Fatigue crack growth for straight-fronted edge crack in a round bar

Transcription

1 International Journal of Fatigue 28 (2006) Fatigue crack growth for straight-fronted edge crack in a round ar FengPeng Yang a, ZhenBang Kuang a, *, V.N. Shlyannikov a epartment of Engineering Mechanics, Shanghai Jiaotong University, Shanghai , China Kazan State Power Engineering Institute, 51 Krasnoselskaya street, Kazan, Russia Federation Received 12 August 2004; received in revised form 12 May 2005; accepted 5 July 2005 Availale online 29 Septemer 2005 Astract In this paper, firstly fatigue crack growth for a straight-fronted edge crack in an elastic ar of circular cross-section is studied through experiments under pure fatigue axial loading. Three different initial notch depths are discussed. The relations etween aspect ratio (/c) and relative crack depth (/) are otained, and it is shown that there is great difference in the growth of cracks with different front shapes and initial notch depths. Using the relations, predictions are made of the crack front shape and crack growth rate in the depth direction. Secondly, the variation of crack growth ehavior is also studied under cyclic axial loading with steady torsion. Results show that Mode III loading superimposed on the cyclic Mode I leads to a significant reduction in the crack growth rates. q 2005 Elsevier Ltd. All rights reserved. Keywords: Surface crack; Aspect ratio; Steady torsion; Crack growth rate 1. Introduction The fatigue growth analysis of surface cracks is one of the most important events for the safety operation of a structural component. The prolem is complex and the closed solution is often not availale ecause these flaws are three-dimensional in nature. The circular cylindrical metallic components (ars, wires, olts, shafts, etc.) are commonly used in engineering structures and failure may occur under cyclic loading. The fatigue failure of round ars often develops from surface defects, and thus many analyses have een carried out to determine the stress-intensity factors (SIF) along the front of an edge flaw [1 5]. An actual surface crack may usually e replaced y an equivalent circular arc or an elliptical-arc edge flaw. The stress-intensity factors have een pulished for partcircular, part-elliptical, or straight fronted cracks in a ar. The straight-fronted surface crack is often used in experiments due to easy of manufacturing, and this can e considered as an extreme shape of either the part-circle or part-ellipse surface cracks. The SIF s were otained y using either numerical methods, such as finite element analysis, oundary integral equation and weight function methods, or experimental approaches. Newman and Raju [6] developed the SIF equation * Corresponding author. Tel.: C ; fax: C address: zkuang@mail.sjtu.edu.cn (Z.B. Kuang) /$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi: /j.ijfatigue from a three-dimensional finite-element analysis of a semielliptical surface crack in an elastic finite thickness plate sujected to tension or ending. Forman and Shivakuma [7] assumed the actual cracks to e always part-circles that intersect the free surface of the ar at right angles. Through the studies on the ar and pipe, it was found that the shape of the crack front after some extensions of a straight-edge and a partellipse front are similar with that of a part-circle. Lin and Smith [8] adopted three-dimensional FEA techniques to simulate the development of a fatigue crack shape and showed that the aspect ratio is very sensitive to the initial crack geometry during its early growing stage, ut tended towards a preferred front shape after some extension of the crack. Tschegg [9] studied the fatigue crack growth under Mode III loading and showed that the fracture surface interaction etween the fatigue crack surfaces induced crack closure at the crack tip and a significant reduction of crack growth rates was achieved. Fonte and Freitas [10] analyzed the influence of steady torsion loading on fatigue crack growth rates under rotating or reverse ending. Similar results are otained showing that the steady Mode III loading superimposed on the cyclic Mode I leads to a significant reduction of the crack growth rate. In this paper, experimental results of fatigue crack growth for a straight-fronted edge crack in an elastic ar under axial loading with or without steady torsion are given. Three different initial notch depths are discussed. The relations of aspect ratio and relative crack depth are otained and it is

2 432 F.P. Yang et al. / International Journal of Fatigue 28 (2006) Tale 1 Chemical composition of steel S45 Element C Mn Ni Cr Mo Si P S Gu wt% Fig. 1. etails of the specimen geometry. shown that the growth of the crack fronts is dependent on the initial notch depth. Using the relations, the crack front shape and crack growth rate in the depth direction can e predicted. Under cyclic tension with steady torsion, the crack growth is retarded. This is related to the increase of plastic zone size near the crack front and to some extent may e related to the crack closure effect. 2. Test material and specimens The test material was caron steel S45. Its composition is given in Tale 1. The heat treatment consisted of austenizing at 1123 K for 1.5 h, oil quenching and tempering at 903 K for 1 h. The mechanical properties attained y this heat treatment are summarized as follows: monotonic tensile yield strength s 0 Z MPa, nominal ultimate tensile strength s m Z MPa, true ultimate tensile strength s f Z MPa, reduction of area fz62.87%, Young s modulus EZ206 GPa, true fracture logarithmic strain 3 f Z The outer geometry configuration of a specimen is shown as Fig. 1. The diameter is 12 mm and the length L is 90 mm in the test section. Using linear cutting machine surface edge cracks were cut with three different initial flaw depths 0 : 1.0, 2.0, and 3.0 mm. The geometric parameters of the test section of a specimen and of the crack growing process are shown in Figs. 2 and 3. An equivalent ellipticalarc edge flaw is used to replace the actual part-through crack after some extension. In Fig. 2, 1 ZOB 1 is the current crack depth at time t 1. The crack front may e approximated y an elliptical curve with major axis 2a and minor axis 2. Note that, the length of the major axis of the assumed ellipse 2a is only an estimated value. In the actual experiment, the location of the external surface crack front is measured y h, the distance from the intersected point A to the central axis of symmetry OY and y, the distance from point A to the horizontal axis OX. The crack length in the surface direction can e otained y way of calculating the chord shown in Fig. 3. The front of the initial straight surface crack is denoted y B 0 A 0, and the initial chord from A 0 to o is c 0 Z OA 0. When the crack is developed, the intersection moves to A 1 and the current chord is c 1 ZOA 1, which can e calculated y the p location of the point A i and the formula c 1 Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2 1 Cy2 1. and R are the diameter and radius of the specimen, respectively. Fig. 2. Surface crack geometric parameters. Fig. 3. Crack growth rule in the surface direction.

3 F.P. Yang et al. / International Journal of Fatigue 28 (2006) Tale 2 etails of experiments No. iameter (mm) Initial depth 0 (mm) Max. force F (KN) Stress ratio s min /s max Fracture cycles N f S ,637 S ,216 S ,689 S ,542 S ,521 S ,103 S ,934 S ,025 S ,488 S ,303 S ,064,689 S , Crack growth monitoring All tests are made on a MTS809 servo-hydraulic tensiontorsion test machine and are carried out at 15 Hz and room temperature. For the simple cyclic tension tests, four different applied stress amplitudes with the same stress ratio s min /s max- Z0.1 are used to study the effects of loading. The loading form was sinusoidal and the tests were performed in load control. The detailed data of all experiments are shown in Tale 2. To reduce experimental error, the specimens S01 and S02 are tested in the same condition, maximum fatigue tension 25 kn and stress ratio 0.1. It is shown that the test results have good repeataility. The shape and depth growth of fatigue cracks were monitored using a zoom stereomicroscope and each marking [5,11]. The depth growth of a crack at various cycle intervals was given y the each mark data. uring each test, six to seven each marks were produced on each specimen y reducing the applied load to one-half for several cycles. The typical each markers are shown in Fig. 4(a). The stripes on the cross section of different specimens are clear at different places and it is possile to use these data to draw a diagram as shown in Fig. 4(). From the shape features of cross sections of the specimen otained in this way, the relations etween the relative crack depth / and the surface crack chord length c/ can e measured using a comparison microscope. In addition, ased on measuring the increment of surface crack chord length c with the cycle numers, the curve of surface crack propagation versus cycle numers dc/dn can e otained. Afterwards utilizing the relation of crack depth versus surface crack chord length, it is possile to otain the crack growth rates d/dn in the depth direction. Two different levels of torque 40 and 75 Nm were applied to the specimens in order to study the crack growth under fatigue tension loading with fixed torsion. For these tests, the specimen geometry configurations are chosen so as to e the same as those under pure tension cyclic loading, except that the initial crack is cut 1.2 mm depth. The maximum tension loadings are 25, 28 and 32 kn, respectively. The other parameters are the same as for fatigue tension. 4. Results and discussion 4.1. The evolution of the crack shape The evolution of the crack shape of the straight-fronted edge surface cracks during the tests is determined using each marks and the microscope. The results were oserved for several different tension amplitudes. An example of the shape development is shown in Fig. 4(a). The photograph is for the specimen with the initial crack depth of 1.0 mm and for cyclic Fig. 4. Fatigue crack growth process with initial crack depth 1.0 mm and 25 kn cyclic tension loading. (a) Photograph of the cross section of a specimen. () The sketch of the crack extension at various cyclic numers.

4 434 F.P. Yang et al. / International Journal of Fatigue 28 (2006) tension loading of 25 kn. As is shown in the figure, crack propagation appears first in the deepest point of the cylinder ar. At the eginning of crack growth, the growth rate of the central point of the crack front is faster than that at the intersection with the surface. The reason is that the maximum stress-intensity factor is attained at the deepest point on the crack front for the straight-fronted edge crack (/az0). Consequently, a straight-fronted crack tends to ecome curved, and the flaw aspect ratio /a increases. When /az, the deepest point and the external surface points on the crack fronts, almost have the same stress-intensity factor. After /ao, the crack growth rate at the intersection of the crack with the external surface is higher than that at the deepest point and the curve tends to flatten. The result is also shown in [4]. The sketch of fatigue crack propagation is shown in Fig. 4(). Because the effect of the tension loading amplitudes on the shape evolution is not large, the sketch of the crack growth front curves may e collected from several tested specimens. The fatigue crack developments / with c/ under cyclic tension loading for three initial notch depths 1.0, 2.0, and 3.0 mm are shown in Fig. 5(a) (c). Polynomial functions can e used to express the crack shape development rule as: For 0 Z 1:0 mm ZK0:5917 C3:1204 c K2:1440 c 2 (1) For 0 Z 2:0 mm ZK1:1813 C4:9022 c K3:5062 c 2 (2) For 0 Z 3:0 mm ZK2:1553 C7:4817 c K5:1703 c 2 (3) For other initial flaw depths, the relation function of / versus c/ can e otained through the linear interpolation method from Eqs. (1) (3). Fig. 5(d) gives the contrast of crack growth shape ratio with three initial flaw depths. It can e seen that the crack propagation paths differ with different initial flaws, ut will tend to converge to the same value when the crack depth / is equal to aout. (a) polynomial () polynomial imensionless crackdepth (/) 0.1 imensionless crack depth (/) 0 imensionless arc length (c/) imensionless arclength (c/) (c) (d) imensionless crack depth (/) polynomial imensionless crack depth (/) =1.0mm 0 =2.0mm 0 =3.0mm imensionless arc length (c/) imensionless arc length (c/) Fig. 5. Relationships of the crack growth in the depth and surface directions. (a) 0 Z1.0 mm, () 0 Z2.0 mm, (c) 0 Z3.0 mm and (d) comparison.

5 F.P. Yang et al. / International Journal of Fatigue 28 (2006) (a) 0.9 () =1.0 aspect ratio (/c) 0.7 prediction 0.1 Relative crack depth (/) aspect ratio (/h) 0.7 prediction 0.1 Relative crack depth (/) (c) 0.9 (d) 1.0 aspect ratio (/c) =1.0 0 =2.0 0 =3.0 aspect ratio (/h) 0 =1.0 0 =2.0 0 = Relative crack depth (/) 0.1 Relative crack depth (/) Fig. 6. Fatigue crack growth patterns for different initial surface flaws. 5. Influence of initial crack sizes Fig. 6(a) and () show the fatigue propagation patterns for an initial surface flaw depth 0 Z1.0 mm. Experimental data are given for the relations of the crack aspect ratios (/c and /h) versus relative crack depth (/), and the predicted values otained from Eqs. (1) (3) are also shown. They have good agreement with each other. Fig. 6(c) and (d) show the comparison of fatigue propagation patterns for three surface flaws in round ars. It can also e oserved that the propagation pattern converges to a range of / of aout. 6. Crack growth rate Based on the evolution of the crack shape, a simplified model for otaining the stress-intensity factor K of surface crack under fatigue tension loading is expressed as: pffiffiffiffiffiffi K Z S t pa Fð=a; =; 4Þ (4) Where F is the geometric correction factor for axial loading and S t is the remotely applied tensile stress. According to the elliptical model proposed, F for a point, on the crack front profile, is the function of the major axis of the ellipse a and the minor axis of the ellipse, the diameter of the specimen and a parametric angle f, defining the position of the point at the crack tip. Some writers [3,7,12] analyzed surface cracked shafts using several mathematical techniques including finite element analysis and considered that the correction factor F is the function of relative crack depth /. In this paper, the value of F at the point A with maximum depth and the point B at the intersection with the external surface can e approximated y the polynomial equations [3]. For the notched specimen under crack growth rate [mm/cycle] 1E-3 1E-4 1E-5 d/dn( 0 =1.0) dc/dn( 0 =1.0) d/dn( 0 =2.0) dc/dn( 0 =2.0) d/dn( 0 =3.0) dc/dn( 0 =3.0) SIF[MPaVm] Fig. 7. Fatigue crack growth rates in the depth and surface directions.

6 436 F.P. Yang et al. / International Journal of Fatigue 28 (2006) Cyclic tension(25kn) ten+tor (25KN+40NM) ten+tor (25KN+75NM) Cyclic tension(28kn) ten+tor (28KN+40Nm) ten+tor (28KN+75Nm) arc length c (mm) arc length c (unit:mm) Cycle Numers (*10 3 ) Cycle Numers (*10 3 ) Fig. 8. Fatigue crack growth curves: arc length c versus cycle numers N. (a) Tension 25 knctorsion 0, 40,75 Nm, () tension 28 knctorsion 0, 40, 75 Nm. axial loading, the relations etween F and the ratio of the crack depth to the shaft diameter / are expressed as: At point B : F Z 0:906K1:23 C0:599 2 C12:0 3 (5) 4 K11:2 At point A : F Z 1:12K2:61 C7:34 2 C1:41 3 (6) 4 K4:28 The crack growth rule can adopt the Paris-Erdogan equation under linear elastic fracture mechanics. Fig. 7 shows the crack growth rate of specimens with the initial notch depths 1.0, 2.0 and 3.0 mm. The results show that the crack growth rates in the depth direction and along the external surface direction are similar for different initial depths. Based on the experimental data, the crack growth rates are as follows: d dn Z 1:9037 ek9 ðkþ 3:256 (7) dc dn Z 1:2082 ek9 ðkþ 3:293 (8) 7. Influence of steady torsion In order to study the surface crack growth in shafts under fatigue tension loading with steady torsion, several specimens are tested with an initial flaw depth 1.2 mm. The comining loads are: cyclic tension loads with three amplitudes 25, 28, 32 kn, and steady torsions with two amplitudes 40 and 75 Nm. Fig. 8 shows plots of the extensions of the surface crack chord length c with the numer of cycles N under comined loads. Fig. 8(a) and () are for the specimens under 25 and 28 kn tension loading and 0, 40, 75 Nm steady torsions, respectively. As shown in the figures, steady torsion loading superimposed on cyclic tension leads to a significant retardation of crack growth. With the increase of the value of torsion, the effect is larger. In addition, the retardation effect of Mode III loading is larger under low tension than that under high tension. Fig. 9 shows the relation etween the relative crack depth / of the surface crack and the arc crack length c/. The superposition of steady torsion has also some effect on the crack aspect ratio. The crack front curve under cyclic tension with steady torsion tends to flatten more than that without torsion. The line shown in Fig. 9 is predicted through linear interpolation for pure tension cyclic loading, and is otained from Eqs. (1) and (2). Fig. 10(a) and () show the surface crack growth rate dc/dn versus arc crack length c under single cyclic tension imensionless crack depth (/) cyclic tension tension+torsion(40nm) tension+torsion(75nm) prediction imensionless arc length (c/) Fig. 9. Crack growth shape under cyclic tension with and without steady torsion ( 0 Z1.2 mm).

7 F.P. Yang et al. / International Journal of Fatigue 28 (2006) dc/dn (mm/cycle) 1E-4 1E-5 dc/dn (mm/cycle) 1E-4 Tens=25KN Tens=25KN,Tors=40Nm Tens=25KN,Tors=75Nm ARC CRACK LENGTH c (mm) 1E-5 Tens=32KN Tens=32KN,Tors=40Nm Tens=32KN,Tors=75Nm ARC CRACK LENGTH c(mm) Fig. 10. Crack growth rates dc/dn versus arc crack length c for (a) tension 25 knctorsion 0, 40, 75 Nm, () tension 32 knctorsion 0, 40, 75 Nm. and comined fatigue loading. It can e seen that a significant reduction of the crack growth rates occurs when the Mode III loading is superimposed on the cyclic tension. This is consistent with the results that Fonte et al. otained under ending comined with steady torsion [10]. The Authors also have oserved that the retardation effect of the torsion is reduced when the tension load increases. It is suggested that the retardation effect may e related to a relative increase of plastic zone near the crack tip due to steady torsion. The plastic zone at the crack tip will raise the toughness of the material. Because fatigue occurs at low stress level, the failure may e mainly due to the tension cyclic load. This explanation needs to e tested y further experimental work and theoretical analyses. 8. Conclusions Fatigue crack growth for a straight-fronted edge crack with three different initial notch depths in an elastic ar of circular cross-section is studied. Experiments made under axial loading with and without steady torsion are descried. All the experimental and analytical results are shown: (1) Under pure cyclic tension loading, it can e seen that the crack propagation paths differ with diverse initial flaw depths, ut converge to the same configuration when the crack depth ratio / is larger than aout. (2) The functions of aspect ratio and relative crack depth are otained and y means of the function, the crack front shape and crack growth rate can e predicted well. (3) A significant reduction of the crack growth rates and an increase of fatigue life is found when the Mode III loading is superimposed on the cyclic tension. It is also found that the retardation effect of the torsion is reduced when the tension loading increases. Acknowledgements The financial support from the National Natural Science Foundation of China (Grants No and No ) is gratefully acknowledged. References [1] aoud OEK. Strain energy release rates for a straight-fronted edge crack in a circular ar suject to ending. Eng Fract Mech 1984;19(4): [2] Carpinteri A, Brighenti R. Part-through cracks in round ars under cyclic comined axial and ending loading. Int J Fatigue 1996;18(1):33 9. [3] Thompson K, Sheppard S. Stress intensity factors in shafts sujected to torsion and axial loading. Eng Fract Mech 1992;42(6): [4] Carpinteri A, Carpinteri A. Shape change of surface cracks in round ars under cyclic axial loading. Int J Fatigue 1993;15(1):21 6. [5] McFadyen NB, Bell R, Vosikovsky O. Fatigue crack growth of semielliptical surface cracks. Int J Fatigue 1990;12(1): [6] Newman JC, Raju IS. An empirical stress-intensity factor equation for the surface crack. Eng Fract Mech 1981;15(1-2): [7] Forman RG, Shivakumar V. Growth ehavior of surface cracks in the circumferential plane of solid and hollow cylinders. Fracture Mech 17. ASTM STP 905. ASTM; p [8] Lin XB, Smith RA. Int J Fatigue 1997;19(6): [9] Tschegg EK. A contriution to mode III fatigue crack propagation. Mater Sci Eng 1982;54: [10] Fonte MA, Freitas MM. Semi-elliptical fatigue crack growth under rotating or reversed ending comined with steady torsion. Fatigue Fract Eng Mater Struct 1997;20(6): [11] Mackay TL, Alperin BJ. Stress intensity factors for fatigue cracking in high-strength olts. Eng Fract Mech 1985;21(2): [12] James LA, Mills J. Review and synthesis of stress intensity factor solution applied to cracks in olts. Eng Fract Mech 1988;30: