International Journal of Fatigue

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1 International Journal of Fatigue 31 (9) Contents lists available at ScienceDirect International Journal of Fatigue journal homeage: The effect of load ratio on fatigue life and crack roagation behavior of an extruded magnesium alloy S. Ishihara a, *, A.J. McEvily b, M. Sato a, K. Taniguchi a, T. Goshima a a Deartment of Mechanical Engineering, University of Toyama, 319 Gofuku, Toyama , Jaan b Institute of Materials Science, University of Connecticut, CT, USA article info abstract Article history: Received 12 Setember 8 Received in revised form 26 January 9 Acceted 17 February 9 Available online 27 February 9 Keywords: Fatigue Magnesium alloy Stress ratio S N curves Crack roagation Fatigue exeriments were carried out in laboratory air using an extruded magnesium alloy, AZ31, to investigate the effect of load ratio on the fatigue life and crack roagation behavior. The crack roagation behavior was analyzed using a modified linear elastic fracture mechanics arameter, M. The relation crack roagation rate vs. M arameter was found to be useful in redicting fatigue lives at different R ratios. Good agreement between the estimated and the exerimental results at each stress ratio was obtained. Ó 9 Elsevier Ltd. All rights reserved. 1. Introduction Magnesium alloys are candites as structural materials due to their high secific strength and good ming caacity [1]. For examle, when they are used as car arts, weight saving of the car body and reduction of fuel consumtion can be exected, since they are the lightest among metals. Further, a better driving feeling can be obtained because oscillation of the car body can be reduced due to their good ming caacity. However, before utilizing magnesium alloys for machines and structures, a determination of fatigue lives and crack roagation behavior of the alloys at different load ratios is required. There already have been a number of investigations concerning the fatigue roerties of magnesium alloys. For examle, Goodenberger and Stehens [2] have rovided basic information concerning the fatigue behavior of the cast magnesium alloy, AZ91E-T6 in laboratory air. Hilert and Wagner [3] have carried out exeriments to determine the degree of degration in fatigue resistance due to corrosion of the AZ8 magnesium alloy as a function of various surface treatments such as electro-olishing, grinding, machining and shot eening. Eisenmeier et al. [] have studied the fatigue behavior of AZ91 magnesium alloy that had been rocessed using a vacuum die casting method. They reorted that * Corresonding author. Tel./fax: address: ishi@eng.u-toyama.ac.j (S. Ishihara). the fatigue cracks were initiated at casting defects in the material, and that the crack growth behavior was influenced by the microstructure of the material. Shih et al. [5] erformed fatigue exeriments on the extruded magnesium alloy AZ61A, and reorted that fatigue cracks were initiated at surface or near-surface inclusions, and that the initial crack growth behavior was affected by the microstructure. Until now, emirical relations, Gerber and Goodman diagrams have been utilized for evaluation of the effect of mean stress on fatigue lives, however, theoretical methods to analyze the effect of R ratios on the fatigue lives are very few and limited. McEvily et al. [6] roosed a modified linear elastic fracture mechanics aroach. This method was originally develoed to analyze both the long through and short fatigue crack roagation behavior. Then the method was shown to be able to redict crack growth behavior under multile two-ste loading condition [7] and in different materials [8 9]. In the resent study, fatigue tests were conducted on an extruded magnesium alloy to investigate fatigue lives (S N curves) and crack roagation behavior under different stress ratios, R. The modified linear elastic fracture mechanics aroach was alied to analyze the crack roagation behavior of the extruded magnesium alloy AZ31 and to redict S N curves at different R ratios. Comarison between exerimental and calculated results reveals that the aroach roosed by McEvily et al. [6] is effective and useful in analyzing the short fatigue crack roagation behavior at different R ratios including negative R ratios /$ - see front matter Ó 9 Elsevier Ltd. All rights reserved. doi:1.116/j.ijfatigue.9.2.3

2 S. Ishihara et al. / International Journal of Fatigue 31 (9) Material, secimens and exerimental rocedures 2.1. Material The extruded magnesium alloy AZ31 was used in this investigation. Its chemical comosition is listed in Table 1. A cylindrical billet with a diameter of mm was extruded into a round bar with a diameter of 7 mm, an extrusion ratio of 1. Table 2 lists the extrusion conditions under which the alloy was rocessed. Fig. 1 shows the microstructure of the material used in the resent study. The average grain diameter of the material after extrusion was 9 lm, and its asect ratio was about Exerimental rocedures Secimens and fatigue testing machine In the resent study two tyes of fatigue tests were used. Rotating bending fatigue tests (R = 1) at 3 Hz and axial-load fatigue tests at 1 Hz were conducted in laboratory air at room temerature at stress ratios,, 1, and 2. Fig. 2a and b shows the secimen shaes and dimensions of the rotating bending fatigue tests and axial fatigue tests, resectively. For the rotating bending fatigue tests, round bar secimens with a minimum diameter of 5.6 mm were used (stress concentration factor 1.). For the axial-load tests, round bar secimens with a minimum diameter of mm were used (stress concentration factor 1.6). The secimen surfaces were olished to a mirror-like finish using emery aers and diamond aste rior to fatigue tests to eliminate any effect of the roughness of the secimen surface on the fatigue results, and to facilitate the observations of crack roagation on the secimen surfaces. Monotonic tension and comression tests were also conducted using a similar secimen as shown in Fig. 1b. Its gauge length was 5 mm Fatigue crack roagation tests Short surface crack. The roagation rates of the short surface cracks that initiated and roagated on the secimen surfaces during both the rotating bending fatigue rocess and the axial fatigue rocess were investigated using the relication technique. Fatigue tests were interruted at a constant interval during the fatigue rocess to obtain the relicas of the secimen surfaces. Crack lengths recorded on the relicas were determined with an otical microscoe at a magnification of. For calculations of the stress intensity factor, K, the following exression was used [1] K ¼ Yr ffiffiffiffiffiffi a ; ð1þ where r is stress, a is a half crack length, and Y is a crack-shae correction factor with a value of.73 [11] given by assuming that a crack-shae is semi-circular Long through crack. Besides the crack roagation behavior of the short surface crack, crack roagation behavior of the long Table 1 Chemical comositions of the material used (wt%). Al Zn Mn Fe Si Cu Ni Table 2 Extrusion conditions. Extrusion temerature Extrusion seed Extrusion ratio 573 K 2 m/min 1 Fig. 1. Microstructure of the material AZ31 used in the resent study. through crack was also investigated using the round tye comact tension secimens as shown in Fig. 3. Crack lengths were measured using a travelling microscoe at a magnification of. The following exression [12] was used for calculations of the stress intensity factor, K K ¼ P tw 1=2 ð2 þ aþð:76 þ :8a 11:58a2 þ 11:3a 3 :8a Þ ð1 aþ 3=2 ; ð2þ where a = a/w, a is a crack length, and W is a secimen width. Crack oening stress intensity factor, K o during a loading cycle was determined based on the unloading elastic comliance method [13] by affixing strain gauges ahead of the crack ti. 3. Exerimental results 3.1. Monotonic tension and comression tests Fig. shows the stress vs. strain relationshis for both tension and comression tests. As can be seen from the figure, the yield strength under comression is lower than that for tension. Table 3 lists the mechanical roerties of the material under tension and comression tests. In the comression tests, the secimen (.5 mm in diameter and 8 mm in gauge length) used failed due to buckling at the maximum comressive load. A similar result was also reorted by Muller and Muller [1]. They suggested [1] that twinning is concerned with the above difference between the yield strength in tension and the one in comression, but a detailed mechanism is unknown at resent Fatigue tests S N curves Fig. 5a and b shows S N curves for the different R ratios,, 1 and 2. Those results were obtained from the axial fatigue tests and also from the rotating bending (RB) fatigue tests. In Fig. 5a and b, the maximum stress r max and stress amlitude r a are taken as the vertical axes, resectively. Similar S N curves between the axial fatigue tests and the rotating bending fatigue tests are seen. S N curves for R = 1 and R = 2 bend at cycles. From Fig. 5a, the maximum stress r max at a constant fatigue life decreases with a decrease in the value of R. However, in Fig. 5b, the reverse is true, i.e., the stress amlitude r a at a constant fatigue life increases with a decrease in the R value. The relationshi between stress amlitude r a and mean stress r m was drawn using the ta from Fig. 5. Fig. 6 shows the results.

3 179 S. Ishihara et al. / International Journal of Fatigue 31 (9) (a) Rotating bending fatigue tests (b) Axial fatigue tests Fig. 2. Secimen shaes and dimensions used in the resent study (mm). Fig. 3. Shae and dimensions of the round tye CT secimen (mm). Stress [MPa] Strain [%] Tension Comression Fig.. Stress strain behavior of extruded AZ31 in tension and in comression. Table 3 Mechanical roerties. Materials AZ31 Yield strength (MPa) Tensile strength (MPa) Tension Comression a a Buckling. Elongation (%) We can see that the exerimental ta aear fit better with the Gerber diagram [15] reresented by Eq. (3) rather than with the modified Goodman diagram [15] reresented by Eq. (). To extend this finding, the ta for the magnesium alloy AZEM223 and ZE1A obtained by Hotta and Saruki [16] were also lotted in Fig. 6b, where the vertical and horizontal axes are normalized by the tensile strength of the material, r B. Uon considering all ta it is not ossible to state that the relation r a vs. r m for the extruded magnesium alloy is categorically described by either the Gerber diagram or the modified Goodman diagram in view of the scatter. However our exerimental and theoretical results (Fig. 11) fit better with the Gerber diagram than with the modified Goodman diagram. To confirm this oint, further exerimental studies are needed. r a ¼ r w 1 r! 2 m ð3þ r B r a ¼ r w 1 r m ðþ r B Crack growth behavior Short surface crack growth behavior under rotating bending fatigue rocess, R = 1. Successive observations of the secimens surface during the fatigue rocess were erformed to investigate the crack roagation behavior under the rotating bending fatigue tests of R = 1. The crack length, 2a, as a function of number of cycles N at stress amlitude of 15 MPa is shown in Fig. 7 for the two different cracks (solid and oen circles). The crack reresented by the solid mark coalesced with other crack at about 5, cycles, showing a sudden increase in crack length at the oint. As can be seen from the figure, cracks are initiated at an early fatigue stage, i.e., 5 1% of fatigue life. This result means that the total of fatigue life N f can be aroximated as the crack roagation life, N. Maximum Stress σ max [MPa] Exeriment (;Axial) (R = -1;Axial) (R = -1;RB) (R = -2;Axial) Stress amlitude σ a [MPa] Number of cycles to failure N f [cycle] Number of cycles to failure N f [cycle] (a) σmax vs. N f (b) σa vs. N f Fig. 5. S N curves for different R ratios,, 1, and 2. Exeriment (;Axial) (R = -1;Axial) (R = -1;RB) (R = -2;Axial)

4 S. Ishihara et al. / International Journal of Fatigue 31 (9) (a) Stress amlitude σ a [MPa] 3 R = cycles R = -1 Goodman Gerber Mean stress σm [MPa] R =.5 (b) a / B σ σ Exeriment } Hotta's ta [12] Goodman -1 1 σ σ m / B Gerber Fig. 6. The relation between stress amlitude r a and mean stress r m. In Fig. 6b, the ta reorted by Hotta [16] are also lotted. 5 [ 1 - ] Crack length 2a [m] Exeriment R = -1 R1 σ a = 15 MPa Number of cycles N [cycle] [ 1 5 ] Fig. 7. Crack roagation curves under a stress amlitude of 15 MPa at R = 1. Oen and solid circles in these figures indicate two different cracks observed on the same secimen Long through crack growth behavior,. Fig. 8a shows the relationshi between the rate of crack roagation /dn and stress intensity factor range DK for the long through-thickness crack. In the figure, the relation /dn vs. DK eff is also lotted, where DK eff is the effective range of the stress intensity factor, i.e., K max K o, where K o is the stress intensity factor at the crack oening level. The crack closure behavior was evaluated by using strain gauge attached ahead of the crack ti during fatigue crack roagation test based on the elastic comliance method [11]. From the figure, the threshold value DK effth can be estimated as 1.15 MPa m 1/2. The log log lot between /dn and DK eff DK effth is shown in Fig. 8b. As can be seen from the figure, the relation can be reresented by a straight line with a sloe of 2, and therefore the following relation was obtained. dn ¼ AðDK eff DK effth Þ 2 ð5þ where a is the crack length, A is a material-environmental constant equal to 1 9 (MPa) 2.. Analysis of crack growth behavior based on the modified linear elastic fracture mechanics aroach In order to make use of Eq. (5) in the short crack range, modification is necessary since short fatigue cracks differ from large cracks in the following three asects [17]: (a) In the short crack regime, small scale yielding conditions are not alicable, and a modification for elastic lastic conditions is needed. Irwin [18] roosed that the linear elastic aroach could be extended to include elastic lastic behavior, i.e., those cases where the crack ti lastic zone size is large with resect to the crack length, by increasing the actual crack length, a, by one-half of the lastic zone size. If the lastic zone size is taken to be that as defined by Dugle [19], then the modified crack growth length, a mod, is given as a mod ¼ a þ 1 2 sec 2 r max r Y 1 a ¼ a 2 sec 2 r max r Y þ 1 ¼ af where r max is the maximum stress in a loading cycle, r Y is the yield strength, and F is termed the elastic lastic correction factor. (b) Crack closure in the wake of a newly formed crack is zero, but as the crack grows, the crack closure level increases to the level associated with a macroscoic crack in a distance of the order of 1 mm. ð6þ /dn [m/cycle] Δ Δ KKeff In air room temerature f = 1Hz ΔK eff th 1.15 /d N [m/cycle] Long through ΔK, ΔK eff [MPam 1/2 ] Δ K eff - Δ K effth [MPam 1/2 ] (a) /dn vs. ΔK, /dn vs. ΔK eff (b) /dn vs. Δ K eff- ΔK effth Fig. 8. Long through crack roagation behavior tested at.

5 1792 S. Ishihara et al. / International Journal of Fatigue 31 (9) The level of crack closure develoed in the wake of a crack varies from zero for a newly formed crack u to K omax for a macroscoic crack. The following exression has been roosed [2] to describe this transient in crack closure behavior: DK o ¼ð1 e kk ÞðK omax K min Þ where DK o is the value of K o K min in the transient range, k is a material constant (units m 1 ) which determines the rate of crack closure develoment, k is the length of the newly formed crack (units m), and K omax is the magnitude of the crack oening level associated with comletion of the transient eriod of growth. The value of k when K omax is reached is generally less than a millimeter. (c) In the very short fatigue crack growth range, the stress for roagation is controlled by the endurance limit of the material rather than by the long crack threshold condition (Kitagawa effect [21]). Irwin [22] has shown that the stress intensity factor, K, is related to the stress concentration factor, K T, by the following equation: K ¼ lim q! r m rffiffiffiffiffiffiffi q ¼ lim q! K T r rffiffiffiffiffiffiffi q where q is the ti radius of the stress concentrator, r m is the maximum stress at the ti of the stress concentrator, and r is the remote stress. In order to achieve the desired transition between the threshold level for fatigue crack growth and the fatigue strength, Eq. () is modified as follows: K ¼ lim q!qe K T r rffiffiffiffiffiffiffi q where q e is a material constant. In the case of a anel containing a central crack under Mode I qffiffi loading, K T is equal to K T ¼ 1 þ 2 a q, and Eq. (5) becomes rffiffiffiffiffiffiffiffi K ¼ þ ffiffiffiffiffiffi a ð1þ q e r The material constant q e is converted to an effective length dimension, r e, by ostulating the following relationshi: rffiffiffiffiffiffiffiffi ¼ ffiffiffiffiffiffiffiffiffiffi 2r e ð11þ q e so that r e is equal to q e /8 in magnitude. (The left hand side of Eq. (7) q is obtained by equating the stress intensity factor r ffiffiffiffiffiffi q e to the LEFM stress intensity factor, r ffiffiffiffiffiffiffiffi 2r. Therefore r e is the distance from the ti of the crack to the oint where the two stress intensity factors are equal). In this modified aroach r e is considered to be the effective length of an inherent flaw. In this interretation of r e, a newly formed crack is only significant when its length exceeds r e, since for crack lengths less than r e, the stress intensity factor associated with r e will be larger. It is ointed out that there is no relationshi between r e and an actual defect. It is merely an adjustable arameter introduced, in order to deal with the Kitagawa effect in a quantitative manner. Table Parameters used for the calculation of M. K omax (MPa m 1/2 ) r e 1 6 (m) DK effth (MPa m 1/2 ) r y (MPa) k (m 1 ) ð7þ ð8þ ð9þ r EL (MPa) The driving force for fatigue crack growth, DK, is now generalized to take into account in geometries other than just the center-cracked anel, as well as ossible elastic lastic effects, as ffiffiffiffiffiffiffiffiffiffiffiffiffi DK ¼ 2r e F þ Y af ffiffiffiffiffiffiffiffiffi ð12þ Dr where the value of Y deends uon the crack-shae. It is assumed that the initial crack-shae is semi-circular, then the value of Y is.73 [11]. The magnitude of r e is of the order of one micron. Its value is determined by setting a equal to r e, DK equal to the effective range of the stress intensity factor at the threshold level, DK effth, which is taken to corresond to a crack growth rate of 1 11 m/cycle, and Dr equal to the stress range at the fatigue strength level, Dr EL (1 7 cycles), i.e., 2 3 r e ¼ DK 2 effth 1 Dr EL 2F 1 þ ffiffiffi 5 ð13þ 2 Y þ :5Y 2 With these modifications, Eq. (5) is written as h dn ¼ A ffiffiffiffiffiffiffiffiffiffiffiffiffi 2r e F þ Y ffiffiffiffiffiffiffiffiffi i 2 af Dr ð1 e kk ÞðK omax K min Þ DK effth Eq. (1) can be written in simlified form as dn ¼ AM2 ð1þ ð15þ where M, the net driving force for fatigue crack roagation, is the quantity within brackets in Eq. (6). Eq. (15) will be used in the following analyzes. The rate of fatigue crack roagation in the resent extruded magnesium alloy was analyzed using the M arameter in Eq. (15). The values of arameters used for calculating M are listed in Table. The values of K omax, DK effth, and k were determined on the exerimental ta on the resent extruded magnesium alloy AZ31 for the long through crack. The value of tensile strength r B was used instead of the yield strength r y for the calculations of M at high R ratios. Fig. 8 shows the log log lot of the relation of /dn and M for the short surface cracks. The ta were obtained by the axial fatigue tests (R = 1 and 2) as well as the rotating bending fatigue tests (R = 1). For the case of R = 2, the values of M were calculated using the comressive yield strength, r y = 123 MPa instead of the tensional yield strength, since the mean stress is comression for the case. In the figure, /dn vs. DK eff DK effth relation for the long through crack is also lotted for a comarison. As can be seen from this figure, no difference between the both relations can be seen (see Fig. 9). In accord with Eq. (15), the value of the sloe was set at 2. The relationshi between /dn and M can be exressed as dn ¼ 1 9 M 2 ð16þ 5. Estimation of S N curves on the basis of the relation /dn M It is ossible to regard crack roagation life N as equal to that of total fatigue life N f, because the fatigue crack initiates early in the fatigue rocess. Crack roagation life can be estimated by numerically integrating Eq. (16) from an initial crack length 2r e to a final crack length of mm. However, when we calculated the M arameter for the case of R = 2 using the yield strength in tension rather than the yield strength in comression, the redictions disagreed with the exerimental values. This disagreement between the estimated fatigue lives and exerimental values is due to the large difference

6 S. Ishihara et al. / International Journal of Fatigue 31 (9) between the tensional yield strength and the comressive yield strength, as shown in Fig.. Similarly, during fatigue cycling, the cyclic yield strength is also exected to be lower in the negative R ratio region due to the occurrences of the twinned crystals introduced under comression. Better agreement was obtained when the comressive yield strength was used. Fig. 1 comares the redicted and exerimental behavior. As seen from the figure, the redicted S N curves agree well with the exerimental ta for a wide range of R ratios, R = 2 to.1, with the yield strength in comression being used for R = 2. The yield strength in tension was used for other R values. Comarisons of the redicted results with the exerimental ones were also made with resect to r m vs. r a, where r m indicates mean stress, and r a means stress amlitude. The results are shown Stress amlitude σ a [MPa] R = -1 R = Number of cycles to failure N f [cycle] Exeriment (;Axial) (R = -1;Axial) (R = -1;RB) (R = -2;Axial) Fig. 1. Estimated and exerimental S N curves for different R ratios. Stress amlitude σ a [MPa] /dn [m/cycle] M, ( ΔK eff - ΔK effth ) [MPam 1/2 ] R = cycles R = -1 1 cycles 1 5 cycles Mean stress σm [MPa] Short surface R = -1(RB) R = -2(Axial) R = -1(Axial) Long through Fig. 9. Log log lot of /dn vs. M arameter for short surface crack, and /dn vs. DK eff DK effth for long through crack. R = cycles 1 5 cycles 1 cycles Fig. 11. Comarisons between the estimated r a vs. r m relations and exerimental ones. Three fatigue lives, i.e., cycles, 1 5 cycles, and 1 cycles are taken as arameters in the figure. Solid lines indicate the redictions. in Fig. 11, where three fatigue lives, i.e., cycles, 1 5 cycles and 1 cycles are taken as arameters. Again, we can confirm that the redictions based on the M arameter agree well with the exerimental ta, from the low cycle region, to the high cycle region, In general the use of the modified linear elastic fracture mechanics arameter, M is effective and useful for the analysis of the short fatigue crack roagation behavior as well as for redicting S N curves of the extruded magnesium alloy. However, for the case of the negative R region of the alloy, the yield strength in comression should be used when analyzing short fatigue crack roagation based on the M arameter. Further studies are needed to clarify the crack roagation behavior under the negative R regions. 6. Conclusions Fatigue tests were conducted using the extruded magnesium alloy AZ31 to determine the effect of load ratio on the fatigue lifetime. The following conclusions relating to extruded magnesium alloy AZ31 were reached: (1) Fatigue cracks are initiated at an early fatigue stage, i.e., 5 1% of fatigue life, indicating that total of fatigue lives N f can be aroximated as the crack roagation lives, N. (2) The effect of mean stress on the fatigue strength was given aroximately by the Gerber relationshi. Further exerimental studies are needed to confirm the above result. (3) The modified linear elastic fracture mechanics arameter, M is useful in the analysis of the short fatigue crack roagation behavior and for redicting the S N curves of the AZ31 extruded magnesium alloy. () For the case of the negative R ratios such as R = 1 and R = 2, comressive yield strength should be used in analyzing short fatigue crack roagation and in redicting the S N curves based on the M arameter. (5) Further studies are needed to clarify the crack roagation behavior at the negative R regions. Acknowledgements Authors wish to exress their thanks to Mr. S. Yoshifuji and Mr. S. Koike for their hel in doing exeriments. Thanks also go to Dr. T. Murai for roviding us with the materials used in the resent study. References [1] Kojima Y. Handbook advanced magnesium technology. Tokyo: Kallos Publishing CO., LTD;. [2] Goodenberger DL, Stehens RI. J Eng Mater Technol 1993;115:391. [3] Hilert M, Wagner L. J Mater Eng Perform ;9():2. [] Eisenmeier G, Holzwarth B, Höel HW, Mughrabi H. Mater Sci Eng A 1;319:578. [5] Shih TS, Liu WS, Chen YJ. Mater Sci Eng A 2;325:152. [6] McEvily AJ, Eifler D, Macherauch E. Eng Fract Mech 1991;:571. [7] McEvily AJ, Ishihara S, Endo M. An analysis of multile two-ste fatigue loading. Int J Fatigue 5;27: [8] Ishihara S, McEvily AJ. Int J Fatigue 2;2(11):1169. [9] Ishihara S, McEvily AJ. In: Ravichandran KS, Ritchie RO, Murakami Y, editors. Small fatigue cracks. Elsevier; [1] Nan ZY, Ishihara S, Goshima T, Nakanishi R. Trans Jn Soc Mech Eng A ;7(696):116. [11] McEvily AJ. A method for the analysis of the growth of short fatigue cracks. In: Poklu J, editor. Materials structure and micromechanics of fracture. Switzerland: Trans Tech Publications; ; McEvily AJ. Mater Sci Forum 5;82:3 1. [12] Murakami Y, editor. Stress intensity factors handbook, vol. 1. Pergamon Books; [13] Kikukawa M, Jono M, Tanaka K, Takatani M. J Soc Mater Sci Jn 1976;125:899.

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