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The implications of hysteresis behaviour with regard to fatigue
Scripta METALLURGICA
Vol. 13, pp. 903-905, 1979 Printed in the U.S.A.
THE IMPLICATIONS
Pergamon Press Ltd. All rights reserved.
OF HYSTERESIS BEHAVIOUR WITH
REGARD TO FATIGUE
A. Abel School of Civil Engineering, The University of Sydney 2006, Australia (Received July 9, 1979)
i. Introduction It has been long recognised that the total plastic strain energy required for fatigue fracture can be orders of magnitude higher than that required for failure in the monotonic tension test (i). Morrow (2), pointed out that the total plastic strain energy to cause fatigue fracture could not be considered a constant for a given material but increases with decreasing stress amplitude and Feltner and Morrow (3), suggested that not all of the strain energy is damaging in fatigue. Finney and Laird (4), advanced these ideas by referring to a non-damaging completely reversible plastic strain and to the possibility that irreversibility in some dislocation activity may be the basic element in damage production, such as in fatigue crack initiation. The difficulty is to assess the relative magnitudes of these two kinds of plastic strains (5-6). In most of the attempts the central theme is that the strain fraction which is derived simply from the reverse motion of dislocations during cyclic deformation should be excluded from the cumulative plastic strain since, by its nature, it does not contribute to cyclic hardening and thus to damage. The experimental results analysed in this paper lead to the same conclusion. 2. Experimental
Details
Single slip oriented single crystals of Cu and Cu-Al alloys with aluminium contents of 2, 4, 7, ii and 16 at % were cycled at constant plastic strain amplitudes until fatigue failure occurred. The cyclic response was measured by monitoring the changes in hysteresis loop shape, energy absorption, and peak, average and saturation stresses. The results obtained have been described and discussed in more detail elsewhere (7-8). In this paper variations in the hysteresis loop shape are examined with special reference to fatigue performance. 3. Results and Discussion Most of the tests were conducted at 1.08% constant plastic strain amplitude and the results of these are presented in Fig. i. As constant plastic strain amplitude cycling leads to varying cyclic peak stresses, the number of cycles to failure does not provide a very suitable index of fatigue performance. Accordingly, it has not been plotted in Fig. i, but the maximum attained cyclic peak stress Op, the energy absorption to failure (J/mm3), and values of the parameter BE, which expresses the hysteresis loop shape in a quantitative manner, are shown for each of the six alloys tested.
903
0036-9748/79/090903-03.$02.00/0
Copyright
(c)
1979 Pergamon P r e s s Ltd.
904
H Y S T E R E S I S WITH REGARD TO FATIGUE
Vol.
13, No.
9
For a closed hysteresis loop the energy p a r a m e t e r 8E(9), is c a l c u l a t e d as 8E = (2opAep£-~ods)/~ode where Op is the cyclic peak stress and Aepz is the plastic strain amplitude. Accordingly, when the hysteresis loop shape approaches that of a p a r a l l e l o g r a m the value of 8E approaches zero. Conversely, the more pointed the h y s t e r e s i s loop shape the h i g h e r is the value of the energy parameter. The following d i s c u s s i o n is centred upon two main points. It is shown that there is a good c o r r e s p o n d e n c e between the values of 8E and fatigue p e r f o r m a n c e and second, it is s u g g e s t e d that a physical m e a n i n g can be attached to the hysteresis loop shape or energy parameter. E x p e r i m e n t a l results o b t a i n e d on copper (7), and on the ~Cu-Al single crystals (8), showed that the value of 8E d e c r e a s e d and the fatigue p e r f o r m a n c e d e t e r i o r a t e d as the cyclic amplitude increases. The apparent c o r r e l a t i o n b e t w e e n 8E and fatigue p e r f o r m a n c e is more s i g n i f i c a n t w h e n one takes the alloys one by one as shown in Fig. i. Initially the fatigue performance, a s s e s s e d in terms of the energy absorption to failure, decreases as the alloy content is raised to 2% and to 4% AI. This is p e r f e c t l y m a t c h e d by the v a r i a t i o n in BE. No attempt is made to explain the possible causes of this fatigue b e h a v i o u r but results o b t a i n e d on p o l y - c r y s t a l l i n e copper and Cu-Al alloys (i0) show the same effect. M o v i n g to the Cu-7% A1 alloy, great i m p r o v e m e n t is o b s e r v e d in the energy absorption to failure value in spite of the higher cyclic stresses. The c o r r e s p o n d i n g increase in the value of 8E is considerable. As the d e c r e a s i n g s t a c k i n g fault energies will increase the tendency for planar slip (ii) one may suggest that m o n i t o r i n g the change in the value of BE for various c o m p o s i t i o n s between the 4 and 7% alloys w o u l d provide a method for an e x t e r n a l d e t e r m i n a t i o n of the composition at w h i c h the slip mode changes into a planar one. Between 0 and 7% AI, improvement in fatigue p e r f o r m a n c e is i n d i c a t e d by i n c r e a s i n g total energy a b s o r p t i o n to failure. The fact that b e t w e e n 7 and 16% A1 the total energy a b s o r p t i o n to failure d e c r e a s e d appears at first to suggest a d e t e r i o r a t i o n in fatigue performance. It must be born in mind, however, that the increase in energy a b s o r p t i o n to failure between 0 and 7% A1 o c c u r r e d w i t h very little change in the peak stress value. Between 7 and 16% A1 the peak stress value i n c r e a s e d markedly. Thus, for these alloys, the actual stress amplitude during each test was greater the greater the A1 content. These and other results (8) suggest that the average energy absorption per cycle increases as the A1 content is i n c r e a s e d from 7 to 16 at % and that, if tested with constant stress amplitude, the n u m b e r of cycles to failure w o u l d have shown an increase with A1 content. I n v e s t i g a t i o n s carried out on p o l y c r y s t a l l i n e (12) and single crystal (13) samples of alloys in this c o m p o s i t i o n range have shown that the fatigue p e r f o r m a n c e does, in fact, improve as the A1 content increases and the s t a c k i n g fault energy decreases. Thus the improvement in fatigue p e r f o r m a n c e is m a t c h e d again with i n c r e a s e d 8E values. On this basis it is s u g g e s t e d that materials and cyclic conditions leading to h y s t e r e s i s loop with r e c t i l i n e a r shapes are p o o r e r in fatigue p e r f o r m a n c e than those w h i c h develop more p o i n t e d h y s t e r e s i s loops, that is larger 8E values. This e v i d e n c e suggests that a physical m e a n i n g can be a t t a c h e d to the energy parameter. It seems that the p a r a m e t e r gives i n f o r m a t i o n about the nature of the d e f o r m a t i o n processes involved. Very low 8E values w o u l d indicate d e f o r m a t i o n processes c o r r e s p o n d i n g in a r h e o l o g i c a l m o d e l to a r i g i d - p l a s t i c m a t e r i a l where the energy input of a d e f o r m a t i o n cycle w o u l d be used up in kinetic friction. More p o i n t e d h y s t e r e s i s loop shapes (larger 8E values) w o u l d c o r r e s p o n d w i t h d e f o r m a t i o n p r o c e s s e s where a larger fraction of the energy input is stored in a recoverable manner. The r e c o v e r a b l e energy storage m e c h a n i s m is p r o b a b l y associated with elastic interactions b e t w e e n individual d i s l o c a t i o n s and/or between d i s l o c a t i o n s and the m i c r o s t r u c t u r e . These elastic stresses w h i c h resist the m o v e m e n t of d i s l o c a t i o n s in the forward direction, contribute towards m o v e m e n t during the reverse cycle. There is no need to specify any d i s l o c a t i o n configuration to satisfy this c o n d i t i o n and it can be stated simply that all those dislocations w h i c h raised their p o t e n t i a l for reverse flow during the forward deformation cycle are storing m e c h a n i c a l l y r e c o v e r a b l e elastic energy w h i c h can be u t i l i z e d on load reversal. During u n l o a d i n g and reverse loading the initial departure from a response d e f i n e d by the elastic modulus of the material is a m a n i f e s t a t i o n of this r e c o v e r a b l y stored energy w h i c h is in fact the cause of m e c h a n i c a l hysteresis. The energy parameter, therefore, gives information about
Vol.
13, No.
9
H Y S T E R E S I S WITH REGARD TO FATIGUE
905
the shape of the h y s t e r e s i s loop which, in turn, is affected by the r e c o v e r a b l y stored energy fraction r e s i d i n g in the specimen at the b e g i n n i n g of the load reversal. Thus, fatigue can be monitored, not only to d e t e r m i n e the total p l a s t i c response, but the m a n n e r in w h i c h the work is a b s o r b e d by the material. The ratio of r e c o v e r a b l y stored energy to the energy r e q u i r e m e n t for the i r r e v e r s i b l e processes w i l l develop to a c h a r a c t e r i s t i c value during c y c l i n g w h i c h must be a structure sensitive m a t e r i a l property. It is quite p l a u s i b l e that the two kinds of d e f o r m a t i o n p r o c e s s e s w i l l produce fatigue damage at a different rate. The e v i d e n c e shown above, that is, that b e t t e r fatigue p e r f o r m a n c e is a c c o m p a n i e d by larger 8E values, indicates that the d e f o r m a t i o n p r o c e s s e s a s s o c i a t e d with the r e c o v e r a b l e energy fraction are less damaging. 4. Summary Fatigue p e r f o r m a n c e can be c o r r e l a t e d w i t h the shape of the h y s t e r e s i s loop. R e c t i l i n e a r h y s t e r e s i s loops are a s s o c i a t e d with poor fatigue p e r f o r m a n c e and more p o i n t e d loops are a s s o c i a t e d w i t h b e t t e r fatigue performance. The h y s t e r e s i s loop shape can be q u a n t i t a t i v e l y e x p r e s s e d by a parameter, the value of w h i c h is zero in the case of a r e c t i l i n e a r shape and increases w i t h the i n c r e a s i n g sharpness of the loop. The p r e s e n t results indicate that the value of the p a r a m e t e r increases and the fatigue p e r f o r m a n c e improves as the applied strain amplitude and the s t a c k i n g fault energy of the m a t e r i a l decrease. References G.R. Halford, J o u r n a l of Materials, Vol. I, No. i, 3(1966). J.D. Morrow, A.S.T.M. Spec. Tech. Pub. No. 378, 45(1964). C.E. F e l t n e r and J.D. Morrow, Trans. A.S. Mech. Eng., Vol. 83, Series D, 15(1961). J.M. Finney and C. Laird, Philos. Mag., 31, 339(1975). K. Tanaka, S. Matusuoka, J. Mat. Sci., ii, 656(1976). A.W. Sleeswyk, M.R. James, H.D. P l a t i n g a and W.S.T. Maathuis, Acta Metall., 26, 1265(1978). A. Abel, Mater. Sci. Eng., 36, 117(1978). A. Abel, M. W i l h e l m and V. Gerold, Mater. Sci. Eng., 37, 187(1979). A. Abel, H. Muir, Philos. Mag., 26, 489(1972). A. S a x e n a and S.D. Antolovich, Metall. Trans., 6A, 1809(1965) P. Lukas and M. Klesnil, Mater. Sci. Eno., ii, 345(1973). D.H. Avery and W.A. Backofen, Fracture of Solids, Wiley, New York, 339(1963). H. Ishii and J. Weertman, Metall. Trans., 2, 3441(1971).
i.
2. 3. 4.
5. 6. 7.
8. 9. i0. ii. 12. 13.
Acknowledgement N/mm 2
P120 110 ~E
J/rnm 3
0.4
40
0"3
30
0.2
20
0-1
10
90 bSO
f~/ :
Averoge energy
The e x p e r i m e n t a l work was c a r r i e d out at the MaxP l a n c k - I n s t i t u t fur M e t a l ~ forschung, Institut fur Werkstoffwissenschaften, Stuttgart, F.R.G. Thanks and gratitude are expressed for the financial support p r o v i d e d by a M a x - P l a n c k Fellowship.
porometer FIG. 1 C o m p a r a t i v e fatigue results
O Cu
2
4
7
11
at % AI
16
Vol. 13, pp. 903-905, 1979 Printed in the U.S.A.
THE IMPLICATIONS
Pergamon Press Ltd. All rights reserved.
OF HYSTERESIS BEHAVIOUR WITH
REGARD TO FATIGUE
A. Abel School of Civil Engineering, The University of Sydney 2006, Australia (Received July 9, 1979)
i. Introduction It has been long recognised that the total plastic strain energy required for fatigue fracture can be orders of magnitude higher than that required for failure in the monotonic tension test (i). Morrow (2), pointed out that the total plastic strain energy to cause fatigue fracture could not be considered a constant for a given material but increases with decreasing stress amplitude and Feltner and Morrow (3), suggested that not all of the strain energy is damaging in fatigue. Finney and Laird (4), advanced these ideas by referring to a non-damaging completely reversible plastic strain and to the possibility that irreversibility in some dislocation activity may be the basic element in damage production, such as in fatigue crack initiation. The difficulty is to assess the relative magnitudes of these two kinds of plastic strains (5-6). In most of the attempts the central theme is that the strain fraction which is derived simply from the reverse motion of dislocations during cyclic deformation should be excluded from the cumulative plastic strain since, by its nature, it does not contribute to cyclic hardening and thus to damage. The experimental results analysed in this paper lead to the same conclusion. 2. Experimental
Details
Single slip oriented single crystals of Cu and Cu-Al alloys with aluminium contents of 2, 4, 7, ii and 16 at % were cycled at constant plastic strain amplitudes until fatigue failure occurred. The cyclic response was measured by monitoring the changes in hysteresis loop shape, energy absorption, and peak, average and saturation stresses. The results obtained have been described and discussed in more detail elsewhere (7-8). In this paper variations in the hysteresis loop shape are examined with special reference to fatigue performance. 3. Results and Discussion Most of the tests were conducted at 1.08% constant plastic strain amplitude and the results of these are presented in Fig. i. As constant plastic strain amplitude cycling leads to varying cyclic peak stresses, the number of cycles to failure does not provide a very suitable index of fatigue performance. Accordingly, it has not been plotted in Fig. i, but the maximum attained cyclic peak stress Op, the energy absorption to failure (J/mm3), and values of the parameter BE, which expresses the hysteresis loop shape in a quantitative manner, are shown for each of the six alloys tested.
903
0036-9748/79/090903-03.$02.00/0
Copyright
(c)
1979 Pergamon P r e s s Ltd.
904
H Y S T E R E S I S WITH REGARD TO FATIGUE
Vol.
13, No.
9
For a closed hysteresis loop the energy p a r a m e t e r 8E(9), is c a l c u l a t e d as 8E = (2opAep£-~ods)/~ode where Op is the cyclic peak stress and Aepz is the plastic strain amplitude. Accordingly, when the hysteresis loop shape approaches that of a p a r a l l e l o g r a m the value of 8E approaches zero. Conversely, the more pointed the h y s t e r e s i s loop shape the h i g h e r is the value of the energy parameter. The following d i s c u s s i o n is centred upon two main points. It is shown that there is a good c o r r e s p o n d e n c e between the values of 8E and fatigue p e r f o r m a n c e and second, it is s u g g e s t e d that a physical m e a n i n g can be attached to the hysteresis loop shape or energy parameter. E x p e r i m e n t a l results o b t a i n e d on copper (7), and on the ~Cu-Al single crystals (8), showed that the value of 8E d e c r e a s e d and the fatigue p e r f o r m a n c e d e t e r i o r a t e d as the cyclic amplitude increases. The apparent c o r r e l a t i o n b e t w e e n 8E and fatigue p e r f o r m a n c e is more s i g n i f i c a n t w h e n one takes the alloys one by one as shown in Fig. i. Initially the fatigue performance, a s s e s s e d in terms of the energy absorption to failure, decreases as the alloy content is raised to 2% and to 4% AI. This is p e r f e c t l y m a t c h e d by the v a r i a t i o n in BE. No attempt is made to explain the possible causes of this fatigue b e h a v i o u r but results o b t a i n e d on p o l y - c r y s t a l l i n e copper and Cu-Al alloys (i0) show the same effect. M o v i n g to the Cu-7% A1 alloy, great i m p r o v e m e n t is o b s e r v e d in the energy absorption to failure value in spite of the higher cyclic stresses. The c o r r e s p o n d i n g increase in the value of 8E is considerable. As the d e c r e a s i n g s t a c k i n g fault energies will increase the tendency for planar slip (ii) one may suggest that m o n i t o r i n g the change in the value of BE for various c o m p o s i t i o n s between the 4 and 7% alloys w o u l d provide a method for an e x t e r n a l d e t e r m i n a t i o n of the composition at w h i c h the slip mode changes into a planar one. Between 0 and 7% AI, improvement in fatigue p e r f o r m a n c e is i n d i c a t e d by i n c r e a s i n g total energy a b s o r p t i o n to failure. The fact that b e t w e e n 7 and 16% A1 the total energy a b s o r p t i o n to failure d e c r e a s e d appears at first to suggest a d e t e r i o r a t i o n in fatigue performance. It must be born in mind, however, that the increase in energy a b s o r p t i o n to failure between 0 and 7% A1 o c c u r r e d w i t h very little change in the peak stress value. Between 7 and 16% A1 the peak stress value i n c r e a s e d markedly. Thus, for these alloys, the actual stress amplitude during each test was greater the greater the A1 content. These and other results (8) suggest that the average energy absorption per cycle increases as the A1 content is i n c r e a s e d from 7 to 16 at % and that, if tested with constant stress amplitude, the n u m b e r of cycles to failure w o u l d have shown an increase with A1 content. I n v e s t i g a t i o n s carried out on p o l y c r y s t a l l i n e (12) and single crystal (13) samples of alloys in this c o m p o s i t i o n range have shown that the fatigue p e r f o r m a n c e does, in fact, improve as the A1 content increases and the s t a c k i n g fault energy decreases. Thus the improvement in fatigue p e r f o r m a n c e is m a t c h e d again with i n c r e a s e d 8E values. On this basis it is s u g g e s t e d that materials and cyclic conditions leading to h y s t e r e s i s loop with r e c t i l i n e a r shapes are p o o r e r in fatigue p e r f o r m a n c e than those w h i c h develop more p o i n t e d h y s t e r e s i s loops, that is larger 8E values. This e v i d e n c e suggests that a physical m e a n i n g can be a t t a c h e d to the energy parameter. It seems that the p a r a m e t e r gives i n f o r m a t i o n about the nature of the d e f o r m a t i o n processes involved. Very low 8E values w o u l d indicate d e f o r m a t i o n processes c o r r e s p o n d i n g in a r h e o l o g i c a l m o d e l to a r i g i d - p l a s t i c m a t e r i a l where the energy input of a d e f o r m a t i o n cycle w o u l d be used up in kinetic friction. More p o i n t e d h y s t e r e s i s loop shapes (larger 8E values) w o u l d c o r r e s p o n d w i t h d e f o r m a t i o n p r o c e s s e s where a larger fraction of the energy input is stored in a recoverable manner. The r e c o v e r a b l e energy storage m e c h a n i s m is p r o b a b l y associated with elastic interactions b e t w e e n individual d i s l o c a t i o n s and/or between d i s l o c a t i o n s and the m i c r o s t r u c t u r e . These elastic stresses w h i c h resist the m o v e m e n t of d i s l o c a t i o n s in the forward direction, contribute towards m o v e m e n t during the reverse cycle. There is no need to specify any d i s l o c a t i o n configuration to satisfy this c o n d i t i o n and it can be stated simply that all those dislocations w h i c h raised their p o t e n t i a l for reverse flow during the forward deformation cycle are storing m e c h a n i c a l l y r e c o v e r a b l e elastic energy w h i c h can be u t i l i z e d on load reversal. During u n l o a d i n g and reverse loading the initial departure from a response d e f i n e d by the elastic modulus of the material is a m a n i f e s t a t i o n of this r e c o v e r a b l y stored energy w h i c h is in fact the cause of m e c h a n i c a l hysteresis. The energy parameter, therefore, gives information about
Vol.
13, No.
9
H Y S T E R E S I S WITH REGARD TO FATIGUE
905
the shape of the h y s t e r e s i s loop which, in turn, is affected by the r e c o v e r a b l y stored energy fraction r e s i d i n g in the specimen at the b e g i n n i n g of the load reversal. Thus, fatigue can be monitored, not only to d e t e r m i n e the total p l a s t i c response, but the m a n n e r in w h i c h the work is a b s o r b e d by the material. The ratio of r e c o v e r a b l y stored energy to the energy r e q u i r e m e n t for the i r r e v e r s i b l e processes w i l l develop to a c h a r a c t e r i s t i c value during c y c l i n g w h i c h must be a structure sensitive m a t e r i a l property. It is quite p l a u s i b l e that the two kinds of d e f o r m a t i o n p r o c e s s e s w i l l produce fatigue damage at a different rate. The e v i d e n c e shown above, that is, that b e t t e r fatigue p e r f o r m a n c e is a c c o m p a n i e d by larger 8E values, indicates that the d e f o r m a t i o n p r o c e s s e s a s s o c i a t e d with the r e c o v e r a b l e energy fraction are less damaging. 4. Summary Fatigue p e r f o r m a n c e can be c o r r e l a t e d w i t h the shape of the h y s t e r e s i s loop. R e c t i l i n e a r h y s t e r e s i s loops are a s s o c i a t e d with poor fatigue p e r f o r m a n c e and more p o i n t e d loops are a s s o c i a t e d w i t h b e t t e r fatigue performance. The h y s t e r e s i s loop shape can be q u a n t i t a t i v e l y e x p r e s s e d by a parameter, the value of w h i c h is zero in the case of a r e c t i l i n e a r shape and increases w i t h the i n c r e a s i n g sharpness of the loop. The p r e s e n t results indicate that the value of the p a r a m e t e r increases and the fatigue p e r f o r m a n c e improves as the applied strain amplitude and the s t a c k i n g fault energy of the m a t e r i a l decrease. References G.R. Halford, J o u r n a l of Materials, Vol. I, No. i, 3(1966). J.D. Morrow, A.S.T.M. Spec. Tech. Pub. No. 378, 45(1964). C.E. F e l t n e r and J.D. Morrow, Trans. A.S. Mech. Eng., Vol. 83, Series D, 15(1961). J.M. Finney and C. Laird, Philos. Mag., 31, 339(1975). K. Tanaka, S. Matusuoka, J. Mat. Sci., ii, 656(1976). A.W. Sleeswyk, M.R. James, H.D. P l a t i n g a and W.S.T. Maathuis, Acta Metall., 26, 1265(1978). A. Abel, Mater. Sci. Eng., 36, 117(1978). A. Abel, M. W i l h e l m and V. Gerold, Mater. Sci. Eng., 37, 187(1979). A. Abel, H. Muir, Philos. Mag., 26, 489(1972). A. S a x e n a and S.D. Antolovich, Metall. Trans., 6A, 1809(1965) P. Lukas and M. Klesnil, Mater. Sci. Eno., ii, 345(1973). D.H. Avery and W.A. Backofen, Fracture of Solids, Wiley, New York, 339(1963). H. Ishii and J. Weertman, Metall. Trans., 2, 3441(1971).
i.
2. 3. 4.
5. 6. 7.
8. 9. i0. ii. 12. 13.
Acknowledgement N/mm 2
P120 110 ~E
J/rnm 3
0.4
40
0"3
30
0.2
20
0-1
10
90 bSO
f~/ :
Averoge energy
The e x p e r i m e n t a l work was c a r r i e d out at the MaxP l a n c k - I n s t i t u t fur M e t a l ~ forschung, Institut fur Werkstoffwissenschaften, Stuttgart, F.R.G. Thanks and gratitude are expressed for the financial support p r o v i d e d by a M a x - P l a n c k Fellowship.
porometer FIG. 1 C o m p a r a t i v e fatigue results
O Cu
2
4
7
11
at % AI
16