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Parameters and micromechanisms of fatigue crack growth in sheet magnesium alloy samples
Parameters and micromechanisms of fatigue crack growth in sheet magnesium alloy samples N. M. G r i n b e r g a n d V. A. S e r d y u k Growth rates of part-through and through fatigue cracks have been measured for two magnesium alloys - MAI 2 and IMV6 - and the micromechanisms of fatigue fracture were studied at all stages of growth. Conclusions about the peculiarities of the kinetics and micromechanisrns of part-through and through crack growth, depending on the applied stress amplitudes and alloy structure, are made from a comparison of the results obtained. The growth of existing through cracks can be considered by experimental determination of fatigue crack growth rates and by plotting the kinetic diagram for fatigue fracture. In reality, however, it is the growth of surface (part-through) sub-microscopic cracks which dominates the fatigue process, and the mechanism of part-through crack growth differs from that for through fatigue cracks. 1-5 Using linear mechanics equations, Pearson 1 found that the growth rates of very short cracks were higher than those for through cracks. Hagiwara et al 2 have shown experimentally differences in the growth rates of surface and internal cracks and related these differences to the structural characteristics of the material. It was also shown3 that for surface cracks of approximately 0.5 mm length, the transient condition of non-propagation of a very short crack is at stresses equal to the fatigue limit, but not the threshold value of stress intensity factor. Investigations of fatigue crack growth rates in sheet magnesium alloy samples 6-8 have indicated that this process could be described by the Paris equation d/ dN
n
- CKma x
(1)
where C and n are coefficients which depend on the region of stress intensity factor change. This paper is the result of investigations of fatigue crack growth rates in magnesium alloy sheets which aimed to study the parameters and micromechanisms of crack growth in different alloys and the structural state during microcrack development, beginning with their surface nucleation. Further, it was intended to examine the influence of stress amplitude on the growth rate of microcracks as it has been shown that the K parameters are an ambiguous description of the function of fatigue crack growth rate. 9,10
MA TERIA L AND TECHNIQUE Two magnesium alloys MA12 and IMV6 were investigated. Chemical compositions, heat treatments and mechanical properties of the alloys are given in Table 1. After annealing the MA12 alloy has an equiaxial structure of a-phase on a magnesium base with large intermetallic Mg9Nd precipitates distributed within the grains and on their boundaries. The average grain diameter of this alloy is 20/~m and the inclusion size ranges from 1 to 15 #m. The IMV6 alloy after hot-pressing comprises a-phase with an average grain diameter of 40/~m and finely dispersed
Table 1. Characteristics and heat treatment of MA12 and IMV6 magnesium alloys Tensile properties Composition (weight%)
Heat treatment
Of
MA12
2.9 Nd 0.44 Zr rest Mg
1 h anneal at 623K cooled in air
209
128
18.5
I MV6
7.8 ¥ 0.12 AI 0.55 Mn 0.49 Cd 0.11 Ce rest Mg
Hot pressed with no other treatment
304
240
16.0
Alloy
GO.2
(~
(MPa) (MPa) (%)
intermetallic Mg24Y5 precipitate. Under cyclic load, the IMV6 alloy displays structural instability which results in its softening, 11 whereas the MAI 2 has a stable structure. 12 Samples were stamped from a sheet, and then mechanically, and electrolytically polished and notched to fix a site for crack nucleation and growth. 11 These were subjected to cyclic symmetric cantilever bending at a constant strain amplitude and a frequency of 12.5 Hz in a device described elsewhere 12 and the fatigue crack nucleation and growth were observed using an MI-I optical microscope. The crack length increments were read at invervals of 0.01 mm from the initial length of 0.05 mm to 0.5 mm with an accuracy of about -+ 2%. Values of stress intensity factor K were calculated as K = a V r ~ Y, where e is the nominal brutto-weight stress of a cycle, l is the crack length and Y is a coefficient which accounts for the sample and crack shapes. The values of Y were determined experimentally by the compliance method. 8 The threshold value Kth was obtained by a stepwise decrease of the load by not more than 3% corresponding to the crack length. 13 Fractographic electron microscopy investigations were carried out using UEMV-100V Electron Microscope and employing the method of two-stage plastic/carbon replicas. Magnifications of x 7000 to x 20 000 were used to study the fracture surface morphologies.
STRESS INTENSITY DETERMINATION The stress intensity factor for bending has been calculated
0142--1123/81/030143--06 $02.00 © IPC Business Press Limited 1981
INT, J. FATIGUE July 1981 143
using the expression 24 K-
6M vT Y tb 2
where M is the bending moment, b is the sample width, I is the crack length and Y is the dimensionless coefficient depending on the l/b ratio. Since the tests were carried out with a constant strain amplitude the values of applied load (and moment M ) decreased with the increase in crack length such that P = P0 41 where ~ 1 is a dimensionless coefficient which accounts for the load fall with increasing 1 and P and P0 are the current and initial load amplitudes respectively. Then K=
(3)
tb 2
-dU -
p2 d(£/P)
(4)
-
dF
2
dF
where f is the sample inflection along the line of load application and F is the crack surface area. The stress intensity factor is given by E K 2 =-- a
(5)
n where E is the Young's modulus and n = 1 for plane stress conditions and n = 1 -/~2 for the plane strain state. Equation (2) can be transformed into K = abl/2 Yl where Y1 = Y(I/b) 112 and o = 6M/tb 2. Then
K = ~OObl/2 • Y1
(6)
To determine the dimensionless coefficient Y1 Equations (5) and (6) can be equated and used with Equation (4) to give
Et2b 3 (d(f/P)~ 1/2
y ~ - - ~
72nL 2 \ - - ~
/
where L is the distance between the point of force application and the calculated section. RESULTS AND DISCUSS~ON
It can be seen in Fig. 1 that the di/dN = f(Kmax) relationship plotted in logarithmic coordinates has two linear sections for MA12 and three for IMV6. In each of them the crack growth rate is described by the Paris equation with different C and n values (Table 2). The first section - of very slow crack growth where Kmax ~
144
,r#
E
i(53
16"
6M0¢ VT Y
where 4 = 41 x 42 and ~b2 is a factor which accounts for the stress distribution around the notch and is determined according to Neuber 25 Changes of load P corresponding to an increased crack length were determined by means of strain gauges. The experimental 41 = f(l/b) dependence was plotted for samples of both alloys. Values of Y were obtained using elastic compliance measurements made on the notched specimens in conjunction with the well known expression 26 a
/
(2)
- -
INT. J. FATIGUE July 1981
I i I I
I
I I I 5
I I
~"-
=1
II I0
Fig. 1 Fatigue crack growth rate v$ stress intensity factor for magnesium (1) MA12 and (2) IMV6 alloys
The second section - of slow crack development ends when the threshold value of stress intensity factor, K 2 is reached and this corresponds to crack propagation throughout the sample thickness. Crack length at the surface reaches 2 to 2.5 mm at all the stresses investigated. Hence the ratio of the crock increment in depth to the increment in surface length (db/di) remains equal to 0.2 to 0.25 for both sections. This confirms the earlier result 14 showing that the surface crack has a half-eliptical form with its major axis along the sample width and minor axis along the thickness. The third linear section - where K 2 ~
Table 2. C and n coefficients f o r d i f f e r e n t sections of stress intensity factors Initial stress amplitude (MPa)
Section I (Krnax ~< K1 ) C1
n1
C2
n2
C3
n3
MA12
137 110 82
--
-
4.8 x 10 -7 2.8 x 10 -7 1.4 x 10 -7
2.2 2.2 2.2
4.5 x 10 -1° 4.5 x 10 -1° 4.5 x 10 -1°
4.2 4.2 4.2
IMV6
172
8
1.2 1.2 1.2
5.8 x 10 -8 2.8 x 10-8 1.8 x 10-8
2.8 2.8 2.8
2.1 x 10 -1° 2.1 x l 0 - ; ° 2.1 x 10 -;°
4.4 4.4 4.4
Alloy
S e c t i o n II ( K , ~< Kmax ~< K2)
x l 0 -~ 5 x l 0 -~ 4 . 4 x 1 0 -~
133 118
/
results show that over the linear section, (Kmax ~ K3) the growth rate of both through and part-through cracks can be described by the Paris equation. However, for part-through cracks the factor C is not a constant, but varies with the stress amplitude. It should be noted that the threshold values of stress intensity factor K 2, corresponding to the transition of a part-through to a through crack, increase with stress increase and approach the threshold value K 3. This implies that with the increase in stress amplitude, the range of/
#
I t -
o.
I I
2
I
ii
I K m (MPo~/'~)
I I
I
IO
S e c t i o n III (K2 ~ Kmax ~ K3)
#
#
E E
(7) 7.
I II
It
II
I
I
d
t II II
4'
I I
Km (MP0~rm) Fig. 2 Growth rate of part-through (1 --3, 5--7) and through (4, 8) fatigue cracks vs stress intensity factor for magnesium (a) MA12 and (b) IMV6 alloys at different stress amplitudes: (1) o = 137MPa (2) o = 110MPa, (3) o = 82 MPa, (5) o = 172 MPa, (6) o = 133 MPa, (7) o = 118 MPa. In (b), K'L = 3.8 MPa~/m when dlldN = 3.3 x 1O- s m/cycle, K'~' = 4.3 MPa ~/m when dl/dN = 6 x 1 0 - ' m/cycle and K;" = 4.5 MPa ~/m when dl/dN = 8.6 x 10 -s m/cycle
where A and m are constants. The region K 0 ~/
I N T . J. F A T I G U E J u l y 1981
145
rates in a plate were determined in tension and bending has shown that the lack of coincidence between the growth rates of part-through and through cracks exists for the both types of deformation. The electron fractographic investigation of fatigue failed samples revealed some correlation between the crack growth rate in separate sections of the curve and the fracture micromechanism depending on the alloy structural state. For IMV6, in the zone corresponding to the first section of the surface crack growth rate curve (Kmax ~
structure, for example, the titanium Ti-6A1-4V alloy. In the second zone (K 1 ~< Kmax ~< K 2) facets of regular striated structure with small spaces between the striations are observed (Fig. 3c) together with the cleavage regions. In the zone of through crack growth striation spacing increases and as the stress intensity factor approaches the threshold value K 3 ductile cleavage regions and a dimple structure appear together with striations (Fig. 3e). This structure of ductile fracture dominates in the zone of final fracture Kmax/> K3. The fracture structures described are typical for IMV6 alloy at all the stresses studied. Fracture by transgrain quasi-cleavage - with a river structure (Fig. 3f) - dominates in MA12 alloy in the zone of part-through crack growth at low stress amplitudes and regions showing weak striations occur seldom (Fig. 3i). As the stress amplitude increases many regions of intergrain fracture appear (Fig. 3j). The quasi-cleavage structure with
i
Fig. 3 Microfra¢tographs of fatigue failure in magnesium (a--e) I M V 6 and (f--k) M A 1 2 alloys: a-river structure and flat surface of transgranular fracture in region o f part-through crack growth (Kmax < K1 ), b- river structure f r o m grain boundary in same region, c- striations in region of part-through crack growth at K 1 < Kma x < K2, d o striations in region o f part-through crack growth at K 2 < Kma x < K 3, e- dimples in region Kma x -~ K3, f- structure and cleavage in region of part-through crack growth at Kma x < K 2 (o = 82 MPa), g- river structure in region of partthrough crack growth at Kma x < K 2 (o = 82 MPa), h- river structure in region o f through crack growth at K 0 < Krnax < K3 (a = 82 MPa), i- striations in region of part-through crack growth at Kma x < K 2 (o = 82 MPa), j- intergranular fracture w i t h particle traces in region of partthrough crack growth at Kma x < K 2 (a = 137 MPa), k- dimples in region of through crack growth at Kma x > / K 3. Magnification x 7 0 0 0
146
INT. J. FATIGUE July 1981
a few striated facets is also typical for a through crack in its stable growth zone and dimples are observed in the final fracture zone (Fig. 3k). The above results show, that the growth of surface microcrecks in both alloys occurs with the formation of more brittle components than the growth of through microcracks and this conforms with the higher growth rate of part-through cracks compared with the through cracks. The results also justify the conclusion that the fracture micromechanisms for the two alloys are different both for part-through and through cracks. The more brittle fracture in MA12 alloy (presence of facets with inter- and transgraJn cleavage) compared with the fracture of IMV6 alloy (formation of striated fracture) indicates, that the fatigue microcreck in the former develops with lower plastic deformation in the crack zone. However, the change of growth rate for part-through microcracks, depending on stress amplitude and the greater growth rate compared with the through crack, is typical for both alloys irrespective of fracture mechanism. The stress amplitude dependence of part-through fatigue crack growth rate may be explained as follows. The initial microcrack in a sample, having a hemispherical notch, penetrates to a great degree the subsurface layer - where the plastic deformation prevails - compared with the sample volume. 11,20 The forming dislocation structure depends on the stress amplitude, for example, in copper the cell diameter is inversely proportional to the stress value. 2t Transition from the high-amplitude to the lowamplitude region changes the fatigue mechanism. 22 As the observation showed, the minimum stress amplitude corresponds to the low-amplitude and the maximum stress amplitude to the high-amplitude region of the e/N diagram of the alloys investigated. In MA12 alloy transition from the first stress region to the second changes the part-through crack growth mechanism from mainly transgrain to intergrain. In the low-amplitude region the IMV6 alloy is characterized by the fatigue fracture mechanism with the formation of persistent slip bands in one slip system due to softening 11 similar to that described for MA12 (T6) alloy. 12,13 In the high-amplitude region the second slip system activates and promotes a more rapid work-hardening and thus greater crack growth. Differences in dislocation structures formed at various stresses in the process of hardening or softening prior to crack formation and its transition to a through crack stimulated the effect that these stresses had on the growth rate of part-through cracks at the same value of Kmax- The higher growth rate of partthrough cracks compared with through cracks at the same value of Kmax seems to be explained by the different extent of plastic deformation and stress field in the zones in front of the cracks. This assumption is confirmed by the results of Pratt 23 which show the delay in through crack growth rate when its length is short under the plastic zone from notch.
3)
4) 5)
REFERENCES 1.
Pearson, $. 'Initiation and fatigue cracks in commercial aluminium alloys and subsequent propagation of very short cracks' Engng Fracture Mech 7 (1975) pp 235--247
2.
Hagiwara, Y., Yoshino, T. and Kunio, T. 'Difference between the surface and inner fatigue crack propagation behaviours. Effect of material properties'. Proc 19th Japan Conf Materials Research (1976) pp 22--27
3.
Kitagawa, H. and Takahashi, S. 'Application of fracture mechanics to very small cracks in early stage', 2nd Confon Mechanical Behaviour o f Materials (Boston MA, August 1976) pp 627--630
4.
Collipriest, J. E. 'An experimentalist's view of the surface flaw problem' The surface crack: physical problems and computational solutions (edited by Swedlow, ASME, 1972) pp 43--61
5.
Nair, P. K. 'Fatigue crack growth model for part-through flaws in plates and pipes', Trans ASME J Mater and Tech 101 No 1 (1971) pp 53--58
6.
Grinberg, N. M., Sardyuk, V. A., Zmeevets, S. G., Ostapenko, I. L., Malinkina, T. I. and Kamyshkov, A. S. 'Rost ustalostnykh treshchin v magniyevom splave MA12 na vozdukhe i v vakuume' Problemyprochnosti No 3 (1978) pp 12--16
7.
Grinberg, N. M., Serdyuk, V. A. and Zmeevets, S. G. 'Vliyanie strukturnogo sostoyaniya na ustalostnoye razrusheniye magniyevogo splave MA12 na vozdukhe i v vakuume. Soobshcheniye 2. Mikroskopicheskie i makroskopicheskiye osobennosti rosta usta!ostnykh treshchin na vozdukhe' Problemy prochnosti No 10 (1978) pp 46--52
8.
Serdyuk, V. A. 'lssledovanie skorosti testa ustalostnykh treshchin v magniyevykh splavakh pri komnatnoi i nizkoi temperaturakh' Problemyprochnosti No 11 (1980) pp 18--23
9.
Gudkov, Ao A. and Zoteev, V. S. 'Vliyanie prilozhennogo napryazheniya na skorost rasprostraneniya ustalostnoi treshchiny pri pulsiruyushmetallov, M'. Metallurgiya No 4 (1977) pp 17--28
CONCL USIONS
1)
2)
The growth rate of part-through fatigue cracks in magnesium alloy sheet samples is described by the Paris equation similarly to the midamplitude region of a kinetic diagram for a througri crack, but the coefficient C for part-through cracks depends on the stress amplitude. The part-through crack growth rate in log (d//dN) = log f (Kmax) coordinates is characterized, depending
on the alloy structural state, by one or two linear sections. The first region is observed only in a softening alloy and corresponds to the growth of surface cracks in persistent slip bands. The second region corresponds to the growth of elliptical part-through cracks by a stage II mechanism. The part-through crack growth rate is higher than that for through cracks at the same Kmax value, and the intensity of its change is greater for through than for part-through cracks. The full kinetic diagram for magnesium alloy fatigue has an S shape for through cracks and may be approximated by Equation (7). Micromechanisms of magnesium alloy fatigue fracture depend on the alloy structural state, Kmax value, and in the case of part-through cracks on stress amplitude as well. More brittle fracture is typical for surface cracks compared with the internal cracks. More plastic micromechanisms of fatigue crack growth are observed for the IMV6 alloy compared with the MAI 2 alloy.
10.
Kishkina, S. I. and Starova, E. N. 'Nekotoryye osobennosti rosta treshchin ustalosti v tonkikh plastinakh aluminievykh splavov, v Sb. Problemy metallovedeniya tsvetnykh splevov, M'. Nauka (1978) pp 166--172
11.
Grinberg, N. M. and Serdyuk, V. A. 'Razuprochneniye magnievogo splave IMV6 v protsesse ustalosti' Problem)/ prochnosti No 1 (1980) pp 35--39
12.
Grinberg, N. M., Serdyuk, V. A. and Zmeevets, S. G. 'Vliyanie strukturnogo sostoyaniya na ustalostnoe
I N T . J. F A T I G U E J u l y 1981
147
rasrushenie magnievogo splava MA12 na vozdukhe i v vakuume. Soobshcheniye I. Dolgovechnost i zarozhdenie ustalostnykh treshchin' Problemy prochnosti No 9 (1978) pp 32--38
i vnutrnnikh sloev polikristallicheskogo zheleza pri ustalostnom nagruzhenii' DAN SSSR 205 No 4 (1972) pp 8 1 2 814
13.
Grinberg, N. M., Serdyuk, V. A., Yakovenko, L. F. Malinkine, T. I. and Kamyshkov, A. S. 'Kinetika i mekhanism ustalostnogo razrusheniya magnievykh splavov MA2-1 i MA12' Problemy provhnosty no 8 (1977) pp 40--45
21.
Pratt, J. J. 'Dislocation substructure in strain-cycled copper as influenced by temperature' Acta Met 15 No 5 (1967) pp 319--327
22.
14.
Yakovenko, L. F. 'Ustanovka dlya ustalostnykh ispytanii v vakuume pri komnatnoi i nizkoi temperaturakh' Zavodskaya lab No 2 (1971) pp 232--234
Wood, W. A., Cousland, S. Mck. and Sargant, K. R. 'Systematic microstructural changes peculiar to fatigue deformation' Acta Met 11 No 7 (1963) pp 643--652
23.
15.
Metodicheskie ukazaniya. Raschety i ispytaniya na prochnost v mashinostroenii. Metody mekhanicheskikh ispytanii metallov. Opredelenie kharakteristik soprotivleniya razvitiyu treshchiny (treshchinostoikost) pri tsiklicheskom nagruzhenii -- Lvov: VN I IMash Gosstandarta SSSR, FM I AN USSR (1979) 125 p
Plumbridga, W. J. 'Problems associated with early stage fatigue crack growth' Metal Sci 12 No 5 (1978) pp 251--256
24.
Brown, W. F. Jr. and Srawley, J. E. 'Plane strain toughness testing of high strength metallic materials' ASTM STP 410 (Philadelphia, 1967)
25.
Neuber, H. "Theory o f notch stress" (J. S. Edwards, Ann Arbor, Michigan, 1946)
26.
Irwin, G. R. 'Fracture' Encyclopedia o f Physics 6 (1958) pp 551--590
16.
Burk, L. H. 'Fatigue Growth of Surface Cracks in Bending' Engng Fracture Mech 9 (1977) pp 389--395
17.
Burk, L. N., Sullivan, C. P. and Wells, C. H. 'Fatigue of a glass bead blasted nickel-base superalloy' Met Trans 1 (1970) pp 1595--1600
18.
Yarema, S. Ya. and Mikitishin, S. I. 'Analiticheskoe opisanie diagrammy ustalostnogo razrusheniya materialov' Fiziko-khimicheskaya mekhanika materialov No 6 (1975) pp 47--54
19.
Nagai, A., Toyo~Kla, M. and Okamoto, T. 'A study on the fatigue crack growth in 9% Ni steel plate' Engng Fracture Mech 7 (1975) pp 481--490
20.
Goritskii, V. M., Ivanova, V. S., Orlov, L. G. and Terentev, V. F. 'O razlichii plasticheskoi deformatsii poverkhnostnykh
148
I N T . J. F A T I G U E J u l y 1981
AUTHORS The authors are with the Physico-Technical Institute of Low Temperatures which is part of the Ukrainian Academy of Sciences. Inquiries in the first instance should be directed to Dr N. M. Grinberg at the following address: PhysicoTechnical Institute of Low Temperatures, Ukrainian Academy of Sciences, Lenin's Prospect 47, Kharkov 164, USSR.
n
- CKma x
(1)
where C and n are coefficients which depend on the region of stress intensity factor change. This paper is the result of investigations of fatigue crack growth rates in magnesium alloy sheets which aimed to study the parameters and micromechanisms of crack growth in different alloys and the structural state during microcrack development, beginning with their surface nucleation. Further, it was intended to examine the influence of stress amplitude on the growth rate of microcracks as it has been shown that the K parameters are an ambiguous description of the function of fatigue crack growth rate. 9,10
MA TERIA L AND TECHNIQUE Two magnesium alloys MA12 and IMV6 were investigated. Chemical compositions, heat treatments and mechanical properties of the alloys are given in Table 1. After annealing the MA12 alloy has an equiaxial structure of a-phase on a magnesium base with large intermetallic Mg9Nd precipitates distributed within the grains and on their boundaries. The average grain diameter of this alloy is 20/~m and the inclusion size ranges from 1 to 15 #m. The IMV6 alloy after hot-pressing comprises a-phase with an average grain diameter of 40/~m and finely dispersed
Table 1. Characteristics and heat treatment of MA12 and IMV6 magnesium alloys Tensile properties Composition (weight%)
Heat treatment
Of
MA12
2.9 Nd 0.44 Zr rest Mg
1 h anneal at 623K cooled in air
209
128
18.5
I MV6
7.8 ¥ 0.12 AI 0.55 Mn 0.49 Cd 0.11 Ce rest Mg
Hot pressed with no other treatment
304
240
16.0
Alloy
GO.2
(~
(MPa) (MPa) (%)
intermetallic Mg24Y5 precipitate. Under cyclic load, the IMV6 alloy displays structural instability which results in its softening, 11 whereas the MAI 2 has a stable structure. 12 Samples were stamped from a sheet, and then mechanically, and electrolytically polished and notched to fix a site for crack nucleation and growth. 11 These were subjected to cyclic symmetric cantilever bending at a constant strain amplitude and a frequency of 12.5 Hz in a device described elsewhere 12 and the fatigue crack nucleation and growth were observed using an MI-I optical microscope. The crack length increments were read at invervals of 0.01 mm from the initial length of 0.05 mm to 0.5 mm with an accuracy of about -+ 2%. Values of stress intensity factor K were calculated as K = a V r ~ Y, where e is the nominal brutto-weight stress of a cycle, l is the crack length and Y is a coefficient which accounts for the sample and crack shapes. The values of Y were determined experimentally by the compliance method. 8 The threshold value Kth was obtained by a stepwise decrease of the load by not more than 3% corresponding to the crack length. 13 Fractographic electron microscopy investigations were carried out using UEMV-100V Electron Microscope and employing the method of two-stage plastic/carbon replicas. Magnifications of x 7000 to x 20 000 were used to study the fracture surface morphologies.
STRESS INTENSITY DETERMINATION The stress intensity factor for bending has been calculated
0142--1123/81/030143--06 $02.00 © IPC Business Press Limited 1981
INT, J. FATIGUE July 1981 143
using the expression 24 K-
6M vT Y tb 2
where M is the bending moment, b is the sample width, I is the crack length and Y is the dimensionless coefficient depending on the l/b ratio. Since the tests were carried out with a constant strain amplitude the values of applied load (and moment M ) decreased with the increase in crack length such that P = P0 41 where ~ 1 is a dimensionless coefficient which accounts for the load fall with increasing 1 and P and P0 are the current and initial load amplitudes respectively. Then K=
(3)
tb 2
-dU -
p2 d(£/P)
(4)
-
dF
2
dF
where f is the sample inflection along the line of load application and F is the crack surface area. The stress intensity factor is given by E K 2 =-- a
(5)
n where E is the Young's modulus and n = 1 for plane stress conditions and n = 1 -/~2 for the plane strain state. Equation (2) can be transformed into K = abl/2 Yl where Y1 = Y(I/b) 112 and o = 6M/tb 2. Then
K = ~OObl/2 • Y1
(6)
To determine the dimensionless coefficient Y1 Equations (5) and (6) can be equated and used with Equation (4) to give
Et2b 3 (d(f/P)~ 1/2
y ~ - - ~
72nL 2 \ - - ~
/
where L is the distance between the point of force application and the calculated section. RESULTS AND DISCUSS~ON
It can be seen in Fig. 1 that the di/dN = f(Kmax) relationship plotted in logarithmic coordinates has two linear sections for MA12 and three for IMV6. In each of them the crack growth rate is described by the Paris equation with different C and n values (Table 2). The first section - of very slow crack growth where Kmax ~
144
,r#
E
i(53
16"
6M0¢ VT Y
where 4 = 41 x 42 and ~b2 is a factor which accounts for the stress distribution around the notch and is determined according to Neuber 25 Changes of load P corresponding to an increased crack length were determined by means of strain gauges. The experimental 41 = f(l/b) dependence was plotted for samples of both alloys. Values of Y were obtained using elastic compliance measurements made on the notched specimens in conjunction with the well known expression 26 a
/
(2)
- -
INT. J. FATIGUE July 1981
I i I I
I
I I I 5
I I
~"-
=1
II I0
Fig. 1 Fatigue crack growth rate v$ stress intensity factor for magnesium (1) MA12 and (2) IMV6 alloys
The second section - of slow crack development ends when the threshold value of stress intensity factor, K 2 is reached and this corresponds to crack propagation throughout the sample thickness. Crack length at the surface reaches 2 to 2.5 mm at all the stresses investigated. Hence the ratio of the crock increment in depth to the increment in surface length (db/di) remains equal to 0.2 to 0.25 for both sections. This confirms the earlier result 14 showing that the surface crack has a half-eliptical form with its major axis along the sample width and minor axis along the thickness. The third linear section - where K 2 ~
Table 2. C and n coefficients f o r d i f f e r e n t sections of stress intensity factors Initial stress amplitude (MPa)
Section I (Krnax ~< K1 ) C1
n1
C2
n2
C3
n3
MA12
137 110 82
--
-
4.8 x 10 -7 2.8 x 10 -7 1.4 x 10 -7
2.2 2.2 2.2
4.5 x 10 -1° 4.5 x 10 -1° 4.5 x 10 -1°
4.2 4.2 4.2
IMV6
172
8
1.2 1.2 1.2
5.8 x 10 -8 2.8 x 10-8 1.8 x 10-8
2.8 2.8 2.8
2.1 x 10 -1° 2.1 x l 0 - ; ° 2.1 x 10 -;°
4.4 4.4 4.4
Alloy
S e c t i o n II ( K , ~< Kmax ~< K2)
x l 0 -~ 5 x l 0 -~ 4 . 4 x 1 0 -~
133 118
/
results show that over the linear section, (Kmax ~ K3) the growth rate of both through and part-through cracks can be described by the Paris equation. However, for part-through cracks the factor C is not a constant, but varies with the stress amplitude. It should be noted that the threshold values of stress intensity factor K 2, corresponding to the transition of a part-through to a through crack, increase with stress increase and approach the threshold value K 3. This implies that with the increase in stress amplitude, the range of/
#
I t -
o.
I I
2
I
ii
I K m (MPo~/'~)
I I
I
IO
S e c t i o n III (K2 ~ Kmax ~ K3)
#
#
E E
(7) 7.
I II
It
II
I
I
d
t II II
4'
I I
Km (MP0~rm) Fig. 2 Growth rate of part-through (1 --3, 5--7) and through (4, 8) fatigue cracks vs stress intensity factor for magnesium (a) MA12 and (b) IMV6 alloys at different stress amplitudes: (1) o = 137MPa (2) o = 110MPa, (3) o = 82 MPa, (5) o = 172 MPa, (6) o = 133 MPa, (7) o = 118 MPa. In (b), K'L = 3.8 MPa~/m when dlldN = 3.3 x 1O- s m/cycle, K'~' = 4.3 MPa ~/m when dl/dN = 6 x 1 0 - ' m/cycle and K;" = 4.5 MPa ~/m when dl/dN = 8.6 x 10 -s m/cycle
where A and m are constants. The region K 0 ~/
I N T . J. F A T I G U E J u l y 1981
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rates in a plate were determined in tension and bending has shown that the lack of coincidence between the growth rates of part-through and through cracks exists for the both types of deformation. The electron fractographic investigation of fatigue failed samples revealed some correlation between the crack growth rate in separate sections of the curve and the fracture micromechanism depending on the alloy structural state. For IMV6, in the zone corresponding to the first section of the surface crack growth rate curve (Kmax ~
structure, for example, the titanium Ti-6A1-4V alloy. In the second zone (K 1 ~< Kmax ~< K 2) facets of regular striated structure with small spaces between the striations are observed (Fig. 3c) together with the cleavage regions. In the zone of through crack growth striation spacing increases and as the stress intensity factor approaches the threshold value K 3 ductile cleavage regions and a dimple structure appear together with striations (Fig. 3e). This structure of ductile fracture dominates in the zone of final fracture Kmax/> K3. The fracture structures described are typical for IMV6 alloy at all the stresses studied. Fracture by transgrain quasi-cleavage - with a river structure (Fig. 3f) - dominates in MA12 alloy in the zone of part-through crack growth at low stress amplitudes and regions showing weak striations occur seldom (Fig. 3i). As the stress amplitude increases many regions of intergrain fracture appear (Fig. 3j). The quasi-cleavage structure with
i
Fig. 3 Microfra¢tographs of fatigue failure in magnesium (a--e) I M V 6 and (f--k) M A 1 2 alloys: a-river structure and flat surface of transgranular fracture in region o f part-through crack growth (Kmax < K1 ), b- river structure f r o m grain boundary in same region, c- striations in region of part-through crack growth at K 1 < Kma x < K2, d o striations in region o f part-through crack growth at K 2 < Kma x < K 3, e- dimples in region Kma x -~ K3, f- structure and cleavage in region of part-through crack growth at Kma x < K 2 (o = 82 MPa), g- river structure in region of partthrough crack growth at Kma x < K 2 (o = 82 MPa), h- river structure in region o f through crack growth at K 0 < Krnax < K3 (a = 82 MPa), i- striations in region of part-through crack growth at Kma x < K 2 (o = 82 MPa), j- intergranular fracture w i t h particle traces in region of partthrough crack growth at Kma x < K 2 (a = 137 MPa), k- dimples in region of through crack growth at Kma x > / K 3. Magnification x 7 0 0 0
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a few striated facets is also typical for a through crack in its stable growth zone and dimples are observed in the final fracture zone (Fig. 3k). The above results show, that the growth of surface microcrecks in both alloys occurs with the formation of more brittle components than the growth of through microcracks and this conforms with the higher growth rate of part-through cracks compared with the through cracks. The results also justify the conclusion that the fracture micromechanisms for the two alloys are different both for part-through and through cracks. The more brittle fracture in MA12 alloy (presence of facets with inter- and transgraJn cleavage) compared with the fracture of IMV6 alloy (formation of striated fracture) indicates, that the fatigue microcreck in the former develops with lower plastic deformation in the crack zone. However, the change of growth rate for part-through microcracks, depending on stress amplitude and the greater growth rate compared with the through crack, is typical for both alloys irrespective of fracture mechanism. The stress amplitude dependence of part-through fatigue crack growth rate may be explained as follows. The initial microcrack in a sample, having a hemispherical notch, penetrates to a great degree the subsurface layer - where the plastic deformation prevails - compared with the sample volume. 11,20 The forming dislocation structure depends on the stress amplitude, for example, in copper the cell diameter is inversely proportional to the stress value. 2t Transition from the high-amplitude to the lowamplitude region changes the fatigue mechanism. 22 As the observation showed, the minimum stress amplitude corresponds to the low-amplitude and the maximum stress amplitude to the high-amplitude region of the e/N diagram of the alloys investigated. In MA12 alloy transition from the first stress region to the second changes the part-through crack growth mechanism from mainly transgrain to intergrain. In the low-amplitude region the IMV6 alloy is characterized by the fatigue fracture mechanism with the formation of persistent slip bands in one slip system due to softening 11 similar to that described for MA12 (T6) alloy. 12,13 In the high-amplitude region the second slip system activates and promotes a more rapid work-hardening and thus greater crack growth. Differences in dislocation structures formed at various stresses in the process of hardening or softening prior to crack formation and its transition to a through crack stimulated the effect that these stresses had on the growth rate of part-through cracks at the same value of Kmax- The higher growth rate of partthrough cracks compared with through cracks at the same value of Kmax seems to be explained by the different extent of plastic deformation and stress field in the zones in front of the cracks. This assumption is confirmed by the results of Pratt 23 which show the delay in through crack growth rate when its length is short under the plastic zone from notch.
3)
4) 5)
REFERENCES 1.
Pearson, $. 'Initiation and fatigue cracks in commercial aluminium alloys and subsequent propagation of very short cracks' Engng Fracture Mech 7 (1975) pp 235--247
2.
Hagiwara, Y., Yoshino, T. and Kunio, T. 'Difference between the surface and inner fatigue crack propagation behaviours. Effect of material properties'. Proc 19th Japan Conf Materials Research (1976) pp 22--27
3.
Kitagawa, H. and Takahashi, S. 'Application of fracture mechanics to very small cracks in early stage', 2nd Confon Mechanical Behaviour o f Materials (Boston MA, August 1976) pp 627--630
4.
Collipriest, J. E. 'An experimentalist's view of the surface flaw problem' The surface crack: physical problems and computational solutions (edited by Swedlow, ASME, 1972) pp 43--61
5.
Nair, P. K. 'Fatigue crack growth model for part-through flaws in plates and pipes', Trans ASME J Mater and Tech 101 No 1 (1971) pp 53--58
6.
Grinberg, N. M., Sardyuk, V. A., Zmeevets, S. G., Ostapenko, I. L., Malinkina, T. I. and Kamyshkov, A. S. 'Rost ustalostnykh treshchin v magniyevom splave MA12 na vozdukhe i v vakuume' Problemyprochnosti No 3 (1978) pp 12--16
7.
Grinberg, N. M., Serdyuk, V. A. and Zmeevets, S. G. 'Vliyanie strukturnogo sostoyaniya na ustalostnoye razrusheniye magniyevogo splave MA12 na vozdukhe i v vakuume. Soobshcheniye 2. Mikroskopicheskie i makroskopicheskiye osobennosti rosta usta!ostnykh treshchin na vozdukhe' Problemy prochnosti No 10 (1978) pp 46--52
8.
Serdyuk, V. A. 'lssledovanie skorosti testa ustalostnykh treshchin v magniyevykh splavakh pri komnatnoi i nizkoi temperaturakh' Problemyprochnosti No 11 (1980) pp 18--23
9.
Gudkov, Ao A. and Zoteev, V. S. 'Vliyanie prilozhennogo napryazheniya na skorost rasprostraneniya ustalostnoi treshchiny pri pulsiruyushmetallov, M'. Metallurgiya No 4 (1977) pp 17--28
CONCL USIONS
1)
2)
The growth rate of part-through fatigue cracks in magnesium alloy sheet samples is described by the Paris equation similarly to the midamplitude region of a kinetic diagram for a througri crack, but the coefficient C for part-through cracks depends on the stress amplitude. The part-through crack growth rate in log (d//dN) = log f (Kmax) coordinates is characterized, depending
on the alloy structural state, by one or two linear sections. The first region is observed only in a softening alloy and corresponds to the growth of surface cracks in persistent slip bands. The second region corresponds to the growth of elliptical part-through cracks by a stage II mechanism. The part-through crack growth rate is higher than that for through cracks at the same Kmax value, and the intensity of its change is greater for through than for part-through cracks. The full kinetic diagram for magnesium alloy fatigue has an S shape for through cracks and may be approximated by Equation (7). Micromechanisms of magnesium alloy fatigue fracture depend on the alloy structural state, Kmax value, and in the case of part-through cracks on stress amplitude as well. More brittle fracture is typical for surface cracks compared with the internal cracks. More plastic micromechanisms of fatigue crack growth are observed for the IMV6 alloy compared with the MAI 2 alloy.
10.
Kishkina, S. I. and Starova, E. N. 'Nekotoryye osobennosti rosta treshchin ustalosti v tonkikh plastinakh aluminievykh splavov, v Sb. Problemy metallovedeniya tsvetnykh splevov, M'. Nauka (1978) pp 166--172
11.
Grinberg, N. M. and Serdyuk, V. A. 'Razuprochneniye magnievogo splave IMV6 v protsesse ustalosti' Problem)/ prochnosti No 1 (1980) pp 35--39
12.
Grinberg, N. M., Serdyuk, V. A. and Zmeevets, S. G. 'Vliyanie strukturnogo sostoyaniya na ustalostnoe
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rasrushenie magnievogo splava MA12 na vozdukhe i v vakuume. Soobshcheniye I. Dolgovechnost i zarozhdenie ustalostnykh treshchin' Problemy prochnosti No 9 (1978) pp 32--38
i vnutrnnikh sloev polikristallicheskogo zheleza pri ustalostnom nagruzhenii' DAN SSSR 205 No 4 (1972) pp 8 1 2 814
13.
Grinberg, N. M., Serdyuk, V. A., Yakovenko, L. F. Malinkine, T. I. and Kamyshkov, A. S. 'Kinetika i mekhanism ustalostnogo razrusheniya magnievykh splavov MA2-1 i MA12' Problemy provhnosty no 8 (1977) pp 40--45
21.
Pratt, J. J. 'Dislocation substructure in strain-cycled copper as influenced by temperature' Acta Met 15 No 5 (1967) pp 319--327
22.
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Yakovenko, L. F. 'Ustanovka dlya ustalostnykh ispytanii v vakuume pri komnatnoi i nizkoi temperaturakh' Zavodskaya lab No 2 (1971) pp 232--234
Wood, W. A., Cousland, S. Mck. and Sargant, K. R. 'Systematic microstructural changes peculiar to fatigue deformation' Acta Met 11 No 7 (1963) pp 643--652
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Plumbridga, W. J. 'Problems associated with early stage fatigue crack growth' Metal Sci 12 No 5 (1978) pp 251--256
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Brown, W. F. Jr. and Srawley, J. E. 'Plane strain toughness testing of high strength metallic materials' ASTM STP 410 (Philadelphia, 1967)
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Neuber, H. "Theory o f notch stress" (J. S. Edwards, Ann Arbor, Michigan, 1946)
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Burk, L. H. 'Fatigue Growth of Surface Cracks in Bending' Engng Fracture Mech 9 (1977) pp 389--395
17.
Burk, L. N., Sullivan, C. P. and Wells, C. H. 'Fatigue of a glass bead blasted nickel-base superalloy' Met Trans 1 (1970) pp 1595--1600
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Yarema, S. Ya. and Mikitishin, S. I. 'Analiticheskoe opisanie diagrammy ustalostnogo razrusheniya materialov' Fiziko-khimicheskaya mekhanika materialov No 6 (1975) pp 47--54
19.
Nagai, A., Toyo~Kla, M. and Okamoto, T. 'A study on the fatigue crack growth in 9% Ni steel plate' Engng Fracture Mech 7 (1975) pp 481--490
20.
Goritskii, V. M., Ivanova, V. S., Orlov, L. G. and Terentev, V. F. 'O razlichii plasticheskoi deformatsii poverkhnostnykh
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AUTHORS The authors are with the Physico-Technical Institute of Low Temperatures which is part of the Ukrainian Academy of Sciences. Inquiries in the first instance should be directed to Dr N. M. Grinberg at the following address: PhysicoTechnical Institute of Low Temperatures, Ukrainian Academy of Sciences, Lenin's Prospect 47, Kharkov 164, USSR.