed.), Proc. Symp. on Work-hardening in Tension and Fatigue, Cincinnati, Ohio, 1975, Metallurgical Society of AIME, Warrendale, PA, 1975, p. 206. 4 k. M. Brown and W. M. Stobbs, Philos. Mag., 23 ( 1971 ) 1185.
5 D.V. Wilson, Acta Metall., 13 (1965) 807. 6 l). V. Wilson and Y. A. Konnen, Acta Metall. 12 (1964) 617. 7 Z. Wang, PhD. Thesis, Polytechnic Institute of New York, 1986. 8 A. Lasalmonie and J. W. Martin, Scr. Metall., 8 (1974) 377.
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