Modeling of fatigue contact damages formation in rolling bodies and assessment of their lifetime

Wear 271 (2011) 186–194

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Modeling of fatigue contact damages formation in rolling bodies and assessment of their lifetime O.P. Datsyshyn ∗ , V.V. Panasyuk, A.Yu. Glazov Karpenko Physico-Mechanical Institute, National Academy of Science of Ukraine, Naukova Str., 5, Lviv 79601, Ukraine

a r t i c l e

i n f o

Article history: Received 27 August 2010 Accepted 3 October 2010 Available online 14 October 2010 Keywords: Rolling contact fatigue Crack growth paths Damages formation Pitting Rail steels Lifetime

a b s t r a c t Within the framework of materials fracture mechanics the computational model for investigation of fracture processes and prediction of rolling bodies residual lifetime under their cyclic contact has been formulated. The step-by-step calculation of crack propagation paths using solutions of the singular integral equations of two-dimensional contact problems for solids with curvilinear cracks and also local fracture criteria under complex stress–strain state are the basis of the model; calculation takes into account the fatigue crack growth resistance characteristics of materials and tribojoints performance parameters. The algorithms of cracks propagation paths construction in the contact zone take into account a change of the stress–strain state, that is caused both by the cracks extension and a counterbody movement (load change) in a contact cycle. They also take into consideration the possible change of fracture mechanism in the cracks propagation process and friction between cracks faces. Taking into account the published experimental data on crack initiation and propagation under cyclic contact of solids, two criteria of material local fracture have been included in the calculation model: the criterion of generalized normal opening (  -criterion) and the criterion of generalized transverse shear. Conditions for evaluation of pitting particles sizes in dependence on the service parameters and characteristics of fatigue crack growth resistance have been formulated. Realization of the model is given for a wheel–rail pair under boundary lubrication in contact. Using the fatigue crack growth resistance characteristics of RSB 12 and 75KhGST (75ХCT) rail steels, their residual lifetime has been evaluated by pitting development. The corresponding curves of contact fatigue have been constructed. © 2010 Elsevier B.V. All rights reserved.

1. Introduction A majority of damages of rolling pair elements (wheel–rail systems, bearings, gears, mill rollers) under their cyclic contact are formed by crack propagation. In general, under complex loading in the contact region, cracks propagate along curvilinear paths. In this case one crack or a system of cracks grow, in dependence on service conditions, forming different types of damages of the rolling body surface layer: pitting, spalling, cracking, squat or dark-spot (surface darkening and sinking due to the nearsurface crack branching), etc. These damages often cause the growth of main cracks thus leading to the body fracture. This is particularly the case for the dark-spot defects typical of rails. In this context, many well-known scientists propose (see e.g. [1,2]) to determine durability of rolling pairs by the time (number of rolling cycles) to the mentioned damages formation (in particular pitting and spalling). That is why there is a need to develop models that can describe these processes qualitatively and quantitatively. Among the works devoted to the models

∗ Corresponding author. Tel.: +380 32 2540065; fax: +380 32 2649427. E-mail address: [email protected] (O.P. Datsyshyn). 0043-1648/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.10.023

of durability evaluation and formation of damages in rolling bodies papers [3–14] have to be mentioned. In particular, the models of pitting formation and evaluation of rolling surface durability under boundary lubrication conditions are considered in [3,4,8–14]. In the majority of these models the path of crack propagation and durability are predicted by an increment of a shear macrocrack (an edge or a subsurface) length in the contact region. Later when the crack goes in the region of tension the crack propagation path and durability are predicted in a simplified way by the direction (angle) of its initial deviation in an elastic half plane under action on its boundary of the model load. Only in [10,14] the authors evaluate the durability of gear teeth by the development of micropitting, using curvilinear crack propagation paths constructed by them. It should be also added that the results of performed investigations are given in recent review papers [13–16]. In this paper a model which is based on the approaches of fracture mechanics is considered. The elements of the model have already been presented to a certain extent in [17–19]. Thus, within the framework of materials fracture mechanics the computational model for investigation of fracture processes and the prediction of solids residual lifetime under their cyclic contact has been formulated. The step-by-step calculation of crack propagation paths using

O.P. Datsyshyn et al. / Wear 271 (2011) 186–194

Fig. 1. Computational scheme of the model.

solutions of the singular integral equations of two-dimensional contact problems for solids with curvilinear cracks (Fig. 1) and also local fracture criteria under complex stress–strain state are the basis of the model; calculation takes into account the fatigue crack growth resistance characteristics of materials and tribojoints performance parameters. The algorithms of cracks propagation paths construction in the contact zone take into account a change of the stress–strain state, which is caused both by the crack extension and a countrbody movement (load change) in a contact cycle. They also take into consideration the possible change of fracture mechanism in the cracks propagation process and friction between cracks faces. Taking into account the published experimental data on crack initiation and propagation under cyclic contact of solids, two criteria of material local fracture have been included in the calculation model: the criterion of generalized normal opening (  -criterion) and the criterion of generalized transverse shear. Conditions for evaluation of pitting particle sizes in dependence on service parameters and characteristics of fatigue crack growth resistance of materials have been formulated. On the basis of the formulated above model the propagation paths of one arbitrarily oriented edge [17–22] and a subsurface [19,23] cracks under dry friction, wetting and boundary lubrication at the stage of their propagation mainly by opening mode mechanism of fracture have been investigated. Under the same conditions in [19,22,24] the evolution of a system of the edge parallel and branched cracks has been investigated. Based on the comparison of the results obtained by this model and of the known experimental data some causes and regularities of such typical damages formation in rolling bodies as pitting, spalling, dark-spot, cracking, etc. have been established. It is shown that during rolling the values of the friction coefficient in the contact between bodies, dimensions of the contact region, contact pressure value, fatigue crack growth resistance characteristics of materials under mode I and mode II fracture, presence of service or random elements in the contact zone (water, lubricant, abrasive particles, etc.) which can be trapped by a crack, have a great influence on the character of contact surface fracture and sizes of crumbling particles. In this paper the model realization is illustrated with the example of a wheel–rail pair under boundary lubrication between the pair elements. Durability of the rail steel has been evaluated by the time of pitting formation with account of the periods of an edge macrocrack growth at first by the mechanism of shear (mode II) and then by opening mechanism (mixed mode I + II, with prevailing mode I). 2. Computational model Now the case of contact rolling interaction of two bodies one of which is damaged by cracks is considered. This body is modeled by a half plane weakened by a system of cracks (Fig. 1). The second body (counterbody) is modeled in the form of normal p(x, ) and

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tangential q(x, f, ) forces (tractions) distributed on the half plane boundary. These forces depend on the shape and dimensions of the counterbody, mechanical characteristics of the materials and contact surfaces, and specific features of contact interaction in contact cycles. Concentrated forces, elliptic (Hertz) distribution of forces, etc. are the most widely used schemes of model loading p(x). The slip/stick friction forces are taken into account in the form of tangential forces q(x, f, ). The simplest possible representation of the distribution of tangential forces is given by their relationship with the normal forces specified by the Amonton–Coulomb law q(x) = fp(x) under the condition of complete sliding between the bodies, where f is the coefficient of friction. It is known [25] that, in the fatigue fracture mechanics of materials, the fatigue life (number of loading cycles) N of damaged bodies is found by assessing its two basic components, namely, the period Ni to the initiation of the macrocrack of length l0 and the period Ng of the macrocrack growth from the initial length l0 to the critical (admissible) length lc . The period Ng is called residual service lifetime. Consider now this period. On the basis of the experimental data analysis, it can be concluded that the macrocracks (edge and subsurface) formed in the contact zone at first develop almost rectilinearly according to the mechanism of transverse shear (mixed mode I + II with prevailing mode II) and then curvilinearly according to the opening mechanism (mixed mode I + II with prevailing mode I). That is why the component Ng is represented in the form Ng = Ng + Ng ,

(1)

where



lc

Ng = l0 lc

v−1 (K (l), Cr ) dl,

 Ng =

(2)

v−1 (K (l), Cs ) dl,

l0

Ng

and Ng are the durations of the periods of macrocrack propagation according to the mechanisms of shear and opening, respectively, l0 , l0 , lc , lc are the initial and critical (admissible) lengths of the macrocrack in the stages of shear and opening, v = dl/dN is the crack growth rate, and l is the crack length. The parameter of the stress–strain state K(l, , ) responsible for fracture at the crack tip is chosen in agreement with the probable mechanism of fracture. In general, the dependences v(K) are established experimentally in the form of fatigue fracture diagrams (FFD) [25]. These diagrams are also used to find the constants Cr , Cs (r, s = 1, 2, . . .) characterizing the fatigue crack growth resistance of the material. Within the framework of linear fracture mechanics, the parameter K(l, , ) is determined in terms of the stress intensity factors (SIF) KI and KII by using the relations of the corresponding local fracture criterion: ¯ I (l, ), KII (l, ),  ∗ (l, )], K = K(l, ,  ∗ ) = K[K ¯ I (l, ), KII (l, )],  ∗ = F(l, ) = F[K

(3)

where  is a polar angle measured from the tangent to the crack at its tip and  * is the angle at which the parameter K attains its extreme value (or maximum in modulus) for fixed l and . Because the stresses and, hence, the quantities K(l, ,  * ) and  * (l, ) vary in rolling cycles, we assume that the crack grows in a cycle only when attaining the extreme value of the parameter K(l, ,  * ) (for  = * ) both by angle  and by the argument . Then the direction of crack growth at the point A (Fig. 1) is determined by the angle  ∗∗ =  ∗ (l, ∗ ). In this case, the range of the parameter K in a contact cycle must be larger that the range of the threshold of fatigue crack growth in the material Kth , i.e., the following conditions must

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be satisfied:(4)max|K(l, , *)| = |K(l, *,  ** )|, (5)K = maxK(l, ,  ** ) − minK(l, ,  ** ) > Kth . It is assumed that the maximum shear stresses are responsible for fracture at the crack tip in the stage of shear growth of the macrocrack. For the conditions of complex stress–strain state the criterion of generalized transverse shear is applied. In this case, relations (3) for the corresponding criterion take the form [4] ∗ 1 cos [KI (l, ) sin  ∗ + KII (l, )(3 cos  ∗ − 1)], 2 2 ∗ ∗  ∗ 2KII (l, )tg3 − 2KI (l, )tg2 − 7KII (l, )tg + KI (l, ) = 0. 2 2 2 (6)

K = KII (l, ,  ∗ ) =

The conditions of crack growth (4) and (5) are specified as follows:(7)max|KII (l, , *)| = |KII (l, *, *)|, (8)maxKII (l, ,  ** ) − minKII (l, ,  ** ) ≥ KIIth , where KIIth is the macrocrack fatigue growth threshold under the conditions of transverse shear (pure mode II). If the sign of equality is fulfilled, relationship (8) describes the condition of transition from the stage of the macrocrack initiation to the stage of its shear growth and, hence, can be used for the evaluation of the initial (incipient) crack length l0 . The maximum tensile normal circular stresses are responsible for fracture in the second stage of the macrocrack growth according to the opening mechanism. In this case, the parameter K(l, ,  * ) is described by the relations of the generalized normal opening criterion of fracture (  -criterion) as follows [26]: K = KI (l, ,  ∗ ) = cos3

 ∗ = 2arctg

KI (l, ) −

∗ 2



KI (l, ) − 3KII (l, )tg

∗ 2



,

(9)



KI2 (l, ) + 8KII2 (l, )

4KII (l, )

.

KI (l0 ) = KI,vth

or or

KII (l0 ) = KII,1−2 , KI (l0 ) = KI,1−2 ,

y1,j+1 (x1 ) = aj+1 (x1 − qj )3 + bj+1 (x1 − qj )2 + cj+1 (x1 − qj ) + dj+1 , qj = x1j − ıj ,

(13) (14)

where Kvth = K10−10 is the conventional threshold SIF range [25] corresponding to the fatigue crack growth rate vth = 10−10 m/cycle and K1–2 is the SIF range corresponding to the lower bound of the Paris rectilinear section of the fatigue fracture diagram (FFD) of the material. The path of the macrocrack growth is constructed by using a step-by-step procedure based on the algorithm proposed in [17]. To construct the path, two types of steps are introduced, namely, the main step connected with crack growth and the auxiliary step connected with the load movement in a contact cycle. In each stage

h , 2 j = 1, 2, . . . .

ıj ≥ 0, ıj ≈

qj ≤ x1 ≤ x1,j+1 ,

(15) The coefficients appearing in these equations are determined from the criterion relations of crack growth and the conditions of smooth conjugation of the path adjacent sections. After j steps of the path construction, the equations of its contour in the local coordinate system х1 O1 y1 (Fig. 1) can be written in the form



y1 (x1 ) =

y1k (x1 ) y1j (x1 )

qk−1 ≤ x1 ≤ qk+1 qj ≤ x1 ≤ x1,j

k = 1, . . . , j − 1 .

(16)

Now the residual service lifetime is determined from Eqs. (2) as follows: Ng ≈

The corresponding crack growth conditions take the form:(10)maxKI (l, , *) = KI (l, *,  ** ), (11)maxKI (l, ,  ** ) − minKI (l, ,  ** ) ≥ KIth , where KIth is the threshold of fatigue growth of the mode-I macrocrack. When the sign of equality is fulfilled, condition (11) can also be used as the condition of transition from the stage of the macrocrack growth according to the mechanism of shear to the stage of its growth according to the opening mode mechanism and, hence, for the evaluation of the critical length of the shear macrocrack lc . In the general case, the shear crack can grow even if condition (11) is satisfied. In this case, the indicated inequality represent the branching condition. Thus, it is also assumed that l0 = lc and the following condition to determine the critical length lc is used:(12)KI = KIfc , where KIfc is the critical range of the SIF KI upon attainment of which the process of crack growth becomes spontaneous. In the numerical calculations, the transition conditions (8) and (11) used to find the crack lengths l0 , lc and l0 depending on the structure of the formulas, which describe the diagram of fatigue crack growth rate in the material in relations (2), are represented in a somewhat different form, namely, KII (l0 ) = KII,vth

of the path construction, the step of crack path increment h is plotted from the crack tip in the direction specified by the angle  =  ** (Fig. 1). The auxiliary step  is used to find the extremes and the range of the parameter K in the contact cycle. At each path construction step, the quantities * ,  ** and K are regarded as constant and the SIF KI and KII are determined from the solution of the singular integral equations of the static (in the general case, contact) problem of the theory of elasticity for a half plane containing a curvilinear crack of different configurations. Each increment of the path is approximated by a third-degree polynomial described in a local coordinate system х1 O1 y1 by the following equation [at the (j + 1)-th step]:

jc 

lk vk–1 [K(l), Cr , Cs ],

(17)

k=1

where jc is the total number of crack growth steps prior to the attainment of the critical length; lk and vk are, respectively, the increment of crack length and the rate of its tip propagation at the k-th step. In constructing the crack growth paths for a system of cracks, i.e., from М several tips simultaneously, the increments of crack length hm are related to the velocities of motion of their tips [22,23]: hm /h1 = vm /v1 (m = 2,. . .,M). 3. Evaluation of rail steels lifetime by pitting formation It is known that service durability of a rail is limited greatly by damages caused by contact-mechanical fatigue on its surface, namely: the appearance of pitting and spalling. Already in 1935 Way [27] assumed that pitting was formed only in the presence of lubricant in contact between the rolling bodies. Therefore, the growth of edge cracks in rolling bodies under boundary lubrication conditions is considered below. As mentioned above in the introduction, results of previous investigations of this phenomenon are presented in the recent review papers [13–16], while in publications [3,4,8–14] some models of the pitting process in the rolling bodies are described. Thus, in study [3] it is assumed that pits formation under unidirectional rolling takes place first of all due to contact forces between bodies. That is, as a result of cyclic shear stresses action in the contact region the initial shear edge cracks propagate mainly linearly with a possible later curving of the crack path under the influence of tensile stresses and propagation by the opening mechanism. A role of the liquid (lubricant) is to decrease friction between crack faces [3]. A possible pressure of the lubricant on the crack faces does not cause crack growth to the rolling body surface with subsequent pitting of it. The authors of papers [9,8,11] consider that macrocracks can initiate both on the surfaces of a follower and a following bodies of a rolling pair but pitting is realized only on the follower surface in the presence of a lubricant,

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show [28] microcracks initiate on contact surfaces of rolling bodies (including rails), that is if attaining a certain length, they transform into macrocracks which in their turn grow almost rectilinearly by shear mechanism of fracture (predominantly by mode II). Under boundary lubrication a lubricant (or some other liquid) penetrates into these macrocracks, decreasing the friction between their faces and facilitating their mutual slipping. Further, during rolling the lubricant can divide contacting faces and wedge a crack. As a result the crack growth mechanism changes from shear to opening and, according to calculations of these cracks path development, this change causes the rolling surface pitting. So, a rail damaged by an edge macrocrack is modeled by an elastic half plane with an edge cut (Fig. 2a). Contact influence from the wheel (counterbody) side in each rolling cycle is  modeled by recip-

Fig. 2. Computational schemes.

if the initial shear crack is inclined to the surface with a sharp angle in the direction of a counterbody motion. Under such conditions the liquid penetrates into a crack. In this case it is assumed as in [3] that the initial macrocrack propagates at first linearly by shear mechanism of fracture. The length of the shear macrocrack can be rather large [11,28], commensurable with the size (length) of a contact region. Later, due to the lubricant pressure on the crack faces the opening fracture mechanism starts to act and the crack turns back to the surface. Note, that authors of the above papers [3,8,9,11] deal only with the evaluation of the stress intensity factors (SIF) and the 0∗ angles of initial inclination (Fig. 2) for edge rectilinear (or plane) cracks. Using these data they make a prediction of the crack path at the second curvilinear stage of its growth during which the pitting of the crack surface takes place or does not. In [3,5,8] attempts have been performed to evaluate a contact durability of the nearsurface layer of rolling bodies. However, such a prediction of the damage size and shape as well as durability can be false. It should be added here that in the authors’ papers [17–20] the edge crack growth paths are constructed in its opening stage under conditions of boundary lubrication in contact, while in [17–19] attempts are undertaken to evaluate residual lifetime under such conditions. In this work within the framework of the above formulated model a contact durability (lifetime) of rail steels (RSB12, 75KhGST) in rolling with boundary lubrication and under pitting development is evaluated. Lifetime (number of rolling cycles up to pitting appearance) is evaluated in the shear and opening stages with account of such important parameters as: fatigue crack growth resistance of steels, friction coefficient in contact between a wheel and a rail, the angle of the shear crack initial inclination, intensity and distribution of lubricant pressure, entrapped by the crack, on its faces. Two cases of pressure distribution are considered: uniform (Fig. 2a) and increasing linearly from the crack tip to its mouth (Fig. 2b). 3.1. Computation model elements The unidirectional cyclic rolling with uniform rate under boundary lubrication (friction) conditions is considered. Such conditions are realized when the oil interlayer thickness equals zero and it does not divide the rolling bodies, but at the same time lowers significantly the friction between them. As experimental results

rocal motion of Hertzian contact forces p(x) = p0 a2 − (x − x0 )2 /a in one direction along the half plane boundary with maximum pressure p0 in the centre of the contact region of length 2a. Friction forces arising under bodies slipping are taken into account by using tangential forces q(x) = fp(x), related with normal ones by a Coulomb’s law of friction coefficient f. Since a rail in a wheel–rail pair is a follower, the tangential forces act in the direction opposite to the wheel (counterbody) motion. Lubricant is considered to be a non-compressing liquid and its pressure on the crack faces, if in a contact cycle the counterbody closes its mouth, is modeled by two ways (Fig. 2): by linearly distributed pressure on the crack faces  rp  0 p1 (t) = (l − t) 1 − 2 , t ∈ L, (18) l where  = x0 /a, 0 ≤ r ≤ 1, l is current crack length; by uniformly distributed pressure on the crack faces p1 = rp0



1 − 2 .

(19)

It is assumed also that in each rolling cycle a crack increases only at such a location of a counterbody (at such  = * ), that provides the maximum value of the tensile circular stress intensity factor KI in the rolling cycle (in opening stage) or a modulus of the intensity factor of shear stresses |KII | (in shear stage) i.e. conditions (7) or (10) are fulfilled. Note that for many configurations of parameter values ˇ, ε, r, f, fc conditions (8) and (11) can be fulfilled simultaneously, thus indicating the possibility of the crack branching. However, for simplicity, consider below a non-branched crack, assuming that a shear branch after formation of a opening branch, slightly affects the last one, which becomes a dominat. It should be added that some elements of this model are presented in [18,19]. 3.2. Numerical results All results in this paper are obtained by numerical solution of singular integral equations of corresponding elasticity theory contact problems for a half plane with an edge curvilinear crack under action of moving Hertzian contact load on the half plane boundary, while crack faces are either in contact [29] or are subjected to action of normal pressure (see Fig. 2) [17,21]. 3.2.1. Stress intensity factors and edge crack growth paths At first the SIFs KI , KII , KI , KII are investigated for an initially rectilinear edge crack in an elastic half plane along which boundaries the contact load moves (Fig. 2). In this case it is assumed that the crack faces can be in contact with friction in some regions. Solve numerically by the method of mechanical quadratures (the Gauss–Chebyshev method) the singular integral equation of the corresponding contact problem [29]. Calculations are done for a rather wide range of parameters typical of the wheel–rail system,

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namely: friction coefficient between a wheel and a rail f = 0–0.2, relative crack length ε = l0 –a = 0.1–3.0, friction coefficient between crack edges fc = 0–0.2, crack inclination angle to the boundary (to the direction of tangential forces) ˇ = 135–160◦ . The process of the crack opening-closure in a rolling cycle (with the change of ) has been investigated. It has been found from the analysis of the obtained data that maximum values of SIF |KII | are attained when contact load moves near the crack mouth. The crack tip in this case is closed (KI = 0) and, according to the second relation (6), the angle  ** = 0. Thus, a shear crack will grow rectilinearly, what agrees well with experimental data [28] and proves the statement of the computational model. Besides, for a counterbody locations, if max|KII | is realized, crack faces are in contact practically along its all length. This allows us to simplify computations in the future, including also the lifetime evaluation at the shear stage of crack growth, using the solutions of exactly this problem. The analysis of numerical results illustrates that with the increase of friction between crack faces (in case of their contact along the whole crack) the SIF maximum values|KII | decrease (Fig. 3a) while with the increase of friction slipping between contact bodies the max|KII | significantly increase (Fig. 3b) for short cracks (ε ≤ 1.0) and remain almost the same for the long ones (1.0 ≤ ε < 3.0) [29]. In this case the change of the friction coefficients f and fc practically has no influence on the locations of a counterbody at which extremums are realized. With the crack length growth these locations change (Fig. 3c) and especially vary (increase) the extremum

values of SIF KII , and also of KII . However, additional calculations show that for ε > 2.0 the value of max|KII | decrease (as is naturally expected). As it is seen from Fig. 3d the crack orientation in the considered range of its change has a rather significant influence on the counterbody position at which |KII | extremums are realized, and also on their value. To find out the most favorable angles for shear edge cracks development (arising) the curves for the dependence of the range FII (ε) for different angles ˇ = ␲/9–8␲/9 (ε = 0.1–3.0) were constructed. Curves for angles ˇ ≈ 5␲/6–8␲/6 were located above all other curves [19,29]. This agrees well with experimental and engineering data [3,28] that initial edge shear cracks in a follower, in particular in rails, are formed at an angle of 20–30◦ to the rolling surface (in direction of a following counterbody movement). To evaluate SIF and to construct the edge crack paths in a half plane under action of a moving model contact load on its boundary and also under simultaneous crack edges wedging by a normal pressure (liquid), a singular integral equation [17,21] of the first basic problem of the elasticity theory has to be solved. Calculated results of SIF KI that is responsible for the opening mechanism of crack propagation, are presented in Fig. 4. It is seen in this figure that in all four cases (Fig. 4a–d) substitution of the uniform pressure on the crack faces (dashed lines) by the linear one (solid lines) decreases the SIF value significantly, especially their maximum values which decrease almost in two times. Note also, that max FI is realized predominantly as soon as contact load closes the crack

√ Fig. 3. Dependence of normalized SIF FII = KII /(p0 a) on a counterbody location with respect to the crack mouth for different: (a) friction coefficients fc between a crack faces; (b) friction coefficients f between a wheel and a rail; (c) relative lengths of the initial crack ε = l0 /a; (d) angles ˇ of initial rectilinear crack inclination.

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√ Fig. 4. Dependence of normalized SIF FI = KI /(p0 a) on a counterbody location with respect to the crack mouth for different: (a) angles ˇ of initial rectilinear crack inclination; (b) relative lengths of the initial crack ε = l0 /a; (c) friction coefficients f between a wheel and a rail; (d) values of parameter r of liquid pressure intensity on the crack faces. — linear pressure on the crack faces; – – – uniform pressure on the crack faces.

mouth, moving from right to left. For the relative length of the initial crack in the range ε = 0.1–0.5 max FI is realized at  = 0.8–0.9. For longer cracks this moment occurs later (see Fig. 4b). For ε = 1.0 max FI is realized at  = 0.7. It should be added here that the obtained SIF data agree well with the results of [8,9]. The growth paths of the initially rectilinear edge crack by opening mechanism during rolling contact under action of liquid pressure (water or lubricant) on the crack faces are considered now. The paths were constructed stepwise using the   -criterion of local fracture (see Eq. (9)) in dependence on the change of the initial crack inclination angle ˇ, its relative length ε and parameter r, that characterizes the intensity of lubricant pressure on the crack faces. Fig. 5 shows the paths appearing under uniform pressure on the crack faces (dashed lines) and paths which are formed under linear distribution of the pressure (solid lines). As shown in (Fig. 5a and b) if the pressure on the crack faces changes from the uniform to the linearly distributed pressure, the path length decreases and the crack grows more sharp to the boundary at different inclination angles and relative lengths of the initial crack (here r = 1.0). However, in the case of the lubricant pressure intensity decrease (r = 0.1; 0.2), the crack length for a linear distribution is much greater than under uniform distribution (Fig. 5c). Please, note that the SIF KI (l) along the crack paths for all values of r (0 < r ≤ 1.0) is less for the linear distribution of the pressure compared with the uniform one.

It should be added here that for the construction of plots FI (ε) (Fig. 4) and when constructing the crack propagation paths by a opening fracture mechanism, the steps of the crack construction are in most cases as follows: h1 = h/l0 = 0.03 and  = 0.01. A maximum order of a system of complex algebraic equations to which singular integral equations of the elasticity theory problems have been reduced by the method of mechanical quadratures, is M = 100. 3.2.2. Lifetime Applying the computation model presented in Section 2, the residual lifetime of the rail steel by pitting development under rolling with lubrication is evaluated. So, a number of rolling cycles during which the initial crack increases from length l0 to the critical length lc may be established. The values of length l0 , lc = l0 are found on the basis of the first relations in Eqs. (13) and (14), and lc —from condition (12). Calculations are done for the rail steel with a lamellar pearlite structure: for shear – by the fatigue fracture diagram for RSB12 steel [28], for opening – by the diagram for 75KhGST steel [30]. Both diagrams are described by the Yarema–Mykytyshyn formula [25]:



v = v0

K − Kth Kfc − K

q

,

Kth ≤ K ≤ Kfc .

(20)

As a result the following characteristics of fatigue crack growth resistance of steels are obtained (Table 1).

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O.P. Datsyshyn et al. / Wear 271 (2011) 186–194 Table 1 Characteristics of fatigue crack growth resistance of the rail steels. Steel grade

Fig. 5. Paths of an edge crack propagation versus the change of: (a) initial inclination of crack angle ˇ; (b) relative length of the initial crack ε = l0 /a; (c) intensity of lubricant pressure on crack faces; f = 0.1. — linear pressure on the crack faces; – – – uniform pressure on the crack faces.

K1−2

v0

Kth √ MPa m

Kfc

q

Mode II RSB12

13.01

92.4

16.5

3.84 × 10−7

1.41

Mode I 75KhGST

4.3

37.0

5.2

1.06 × 10−7

1.26

m/cycle

In the calculations a half length of the contact region a = 7 mm, f = 0.1, ˇ = 5␲/6, fc = 0.1 was assumed. In Tables 2 and 3 the calculation results of the durability for both cases of liquid pressure distribution along the crack faces are given. The maximum values of contact pressure p0 were chosen according to the service data for a wheel–rail system. Using the data in Tables 2 and 3 the contact fatigues curves are plotted in Fig. 6 for both distributions of the liquid pressure on the crack faces. When comparing the two curves (Fig. 6) the conclusion is drawn that in the range of p0 values 800–1500 MPa under uniform distribution of liquid pressure on the crack faces the durability decreases by 45% on average compared with the linear distribution with equal values of other parameters. Fig. 6 also presents generalized experimental data obtained in papers [31,32], that correlate with our data (if to assume that the period of edge cracks initiation in rolling bodies with lubrication is by one or two orders lower than the period of their growth to crumbling of the rolling surface). Finally, it can be added that based on the macrocrack length values l0 , lc = l0 , lc (Tables 2 and 3) and the angle ˇ it is easy to evaluate the crumbling particles sizes. For example, at p0 = 1000 MPa under linear distribution of pressure the crumbling pit depth is 3.2 mm, and its length is 11.8 mm. Under uniform distribution we find correspondingly that the pit depth is 2.2 mm, and the length is 8.4 mm. For comparison the experimental val-

Table 2 Dependence of residual lifetime on maximum value p0 of contact pressure under linear distribution of lubricant pressure along the crack faces. p0 (MPa)

l0 (mm)

l0 (mm)

lc (mm)

Ng × 10−6

Ng × 10−6

Ng × 10−6

600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800

3.038 2.328 1.831 1.515 1.279 0.966 0.824 0.723 0.627 0.567 0.522 0.485 0.441

11.07 8.71 7.07 6.02 5.20 4.49 3.98 3.58 3.27 3.01 2.80 2.62 2.47

16.84 13.31 11.11 9.66 8.17 7.30 6.69 6.19 5.39 5.06 4.80 4.58 4.38

2.3913 1.6405 1.6161 0.9267 0.8098 0.8628 0.6250 0.5290 0.4485 0.3621 0.3003 0.2719 0.2894

1.6265 1.5356 1.4780 1.477 1.4103 1.3050 1.1963 1.1194 1.0403 0.9882 0.9290 0.8820 0.8292

4.0178 3.1761 3.0942 2.4046 2.2201 2.1678 1.8212 1.6484 1.4888 1.3503 1.2293 1.1539 1.1186

Table 3 Dependence of residual lifetime on maximum value p0 of contact pressure under uniform distribution of lubricant pressure along the crack faces. p0 (МPа)

l0 (mm)

l0 (mm)

lc (mm)

Ng × 10−6

Ng × 10−6

Ng × 10−6

600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800

3.038 2.328 1.831 1.515 1.279 0.966 0.824 0.723 0.627 0.567 0.522 0.485 0.441

5.629 4.628 3.931 3.447 3.092 2.715 2.424 2.192 2.002 1.844 1.709 1.592 1.489

9.18 7.38 6.33 5.87 5.14 4.62 4.18 3.83 3.51 3.24 3.14 3.04 2.93

1.7282 1.3837 1.4772 0.8421 0.7536 0.8197 0.5881 0.4520 0.3802 0.3395 0.3065 0.2885 0.2742

0.7583 0.6402 0.5771 0.5345 0.4837 0.4266 0.3897 0.3608 0.3381 0.3195 0.3048 0.2929 0.2832

2.4865 2.0238 2.0542 1.3757 1.2373 1.2462 0.9778 0.8129 0.7183 0.6590 0.6113 0.5814 0.5575

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ucts, etc.), propagate to the contact surface and cause its pitting (Fig. 5). 5. Theoretical (prediction) curves of contact fatigue (dependences of the number of rolling cycles Ng on maximum values of contact Hertzian pressure p0 (Fig. 6)) for rail steels by the pitting formation criterion, using the crack growth resistance characteristics of these steels in shear and opening mode of fracture have been constructed. These curves correlate with experimental data. 6. Conditions for establishing of the pitting particle sizes, with account of crack growth resistance characteristics and service parameters of rolling pairs have been formulated. Thus established sizes of crumbling particles/pits for rails agree well with engineering and experimental data. 7. The proposed model allows us to predict the processes of crack propagation, contact lifetime of the rolling pair elements, sizes and shape of the particles of their damage, using the computer experiment only. In particular, in the case of a wheel–rail pair in rolling under boundary lubrication the proposed calculation algorithm allows to establish easily the above values for a wide range of basic service parameters (f, fc , r, p0 , a, ˇ). Fig. 6. Contact fatigue curves for the case of uniform (dashed line) and linear (solid line) pressure on the crack faces.

ues in [16] are presented: the depth is 0.5–5 mm, and the length is 10–12 mm. If can be seen that the higher contact load (higher p0 ) is the more quickly pitting arises (the lower is durability) and the finer crumbling particles will be.

Acknowledgements This work was financially supported partly by the National Academy of Science of Ukraine. Also authors are very grateful to the Rostyslaw J. Lewyckyj from USA (North Carolina) for financial support of this work. References

4. Conclusions 1. The calculation model for investigation of the processes of fracture and assessment (prediction) of the rolling bodies lifetime under their cyclic contact has been proposed within the framework of fracture mechanics. The construction of the crack propagation paths in the contact region with account of local fracture criteria, cyclic crack growth resistance characteristics of material and service parameters of a rolling pair is the key element of the model. Two criteria of local fracture are used: a criterion of generalized opening (  -criterion) and a criterion of generalized transverse shear. 2. The model is developed in detail for the case of rolling under boundary lubrication. Residual contact lifetime (with no account of the original macrocrack initiation stage) is assessed by the criterion of a pitting formation on the rolling surface. It is assumed that in the first stage of pitting formation an edge macrocrack in a follower body propagates by the transverse shear mode in the direction of a counterbody motion, while in the second stage, under pressure of the liquid filling the crack – by the opening fracture mechanism. The liquid pressure is modeled in two ways: by uniformly distributed normal pressure along the crack edges or by linearly distributed pressure. 3. Numerical calculations are presented for a wheel–rail system, assuming that a rail in this pair is a damaged follower body. Calculations have proved (Tables 2 and 3) the expectation that the increase of friction between the edge crack faces significantly decelerates its growth and causes the increase of the rail lifetime (owing to the shear stage of the crack propagation). The augment of the liquid pressure on the crack edges accelerates its growth and decreases the rail durability (owing to the opening stage of fracture); an angle of the crack initial orientation has also a significant effect on the rail durability. 4. The known Way’s hypothesis [27] according to which the main cause of pitting formation is the presence of lubricant in the contact region has been proved theoretically: the edge inclined cracks, wedged by any type of pressure (water, oil, wear prod-

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