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A verification of the assumption of anti-fatigue design
International Journal of Fatigue 23 (2001) 271–277 www.elsevier.com/locate/ijfatigue
Technical note
A verification of the assumption of anti-fatigue design Weixing Yao *, Bin Ye, Lichun Zheng Department of Aircraft Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China Received 1 March 2000; received in revised form 29 July 2000; accepted 4 September 2000
Abstract In this paper, the approaches of anti-fatigue design are briefly reviewed and are classified into three groups: nominal stress approach, local stress–strain approach and stress field intensity approach. Two types, each with two notches, are specially designed and a total of 30 specimens with 4 dimensions are tested under different constant–amplitude loading. These specimens are also analysed with elasto-plastic FEM to obtain stress–strain distributions and stress concentration factors. Fatigue life is estimated based on the different approaches. Experimental and analytical results show that the assumption of stress field intensity is more reasonable than that of the other approaches. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Design against fatigue; Nominal stress approach; Local stress–strain approach; Stress field intensity approach; Fatigue experiment
1. Introduction Metal fatigue is a complicated phenomenon which depends on many factors, and up to now the description of fatigue crack initiation has been left for further investigation. Notches in engineering structures are always the critical places for fatigue failures, so the prediction of fatigue life or fatigue strength of notched elements is the key point of the structure design against fatigue. In the history of fatigue research, many approaches [1,2] for the prediction of fatigue life of notched elements have been developed. Most of the approaches can been classified into three types according to their assumptions: nominal stress approach (NSA), local stress–strain approach (LSSA) and stress field intensity approach (SFIA). In this paper two types of specimens, each with two notches, are designed and tested under different constant–amplitude loading to see which assumption of the approaches is the most reasonable.
Most of the approaches can be classified into the following three groups: NSA, LSSA and SFIA. 2.1. Nominal stress approach (NSA) The assumption of the NSA is that the fatigue life of elements with different notches is the same if these notches have the same theoretical stress concentration factors and are under the same loading history (Fig. 1). Obviously the nominal stress S and theoretical stress concentration factor KT are the control parameters in this approach. But in practice, the fatigue life predicted by this approach is far removed from the fatigue life obtained by fatigue experiments in most cases. Many
2. A brief review of the approaches The approaches for prediction of fatigue life have been reviewed in detail in some of the literature [1–3]. * Corresponding author. Tel.: +86-25-489-2177; fax: +86-25-4891422. E-mail address: [email protected] (W.X. Yao). 0142-1123/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 8 3 - 9
Fig. 1.
Nominal stress approach.
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2.3. Stress field intensity approach (SFIA) The assumption of the SFIA is that the fatigue life of the elements with different notches is the same if the stress field intensity histories are the same (Fig. 3) [7–9]. SFIA defines the stress field intensity sFI as: sFI⫽
冕
1 → f(sij )j( r )dv V
(1)
⍀
where ⍀ is the fatigue failure region and V is the volume of ⍀ which only depends on the material, f(sij ) is the →
failure function and j( r ) is the weight function. A brief discussion of Eq. (1) follows. Fig. 2. Local stress–strain approach.
other models, such as the stress severity factor (SSF), effective stress method, and detailed fatigue rating (DFR), belong to this approach [1]. 2.2. Local stress–strain approach (LSSA) The assumption of the LSSA is that the fatigue life of elements with different notches is the same if the local stress–strain histories at the roots of the notches are the same (Fig. 2). The LSSA is a point criterion. The local stress and local strain are the control parameters of fatigue life of notched elements [4]. Many kinds of LSSA have been formed with the combination of the cyclic stress–strain relationship of the material, the strain–life curve of the material, the fatigue damage cumulative rule and the solution of the local stress–strain at the notch. In practice, the fatigue life estimated based on the real local stress and strain is lower than the fatigue life obtained by the fatigue test, so usually the fatigue notch factor Kf is used to calculate the local stress and strain in Neuber’s equation instead of theoretical stress concentration factor KT. Many researchers [5] have made efforts to estimate the Kf accurately, and Peterson’s equation is the most popular method [6].
2.3.1. The fatigue failure region ⍀ From the point of view of the fatigue mechanism, crack initiation generally occurs within the local region near the surface of the specimen where stress increases. This region is several grain sizes in extent. So the size and form of ⍀ depend on the material. 2.3.2. The failure function f(sij ) f(sij ) is the function to describe crack initiation under multiaxial stress, and may be different for different materials. For example, Von Mises’ equivalent stress is used for elasto-plastic materials, such as carbon steel, aluminum alloy and titanium, and maximum major stress is used for materials such as cast iron and cast steel. f(sij ) contains the effect of multiaxial stress near a notch, so the SFIA can deal with fatigue strength under multiaxial loading. →
2.3.3. The weight function j( r ) →
The weight function j( r ) physically means the contri→
bution of stress at Q to the peak stress at 兩 r 兩=0. →
→
1. 0ⱕj( r )ⱕ1 and j( r ) is a generalised monotonically →
decreasing function about 兩 r 兩; 2. j(0)⬅1 which means that the contribution of the stress at the notch root is maximum; →
3. when the stress gradient G=0, j( r )⬅1, which is consistent with the condition of smooth specimens. →
j( r ) is not only related to notch geometry and loading type for isotropic materials, but also to material properties for anisotropic materials. It can be obtained analytically or numerically and as an approximation of first order the following expression is used in this paper: →
j( r )⫽1⫺cr(1⫹sinq) Fig. 3.
Stress field intensity approach.
where c is a factor related to stress gradient.
(2)
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273
Table 1 A brief comparison among the three approaches Failure place Stress concentration NSA
Notch root LSSA
Local region near notch SFIA
Control parameters
KT and nominal stress KTS
Stress field intensity sFI
Material properties
S–N curves under different KT and R
Local stress and local strain KfS and smax Cyclic s–e relationship e–Nf curve (KT=1)
Cyclic s–e relationship S–N or e–Nf curve under KT=1
Table 2 The chemical and mechanical properties of aluminum alloy LY12-CZ (a) Chemical properties of aluminum alloy LY12-CZ Elements Wt%
Cu 4.61
Mg 1.54
Mn 0.58
Fe 0.29
Si 0.26
Zn 0.10
Ni 0.024
(b) Mechanical properties of aluminum alloy LY12-CZ Young’s Modulus E (MPa) 71022
Ultimate strength sb (MPa) 466
Yield strength sY (MPa) 343
2.3.4. The failure criterion →
For a smooth specimen, KT=1, G=0 and then j( r )=1. The stresses in the smooth specimen are equal everywhere, i.e. f(sij )⬅constant. According to Eq. (1), sFI=f(sij). Obviously, when sFI=f(sij)ⱖsf, fatigue failure occurs for a smooth specimen. Because Eq. (1) is universally applicable to notched specimens as well as to smooth ones, the failure criterion can be written as: sFIⱖsf
(3)
2.4. A brief comparison among the three approaches From the above discussion it can been seen that the assumptions of the three approaches are different, and the material properties required for each approach are also different. Table 1 gives a brief comparison of the three approaches.
3. Experiments 3.1. Specimens Two types of specimens are specially designed and tested under different constant–amplitude loadings to verify the assumptions of the approaches. Specimens are made from aluminum alloy LY12-CZ plate with thickness of 2 mm. The chemical and mechanical properties are listed in Table 2. They are cut longi-
Fig. 4. (a) Double-hole specimen (DH). (b) Double-notch specimen (DN-A). (c) Double-notch specimen (DN-B).
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Table 3 Experimental results of the double-hole specimens The place where the crack initiated
Specimen
Loading (MPa)
Frequency (Hz)
Fatigue life
DH-A01 DH-A02 DH-A03 DH-A04 DH-A05 DH-A06 DH-A07 DH-A08 DH-A09 DH-A10
80 80 80 80 80 80 150 150 150 150
12 12 12 12 12 12 6 6 6 6
122 734 113 000 169 235 144 708 122 469 143 164 15 900 18 952 18 518 16 426
DH-A11
150
6
15 649
DH-A12 DH-B01 DH-B02 DH-B03
150 98 98 98
6 14 14 14
17 741 136 600 181 890 166 300
One side of the hole One side of the hole One side of the hole One side of the hole One side of the hole One side of the hole One side of the hole One side of the hole Both the side of the hole One side of the slot One side of the hole and one side of the slot Both the side of the hole One side of the slot One side of the slot One side of the slot
Table 4 Experimental result of the double-notch specimens Specimen
Loading (MPa)
Frequency (Hz)
Cycles to failure
The places where cracks initiate
DN-A01 DN-A02 DN-A03 DN-B01 DN-B02 DN-B03
78.4 78.4 78.4 80 80 80
20 22 22 8 8 8
335 530 143 440 192 950 73 320 74 962 76 178
DN-B04
80
8
67 342
DN-B05 DN-B06 DN-B07 DN-B08 DN-B09 DN-B10 DN-B11 DN-B12
80 80 100 100 100 100 100 100
8 8 6 6 6 6 6 6
74 694 90 205 33 241 33 919 26 285 47 784 19 336 25 500
One edge notch One edge notch One edge notch One side of the hole One side of the hole Both the sides of the hole One side of the hole and one notch One side of the hole Both the sides of the hole One side of the hole One side of the hole Both the sides of the hole One side of the hole Both the sides of the hole One side of the hole
Table 5 Cyclic stress–strain curve of aluminium alloy LY12-CZ Stress (MPa)
Strain
Stress (MPa)
0 249 309 327 337 348 355 361
0 0.00356 0.00450 0.00500 0.00550 0.00700 0.00850 0.01000
371 382 396 415 451 462 466 470
Strain 0.01300 0.01800 0.02500 0.04000 0.08000 0.10000 0.11500 0.01510
tudinally from the same plate to diminish scatter. A double-hole specimen has two separate notches, one is a hole and the other is a slot [Fig. 4(a)]. A double-notch specimen has three separate notches, one is a central hole and the others are edged notches [Fig. 4(b) and (c)]. 3.2. Experiment The experiment is carried out at the MTS 810. For the 12 DH-A specimens, 6 specimens are tested under a nominal stress of 80 MPa, other specimens under 150 MPa, and 3 DH-B specimens are tested under 98 MPa. The experimental results are listed in Table 3. Three specimens labelled DN-A are tested under a constant–
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275
Table 6 Analytical results of the specimens (stress: MPa) Specimen
KT (gross)
KT (net)
K fa
Loading S
smax
sFI
The hole of specimen DH-A
3.61
2.27
2.19
The slot of specimen DH-A
3.96
3.42
2.68
The hole of DH-B The slot of DH-B Hole of DN-A Edge-notch of DN-A Hole of DN-B
3.42 3.50 1.64 1.74 2.04
2.44 2.75 1.64 1.74 2.04
2.34 2.43 1.51 1.57 1.99
Edge-notch of DN-B
2.08
2.08
1.71
80 150 80 150 98 98 78.4 78.4 80 100 80 100
286.11 363.55 297.66 366.55 316.64 341.40 253.47 266.53 303.91 378.86 309.31 386.65
247.62 320.06 239.33 319.24 276.06 279.23 224.12 230.10 261.89 298.93 250.45 292.40
a
Kf is calculated based on Peterson’s equation, and the constant a0=0.63 [11].
Table 7 Estimated and experimental fatigue life Failure place The hole of DH-A The slot of DH-B Edge-notch of DN-A Hole of DN-B
LSSA (Neuber) S=80 MPa S=150 MPa S=98 MPa S=78.4 MPa S=80 MPa S=100 MPa
42 053 1963 46 509 78 487 33 302 11 926
amplitude loading of 78.4 MPa. For the 12 specimens labelled DN-B, 6 specimens are tested under a nominal stress of 80 MPa, and the other specimens under 100 MPa. The experimental results are listed in Table 4. The fatigue life listed in the Tables 3 and 4 is the number of cycles to a crack size of about 0.5 mm, which is determined by microscopic monitoring during the experiment, and fractography after the experiment. 3.3. Analysis NASTRAN was employed for elasto-plastic FE analysis to calculate the stress–strain distribution and stress field intensity sFI of the specimens. The cyclic stress– strain curve, which is listed in Table 5, is used in the elasto-plastic FE analysis. The analytical results are listed in Table 6. Von Mises effective stress is taken as the fatigue failure function f(sij) and the size of the fatigue failure region for LY12-CZ aluminum is 0.185 mm [10] in calculating SFI sFI with Eq. (1). Fig. 5 presents Von Mises stress distribution along the line at the minimum width of the double-hole specimens. It can be seen that the maximum stress is at the edge of the slot, but the stress decreases faster, so the stress field intensity of the hole is larger than that of the slot. Fig. 6 presents Von Mises stress distribution along the line at the minimum width of the double-notch specimens.
NSA
SFIA
Experiment (average)
706 891 46 212 780 981 495 669 329 583 124 903
202 770 32 773 96 746 295 802 146 617 58 487
135 885 17 352 161 597 223 973 76 117 31 011
The maximum stress occurs at the edged notches, but the stress decreases faster, which means that the stress field intensity of the central hole is larger. Table 7 presents the fatigue life predicted by the different approaches. Miner’s rule is used in the prediction. The S–N curve for the NSA is listed in Table 8 and the strain–life curve e⫺Nf for LSSA is expressed by Eq. (4). logNf⫽A0⫹A1tan h−1
再
log(euee/e2eq) log(eu/ee)
冎
(4)
冉 冊
smax 1−m , smax is E maximum local stress, m=0.4, A0=3.4328, A1=2.7551, eu=0.053, and ee=0.0135 are material constants. where effective strain eeq=(2ea)m
4. Discussion Comparing the analytical results with the experimental results, it can be seen that stress gradient is a very important parameter for the prediction of fatigue life, as well as the peak stress. In fact this argument had been put forward several decades earlier [12] and many models developed to find a way to take into consideration the two factors properly. The SFIA is one new such model which gives a parameter sFI as the intensity of
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Fig. 6. (a) The stress distribution at the minimum width of the specimen DN-A. (b) The stress distribution at the minimum width of the specimen DN-B.
Table 8 S–N curve of aluminum alloy LY12-CZ (R=0.1)
Fig. 5. (a) The stress distribution at the minimum width of the specimen DH-A (S=80 MPa). (b) The stress distribution at the minimum width of the specimen DH-A (S=150 MPa). (c) The stress distribution at the minimum width of the specimen DH-B (S=98 MPa).
the stress field near the notch. For the double-hole specimens DH-A and DH-B, the maximum stress at the roots of the slots is larger than that at the roots of the holes (Table 6), but the fatigue crack initiated at the root of the hole (Table 3). For the double-notch specimens DNA and DN-B, the same conclusion can be obtained from Tables 4 and 6. For specimen DH-A, the SFI of the hole is nearly the same as that of the slot under load S=150 MPa, which means that the fatigue cracks initiated at the slot of two specimens and at the hole of five specimens. Among them, fatigue cracks initiated at both the hole and the slot almost simultaneously for specimen DH-A11. The fatigue life predicted by the SFIA is closest to
Stress S (MPa)
Cycles to failure N
400 350 300 250 200 180 170 160
2560 19 100 83 976 311 050 572 296 600 260 846 313 1 309 410
the experimental results on the average because the S– N curve and the cyclic stress–strain curve are with 50% survival. The prediction of fatigue life by the LSSA is good for low-cycle fatigue, and prediction by the NSA is much more conservative.
5. Conclusions The following conclusions can be drawn according to the above experiments and analysis. 1. The maximum stress at the root of the notch is not the control parameter of fatigue crack initiation, because the stress gradient plays a very important role. 2. Stress field intensity sFI is the control parameter of fatigue crack initiation since the peak stress and the
W. Yao et al. / International Journal of Fatigue 23 (2001) 271–277
stress gradient are properly considered in this approach. 3. KTS is not the control parameter of fatigue crack initiation. 4. The fatigue life predicted by the SFIA is more accurate than the other approaches.
[5] [6] [7] [8] [9]
References [1] Zheng LC, Yao WX. A review of approaches for predicting fatigue crack life. Mechanics and Practice 1996;18(4):9–15 [in Chinese]. [2] Duggan TV, Byrne J. Fatigue as a design criterion. Macmillan Press Ltd, 1977. [3] Suresh S. Fatigue of materials. Cambridge, 1998. [4] Dowling NE, Brose WR, Wilson WK. Notched member fatigue
[10]
[11]
[12]
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life prediction by the local stress strain approach. In: Wetezl RM, editor. Fatigue under complex loading — analysis and experiments., 1977:55–84. Yao WX, Xia KQ, Gu Y. On the fatigue notch factor Kf. Int J Fatigue 1995;17(4):245–51. Peterson RE. Notch sensitivity. In: Sines G, editor. Metal fatigue. McGraw-Hill, 1959:293–306. Yao WX. Stress field intensity approach for prediction of fatigue life. Int J Fatigue 1993;15(3):243–5. Yao WX. On the notched strength of composite laminates. Composite Sci Technol 1992;45:105–10. Yao WX. The prediction of fatigue behaviour by stress field intensity approach. Acta Mech Sol Sin 1996;9(4):337–49. Zheng LC. Research on the fatigue life of crack initiation based on stress field intensity approach, Master Thesis. P.R. China: Nanjing University of Aeronautics and Astronautics; 1996. Neuber H. Theory of stress concentration for shear strained prismatical bodies with arbitrary nonlinear stress–strain law. J Appl Mech ASME 1961;28(4):544–50. Jhansale HR. A unified approach for modeling inelastic behaviour of structural metals under complex cyclic loadings. ADA040741, 1977.
Technical note
A verification of the assumption of anti-fatigue design Weixing Yao *, Bin Ye, Lichun Zheng Department of Aircraft Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China Received 1 March 2000; received in revised form 29 July 2000; accepted 4 September 2000
Abstract In this paper, the approaches of anti-fatigue design are briefly reviewed and are classified into three groups: nominal stress approach, local stress–strain approach and stress field intensity approach. Two types, each with two notches, are specially designed and a total of 30 specimens with 4 dimensions are tested under different constant–amplitude loading. These specimens are also analysed with elasto-plastic FEM to obtain stress–strain distributions and stress concentration factors. Fatigue life is estimated based on the different approaches. Experimental and analytical results show that the assumption of stress field intensity is more reasonable than that of the other approaches. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Design against fatigue; Nominal stress approach; Local stress–strain approach; Stress field intensity approach; Fatigue experiment
1. Introduction Metal fatigue is a complicated phenomenon which depends on many factors, and up to now the description of fatigue crack initiation has been left for further investigation. Notches in engineering structures are always the critical places for fatigue failures, so the prediction of fatigue life or fatigue strength of notched elements is the key point of the structure design against fatigue. In the history of fatigue research, many approaches [1,2] for the prediction of fatigue life of notched elements have been developed. Most of the approaches can been classified into three types according to their assumptions: nominal stress approach (NSA), local stress–strain approach (LSSA) and stress field intensity approach (SFIA). In this paper two types of specimens, each with two notches, are designed and tested under different constant–amplitude loading to see which assumption of the approaches is the most reasonable.
Most of the approaches can be classified into the following three groups: NSA, LSSA and SFIA. 2.1. Nominal stress approach (NSA) The assumption of the NSA is that the fatigue life of elements with different notches is the same if these notches have the same theoretical stress concentration factors and are under the same loading history (Fig. 1). Obviously the nominal stress S and theoretical stress concentration factor KT are the control parameters in this approach. But in practice, the fatigue life predicted by this approach is far removed from the fatigue life obtained by fatigue experiments in most cases. Many
2. A brief review of the approaches The approaches for prediction of fatigue life have been reviewed in detail in some of the literature [1–3]. * Corresponding author. Tel.: +86-25-489-2177; fax: +86-25-4891422. E-mail address: [email protected] (W.X. Yao). 0142-1123/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 8 3 - 9
Fig. 1.
Nominal stress approach.
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2.3. Stress field intensity approach (SFIA) The assumption of the SFIA is that the fatigue life of the elements with different notches is the same if the stress field intensity histories are the same (Fig. 3) [7–9]. SFIA defines the stress field intensity sFI as: sFI⫽
冕
1 → f(sij )j( r )dv V
(1)
⍀
where ⍀ is the fatigue failure region and V is the volume of ⍀ which only depends on the material, f(sij ) is the →
failure function and j( r ) is the weight function. A brief discussion of Eq. (1) follows. Fig. 2. Local stress–strain approach.
other models, such as the stress severity factor (SSF), effective stress method, and detailed fatigue rating (DFR), belong to this approach [1]. 2.2. Local stress–strain approach (LSSA) The assumption of the LSSA is that the fatigue life of elements with different notches is the same if the local stress–strain histories at the roots of the notches are the same (Fig. 2). The LSSA is a point criterion. The local stress and local strain are the control parameters of fatigue life of notched elements [4]. Many kinds of LSSA have been formed with the combination of the cyclic stress–strain relationship of the material, the strain–life curve of the material, the fatigue damage cumulative rule and the solution of the local stress–strain at the notch. In practice, the fatigue life estimated based on the real local stress and strain is lower than the fatigue life obtained by the fatigue test, so usually the fatigue notch factor Kf is used to calculate the local stress and strain in Neuber’s equation instead of theoretical stress concentration factor KT. Many researchers [5] have made efforts to estimate the Kf accurately, and Peterson’s equation is the most popular method [6].
2.3.1. The fatigue failure region ⍀ From the point of view of the fatigue mechanism, crack initiation generally occurs within the local region near the surface of the specimen where stress increases. This region is several grain sizes in extent. So the size and form of ⍀ depend on the material. 2.3.2. The failure function f(sij ) f(sij ) is the function to describe crack initiation under multiaxial stress, and may be different for different materials. For example, Von Mises’ equivalent stress is used for elasto-plastic materials, such as carbon steel, aluminum alloy and titanium, and maximum major stress is used for materials such as cast iron and cast steel. f(sij ) contains the effect of multiaxial stress near a notch, so the SFIA can deal with fatigue strength under multiaxial loading. →
2.3.3. The weight function j( r ) →
The weight function j( r ) physically means the contri→
bution of stress at Q to the peak stress at 兩 r 兩=0. →
→
1. 0ⱕj( r )ⱕ1 and j( r ) is a generalised monotonically →
decreasing function about 兩 r 兩; 2. j(0)⬅1 which means that the contribution of the stress at the notch root is maximum; →
3. when the stress gradient G=0, j( r )⬅1, which is consistent with the condition of smooth specimens. →
j( r ) is not only related to notch geometry and loading type for isotropic materials, but also to material properties for anisotropic materials. It can be obtained analytically or numerically and as an approximation of first order the following expression is used in this paper: →
j( r )⫽1⫺cr(1⫹sinq) Fig. 3.
Stress field intensity approach.
where c is a factor related to stress gradient.
(2)
W. Yao et al. / International Journal of Fatigue 23 (2001) 271–277
273
Table 1 A brief comparison among the three approaches Failure place Stress concentration NSA
Notch root LSSA
Local region near notch SFIA
Control parameters
KT and nominal stress KTS
Stress field intensity sFI
Material properties
S–N curves under different KT and R
Local stress and local strain KfS and smax Cyclic s–e relationship e–Nf curve (KT=1)
Cyclic s–e relationship S–N or e–Nf curve under KT=1
Table 2 The chemical and mechanical properties of aluminum alloy LY12-CZ (a) Chemical properties of aluminum alloy LY12-CZ Elements Wt%
Cu 4.61
Mg 1.54
Mn 0.58
Fe 0.29
Si 0.26
Zn 0.10
Ni 0.024
(b) Mechanical properties of aluminum alloy LY12-CZ Young’s Modulus E (MPa) 71022
Ultimate strength sb (MPa) 466
Yield strength sY (MPa) 343
2.3.4. The failure criterion →
For a smooth specimen, KT=1, G=0 and then j( r )=1. The stresses in the smooth specimen are equal everywhere, i.e. f(sij )⬅constant. According to Eq. (1), sFI=f(sij). Obviously, when sFI=f(sij)ⱖsf, fatigue failure occurs for a smooth specimen. Because Eq. (1) is universally applicable to notched specimens as well as to smooth ones, the failure criterion can be written as: sFIⱖsf
(3)
2.4. A brief comparison among the three approaches From the above discussion it can been seen that the assumptions of the three approaches are different, and the material properties required for each approach are also different. Table 1 gives a brief comparison of the three approaches.
3. Experiments 3.1. Specimens Two types of specimens are specially designed and tested under different constant–amplitude loadings to verify the assumptions of the approaches. Specimens are made from aluminum alloy LY12-CZ plate with thickness of 2 mm. The chemical and mechanical properties are listed in Table 2. They are cut longi-
Fig. 4. (a) Double-hole specimen (DH). (b) Double-notch specimen (DN-A). (c) Double-notch specimen (DN-B).
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Table 3 Experimental results of the double-hole specimens The place where the crack initiated
Specimen
Loading (MPa)
Frequency (Hz)
Fatigue life
DH-A01 DH-A02 DH-A03 DH-A04 DH-A05 DH-A06 DH-A07 DH-A08 DH-A09 DH-A10
80 80 80 80 80 80 150 150 150 150
12 12 12 12 12 12 6 6 6 6
122 734 113 000 169 235 144 708 122 469 143 164 15 900 18 952 18 518 16 426
DH-A11
150
6
15 649
DH-A12 DH-B01 DH-B02 DH-B03
150 98 98 98
6 14 14 14
17 741 136 600 181 890 166 300
One side of the hole One side of the hole One side of the hole One side of the hole One side of the hole One side of the hole One side of the hole One side of the hole Both the side of the hole One side of the slot One side of the hole and one side of the slot Both the side of the hole One side of the slot One side of the slot One side of the slot
Table 4 Experimental result of the double-notch specimens Specimen
Loading (MPa)
Frequency (Hz)
Cycles to failure
The places where cracks initiate
DN-A01 DN-A02 DN-A03 DN-B01 DN-B02 DN-B03
78.4 78.4 78.4 80 80 80
20 22 22 8 8 8
335 530 143 440 192 950 73 320 74 962 76 178
DN-B04
80
8
67 342
DN-B05 DN-B06 DN-B07 DN-B08 DN-B09 DN-B10 DN-B11 DN-B12
80 80 100 100 100 100 100 100
8 8 6 6 6 6 6 6
74 694 90 205 33 241 33 919 26 285 47 784 19 336 25 500
One edge notch One edge notch One edge notch One side of the hole One side of the hole Both the sides of the hole One side of the hole and one notch One side of the hole Both the sides of the hole One side of the hole One side of the hole Both the sides of the hole One side of the hole Both the sides of the hole One side of the hole
Table 5 Cyclic stress–strain curve of aluminium alloy LY12-CZ Stress (MPa)
Strain
Stress (MPa)
0 249 309 327 337 348 355 361
0 0.00356 0.00450 0.00500 0.00550 0.00700 0.00850 0.01000
371 382 396 415 451 462 466 470
Strain 0.01300 0.01800 0.02500 0.04000 0.08000 0.10000 0.11500 0.01510
tudinally from the same plate to diminish scatter. A double-hole specimen has two separate notches, one is a hole and the other is a slot [Fig. 4(a)]. A double-notch specimen has three separate notches, one is a central hole and the others are edged notches [Fig. 4(b) and (c)]. 3.2. Experiment The experiment is carried out at the MTS 810. For the 12 DH-A specimens, 6 specimens are tested under a nominal stress of 80 MPa, other specimens under 150 MPa, and 3 DH-B specimens are tested under 98 MPa. The experimental results are listed in Table 3. Three specimens labelled DN-A are tested under a constant–
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275
Table 6 Analytical results of the specimens (stress: MPa) Specimen
KT (gross)
KT (net)
K fa
Loading S
smax
sFI
The hole of specimen DH-A
3.61
2.27
2.19
The slot of specimen DH-A
3.96
3.42
2.68
The hole of DH-B The slot of DH-B Hole of DN-A Edge-notch of DN-A Hole of DN-B
3.42 3.50 1.64 1.74 2.04
2.44 2.75 1.64 1.74 2.04
2.34 2.43 1.51 1.57 1.99
Edge-notch of DN-B
2.08
2.08
1.71
80 150 80 150 98 98 78.4 78.4 80 100 80 100
286.11 363.55 297.66 366.55 316.64 341.40 253.47 266.53 303.91 378.86 309.31 386.65
247.62 320.06 239.33 319.24 276.06 279.23 224.12 230.10 261.89 298.93 250.45 292.40
a
Kf is calculated based on Peterson’s equation, and the constant a0=0.63 [11].
Table 7 Estimated and experimental fatigue life Failure place The hole of DH-A The slot of DH-B Edge-notch of DN-A Hole of DN-B
LSSA (Neuber) S=80 MPa S=150 MPa S=98 MPa S=78.4 MPa S=80 MPa S=100 MPa
42 053 1963 46 509 78 487 33 302 11 926
amplitude loading of 78.4 MPa. For the 12 specimens labelled DN-B, 6 specimens are tested under a nominal stress of 80 MPa, and the other specimens under 100 MPa. The experimental results are listed in Table 4. The fatigue life listed in the Tables 3 and 4 is the number of cycles to a crack size of about 0.5 mm, which is determined by microscopic monitoring during the experiment, and fractography after the experiment. 3.3. Analysis NASTRAN was employed for elasto-plastic FE analysis to calculate the stress–strain distribution and stress field intensity sFI of the specimens. The cyclic stress– strain curve, which is listed in Table 5, is used in the elasto-plastic FE analysis. The analytical results are listed in Table 6. Von Mises effective stress is taken as the fatigue failure function f(sij) and the size of the fatigue failure region for LY12-CZ aluminum is 0.185 mm [10] in calculating SFI sFI with Eq. (1). Fig. 5 presents Von Mises stress distribution along the line at the minimum width of the double-hole specimens. It can be seen that the maximum stress is at the edge of the slot, but the stress decreases faster, so the stress field intensity of the hole is larger than that of the slot. Fig. 6 presents Von Mises stress distribution along the line at the minimum width of the double-notch specimens.
NSA
SFIA
Experiment (average)
706 891 46 212 780 981 495 669 329 583 124 903
202 770 32 773 96 746 295 802 146 617 58 487
135 885 17 352 161 597 223 973 76 117 31 011
The maximum stress occurs at the edged notches, but the stress decreases faster, which means that the stress field intensity of the central hole is larger. Table 7 presents the fatigue life predicted by the different approaches. Miner’s rule is used in the prediction. The S–N curve for the NSA is listed in Table 8 and the strain–life curve e⫺Nf for LSSA is expressed by Eq. (4). logNf⫽A0⫹A1tan h−1
再
log(euee/e2eq) log(eu/ee)
冎
(4)
冉 冊
smax 1−m , smax is E maximum local stress, m=0.4, A0=3.4328, A1=2.7551, eu=0.053, and ee=0.0135 are material constants. where effective strain eeq=(2ea)m
4. Discussion Comparing the analytical results with the experimental results, it can be seen that stress gradient is a very important parameter for the prediction of fatigue life, as well as the peak stress. In fact this argument had been put forward several decades earlier [12] and many models developed to find a way to take into consideration the two factors properly. The SFIA is one new such model which gives a parameter sFI as the intensity of
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Fig. 6. (a) The stress distribution at the minimum width of the specimen DN-A. (b) The stress distribution at the minimum width of the specimen DN-B.
Table 8 S–N curve of aluminum alloy LY12-CZ (R=0.1)
Fig. 5. (a) The stress distribution at the minimum width of the specimen DH-A (S=80 MPa). (b) The stress distribution at the minimum width of the specimen DH-A (S=150 MPa). (c) The stress distribution at the minimum width of the specimen DH-B (S=98 MPa).
the stress field near the notch. For the double-hole specimens DH-A and DH-B, the maximum stress at the roots of the slots is larger than that at the roots of the holes (Table 6), but the fatigue crack initiated at the root of the hole (Table 3). For the double-notch specimens DNA and DN-B, the same conclusion can be obtained from Tables 4 and 6. For specimen DH-A, the SFI of the hole is nearly the same as that of the slot under load S=150 MPa, which means that the fatigue cracks initiated at the slot of two specimens and at the hole of five specimens. Among them, fatigue cracks initiated at both the hole and the slot almost simultaneously for specimen DH-A11. The fatigue life predicted by the SFIA is closest to
Stress S (MPa)
Cycles to failure N
400 350 300 250 200 180 170 160
2560 19 100 83 976 311 050 572 296 600 260 846 313 1 309 410
the experimental results on the average because the S– N curve and the cyclic stress–strain curve are with 50% survival. The prediction of fatigue life by the LSSA is good for low-cycle fatigue, and prediction by the NSA is much more conservative.
5. Conclusions The following conclusions can be drawn according to the above experiments and analysis. 1. The maximum stress at the root of the notch is not the control parameter of fatigue crack initiation, because the stress gradient plays a very important role. 2. Stress field intensity sFI is the control parameter of fatigue crack initiation since the peak stress and the
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stress gradient are properly considered in this approach. 3. KTS is not the control parameter of fatigue crack initiation. 4. The fatigue life predicted by the SFIA is more accurate than the other approaches.
[5] [6] [7] [8] [9]
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