International Journalof Fatigue

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1 International Journal of Fatigue 29 (2007) International Journalof Fatigue Variable amlitude fatigue crack growth, exerimental results and modeling R. Hamam a, S. Pommier a, *, F. Bumbieler b a Laboratory of Mechanics and Technology Cachan, 61, Avenue du Prés. Wilson, Cachan, France b French Railway Agency (AEF-SNCF), 21, Avenue Salvador Allende, Vitry sur Seine, France Received 13 Setember 2006; received in revised form 31 January 2007; acceted 2 February 2007 Available online 16 February 2007 Abstract An incremental model was develoed so as to redict the growth of fatigue cracks under comlex load sectra. It contains a crack roagation law (da/dt = a dq/dt ) and a cyclic elastic lastic constitutive law for the cracked structure dq=dt ¼ f ð/ / J; / c X ; /c th ; / m X ; /m thþ. The crack growth rate da/dt is a rate of creation of cracked area er unit length of crack front. The lastic flow intensity factor rate dq/dt is function of the loading level / and of the thresholds for lastic deformation either within the monotonic or within the cyclic lastic zone. Two internal variables are introduced so as to define each threshold, the first one / X is associated with internal stresses, while the second one, / th, measures the effective threshold for lastic deformation in the crack ti region. The material arameters in the equations are determined using the finite element method. This identification was erformed for a 0.48%C carbon steel. Then various fatigue crack growth exeriments have been erformed in order to validate the model, monotonic fatigue crack growth exeriments at different stress ratios from R = 1 tor = 0.4, single overloads with overloads factor between 1.5 and 1.8, and bloc loads with X overloads every Y cycles, X and Y varying from one exeriment to another. The redictions of the model reroduce well exerimental results. Finally the model was alied to an industrial roblem: the growth of a semi ellitical crack at the surface of a train wheel. For this urose, load sectra were measured in situ on a train wheel, it came out that the model had to be extended to biaxial tension-comression and bending loading conditions, which was done. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Variable amlitude fatigue; Overloads; Stress ratio; Sectrum 1. Introduction Predicting fatigue crack growth in metals under random loadings is a difficult task, in articular because of load history effects, which are known for decades to stem from lastic deformation in the vicinity of the crack ti [1 6]. In addition, history effects are closely related to the elastic lastic behaviour of the material [6 9]. Let consider for instance the effect of a single overload. The material ahead of the crack ti yields during the alication of the overload. As long as the lastic zone of the overload is fully * Corresonding author. Tel.: ; fax: address: ommier@lmt.ens-cachan.fr (S. Pommier). constrained inside the elastic bulk, comressive residual stresses aear at unloading as a consequence of lastic deformation and limit the efficiency of subsequent fatigue cycles. This is at the origin of the well-known overload retardation effect. Besides, the Bauschinger effect of a material makes reverse lasticity within the overload lastic zone easier. In such a case, the overload retardation effect is less marked and somehow delayed [9]. Hence the need to estimate internal stresses in the crack ti region as recisely as ossible for each material. For this urose, a local aroach (FEM) is required so as to cature the very details of the elastic lastic cyclic deformation of the material in the crack ti region. But, in ractice, it is common to encounter fatigue lives that reach tens of millions of cycles. Global aroaches are then usually referred to /$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi: /j.ijfatigue

2 R. Hamam et al. / International Journal of Fatigue 29 (2007) local (and time consuming) ones. A method was therefore roosed so as to caitalize the advantages of both local and global aroaches, i.e. the quality and the recision of the comutation on the one hand and the reduced number of degrees of freedom on the other hand [12]. The idea of that method is as follows. The mode I dislacement field in the local coordinate system attached to the crack front is assumed to be deicted by its artition into an elastic field and a lastic field. Then, each art of the dislacement field is also assumed to be the roduct of a known reference field, function of sace coordinates only, and of an intensity factor, function of the loading conditions. This aroximation is only valid in the near crack ti region. The main advantage of this aroach is that it reduces drastically the number of degrees of freedom in the roblem. As a matter of fact, since the reference fields are assumed to be known, the remaining degrees of freedom are the osition of the crack ti and the intensity factors of the elastic and lastic arts of the dislacement field, which are corresondingly labelled the effective stress intensity factor ek I and the lastic flow intensity factor q. To summarize briefly the rocedure, the dislacement field is comuted using the finite element method and a suitable constitutive law for the material. Then the two intensity factors are extracted using the least square method. The results are analysed, at the global scale, by lotting the effective stress intensity factor versus the lastic flow intensity factor. Using this tye of results, just as one would use a stress versus lastic strain curve, it was ossible to build u an emirical model for the cyclic elastic lastic behaviour of the crack ti region at the global scale [12]. In this model, internal variables had to be defined, which measure the level of internal stresses and the effective thresholds for lasticity in the crack ti region either within the cyclic lastic zone or within the monotonic lastic zone [12 14]. Emirical equations are roosed for their evolutions with resect to the intensity factor of the lastic art of the dislacement field and with resect to crack growth. Using the finite element method and the aroximation discussed above, a cyclic elastic lastic constitutive model was build for the crack ti region at the global scale. This model rovides the lastic flow intensity factor rate as a function of the loading level and of the current values of the internal variables [12]. Besides, with the rough assumtion that the crack grows because of crack ti lasticity, a linear relation is assumed between the crack growth rate and the lastic flow intensity factor. If necessary, this equation can be modified to account for the effect of the environment, but this is not the object of the resent aer. In this aer, it is discussed how the model can be alied to an industrial roblem: the growth of a semiellitical crack at the surface of a train wheel. For this urose, load sectra were measured in situ on a train wheel. The model was identified for the material constituting the train wheel and the redictions of the model are comared to fatigue crack growth exeriments under variable amlitude loadings conditions. It came out that the model should be extended to biaxial tension-comression and bending loading conditions. These evolutions are discussed in the following. 2. Position of the industrial roblem Three asects are indisensable to master railway systems: design, manufacture and maintenance. Maintenance defines surveillance intervals which are defined at resent from field exerience. The aim of the French Railway Agency (AEF-SNCF) is to use fracture mechanic in order to satisfy to Euroean interoerability imerative of railway stock. The initiation of cracks at the surface of a wheel is usually attributed to foreign object damage (ballast imact). In such a case, cracks are semi ellitic and their growing ath is normal to the radial direction of the wheel. The aim of this research is to define the insection intervals of wheels, i.e. the number of kilometers necessary for a crack to grow from the smallest dimension detectable by non destructive technique (NDT) and the critical crack length. The diameter and the thickness of a wheel are in the order of one meter and 30 mm; whereas the initial crack length is in the order of 1 mm. A crack initiated at the surface of a wheel can therefore be considered as a semi-ellitical crack growing at the surface of a semi-infinite media, with a finite thickness. So as to characterize the local loading conditions of the train wheel, in situ measurements have been erformed by the French Railway Agency. First of all, a 3D finite element model of the wheel was built so as to identify the critical areas of the wheel under various loading conditions (running of the train along a straight line, along a curve...) (Fig. 1b). Then, strain gauges were glued symmetrically onto the internal and external faces of the wheel. Different radius and different angular ositions were instrumented, among which the sots identified as the most critical ones. Fig. 1. (a) Schematic of the mounting of a train wheel, (1) the rail, (2) the wheel and (3) the axle. (b) Finite element model of the wheel. Load case: running of the train along a curve.

3 1636 R. Hamam et al. / International Journal of Fatigue 29 (2007) The strain gauges signals were recorded as a function of time. These measurements allowed extracting the radial, hoo and shear comonents of the stress tensor along each side of the wheel during the train running. A short extract of the evolution of the radial stress on each side of the wheel is lotted in Fig. 2a. The train was running along a straight line at 120 km/h which is the least severe situation. In such a case, the external face is mostly in tension while the internal face is under comression in the critical area of the wheel. The finite element analysis of the wheel showed that the stress field across the wheel s thickness is not far from being linear in the area of concern. Therefore, it is ossible to extract the bending and the tension-comression comonents of the stress field across the wheel s thickness (Fig. 2b and c) in the critical area using in situ measurements. The tension-comression art of the stress field is close to constant amlitude loading with a stress ratio equal to 1 (Fig. 2b). But the bending art of the signal is highly variable, even when the train is simly running along a straight line (Fig. 2c). The same analysis can be erformed with the other comonents of the stress tensor. It is found that the shear comonent is usually small comared to the radial and hoo ones (Fig. 3a). And finally, the loading is observed to be almost roortional. The bending art of the hoo stress Fig. 2. Evolution of the radial comonent of the stress tensor during a few revolutions of the wheel. Case of a train running along a straight line. (a) Evolution of the radial stress as measured using strain gauges glued symmetrically onto the internal and the external faces of the wheel. (b) tension-comression art r rr ¼ðr ext rr r rr ¼ðr ext rr r int rr þ r int rr Þ=2 and (c) bending art Þ=2 of the radial stress along the thickness of the wheel. Fig. 3. Bending art r i ¼ðr ext i r int i Þ=2 of the measured stress comonents. (a) Comarison of the radial, shear and hoo stresses. The shear stress is negligible. (b) Hoo stress versus radial stress, the loading is more or less roortional.

4 R. Hamam et al. / International Journal of Fatigue 29 (2007) is around one half of the bending art of the radial stress (Fig. 3b). Finally, the roblem can be sketched out as follows. The crack s geometry can be aroached by that of a semi-ellitical crack in a semi-infinite late subjected to biaxial bending and tension-comression loadings [15,16]. The two bending comonents are roortional. During the running of a train, the bending art of the loading is highly variable. The variable nature of the loading arises from running conditions which vary among straight line, curve or slit switch. For a given running condition (straight line for instance) the correlation length in the stress versus time signal is well over a thousands wheel s revolutions. Finally, a train endures around two millions wheel s revolutions er day. The aim of this study is thus to simulate efficiently the growth of a semi-ellitical crack under variable biaxial tension-comression and bending. For this urose the arameters of the incremental fatigue crack growth model roosed by Pommier et al. [7,12] were identified for the material of the wheel. Then, the model was validated using the results of various variable amlitude fatigue crack growth exeriments. But the model had to be enriched so as to account for the high comression levels that are reached in a wheel disk and for the biaxial loading condition that arise from the axisymmetric geometry of the wheel. 3. Model 3.1. Background The mode I dislacement field in the near crack ti region is assumed to be deicted by its artition into an elastic field and a lastic field. Then, each art of the dislacement field is also assumed to be the roduct of a reference field, function of sace coordinates only, and of an intensity factor, function of the boundary conditions (loading, crack geometry...). The main advantage of this aroach is that it reduces drastically the number of degrees of freedom in the roblem. As a matter of facts, the reference fields are given in the coordinate system attached to the crack front. Therefore, the degrees of freedom in the roblem are reduced to: the location of the crack ti and the values of the two intensity factors (Eq. (1)), which are corresondingly labelled, the effective stress intensity factor ek I and the lastic flow intensity factor q: From a ractical oint of view, it is not necessary to have at ones disosal an analytical exression for the fields u e (x) and u (x). The reference elastic field u e (x) is the numerical solution of a revious finite element comutation using the same FE mesh, an elastic behaviour for the material and boundary conditions such as that the nominal stress intensity factor is equal to unity. Similarly a numerical reference solution can be constructed for the lastic field u (x) using a mathematical transform that erforms a artition such as that roosed in Eq. (1) (i.e. the Karhunen Loeve transform). However, analytical solutions are useful for interreting the results. At resent the best aroximation was obtained with the dislacement field around a climb dislocation [10] (Eqs. (2) and (3)): 1 u ðr; hþ x ¼ ðj þ 1Þ ½ðj 1Þ log r 2 cos2 hš u ðr; hþ y ¼ 1 ½ðj þ 1Þh 2coshsin hš ðj þ 1Þ ð3þ This solution corresonds to the insertion of a semi-infinite slit of material of thickness 1 in the half lane y =0,x <0 and it is worth to underline that :u (r,h = ) y = 1. Therefore, q measures the lastic art of the dislacement in the K-dominance area and can be also interreted as the lastic art of the CTOD. Besides, ek I measures the elastic art of the dislacement in the K-dominance area. Provided that the dislacement aroximated by Eq. (1) is taken as the dislacement between consecutive time increments (eulerian aroach), the error associated with this aroximation remains tyically below 10% [12]. This method enables us to generate the evolutions of q versus ek I (Fig. 4). The value of the stress intensity factor ek I calculated using this method differs slightly from the nominal stress intensity factor K 1 I. The difference Ksh I ¼ ek I K 1 I stems from the shielding effect of internal stresses ð2þ u FE ðx; tþ ek I ðtþu e ðxþþqðtþu ðxþ ð1þ where u FE (x,t) is the dislacement field as calculated using the FE method and associated with a very small variation around the current configuration of the crack, where u e (x) is the reference elastic dislacement field and u (x) the reference lastic field. For each time increment in the comutation, the rojection roosed in Eq. (1) is erformed using a ost treatment routine. Fig. 4. Evolution of the calculated value of the lastic flow intensity factor q versus the nominal stress intensity factor K 1 I.

5 1638 R. Hamam et al. / International Journal of Fatigue 29 (2007) at crack ti [11,12]. This difference is also found to vary linearly with q. Therefore, only the evolution of q needs to be modelled. In Fig. 4, the evolution of q is lotted against K 1 I during a cyclically increasing load sequence. After each load s reversal, there is a domain within which q is not varying. Within this domain, the behaviour of the cracked structure can be considered as elastic. The elastic domain of the cracked structure ((C) in Fig. 4) varies both in dimension and osition. Therefore, two internal variables are introduced to define the yield oint of the crack ti region, which are the osition and the dimension of its elastic domain. The equivalent in the continuum theory of lasticity would be kinematics and isotroic hardenings of the material. The dislacement of the elastic domain is a consequence of the develoment of internal stresses, the size of the elastic domain is an effective threshold for crack ti lasticity. Besides, a discontinuity in the evolution of K 1 I versus q is observed at oint D in Fig. 4, when the loading level exceeds the maximum load level reached reviously, or in other words, when it reaches the threshold above which the monotonic lastic zone extends again. Therefore an elastic domain is also defined for the monotonic lastic zone, by its size ((M) in Fig. 4) and its osition. Since, in mode I, crack closure occurs, the closure oint was taken as the definition of the osition of the elastic domain for the monotonic lastic zone. The dislacement of this oint is related to the growth of internal stresses at the scale of the monotonic lastic zone. To these evolutions is associated a model for the elastic lastic behaviour of the crack ti region at the global scale [12] which was build within the framework of dissiative rocesses. Firstly it was discussed that the driving force associated with q, denoted by /, is roortional to J the Rice s integral. Internal variables have to be defined, that measure the level of internal stresses ð/ c X ; /m X Þ and the effective thresholds ð/ c th ; /m thþ for lasticity within the cyclic and the monotonic lastic zones [12 14]. Emirical equations are roosed for their evolutions with resect to the variations of the lastic flow intensity factor oq and to crack growth oa. To summarize briefly, using the finite element method and the aroximation in Eq. (1), a cyclic elastic lastic constitutive model was build for the crack ti region at the global scale. This model rovides the lastic flow intensity factor rate as a function of the loading level and of the current values of the internal variables dq=dt ¼ f ð/ / J; / c X ; /c th ; /m X ; /m th Þ [12]. Besides, it is assumed that there is a linear relation between the crack growth rate and crack ti lasticity: da/dt = a dq/dt, where a is a constant. It is worth to mention that the crack growth rate da/dt, corresonds to the rate of creation of cracked area er unit length of the crack front. The equations of the model are not rovided here, but can be found in other ublications [12] Identification for a 0.48%C mild steel So as to identify the arameters of the model using the method described in Section 3.1, it is required to identify the constitutive behaviour of the material of the wheels. For this urose, ush ull secimens were machined from a high-seed train forged wheel, along the radial and along circumferential directions. The material anisotroy is found to be negligible. The material dislays a Lüders eak and a lateau in monotonic tension, but the eak disaears or is significantly reduced in cyclic tests. Therefore, the eak was not taken into account in the constitutive behaviour of the material. Finally, the constitutive model of the material includes three non-linear kinematics hardening terms and one non-linear isotroic hardening term (Fig. 5 and Table 1). Each non-linear kinematics hardening term obeys the Armstrong Frederick hardening rule (Eq. (4)): _X i ¼ 2 3 C i_e c i X i _ And the isotroic hardening is as follows (Eq. (5)): _R ¼ bðr 0 þ Q RÞ_ with Rj þ0 ¼ R 0 ð5þ With: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 _ ¼ 3 _e : _e ð6þ A very good agreement is obtained between the exerimental and the simulated stress strain curves (Fig. 5). ð4þ Fig. 5. cyclic-stress strain curves as measured on a 0.48%C steel, comarison between the exeriments and the simulation. Table 1 Material arameters E m R 0 Q b C 1 c 1 C 2 c 2 C 3 c 3 190, , , ,397 29

6 R. Hamam et al. / International Journal of Fatigue 29 (2007) Then various finite element analyses were erformed so as to identify the cyclic lastic behaviour of the cracked structure from the knowledge of the cyclic lastic behaviour of the material. A refined finite element model was build for the cracked structure and loading unloading sequences were simulated. The artition roosed in Eq. (1) was erformed at each increment of the comutation. And the evolution of q versus K 1 I is determined for the 0.48%C steel emloyed for the wheels. An examle of such a comutation is given in Fig. 6. During an unloading sequence from a oint (q 0,K 0 ), q is found to remain constant if K 0 K(t) is below a threshold b c, then the evolution is found to be roughly roortional to the square root of q 0 q. The same evolution was found whatever the value of the initial oint (q 0,K 0 ) either for a loading or an unloading sequence. This enables us to identify two arameters a c and b c emloyed in the evolution c X =@q of the osition /c X of the elastic domain of the cyclic lastic zone with resect to the variations of q, where / i ¼ðð1 m 2 Þ=2EÞK 2 i. It is useful to underline that if such a relation between q and K I is chosen, with the assumtion that the crack growth rate is roortional to the lastic flow intensity factor rate (Eq. (7)), and without taking into account any effects related to the monotonic lastic zone ð/ m X ; /m thþ, the crack growth rate obeys a relation which is a classical alternative to the Paris law (Eq. (8)) [17], where b c is laying the role of an effective threshold for fatigue crack growth: da dt ¼ a dq dt ð7þ DK ¼ a c Dq þ b c & Eq: ð7þ ) da dn ¼ 2a ðdk b a 2 c Þ 2 /ðdk DK eff th Þ2 ð8þ c An automated rotocol was set u so as to identify the entire set of arameters in the model from finite element comutations. The set of arameters identified for the 0.48%C mild carbon steel and the set of equations in the model are rovided in Aendix 1. At this stage, the constitutive model dq=dt ¼ f ð/ / J; / c X ; /c th ; /m X ; /m thþ of lasticity in the crack ti region is fully identified. Then, a constant amlitude fatigue crack growth exeriment is necessary to identify the tuning arameter a in Eq. (7). For this urose, the exeriment was simulated using the model and the results are lotted in a Paris diagram. First, it is observed that the calculated fatigue crack growth rate obeys the Paris law. Besides, the tuning arameter a modifies rimarily the coefficient of the Paris law and not its exonent. This exonent is close to 3. The coefficient a is then adjusted so as to obtain the best agreement between the simulation and the exeriment (see Fig. 7) Validation The model is identified using ush ull tests (and FE comutations) and constant amlitude fatigue crack growth tests. Then, once the model is identified, it also needs to be validated. For this urose, variable amlitude fatigue crack growth exeriments were conducted and comared with the simulations. CCT samle were used, with a thickness of 5 mm. The ositions of the two tis of the crack, on each side of the samle were determined otically. The measurement of the total crack length, 2a, was erformed every 1 mm. The following exeriments were erformed. First of all (Fig. 8a) the crack was grown u to a length of 2a =23 mm under a stress ratio R = 0 and a maximum stress level R max = 100 MPa. Then an overload either at R eak = 150 or 180 MPa was alied. And the crack was grown again at R = 0 and R max = 100 MPa. A very tyical delayed retardation effect is observed in both cases. A reasonable Fig. 6. Simulation of the very beginning of the unloading of the cracked structure from a maximum alied stress intensity factor K 0 = 21 MPa m 1/2 and an initial lastic flow intensity factor equal to q 0 =1lm. Fig. 7. Comarison between simulations and a constant amlitude fatigue crack growth exeriment at R = 0 in the 0.48%C mild carbon steel emloyed for the wheel s. Adjustment of the tuning arameter a in Eq. (7).

7 1640 R. Hamam et al. / International Journal of Fatigue 29 (2007) Effect of a comression hase below the closure oint The incremental model was identified and validated using a set of variable amlitude fatigue crack growth exeriments. However, the model had to be adated for the roblem of the train s wheel. As a matter of fact, the stress ratio is often below 1. It was therefore necessary to examine the effect of the comression hase below the contact oint between the crack faces Finite elements analyses and modelling The finite element method was emloyed so as to analyse the effect of a comression hase on the lastic flow intensity factor. For this urose various simulations have been erformed. For instance, the crack was oened, unloaded down to the closure oint and reloaded immediately, or reloaded after a significant comression hase below that closure oint. The differences in the (q,k I ) curves after the re-oening of the crack enables to discuss the effect of the comression hase on the internal variables of the model. Fig. 8. (a) retardation effect after a single overload. (b) Mean retardation effect of 1% of overloads at R eak = 1.5 R max within a block of either 100 or 1000 cycles. agreement is found between the simulations and the exeriments. Secondly, (Fig. 8b) the crack was grown under blocks of cycles with 1% of consecutive overloads at R eak = 1.5 R max either in blocks of 100 or in blocks of 1000 cycles. The mean retardation effect is maximum when the block length is of thousand overloads. A good agreement is found between the simulations and the exeriments. An exeriment was also conducted with 1% of overloads er block of ten thousand cycles. This exeriment is not shown here. The mean crack growth rate in that exeriment suerimoses with that obtained for 1% of overloads er block of 1000 cycles, and the simulations coincide with the exeriments. Thirdly, the simulation of such exeriments requires less than 2 min. This demonstrate that the model is alicable in an industrial context. Fig. 9. (a) Comuted evolutions of K I and q, with or without a comression hase below the closure oint. (b) idem, from the re-oening oint in a (K K 0,(q q 0 ) 1/2 ).

8 R. Hamam et al. / International Journal of Fatigue 29 (2007) An examle of such comutations is given in Fig. 9. Three main conclusions can be drawn from these comutations. First of all, there is a very significant effect of the comression hase below the closure oint on the evolution of q after the crack s re-oening (Fig. 9a). For the same variation of the stress intensity factor above the reoening oint the variation of q is at least 20% larger if a comression down to R min = 100 MPa is alied. However, if the crack is unloaded, the closure oint is not modified (Fig. 9a). This is the second imortant result: a comression hase below the closure oint does not modify the osition of this closure oint. The third result is that the threshold at which the monotonic lastic zone extends itself is not modified (Fig. 9b). And finally, though it is not obvious in Fig. 9, it is also found that the size of the elastic domain of the cyclic lastic zone is not modified by a comression hase below the closure oint. As a consequence, neither the osition / m X (closure oint) and the size /m th of the monotonic lastic zone, nor the size of the cyclic lastic zone / c th vary during a comression hase. The only internal variable that is varying is the osition / c X of the cyclic lastic zone. It is also shown through these comutations that the osition of the elastic domain of the cyclic lastic zone is merely varying like K min Comarison with exerimental results Therefore, in the model, the elastic domain for the cyclic lastic zone is allowed to be dislaced below the closure oint during the comression hase, this dislacement simly follows the alied stress intensity factor K min. Because of that dislacement, the evolution law of the lastic flow Fig. 10. Comarison between exerimental results (symbols) and simulations (lines) for constant amlitude fatigue crack growth exeriments at stress ratios R varying between 0.4 and 1. intensity factor above the closure oint is different. This simle modification of the model is sufficient to reroduce successfully the exerimental results (Fig. 10). The hysical meaning of the dislacement of the elastic domain of the cyclic lastic zone below the contact oint is not obvious. It is ossible to argue that closure corresonds to the first contact between the crack faces, which occurs at a rather large distance from the crack ti. But, in the near crack ti region, a bum remains behind the crack ti. When a comression hase is alied below the closure oint, this bum is erased. This imlies a modification of the state of internal stresses in the cyclic lastic zone. In our modelling, such a modification is modelled by a dislacement of the elastic domain of the cyclic lastic zone. 5. Biaxial loading conditions Now, the second feature of the industrial roblem is that the crack is subjected to biaxial loading. Let consider the case of a through thickness crack in an infinite sheet subjected to a biaxial loading (S x,s y ), the stress intensity ffiffiffiffiffi factor and the T-stress are exressed as follows: K I ¼ S y a and T = S x S y. The idea is to show that it is ossible to account for the effect of biaxial loading through the T-stress. For this urose, finite element comutations were erformed so as to study the effect of the T-stress on the (q,k I ) curves Finite element analyses For this analysis the material constitutive model that was emloyed was elastic, ideally lastic (E = 200 GPa, m = 0.3, Re = 400 MPa). Two finite element meshes were built with the same refinement at crack ti, but with two different crack dimensions 2a =12mm and 2a = 24 mm. The same method was emloyed in both cases so as to determine q from the dislacement fields. The two cracks were loaded u to the same value of the maximum stress intensity factor K I and then unloaded. In such a case the T-stress is lower at the ti of the smallest crack, since T ¼ S y ¼ K I = ffiffiffiffiffi a. The results of this comutation is lotted in Fig. 11a. It is obvious that the two cracks behave in a different manner. The lastic flow intensity factor q is larger for the smallest crack. If now the same values of K I and of T are alied, by alying a suitable biaxial loading state, the same evolution of q is obtained for the two cracks (Fig. 11b). Two conclusions arise from these results. First of all, there is a significant effect of a biaxial stress loading condition on the evolution of the lastic flow intensity factor and secondly this effect can be characterized by the arameter (T/K I ) (see Fig. 12). In the following, the biaxiality ratio is defined by the arameter (T/K I ). Since an automated rotocol was built u, so as to identify automatically the arameters of the model using finite element comutations, it is easy to erform a set of

9 1642 R. Hamam et al. / International Journal of Fatigue 29 (2007) crack in the order of 1 10 mm, growing in a semi-infinite sheet subjected to a biaxial stress (S x = S y /2). It is worth to underline that the constitutive equations of the model have been chosen so as to fit the finite element results but also so as to ensure that a unique set of arameter is identified in each case. Therefore, it is ossible to interolate the set of identified arameters with resect to the ratio (T/K I ) using for instance olynomials. The results are rovided in Aendix 2. The exressions of the evolutions of the arameters of the model with resect to the biaxiality ratio (T/K) were imlemented in the model so as to simulate the effect of a biaxial loading condition on fatigue crack growth Exerimental results Fig. 11. Comuted evolutions of K I and q, for two cracks (a = 6 mm and a = 12 mm) (a) subjected to the same value of K I, under a remote uniaxial loading condition and (b) to the same K I and the same T(T = 0). So as to characterize the effect of the T-stress, comlementary fatigue crack growth exeriments have been erformed on CT secimens. These secimens were machined from the train wheel. The same thickness (5 mm) was used for the CT and the CCT secimens (the exression of K I and T for both secimens are given in Aendix 3). Attention was aid to machine them with their crack lane lying in the same lane as that of CCT secimens. CT and CCT secimen dislay a very different T/K I ratio. While this ratio is negative in CCT secimen, it becomes ositive in CT secimens (see Aendix 3). In the exeriment it is observed that the fatigue crack growth rate is nearly 1.7 higher in CCT secimen in comarison with the crack growth rate in CT secimens. This confirms the role of the T-stress on the fatigue crack growth rate. Besides, this difference is observed both at R = 0.4 and R = 0, but is larger at R = 0.4. The simulations are in reasonable agreement with the exerimental results. Both the effect of the T-stress and the fact that this effect is larger at R = 0.4 than at R = 0 is reroduced by the model. 6. Conclusions Fig. 12. fatigue crack growth rate at R = 0 and R = 0.4 in a CT and in a CC. Both secimens have the same thickness (5 mm) and were machined with the crack lane lying in the same direction of the train wheel. Comarison with the simulations. identifications for a set of biaxiality ratios. This set of identifications was erformed for the tyical set of (T/K I ) ratios that are observed along the crack front of a semi-ellitical An incremental model for fatigue crack growth under comlex loading conditions, which was roosed in revious ublications, was identified, validated and modified so as to be alied to the roblem of a semi-ellitical crack growing at the surface of a train s wheel. For this urose, ush ull exeriments were conducted so as to identify the stress strain behaviour of the medium carbon steel emloyed for the wheels. The material is found to be isotroic and its behaviour was modelled by three non-linear kinematics hardening and one non-linear isotroic hardening. Then finite element analyses were conducted so as to comute the detail of the cyclic stress strain behaviour in the crack ti region. A ost-treatment routine was alied so as to identify the cyclic behaviour of the crack structure at the global scale from local FE comutations. More

10 R. Hamam et al. / International Journal of Fatigue 29 (2007) recisely, the evolutions of the lastic flow intensity factor were comuted as a function of the alied stress intensity factors in various cases. Then, an emirical model was associated to these evolutions, for which the arameters were identified for the material of the wheel. The model was imlemented and allows redicting the fatigue crack growth rate with the assumtion that the fatigue crack growth rate is directly roortional to the lastic flow intensity factor rate. Fatigue crack growth exeriments including overloads, and block loadings were comared to the redictions of the model and the results are satisfactory. However, so as to aly this model to the industrial case of the train wheel, some develoments had to be done. As a matter of fact, a train wheel was instrumented so as to measure strains in situ. It was found that, in the most critical area, the stress field across the wheels thickness can be considered as the sum of a biaxial bending loading case and of a biaxial tension-comression loading case. Therefore, the role of comressive stresses and of a biaxial stress state had to be considered. The biaxial stress state was accounted for through the biaxiality arameter (T/K). The comarison between exerimental fatigue crack growth rates within either CCT secimens or CT secimens, both at R = 0 and R = 0.4, showed that the T-stress has a significant effect on fatigue crack growth. The finite element method was emloyed so as to determine the evolutions of the arameters of the fatigue crack growth model as a function of the biaxiality ratio (T/K). These evolutions were imlemented in the model and it was found that exeriments and simulations are in good agreement. The effect of comressive stresses was studied using the finite element method and fatigue crack growth exeriments at various stress ratios, with R varying between 1 and 0.5. It was shown that neither the crack closure oint, neither the size of the monotonic lastic zone, nor the size of the cyclic lastic zone are modified by a comressive overload below the closure oint. However, the osition of the elastic domain of the cyclic lastic zone is dislaced. The model was modified so as to take into account this dislacement, and a good agreement is obtained between the simulations and the exeriments. Aendix 1. Set of equations We use the following relationshis between the stress intensity factor and u: rffiffiffiffiffiffiffiffiffiffiffiffi / ¼ A 2 K 2 1 m signeðkþ with A ¼ 2 2E Extension of the monotonic lastic zone: f m ¼ 0and d/ dt > 0 Cracking law: da ¼ a dq qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt 2 dt Plastic criterion ð/ xm < /Þ : f m ¼ ð/ / xm Þ 2 / m Flow rule: dq ¼ Consistence: f m ¼ 0; dfm ¼ 0 dt Evolution xm ¼ a xm / xm ffiffiffiffiffiffiffiffiffiffiffiffi ¼ A2 a 2 / m þ/ ffiffiffiffiffiffiffiffiffiffiffiffi xm m ¼ a / m / m þ/ xm Ab ¼ k a / xm þ k b :/ m with A 2 ¼ ð1 m2 Þ 2E Material arameters (a m, b m, a xm, a, k a, k b ) Initial values: / m0 ¼ A 2 b 2 m ; / x0 ¼ 0 Extension of the cyclic lastic zone: f c =0 Cracking law: da ¼ a dt 2 Plastic criterion: f c ¼ Flow rule: dq ¼ dq dt q Consistence: f c ¼ 0; dfc ¼ 0 dt Evolution c c xc @a xc A2 a 2 c / xc þd/ ffiffiffiffiffiffiffiffiffiffiffiffiffi c d / xc þd/ c / c m a / ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð/ / xc Þ 2 Abc 2 / c with d ¼ sign dq dt ¼ k a / xm þ k b / m with A 2 ¼ ð1 m2 Þ 2E Material arameters (a c, b c ) Evolutions of / c and / xc when the monotonic lastic zone is activated: ffiffiffi / ¼ / c þ / xc and / c ¼ A2 b c / 2 2 A b c Set of material arameters for T = 0: a m ¼ 17:3 MPa ffiffiffiffi m =lm b m ¼ 4:6 MPa ffiffiffiffi m a cf ¼ 38:2 MPa ffiffiffiffi m =lm b cf ¼ 5:8MPa ffiffiffiffi m a xm ¼ 1:5MPa ffiffiffiffi m =lm a ¼ 0:00093 MPa ffiffiffiffi m =lm k a ¼ 0:019 MPa ffiffiffiffi m =lm k b ¼ 0:0025 MPa ffiffiffiffi m =lm Aendix 2. Evolution of the material arameters for various values of the biaxiality ratio (T/K), and the fitted olynomials imlemented in the model.

11 1644 R. Hamam et al. / International Journal of Fatigue 29 (2007)

12 R. Hamam et al. / International Journal of Fatigue 29 (2007) Aendix 3. CCT secimen Exression of K [18]: K ¼ FY with F > 0 BW 1 2 Exression of K [18]: K ¼ FY if F > 0 BW 1 2 K ¼ 0 if F 6 0 Y ¼ With: B: secimen thickness W: secimen width F: alied force N Y: shae factor given by: 1 h 2ð0:707 0:007h 2 þ 0:007h 4 Þ and h ¼ a cos h 2W Exression of T [19]: T ¼ S y 0:997 þ 0:283 a w 3:268 a 2 a 3 þ 6:622 w w 5:995 a 4 w With: Sy: Alied stress MPa. CT secimen With: B: secimen thickness W: secimen width F: alied force N Y: shae factor given by: Y ¼ð2þhÞ 0:886 þ 4:64h 13:32h2 þ 14:72h 3 5:6h 4 ð1 hþ 1:5 and h ¼ a W Exression of T [19]: T ¼ S y 1:996 þ 10:169 a w þ 10:546 a 2 w With: Sy: Alied stress MPa given by S y ¼ F W B. References [1] Forman RG, Kearney VE, Engle RM. Numerical analysis of crack roagation in cyclic loaded structures. J Basic Eng 1967;89: [2] Pelloux RMN. Mechanisms of formation of ductile fatigue striations. Trans ASM 1969;62: [3] Elber W. The significance of fatigue crack closure. ASTM STP 1971;468:

13 1646 R. Hamam et al. / International Journal of Fatigue 29 (2007) [4] Wheeler O. Sectrum loading and crack growth. J Basic Eng 1972;94: [5] Schijve J. The significance of fractograhy for investigations of fatigue crack growth under variable amlitude fatigue. Fatigue Fract Eng Mater Struct 1999;22: [6] Skorua M. Load interaction effects during fatigue crack growth under variable amlitude loading a literature review Part II: qualitative interretation. Fatigue Fract Eng Mater Struct 1999;22(10): [7] Pommier S, Risbet M. Time-derivative equations for fatigue crack growth in metals. Int J Fract 2005;131(1): [8] Pommier S. Cyclic lasticity and variable amlitude fatigue. Int J Fatigue 2003;25(9-11): [9] Pommier S, De Freitas M. Effect on fatigue crack growth of interactions between overloads. Fatigue Fract Eng Mater Struct 2002;25:709. [10] Dundurs J. Elastic interaction of dislocations with inhomogeneities. In: Mura T, editor. Mathematical theory of dislocations. New York: ASME; [11] Lakshmanan V, Li JCM. Edge dislocations emitted along inclined lanes from a mode I crack. Mater Sci Eng A 1988;104(10): [12] Pommier S, Hamam R. Incremental model for fatigue crack growth base on a rojection hyothesis for mode I elastic lastic dislacements fields. FFEMS, in ress. [13] Pommier S. Cyclic lasticity of a cracked structure submitted to mixed mode loading. In: Khan AS, Kazmi R, editors. Proceedings of PLASTICITY 06, Halifax, Canada, Juillet, [14] Pommier, S. Cyclic lasticity of a cracked structure submitted to mixed mode loading. Plasticity 06 12th International Symosium on Plasticity and its Current Alications. Halifax, Nova Scotia, Canada, July; [15] Newman JC, Raju IS. An emirical stress-intensity factor equation for the surface crack. Eng Fract Mech 1981;15: [16] Wang X. Elastic T-stress solutions for semi-ellitical surface cracks in finite thickness lates. Eng Fract Mech 2003;70: [17] McClintock FA. Discussion to C. Laird s aer The influence of metallurgical microstructure on the mechanisms of fatigue crack roagation. Fatigue Crack Proagation, ASTM STP [18] Normalisation française, June 1991, Pratique des essais de vitesse de roagation de fissure en fatigue, A [19] Sherry AH, France CC, Goldthore MR. Comendium of T-stress solution for two and three dimensional cracked geometries. Fatigue Fract Eng Mater Struct 1995;18: