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ARTICLE IN PRESS Engineering Fracture Mechanics xxx (2006) xxx–xxx www.elsevier.com/locate/engfracmech Fatigue crack propagation from a cold-worked hole S. Pasta * Department of Mechanics, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy Received 8 February 2006; received in revised form 1 August 2006; accepted 7 August 2006 Abstract The cold expansion process is widely used to enhance the fatigue life of structures with fastener holes. Various studies assert that the cold expansion improves the fatigue strength of fastener holes; however, the improvement of fatigue life is diﬃcult to quantify. Therefore, the inﬂuence on fatigue life of cold-worked process was studied by numerical and experimental tests. Then, a parametric study on material hardening behavior and Bauschinger’s parameter was performed for several loading conditions in order to determine their eﬀect on crack growth propagation. The results of the numerical tests have exhibited a good prediction of the fatigue life of the component. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Cold expansion; Fatigue; Crack growth analysis 1. Introduction Fatigue crack growth in aircraft and naval components originates from stress concentration such as that produced by a fastener hole. Consequently, a cold-worked process has been used for over 25 years as standard technique to delay the propagation of fatigue cracks. The cold-worked process introduces beneﬁcial residual circumferential stresses into an annular region around the hole, and the presence of this compressive residual stress inhibits the growth and propagation of cracks. In fact, the eﬀect of residual stress may be explained using a fatigue crack closure model in which compressive residual stress reduces the eﬀective stress intensity factor, consequently reducing the crack growth rate. The cold-worked hole is modelled with diﬀerent techniques, but generally, the process is obtained by using increased pressure to plasticize an annular zone around the hole. The pressure on the surrounding material is realized by generating interference between the material of the drilled plates and the pressuring element, i.e. the mandrel. Such interference causes a stress state which decreases with increasing distance from the edge of the hole. When the mandrel is removed and superﬁcial pressure on the hole is erased, a residual stress ﬁeld is created due to diﬀerent elastic release out of all unequally deformed sheets. Therefore, a careful prediction of residual stresses is necessary to estimate the fatigue life of cracks. * Tel.: +39 0916657170; fax: +39 091484334. E-mail address: spasta@dima.unipa.it 0013-7944/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.08.006 Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS 2 S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx Nomenclature a b E I H n Pmax R KI Ua a m r radius of the hole outside radius of the plate elastic modulus interference Bauschinger’s parameter strain hardening exponent maximum load level of plastic anisotropy stress intensity factor radial displacement of the plate Boudiansky’s parameter Poisson’s ratio stress eﬀective Subscripts h refers m refers f refers y refers r refers to to to to to the hoop direction the mandrel the hole edge initial yielding the ultimate tensile stress Superscripts – refers to the unloading step The residual stress ﬁeld can be determined by means of analytical models and experimental procedures. The most important experimental technique is Sach’s boring [1]; in contrast, non-destructive methods such as X-ray are less accurate in predicting the reverse yielding zone. The literature presents ﬁve most representative theories for the evaluation of residual stress ﬁeld, and these analytical models are an example of the historical development of plasticity theory [2–6]. One of these theories has been developed by Boudiansky [7] and further modiﬁed by Guo Wanlin [2] in order to obtain the exact solution for a cold-worked hole considering a nonlinear response during the unloading step for a sheet having ﬁnite dimensions. In order to predict numerically the crack growth propagation, it is necessary to know the stress intensity factor for several crack lengths. The evaluation of the stress intensity factor is correlated to the presence of the residual stress ﬁeld, and in the literature, some solutions are proposed. Hsu [8] has found the stress intensity factor for two cracks growing from a hole using an integration process based on weight function. Then, he compared his solution with the results estimated by Newman [9] for the same geometry. In contrast, other authors proposed an approach based on the Green’s function [10]. The Grandt and Kullgren’s method [11] is founded on an early analysis carried out by Grandt [12] and supported by the work of Bueckner [13] and Rice [14]. Their model provides a solution regarding the stress intensity factor for single and double cracks through the thickness and corner cracks emanating from open holes. Finally, it may be asserted that these mentioned analytical techniques are very useful if combined with a computer program such as AFGROW to predict the life of fatigue crack propagation, which is especially helpful at the component design stage. The objective of the current investigation is to characterize the fatigue behavior of a cold-worked 5083H321. The fatigue life and the crack growth rate curves have been deduced for several cold expansion levels in order to determine the optimal value of cold expansion level. Then, experimental results have been compared with the numerical crack growth curves, verifying the validity of model to predict fatigue life. Moreover, the eﬀect of Bauschinger’s parameter and the strain hardening exponent on crack growth propagation has been studied by means of numerical prediction for several combinations of stress levels, expansion levels, Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx 3 and stress ratios. Such analyses were performed using the residual stress ﬁeld provided by the Guo’s elastic– plastic theory and the residual stress intensity factor achieved through the Grandt and Kullgren’s weight function. 2. Experimental procedure 2.1. Material properties A 5083-H321 aluminium alloy was chosen for this study. The amount of magnesio confers to the alloy the mechanical properties of hardness and strength. The material was treated with the H321 annealing treatment in order to remove the residual stress. This treatment represents a strain-hardened and stabilized condition, with a quarterhard condition after the thermal stabilization treatment. The plates were 5 mm thick with a 5.8 mm-diameter drilled hole. A set of tensile specimens with kt = 1 was loaded by a 20 kN uniaxial machine in order to obtain the tensile strength. From the stress-strain curve, the yield stress was obtained using the 0.2% proof stress, and the strain hardening exponent was obtained using the average values of various tensile tests. Table 1 shows the material properties of 5083-H321 aluminum alloy. 2.2. Cold expansion process A tapered, cylindrical mandrel for the cold-worked process was adopted. The mandrel taper is 1:100 permitting a gradual application of loading pressure. Such mandrel geometry was used in order to realize an equal pressure on the plate surface at the greatest value of interference. A split sleeve was introduced to protect the material at the hole edge from tearing or generally to protect the hole from strong friction caused by high interference. The introduction of the split sleeve was followed by the application of lubricating ﬂuid in order to reduce the friction of the element itself. The process consisted of pulling the mandrel through the pre-drilled hole; then, a ﬁnal reaming was carried out to obtain a circular hole after the expansion process. Three diﬀerent expansion levels were carried out by means of a set of three mandrels in order to pressurize 15 specimens. The expansion levels were 2.5%, 4%, and 7% of nominal interference, deﬁned as the ratio of the interference value to the hole radius. 2.3. Fatigue tests Fatigue crack growth tests were performed by an Instron 1603 resonance testing machine with a 5 kN load cell. The fatigue specimens were obtained from a 5 mm thick alloy plate, the dimensions of which are shown in Fig. 1. For fatigue tests, the loading frequency was approximately 120 Hz, while the stress ratio was R = 0.1. Crack growth length was monitored on both sides of the hole edge by means of penetrating liquids and high-resolution digital images. This method employs penetrating liquids which are applied over the surface of component. The ﬂuid enters the discontinuity, improving the quality of contrast of the digital image. At the same time, a Nikon camera monitors the crack growth propagation step by step. The crack growth rates were determined by measuring the ﬁrst crack length recognizable from digital image. The Wholer’s curves were achieved for the cold-worked and for the plain hole. Fig. 2 shows the fatigue life for the three expansion levels and for the plain hole. Table 1 Material properties of 5083-H321 aluminium alloy and parameters of numerical analysis Material properties n 6.93 R 1 b (mm) 25 rr (MPa) 360 ry (MPa) 255 m 0.33 E (GPa) 70.2 Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS 4 S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx Fig. 1. Shape and dimensions of fatigue test specimen (dimensions in mm). 10000 Stress range [MPa] Plain hole I=2.5% I=4% I=7% 1000 100 104 105 106 Number of cycles to fracture [cycles] Fig. 2. Fatigue curve for hole-cold expansion and plain hole. In all considered cases, the cold-worked process always improves the fatigue life, compared to the fatigue strength of the plain hole. Moreover, the eﬀect of cold working is highlighted at a higher number of cycles. The number of cycles to fracture of the plain hole at the same stress level is always lower than the number of cycles to fracture of cold-worked holes. In particular, it was observed that the fatigue resistance improves as the expansion levels increase from 2.5% to 4%, while the results obtained for 7% expansion levels are lower than the previous expansion levels. 2.4. Fracture surface analysis The fracture surfaces were analyzed in order to determine the crack front and in order to establish an eventual inﬂuence of mandrel entry direction on fracture. For a plain hole, the fracture surface shows two cracks propagating symmetrically and perpendicularly to loading direction (Fig. 3). Fig. 4 illustrates an example of the fracture surface for a cold-worked hole. Fig. 4 reveals that the fatigue crack initiates at the hole edge and near the mandrel entrance. Such a result conﬁrms that drilling direction inﬂuences the beginning of fatigue crack propagation according to various Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx 5 Fig. 3. Example of fracture surface for plain hole: the arrows indicate the direction of the crack front. Fig. 4. Example of fracture surface for 4% hole-cold expanded: the corner crack is showed on mandrel entrance. studies presented in the literature [15,16]. Moreover, the initial shape of the crack is a corner disk; therefore, this shape of corner crack is assumed for the next numerical analysis. 2.5. Crack propagation The crack length as a function of the number of cycles was derived by processing the digital images obtained through fatigue tests. Two diﬀerent cracks develop on the right (dx) and on the left (sx) of the hole; thus the crack length curves for both cracks are showed separately. The crack length curves are shown for each expansion level and for the plain hole at several loads Pmax. Fig. 5 shows an improvement of stable crack propagation for all stress levels before fracture. The variation of the slope of the crack length curve is correlated with the values of the stress intensity factors present at the crack tip. In fact, the compressive residual stress diminishes the eﬀective stress intensity factors, i.e. reduces the crack growth rate. When the crack propagates within the tensile residual stress ﬁeld, the residual stress intensity factor cumulates with the applied stress intensity factor, considerably enhancing the slope of the crack length curve. In Fig. 6 are shown the crack growth rate curves vs. DKI stress intensity factor range for a plain hole and for each expansion level. Fig. 6 illustrates that crack growth rate diminishes as the expansion level increases, and it is always lower than the crack growth rate of the plain hole. Moreover, when DKI is near to fracture toughness, the crack growth rate curves tend to approach a common value. 3. Analytical model 3.1. Residual stress model The residual stress distribution, which is sketched in Fig. 7, has been estimated with Guo’s model, which takes into account a non-linear response during the unloading step due to the removal of the mandrel. The assumption of the material’s yield limit in reversed loading was considered by Guo using the Bauschinger’s eﬀect [17–19]. Based on Budiansky’s solution, Guo found the closed-form solution for a generic plate of ﬁnite dimension. For the loading step, the residual stresses are explicated in terms of generic a, Budiansky’s parameter. On the other hand, for points which undergo plastic deformation caused by reversed loading, Guo Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS 6 S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx a 14 12 Crack length [mm] 10 8 6 4 2 0 50000 100000 Plain hole dx Plain hole sx I=2.5% dx I=2.5% dx I=4% dx I=4% sx 150000 200000 250000 300000 Number of cycles [cycles] b Crack length [mm] 14 12 10 8 6 4 2 0 40000 60000 80000 100000 120000 140000 160000 180000 200000 220000 I=2.5% dx I=2.5% dx I=4% dx I=4% sx I=7% dx I=7% sx Number of cycles [cycles] c 14 Plain hole dx Plain hole sx I=2.5% dx I=2.5% sx I=4% dx I=4%sx 12 10 Crack length [mm] 8 6 4 2 0 20000 40000 60000 80000 100000 120000 140000 Number of cycles [cycles] Fig. 5. Crack length curves for each expansion level and plain hole: (a) 30 kN, (b) 32 kN and (c) 35 kN maximum load. Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx 10-5 ( da dN da dN da dN da dN = 4 . 52 × 10—9 (ΔKI 4.82 ( = 3 . 46 × 10—11(ΔKI 3.47 3.84 7 Crack growth rate [m/cycles] 10-7 Plain hole 10-8 1 10 100 1/2 Stress intensity factor range [MPa m ] Fig. 6. Crack growth rate curves for plain holes and for 2.5%, 4% and 7% expansion levels. Fig. 7. Sketch of hole subjected to elastic–plastic deformation (I = 4%). considered the reverse yield criterion based on Ball’s model. Moreover, this model assumes material behavior following the Ramberg–Osgood relationship and yield model governed by the Mises–Hencky criterion. The parameters used for the analytical determination of residual stress proﬁles are given in Table 1. 3.2. Stress intensity factor model The weight function method proposed by Grandt [11] and others [12–14] simpliﬁes substantially the determination of the stress intensity factor (SIF) for a fastener hole. If the weight function is known for a cracked member, the SIF caused by an arbitrary applied load can be calculated by using the same weight function. It Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ( 10-6 = 5 . 86 × 10—13 (ΔKI ( = 2 . 47 × 10—12 (ΔKI 4.2 I=2.5% I=4% I=7% ARTICLE IN PRESS 8 16 14 S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx Stress intensity factor [MPa m1/2] 12 10 8 6 4 2 0 0.0 0.5 1.0 1.5 2.0 2.5 KI without cold-worked KI I=2.5% KI I=4% Distance from hole edge [mm] Fig. 8. Stress intensity factor for 4% and 2.5% expansion levels and for plain hole at 30 kN maximum load. can be shown [20] that the SIF for a cracked member subjected to the arbitrary applied load is the same as the SIF in a geometrically identical member with the local stress ﬁeld applied to the crack faces. The local stress ﬁeld is solved for an uncracked member that makes the stress analysis relatively easy. The eﬀective stress intensity factors vs. the crack length are shown in Fig. 8 for 2.5% and 4% expansion levels and for a plain hole. For a cold-worked hole, the stress intensity factor proﬁle displays an interesting form. First, it rises to a peak, then falls to a minimum, and then rises again to approach the ‘‘no residual stress’’ curve asymptotically. Such a proﬁle is due to the residual stress ﬁeld in the reverse yielding zone. In fact, the stress intensity factor of a cold-worked hole decreases as the amplitude of compressive residual stress increases. When the amplitude of compressive residual stress diminishes, the stress intensity factor for a cold-worked hole rises. 4. Analysis of crack propagation with AFGROW AFGROW is software developed by Harter [21] based on the concept of linear elastic fracture mechanics. The initial crack length was assumed from the knowledge of ﬁrst crack length observed in experimental activity, while the model geometry is a planar crack at the hole in the shape of a 90-degree sector of a disk. The material properties, the specimen’s dimension, and the load were introduced as used by experimental data. The Nasgro’s model developed by Newman and Forman [22] was used for the numerical analysis with the parameters of the AFGROW database. That is considered permissible because the prediction performed on plain holes have shown a good agreement between experimental and numerical crack length curves, conﬁrming the validity of the model. The residual stress intensity factor proﬁle was introduced as a function of the crack length. The values of residual stress were not introduced directly in AFGROW because the software overestimates the stress intensity factor. Crack propagation tests were performed for 2.5%, 4%, and 7% cold-worked holes. Fig. 9 shows the comparison between experimental and numerical propagation curves for a 4% expansion level and for each loading level. Fig. 9 shows that the numerical model is able to predict the crack propagation curve with good approximation either in the shape or in the number of cycles. Moreover, the stress intensity factor valued by means of Grandt and Kullgren’s weight function estimates with good quality the eﬀective stress at the crack tip. In particular, Fig. 10 compares the experimental and numerical crack length curves for 32 kN stress level. The numerical curve can be split into two zones, each of which represents the crack growth into compressive and tensile residual stress zones. The high magnitude of the compressive residual stress in the reverse Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx 12 9 10 42kN Exp. 35kN Exp. 35kN AFGROW 6 Crack length [mm] 8 42kN AFGROW 4 2 32kN Exp. 32kN AFGROW 30kN Exp. 30kN AFGROW 0 50000 100000 150000 200000 250000 Number of cycles [cycles] Fig. 9. Comparison between experimental (Exp.) and numerical (AFGROW) crack length curves for 4% expansion level and for several values of maximum load Pmax. 12 32kN AFGROW 10 32kN Exp. Crack length [mm] 8 6 4 Compressive residual stress 2 Positive residual stress 0 180000 190000 200000 210000 Number of cycles [cycles] Fig. 10. Experimental and numerical crack length curves for 32 kN at the 4% expansion level. yielding zone produces a low crack growth rate, and so this ﬁrst part signiﬁcantly delays the propagation of the crack, i.e. improves the fatigue life of the component. When the crack grows in the plastic zone, the residual stresses are lower and crack length is more evident, so that the delay is lower than the crack propagation in the reverse yield zone. When the crack propagates in the tensile residual stress zone, the crack growth rate increases, and the crack is not delayed. These zones are clearly distinguished along the experimental curve. Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS 10 S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx 5. Parametrical analysis The parametrical analysis was performed in order to estimate the inﬂuence of Bauschinger’s parameter and strain hardening exponent on crack growth propagation. The study was carried out with AFGROW for several stress levels and stress ratios at 2.5% and 4% expansion levels. In particular, the stress levels were 30 kN and 35 kN, while the stress ratios were R = 0.1 and R = 0.5. The numerical tests were performed as described in the numerical analysis presented in Section 4. In order to perform the numerical prediction of crack growth propagation, the residual stress curves were found for 2.5% and 4% expansion levels at several values of Baushinger’s parameter (H = 0, H = 0.5, and H = 1) and strain hardening exponent (n = 5, n = 8.5, and n = 6.93). Figs. 11 and 12 show the eﬀect of H and n parameters on the residual stress proﬁle at 2.5%, and 4% expansion levels. Fig. 11 shows that the H Baushinger’s parameter inﬂuences the magnitude of the compressive residual stress following the reverse yielding. Actually, the hoop stress values after the reverse yielding increase with 200 a 100 Hoop residual stress [MPa] 0 H=0 -100 H=0.5 -200 H=1 R=1 n=6.93 I=2.5% -300 -400 0 1 2 3 4 5 6 Distance from hole edge [mm] b Hoop residual stress [MPa] 200 100 0 -100 H=0.5 -200 H=1 H=0 R=1 n=6.93 I=4% -300 -400 0 1 2 3 4 5 6 Distance from hole edge [mm] Fig. 11. Residual stress curves for diﬀerent values of H Baushinger’s parameter: (a) 2.5% expansion level, (b) 4% expansion level. Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx 200 11 100 Hoop residual stress [MPa] 0 n=5 -100 n=6.93 n=8.5 -200 R=1 H=1 I=2.5% -300 -400 0 1 2 3 4 5 6 Distance from hole edge [mm] 200 100 Hoop residual stress [MPa] 0 -100 n=5 n=6.93 -200 n=8.5 R=1 H=1 I=4% -300 -400 0 1 2 3 4 5 6 Distance from hole edge [mm] Fig. 12. Residual stress curves for diﬀerent values of n strain hardening exponent: (a) 2.5% expansion level, (b) 4% expansion level. the variation of H from kinematic hardening (H = 1) to isotropic hardening (H = 0). Also, Bauschinger’s parameter modiﬁes the reverse yielded area which is reduced for decreasing H values. Nevertheless, Bauschinger’s parameter does not change the elastic–plastic boundary, i.e. the plastic radius rp. Fig. 12 shows that the n strain hardening exponent strongly aﬀects the residual stress proﬁle. The material hardening behavior changes either the reverse yielding or the elastic–plastic boundary, modifying the shape of the residual stress proﬁle. Above all, the magnitude of compressive residual stress corresponding to the reverse yielding increases for increasing values of the strain hardening exponent, while the plastic radius decreases because the plastic domain is smaller. When the residual stress proﬁles are known, the crack length curves as a function of the number of cycles can be achieved. Figs. 13 and 14 show the inﬂuence of H and n material parameters on crack growth propagation for several combinations of stress levels, expansion levels, and stress ratios. Fig. 13 illustrates the crack length curves of residual stress proﬁles of Figs. 11a and 12a at 2.5% expansion level for 35 kN and 30 kN stress levels and R = 0.1 and R = 0.5 stress ratios. It can be seen that the curves with Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS 12 14 35kN 12 H=1 n=6.93 H=0.5 n=6.93 H=0 n=6.93 H=1 n=5 H=1 n=8.5 R=0.1 I=2.5% 30kN H=1 n=6.93 H=0.5 n=6.93 H=0 n=6.93 H=1 n=5 H=1 n=8.5 S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx 10 Cracklength [mm] 8 6 4 2 0 80000 100000 120000 140000 160000 180000 Number of cycles [cycles] 14 35kN 12 H=1 n=6.93 H=0.5 n=6.93 H=0 n=6.93 H=1 n=5 H=1 n=8.5 R=0.5 I=2.5% 30kN H=1 n=6.93 H=0.5 n=6.93 H=0 n=6.93 H=1 n=5 H=1 n=8.5 10 Crack length [mm] 8 6 4 2 0 80000 100000 120000 140000 160000 180000 Number of cycles [cycles] Fig. 13. Crack length curves for 2.5% expansion level and for 30 kN and 35 kN stress levels at (a) R = 0.1 stress ratio and (b) R = 0.5 stress ratio. longer fatigue propagation always happen for H = 0, n = 6.93, for which the magnitude of compressive residual stress is largest. Such a result occurs for all stress levels and stress ratios. The diﬀerence in the number of cycles is less than 25% between H = 0, n = 6.93 and H = 1, n = 8.5 curves. In addition, the order of crack length curves does not change from high stress level to low stress level and for diﬀerent stress ratios. Fig. 14 shows the crack length curves of residual stress proﬁles of Figs. 11b and 12b at 4% expansion level. The best propagation curve happens for H = 1, n = 5 at all combinations of stress levels and stress ratios. For the 4% expansion level, the eﬀect of H and n parameters determines a higher variation in the numbers of cycles than for the 2.5% nominal interference. In fact, the value of such variation is 33%, occurring between H = 0, n = 6.93 and H = 1, n = 5 curves. Finally, it can be asserted that the longest fatigue crack propagation is achieved for high values of the difference between the plastic radius rp and the plastic radius of reverse yielding rÀ , i.e. for one more extended p zone of compressive residual stress. An even more important result is that a remarkable variation of Bauschinger’s parameter does not considerably change the fatigue life. This result can be considered interesting Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx 14 35kN 12 H=1 n=6.93 H=0.5 n=6.93 H=0 n=6.93 H=1 n=5 H=1 n=8.5 R=0.1 I=4% 30kN H=1 n=6.93 H=0.5 n=6.93 H=0 n=6.93 H=1 n=5 H=1 n=8.5 13 10 Crack length [mm] 8 6 4 2 0 100000 150000 200000 250000 300000 Number of cycles [cycles] 14 35kN 12 H=1 n=6.93 H=0.5 n=6.93 H=0 n=6.93 H=1 n=5 H=1 n=8.5 R=0.5 I=4% 35kN H=1 n=6.93 H=0.5 n=6.93 H=0 n=6.93 H=1 n=5 H=1 n=8.5 10 Crack length [mm] 8 6 4 2 0 100000 150000 200000 250000 300000 Number of cycles [cycles] Fig. 14. Crack length curves for 4% expansion level and for 30 kN and 35 kN stress levels at (a) R = 0.1 stress ratio and (b) R = 0.5 stress ratio. because this parameter must be determined with diﬃcult experimental test. Moreover, the worst combination of the considered parameters can cause signiﬁcant diﬀerences on the number of cycles, up to 33% as occurring for 4% expansion level. 6. Conclusions The fatigue behavior of a crack emanating from a cold–worked hole has been characterized for a 5083-H321 aluminium alloy at several expansion levels. The fatigue life was enhanced through the coldworked process, with the optimum fatigue strength evident at the 4% expansion level. In order to perform the numerical prediction, the compressive residual stress distribution was determined with Guo’s analytical model, and the residual stress intensity factor was evaluated using Grandt and Kullgren’s weight function. The crack propagation was evaluated with good approximation by AFGROW using a predicting model in which compressive residual stress acts to reduce the crack opening stress intensity Please cite this article in press as: Pasta S, Fatigue crack propagation from a cold-worked hole, Eng Fract Mech (2006), doi:10.1016/j.engfracmech.2006.08.006 ARTICLE IN PRESS 14 S. Pasta / Engineering Fracture Mechanics xxx (2006) xxx–xxx factor. 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