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Experimental method for the approximate determination of dislocation density in fatigue
Scripta METALLURGICA
Vol, II, pp, 529-532, 1977 Printed in the United States
Pergamon Press, Inc,
EXPERIMENTAL METHOD FOR THE APPROXIMATEDETERMINATIONOF DISLOCATION DENSITY IN FATIGUE Parviz Dadras Department of M e t a l l u r g i c a l Engineering Arya-Mehr U n i v e r s i t y of Technology Tehran, Iran
(Received January 31, 1977) (Revised May 20, 1977)
Under the influence of an a l t e r n a t i n g stress f i e l d , a d i s l o c a t i o n segment can o s c i l l a t e between two pinning points provided by jogs, point defects, and other d i s l o c a t i o n s in a manner s i m i l a r to a s t r i n g v i b r a t i n g in a viscous medium ( I ) . For the high stress levels leading to fatigue deformation and f r a c t u r e , some of the d i s l o c a t i o n s can break away from the weak pinning points and thus accommodate l a r g e r s t r a i n s by producing longer segments of o s c i l l a t i n g dislocat i o n s . The conbributions that the d i s l o c a t i o n s make to the overall magnitude of s t r a i n as they bow out ~n t h e i r g l i d e planes under the influence of an o s c i l l a t i n g stress c o n s t i t u t e s a reversi b l e e l a s t i c s t r a i n of the magnitude (2) 6E = A A b where A is the area swept out by the d i s l o c a t i o n , b is the Burgers' vector and A is the number of active d i s l o c a t i o n s in a u n i t volume. The e l a s t i c s t r a i n ~e reduces the apparent e l a s t i c modulus, E, by 6 E in such a way t h a t : 6 = EE = (E + ~E) (~ + 6c) Using the expression f o r A, as calculated by Mott (3), one finds t h a t : ~E E - c A L3
(Eq. I )
where c is a constant and L is the average free length of the d i s l o c a t i o n s c o n t r i b u t o r y to modulus defect, 6E . E A r e l a t i o n s h i p between the modulus defect and the change in resonant frequency f o r the case where the specimen is vibrated in a f l e x u r e mode of deformation can be developed. Based on the f i n d i n g of Klein (4), the natural frequency of a c a n t i l e v e r plate of a r b i t r a r y plan form can be expressed as : Fr = K1 E_l i ~ ]
I/2
where K1 is a constant dependent on the form, mass, and thickness of the plate. Since Poisson's r a t i o , v, is not s t r u c t u r e s e n s i t i v e , i t can be assumed that i t s value does not change appreciably as a r e s u l t of c y c l i c deformation. Therefore, Fr = K2 ( E ) I / 2 which leads to: 2 6 Fr = 6E Fr
E
529
(Eq. 2)
530
DISLOCATION DENSITY IN FATIGUE
Vol. ii, No. 7
Moreover, in cyclic deformation, the flow stress can be related to the length of the active dislocation segment L, by T
:
Gb WL
-
(Eq. 3)
-
where w usually has values between 3 and 5 (5). Taking the value of w as 5 (6), the resulting decrease in L can be related to the changes in the flow stress for the increasing number of cycles. Furthermore, i f L, as determined by this method, is a reasonable approximation of the average length of dislocation segments which lead to the modulus change, then A can also be calculated. Thus, combining equations l and 2 using an approximation for the constant C (7): A -
6
~Fr
L3
Fr
(Eq. 4)
The number of active dislocations per unit volume of length L multiplied by the lengths of the dislocation segments equals the density of dislocations active in fatiguey p*. In the present investigation, the necessary experimental data are obtained which f a c i l i t a t e the approximate determination of the length and density of dislocations which are active in a fatigue deformation process. To perform the cyclic tests, a resonant bending fatigue apparatus was developed. An alternating f i e l d was produced by two electromagnets as shown in Figure I. Constant-bendingstress specimens with permanent magnets clamped to their ends were placed in the alternating force field of the electromagnets and forcing functions of equal amplitude were assured in all the tests. A non-contacting inductive displacement transducer supplied the feedback signal to the servo controller ( 8 ) . In these constant-force resonance tests, the frequency was maintained at resonance while stress and amplitude were allowed to change. A total of five single crystal copper specimens were tested. The test variables were continually monitored for all the specimens, but accurate measurements of frequency and stress were made only for one typical sample. The crystals, all of the same orientation, were grown from a copper stock of 99.999+% purity and, prior to testing, they were all subjected to an annealing treatment in a vacuum of 2 x lO-b torr of 850oc for 24 hours.
~ ~
I SCOPE I t
-
~
~ 4 I I ....... I rnlINTFRI IX*YRECORDER~ L. . . . . . .
___1__'
........
STRAIN ~ ~ STRAIN I- ~ I N D I C A T O R ~ ' - - ' ~ GAGE COMPSG~ ~SPEC~
"3,
"
R I ECORDER I ~F.W.RECTIFIEBLL~ AMPLIFIER ~ I
,
I
, LI.J:RA,S CER
, ,
1 "
I,DUi ,vE-"½
TRANSDUCER/~°"I L,.~l~'i
,½ i I~IPLIFIER I
,F:O o,,v i I
I
I~--._~J-..6,,P"
-[CONTROLLER I
FIG. l Schematic of apparatus. Figure 2 demonstrates the changes in resonant frequency, end deflection and stress. The resonant frequency, after a rapid decrease from an i n i t i a l value of 22.40 hz to a minimum value of 18.68 hz at 4400 cycles, increases gradually and reaches saturation at about 4 x lO4 cycles. The stress and end deflection gradually increase and simultaneously reach saturation at 2.5 x I05 cycles. The increase in end deflection as the number of cycles advances is due only to the structural changes in the cyrstal and not to any influence of the apparatus. Since the
Vol.
11,
No.
7
DISLOCATION DENSITY IN FATIGUE
531
input energy to vibrate the specimen was constant, the small i n i t i a l end deflection is only the result of the large energy dissipation by hysteresis losses. As these losses decrease (8), the amplitude of deflection increases.
23
22 0.3
u.-~21 0.2 ~" 20 0.1 19 2
4
6
8
10
12
14
NUMBER OF CYCLES, 104
FIG. 2 Relationship between stress, end deflection and resonant frequency. From the data in Figure 2 and using equations 3 and 4, the parameters L, A, and p* were evaluated. The results are given in Table l and plotted in Figure 3. The active length L steeply decreases during the f i r s t lO00 cycles and approaches a value of approximately O.lum. At saturation, p* is 9.6 x I09 cm-2 and the resolved shear stress is 2.78 kg-mm-2. This stress level is very close to that previously reported for the development of a two-phase dislocation structure (matrix and wall) in single crystals of copper fatigued to saturation at low strain amplitudes (9, lO). In such a structure, the fatigue deformation concentrates into the persustent s l i p bands (which correspond to the wall structures) and the crystal accommodates the applied strain by adjusting the relative amounts of the two phases, thus maintaining a constant strain amplitude in each phase irrespective of the average applied strain ( l l ) . This implies that at a constant saturation stress, and independent of the applied plastic strain amplitude, the configuration and the density of the dislocations within the two phases w i l l remain the same. Therefore, in a flexure test, in spite of a non-homogeneous strain f i e l d , and at least to the depths that the two-phase structure extends, the dislocation parameters would be comparable to those in a push-pull test. The result of TEM studies (12) for push-pull low amplitude tests indicates that at saturation, the average density of dislocations for the matrix structure is 8.5 x lOlOcm-2 and for the wall structure i t is 5.6 x lOlO cm-2.
NUMBER OF CYCLES(104)
FIG. 3 Changes in L, A and p* as a function of number of cycles.
532
DISLOCATION
DENSITY
IN FATIGUE
Vol,
Ii, No,
7
The'average density of mobile dislocations at saturation is found to be 9.6 x I09 cm-2. Therefore, i t is estimated that about 13% of the dislocations are active in the low amplitude fatigue deformation process.
TABLE l N
Fr
~Fr
~SFr Fr T~-
p*
T
k9 cm2
(10 4 )
(hz)
(hz)
0.4
18.80
3.60
l .161
130
O.191
l .34
2.62
l
19.28
3.12
0.139
166
0.149
2.52
3.75
2
19.50
2.90
0.129
206
0.120
4.48
5.38
4
19.56
2.84
0.127
242
0.I02
7.18
7.32
6
"
258
O.096
8.61
8.27
ii
267
O.093
9.47
8.81
272
0.091
lO.ll
9.20
8
ii
(.m)
s II
(I 014/cm3)
(109/cm2)
lO
"
in
12
"
Ii
276
O.090
I0.45
9.41
14
"
no
278
O.089
lO.81
9.62
References I.
A. V.~anatoand K. Lucke, J. of Appl. Phys., Vol. 27, p. 583 and 789 (1956).
2.
J. Friedel, Dislocations, Pergamon Press, N. Y. , p. 235 (1963).
3.
N. F. Mott, Phil. Mag., Vol. 43, p. ll51 (1952).
4.
B. Klein, J. of Royal Aeron. Soc., Vol. 60, p. 282 (1956).
5.
F. R. N. Nabarro, Z. S. Basinski and D. B. Holt, Advances in Physics, Vol. 13, No. 50, p. 193 (1964).
6.
H. Wiedersich, Journal of Metals, p. 425 (May 1964).
7.
F. R. N. Nabarro, Theory of Crystal Dislocations, Oxford Univ, Press (1967).
8.
P. Dadras, Ph.D. Dissertation, MAE Dept. Univ. of Del. (1972).
9.
P. J. Woods, Phil. Mag., Vol. 28, p. 155 (1973).
I0. A. T. Winter, Phil. Mag., Vol. 30, p. 719 (1974). I I . J. M. Finney and C. Laird, Phil. Mag., Vol. 31, p. 339 (1975). 12. J. G. Antonopoulos and A. T. Winter, Phil. Mag., Vol. 33, p. 87 (1976).
Vol, II, pp, 529-532, 1977 Printed in the United States
Pergamon Press, Inc,
EXPERIMENTAL METHOD FOR THE APPROXIMATEDETERMINATIONOF DISLOCATION DENSITY IN FATIGUE Parviz Dadras Department of M e t a l l u r g i c a l Engineering Arya-Mehr U n i v e r s i t y of Technology Tehran, Iran
(Received January 31, 1977) (Revised May 20, 1977)
Under the influence of an a l t e r n a t i n g stress f i e l d , a d i s l o c a t i o n segment can o s c i l l a t e between two pinning points provided by jogs, point defects, and other d i s l o c a t i o n s in a manner s i m i l a r to a s t r i n g v i b r a t i n g in a viscous medium ( I ) . For the high stress levels leading to fatigue deformation and f r a c t u r e , some of the d i s l o c a t i o n s can break away from the weak pinning points and thus accommodate l a r g e r s t r a i n s by producing longer segments of o s c i l l a t i n g dislocat i o n s . The conbributions that the d i s l o c a t i o n s make to the overall magnitude of s t r a i n as they bow out ~n t h e i r g l i d e planes under the influence of an o s c i l l a t i n g stress c o n s t i t u t e s a reversi b l e e l a s t i c s t r a i n of the magnitude (2) 6E = A A b where A is the area swept out by the d i s l o c a t i o n , b is the Burgers' vector and A is the number of active d i s l o c a t i o n s in a u n i t volume. The e l a s t i c s t r a i n ~e reduces the apparent e l a s t i c modulus, E, by 6 E in such a way t h a t : 6 = EE = (E + ~E) (~ + 6c) Using the expression f o r A, as calculated by Mott (3), one finds t h a t : ~E E - c A L3
(Eq. I )
where c is a constant and L is the average free length of the d i s l o c a t i o n s c o n t r i b u t o r y to modulus defect, 6E . E A r e l a t i o n s h i p between the modulus defect and the change in resonant frequency f o r the case where the specimen is vibrated in a f l e x u r e mode of deformation can be developed. Based on the f i n d i n g of Klein (4), the natural frequency of a c a n t i l e v e r plate of a r b i t r a r y plan form can be expressed as : Fr = K1 E_l i ~ ]
I/2
where K1 is a constant dependent on the form, mass, and thickness of the plate. Since Poisson's r a t i o , v, is not s t r u c t u r e s e n s i t i v e , i t can be assumed that i t s value does not change appreciably as a r e s u l t of c y c l i c deformation. Therefore, Fr = K2 ( E ) I / 2 which leads to: 2 6 Fr = 6E Fr
E
529
(Eq. 2)
530
DISLOCATION DENSITY IN FATIGUE
Vol. ii, No. 7
Moreover, in cyclic deformation, the flow stress can be related to the length of the active dislocation segment L, by T
:
Gb WL
-
(Eq. 3)
-
where w usually has values between 3 and 5 (5). Taking the value of w as 5 (6), the resulting decrease in L can be related to the changes in the flow stress for the increasing number of cycles. Furthermore, i f L, as determined by this method, is a reasonable approximation of the average length of dislocation segments which lead to the modulus change, then A can also be calculated. Thus, combining equations l and 2 using an approximation for the constant C (7): A -
6
~Fr
L3
Fr
(Eq. 4)
The number of active dislocations per unit volume of length L multiplied by the lengths of the dislocation segments equals the density of dislocations active in fatiguey p*. In the present investigation, the necessary experimental data are obtained which f a c i l i t a t e the approximate determination of the length and density of dislocations which are active in a fatigue deformation process. To perform the cyclic tests, a resonant bending fatigue apparatus was developed. An alternating f i e l d was produced by two electromagnets as shown in Figure I. Constant-bendingstress specimens with permanent magnets clamped to their ends were placed in the alternating force field of the electromagnets and forcing functions of equal amplitude were assured in all the tests. A non-contacting inductive displacement transducer supplied the feedback signal to the servo controller ( 8 ) . In these constant-force resonance tests, the frequency was maintained at resonance while stress and amplitude were allowed to change. A total of five single crystal copper specimens were tested. The test variables were continually monitored for all the specimens, but accurate measurements of frequency and stress were made only for one typical sample. The crystals, all of the same orientation, were grown from a copper stock of 99.999+% purity and, prior to testing, they were all subjected to an annealing treatment in a vacuum of 2 x lO-b torr of 850oc for 24 hours.
~ ~
I SCOPE I t
-
~
~ 4 I I ....... I rnlINTFRI IX*YRECORDER~ L. . . . . . .
___1__'
........
STRAIN ~ ~ STRAIN I- ~ I N D I C A T O R ~ ' - - ' ~ GAGE COMPSG~ ~SPEC~
"3,
"
R I ECORDER I ~F.W.RECTIFIEBLL~ AMPLIFIER ~ I
,
I
, LI.J:RA,S CER
, ,
1 "
I,DUi ,vE-"½
TRANSDUCER/~°"I L,.~l~'i
,½ i I~IPLIFIER I
,F:O o,,v i I
I
I~--._~J-..6,,P"
-[CONTROLLER I
FIG. l Schematic of apparatus. Figure 2 demonstrates the changes in resonant frequency, end deflection and stress. The resonant frequency, after a rapid decrease from an i n i t i a l value of 22.40 hz to a minimum value of 18.68 hz at 4400 cycles, increases gradually and reaches saturation at about 4 x lO4 cycles. The stress and end deflection gradually increase and simultaneously reach saturation at 2.5 x I05 cycles. The increase in end deflection as the number of cycles advances is due only to the structural changes in the cyrstal and not to any influence of the apparatus. Since the
Vol.
11,
No.
7
DISLOCATION DENSITY IN FATIGUE
531
input energy to vibrate the specimen was constant, the small i n i t i a l end deflection is only the result of the large energy dissipation by hysteresis losses. As these losses decrease (8), the amplitude of deflection increases.
23
22 0.3
u.-~21 0.2 ~" 20 0.1 19 2
4
6
8
10
12
14
NUMBER OF CYCLES, 104
FIG. 2 Relationship between stress, end deflection and resonant frequency. From the data in Figure 2 and using equations 3 and 4, the parameters L, A, and p* were evaluated. The results are given in Table l and plotted in Figure 3. The active length L steeply decreases during the f i r s t lO00 cycles and approaches a value of approximately O.lum. At saturation, p* is 9.6 x I09 cm-2 and the resolved shear stress is 2.78 kg-mm-2. This stress level is very close to that previously reported for the development of a two-phase dislocation structure (matrix and wall) in single crystals of copper fatigued to saturation at low strain amplitudes (9, lO). In such a structure, the fatigue deformation concentrates into the persustent s l i p bands (which correspond to the wall structures) and the crystal accommodates the applied strain by adjusting the relative amounts of the two phases, thus maintaining a constant strain amplitude in each phase irrespective of the average applied strain ( l l ) . This implies that at a constant saturation stress, and independent of the applied plastic strain amplitude, the configuration and the density of the dislocations within the two phases w i l l remain the same. Therefore, in a flexure test, in spite of a non-homogeneous strain f i e l d , and at least to the depths that the two-phase structure extends, the dislocation parameters would be comparable to those in a push-pull test. The result of TEM studies (12) for push-pull low amplitude tests indicates that at saturation, the average density of dislocations for the matrix structure is 8.5 x lOlOcm-2 and for the wall structure i t is 5.6 x lOlO cm-2.
NUMBER OF CYCLES(104)
FIG. 3 Changes in L, A and p* as a function of number of cycles.
532
DISLOCATION
DENSITY
IN FATIGUE
Vol,
Ii, No,
7
The'average density of mobile dislocations at saturation is found to be 9.6 x I09 cm-2. Therefore, i t is estimated that about 13% of the dislocations are active in the low amplitude fatigue deformation process.
TABLE l N
Fr
~Fr
~SFr Fr T~-
p*
T
k9 cm2
(10 4 )
(hz)
(hz)
0.4
18.80
3.60
l .161
130
O.191
l .34
2.62
l
19.28
3.12
0.139
166
0.149
2.52
3.75
2
19.50
2.90
0.129
206
0.120
4.48
5.38
4
19.56
2.84
0.127
242
0.I02
7.18
7.32
6
"
258
O.096
8.61
8.27
ii
267
O.093
9.47
8.81
272
0.091
lO.ll
9.20
8
ii
(.m)
s II
(I 014/cm3)
(109/cm2)
lO
"
in
12
"
Ii
276
O.090
I0.45
9.41
14
"
no
278
O.089
lO.81
9.62
References I.
A. V.~anatoand K. Lucke, J. of Appl. Phys., Vol. 27, p. 583 and 789 (1956).
2.
J. Friedel, Dislocations, Pergamon Press, N. Y. , p. 235 (1963).
3.
N. F. Mott, Phil. Mag., Vol. 43, p. ll51 (1952).
4.
B. Klein, J. of Royal Aeron. Soc., Vol. 60, p. 282 (1956).
5.
F. R. N. Nabarro, Z. S. Basinski and D. B. Holt, Advances in Physics, Vol. 13, No. 50, p. 193 (1964).
6.
H. Wiedersich, Journal of Metals, p. 425 (May 1964).
7.
F. R. N. Nabarro, Theory of Crystal Dislocations, Oxford Univ, Press (1967).
8.
P. Dadras, Ph.D. Dissertation, MAE Dept. Univ. of Del. (1972).
9.
P. J. Woods, Phil. Mag., Vol. 28, p. 155 (1973).
I0. A. T. Winter, Phil. Mag., Vol. 30, p. 719 (1974). I I . J. M. Finney and C. Laird, Phil. Mag., Vol. 31, p. 339 (1975). 12. J. G. Antonopoulos and A. T. Winter, Phil. Mag., Vol. 33, p. 87 (1976).