Scripta Materialia 47 (2002) 225–229 www.actamat-journals.com A microstructural model for the prediction of high cycle fatigue life J.S. Park a, S.H. Park b, C.S. Lee a a,* Department of Materials Science and Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, South Korea b POSCO Technical Research Laboratories, Pohang 790-785, South Korea Received 13 February 2002; accepted 2 April 2002 Abstract In this study, a model predicting the high cycle fatigue life was developed in relation to the microstructural variable, especially, the grain size. The concept of small crack propagation theory was adopted to reflect the influence of the microstructural variables. Reasonable agreement was found between the experimental data and the predicted curve. Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: High cycle fatigue; Life prediction; Small crack theory; Microstructure 1. Introduction High cycle fatigue properties are generally represented by the S–N (stress–life) curves. For obtaining a single S–N curve, it usually takes long time and tedious efforts so that there have been several attempts to develop the appropriate fatigue life prediction models. Most of earlier models based on the continuum damage mechanics contain various kinds of fatigue damage parameters [1]. Though the earlier models utilizing the macrovariables such as strain energy and elastic modulus are useful for predicting high cycle fatigue life in a macroscopic sense, they are lacking in the influence of microstructures. Previous works on fatigue * Corresponding author. Tel.: +82-562-279-2141; fax: +82562-279-2399. E-mail address: cslee@postech.ac.kr (C.S. Lee). behavior of materials [2,3] have evidenced that the fatigue life is greatly influenced by the microstructural variables such as the grain size, morphology, size and distribution of second phase particles. It is, therefore strongly required to develop a model reflecting the influence of microstructural parameters in the high cycle fatigue. The small crack theory, originally developed by Miller et al., has received much attention as a candidate to achieve this goal for its unique concept of microstructural barriers. However, previous studies on the small crack theory have been focused on the propagation behavior [4,5] and there have been few attempts to introduce the small crack concept into the HCF prediction model. Therefore, in this study, the concept of small crack theory was utilized not only to estimate the overall life spent during high cycle fatigue but also to incorporate the microstructural influence into 1359-6462/02/$ - see front matter Ó 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 6 2 ( 0 2 ) 0 0 1 1 5 - X 226 J.S. Park et al. / Scripta Materialia 47 (2002) 225–229 the model. Also, the effect of crack arrest on HCF life was considered quantitatively based on the concept of strain accumulation. Particular interest was given to develop the life prediction model with respect to the grain size. 2. Development of life prediction model for high cycle fatigue It is well known that the fatigue crack initiation takes most of the high cycle fatigue life. In the small crack theory, the fatigue crack initiation process is thought to be composed of three stages, such as crack nucleation, microcrack propagation and propagation through a barrier [6]. However, the crack nucleation stage is considered negligible in the entire life, since it has been reported that the crack nucleation occurs within small cycles [4,7]. Therefore, the fatigue crack initiation stage can be thought as the sum of each period for microcrack propagation and propagation through a barrier. Also, there were many works reporting that most of the life (about 70–90%) is spent for propagating the first grain after crack nucleation [4,7]. As such, in this model, it is assumed that the entire life is proportional to the cycles spent in one grain and one barrier, as shown in Fig. 1. If the barrier is a grain boundary, N3f can be neglected because the width of grain boundary is very small. Therefore, the entire life can be expressed by the sum of N1f , the propagation period through a grain and N2f , the arrest period at grain boundary. As a first approximation to construct the N1f , small crack Fig. 1. Schematic diagram showing the concept of modeling: 1 grain þ 1 barrier. growth law (Eq. (1)) suggested by Miller [5] was adopted. da=dN ¼ CðDcÞa ðd  aÞ ð1Þ where Dc is the resolved shear strain range, d is the grain size, a is the crack length, and C and a are material constants. The Dc in the above equation can be converted to the resolved shear stress range (Ds) using a following relationship, n Ds ¼ KðDcÞ , as shown in Eq. (2). m da=dN ¼ AðDsÞ ðd  aÞ ð2Þ where A and m are material constants. Since the microstructural variables influence on the resolved shear stress, modification for Ds can be made by considering the microstructural effect on the driving force as shown in Eq. (3). Ds ¼ Dsapp  Dsf  Dsb ð3Þ where Dsapp , Dsf and Dsb indicate the applied stress, the friction stress due to solute atom and the friction stress due to the back stress arising from the dislocation pile-up, respectively. Here, the Dsb can be expressed by Eq. (4), where d is the grain size and Cb is a constant [8,9]. Dsapp Dsb ¼ Cb pffiffiffi d ð4Þ When substituting Eqs. (3) and (4) into Eq. (2), the modified small crack growth rate can be obtained by Eq. (5).  m da Dsapp ¼ A Dsapp  Dsf  Cb pffiffiffi ðd  aÞ ð5Þ dN d Integrating from a ¼ 0 to acr (the length of crack when the small crack is arrested in front of a barrier) will result N1f (the cycles required to propagate a crack through a grain) with the form of Eq. (6).   d log d  acr m ð6Þ N1f ¼  Dsapp A Dsapp  Dsf  Cb pffiffiffi d Cumulative strain criterion has been used to calculate N2f , the cycles spent for the arrest at a barrier. When the crack is arrested in front of a barrier, the crack does not advance with continued J.S. Park et al. / Scripta Materialia 47 (2002) 225–229 cycles. However, the strain resulted from the pileup dislocations will be accumulated during the cyclic loading. With continued cycling, the accumulated strain reaches to a critical value, and then the barrier is broken. In general, when s is applied on the slip band having the length of L, the pile-up stress sp can be expressed by Eq. (7) where m is Poisson ratio and l is the shear modulus [10]. sp ¼ ð1  mÞps2 L 4l ð7Þ During cyclic loading, Eq. (8) can be obtained. 2 Dsp ¼ ð1  mÞpðDsÞ L 4l ð8Þ Using Eqs. (3) and (4), Eq. (8) can be expressed as follow. 2  ð1  mÞp Dsapp Dsp ¼ Dsapp  Dsf  Cb pffiffiffi ðd  acr Þ 4l d ð9Þ Using Eq. (9) and the cyclic stress–strain relation, the strain accumulated on the barrier in a cycle, Dc can be obtained. If the Dc is the critical strain required to overcome the barrier, then N2f , the cycles spent at crack arrest can be expressed as Dc divided by Dc. N2f ¼ 2=n  Dc Dsapp ½d  acr ¼ f Dsapp  Dsf  Cb pffiffiffi Dc d 1=n where f ¼ Dc   ð1  mÞp 4Kl 1=n ð10Þ The total life, the sum of N1f and N2f , can be given by Eq. (11).  d  log da cr NT ¼  m Ds ffiffi A Dsapp  Dsf  Cb papp d  where Dsapp þ f Dsapp  Dsf  Cb pffiffiffi d f ¼ Dc  ð1  mÞp 4Kl 1=n 2=n ½d  acr 1=n ð11Þ 227 3. Experimental procedure A stress–life (S–N ) test was conducted using steel specimens with various grain sizes. The chemical composition of the tested steel includes 0.17C, 0.94Mn in wt.% and the balanced Fe. To obtain the specimens with various grain sizes, as received specimens (average grain size 4.6 lm) were heat-treated at 800 and 1000 °C for 30 min, respectively, followed by air cooling. The average grain sizes of the heat-treated steels were 9.3 and 13.3 lm respectively. Hourglass specimens with the length of 160 mm and the smallest section area of 30 mm2 were machined for the HCF test. Before mechanical testing the specimens were polished using abrasive papers up to no. 2000. S–N test was performed under the uni-axial loading condition. The sine wave form was adopted with the stress ratio (R) of zero and the frequency of 30 Hz. 4. Verification of the model with experimental results Fig. 2 shows the experimental S–N data for the steels with grain size of 4.6 and 9.3 lm. For each steel, the unknown constants (Dsf , Cb , m, acr , A, f ) in Eq. (11) were determined by some reasonable calculation and iteration method, as follows. Here, the friction force (Dsf ) arising from the presence of the solute atoms in the grain interior was assumed to be constant for the steels with the same chemical composition. The value of Dsf was obtained as 82.1 MPa by an empirical equation developed by Pickering [11] for carbon steels, as shown in Eq. (12). rfriction ¼ 15:4½3:5 þ 2:1ð%MnÞ þ 5:4ð%SiÞ ð12Þ Wilson and Bate [12] proposed a method to estimate the back stress by calculating half the difference between the lower yield stresses observed in Bauschinger test. That is, rb 1=2ðryf  ryr Þ ð13Þ where ryf and ryr are yield stresses in forward and reversed loading respectively. To obtain the correct Cb value, Baushinger test was carried out in this study under the strain rate of 0.001 s1 for the 228 J.S. Park et al. / Scripta Materialia 47 (2002) 225–229 Fig. 2. S–N curves for the steels with grain size of (a) 4.6 and (b) 9.3 lm. each steel. The applied strains were 0.04, 0.08 and 0.12. Fig. 3 shows the variation of back stress with respect to the value of Drapp =d 1=2 . From Fig. 4 and Eq. (4), the value of Cb was determined as 0.368 lm1=2 . The m value, which was related to the propagation velocity of small crack through the grain interior and was calculated from the work of Ibrahim and Miller [7]. The n value (strain hardening exponent) was determined from the results of tensile test for each steel. Table 1 summarizes the values of unknown constants in Eq. (11). Using the known constants of steels with the grain sizes of 4.6 and 9.3 lm, the predicted life for specimen with the grain size of 13.3 lm was calculated. Calculation was based on the assumption that the iterated constants (acr , A and f) show the linear relationship with the grain size. The calculated and measured constants for the steel with the grain size of 13.3 lm were listed in Table 2. After substituting the constants into Eq. (11), the predicted curve for the steel with grain size of 13.3 lm was compared with the experimental S–N data. As shown in Fig. 4, reasonable agreement was found between the experimental data and the predicted curve. However, the deviation was noted near the short fatigue life region (<5 105 cycles), where high stress range was applied. It was thought that the deviation was partly resulted from the assumption that several constants (acr , A and f) vary linearly with the variation of grain size. Also, as the fatigue life reduces to the shorter value, the Fig. 3. The variation of back stress with the value of applied stress and grain size. Fig. 4. A plot showing the experimental data and predicted curve for the steel with grain size of 13.3 lm. 229 J.S. Park et al. / Scripta Materialia 47 (2002) 225–229 Table 1 Constants of Eq. (11) for the steels with the grain size of 4.6 and 9.3 lm Steels Calculated As received 800 °C Measured Iterated 1031 Dsf Cb m d n acr A 84.3 84.3 0.369 0.369 10.98 10.98 4.6 9.3 0.182 0.235 4.53 9.10 1.39 1.35 1022 f 45.5 19.2 Table 2 Constants of Eq. (11) for the steels with the grain size of 13.3 lm Steel 1000 °C Calculated (identical values in Table 1) Measured Calculated from Table 1 (assume linear relation with GS) Dsf Cb m d n acr A 84.3 0.369 10.98 13.3 0.267 12.89 1.29 portion of crack propagation becomes more significant, which was considered negligible in this study, since our interest was focused on the regime of high cycle fatigue. Therefore, our model using the concept of small crack theory is valid in the high cycle fatigue region and another model for the prediction of fatigue life at the high stress range is needed. 1031 f 1022 1.2 Acknowledgements This work was supported by the Korean Ministry of Education under the grant no. 2001-041E00432, and also partly supported by Pohang Steel Company, Korea. References 5. Summary This study aims to develop a model predicting the high cycle fatigue life in relation to the microstructural variable, especially, the grain size. The concept of small crack propagation suggested by Miller was adopted and modified to reflect the influence of the microstructural variables. It was assumed that the whole fatigue crack initiation process can be represented by the microcrack propagation consisting of two stages, the propagation through one grain and the arrest at a barrier. Reasonable agreement was found between the experimental data and the predicted curve. [1] Fatemi A, Yang L. Int J Fatigue 1998;20:9. [2] Landgraf RW. In: Meshii M, editor. Fatigue and microstructures. Ohio: ASM; 1979. p. 439. [3] Park KJ, Lee CS. Scripta Metall Mater 1996;34:215. [4] Newmann Jr JC. Prog Aerospace Sci 1998;34:347. [5] Miller KJ. Fatigue Fract Engng Mater Struct 1987;10:93. [6] Kujawski D, Ellyin F. In: Short fatigue cracks. London: Mechanical Engineering Publications Ltd.; 1992. p. 3915. [7] Ibrahim MFE, Miller KJ. Fatigue Engng Mater Struct 1980;2:351. [8] Lin MR, Fine ME, Mura T. Acta Metall 1986;34:619. [9] Johnston TL, Feltner CE. Metall Trans A 1970;1:1161. [10] Hirth JP, Loathe J. In: Theory of dislocations. New York: John Wiley & Sons, Inc.; 1982. p. 771. [11] Leslie William C. In: The Physical Metallurgy of Steels. New York: Hemisphere Corp.; 1981. p. 147. [12] Wilson DV, Bate PS. Acta Metall 1986;34:1107.