- Email: [email protected]
A model relating grain size to the endurance limit and non-propagating crack behaviour
ELSEVIER
Materials
Science
and Engineering
A234&236
(1997)
636-638
A model relating grain size to the endurance limit and non-propagating crack behaviour A. Plumtree * Department
of Mechanical Received
Engineering, 5 February
Unitiersity
of Waterloo,
1997; received
in revised
Waterloo, form
3 April
ON N2L
3G1, Canada
1997
Abstract A model is presented for determining the endurance limit and depth of non-propagating cracks in smooth aluminum specimens. The approach is based on the non-uniformity of surface strains and crack closure development. The model is used to explain the effect of decreasing endurance limit stress with increase in grain size for commercially pure aluminum. The predicted limits are in very good agreement with reported values. Likewise, the agreement is also very good for an aluminum alloy with pancake-shaped grains, provided that the smallest grain size is considered. At stresses below the endurance limit but above the nominal threshold stress non-propagating cracks form whose lengths may be predicted by the model. 0 1997 Elsevier Science S.A. Keywords:
Surface
strain
redistribution;
Closure;
Fatigue
life prediction;
1. The model
Non-propagating
cracks
where (2, is the strain redistribution factor (Q, = Ae/Ae, where AE is the local strain in the favourably oriented grain and Ae is the nominal strain), q is a constant whose value is 5.3, a is the decay constant (OZ= /3D-‘,
where /3 is a constant depending upon deformation character), a is the projected crack length and D is the grain size. The nominal threshold stress range, AS,,, at any crack depth, a, can be divided into its opening and intrinsic stress components. The opening stresscomponent is the nominal stress range required to open the crack. The nominal intrinsic stresscomponent, ASith, is defined as the minimum stress range that will cause a fully open crack to grow. At a high stressratio (i.e. no closure) the intrinsic component of the nominal threshold stressrange ASith can be determined from the intrinsic threshold strain intensity factor range [2]. Small cracks growing in the surface-reversed plastic zone (Q,) are expected to stay open for most of the stress cycle [4]. It is assumed that closure starts to develop when the crack tip reaches the centre of the surface-reversed plastic zone, thereby introducing an effective crack depth, a, (a, = a - Q,/2), measured from the centre of the plastic zone. The increase in crack opening stress with effective crack depth, a,, can be expressed by modifying the equation suggestedby Minakawa et al. [5] and the crack closure development can be described by a parameter, &, given as follows:
* Tel.: 8851211; + 1 519 [email protected]
+ 1
H = ASit, + ASopmax Cl ASi,h + B’ASopmax
0921-5093/97/$17.00
Science
The present paper extends a recently developed short crack model [l-3] to predict the fatigue life of smooth specimens of different grain size and to explain the dominant grain dimension in an aluminum alloy containing pancake-shaped grains. In addition, the relative depths of non-propagating cracks in smooth and notched specimenswill be determined. When a material is subjected to a nominal cyclic strain range (Ae), the strain in each surface layer is accommodated by lower restraint. Favourably oriented grains experience the largest amount of surface deformation. The local strain (AC) decreaseswith depth from the surface, eventually approaching the nominal strain range. In order to satisfy the surface and interior conditions, the strain redistribution factor may be expressed by: Qc = 1 +
q
exp( - xa)
PIISO921-5093(97)00246-3
0 1997 Elsevier
(1)
519
8886197;
S.A. All rights
e-mail:
reserved.
(2)
A. Plumtree
/Materials
Science
and Engineering
where B’=exp(-kka,) for a,>0 and 8’=1 for a,
A&, = Wtdb
(3)
For small cracks with a < Qp, H,, is approximately unity and AS,, = ASit,,, indicating that the crack is fully open. For long cracks, however, the steady state value of H,, is invariant with crack length and its magnitude increases as the stress ratio decreases.
A234-236
(1997)
Turnbull and De 10s Rios [6] conducted fully reversed (R = - 1) fatigue tests on polycrystalline commercially pure aluminum samples with grain sizes of 2 1, 40 and 135 urn. The values of the nominal threshold stress range at different crack depths and stress ratios were estimated using the model with A&, = 1.30 MPaJm [Topper, T.H., Private Communication, 19971. Fig. 1 shows the relationship between the nominal threshold stress range and crack depth for the three different grain sizes at a stress ratio of R = - 1. For long cracks, the log-log relationship between AS,, and crack depth is linear with a slope of - 0.5. In the short-crack region the threshold component of the stress range exhibits a maximum representing the stress range required for continuous crack growth, which defines the endurance limit. The estimated endurance ranges for the 21, 40 and 135 urn grain size samples were 98, 80 and 56 MPa, respectively, and are in very
1000
1
101
.A
. 1
CRACK
Fig.
1. The
effect
1000
100
IO
of grain
DEPTH
size on the
10000
dpm)
threshold
stress
for
pure
637
good agreement with the experimentally determined data of 96, 80, and 56 MPa, respectively [6]. Plumtree and Varvani-Farahani [7] conducted fully reversed (R = - 1) tests on aluminum alloy 2024-T351. The average grain sizes in the longitudinal (L-), transverse (T-) and short transverse (S-) directions were approximately 415 urn, 158 urn and 34 urn, respectively. Again the values for the nominal stress range were estimated using the model. When the smallest average grain dimension (34 urn) was considered, the predicted endurance limit was in very good agreement with the experimental value. In this instance, fewer dislocations were required to form a microcrack in a slip band [8].
3. Non-propagating 2. Endurance limits
636-638
cracks
When the applied stress range exceeds AS,,, yet is less than the endurance range AS,,, fatigue cracks initiate and propagate to a limiting length. Once they reach this value, given by the intersection of the applied stress with the threshold stress curve, they stop and become non-propagating. Microscopic observation of the 2024-T35 1 aluminum alloy [7] revealed that the LS and LT specimens (specimen orientation first followed by crack depth direction) contained non-propagating crack lengths ranging from 2 to 435 urn and from 4 to 70 urn, respectively, as the stress amplitude was increased from AS,, to A&r which, despite the presence of cracked particles, was just below that for the other specimens. The ST and SL specimens contained non-propagating cracks with lengths of 2-17 and 4-67 urn, respectively, and those in the TS and TL specimens were 668 and 5-77 urn, respectively, as the stress level was increased from the initiation level to the endurance limit. Grain boundaries were the major barriers to the growth of all non-propagating cracks. The length of most of these cracks was generally less than one-grain and only a few cracks were as long as, or exceeded, two grains. Using a confocal scanning laser microscope (CSLM), the aspect ratio (crack depth, u, to total surface crack length, 2c) was found to be constant at 0.42 and independent of specimen direction for cracks within the first surface grain [9]. This result was verified by successively removing a known surface layer thickness and measuring the corresponding crack length. Using the model the values of the threshold stress amplitude with depth was determined for a stress ratio of R = - 1 and plotted in Fig. 2. For a minimum crack depth of 3 urn the nominal threshold stress amplitude (128 MPa) for the smallest grain dimension of 34 urn was approximately the same as that experimentally observed (121- 134 MPa) when cracked particles were not instrumental in fatigue crack initiation. An endurance stress amplitude of 146 MPa for R = - 1,
A. Plumtree
638
/Materials
Science
and Engineering
A234-236
(1997)
636-638
shows that the endurance limit stressis dependent upon the smallest grain size. Persistent slip bands were narrower and early microcracking occurred. (3) The model successfully predicts the presence of non-propagating fatigue cracks when the applied stress level is less than the endurance limit stress. Acknowledgements 0 TS M TL
0A STsi 9 LT m LS
L
I I IIII
1 I
I I #III
1
1 5 10 50 100 PROJECTED CRACK DEPTH, a (urn) Fig. 2. Variation T351.
of crack
corresponding to grain diameters) sults were in very 138, LS 138, TL
depth
with
stress amplitude
The author would like to thank Dr Guangxu Cheng, Xi’an Jiatong University, P.R. China, presently visiting the Department of Mechanical Engineering, University of Waterloo, for many interesting discussions. Also, thanks are due to Marlene Dolson for typing the manuscript. This research has been supported by the Natural Sciences and Research Council of Canada through grant OGP 000 2770.
for Al 2024-
References a crack depth of 135 pm (about four was predicted. The experimental regood agreement (ST 138, SL 148, LT 141, TS 146 MPa).
[l] [2]
[3] [4]
4. Conclusions (1) Based on microstructurally dependent local strain distribution and crack closure development near the free surface a model has been applied to successfully predict the endurance limits of pure aluminum specimens with different grain sizes. (2) When considering an aluminum alloy with an orthogonal (pancake) grain morphology the model
H. Abdel-Raouf, 25 (1991)
[5] [6] [7] [8] [9]
T.H.
Topper,
A. Plumtree,
Ser. Metall.
Mater.
597-602.
H. Abdel-Raouf, T.H. Topper, A. Plumtree, Fatigue Fract. Eng. Mater. Struct. 15 (1992) X95-909. D.L. DuQuesnay, H.A. Abdel-Raouf, T.H. Topper, A. Plumtree, Fatigue, Fract. Eng. Mater. Struct. 15 (1992) 979-993. DuQuesnay, D.L., PhD thesis, University of Waterloo, Waterloo, ON, Canada, 1991. K. Minakawa, H. Nakamura, A.J. McEvily, Ser. Metall. 18 (1984) 1371. A. Turnbull, E.R. De 10s Rios, Fatigue Fract. Mater. Struct. 18 (1995) 145551468. A. Plumtree, A. Varvani-Farahani, Sci. Forum 2171222 (1996) 1377-1382. A. Zabett, A. Plumtree, Fatigue Fract. Eng. Mater. Struct. 18 (1995) 801-809. A. Varvani-Farahani, T.H. Topper, A. Plumtree, Fatigue Fract. Eng. Mater. Struct. 19 (1996) 1153-1159.
Materials
Science
and Engineering
A234&236
(1997)
636-638
A model relating grain size to the endurance limit and non-propagating crack behaviour A. Plumtree * Department
of Mechanical Received
Engineering, 5 February
Unitiersity
of Waterloo,
1997; received
in revised
Waterloo, form
3 April
ON N2L
3G1, Canada
1997
Abstract A model is presented for determining the endurance limit and depth of non-propagating cracks in smooth aluminum specimens. The approach is based on the non-uniformity of surface strains and crack closure development. The model is used to explain the effect of decreasing endurance limit stress with increase in grain size for commercially pure aluminum. The predicted limits are in very good agreement with reported values. Likewise, the agreement is also very good for an aluminum alloy with pancake-shaped grains, provided that the smallest grain size is considered. At stresses below the endurance limit but above the nominal threshold stress non-propagating cracks form whose lengths may be predicted by the model. 0 1997 Elsevier Science S.A. Keywords:
Surface
strain
redistribution;
Closure;
Fatigue
life prediction;
1. The model
Non-propagating
cracks
where (2, is the strain redistribution factor (Q, = Ae/Ae, where AE is the local strain in the favourably oriented grain and Ae is the nominal strain), q is a constant whose value is 5.3, a is the decay constant (OZ= /3D-‘,
where /3 is a constant depending upon deformation character), a is the projected crack length and D is the grain size. The nominal threshold stress range, AS,,, at any crack depth, a, can be divided into its opening and intrinsic stress components. The opening stresscomponent is the nominal stress range required to open the crack. The nominal intrinsic stresscomponent, ASith, is defined as the minimum stress range that will cause a fully open crack to grow. At a high stressratio (i.e. no closure) the intrinsic component of the nominal threshold stressrange ASith can be determined from the intrinsic threshold strain intensity factor range [2]. Small cracks growing in the surface-reversed plastic zone (Q,) are expected to stay open for most of the stress cycle [4]. It is assumed that closure starts to develop when the crack tip reaches the centre of the surface-reversed plastic zone, thereby introducing an effective crack depth, a, (a, = a - Q,/2), measured from the centre of the plastic zone. The increase in crack opening stress with effective crack depth, a,, can be expressed by modifying the equation suggestedby Minakawa et al. [5] and the crack closure development can be described by a parameter, &, given as follows:
* Tel.: 8851211; + 1 519 [email protected]
+ 1
H = ASit, + ASopmax Cl ASi,h + B’ASopmax
0921-5093/97/$17.00
Science
The present paper extends a recently developed short crack model [l-3] to predict the fatigue life of smooth specimens of different grain size and to explain the dominant grain dimension in an aluminum alloy containing pancake-shaped grains. In addition, the relative depths of non-propagating cracks in smooth and notched specimenswill be determined. When a material is subjected to a nominal cyclic strain range (Ae), the strain in each surface layer is accommodated by lower restraint. Favourably oriented grains experience the largest amount of surface deformation. The local strain (AC) decreaseswith depth from the surface, eventually approaching the nominal strain range. In order to satisfy the surface and interior conditions, the strain redistribution factor may be expressed by: Qc = 1 +
q
exp( - xa)
PIISO921-5093(97)00246-3
0 1997 Elsevier
(1)
519
8886197;
S.A. All rights
e-mail:
reserved.
(2)
A. Plumtree
/Materials
Science
and Engineering
where B’=exp(-kka,) for a,>0 and 8’=1 for a,
A&, = Wtdb
(3)
For small cracks with a < Qp, H,, is approximately unity and AS,, = ASit,,, indicating that the crack is fully open. For long cracks, however, the steady state value of H,, is invariant with crack length and its magnitude increases as the stress ratio decreases.
A234-236
(1997)
Turnbull and De 10s Rios [6] conducted fully reversed (R = - 1) fatigue tests on polycrystalline commercially pure aluminum samples with grain sizes of 2 1, 40 and 135 urn. The values of the nominal threshold stress range at different crack depths and stress ratios were estimated using the model with A&, = 1.30 MPaJm [Topper, T.H., Private Communication, 19971. Fig. 1 shows the relationship between the nominal threshold stress range and crack depth for the three different grain sizes at a stress ratio of R = - 1. For long cracks, the log-log relationship between AS,, and crack depth is linear with a slope of - 0.5. In the short-crack region the threshold component of the stress range exhibits a maximum representing the stress range required for continuous crack growth, which defines the endurance limit. The estimated endurance ranges for the 21, 40 and 135 urn grain size samples were 98, 80 and 56 MPa, respectively, and are in very
1000
1
101
.A
. 1
CRACK
Fig.
1. The
effect
1000
100
IO
of grain
DEPTH
size on the
10000
dpm)
threshold
stress
for
pure
637
good agreement with the experimentally determined data of 96, 80, and 56 MPa, respectively [6]. Plumtree and Varvani-Farahani [7] conducted fully reversed (R = - 1) tests on aluminum alloy 2024-T351. The average grain sizes in the longitudinal (L-), transverse (T-) and short transverse (S-) directions were approximately 415 urn, 158 urn and 34 urn, respectively. Again the values for the nominal stress range were estimated using the model. When the smallest average grain dimension (34 urn) was considered, the predicted endurance limit was in very good agreement with the experimental value. In this instance, fewer dislocations were required to form a microcrack in a slip band [8].
3. Non-propagating 2. Endurance limits
636-638
cracks
When the applied stress range exceeds AS,,, yet is less than the endurance range AS,,, fatigue cracks initiate and propagate to a limiting length. Once they reach this value, given by the intersection of the applied stress with the threshold stress curve, they stop and become non-propagating. Microscopic observation of the 2024-T35 1 aluminum alloy [7] revealed that the LS and LT specimens (specimen orientation first followed by crack depth direction) contained non-propagating crack lengths ranging from 2 to 435 urn and from 4 to 70 urn, respectively, as the stress amplitude was increased from AS,, to A&r which, despite the presence of cracked particles, was just below that for the other specimens. The ST and SL specimens contained non-propagating cracks with lengths of 2-17 and 4-67 urn, respectively, and those in the TS and TL specimens were 668 and 5-77 urn, respectively, as the stress level was increased from the initiation level to the endurance limit. Grain boundaries were the major barriers to the growth of all non-propagating cracks. The length of most of these cracks was generally less than one-grain and only a few cracks were as long as, or exceeded, two grains. Using a confocal scanning laser microscope (CSLM), the aspect ratio (crack depth, u, to total surface crack length, 2c) was found to be constant at 0.42 and independent of specimen direction for cracks within the first surface grain [9]. This result was verified by successively removing a known surface layer thickness and measuring the corresponding crack length. Using the model the values of the threshold stress amplitude with depth was determined for a stress ratio of R = - 1 and plotted in Fig. 2. For a minimum crack depth of 3 urn the nominal threshold stress amplitude (128 MPa) for the smallest grain dimension of 34 urn was approximately the same as that experimentally observed (121- 134 MPa) when cracked particles were not instrumental in fatigue crack initiation. An endurance stress amplitude of 146 MPa for R = - 1,
A. Plumtree
638
/Materials
Science
and Engineering
A234-236
(1997)
636-638
shows that the endurance limit stressis dependent upon the smallest grain size. Persistent slip bands were narrower and early microcracking occurred. (3) The model successfully predicts the presence of non-propagating fatigue cracks when the applied stress level is less than the endurance limit stress. Acknowledgements 0 TS M TL
0A STsi 9 LT m LS
L
I I IIII
1 I
I I #III
1
1 5 10 50 100 PROJECTED CRACK DEPTH, a (urn) Fig. 2. Variation T351.
of crack
corresponding to grain diameters) sults were in very 138, LS 138, TL
depth
with
stress amplitude
The author would like to thank Dr Guangxu Cheng, Xi’an Jiatong University, P.R. China, presently visiting the Department of Mechanical Engineering, University of Waterloo, for many interesting discussions. Also, thanks are due to Marlene Dolson for typing the manuscript. This research has been supported by the Natural Sciences and Research Council of Canada through grant OGP 000 2770.
for Al 2024-
References a crack depth of 135 pm (about four was predicted. The experimental regood agreement (ST 138, SL 148, LT 141, TS 146 MPa).
[l] [2]
[3] [4]
4. Conclusions (1) Based on microstructurally dependent local strain distribution and crack closure development near the free surface a model has been applied to successfully predict the endurance limits of pure aluminum specimens with different grain sizes. (2) When considering an aluminum alloy with an orthogonal (pancake) grain morphology the model
H. Abdel-Raouf, 25 (1991)
[5] [6] [7] [8] [9]
T.H.
Topper,
A. Plumtree,
Ser. Metall.
Mater.
597-602.
H. Abdel-Raouf, T.H. Topper, A. Plumtree, Fatigue Fract. Eng. Mater. Struct. 15 (1992) X95-909. D.L. DuQuesnay, H.A. Abdel-Raouf, T.H. Topper, A. Plumtree, Fatigue, Fract. Eng. Mater. Struct. 15 (1992) 979-993. DuQuesnay, D.L., PhD thesis, University of Waterloo, Waterloo, ON, Canada, 1991. K. Minakawa, H. Nakamura, A.J. McEvily, Ser. Metall. 18 (1984) 1371. A. Turnbull, E.R. De 10s Rios, Fatigue Fract. Mater. Struct. 18 (1995) 145551468. A. Plumtree, A. Varvani-Farahani, Sci. Forum 2171222 (1996) 1377-1382. A. Zabett, A. Plumtree, Fatigue Fract. Eng. Mater. Struct. 18 (1995) 801-809. A. Varvani-Farahani, T.H. Topper, A. Plumtree, Fatigue Fract. Eng. Mater. Struct. 19 (1996) 1153-1159.