Effect of steady torsion on fatigue crack initiation and propagation under rotating bending: Multiaxial fatigue and mixed-mode cracking

Engineering Fracture Mechanics 78 (2011) 826–835

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Effect of steady torsion on fatigue crack initiation and propagation under rotating bending: Multiaxial fatigue and mixed-mode cracking M. de Freitas a,*, L. Reis a, M. da Fonte b, B. Li a a b

Department of Mechanical Engineering, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Department of Mechanical Engineering, Nautical School, 2780-058 Paço de Arcos, Portugal

a r t i c l e

i n f o

Article history: Received 30 June 2009 Received in revised form 22 December 2009 Accepted 24 December 2009 Available online 4 January 2010 Keywords: Mixed-mode Multiaxial fatigue Fatigue crack growth Axles Shafts

a b s t r a c t Axles and shafts are of prime importance concerning safety in transportation industry and railway in particular. Knowledge of fatigue crack growth under typical loading conditions of axles and shafts with rotating bending and steady torsion is therefore essential for design and maintenance purposes. The effect of a steady torsion on both small and long crack growth under rotating bending is focused in this paper. For small crack growth, a modified effective strain-based intensity factor range is proposed as the parameter that correlates small fatigue crack growth data under proportional or non-proportional multiaxial loading conditions. Results show that this parameter is appropriate for determining fatigue crack growth of small cracks on rotating bending with an imposed steady torsion. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Failure of railway axles has been a matter of concern since the development of railway industry and some dramatic failures have been reported from the early decades of the XIX century. Since Wohler studies on fatigue failure of railway axles in the middle of the nineteen century, endurance curves determined under rotating bending tests constitute the most important material data when designing against fatigue failure and avoid crack initiation. With the development of modern high speed trains, railway axles are submitted to more severe loading conditions and it is still a challenge for the test and design engineers to accurately determine the fatigue lifetime of railway components [1]. Fatigue failure comprises crack initiation (including early crack nucleation and small crack growth), long crack growth and final failure. Therefore the knowledge of all crack growth stages is of prime importance in order to monitor an adequate fatigue life prediction and maintenance procedures. For long cracks, linear elastic fracture mechanics (LEFM) is a powerful tool to predict fatigue crack growth that has been proved with excellent results in aeronautic industry and has been the main tool to perform damage tolerance analysis [2]. To predict fatigue crack growth through LEFM, a stress intensity factors solution for the specific geometry is needed together with materials fatigue crack growth data. However, the LEFM approach is inappropriate for small crack growth. Therefore, various approaches have been proposed based on the effective strain range, the range of J-integral approach, micro-structural effects [3], etc. Besides, shafts are also widely used in railway, automotive, aeronautic and energy industries and in this mechanical component both a torsion and bending load are combined. Generally, fatigue failures in rotor power shafts have origin on surface cracks that grow with semi-elliptical shape under cyclic bending, mode I (DKI), combined with steady torsion, mode III (KIII) [4]. Large number of power plant systems run with a general steady torsion combined with cyclic bending stress either due * Corresponding author. Tel.: +351 218417459; fax: +351 218474045. E-mail address: [email protected] (M. de Freitas). 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.12.012

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to the self-weight bending during the rotation or possible misalignment between journal bearings. In case of rotor shafts such as those used in electric power plants, such as propeller shafts of screw ships, or any other rotary load-transmission devices, the lifetime spent between crack initiation and final fracture is of capital importance to improve the inspection intervals and maintenance procedures. Many researchers have studied the influence of a static mode III loading on cyclic mode I for circumferential notched shafts mainly since 80s, such as Akhurst and Lindley [4], Hourlier et al. [5], Pook [6], Yates and Miller [7] and so on. Tschegg et al. [8,9] has made an important contribution to clarify the influence of steady or cyclic mode III on fatigue crack propagation behavior when combined with mode I loading. Fonte and Freitas [10] have studied the influence of steady torsion on fatigue crack growth of semi-elliptical cracks in shafts under rotating bending. Despite the relevant and practical importance of mixed-mode fatigue, the applications of fracture mechanics have been concentrated on crack growth problems in mode I. Several solutions have been proposed for surface cracks in round bars. Raju and Newman [11], Shiratori et al. [12] and Carpinteri et al. [13,14] have presented stress intensity factors (SIF) for semi-elliptical cracks in round bars under pure bending. Fonte and Freitas [15] have presented stress intensity factors (SIF) for semi-elliptical cracks in round bars under pure bending but also under pure torsion. Freitas and Francois [1] carried out experimental work in rotary bending specimens to simulate railway axles, and it was shown that the proposed SIF are not directly applicable to the fatigue crack growth analysis because the position of fatigue crack front changes continuously during a load cycle. The maximum value of SIF is not reached at the same rotation angle for all points along the crack tip profile [1,16], and it is well-known that crack initiation process in rotor shafts has generally origin on the surface and grows with semi-elliptical shape. In this paper, the effect of a steady torsion load on fatigue crack growth due to cyclic bending in bars (axles or shafts) is studied. Both small and long fatigue crack growth data obtained from rotating or alternate cyclic bending tests on a medium carbon steel are presented and the results are discussed. For small crack growth, an effective strain-based intensity factor range is proposed as the parameter which correlates small fatigue crack growth data under proportional or non-proportional multiaxial loading conditions.

2. Experimental procedure 2.1. Material and specimens The material used in this study was the Ck45 steel in the normalized condition. The chemical composition is presented in Table 1 and mechanical properties, monotonic and cyclic, are shown in Table 2. Two different specimens were used on this study: a classic cylindrical hourglass shaped rotating bending specimen that was used on long crack growth tests, where the geometry and dimensions used can be found in [1,10]; for small crack growth tests, a cylindrical notched specimen was used, as shown in Fig. 1. All specimens were machined from round bars. After machining, all specimens with notch (small crack study) were electro-polished, with the aim of relieving some of the residual stresses. The other specimens were classical polished to achieve a smooth surface.

Table 1 Chemical composition of Ck45 steel (%). C

Si

Mn

P

S

Cr

Ni

Mo

Cu

0.45

0.21

0.56

0.018

0.024

0.10

0.05

0.01

0.10

Table 2 Monotonic and cyclic properties of Ck45 steel. Tensile strength Yield strength Young’s modulus Elongation Cyclic yield strength Strength coefficient Strain hardening exponent Fatigue strength coefficient Fatigue strength exponent Fatigue ductility coefficient Fatigue ductility exponent Additional hardening

Ru (MPa) Rp0.2 (MPa) E (GPa) A (%) Rp0.2,cyclic (MPa) K0 (MPa) n0 rf0 (MPa) b

ef0 c

a

660 410 206 23 340 1206 0.20 948 0.102 0.17 0.44 0.3

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Fig. 1. Specimen for small crack growth tests (mm).

2.2. Testing conditions 2.2.1. Long cracks Experiments for long crack growth were performed in two fatigue testing machines which were designed and constructed to model the real conditions of shafts in service [10]. One fatigue machine performed rotary bending with or without steady torsion and another one performed alternating bending under controlled displacement. A chordal notch was made on the cylindrical surface, approximately 3 mm length and 0.2 mm depth and specimens were pre-cracked under constant amplitude cyclic bending stress for crack growth till the crack had a minimum length of 4 mm. With the 200 MPa bending stress loading and the geometry conditions the stress intensity factor range at the crack tip near the surface, calculated as described p in the next section is around 8 MPa m which is greater than the threshold for this steel [1]. Fatigue crack initiation and crack length measurements were carried out with optical microscope at a magnification of 80X and a stroboscope with the light at the same frequency of testing. Measurements of crack length were obtained in continuous process at constant intervals till the crack depth reaches approximately 3/4 r. Fatigue tests under rotary and alternating bending under controlled deflection were carried out at 24 Hz and 12 Hz, respectively, at room temperature. Three torsion stress levels and two specimen diameters were used in order to determine the effect of torsion on fatigue crack

Fig. 2. Fracture surface of rotating bending specimens (a) d = 40 mm; (b) d = 12 mm.

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growth rates [10]. Fig. 2 shows two examples of the fracture surface of two different specimens with different diameters. In some selected specimens the crack front shape was marked through beach marks, using a lower bending stress level in order to obtain experimentally the crack shape, i.e. crack length vs. crack depth. 2.2.2. Small cracks For the experimental work in small cracks a frame work was designed and assembled on a servo-hydraulic testing machine which performed alternating bending under load control. Steady torsion was imposed through a lever arm connected to the specimen. A replica was taken to monitor the crack initiation and small crack growth on the notched specimens subjected to multiaxial loading cyclic bending and steady torsion. This method with an optical microscopy at a magnification between 50 and 1000 times allows the detection of very small cracks. In this study, the fatigue life spent to a crack length of 0.1 mm will be considered as crack initiation life and the propagation of cracks from the size of 0.1 mm up to 2.0 mm is considered as small crack growth. 3. Results 3.1. Fatigue crack growth for long cracks Fatigue crack growth tests were performed under two loading conditions respectively: (1) cyclic bending; (2) cyclic bending with a steady torsion where cyclic bending was achieved either through rotating or alternate bending. The geometric parameters of cylindrical specimens and semi-elliptical crack dimension are characterized according to Shiratori nomenclature [8] as shown in Fig. 3, i.e. the semi-arc crack length is denoted by s and the minor semi-ellipse axis corresponding to the maximum crack depth is denoted by b. The semi-elliptical crack depth b was related with the total arc crack length 2s and agrees with the equation b = 2s/p already found before in previous experimental tests for pure rotary bending, as shown in Fig. 2 and mixed-mode [1,10]. Some deviation for the largest applied torques was found, however for the crack depth b up to r/2 the equation is still valid. Fig. 4 plots the fatigue crack growth rate (d(2s)/dN) data vs. total arc crack length (2s). A significant effect of steady torsion on fatigue crack growth rate can be observed for the levels of torsion presented. Results show that fatigue crack growth rate

d(2s)/dN [mm/cycle]

Fig. 3. Geometric parameters of semi-elliptical surface cracks.

-4

10

FIT line Mode I Mode I Mode I Mode (I+III) Mode (I+III) FIT line (I+III)

Mode I

Mode (I+III)

-5

10

10

arc crack length 2s [mm] Fig. 4. Fatigue crack growth rate d(2s)/dN vs. arc crack length 2s in rotating bending.

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in rotating bending decreases in the presence of a steady torsion and this retardation effect increases with the level of steady torsion [10]. 3.2. Fatigue crack growth for small cracks On small crack growth due to alternate bending with or without steady torsion, it was observed that one or several microcracks usually appeared on the specimen with one of them later becoming dominant. Fig. 5 shows three examples of fatigue crack paths obtained by the replica’s method. The higher the applied stresses (both bending and torsion), the more number of micro-cracks were observed during crack initiation, as shown on Fig. 5c). Linking of micro-cracks sometimes leads to a sudden increase in their surface length and this usually grew to be the major crack to failure. Once monitored the crack initiation and small crack growth, then the small crack growth rate was evaluated. In Fig. 6, the measured crack lengths, in small crack regime are plotted vs. the number of cycles for three levels of cyclic bending load and one torsion load. It is clear from Fig. 6 that the bending stress level has a strong influence on both fatigue crack growth and on the differentiation regime between small and long crack. It is evident that an acceleration on crack growth is observed at 1 mm of crack length for the lower stress levels and less than 1 mm for the higher stress levels. Fig. 7 shows the arc crack length 2s as a function of the number of cycles for small cracks (<2 mm) and long cracks. For long crack growth data a cyclic bending stress of 200 MPa and two static torsion stress levels, 100 and 200 MPa were used. For small crack growth tests three cyclic bending stress levels (480, 450 and 420 MPa) and one static torsion stress levels of

Fig. 5. Fatigue crack path by replica method (a) 2s = 0.3 mm; (b) 2s = 0.75 mm; (c) multiple crack initiation.

Fig. 6. Fatigue crack growth in small crack regime.

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10

-4

10

-5

Mode (I), Bs=200 MPa Mode (I+III), Bs=200,Ts=100 MPa Mode (I+III), Bs=200,Ts=200 MPa Mode (I+III), Bs=420,Ts=205 MPa Mode (I+III), Bs=420,Ts=205 MPa Mode (I+III), Bs=450,Ts=205 MPa

Mode I Mode(I+III)

d(2s)/dN [mm/cycle]

10

831

Mode (I+III)

10

-6

0.1

1

10

arc crack length 2s [mm] Fig. 7. FCG rate vs. total arc crack length 2s under alternating bending + torsion for small and long cracks.

205 MPa were used. Despite that the small and long crack growth data have been obtained for different stress levels and different geometries, therefore presenting some scatter on the data, the correlation between the small crack and the long crack growth rate data presented is very consistent. A continuous trend on crack growth rate between small and long crack data is observed. The correlation between both short and long crack growth data can only be carried out through the stress or strain intensity factors that will be explained in the next section. 4. Discussion 4.1. Stress intensity factors Assuming that cracks have always a semi-elliptical shape as shown in Fig. 3, the points on crack front at deepest crack depth (A) and where the crack intersects the external surface (points B and C) are the most important ones. The semi-ellipse axes are obtained according to Eq. b = 2s/p experimentally obtained [1,10], and Eq. (1) was deducted.

r  senh a ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1  br 2 ð1  cos hÞ2

ð1Þ

The major research field of fracture mechanics is the development for determining the stress intensity factors (SIF). In this study, SIF for mode I and mode III along the crack front of semi-elliptical surface cracks normal to the axis in shafts subjected to bending, torsion and bending-torsion simultaneously, determined by a three-dimensional finite-element analysis [15,16] will be used. The total arc crack length (2s) was chosen as the experimentally obtained crack parameter in fatigue crack growth tests and according to Shiratori et al. [8] and Fonte and Freitas [10]. Therefore the mode I stress intensity factor KI for any point along the surface crack was obtained by the equation:

KI ¼ FI

  pffiffiffiffiffiffi b b ; ; / B s ps r a

ð2Þ

where BS is the remote applied bending stress, s is the half arc crack length and FI is the boundary correction factor or the dimensionless SIF for mode I which is a function of the semi-elliptical crack shape (b/a), relative crack depth (b/r) and of the position points along the crack front defined by the parametric angle U, Fig. 3, which was determined taking into account the FEM calculations presented in [15,16]. As indicated before, limited results for the SIF of semi-elliptical surface cracks in round bars subjected to torsion stress levels are available, and the results presented in [15,16] are used in this paper. Therefore, SIF KIII will be obtained using the same geometric parameters that were used for mode I (bending):

K III ¼ F III

  pffiffiffiffiffiffi b b ; ; / T s ps r a

ð3Þ

where Ts is the remote applied torsion stress on the outer surface of the shaft, s is the half arc crack length and FIII is the boundary correction factor for mode III which is a function of the semi-elliptical crack shape (b/a), relative crack depth ( b/r) and of the position along the crack front determined by the referred parametric angle U in Fig. 3 that was determined taking into account the FEM calculations described in [15,16].

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4.2. Mixed-mode crack propagation under multiaxial loading 4.2.1. Long cracks Most of fatigue crack growth studies have been done on single-mode loading and usually are performed under mode I loading condition. However, single-mode loading seldom occurs in practice, and in many cases cracks are not normal to the maximum principal stress direction. The multiaxial mixed-mode fatigue crack growth is a common problem in many engineering structures and components. For long cracks, the propagation mechanisms are often analyzed using linear elastic fracture mechanics approach. Under combined axial and torsion loading, the surface corner of the semi-elliptical crack has mixed mode I + mode II, but at maximum depth of the semi-elliptical cracks has only mixed mode I + mode III [15,16]. For mixed-mode loading, one can assume that the fatigue crack growth rate may be expressed by the Paris law where the stress intensity factor range is replaced by an equivalent SIF range, DKeq:

da ¼ CðDK eq Þm dN

ð4Þ

There are many approaches proposed to define the equivalent SIF range DKeq for mixed-mode loadings. One of them is based on the addition of the Irwin’s elastic energy release rate, G parameters for the three loading modes:

h i1=2 DK eq ¼ ðGtotal EÞ1=2 ¼ DK 2I þ DK 2II þ ð1 þ v ÞDK 2III

ð5Þ

where DKI, DKII and DKIII are the stress intensity factors for mode I, II and III respectively and m is the Poisson’s ratio. Therefore, Eq. (5) can be used to correlate the fatigue long crack growth data in a mixed-mode loading condition. Under rotating or alternate bending a stress ratio R = 1 exists, therefore rmax = rmin and according to the recommendations specified on ASTM E647, in this study DKI = KImax will be used. For static torsion DKIII = 0. Fig. 8 shows the fatigue crack growth rate as a function of the mixed mode equivalent energy release range, determined as mentioned before, for several loading conditions: rotary bending, alternate bending and these bending conditions combined with steady torsion. Under cyclic bending conditions (rotating or alternate) the correlation between crack growth rate and stress intensity factor range is ds/dN = 5.1  1012(KImax)3.2. As observed in Fig. 8, a significant effect of steady torsion on FCG rate is observed. Results show that fatigue crack growth in rotating or alternate bending decreases with the level of steady torsion. For the final crack lengths measured, when crack depth approaches the radius of the specimen, a poorest correlation is obtained. This is due to the absence of small scale yielding conditions. This effect is much more visible under rotating bending than in alternate bending because the crack front at the surface of the specimen on rotating bending has different positions during one cycle (one rotation of the specimen). 4.2.2. Small cracks For small cracks, the mixed-mode models proposed for long cracks are inappropriate, because the SIF, K, can only be used in the case where the plastic zone size or the influence of the micro-structure is negligible. Therefore, substantial effort has been directed to model the small crack growth behavior [17–22], where the elastic–plastic loading conditions may be significant. Vormwald and Döring [20,21] proposed to use the range of cyclic J-integral as the driving force parameter for small crack growth:

da ¼ CðDJ eff Þm dN

ð6Þ

ds/dN [m/cycle]

10

10

10

10

-6

-7

-8

Mode I (rotary) Mode I (alternate) FIT Mode I (rotary) Mode (I+III)(altern) FIT Mode(I+III) Mode I (rotary)

-9

10

15

20

25

30

1/2

KImax [MPa m ] Fig. 8. Fatigue crack growth rates vs. KImax in rotating and alternating bending.

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da where dN is the crack growth rate, DJ eff is the cyclic J-integral, and C and m are fitting material parameters which were determined by crack growth tests on specimens containing long (several millimeters) cracks. In multiaxial non-proportional cycling, the crack experiences mixed-mode loading. At the stress-free surface, only mode I and mode II states exist. For a given cycle, the simplest mixed mode hypothesis is applied:

1=m m DJeff ¼ ðDJm I;eff þ DJ II;eff Þ

ð7Þ

As it is not possible to calculate the J-integrals in each of several thousands of cycles, an approximation is used by postulating a proportionality between J-integral and decomposed virtual strain energy density:

DJI;eff ¼ 2p  Y 2I  DW I  a DJII;eff ¼

p 1þm

 ðY II  U II;eff Þ2  DW II  a

ð8Þ ð9Þ

where YI and YII are the geometry functions of the stress intensity factor, DWI and DWII are the effective mode I and mode II virtual strain energy densities, respectively. In Eq. (9), the factor of UII,eff is between 0 and 1 and incorporates indentation and friction of the crack faces, which considers both the micro-structural effects and the influence of a normal stress on shear crack growth [21]. For elastic–plastic loading conditions, a strain-based intensity factor, DKI(e), is often used as shown in Ref. [22]. Any equivalent stress intensity mode I can be used as strain intensity models with the appropriate substitution of EDe or GDc for the normal stress and shear stress terms, respectively. This allows that an effective-strain intensity factor, DKeq(e), could be written in terms of the strain amplitudes and crack geometry factors. For example, Reddy and Fatemi [22] proposed the parameter DKCPA as a measure of crack growth rate in the form of an effective strain-based intensity factor range:

  rn;max pffiffiffiffiffiffi DK CPA ¼ YGDc 1 þ k pa

ry

ð10Þ

The crack growth rate can be expressed by the same form of Paris´ law:

da ¼ CðDK CPA Þm dN

ð11Þ

where C and m are material dependent constants. An alternative method of calculating the effective-strain intensity factor is to use the ASME code approach for multiaxial loading. The ASME Code Case N-47-23 is based on the von Mises hypothesis, but employs the strain difference Dei between time t1 and t2, where the equivalent strain range Deeq is maximized with respect to time:

Deeq ¼

 12 3 pffiffiffi ðDex  Dey Þ2 þ ðDey  Dez Þ2 þ ðDez  Dex Þ2 þ ðDc2xy þ Dc2yz þ Dc2xz Þ 2 ð1 þ mÞ 2 1

ð12Þ

However in certain conditions, Eq. (12) gives a lower equivalent strain range for out-of-phase loading than for in-phase loading, which estimates unconservative life for non-proportional fatigue. In this paper, an improved model based on the ASME code approach is proposed by considering the additional cyclic hardening due to non-proportional loading and correcting the strain range parameter for non-proportional loading path:

DeNP ¼ ð1 þ aF NP ÞDeeq

ð13Þ

where FNP is the non-proportionality factor of the loading path calculated by the MCE (Minimum Circumscribed Ellipse) approach [23], a is a material constant for additional cyclic hardening [24], Deeq is the strain range parameter calculated by Eq. (12), and DeNP is the corrected strain range parameter. The non-proportionality factor FNP of the loading path was computed as FNP = Rb/Ra, where Ra and Rb are the major and minor semi-axis of the minimum ellipse circumscribed to the loading path as described in [23]. In this paper, Eq. (13) is used for correlating small crack growth rates. Similar to Eqs. (10) and (13) can also be extended as a measure of crack growth rate and effective strain-based intensity factor range:

DK NP ¼ Eð1 þ aF NP ÞDeeq

pffiffiffiffiffiffi pa

ð14Þ

From the measured crack length vs. the number of cycles shown in Figs. 5 and 6, the crack growth rates can be calculated and correlated by the effective-strain intensity factor range expressed by Eq. (14), which is proposed in this paper, yielding the Eq. (15):

da ¼ CðDK NP Þm dN

ð15Þ

The equivalent strain range parameter was determined from a stabilized cycle computed by Finite Element Modelling of the geometry, taking into account the multiaxial loading conditions and the cyclic behavior of the material, including additional cyclic hardening due to multiaxial loading [25]. With this methodology it must be referred that both cyclic bending

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Fig. 9. Fatigue crack growth rates vs. DKNP in alternating bending and steady torsion for small cracks.

and static torsion influence the calculation of the strain intensity factor range, due to the strain effect on Eq. (12). For constant amplitude cyclic bending and static torsion load path, the non-proportional factor FNP = 0 and for normalized Ck45 used on this study the cyclic additional strain hardening parameter a = 0.30. Fig. 9 shows the plot of the small crack growth rate ( ds/dN) as a function of the strain intensity factor range (DKNP), for the data obtained with the specimens tested at the three cyclic bending stress levels 480, 450 and 420 MPa respectively and the static torsion stress 205 MPa as shown in Fig. 7. Some scatter of the data is observed which is a usual characteristic of small crack growth data and that can be explained due to factors such as the influence of grain boundaries for very short cracks and low stress levels or due to the existence of multiple small cracks for higher stress levels that when they link each other leads to an acceleration effect affecting the linearity of crack growth. Nevertheless fatigue crack growth parameters with a reasonable correlation can be calculated for small crack growth rate using Eq. (15) and the following correlated parameters for the Ck45 steel are C = 2  1011 and m = 1.8 were p achieved, for crack growth in m/cycle and DKNP in MPa m. 5. Conclusions The effects of steady torsion on fatigue crack growth under cyclic bending were studied for both small and long crack growth; the replica’s method was used to monitor crack initiation and small crack growth of the Ck45 steel specimens under multiaxial fatigue loading conditions. For long cracks a superimposed steady mode III loading to a crack growing in cyclic mode I leads to significant fatigue crack growth retardation. Crack growth rates decrease with increasing mode III for cyclic mode I (DKI) + static mode III (KIII) loading. An effective-strain intensity factor range based on the corrected strain range parameter through the use of non-proportional loading effect and additional cyclic hardening coefficient is proposed. This parameter was used to correlate the small crack growth rate data. The correlation achieved is reasonable and consistent with long crack data and can be considered a useful tool for calculation of fatigue life. References [1] de Freitas M, François D. Analysis of fatigue crack growth in rotary bend specimens and railway axles. Fat Fract Engng Mater Struct 1995;18:171–8. [2] Newman JC, Philips EP, Swain MH, Everett RA. Fatigue mechanics: an assessment of a unified approach to life prediction. ASTP STP 1122; 1992. pp. 5– 27. [3] Miller KJ. Metal fatigue – a new perspective. In: Argon AS, editor. Topics in fracture and fatigue. Springer-Verlag; 1992. p. 309–30. [4] Akhurst KN, Lindley TC, Nix KJ. The effect of mode III loading on fatigue crack growth in a rotating shaft. Fatigue Engng Struct 1983;6:345–8. [5] Hourlier F, McLean D, Pineau A. Fatigue crack growth behaviour of Ti–5Al–2.5Sn alloy under complex stress (mode I + steady mode III). Met Technol 1978:154–8. [6] Pook LP. Comments on fatigue crack growth under mixed modes I and III and pure mode III loading. In: Miller KJ, Brown MW, editors. Multiaxial fatigue, ASTM STP 853; 1985. p. 259. [7] Yates JR, Miller KJ. Mixed mode (I+III) fatigue thresholds in a forging steel. Fatigue Engng Mater Struct 1989;12:259–70. [8] Tschegg EK, Stanzl S, Mayer H, Czegley M. Crack face interactions near threshold fatigue crack growth. Fatigue Fract Engng Mater Struct 1983;16:71–83. [9] Tschegg EK. Mode III and mode I fatigue crack propagation behaviour under torsional loading. J Mater Sci 1983;18:1604–14. [10] Fonte M, Freitas M. Semi-elliptical crack growth under rotating or reversed bending combined with steady torsion. Fat Fract Engng Mater Struct 1997;20:895–906.

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