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High temperature nonproportional low cycle fatigue using fifteen loading paths
Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
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High temperature nonproportional low cycle fatigue using fifteen loading paths Naomi Hamada a,1, Masao Sakane b,⇑, Takamoto Itoh b,2, Hideyuki Kanayama b,2 a b
Department of Mechanical Engineering, Faculty of Engineering, Hiroshima Kokusai Gakuin University, 20-1 Nakano, 6-Chome, Aki-ku, Hiroshima 739-0321, Japan Department of Mechanical Engineering, College of Science and Engineering, Ritsumeikan University, 1-1-1 Noji-higasi, Kusatu-shi, Shiga 525-8577, Japan
a r t i c l e
i n f o
Article history: Available online 30 July 2014 Keywords: Multiaxial fatigue Nonproportional loading High temperature fatigue Low cycle fatigue Fatigue life prediction Additional hardening
a b s t r a c t This paper discusses effects of nonproportional loading on low cycle fatigue lives for Type 304 stainless steel hollow cylinder specimens at 923 K. Strain controlled axial–torsion low cycle fatigue tests were performed using fifteen proportional and nonproportional strain paths at a strain rate of 0.1%/s. Nonproportional straining significantly reduced fatigue lives of the steel at 923 K. Reduction in fatigue life due to nonproportional straining reached about 80% compared with the proportional straining in specific strain paths. The nonproportional strains, DeNP and DeNP , and energy parameter, DrIDeI, successfully correlated proportional and nonproportional fatigue lives but ASME equivalent strain gave significantly unconservative estimates. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction High temperature applications sometimes undergo multiaxial low cycle fatigue (LCF) damage rather than uniaxial fatigue damage in the combination of mechanical and thermal loadings. Many studies have been performed for assessing high temperature nonproportional low cycle fatigue damage [1–13] and the equivalent strain defined in ASME Code Case (ASME strain) [14] is a typical parameter for nonproportional life assessment. However, the ASME strain has been reported to underestimate the fatigue lives in some nonproportional strain histories [3,15]. One of the key issues of nonproportional fatigue is that there exist many influential factors on nonproportional fatigue life so that a nonproportional parameter suitable for describing the damage under some strain histories is not always suitable for other strain histories. Thus, nonproportional parameters should be demonstrated to be successfully applicable to the experimental data that include the various influential factors on nonproportional fatigue life. Shortage of such extensive nonproportional data makes the development of suitable parameters difficult. The objective of this study is to develop a nonproportional strain parameter suitable for estimating high temperature nonproportional LCF lives. Axial–torsion multiaxial LCF tests were
⇑ Corresponding author. Tel.: +81 77 561 2746; fax: +81 77 561 2665. 1 2
E-mail address: [email protected] (M. Sakane). Tel.: +81 82 820 2670; fax: +81 82 820 2660. Tel.: +81 77 561 4965; fax: +81 77 561 2665.
http://dx.doi.org/10.1016/j.tafmec.2014.07.006 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.
performed on Type 304 stainless steel under two proportional and thirteen nonproportional strain histories. The temperature dependence of nonproportional LCF lives will be discussed in relation with the amount of additional hardening of the material.
2. Experimental procedure Fig. 1 shows the shape and dimensions of the specimen used here. The specimen is a hollow cylinder of Type 304 austenitic stainless steel with 12 mm O.D., 9 mm I.D. and 5 mm gage length. The specimen has a short gage length of 5 mm but this short gage length is for avoiding buckling of the specimen under severe nonproportional loading. The material underwent a solution treatment at 1373 K before machining. Axial-torsion multiaxial low cycle fatigue tests were performed using the fifteen strain paths shown in Fig. 2, where the various factors are programmed in these strain waves. Principal strain direction change effect is examined by comparing the LCF life between Cases 0, 1, 2, 3, 4 and 5 and zero-to-peak strain/full reversed loading effect by Cases 1, 2 and 3, 4. The effect of step length is studied by comparing Cases 5–9. Rotating principal strain direction effect is inspected from Cases 1, 2, 11, 12 and 5, 8, 9, 10 and direction of loading path from 8, 9, 10 and 12, 13. From Case 1 to 14, the axial strain range same amplitude as the pffiffiffi De has the p ffiffiffi shear strain range of Dc= 3, i.e., De ¼ Dc= 3. LCF tests were performed at a strain rate of 0.1%/s on Mises base at 923 K. The total strain range employed was 0.4%, 0.5% and 0.7%.
N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
137
Nomenclature d fNP
f NP m
a b /
c k M B
Leibniz’s notation for differentiation nonproportional factor based on maximum principal strain nonproportional factor based on COD strain constant of COD strain independent of material (m = 1.66) material constant expressing the degree of additional hardening constant of COD strain independent of material (b = 1.83) principal strain ratio shear strain in the normal plane to specimen axis principal stress ratio constant of COD stress independent of material (M = 0.5) constant pffiffiffi of COD stress independent of material ðB ¼ 1= 2Þ
Experimental apparatus used was an axial-torsion servohydraulic machine which can apply an axial load in combination with a torsional load. The maximum axial load and torsional load of the experimental apparatus are 49 kN and 196 N m, respectively. The specimen was gripped by separated collars to pull rod tightly. The axial and the shear strains were measured by a cantilever type extensometer with eddy current sensors which can measure axial and torsional displacement separately. The specimen was heated by an electric resistant furnace and the temperature distribution along the gage length of the specimen was within ±2 K. The number of cycles to failure was defined as the cycle of 5% axial stress range drop from the saturation. The variations of stress ranges with cycles are shown in Fig. 3 for Case 13. The stress range sharply dropped at the final stage in LFC process. So there is no significant effect on the number of cycle to failure if another percentage is employed other than 5%. One cycle was defined as a full straining for both axial and shear straining shown in Fig. 2. Following to this definition, Cases 3, 4 and 13 were counted as two cycles for a full loading along the strain path shown in Fig. 2 and the other cases were counted as one cycle. 3. Experimental results and discussion 3.1. Correlation of nonproportional fatigue lives with strain parameters ASME Code Case [14] defines an equivalent strain to express the nonproportional fatigue damage. The equivalent strain is originated from the Mises strain but it was modified to have a maximum value taking any times C and D in Eq. (1) (ASME strain range).
Fig. 1. Shape and dimensions of test specimen (mm).
De DeASME Deeq DeI DeI DeNP
DeNP Depath Dreq DrI DrI DrIpre DrIP
axial normal strain range strain range defined in ASME code case Mises’ equivalent strain range maximum principal strain range maximum strain range based on COD strain nonproportional strain range based on maximum principal strain nonproportional strain range based on COD strain pffiffiffi strain range based on strain path length on e c= 3 plot Mises’ equivalent stress range maximum principal stress range maximum stress range based on COD stress maximum principal stress range predicted maximum principal stress range under proportional loading
DeASME ¼ max
" 1=2 # 1 ðeC eD Þ2 þ ðcC cD Þ2 3
ð1Þ
eC, cC, eD and cD are the axial and shear strains at the times C and D to maximize the strain in the bracket. Table 1 summarizes the experimental data generated in this study and Fig. 4 correlates the number of cycles to failure with ASME strain range (DeASME). Numbers attached to the data in the figure are the case numbers of strain paths shown in Fig. 2. Fatigue lives in Case 0 are plotted with solid circles and the narrow lines in the figure are a factor of two band based on the solid bold line of Case 0 data. ASME strain range does not correlate the fatigue lives under fifteen strain paths within a factor of two scatter band. Especially, fatigue lives in Cases 1, 2, 10, 12, 13, and 14 are underestimated by more than a factor of two. The smallest fatigue lives are found in Cases 2, 12 and 14 that are smaller by a factor of 4 compared with the proportional data. To discuss the fatigue lives in respective strain paths, the ratio of the nonproportional fatigue life (Nf) to the proportional life in Case 0 (Nf Case 0) at 923 K in each strain path is shown with a bar graph by the case number, where the ratio at room temperature is also shown in Fig. 5 [15]. Cases 1 and 2 which are the cruciform straining along normal and shear strains have a ratio of 0.3 but Cases 3 and 4 having the same cruciform straining with rotating 45° give the values around 0.8 at 923 K. Considering that Cases 1–4 all have the similar ratio around 0.2 at room temperature, the larger ratios in Cases 3 and 4 at 923 K may result from the thermal activation. Comparing the ratio between Cases 5 and 9 shows the effect of step length in nonproportional loading. Since the ratio in Case 6 is about 1.5 times smaller than that in Case 5, even small steps decrease LCF life. The ratio decreases as the step length increases as shown in Cases 7 and 8. Tensile mean straining gives a smaller ratio than the compressive mean strain, Cases 8 and 9. The ratio decreases as the phase difference between normal and shear strain increases, which is understood from the comparison of the ratio between Cases 5, 11 and 12. Box loadings in Cases 10–13 all have the similar ratio whereas Case 12 has a slightly smaller ratio than the other box loadings. The circular straining, Case 14, shows the smallest ratio as Case 12. This indicates that the 90° phase difference gives the largest nonproportional fatigue damage. The general trend of the nonproportional data at 923 K shown in Fig. 5 does not significantly differ from that at room temperature except for Cases 3 and 4 and Cases 7 and 8 as previously mentioned, but the life ratios at room temperature are smaller than
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N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
Fig. 2. Fifteen strain paths used in experiments.
SUS304, T=923K
n(t) is schematically shown in Fig. 6. The angle (n(t)) in the figure has twice magnitude of the actual angle in a specimen. The maximum principal strain range (DeI) in nonproportional straining is defined as the maximum value of the bracket in Eq. (2). The maximum principal strain range did not correlate the proportional LCF lives within a small scatter band for Type 304 stainless steel [20] and Cr–Mo–V steel [21] at high temperatures. The strain gave too conservative estimates for torsion data compared with axial data beyond a factor of two scatter band. In this regard, the authors have proposed the equivalent strain based on crack opening displacement (COD strain) for correlating axial and torsion proportional low cycle fatigue data within a small scatter band [20–23]. The COD strain (eI ðtÞ) in nonproportional straining is expressed as a function of maximum principal strain (eI(t)) [15].
Axial stress range Δσ Shear stress range Δτ , MPa
600 500
5% drop
400 300 Nf =440
200 Δε =0.7% Δγ =1.21% Case 13
Δσ
100
Δτ
0 101
102
103
Number of cycles N
e ðtÞ ¼ bð2 /ðtÞÞm eI ðtÞ
Fig. 3. Definition of number of cycles to failure in Case 13.
those at 923 K in most cases. Therefore, the nonproportional loading at room temperature gives larger damage than at high temperature. Conversely, this may result from the thermal activation at 923 K which may reduce the additional hardening effect, strongly evident at room temperature. The main cause of the additional hardening of Type 304 stainless steel is severe interactions of slip planes resulting in smaller cell formation [16], [17]. The thermal activation at high temperatures may weaken the interactions and firm cell formation. Other than ASME strain range, couples of strain ranges were applied to correlate the nonproportional LCF lives. The strains applied are briefly explained below. In nonproportional straining, the maximum principal strain direction changes with time, so the definition of the maximum principal strain range should be clearly defined. Itoh et al. [15] proposed a definition of the maximum principal strain range (DeI) for nonproportional straining as, [18,19]
DeI ¼ max½eImax cos nðtÞeI ðtÞ
ð2Þ
eImax, eI(t) and n(t) are the maximum principal strain in a cycle, the maximum principal strain at time (t) and the angle between the eImax direction and the eI(t) direction, respectively. The polar figure indicating the relationship between the two strains and the angle
ð3Þ
b and m are the constants independent of materials and take the values of 1.83 and 1.66, respectively, which were confirmed numerically by FEM analysis using many constitutive relationships [15]. /(t) is the principal strain ratio at time (t) defined as e3/e1 for |e3(t)| |e1(t)| and e1/e3 for |e3(t)| > |e1(t)|, where e1(t) and e3(t) are maximum and minimum principal strains at time t respectively. Maximum COD strain range is defined similar to Eq. (2) as following.
DeI ¼ max½eImax cos nðtÞe ðtÞ
ð4Þ ⁄
⁄
e is the maximum value of e (t) in a cycle and e (t) is the COD strain to maximize the value in the bracket. Path strain range [4] is a strain range defined as a half-length of the strain path shown in Fig. 2 and is expressed by the following equation. Imax
Depath
1 ¼ 2
Z cycle
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d
e2 þ
c2 3
ð5Þ
Depath, e, c and d are the path strain range, the axial strain, the shear strain and the Leibniz’s notation for differentiation, respectively. Figs. 7–9 correlates the nonproportional data with the three strains mentioned above. The correlation with the maximum principal strain range underestimates about a half of the data by more
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N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143 Table 1 Summary of the test results. Failure cycle Nf, cycles
Axial strain range De (%)
Shear strain range Dc (%)
ASME strain range DeASME (%)
Principal strain range DeI (%)
COD strain range DeI (%)
Path strain range Depath (%)
Axial stress range Dr (MPa)
0 0 0 0 0 0
5450 7200 3000 1500 400 160
0.40 0.40 0.50 0.70 1.00 1.50
0.00 0.00 0.00 0.00 0.00 0.00
0.40 0.40 0.50 0.70 1.00 1.50
0.40 0.40 0.50 0.70 1.00 1.50
0.40 0.40 0.50 0.70 1.00 1.50
0.40 0.40 0.50 0.70 1.00 1.50
419 401 456 477 531 580
0 0 0 0 0 0
419 401 456 477 531 580
5 6 7 10 11 12 13 14
3000 3200 3700 1200 3000 3200 2600 1200
0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69
0.57 0.57 0.57 0.57 0.45 0.40 0.40 0.40
0.56 0.56 0.56 0.56 0.45 0.40 0.40 0.40
0.54 0.54 0.54 0.54 0.44 0.40 0.40 0.40
0.57 0.80 0.80 0.80 0.57 0.57 0.57 0.63
383 388 399 447 428 473 475 497
148 184 215 284 248 300 257 319
431 444 456 493 446 476 476 499
1 2 3 4 5 6 7 7 8 8 9 10 11 12 13 14
1100 1300 1400 1100 1720 1150 1000 750 1170 1100 710 500 900 830 1300 920
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87
0.50 0.50 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.56 0.50 0.50 0.50
0.50 0.50 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.56 0.50 0.50 0.50
0.50 0.50 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.55 0.50 0.50 0.50
1.00 1.00 0.71 0.71 0.71 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.71 0.71 0.71 0.79
612 497 501 374 393 388 424 461 554 514 532 582 492 524 547 531
377 320 278 257 178 203 262 267 375 355 349 342 311 345 354 350
616 509 610 566 461 457 480 512 580 549 548 617 526 532 557 535
1 2 3 4 5 6 7 9 10 11 12 13 14
575 280 608 581 990 450 410 242 250 650 310 440 335
0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70
1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21
0.70 0.70 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.78 0.70 0.70 0.70
0.70 0.70 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.78 0.70 0.70 0.70
0.70 0.70 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.78 0.70 0.70 0.70
1.40 1.40 0.99 0.99 0.99 1.40 1.40 1.40 1.40 0.99 0.99 0.99 1.10
658 615 585 456 441 457 501 – 480 495 583 575 593
395 353 335 276 196 242 296 – 315 343 374 378 391
664 624 692 607 511 494 533 – 630 541 598 606 600
Strain path number (Case)
than a factor of two, Fig. 7. The most unconservatively estimated lives are the lives in Cases 2 and 12 at DeI = 0.7%. These lives are around one fifth of the Case 0 lives at the same principal strain range. The results of the correlation with the maximum principal strain are similar to those with ASME strain. The maximum COD strain range also underestimates a half of the data by more than a factor of two, Fig. 8. The correlation with the maximum COD strain range shows almost the same trend as that with the maximum principal strain. The path strain range appears to correlate the LCF data with a small scatter, Fig. 9, whereas some data are overestimated by more than a factor of two. In the correlation of the LCF data at room temperature with the path strain range, the path strain range overestimated some data and also underestimates other data by more than a factor of two [15]. This means that the path strain range does not cover the temperature dependency of the data correlation. The question of the path strain range is that it predicts the same LCF life for the strain waveform that has the same path length, but the LCF life in experiments varied with step length having the same path length, which is understood from the different LCF live in Cases 6, 7 and 8.
Shear stress range Ds (MPa)
Principal stress range DrI (MPa)
3.2. Correlation of nonproportional fatigue lives with stress parameters LCF lives under nonproportional straining are generally smaller than those of under proportional loading for stainless steels. The smaller fatigue lives are attributed to the significant additional hardening due to nonproportional loading. Therefore, it is meaningful to discuss the applicability of stress parameter to the LCF lives. The stress parameters discussed are the maximum principal stress range, the equivalent stress range based on crack opening displacement (COD stress) and the Mises stress range. The maximum principal stress range is defined similarly to the maximum principal strain as [18,19],
DrI ¼ max½rImax cos nðtÞrI ðtÞ
ð6Þ
rImax and rI(t) are the maximum principal stress in a cycle and the principal stress at time t, respectively. n(t) is the angle between the rImax direction and the rI(t) direction. The COD stress is also similarly defined as [24–27],
r ðtÞ ¼ Bð2 kðtÞÞM rI ðtÞ
ð7Þ
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SUS304, T=923K Factor of 2
1.0
0.5
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0 0.1 102
103
104
Maximum principal strain range Δε I , %
ASME strain range ΔεASME, %
SUS304, T=923K
Number of cycle to failure Nf
1.0
0.5 Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0.1 102
103
104
Number of cycle to failure N f Fig. 7. Correlation of nonproportional fatigue lives with maximum principal strain range.
1.4
Room Temp.
1.0 0.8 0.6 0.4 0.2 0
SUS304, T=923K
T=923K
1.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Case Fig. 5. Ratio pffiffiffi of nonproportional fatigue life to push–pull fatigue life (Case 0) at De ¼ Dc= 3 ¼ 0:5% for the fourteen cases.
Maximum COD strain range Δε I*, %
Ratio of non-proportional fatigue life to proportional fatigue life Nf /Nf Case 0
Fig. 4. Correlation of nonproportional fatigue lives with ASME strain range.
Factor of 2
Factor of 2 1.0
0.5 Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0.1 102
103
104
Number of cycle to failure Nf Fig. 8. Correlation of nonproportional fatigue lives with maximum COD strain range.
SUS304, T=923K
Fig. 6. Definitions of the maximum principal strain, the principal strain range and the angle of n(t) in a polar figure.
B and M arepthe ffiffiffi constants independent of materials and take the values of 1= 2 and 0.5, respectively, which were derived numerically by FEM analysis using many constitutive relationships as well as the COD strain. kðtÞ is the maximum principal stress ratio similarly defined as for the case of the maximum principal strain ratio, replacing the principal strains by the principal stresses. The
Path strain range Δεpath, %
1.0
0.5
Factor of 2
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0.1 102
103 Number of cycle to failure Nf
104
Fig. 9. Correlation of nonproportional fatigue lives with path strain range.
maximum COD stress range is defined as a function of the COD stress and n(t) as
DrI ¼ max½rmax cos nðtÞr ðtÞ
ð8Þ
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N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
Mises stress range is defined as Eq. (9) similar to ASME strain range as
ð9Þ
where rC, sC, rD and sD are the axial and shear stresses at times C and D to maximize the strain in the bracket. The correlation of LCF lives with the three stresses is shown in Figs. 10–12. The maximum principal stress gives almost the same correlation as the maximum COD stress, Figs. 10 and 11. The two stress parameters overestimate the nonproportional fatigue lives for some strain histories by more than a factor of two. The scatter becomes worse in the correlation with the Mises stress, Fig. 12. Mises stress also overestimates the nonproportional fatigue lives for most of the strain histories. The unconservative estimates in the correlation of the LCF lives with the strain parameters like the Mises strain and the conservative estimates with the stress parameters like the Mises stress indicate the followings. In the correlation with the strain parameters, the material yielded an additional hardening under nonproportional loadings and larger stresses were applied to the specimen compared with proportional loading resulting in smaller fatigue lives under nonproportional loading. However, in the correlation with the stress parameters, the correlation became too conservative that means that the LCF lives were not reduced as the amount of additional hardening under nonproportional loading. Therefore, the correlation of the LCF lives with an energy parameter was examined where the product of the maximum principal strain and stress is used here as a typical parameter. The results of the correlation are shown in Fig. 13. The correlation shown in the figure is quite improved from that with the stress parameters and most of the data are collapsed into a factor of two scatter band, which means that the energy parameter is the most suitable parameter to describe the LCF damage under nonproportional loading. The similar trend of the correlation was also found at room temperature [15]. However, the energy parameter that needs both strain and stress is not so convenient in practical design of components and structures. 3.3. Correlation of LCF fatigue lives with nonproportional strain
Maximum principal stress range Δσ I , MPa
As shown in Fig. 13, the energy parameter correlates the nonproportional lives with a small scatter. Energy parameters, however, need both the stress and strain ranges for the estimates of LCF lives. It is not easy to obtain cyclic stress ranges under nonproportional loadings utilizing a constitutive modeling because the constitutive modeling for nonproportional loading is still under
SUS304, T=923K Factor of 2
COD stress range Δσ I* , MPa
1=2
Factor of 2 800
400
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0
102
103
104
Number of cycle to failure N f Fig. 11. Correlation of nonproportional fatigue lives with COD stress range.
SUS304, T=923K Mises stress range Δσ eq , MPa
Dreq ¼ max½fðrC rD Þ2 þ 3ðsC sD Þ2 g
SUS304, T=923K
Factor of 2 800
400
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0
10
2
103
104
Number of cycle to failure N f Fig. 12. Correlation of nonproportional fatigue lives with Mises stress range.
research process or requires many material constants in the reliable models proposed. Therefore, a parameter written with only strain is preferable in practical design for applications. Itoh et al. [15] proposed the nonproportional strain (DeNP) which is an equivalent strain taking account of additional hardening of materials under nonproportional loadings. They applied the strain to the nonproportional LCF lives of Type 304 stainless steel fatigued at room temperature and demonstrated the applicability of the strain. Their application, however, limited to the LCF data at room temperature and the applicability of the parameter to LCF lives at high temperatures is still an open question. This paper examines the applicability of the nonproportional strain to the high temperature LCF data presented in this paper. The nonproportional strain is written as,
800
DeNP ¼ ð1 þ afNP ÞDeI
400
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0 102
103 Number of cycle to failure Nf
104
Fig. 10. Correlation of nonproportional fatigue lives with maximum principal stress range.
ð10Þ
a is the material constant which expresses the amount of additional hardening under nonproportional loading and is calculated as the ratio of stress range under circular straining to that in Case 0 in a r1 e1 plot at the same strain amplitude. The value of a for Type 304 stainless steel becomes 0.3 at 923 K whereas it was 0.9 at room temperature [15], which means that the amount of the additional hardening under circular path at 923 K is 1/3 of that at room temperature. Thermal activation may reduce the amount by thermally assisted recovery system of dislocations [16,28]. fNP in Eq. (10) is a nonproportional factor that expresses the intensity of nonproportionality. The experimental results in Table 1 and Fig. 5 tell us that
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N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
Nonproportional strain range ΔεNP, %
Energy parameter Δε I Δσ I , MPa
SUS304, T=923K 10
Factor of 2
5
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
1 103
102
104
SUS304, T=923K Factor of 2 1.0
0.5 Δε = 0.7% Δε = 0.5% Δε = 0.4%
0.1 102
The nonproportional factor should be a function of angle n(t) from the first factor and should have an integral form from the second factor. Therefore, the nonproportional factor is expressed as, T
ðj sin nðtÞjeI ðtÞÞdt
ð11Þ
¼ ð1 þ a
f NP ÞD I
e
ð12Þ
fNP is nonproportional factor based on the COD strain and is expressed as,
fNP
1:66 ¼ T eImax
Factor of 2 1.0
0.5 Δε = 0.7% Δε = 0.5% Δε = 0.4%
Δε
Case 0
Z
T
ðj sin nðtÞje ðtÞÞdt
ð13Þ
0
Figs. 14 and 15 correlates the LCF lives generated in this study with the nonproportional strains shown in Eqs. (10) and (12), respectively. The correlations are quite satisfactory and all the data except only a couple of data are collapsed within a factor of two scatter band. There is no distinguished difference in the correlation between the nonproportional strain and the nonproportional strain based on COD, because the pure shear strain path is not included in the strain path shown in Fig. 2. The difference in the correlation would be more pronounced if the shear strain path is included in this paper. The principal strain give a trend to overestimate torsion proportional LCF lives for Type 304 stainless steel if the estimate is based on push–pull LCF lives [20,21]. To improve the estimate for torsion LCF lives, the equivalent strain based on COD was proposed [23]. Considering the successful correlation with the energy parameter and the nonproportional strains expressed with Eqs. (10) and (12), the nonproportional factor will correspond with the amount of additional hardening under nonproportional loading. To examine the relationship between the nonproportional factor and the additional hardening, the stress range calculated by the following
0.1
*=
(1+α fNP*) Δε I*
102
104
103
Number of cycle to failure Nf Fig. 15. Correlation of nonproportional fatigue lives with nonproportional COD strain.
equation is compared with the experimental stress range. The equation is the same equation as the nonproportional strain but the principal strain range is substituted by the principal stress range.
DrIpre ¼ ð1 þ afNP ÞDrIP
ð14Þ
DrIP is the maximum principal stress range under the proportional loading. Constant a has the value of 0.3 as mentioned above. Fig. 16 shows the relationship between the predicted maximum principal stress and experimental stress ranges under nonproportional loading. The predicted stress ranges by Eq. (14) are in good Predicted stress range Δσ I pre, MPa
De
SUS304, T=923K
0
T is the time per cycle and the constant 1.57 is determined so as to the value of fNP being unity for the circular strain path. fNP takes null value for proportional loading and positive values from 0 to 1 for nonproportional loading. The nonproportional strain and nonproportional factor described above are based on the principal strain. Those based on the COD strain are also definable as follows. NP
Fig. 14. Correlation of nonproportional fatigue lives with nonproportional strain.
Nonproportional COD strain range ΔεNP*, %
the nonproportional factor should take account of the following factors. Fatigue life reduces more significantly when the principal strain direction changes at larger angle. Fatigue life reduces more significantly when the path length is larger after the principal strain direction change.
1:57 T eImax
104
Number of cycle to failure Nf
Fig. 13. Correlation of nonproportional fatigue lives with energy parameter.
fNP ¼
= (1+α fNP) Δε I
103
Number of cycle to failure Nf
Z
Δε
Case 0
800
α =0.3
20%
400
Δε = 0.7% Δε = 0.5% Δε = 0.4%
0
400
800
Experimental stress range Δσ I , MPa Fig. 16. Comparison of predicted stress ranges by Eq. (14) with experimental stress ranges.
N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
agreement with the experimental stress ranges within a factor of 1.20, so the nonproportional factor shown in Eq. (14) properly estimates the amount of the additional hardening under nonproportional loadings at 923 K. 4. Conclusions (1) Nonproportional straining drastically decreased low cycle fatigue lives of Type 304 stainless steel. Nonproportional low cycle fatigue lives depended on the strain history. Most damaging nonproportional strain path was the strain path with 90° phase difference with triangular and sinusoidal strain waves. (2) The equivalent strain range defined in ASME Code Case and the maximum principal strain underestimated the nonproportional fatigue lives in some strain histories. The equivalent strain based on the path length overestimated the nonproportional fatigue lives in some strain histories. The maximum scatter in the correlation with these strain parameters was a factor of four that is smaller the scatter of the data at room temperature previously reported. (3) Constant a that expresses the amount of additional hardening was 0.3 at 923 K in this study and was smaller than 0.9 at room temperature. Thermal activation was expected to reduce the amount of additional hardening at high temperature. (4) The energy parameter of the product of principal strain and stress correlated the nonproportional fatigue lives within a factor of two scatter band. (5) Nonproportional fatigue strains, DeNP and DeNP , which take account of the amount of additional hardening were suitable parameters to correlate the nonproportional fatigue lives of Type 304 stainless steel at 923 K.
References [1] S. Taira, T Inoue, M. Takahashi, Low cycle fatigue under multiaxial stresses, in: Proceedings of the 11th Japan Congress on Materials Research, 1968, pp. 60– 65. [2] M. Ohnami, N. Hamada, Biaxial low-cycle fatigue of a SUS304 stainless steel at elevated temperature, in: Proceedings of the 25th Japan Congress on Materials Research, Society of Materials Science, Japan, 1982, pp. 93–99. [3] M. Ohnami, M. Sakane, N. Hamada, Effect of changing principal stress axes on low-cycle fatigue life in various strain wave shapes at elevated temperature, ASTM STP 853 (1985) 622–634. [4] M. Kitade, T. Ikari, M. Yamanouchi, K. Nakajima, Cyclic deformation and failure of 21/4Cr–Mo steel under biaxial stresses, in: Proc. 25th Symp. on Strength of Mater. At High Temp., Society of Materials Science, Japan, 1987, pp. 97–101. [5] A. Nitta, T. Ogata, K. Kuwabara, The effect of axial–torsional strain phase on elevated-temperature biaxial low-cycle fatigue life in SUS304 stainless steel, Soc. Mater. Sci., Jpn. 36 (1987) 376–382. [6] A. Nitta, T. Ogata, K. Kuwabara, Relationship between fracture mode and fatigue life under biaxial loading at 550 °C in SUS304 stainless steel, Soc. Mater. Sci., Jpn. 37 (1988) 334–339.
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[7] A. Nitta, T. Ogata, K. Kuwabara, Fracture modes and fatigue life evaluation of SUS304 stainless steel under nonproportional biaxial loading condition at 550 °C, Soc. Mater. Sci., Jpn. 38 (1989) 416–422. [8] R.M. Andrews, M.W. Brown, Elevated temperature out-of-phase fatigue behavior of a stainless steel, in: M.W. Brown, K.J. Miller (Eds.), Biaxial and Multiaxial Fatigue, EGF3, Mechanical Engineering Pub., London, 1989, pp. 641– 658. [9] E.R. de los Rios, R.M. Andrews, M.W. Brown, K.J. Miller, Elevated temperature out-of-phase fatigue behavior of a stainless steel, in: M.W. Brown, K.J. Miller (Eds.), Biaxial and Multiaxial Fatigue, EGF3, Mechanical Engng. Pub., London, 1989, pp. 659–682. [10] T. Ogata, A. Nitta, K. Kuwabara, Biaxial low-cycle fatigue of Type 304 stainless steel under in-phase and out-of-phase straining conditions, in: K. Kussmaul, D. McDiarmid, D. Socie (Eds.), Fatigue Under Biaxial and Multiaxial Loading, ESIS10, Mechanical Engng. Pub., London, 1991, pp. 377–392. [11] T. Itoh, M. Sakane, M. Ohnami, Y. Takahashi, T. Ogata, Non-proportional multiaxial low cycle fatigue using cruciform specimen, Creep: Characterization, Damage and life Assessments, in: Woodford et al., (Eds.), Proc. Fifth Int. on Creep of Mater., FL, 1992, pp. 331–339. [12] H. Frenz, J. Meersmann, J. Ziebs, H.J. Kühn, R. Sievert, J. Olschewski, Hightemperature behaviour of IN 738 LC under isothermal and thermo-mechanical cyclic loading, Mater. Sci. Eng.: A 230 (1997) 49–57. [13] G. Kang, Q. Gao, L. Cai, Y. Sun, Experimental study on uniaxial and nonproportionally multiaxial ratcheting of SS304 stainless steel at room and high temperatures, Nucl. Eng. Des. 216 (2002) 13–26. [14] ASME, Boiler and Pressure Vessel Code Section III, Division 1 NH, 2004. [15] T. Itoh, M. Sakane, M. Ohnami, D. Socie, Nonproportional low cycle fatigue criterion for type SUS304 stainless steel ASME, J. Eng. Mater. Technol. 117 (1995) 285–292. [16] S. Nishino, N. Hamada, M. Sakane, M. Ohnami, N. Matsumura, M. Tokizane, Microstructural study of cyclic strain hardening behavior in biaxial stress states at elevated temperature, Fatigue Fract., Eng. Mater. Struct. 9 (1986) 65– 77. [17] S. Kida, T. Itoh, M. Sakane, M. Ohnami, F. Socie, Dislocation structure and nonproportional hardening of Type 304 stainless steel, Fatigue Fract., Eng. Mater. Struct. 20 (1997) 1375–1386. [18] T. Itoh, M. Sakane, K. Ohsuga, Multiaxial low cycle fatigue life under nonproportional loading, Int. J. Press. Ves. Pip. 110 (2013) 50–56. [19] T. Itoh, M. Sakane, Y. Shimizu, Definition of stress and strain ranges for multiaxial fatigue life evaluation under non-proportional loading, J. Soc. Mater. Sci., Jpn. 62 (2013) 117–124. [20] M. Sakane, M. Ohnami, M. Sawada, Fracture modes and low cycle biaxial fatigue life at elevated temperature, ASME J. Eng. Mater. Technol. 109 (1987) 236–243. [21] M. Sakane, M. Ohnami, M. Sawada, Biaxial low cycle fatigue of unaged and aged 1Cr–1Mo–1/4V steels at elevated temperature, ASME J. Eng. Mater. Technol. 113 (1991) 244–253. [22] N. Hamada, M. Sakane, M. Ohnami, A study of high temperature low cycle fatigue criterion in biaxial stress state, Bull. Soc. Mech. Eng., Jpn. 28 (1985) 1341–1347. [23] T. Itoh, M. Sakane, M. Ohnami, High temperature multiaxial low cycle fatigue of cruciform specimen, ASME J. Eng. Mater. Technol. 116 (1994) 90–98. [24] N. Hamada, M. Sakane, M. Ohnami, Creep-fatigue studies under a biaxial stress state at elevated temperature, Fatigue Eng. Mater. Struct. 7 (1984) 85–96. [25] N. Hamada, M. Sakane, M. Ohnami, A study of mechanical parameter controlling high-temperature biaxial low-cycle fatigue, on the effect of constitutive relation of materials, Soc. Mater. Sci. Jpn. 34 (1985) 214–220. [26] N. Hamada, M. Sakane, M. Ohnami, Effect of temperature on biaxial low cycle fatigue crack propagation and failure, in: Proc. of 30th Japan Conger. on Mater. Res., 1987, pp. 69–75. [27] N. Hamada, M. Sakane, M. Ohnami, COD criterion to assess high temperature biaxial low- cycle fatigue, Eur. Struct. Integrity Soc. (1991) 239–247. [28] D. Socie, G. Marquis, Multiaxial Fatigue, Soc. Automotive Eng. Inc., 2000, p. 294.
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
High temperature nonproportional low cycle fatigue using fifteen loading paths Naomi Hamada a,1, Masao Sakane b,⇑, Takamoto Itoh b,2, Hideyuki Kanayama b,2 a b
Department of Mechanical Engineering, Faculty of Engineering, Hiroshima Kokusai Gakuin University, 20-1 Nakano, 6-Chome, Aki-ku, Hiroshima 739-0321, Japan Department of Mechanical Engineering, College of Science and Engineering, Ritsumeikan University, 1-1-1 Noji-higasi, Kusatu-shi, Shiga 525-8577, Japan
a r t i c l e
i n f o
Article history: Available online 30 July 2014 Keywords: Multiaxial fatigue Nonproportional loading High temperature fatigue Low cycle fatigue Fatigue life prediction Additional hardening
a b s t r a c t This paper discusses effects of nonproportional loading on low cycle fatigue lives for Type 304 stainless steel hollow cylinder specimens at 923 K. Strain controlled axial–torsion low cycle fatigue tests were performed using fifteen proportional and nonproportional strain paths at a strain rate of 0.1%/s. Nonproportional straining significantly reduced fatigue lives of the steel at 923 K. Reduction in fatigue life due to nonproportional straining reached about 80% compared with the proportional straining in specific strain paths. The nonproportional strains, DeNP and DeNP , and energy parameter, DrIDeI, successfully correlated proportional and nonproportional fatigue lives but ASME equivalent strain gave significantly unconservative estimates. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction High temperature applications sometimes undergo multiaxial low cycle fatigue (LCF) damage rather than uniaxial fatigue damage in the combination of mechanical and thermal loadings. Many studies have been performed for assessing high temperature nonproportional low cycle fatigue damage [1–13] and the equivalent strain defined in ASME Code Case (ASME strain) [14] is a typical parameter for nonproportional life assessment. However, the ASME strain has been reported to underestimate the fatigue lives in some nonproportional strain histories [3,15]. One of the key issues of nonproportional fatigue is that there exist many influential factors on nonproportional fatigue life so that a nonproportional parameter suitable for describing the damage under some strain histories is not always suitable for other strain histories. Thus, nonproportional parameters should be demonstrated to be successfully applicable to the experimental data that include the various influential factors on nonproportional fatigue life. Shortage of such extensive nonproportional data makes the development of suitable parameters difficult. The objective of this study is to develop a nonproportional strain parameter suitable for estimating high temperature nonproportional LCF lives. Axial–torsion multiaxial LCF tests were
⇑ Corresponding author. Tel.: +81 77 561 2746; fax: +81 77 561 2665. 1 2
E-mail address: [email protected] (M. Sakane). Tel.: +81 82 820 2670; fax: +81 82 820 2660. Tel.: +81 77 561 4965; fax: +81 77 561 2665.
http://dx.doi.org/10.1016/j.tafmec.2014.07.006 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.
performed on Type 304 stainless steel under two proportional and thirteen nonproportional strain histories. The temperature dependence of nonproportional LCF lives will be discussed in relation with the amount of additional hardening of the material.
2. Experimental procedure Fig. 1 shows the shape and dimensions of the specimen used here. The specimen is a hollow cylinder of Type 304 austenitic stainless steel with 12 mm O.D., 9 mm I.D. and 5 mm gage length. The specimen has a short gage length of 5 mm but this short gage length is for avoiding buckling of the specimen under severe nonproportional loading. The material underwent a solution treatment at 1373 K before machining. Axial-torsion multiaxial low cycle fatigue tests were performed using the fifteen strain paths shown in Fig. 2, where the various factors are programmed in these strain waves. Principal strain direction change effect is examined by comparing the LCF life between Cases 0, 1, 2, 3, 4 and 5 and zero-to-peak strain/full reversed loading effect by Cases 1, 2 and 3, 4. The effect of step length is studied by comparing Cases 5–9. Rotating principal strain direction effect is inspected from Cases 1, 2, 11, 12 and 5, 8, 9, 10 and direction of loading path from 8, 9, 10 and 12, 13. From Case 1 to 14, the axial strain range same amplitude as the pffiffiffi De has the p ffiffiffi shear strain range of Dc= 3, i.e., De ¼ Dc= 3. LCF tests were performed at a strain rate of 0.1%/s on Mises base at 923 K. The total strain range employed was 0.4%, 0.5% and 0.7%.
N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
137
Nomenclature d fNP
f NP m
a b /
c k M B
Leibniz’s notation for differentiation nonproportional factor based on maximum principal strain nonproportional factor based on COD strain constant of COD strain independent of material (m = 1.66) material constant expressing the degree of additional hardening constant of COD strain independent of material (b = 1.83) principal strain ratio shear strain in the normal plane to specimen axis principal stress ratio constant of COD stress independent of material (M = 0.5) constant pffiffiffi of COD stress independent of material ðB ¼ 1= 2Þ
Experimental apparatus used was an axial-torsion servohydraulic machine which can apply an axial load in combination with a torsional load. The maximum axial load and torsional load of the experimental apparatus are 49 kN and 196 N m, respectively. The specimen was gripped by separated collars to pull rod tightly. The axial and the shear strains were measured by a cantilever type extensometer with eddy current sensors which can measure axial and torsional displacement separately. The specimen was heated by an electric resistant furnace and the temperature distribution along the gage length of the specimen was within ±2 K. The number of cycles to failure was defined as the cycle of 5% axial stress range drop from the saturation. The variations of stress ranges with cycles are shown in Fig. 3 for Case 13. The stress range sharply dropped at the final stage in LFC process. So there is no significant effect on the number of cycle to failure if another percentage is employed other than 5%. One cycle was defined as a full straining for both axial and shear straining shown in Fig. 2. Following to this definition, Cases 3, 4 and 13 were counted as two cycles for a full loading along the strain path shown in Fig. 2 and the other cases were counted as one cycle. 3. Experimental results and discussion 3.1. Correlation of nonproportional fatigue lives with strain parameters ASME Code Case [14] defines an equivalent strain to express the nonproportional fatigue damage. The equivalent strain is originated from the Mises strain but it was modified to have a maximum value taking any times C and D in Eq. (1) (ASME strain range).
Fig. 1. Shape and dimensions of test specimen (mm).
De DeASME Deeq DeI DeI DeNP
DeNP Depath Dreq DrI DrI DrIpre DrIP
axial normal strain range strain range defined in ASME code case Mises’ equivalent strain range maximum principal strain range maximum strain range based on COD strain nonproportional strain range based on maximum principal strain nonproportional strain range based on COD strain pffiffiffi strain range based on strain path length on e c= 3 plot Mises’ equivalent stress range maximum principal stress range maximum stress range based on COD stress maximum principal stress range predicted maximum principal stress range under proportional loading
DeASME ¼ max
" 1=2 # 1 ðeC eD Þ2 þ ðcC cD Þ2 3
ð1Þ
eC, cC, eD and cD are the axial and shear strains at the times C and D to maximize the strain in the bracket. Table 1 summarizes the experimental data generated in this study and Fig. 4 correlates the number of cycles to failure with ASME strain range (DeASME). Numbers attached to the data in the figure are the case numbers of strain paths shown in Fig. 2. Fatigue lives in Case 0 are plotted with solid circles and the narrow lines in the figure are a factor of two band based on the solid bold line of Case 0 data. ASME strain range does not correlate the fatigue lives under fifteen strain paths within a factor of two scatter band. Especially, fatigue lives in Cases 1, 2, 10, 12, 13, and 14 are underestimated by more than a factor of two. The smallest fatigue lives are found in Cases 2, 12 and 14 that are smaller by a factor of 4 compared with the proportional data. To discuss the fatigue lives in respective strain paths, the ratio of the nonproportional fatigue life (Nf) to the proportional life in Case 0 (Nf Case 0) at 923 K in each strain path is shown with a bar graph by the case number, where the ratio at room temperature is also shown in Fig. 5 [15]. Cases 1 and 2 which are the cruciform straining along normal and shear strains have a ratio of 0.3 but Cases 3 and 4 having the same cruciform straining with rotating 45° give the values around 0.8 at 923 K. Considering that Cases 1–4 all have the similar ratio around 0.2 at room temperature, the larger ratios in Cases 3 and 4 at 923 K may result from the thermal activation. Comparing the ratio between Cases 5 and 9 shows the effect of step length in nonproportional loading. Since the ratio in Case 6 is about 1.5 times smaller than that in Case 5, even small steps decrease LCF life. The ratio decreases as the step length increases as shown in Cases 7 and 8. Tensile mean straining gives a smaller ratio than the compressive mean strain, Cases 8 and 9. The ratio decreases as the phase difference between normal and shear strain increases, which is understood from the comparison of the ratio between Cases 5, 11 and 12. Box loadings in Cases 10–13 all have the similar ratio whereas Case 12 has a slightly smaller ratio than the other box loadings. The circular straining, Case 14, shows the smallest ratio as Case 12. This indicates that the 90° phase difference gives the largest nonproportional fatigue damage. The general trend of the nonproportional data at 923 K shown in Fig. 5 does not significantly differ from that at room temperature except for Cases 3 and 4 and Cases 7 and 8 as previously mentioned, but the life ratios at room temperature are smaller than
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N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
Fig. 2. Fifteen strain paths used in experiments.
SUS304, T=923K
n(t) is schematically shown in Fig. 6. The angle (n(t)) in the figure has twice magnitude of the actual angle in a specimen. The maximum principal strain range (DeI) in nonproportional straining is defined as the maximum value of the bracket in Eq. (2). The maximum principal strain range did not correlate the proportional LCF lives within a small scatter band for Type 304 stainless steel [20] and Cr–Mo–V steel [21] at high temperatures. The strain gave too conservative estimates for torsion data compared with axial data beyond a factor of two scatter band. In this regard, the authors have proposed the equivalent strain based on crack opening displacement (COD strain) for correlating axial and torsion proportional low cycle fatigue data within a small scatter band [20–23]. The COD strain (eI ðtÞ) in nonproportional straining is expressed as a function of maximum principal strain (eI(t)) [15].
Axial stress range Δσ Shear stress range Δτ , MPa
600 500
5% drop
400 300 Nf =440
200 Δε =0.7% Δγ =1.21% Case 13
Δσ
100
Δτ
0 101
102
103
Number of cycles N
e ðtÞ ¼ bð2 /ðtÞÞm eI ðtÞ
Fig. 3. Definition of number of cycles to failure in Case 13.
those at 923 K in most cases. Therefore, the nonproportional loading at room temperature gives larger damage than at high temperature. Conversely, this may result from the thermal activation at 923 K which may reduce the additional hardening effect, strongly evident at room temperature. The main cause of the additional hardening of Type 304 stainless steel is severe interactions of slip planes resulting in smaller cell formation [16], [17]. The thermal activation at high temperatures may weaken the interactions and firm cell formation. Other than ASME strain range, couples of strain ranges were applied to correlate the nonproportional LCF lives. The strains applied are briefly explained below. In nonproportional straining, the maximum principal strain direction changes with time, so the definition of the maximum principal strain range should be clearly defined. Itoh et al. [15] proposed a definition of the maximum principal strain range (DeI) for nonproportional straining as, [18,19]
DeI ¼ max½eImax cos nðtÞeI ðtÞ
ð2Þ
eImax, eI(t) and n(t) are the maximum principal strain in a cycle, the maximum principal strain at time (t) and the angle between the eImax direction and the eI(t) direction, respectively. The polar figure indicating the relationship between the two strains and the angle
ð3Þ
b and m are the constants independent of materials and take the values of 1.83 and 1.66, respectively, which were confirmed numerically by FEM analysis using many constitutive relationships [15]. /(t) is the principal strain ratio at time (t) defined as e3/e1 for |e3(t)| |e1(t)| and e1/e3 for |e3(t)| > |e1(t)|, where e1(t) and e3(t) are maximum and minimum principal strains at time t respectively. Maximum COD strain range is defined similar to Eq. (2) as following.
DeI ¼ max½eImax cos nðtÞe ðtÞ
ð4Þ ⁄
⁄
e is the maximum value of e (t) in a cycle and e (t) is the COD strain to maximize the value in the bracket. Path strain range [4] is a strain range defined as a half-length of the strain path shown in Fig. 2 and is expressed by the following equation. Imax
Depath
1 ¼ 2
Z cycle
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d
e2 þ
c2 3
ð5Þ
Depath, e, c and d are the path strain range, the axial strain, the shear strain and the Leibniz’s notation for differentiation, respectively. Figs. 7–9 correlates the nonproportional data with the three strains mentioned above. The correlation with the maximum principal strain range underestimates about a half of the data by more
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N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143 Table 1 Summary of the test results. Failure cycle Nf, cycles
Axial strain range De (%)
Shear strain range Dc (%)
ASME strain range DeASME (%)
Principal strain range DeI (%)
COD strain range DeI (%)
Path strain range Depath (%)
Axial stress range Dr (MPa)
0 0 0 0 0 0
5450 7200 3000 1500 400 160
0.40 0.40 0.50 0.70 1.00 1.50
0.00 0.00 0.00 0.00 0.00 0.00
0.40 0.40 0.50 0.70 1.00 1.50
0.40 0.40 0.50 0.70 1.00 1.50
0.40 0.40 0.50 0.70 1.00 1.50
0.40 0.40 0.50 0.70 1.00 1.50
419 401 456 477 531 580
0 0 0 0 0 0
419 401 456 477 531 580
5 6 7 10 11 12 13 14
3000 3200 3700 1200 3000 3200 2600 1200
0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69
0.57 0.57 0.57 0.57 0.45 0.40 0.40 0.40
0.56 0.56 0.56 0.56 0.45 0.40 0.40 0.40
0.54 0.54 0.54 0.54 0.44 0.40 0.40 0.40
0.57 0.80 0.80 0.80 0.57 0.57 0.57 0.63
383 388 399 447 428 473 475 497
148 184 215 284 248 300 257 319
431 444 456 493 446 476 476 499
1 2 3 4 5 6 7 7 8 8 9 10 11 12 13 14
1100 1300 1400 1100 1720 1150 1000 750 1170 1100 710 500 900 830 1300 920
0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87
0.50 0.50 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.56 0.50 0.50 0.50
0.50 0.50 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.56 0.50 0.50 0.50
0.50 0.50 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.55 0.50 0.50 0.50
1.00 1.00 0.71 0.71 0.71 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.71 0.71 0.71 0.79
612 497 501 374 393 388 424 461 554 514 532 582 492 524 547 531
377 320 278 257 178 203 262 267 375 355 349 342 311 345 354 350
616 509 610 566 461 457 480 512 580 549 548 617 526 532 557 535
1 2 3 4 5 6 7 9 10 11 12 13 14
575 280 608 581 990 450 410 242 250 650 310 440 335
0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70
1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21
0.70 0.70 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.78 0.70 0.70 0.70
0.70 0.70 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.78 0.70 0.70 0.70
0.70 0.70 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.78 0.70 0.70 0.70
1.40 1.40 0.99 0.99 0.99 1.40 1.40 1.40 1.40 0.99 0.99 0.99 1.10
658 615 585 456 441 457 501 – 480 495 583 575 593
395 353 335 276 196 242 296 – 315 343 374 378 391
664 624 692 607 511 494 533 – 630 541 598 606 600
Strain path number (Case)
than a factor of two, Fig. 7. The most unconservatively estimated lives are the lives in Cases 2 and 12 at DeI = 0.7%. These lives are around one fifth of the Case 0 lives at the same principal strain range. The results of the correlation with the maximum principal strain are similar to those with ASME strain. The maximum COD strain range also underestimates a half of the data by more than a factor of two, Fig. 8. The correlation with the maximum COD strain range shows almost the same trend as that with the maximum principal strain. The path strain range appears to correlate the LCF data with a small scatter, Fig. 9, whereas some data are overestimated by more than a factor of two. In the correlation of the LCF data at room temperature with the path strain range, the path strain range overestimated some data and also underestimates other data by more than a factor of two [15]. This means that the path strain range does not cover the temperature dependency of the data correlation. The question of the path strain range is that it predicts the same LCF life for the strain waveform that has the same path length, but the LCF life in experiments varied with step length having the same path length, which is understood from the different LCF live in Cases 6, 7 and 8.
Shear stress range Ds (MPa)
Principal stress range DrI (MPa)
3.2. Correlation of nonproportional fatigue lives with stress parameters LCF lives under nonproportional straining are generally smaller than those of under proportional loading for stainless steels. The smaller fatigue lives are attributed to the significant additional hardening due to nonproportional loading. Therefore, it is meaningful to discuss the applicability of stress parameter to the LCF lives. The stress parameters discussed are the maximum principal stress range, the equivalent stress range based on crack opening displacement (COD stress) and the Mises stress range. The maximum principal stress range is defined similarly to the maximum principal strain as [18,19],
DrI ¼ max½rImax cos nðtÞrI ðtÞ
ð6Þ
rImax and rI(t) are the maximum principal stress in a cycle and the principal stress at time t, respectively. n(t) is the angle between the rImax direction and the rI(t) direction. The COD stress is also similarly defined as [24–27],
r ðtÞ ¼ Bð2 kðtÞÞM rI ðtÞ
ð7Þ
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N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
SUS304, T=923K Factor of 2
1.0
0.5
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0 0.1 102
103
104
Maximum principal strain range Δε I , %
ASME strain range ΔεASME, %
SUS304, T=923K
Number of cycle to failure Nf
1.0
0.5 Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0.1 102
103
104
Number of cycle to failure N f Fig. 7. Correlation of nonproportional fatigue lives with maximum principal strain range.
1.4
Room Temp.
1.0 0.8 0.6 0.4 0.2 0
SUS304, T=923K
T=923K
1.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Case Fig. 5. Ratio pffiffiffi of nonproportional fatigue life to push–pull fatigue life (Case 0) at De ¼ Dc= 3 ¼ 0:5% for the fourteen cases.
Maximum COD strain range Δε I*, %
Ratio of non-proportional fatigue life to proportional fatigue life Nf /Nf Case 0
Fig. 4. Correlation of nonproportional fatigue lives with ASME strain range.
Factor of 2
Factor of 2 1.0
0.5 Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0.1 102
103
104
Number of cycle to failure Nf Fig. 8. Correlation of nonproportional fatigue lives with maximum COD strain range.
SUS304, T=923K
Fig. 6. Definitions of the maximum principal strain, the principal strain range and the angle of n(t) in a polar figure.
B and M arepthe ffiffiffi constants independent of materials and take the values of 1= 2 and 0.5, respectively, which were derived numerically by FEM analysis using many constitutive relationships as well as the COD strain. kðtÞ is the maximum principal stress ratio similarly defined as for the case of the maximum principal strain ratio, replacing the principal strains by the principal stresses. The
Path strain range Δεpath, %
1.0
0.5
Factor of 2
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0.1 102
103 Number of cycle to failure Nf
104
Fig. 9. Correlation of nonproportional fatigue lives with path strain range.
maximum COD stress range is defined as a function of the COD stress and n(t) as
DrI ¼ max½rmax cos nðtÞr ðtÞ
ð8Þ
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N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
Mises stress range is defined as Eq. (9) similar to ASME strain range as
ð9Þ
where rC, sC, rD and sD are the axial and shear stresses at times C and D to maximize the strain in the bracket. The correlation of LCF lives with the three stresses is shown in Figs. 10–12. The maximum principal stress gives almost the same correlation as the maximum COD stress, Figs. 10 and 11. The two stress parameters overestimate the nonproportional fatigue lives for some strain histories by more than a factor of two. The scatter becomes worse in the correlation with the Mises stress, Fig. 12. Mises stress also overestimates the nonproportional fatigue lives for most of the strain histories. The unconservative estimates in the correlation of the LCF lives with the strain parameters like the Mises strain and the conservative estimates with the stress parameters like the Mises stress indicate the followings. In the correlation with the strain parameters, the material yielded an additional hardening under nonproportional loadings and larger stresses were applied to the specimen compared with proportional loading resulting in smaller fatigue lives under nonproportional loading. However, in the correlation with the stress parameters, the correlation became too conservative that means that the LCF lives were not reduced as the amount of additional hardening under nonproportional loading. Therefore, the correlation of the LCF lives with an energy parameter was examined where the product of the maximum principal strain and stress is used here as a typical parameter. The results of the correlation are shown in Fig. 13. The correlation shown in the figure is quite improved from that with the stress parameters and most of the data are collapsed into a factor of two scatter band, which means that the energy parameter is the most suitable parameter to describe the LCF damage under nonproportional loading. The similar trend of the correlation was also found at room temperature [15]. However, the energy parameter that needs both strain and stress is not so convenient in practical design of components and structures. 3.3. Correlation of LCF fatigue lives with nonproportional strain
Maximum principal stress range Δσ I , MPa
As shown in Fig. 13, the energy parameter correlates the nonproportional lives with a small scatter. Energy parameters, however, need both the stress and strain ranges for the estimates of LCF lives. It is not easy to obtain cyclic stress ranges under nonproportional loadings utilizing a constitutive modeling because the constitutive modeling for nonproportional loading is still under
SUS304, T=923K Factor of 2
COD stress range Δσ I* , MPa
1=2
Factor of 2 800
400
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0
102
103
104
Number of cycle to failure N f Fig. 11. Correlation of nonproportional fatigue lives with COD stress range.
SUS304, T=923K Mises stress range Δσ eq , MPa
Dreq ¼ max½fðrC rD Þ2 þ 3ðsC sD Þ2 g
SUS304, T=923K
Factor of 2 800
400
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0
10
2
103
104
Number of cycle to failure N f Fig. 12. Correlation of nonproportional fatigue lives with Mises stress range.
research process or requires many material constants in the reliable models proposed. Therefore, a parameter written with only strain is preferable in practical design for applications. Itoh et al. [15] proposed the nonproportional strain (DeNP) which is an equivalent strain taking account of additional hardening of materials under nonproportional loadings. They applied the strain to the nonproportional LCF lives of Type 304 stainless steel fatigued at room temperature and demonstrated the applicability of the strain. Their application, however, limited to the LCF data at room temperature and the applicability of the parameter to LCF lives at high temperatures is still an open question. This paper examines the applicability of the nonproportional strain to the high temperature LCF data presented in this paper. The nonproportional strain is written as,
800
DeNP ¼ ð1 þ afNP ÞDeI
400
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
0 102
103 Number of cycle to failure Nf
104
Fig. 10. Correlation of nonproportional fatigue lives with maximum principal stress range.
ð10Þ
a is the material constant which expresses the amount of additional hardening under nonproportional loading and is calculated as the ratio of stress range under circular straining to that in Case 0 in a r1 e1 plot at the same strain amplitude. The value of a for Type 304 stainless steel becomes 0.3 at 923 K whereas it was 0.9 at room temperature [15], which means that the amount of the additional hardening under circular path at 923 K is 1/3 of that at room temperature. Thermal activation may reduce the amount by thermally assisted recovery system of dislocations [16,28]. fNP in Eq. (10) is a nonproportional factor that expresses the intensity of nonproportionality. The experimental results in Table 1 and Fig. 5 tell us that
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Nonproportional strain range ΔεNP, %
Energy parameter Δε I Δσ I , MPa
SUS304, T=923K 10
Factor of 2
5
Δε = 0.7% Δε = 0.5% Δε = 0.4% Case 0
1 103
102
104
SUS304, T=923K Factor of 2 1.0
0.5 Δε = 0.7% Δε = 0.5% Δε = 0.4%
0.1 102
The nonproportional factor should be a function of angle n(t) from the first factor and should have an integral form from the second factor. Therefore, the nonproportional factor is expressed as, T
ðj sin nðtÞjeI ðtÞÞdt
ð11Þ
¼ ð1 þ a
f NP ÞD I
e
ð12Þ
fNP is nonproportional factor based on the COD strain and is expressed as,
fNP
1:66 ¼ T eImax
Factor of 2 1.0
0.5 Δε = 0.7% Δε = 0.5% Δε = 0.4%
Δε
Case 0
Z
T
ðj sin nðtÞje ðtÞÞdt
ð13Þ
0
Figs. 14 and 15 correlates the LCF lives generated in this study with the nonproportional strains shown in Eqs. (10) and (12), respectively. The correlations are quite satisfactory and all the data except only a couple of data are collapsed within a factor of two scatter band. There is no distinguished difference in the correlation between the nonproportional strain and the nonproportional strain based on COD, because the pure shear strain path is not included in the strain path shown in Fig. 2. The difference in the correlation would be more pronounced if the shear strain path is included in this paper. The principal strain give a trend to overestimate torsion proportional LCF lives for Type 304 stainless steel if the estimate is based on push–pull LCF lives [20,21]. To improve the estimate for torsion LCF lives, the equivalent strain based on COD was proposed [23]. Considering the successful correlation with the energy parameter and the nonproportional strains expressed with Eqs. (10) and (12), the nonproportional factor will correspond with the amount of additional hardening under nonproportional loading. To examine the relationship between the nonproportional factor and the additional hardening, the stress range calculated by the following
0.1
*=
(1+α fNP*) Δε I*
102
104
103
Number of cycle to failure Nf Fig. 15. Correlation of nonproportional fatigue lives with nonproportional COD strain.
equation is compared with the experimental stress range. The equation is the same equation as the nonproportional strain but the principal strain range is substituted by the principal stress range.
DrIpre ¼ ð1 þ afNP ÞDrIP
ð14Þ
DrIP is the maximum principal stress range under the proportional loading. Constant a has the value of 0.3 as mentioned above. Fig. 16 shows the relationship between the predicted maximum principal stress and experimental stress ranges under nonproportional loading. The predicted stress ranges by Eq. (14) are in good Predicted stress range Δσ I pre, MPa
De
SUS304, T=923K
0
T is the time per cycle and the constant 1.57 is determined so as to the value of fNP being unity for the circular strain path. fNP takes null value for proportional loading and positive values from 0 to 1 for nonproportional loading. The nonproportional strain and nonproportional factor described above are based on the principal strain. Those based on the COD strain are also definable as follows. NP
Fig. 14. Correlation of nonproportional fatigue lives with nonproportional strain.
Nonproportional COD strain range ΔεNP*, %
the nonproportional factor should take account of the following factors. Fatigue life reduces more significantly when the principal strain direction changes at larger angle. Fatigue life reduces more significantly when the path length is larger after the principal strain direction change.
1:57 T eImax
104
Number of cycle to failure Nf
Fig. 13. Correlation of nonproportional fatigue lives with energy parameter.
fNP ¼
= (1+α fNP) Δε I
103
Number of cycle to failure Nf
Z
Δε
Case 0
800
α =0.3
20%
400
Δε = 0.7% Δε = 0.5% Δε = 0.4%
0
400
800
Experimental stress range Δσ I , MPa Fig. 16. Comparison of predicted stress ranges by Eq. (14) with experimental stress ranges.
N. Hamada et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 136–143
agreement with the experimental stress ranges within a factor of 1.20, so the nonproportional factor shown in Eq. (14) properly estimates the amount of the additional hardening under nonproportional loadings at 923 K. 4. Conclusions (1) Nonproportional straining drastically decreased low cycle fatigue lives of Type 304 stainless steel. Nonproportional low cycle fatigue lives depended on the strain history. Most damaging nonproportional strain path was the strain path with 90° phase difference with triangular and sinusoidal strain waves. (2) The equivalent strain range defined in ASME Code Case and the maximum principal strain underestimated the nonproportional fatigue lives in some strain histories. The equivalent strain based on the path length overestimated the nonproportional fatigue lives in some strain histories. The maximum scatter in the correlation with these strain parameters was a factor of four that is smaller the scatter of the data at room temperature previously reported. (3) Constant a that expresses the amount of additional hardening was 0.3 at 923 K in this study and was smaller than 0.9 at room temperature. Thermal activation was expected to reduce the amount of additional hardening at high temperature. (4) The energy parameter of the product of principal strain and stress correlated the nonproportional fatigue lives within a factor of two scatter band. (5) Nonproportional fatigue strains, DeNP and DeNP , which take account of the amount of additional hardening were suitable parameters to correlate the nonproportional fatigue lives of Type 304 stainless steel at 923 K.
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