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Analysis of crack growth under rolling-sliding contact
Accepted Manuscript ANALYSIS OF CRACK GROWTH UNDER ROLLING-SLIDING CONTACT W. Daves, M. Krá čalík, S. Scheriau PII: DOI: Reference:
S0142-1123(18)30611-X https://doi.org/10.1016/j.ijfatigue.2018.12.006 JIJF 4920
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
15 October 2018 3 December 2018 5 December 2018
Please cite this article as: Daves, W., Krá čalík, M., Scheriau, S., ANALYSIS OF CRACK GROWTH UNDER ROLLING-SLIDING CONTACT, International Journal of Fatigue (2018), doi: https://doi.org/10.1016/j.ijfatigue. 2018.12.006
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ANALYSIS OF CRACK GROWTH UNDER ROLLING-SLIDING CONTACT W. Daves1,2,*, M. Kráčalík1, S. Scheriau3 *,1
Materials Center Leoben Forschung GmbH, Roseggerstraße 12, 8700 Leoben Institute of Mechanics, Montanuniversität Leoben, Franz-Josef-Strasse 18/III, 8700 Leoben 3 voestalpine Schienen GmbH, Kerpelystraße 199, 8700 Leoben
2
* [email protected] ABSTRACT
The work focuses on the calculation of shear crack growth with crack driving forces regarding cyclic elastic-plastic material behaviour. The load influence is described by the maximum contact pressure, and slip. The contact pressure and slip distribution in the contact patch are calculated using a 2D finite element model representing a wheel-rail contact. The configurational force concept is used to determine the crack driving forces in a cyclically plastified rail material during the wheel passage. The dependence of the crack driving force and the crack path on the loading conditions is shown. The results are qualitatively compared to experiments.
KEYWORDS Fatigue crack, shear crack, rolling-sliding, wheel-rail, plasticity, crack path
1 INTRODUCTION
Cracks initiated and growing under rolling-sliding contact have to be investigated with new or extended methods. Due to the nonlinearities involved namely, contact mechanics and plastic deformation of the contacting bodies the solutions for crack growth are hard to obtain and produce severe numerical problems. Other influences as surface roughness, wear, cyclic short time and long-time behaviour of the material, and changing loads due to changing contact conditions have to be regarded within the crack models as they may strongly influence the crack initiation time and growth behaviour.
The work aims to present a calculation method for growth of cracks loaded repeatedly by a rollingsliding contact located in a steadily deforming material. This is done using a two-dimensional finite element model where cracks are modelling with given depth, length and inclination to the surface in an undeformed material. It is now - numerically supported - assumed that after some similar cyclic loadings a typical residual stress field and a crack tip plastification will develop near such prescribed cracks and their crack tips. It can be shown that the calculated crack driving force will nearly not change after some initial cycles even the surrounding of the crack will deform further due to repeated
contacts. Based on those findings, a parametric study can be performed which shows how the crack driving force is changing with crack length and depth for varying contact pressure and slip conditions. Such results can be used for the assessment of existing cracks and their crack growth rates and paths.
The influence of contact loads on such cracks has already been analysed in literature [1], [2], [3], [4], [5], [6], [7], [8]. The influence of compressive and residual stresses and friction between the crack faces is investigated in [9]. Recent papers dealing with cracks using similar methods are [10], [11]. A paper dealing with cracks in the anisotropically deformed material is presented in [12]. Practical investigations of cracks in rails are presented in [13]. Crack systems can also be produced in experiments like in a wheel-rail test rig or twin disc tests, see Fig. 1 and [14], [15], [16], [17]. How to deal with cracks in rails in practical applications is presented in [18], [19], and a numerical investigation of crack systems is available in [20]. The actual work extends - in comparison to existing literature – the available work done by regarding the cyclic plastification on crack driving forces and presents an additional insight in the loading of a crack tip during one or more rolling-sliding cycles. A special effort is set to make the rolling-sliding contact loading of a crack more understandable by relating crack driving force peaks to the wheel position.
The work at hand regards cracks in the plastic deforming material in an improved way compared to the conventional J-integral by using the “configurational” or “material” force method, see [21]. It can handle cyclic incremental plastic deformation in the modelling of crack driving forces. The conventional J-integral is defined for proportional loading and non-linear elasticity. With this method, the plasticity influence can be regarded only during the loading of the cracked system. Even the unloading delivers wrong results. Rolling contact involves non-proportional loadings and cyclic plastic material deformation. The use of the configurational force concept allows the investigation of the development of plastic deformation at the crack tip and around the crack, and the use of friction at the crack faces. The disadvantage of the method is that the calculated crack driving forces are now path dependent, and the size of the integration contour influences the results. However, the results are assumed to be more realistic, as the plastic deformation of the material around the crack is regarded for several cycles. Investigations about how the integration contour can be chosen, are future work and some essential findings are already published in [7], and more fundamental in [22].
The contact between the wheel and the rail is included to account for the influence of the rail crack wheel interaction. This is needed to evaluate the influence of the stick-slip behaviour in rolling-sliding contact on the crack driving force. Limitations of the model are the restriction to 2D, even the transfer of contact patches from 3D to 2D is accounted for using the method presented in [6]. Further limitations
are the necessary numerical assumptions in the modelling of the crack face contact and assumption of friction coefficients for wheel-rail and crack face contact. The material of the rail is described by a cyclic plastic kinematic and isotropic hardening model.
There is a lack of experiments on shear cracks and their growth rates under realistic conditions. It is, therefore, necessary to develop models and concepts which can describe in detail the mechanisms of the deformation around and near the crack tip and its influence on the crack driving forces and the resulting crack growth of shear cracks. In a recently published paper of the authors [11], the importance of regarding the plasticity around a shear crack is shown in detail but only for one loading cycle. More cycles and the complex rolling contact loading and its influence on the crack driving force development is discussed in detail in [8], [9]. Parameters and mechanisms which influence the crack driving force and the crack path are numerically evaluated. During a rolling cycle, several maxima of the crack driving force are observed, and a concept is developed, to account for the effect, that each crack tip loading and thus each peak will influence the crack growth rate and the crack growth direction.
The actual work offers a pragmatic, nevertheless, mechanically and thermodynamically founded method to evaluate the influence of rolling-sliding contact related to railway operating conditions on crack driving forces and crack growth directions. It is shown how to calculate from crack driving forces the crack growth. The model can be used for evaluating existing cracks and can be extended to follow numerically the crack growth. It is assumed that the quality of quantitative results for the crack growth direction and the growth rates mainly depends on mixed mode crack experiments and accurate cyclic plasticity (shear ratcheting) data of the investigated materials as, e.g., published recently in [23]. Former work of the authors has shown that plasticity and the resulting residual stresses will significantly change the size and direction of the crack driving forces [7], [11], [24].
This paper is based on the findings that results in [7] have shown that the plastic zone and the residual stresses near the crack tip develop and stabilise within some rolling cycles (less than 10) which support the assumption that a crack has not to be followed from the initiation at the surface. Quantitative validation of our results is in progress, but due to its complexity, it is an on-going experimental research process. The presented results are in agreement with practical experience, and the calculated crack growth rates are qualitatively validated.
2 CRACK MODEL
Fig. 1 shows a typical crack caused by rolling-sliding contact. This crack is assumed to have initiated at the rail surface. After that, the crack is grown through the extremely deformed surface layer of the railhead, which is dominantly produced due to rough surface contact and the local asperity contact load
(Zone 1). Now the crack continues to grow in a less but still heavily deformed region due to the sliding contact between wheel and rail (Zone 2), see [7], [18]. Within the heavily deformed zone there is an influence of the deformation orientation on the crack growth rate, see [25]. The proposed method can be used for crack tips located in zone 1 and 2. For cracks in zone 1 the local contact situation below the asperities has to be regarded. For zone 1 and 2 the material orientation influence on crack growth resistance is active, however, from [25] it can be concluded that the influence of the material deformation in a R260 rail steel is already strongly reduced below values of 70% plastic strain. At higher strains the crack growth rates are changing by high factors depending on the orientation of the material deformation. If the crack tip is located in less than 70% plastically deformed material the growth rates can be calculated as presented in this work, even the plastification during the wheel-rail contact and at the crack tip is still high. In zone 3 the above described restrictions do not apply.
Crack growth of surface shear cracks must be investigated regarding elastic-plastic material under large-scale or general yielding conditions. In practical applications, Rice’s J-integral approach, see [26], [27] is used most often and was developed for deformation plasticity which is, in fact, a non-linear elastic material description. The main disadvantage is that it does not describe irreversible plastic deformation. Furthermore, it is valid for proportional loading only which is not the case for cracks under contact loading. Out of these reasons the configurational force concept is used for the calculation of crack driving forces in our work, because it is based on a thermodynamical formulation which can handle incremental plasticity and is appended as postprocessing procedure for ABAQUS/Standard in the form as described in [28].
As we do not follow the crack from the start, the crack must be introduced in the finite element model with some length. Since crack tips, influenced by a cyclic rolling wheel, are in a standard case always producing a plastic zone, it needs some loadings until a stable plastic zone in front of the crack is developed. Due to this plastification, a residual stress state around the crack tip develops which interacts with the load stress. From the total stress-strain state around the crack tip, the crack driving force and its direction are calculated.
Figure 1: Three assumed mechanisms of crack growth in rails. Crack growth in Zone 1 is local asperity contact load, contact load and shear deformation determined, in Zone 2 it is contact load determined influenced by the sheared material orientation, and in Zone 3 it is contact load determined.
2.1 Crack driving force concept
The configurational force concept can be used as a crack driving force for cyclically non-proportionally loaded cracks and incremental elastic-plastic material description. A detailed description of the concept and its application can be found in [21], [29].
Thermodynamic driving forces that act on defects in materials, e.g. dislocations, precipitations or cracks [30], [31] are called “material” or “configurational” forces. The configurational force vector points in the direction a defect would like to move. The height of the driving force is proportional to the gain in total energy through the movement. The vector points in the direction which minimizes the total energy of the system. The configurational force concept is based on Eshelby´s energy-momentum tensor. The concept enables the derivation of the J-integral for complex constitutive models for elasto-plastic material description, see e.g. [32]. The J-integral is developed based on deformation theory of plasticity which is, in fact, a non-linear elastic description. That means, e.g. regarding a tensile test, the whole area below the stress-strain curve represents the energy put into the system during loading and is available for driving the crack. In an elastic-plastic description, most of the strain energy is dissipated during plastic deformation. This means that only the elastic part of the stress-strain curve is available for driving the crack. Regarding the fact that only the elastic strain energy can drive a crack, the whole strain energy density
below the stress-strain curve is divided into an elastic part
and a plastic part
. The configurational
body force for elastic material properties following [21], [33] is designated as the vector f ,
f C .
(1)
C is the configurational stress tensor expressed as
C I F S, T
where I is the unit tensor,
(2)
is the strain energy density, F is the deformation gradient
tensor and S the first Piola-Kirchhoff stress tensor. F is determined using the displacement gradient
u , and S is calculated from the Cauchy stress T . For plastic material properties f is now defined, see [29],
f
pl
(
I u S ) . T
el
Using the total strain energy = concept within its validity range.
(3)
the method delivers the same results as the J-integral
The configurational force vector is computed on the nodes of a finite element by using
g i N f dV N , X C ij dV , I
I
I
Ve
Ve
(4)
j
I
where g i is the configurational force component at node I by integrating the configurational body I
force f on the element e and N is the matrix that contains the shape functions corresponding to node I
I , N , X is the gradient matrix. [34],[35],[36] j
I
In order to obtain the total configurational force at a node k it is necessary to collect g i from all elements surrounding the node k.
gi gi . k
I
(5)
e
I
The deformation gradient F is computed using the element shape functions N . Then the displacement I
u at the nodes u ( I is the node index of the corresponding element) and its gradient u , X j are defined as:.
I I u N u ,.
(6)
u , X j N , X u [34],[35],[36]. I
I
(7)
j
2.2 Finite element model
The principal scheme of the 2-D finite element (FE) model is drawn in Fig. 2. The model consists of two parts, namely a rail and a wheel part. The wheel part is modelled using elastic material properties. Its radius amounts to 450 mm. The wheel moves from left to right and is accelerated. A rectangular shape represents the rail part. The wheel part is assumed elastic while the rail part is modelled with elastic-plastic material properties near the contact. The out-of-plane thickness of both parts is 10 mm. A plane strain case is assumed. Fixed boundary conditions (red points) are prescribed at the bottom and the side of the rail part. The length of the rail part is chosen in a length that no remarkable change of the results is seen within the calculated 10 cycles if the rail part is enlarged. The size of the rail part is 300x40x10 mm. The elastic-plastic section of the wheel-rail contact zone has a length of 100 mm. The wheel moves along a length of 76 mm.
Figure 2: Scheme of the wheel-rail crack model.
The crack is located in the center of the rail part (in half of the running distance), see Fig. 3. The crack inclination is 30° measured from the rail surface. The model parameters are chosen in the way that a crack is regarded often found in rails. A 30° crack inclination towards the rail surface is used since
usual crack inclinations below the contact found in rails are mostly between 15° and 30°, see [37]. 30° is also often used as the default value in, e.g. eddy current crack measurements. The out of plane thickness of 10 mm is about the length of a contact patch and produces maximum contact pressures in a realistic range of usual wheel loads. The material parameters are measured in cyclic uniaxial tensioncompression tests and are calibrated at the Material Center Leoben. They are given in Table 1.
The chosen depth of the cracks amounts to 0.25, 0.5 and 0.75 mm. Triangle plane strain elements CPE3 are used in the middle part of the contact zone between the wheel and the rail part. The size of the triangle contact zone including the crack amounts to 20x1.4 mm. The numerical comparison of results using triangle and rectangular plane strain elements with similar size has been conducted. No differences in computed stress and strain were noticed. Hence, the triangle plane strain elements are used in the contact zone around the modelled crack. The four node plane strain reduced integration elements CPE4R are used in the rest of the model. The triangle elements around the embedded inclined crack have a size of approximately 0.22x0.12 mm.
The penalty contact formulation is used for wheel-rail contact and between the crack faces. The Coulomb friction coefficient 0.5 is used for both contacts. The value for the wheel-rail contact is chosen with 0.5 as this corresponds to a usual value regarded in literature and corresponds to dry contact. A lower friction (e.g. wet conditions) reduces the traction on the rail’s surface and smaller crack driving forces as calculated can be expected. The value 0.5 as the friction coefficient between the crack faces is chosen since higher values didn’t influence the results remarkably in additional and unpublished calculations. A value of 0.5 seems justified as during the second peak of the crack driving force the crack faces are under high pressure, firmly closed and a rough metallic contact between steel exists. It can be discussed, if rust or other wear particles between the crack faces reduce the friction below 0.5, see also Fig. 1. However, smaller friction values will have big influence on the plastification, residual stresses and thus crack driving forces. Under shear mode, without friction between the crack faces, the wheel contact load acts directly at the crack tips. Lower friction coefficients are realistic if fluids enter the crack. Such fluids can be water, oil or fluid friction modifiers. Experimental evidence supporting this assumption can be found in [38] where it is shown that non-drying friction modifiers will increase wear and crack growth compared to others.
The wheel part uses elastic material behaviour with a Poisson´s ratio of 0.3 and Young´s modulus of 205 GPa. The elastic-plastic rail part is modelled using a combination of kinematic and isotropic hardening material description as implemented in ABAQUS/Standard with the initial yield stress of 340 MPa; the elastic material parameters are same as those for the wheel part. The tangent moduli C i with corresponding exponents Di of the elastic-plastic material model are given in Table 1. Cyclic hardening
parameters are Q = 0.12 GPa and B = 400. The material parameters are measured in cyclic uniaxial tension-compression tests and are calibrated at the Material Center Leoben.
The normal wheel load is varied using 50 kN, 100 kN, 150 kN and 200 kN. The longitudinal slip is defined as s = (R/v – 1)100%. is the angular velocity of the wheel, R is the wheel radius and v is the translational speed. The slip s is taken with 0 %, 0.5 %, 1 % and 1.5 %, where 0 % means the free rolling case.
a)
b)
Figure 3: a) 2-D FE wheel-rail crack contact model. The wheel running direction is from left to right. b) Detail of the mesh around the crack. The black line indicates the embedded crack.
Table 1: The four backstresses used in the elastic-plastic material model
2.3 Crack growth direction
i
Ci [GPa]
Di
1.
20
0
2.
31.8
270
3.
55
776
4.
230
3000
In a homogeneous and isotropic material, the crack path can be determined directly using a thermodynamics-based maximum energy dissipation criterion following [21], [29]. Following these papers the dissipation at the crack tip tip , moving with a velocity v tip is : tip ( f
tip
) v tip max . ,
(8)
where f is the crack driving force. tip This is not the case if an inhomogeneous and strongly deformed material surrounds the crack tip. Then the procedure calculating the crack growth rate must be modified. As long the magnitude of the crack tip velocity, | v tip |, is direction independent (e.g. no material influence) the crack grows into the negative direction of the configurational crack driving force at the crack tip, f . tip The energy dissipated per unit crack extension is equal to J tip e ( f tip ).
(9)
e is the unit vector in crack growth direction. This is identical to Rice’s J-integral notated for a finite
strain setting and expresses the crack driving force at the crack tip.
After decomposition of Jtip into two perpendicular directions in a local coordinate system at the crack tip see, [21], [29], the crack growth angle is calculated using Equation 10: J tip 2 , J tip 1
(10)
arctan
where Jtip1 and Jtip2 are the crack driving force components oriented in the direction of x, respectively y axis in the coordinate system of the crack, see Fig. 4.
The magnitude of the crack driving force is computed by Equation 11: 2 2 J tipM J tip 1 J tip 2 .
(11)
The cyclic crack driving force at the crack tip J tipM is calculated according to [39] as J tipM J tipM max J tipM min 2 J tipM max J tipM min ,
where the J tip
M
max
(12)
is a maximum value calculated during the wheel passes the crack, and the J tip
M
min
is a
minimal value when the wheel passes the crack, see Figs. 5-8. Then the stress intensity factor range at the crack tip K tipM for plane strain conditions is calculated using the relation K tipM
J tipM E
1 2
,
where E is Young´s modulus, is a Poisson number.
(13)
The crack growth rate is calculated using the Walker equation according to [40] as: m
K tipM , da C 1 dN 1 R
(14)
where C , m , are the material parameters taken from [40], see Table 2, and R is the load ratio. The load ratio R is calculated as R
K tipM min . K tipM max
(15)
The crack growth rate computed by Equation 14 takes into account the load ratio.
Figure 4: Local coordinate system at the crack tip. The yellow line denotes the embedded inclined crack. Jtip is divided into a component parallel (Jtip1) and perpendicular (Jtip2) to the surface
Table 2: Material parameters used in equation (14) [40]
C [mm]
4.01e-9
m
3.1319
0.8246
3 RESULTS
The results for the calculated crack driving forces are presented in three parts with a different focus. Firstly, the dependency of the crack driving forces on the value is shown. The four chosen slip values of 0% up 1.5% represent the rolling-sliding behaviour of a wheel from free rolling up to full slip for the chosen wheel load of 150 kN. Secondly, the dependencies on the contact pressure and the slip for three different crack depths are shown. Finally, the crack growth rate is computed for three crack depths.
As the wheel moves from left to right over the modelled rolling distance, it passes the crack at a specific rolling time. During the 10th pass, the wheel center is located just over the crack at 47.5 seconds. During one pass the wheel position and wheel load determines the number, locations, and heights of the computed crack driving force peaks, see Figs. 5-8. The computed crack driving force is shown for a constant normal force of 150 kN and various slip values. A free rolling case (slip 0%) produces two peaks, see Figure 5a. The first peak develops at a position when the wheel just passes the crack. The crack faces are pressed together and slide on each other. Hence, the computed crack growth tendency is showing upwards. The crack driving force vector is sketched at this wheel position, see sketch 1 in Fig. 5b. The second peak is calculated when the wheel has passed the crack. The calculated crack growth tendency shows downwards, see sketch 2 in Fig. 5b.
If a slip of 0.5 % is applied, only one peak is computed, see Fig. 6a. This wheel load represents a partial slip case. The computed peak is calculated when the wheel has passed the crack position, see Fig. 6a,b. The computed magnitude of the crack driving force is much smaller than in the free rolling case, compare Figures 5a and 6a. The crack will likely grow parallel to the crack growth direction, see Fig. 6b. The small crack driving force is a result of the – in this case in the depth of the crack tip - developed residual compression stress around the crack tip during the preceding nine wheel passes. It must be remarked that in the rail in track the plastification due to the rolling-sliding contact will change continuously over millions of cycles. The crack has, in reality, a long time to grow to the investigated length. However, our simulations have shown that nine cycles are necessary to reach an almost steady state of the crack tip plastic zone. Thus no big influence on the driving force from the sudden introduction of the crack can be remarked in the following cycles. This is at least valid for the material parameters given in Table 1.
At higher slips of 1 and 1.5 % the calculated peaks are located in both cases at the same wheel position, see Figs. 7 and 8. The first peak develops before the wheel approaches the crack; the acceleration traction opens the crack. Just at the moment, the wheel passes the crack; the crack faces are pressed together and slide on each other. The crack growth tendency shows downwards when the wheel is in front of the crack. The upward crack growth tendency is calculated when the wheel passes the crack position. A higher crack driving force is calculated at higher slip. This effect results due to the higher traction. The change between the up and downward crack growth tendency is sketched in Figs. 7b and 8b.
a)
b)
Figure 5: a) The computed crack driving force in direction of its unity vector em (m denotes the direction of maximum dissipation) and corresponding crack growth angle over the relative rolling time in a free rolling case (slip 0 %). b) The wheel position and the computed magnitude and direction of the crack driving force vector are shown. The crack tip is located in 0.5 mm depth, and the applied normal force amounts to 150 kN.
a)
b)
Figure 6: a) The computed crack driving force in direction of its unity vector em (m denotes the direction of maximum dissipation) and corresponding crack growth angle over the relative rolling time in a 0.5% slip case (partial slip). b) The wheel position and the computed magnitude and direction of the crack driving force vector are shown. The crack tip is located in 0.5 mm depth, and the applied normal force amounts to 150 kN.
a)
b)
Figure 7: a) The computed crack driving force in direction of its unity vector em (m denotes the direction of maximum dissipation) and corresponding crack growth angle over the relative rolling time in a near full slip case (slip 1 %). b) The wheel position and the computed magnitude and direction of the crack driving force vector are shown. The crack tip is located in 0.5 mm depth, and the applied normal force amounts to 150 kN.
a)
b)
Figure 8: a) The computed crack driving force in direction of its unity vector em (m denotes the direction of maximum dissipation) and corresponding crack growth angle over the relative rolling time in a full slip case (slip 1.5 %). b) The wheel position and the computed magnitude and direction of the crack driving force vector are shown. The crack tip is located in 0.5 mm depth, and the applied normal force amounts to 150 kN.
The influence of the contact pressure and amount of slip on the crack growth tendency is summarized in Figs. 9a-c. Table 3 shows the computed maximum contact pressures in the 10th rolling-sliding loading cycle as a function of the applied normal load. Three crack depths are investigated: 0.25, 0.50 and 0.75 mm, see Figs. 9a,b,c, respectively. It is assumed that only the maximum of the computed crack driving forces during one wheel pass is responsible for crack growth and crack direction. Only the value of crack driving forces with vectors showing downwards are plotted in Figs. 9a,b,c. Within the hatched area the vectors are showing upwards. It can further be assumed, according to [35], that no crack growth for such pearlitic rail steels will occur below a crack driving force value of 500 J/m2 which corresponds to a ΔK threshold value of about 11 MPaˑm1/2 at R=0.11.
The computed crack growth tendencies differ between a crack tip depth of 0.25, 0.50 and 0.75 mm, compare Figs. 9a,b,c. For the case crack depth 0.25 mm, the upward crack growth tendency is computed for a maximum contact pressure higher than 900 MPa and all regarded slip values.
The full picture of the computed crack driving force values is plotted in Figure 10a,b,c, independently if the crack will grow up- or downwards. The maximum value of the computed crack driving force is shown. The computed crack driving forces rise with crack depth, see Figure 10a,b,c. For a slip value of 0.5 %, a compressive residual stress state evolves just in the depth of the crack tip, and lower crack driving forces are computed. The absolute maximum of the computed crack driving force is calculated for the highest investigated maximum contact pressure of approximately 1200 MPa and the highest investigated slip of 1.5 %.
From contact mechanics together with the chosen Coulomb friction law, the traction force at the surface cannot be increased after reaching full slip between wheel and rail applying a higher slip value. Therefore, the crack driving forces will reach nearly their maximum near a value of 1.5% slip.
The smallest crack growth rate is calculated for slip 0.5 % for every investigated crack depth, but the area with the smallest crack growth rate nearly disappears at a crack depth of 0.75 mm and a slip of 0.5% in Fig. 11c. A smaller crack growth rate is calculated for a crack depth of 0.25 mm compared to deeper cracks. The highest crack growth rates are calculated for high maximum contact pressures 1000 – 1200 MPa and slips of 1 and 1.5 %, see Figs 11a,b,c.
a)
b)
c) Figure 9: The computed crack growth tendencies for a) 0.25 mm, b) 0.50 mm and c) 0.75 mm crack depth. The hatched area represents upward crack growth tendency, the colored area downward crack growth.
a)
b)
c)
Figure 10: The computed crack driving forces for a) 0.25 mm, b) 0.50 mm and c) 0.75 mm crack depth.
a)
b)
c)
Figure 11: The computed crack growth rates for a) 0.25 mm, b) 0.50 mm and c) 0.75 mm crack depth.
Table 3: Conversion table between applied normal force and computed maximum contact pressure in the elastic-plastic 10th rolling-sliding cycle
Normal force
Max. contact
[kN]
pressure [MPa]
50
639
100
892
150
1060
200
1206
4 MODEL VALIDATION
The computed crack growth rate of the rail steel R260 is compared with the test rig experiment, see [15]. The average crack length after 100000 wheel passes under dry conditions is 6.98 mm and crack growth angle is 34.4°, see Table 3 in [15]. The cracks were produced on the “head checks affected area” with calculated slips between 0.47 and 1.53 %, see Table 6 in [14]. The normal force used in the test is slightly higher than the maximum normal force used in the crack growth rate simulations (230 kN compared to 200 kN). Simulations with 230 kN using the 2D model resulted in numerical instabilities. Besides that, the lower forces in the simulation can be justified as it is described in [6] that the contact pressures at these high loads are reduced in the test rig after some wheel cycles. Therefore a reduction of the maximum contact pressure from about 1400 MPa to 1200 MPa which corresponds to 200 kN in the 2D model seems reasonable.
The mean computed crack growth depicted from Fig. 11 is shown in Table 4 along with the geometry of the crack. Assuming that the cracks grow from a length of 0.5 to 1 mm, and from 1 to 1.5 mm with the computed rate, see Table 4 and after that approximately with the constant rate of 4.375*10 -5, we get a crack growth length of 4.83 mm after 100000 cycles which is approximately 30 % less than the crack length measured in the test rig experiment, see Table 3 in [15].
As contact conditions are changing during the test rig experiments due to the plastic deformation and wear resulting from wheel-rail contact some deviations in our comparison are expected. The verification of computed crack growth rates under contact conditions needs to our opinion an extensive experimental program at the test rigs as described in the conclusions. However, the presented computed crack growth rate calculation regards the contact loading conditions, non-proportional cyclic materials loading and the plastic deformation near and around the crack.
Table 4: The summarized crack geometry and computed mean crack growth rate (taken from Fig. 11) for maximum contact pressure 1200 MPa (200 kN) and slip 1 %.
Crack
Crack
Crack
Mean crack
depth
inclination
length
growth rate
[mm]
[°]
[mm]
[mm/cycle]
0.5
3.075*10-5
1
6.5625*10-5
1.5
4.375*10-5
0.25 0.5 0.75
30
5 DISCUSSION
The influence of wheel load and slip on the crack driving force is investigated using a two-dimensional finite element model. Two-dimensional modelling of the real three-dimensional (3-D) situation is a simplification; however, it is necessary to detect the main crack driving parameters. The transfer from the 3-D to the 2-D situation can be done taking a cut through the plane of the contact patch and the maximum contact pressure as it is outlined in [7] using the equations in [42]. Making this transfer carefully the errors by the simplification can be reduced to the unavoidable contact mechanical fact, that a 2-D plane strain contact description always describes a line contact, while the real situation is often close to point contact.
The crack driving force is calculated using a configurational force concept. This concept allows cyclic analyses using an incremental elastic-plastic material description. The use of an elastic-plastic material description adds several numerical problems to the contact calculations but changes the stress fields near the crack and its tip severely. The plastic zone around the crack tip can be modelled as well as the entire plastic zone caused by the wheel-rail contact. The interaction of the open crack faces with the rolling wheel is included in our model.
As we introduce cracks into an unloaded material, we have to account for this. Calculations have shown that for the used material parameters about 10 cycles are necessary to establish a constant crack driving force for all investigated contact conditions. This effect happens even though the crack is situated in a steadily plastically deforming field. However, the calculated crack driving forces are strongly influenced by the developing residual stresses near the crack tip after 10 cycles. That means that our calculations include the assumption that wheels contacting at the same rail position always produce
similar loadings, and after some wheel passages, an almost stable characteristic residual stress field develops. The development of the plastic zone is depicted in Figs 12 a,b,c,d in the 1 st, 2nd, 9th, and 10th cycle, respectively. There is a remarkable change from the first to the following cycles. The equivalent accumulated plastic strain (PEEQ) is strongly increasing due to the rolling sliding contact at the crack tip and in the whole wheel/rail contact region. However, the shape of the plastic zone remains the same after a few cycles. The development of the crack driving force is for the case 30°, 150 kN (1060 MPa maximum contact pressure) and 1,5 % slip already stable after two cycles. Other loading cases and crack geometries sometimes need more cycles. In Fig.13, for a similar loading case but 1 mm deep straight crack, the crack driving force (CDF) and the longitudinal residual stresses (S11) around the crack tip are stable after about 7 cycles.
a)
b)
c)
d) st
nd
th
Figure 12: Development of the plastic zone around a crack tip in the a) 1 , b) 2 , c) 9 , and d) 10th cycle. The accumulated equivalent plastic strain (PEEQ) increases, but the shape of the plastic zone remains the same.
Figure 13: Development of the maximum crack driving force J tip during 10 loading cycles for a 1 mm deep straight crack and the horizontal residual stress in the elements surrounding the crack tip after each cycle.
Depending on the normal and tangential wheel load, one, in most cases two and sometimes more crack driving force peaks are calculated while the wheel is passing the crack. As each peak shows a distinct direction of the configurational force vector, it must be assumed that the relative height of the peaks to each other decides the crack growth direction. The peaks will be calculated during Mode 1 or 2 crack loading, and there are only very few validated crack growth rates available related to Mode 2. From experiments, it is not clear which Mode will dominantly influence the crack growth rate. Therefore, it is assumed that the calculated crack driving force direction at the highest peak will decide about the crack direction. This assumption may not be valid as the calculated crack driving forces show in most cases not in the direction of usually observed cracks in rails with angles between 20-40 degrees. This finding supports that both peaks should be regarded for the calculation of crack growth or that cracks in the investigated depths are usually located in the deformed zone where the assumption that the crack velocity is direction independent is not valid. In this case, the crack velocity must be adapted using findings in [25] where for deformed pearlite a strong direction dependent change of these velocities is measured.
Taking the simple highest peak assumption, the results show a crack growth tendency downwards for high normal loads and low slip values. High slip values produce mainly crack growth directions
upwards. The highest growth rates are calculated for high slip values and high loads, however, such cracks show a tendency to grow to the surface. High normal loads and small slip (<0.5 %) may cause crack growth downwards. There is no quantitative validation of our results yet available. The results at hand must be evaluated by practical experience. However, a comparison of calculated to measured crack growth for similar loading conditions and crack geometries delivered a satisfying agreement in growth rates.
Experiments for future verification must be discussed as this is not a straightforward procedure. Surface cracks are typically neighboured by other cracks, and they will surely shield each other. It can be assumed that the calculated crack driving forces will be reduced in most loading cases when crack systems are present.
A possible quantitative verification can be done by full-scale wheel-rail test rig experiments where the contact loading can be calculated at each point of the rail-head in the lateral direction after reaching some steady state. However, the determination of the cycles to crack initiation and afterward measuring the crack length is a challenging task. As test rig experiments are highly reproducible, the test can be stopped at different amount of cycles, and the crack growth rate and direction between these intervals can be determined after metallographic investigations and compared to simulations at each lateral rail position. A similar procedure can be tried in track at specific curves with continuous crack production but this is of course very costly, and the results depend on boundary conditions that are known only to a certain extent, and that cannot be controlled.
6 CONCLUSIONS
The paper offers a new method to evaluate the influence of contact pressure and slip on the crack driving force, crack growth rate and growth direction. The paper is intended to help to understand which contact loading will cause high or low crack growth rates and which loading eventually decides the crack growth direction.
The paper explains how the configurational force method can be used to investigate cracks loaded by rolling-sliding contact.
It is shown how stress intensity values can be calculated from the configurational force results.
It is discussed how crack directions can be derived if multiple crack tip loadings are present during one wheel pass.
Crack growth rates are presented for an inclined crack with different depth in dependence of contact pressure and slip.
All crack driving force calculations include the plastic deformation in the rail and the crack tip plasticity after some cycles thus regarding a characteristic residual stress field due to a given wheel load.
Comparison to literature experiments qualitatively and half-quantitatively verifies the crack growth results.
7 Acknowledgment
The authors gratefully acknowledge the financial support under the scope of the COMET program within the K2 Center “Integrated Computational Material, Process and Product Engineering (ICMPPE)” (Project No 859480). This program is supported by the Austrian Federal Ministries for Transport, Innovation and Technology (BMVIT) and for Digital and Economic Affairs (BMDW), represented by the Austrian research funding association (FFG), and the federal states of Styria, Upper Austria and Tyrol.
References
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Technology: Research, Development and Maintenance, Civil-Comp Press, Stirlingshire, UK, Paper 179, 2016. doi:10.4203/ccp.110.179 [9] Datsyshyn OP, Marchenko HP, Hlazov AY, Levus AB. Influence of compressive stresses on the propagation of surface shear cracks in railroad rails. Mat Sci 2015;51(2):235-243. [10] Brouzoulis J, Ekh M. Crack propagation in rails under rolling contact fatigue loading conditions based on material forces. Int J Fatigue 2012;45:98-105. [11] Kráčalík M, Daves W, Antretter T. Calculation of crack driving forces of surface cracks subjected to rolling/sliding contact. Engng Fract Mech 2016; 152:10-25. [12] Larijani N, Brouzoulis J, Schilke M, Ekh M. The effect of anisotropy on crack propagation in pearlitic rail steel. Wear 2014;314:57-68. [13] Heyder R, Brehmer M. Empirical studies of head check propagation on the DB network. Wear 2014;314:36-43. [14] Stock R, Pippan R. RCF and wear in theory and practice -The influence of rail grade on wear and RCF. Wear 2011;271:125-133. [15] Stock R, Eadie DT, Elvidge D, Oldknow K. Influencing rolling contact fatigue through top of rail friction modifier application – A full scale wheel–rail test rig study. Wear 2011;271:134-142. [16] Seo JW, Kwon SJ, Lee DH, Choi HY. Analysis of contact fatigue crack growth using twin-disc tests and numerical evaluations. Int J Fatigue 2013;55:54-63. [17] Kráčalík M, Trummer G, Daves W. Application of 2D finite element analysis to compare cracking behaviour in twin disc tests and full-scale wheel-rail experiments. Wear 2016;346-347:140-147. [18] Fischer FD, Daves W, Pippan R, Pointner P. Some comments on surface cracks in rails. Fatigue Fract Engng Mater Struct 2006;29 (11):938-948. [19] Zerbst U, Mädler K, Hintze H. Fracture mechanics in railway applications - an overview. Engng Fract Mech 2005;72:163–194. [20] Tillberg J, Larsson F, and Runesson K. A study of multiple crack interaction at rolling contact fatigue loading of rails. Proc. IMechE. Part F: J. Rail and Rapid Transit 2009:223: 319-330. [21] Kolednik O, Predan J, Fischer FD. Reprint of “Cracks in inhomogeneous materials: Comprehensive assessment using the configurational forces concept”. Engng Fract Mech 2010;77:3611-3624. [22] Ochensberger W, Kolednik O. Overload effect revisited – Investigation by use of configurational forces. Int J Fatigue, 2016:83:161-173. [23] Pun CK, Kan Q, Martin PJ, Kang G, Yan W. An efficient computational approach to evaluate the ratcheting performance of rail steel under cyclic rolling contact in service. Int J Mech Sci 2015;101-102:214-226. [24] Pletz M, Daves W, Yao W, Ossberger H. Rolling contact fatigue of three crossing nose materialsmultiscale FE approach. Wear 2014:314(1-2):69-77. [25] Wetscher F, Stock R, Pippan R. Changes in the mechanical properties of a pearlitic steel due to
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Highlights
Shear cracks are investigated under rolling sliding contact and cyclic loading regarding plasticity and residual stresses
Amount and direction of crack driving forces are calculated for full or partial slip
A method is proposed how to relate crack driving forces to crack growth rates for rolling contact
Similar crack growth rates as measured in tests are calculated
S0142-1123(18)30611-X https://doi.org/10.1016/j.ijfatigue.2018.12.006 JIJF 4920
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
15 October 2018 3 December 2018 5 December 2018
Please cite this article as: Daves, W., Krá čalík, M., Scheriau, S., ANALYSIS OF CRACK GROWTH UNDER ROLLING-SLIDING CONTACT, International Journal of Fatigue (2018), doi: https://doi.org/10.1016/j.ijfatigue. 2018.12.006
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ANALYSIS OF CRACK GROWTH UNDER ROLLING-SLIDING CONTACT W. Daves1,2,*, M. Kráčalík1, S. Scheriau3 *,1
Materials Center Leoben Forschung GmbH, Roseggerstraße 12, 8700 Leoben Institute of Mechanics, Montanuniversität Leoben, Franz-Josef-Strasse 18/III, 8700 Leoben 3 voestalpine Schienen GmbH, Kerpelystraße 199, 8700 Leoben
2
* [email protected] ABSTRACT
The work focuses on the calculation of shear crack growth with crack driving forces regarding cyclic elastic-plastic material behaviour. The load influence is described by the maximum contact pressure, and slip. The contact pressure and slip distribution in the contact patch are calculated using a 2D finite element model representing a wheel-rail contact. The configurational force concept is used to determine the crack driving forces in a cyclically plastified rail material during the wheel passage. The dependence of the crack driving force and the crack path on the loading conditions is shown. The results are qualitatively compared to experiments.
KEYWORDS Fatigue crack, shear crack, rolling-sliding, wheel-rail, plasticity, crack path
1 INTRODUCTION
Cracks initiated and growing under rolling-sliding contact have to be investigated with new or extended methods. Due to the nonlinearities involved namely, contact mechanics and plastic deformation of the contacting bodies the solutions for crack growth are hard to obtain and produce severe numerical problems. Other influences as surface roughness, wear, cyclic short time and long-time behaviour of the material, and changing loads due to changing contact conditions have to be regarded within the crack models as they may strongly influence the crack initiation time and growth behaviour.
The work aims to present a calculation method for growth of cracks loaded repeatedly by a rollingsliding contact located in a steadily deforming material. This is done using a two-dimensional finite element model where cracks are modelling with given depth, length and inclination to the surface in an undeformed material. It is now - numerically supported - assumed that after some similar cyclic loadings a typical residual stress field and a crack tip plastification will develop near such prescribed cracks and their crack tips. It can be shown that the calculated crack driving force will nearly not change after some initial cycles even the surrounding of the crack will deform further due to repeated
contacts. Based on those findings, a parametric study can be performed which shows how the crack driving force is changing with crack length and depth for varying contact pressure and slip conditions. Such results can be used for the assessment of existing cracks and their crack growth rates and paths.
The influence of contact loads on such cracks has already been analysed in literature [1], [2], [3], [4], [5], [6], [7], [8]. The influence of compressive and residual stresses and friction between the crack faces is investigated in [9]. Recent papers dealing with cracks using similar methods are [10], [11]. A paper dealing with cracks in the anisotropically deformed material is presented in [12]. Practical investigations of cracks in rails are presented in [13]. Crack systems can also be produced in experiments like in a wheel-rail test rig or twin disc tests, see Fig. 1 and [14], [15], [16], [17]. How to deal with cracks in rails in practical applications is presented in [18], [19], and a numerical investigation of crack systems is available in [20]. The actual work extends - in comparison to existing literature – the available work done by regarding the cyclic plastification on crack driving forces and presents an additional insight in the loading of a crack tip during one or more rolling-sliding cycles. A special effort is set to make the rolling-sliding contact loading of a crack more understandable by relating crack driving force peaks to the wheel position.
The work at hand regards cracks in the plastic deforming material in an improved way compared to the conventional J-integral by using the “configurational” or “material” force method, see [21]. It can handle cyclic incremental plastic deformation in the modelling of crack driving forces. The conventional J-integral is defined for proportional loading and non-linear elasticity. With this method, the plasticity influence can be regarded only during the loading of the cracked system. Even the unloading delivers wrong results. Rolling contact involves non-proportional loadings and cyclic plastic material deformation. The use of the configurational force concept allows the investigation of the development of plastic deformation at the crack tip and around the crack, and the use of friction at the crack faces. The disadvantage of the method is that the calculated crack driving forces are now path dependent, and the size of the integration contour influences the results. However, the results are assumed to be more realistic, as the plastic deformation of the material around the crack is regarded for several cycles. Investigations about how the integration contour can be chosen, are future work and some essential findings are already published in [7], and more fundamental in [22].
The contact between the wheel and the rail is included to account for the influence of the rail crack wheel interaction. This is needed to evaluate the influence of the stick-slip behaviour in rolling-sliding contact on the crack driving force. Limitations of the model are the restriction to 2D, even the transfer of contact patches from 3D to 2D is accounted for using the method presented in [6]. Further limitations
are the necessary numerical assumptions in the modelling of the crack face contact and assumption of friction coefficients for wheel-rail and crack face contact. The material of the rail is described by a cyclic plastic kinematic and isotropic hardening model.
There is a lack of experiments on shear cracks and their growth rates under realistic conditions. It is, therefore, necessary to develop models and concepts which can describe in detail the mechanisms of the deformation around and near the crack tip and its influence on the crack driving forces and the resulting crack growth of shear cracks. In a recently published paper of the authors [11], the importance of regarding the plasticity around a shear crack is shown in detail but only for one loading cycle. More cycles and the complex rolling contact loading and its influence on the crack driving force development is discussed in detail in [8], [9]. Parameters and mechanisms which influence the crack driving force and the crack path are numerically evaluated. During a rolling cycle, several maxima of the crack driving force are observed, and a concept is developed, to account for the effect, that each crack tip loading and thus each peak will influence the crack growth rate and the crack growth direction.
The actual work offers a pragmatic, nevertheless, mechanically and thermodynamically founded method to evaluate the influence of rolling-sliding contact related to railway operating conditions on crack driving forces and crack growth directions. It is shown how to calculate from crack driving forces the crack growth. The model can be used for evaluating existing cracks and can be extended to follow numerically the crack growth. It is assumed that the quality of quantitative results for the crack growth direction and the growth rates mainly depends on mixed mode crack experiments and accurate cyclic plasticity (shear ratcheting) data of the investigated materials as, e.g., published recently in [23]. Former work of the authors has shown that plasticity and the resulting residual stresses will significantly change the size and direction of the crack driving forces [7], [11], [24].
This paper is based on the findings that results in [7] have shown that the plastic zone and the residual stresses near the crack tip develop and stabilise within some rolling cycles (less than 10) which support the assumption that a crack has not to be followed from the initiation at the surface. Quantitative validation of our results is in progress, but due to its complexity, it is an on-going experimental research process. The presented results are in agreement with practical experience, and the calculated crack growth rates are qualitatively validated.
2 CRACK MODEL
Fig. 1 shows a typical crack caused by rolling-sliding contact. This crack is assumed to have initiated at the rail surface. After that, the crack is grown through the extremely deformed surface layer of the railhead, which is dominantly produced due to rough surface contact and the local asperity contact load
(Zone 1). Now the crack continues to grow in a less but still heavily deformed region due to the sliding contact between wheel and rail (Zone 2), see [7], [18]. Within the heavily deformed zone there is an influence of the deformation orientation on the crack growth rate, see [25]. The proposed method can be used for crack tips located in zone 1 and 2. For cracks in zone 1 the local contact situation below the asperities has to be regarded. For zone 1 and 2 the material orientation influence on crack growth resistance is active, however, from [25] it can be concluded that the influence of the material deformation in a R260 rail steel is already strongly reduced below values of 70% plastic strain. At higher strains the crack growth rates are changing by high factors depending on the orientation of the material deformation. If the crack tip is located in less than 70% plastically deformed material the growth rates can be calculated as presented in this work, even the plastification during the wheel-rail contact and at the crack tip is still high. In zone 3 the above described restrictions do not apply.
Crack growth of surface shear cracks must be investigated regarding elastic-plastic material under large-scale or general yielding conditions. In practical applications, Rice’s J-integral approach, see [26], [27] is used most often and was developed for deformation plasticity which is, in fact, a non-linear elastic material description. The main disadvantage is that it does not describe irreversible plastic deformation. Furthermore, it is valid for proportional loading only which is not the case for cracks under contact loading. Out of these reasons the configurational force concept is used for the calculation of crack driving forces in our work, because it is based on a thermodynamical formulation which can handle incremental plasticity and is appended as postprocessing procedure for ABAQUS/Standard in the form as described in [28].
As we do not follow the crack from the start, the crack must be introduced in the finite element model with some length. Since crack tips, influenced by a cyclic rolling wheel, are in a standard case always producing a plastic zone, it needs some loadings until a stable plastic zone in front of the crack is developed. Due to this plastification, a residual stress state around the crack tip develops which interacts with the load stress. From the total stress-strain state around the crack tip, the crack driving force and its direction are calculated.
Figure 1: Three assumed mechanisms of crack growth in rails. Crack growth in Zone 1 is local asperity contact load, contact load and shear deformation determined, in Zone 2 it is contact load determined influenced by the sheared material orientation, and in Zone 3 it is contact load determined.
2.1 Crack driving force concept
The configurational force concept can be used as a crack driving force for cyclically non-proportionally loaded cracks and incremental elastic-plastic material description. A detailed description of the concept and its application can be found in [21], [29].
Thermodynamic driving forces that act on defects in materials, e.g. dislocations, precipitations or cracks [30], [31] are called “material” or “configurational” forces. The configurational force vector points in the direction a defect would like to move. The height of the driving force is proportional to the gain in total energy through the movement. The vector points in the direction which minimizes the total energy of the system. The configurational force concept is based on Eshelby´s energy-momentum tensor. The concept enables the derivation of the J-integral for complex constitutive models for elasto-plastic material description, see e.g. [32]. The J-integral is developed based on deformation theory of plasticity which is, in fact, a non-linear elastic description. That means, e.g. regarding a tensile test, the whole area below the stress-strain curve represents the energy put into the system during loading and is available for driving the crack. In an elastic-plastic description, most of the strain energy is dissipated during plastic deformation. This means that only the elastic part of the stress-strain curve is available for driving the crack. Regarding the fact that only the elastic strain energy can drive a crack, the whole strain energy density
below the stress-strain curve is divided into an elastic part
and a plastic part
. The configurational
body force for elastic material properties following [21], [33] is designated as the vector f ,
f C .
(1)
C is the configurational stress tensor expressed as
C I F S, T
where I is the unit tensor,
(2)
is the strain energy density, F is the deformation gradient
tensor and S the first Piola-Kirchhoff stress tensor. F is determined using the displacement gradient
u , and S is calculated from the Cauchy stress T . For plastic material properties f is now defined, see [29],
f
pl
(
I u S ) . T
el
Using the total strain energy = concept within its validity range.
(3)
the method delivers the same results as the J-integral
The configurational force vector is computed on the nodes of a finite element by using
g i N f dV N , X C ij dV , I
I
I
Ve
Ve
(4)
j
I
where g i is the configurational force component at node I by integrating the configurational body I
force f on the element e and N is the matrix that contains the shape functions corresponding to node I
I , N , X is the gradient matrix. [34],[35],[36] j
I
In order to obtain the total configurational force at a node k it is necessary to collect g i from all elements surrounding the node k.
gi gi . k
I
(5)
e
I
The deformation gradient F is computed using the element shape functions N . Then the displacement I
u at the nodes u ( I is the node index of the corresponding element) and its gradient u , X j are defined as:.
I I u N u ,.
(6)
u , X j N , X u [34],[35],[36]. I
I
(7)
j
2.2 Finite element model
The principal scheme of the 2-D finite element (FE) model is drawn in Fig. 2. The model consists of two parts, namely a rail and a wheel part. The wheel part is modelled using elastic material properties. Its radius amounts to 450 mm. The wheel moves from left to right and is accelerated. A rectangular shape represents the rail part. The wheel part is assumed elastic while the rail part is modelled with elastic-plastic material properties near the contact. The out-of-plane thickness of both parts is 10 mm. A plane strain case is assumed. Fixed boundary conditions (red points) are prescribed at the bottom and the side of the rail part. The length of the rail part is chosen in a length that no remarkable change of the results is seen within the calculated 10 cycles if the rail part is enlarged. The size of the rail part is 300x40x10 mm. The elastic-plastic section of the wheel-rail contact zone has a length of 100 mm. The wheel moves along a length of 76 mm.
Figure 2: Scheme of the wheel-rail crack model.
The crack is located in the center of the rail part (in half of the running distance), see Fig. 3. The crack inclination is 30° measured from the rail surface. The model parameters are chosen in the way that a crack is regarded often found in rails. A 30° crack inclination towards the rail surface is used since
usual crack inclinations below the contact found in rails are mostly between 15° and 30°, see [37]. 30° is also often used as the default value in, e.g. eddy current crack measurements. The out of plane thickness of 10 mm is about the length of a contact patch and produces maximum contact pressures in a realistic range of usual wheel loads. The material parameters are measured in cyclic uniaxial tensioncompression tests and are calibrated at the Material Center Leoben. They are given in Table 1.
The chosen depth of the cracks amounts to 0.25, 0.5 and 0.75 mm. Triangle plane strain elements CPE3 are used in the middle part of the contact zone between the wheel and the rail part. The size of the triangle contact zone including the crack amounts to 20x1.4 mm. The numerical comparison of results using triangle and rectangular plane strain elements with similar size has been conducted. No differences in computed stress and strain were noticed. Hence, the triangle plane strain elements are used in the contact zone around the modelled crack. The four node plane strain reduced integration elements CPE4R are used in the rest of the model. The triangle elements around the embedded inclined crack have a size of approximately 0.22x0.12 mm.
The penalty contact formulation is used for wheel-rail contact and between the crack faces. The Coulomb friction coefficient 0.5 is used for both contacts. The value for the wheel-rail contact is chosen with 0.5 as this corresponds to a usual value regarded in literature and corresponds to dry contact. A lower friction (e.g. wet conditions) reduces the traction on the rail’s surface and smaller crack driving forces as calculated can be expected. The value 0.5 as the friction coefficient between the crack faces is chosen since higher values didn’t influence the results remarkably in additional and unpublished calculations. A value of 0.5 seems justified as during the second peak of the crack driving force the crack faces are under high pressure, firmly closed and a rough metallic contact between steel exists. It can be discussed, if rust or other wear particles between the crack faces reduce the friction below 0.5, see also Fig. 1. However, smaller friction values will have big influence on the plastification, residual stresses and thus crack driving forces. Under shear mode, without friction between the crack faces, the wheel contact load acts directly at the crack tips. Lower friction coefficients are realistic if fluids enter the crack. Such fluids can be water, oil or fluid friction modifiers. Experimental evidence supporting this assumption can be found in [38] where it is shown that non-drying friction modifiers will increase wear and crack growth compared to others.
The wheel part uses elastic material behaviour with a Poisson´s ratio of 0.3 and Young´s modulus of 205 GPa. The elastic-plastic rail part is modelled using a combination of kinematic and isotropic hardening material description as implemented in ABAQUS/Standard with the initial yield stress of 340 MPa; the elastic material parameters are same as those for the wheel part. The tangent moduli C i with corresponding exponents Di of the elastic-plastic material model are given in Table 1. Cyclic hardening
parameters are Q = 0.12 GPa and B = 400. The material parameters are measured in cyclic uniaxial tension-compression tests and are calibrated at the Material Center Leoben.
The normal wheel load is varied using 50 kN, 100 kN, 150 kN and 200 kN. The longitudinal slip is defined as s = (R/v – 1)100%. is the angular velocity of the wheel, R is the wheel radius and v is the translational speed. The slip s is taken with 0 %, 0.5 %, 1 % and 1.5 %, where 0 % means the free rolling case.
a)
b)
Figure 3: a) 2-D FE wheel-rail crack contact model. The wheel running direction is from left to right. b) Detail of the mesh around the crack. The black line indicates the embedded crack.
Table 1: The four backstresses used in the elastic-plastic material model
2.3 Crack growth direction
i
Ci [GPa]
Di
1.
20
0
2.
31.8
270
3.
55
776
4.
230
3000
In a homogeneous and isotropic material, the crack path can be determined directly using a thermodynamics-based maximum energy dissipation criterion following [21], [29]. Following these papers the dissipation at the crack tip tip , moving with a velocity v tip is : tip ( f
tip
) v tip max . ,
(8)
where f is the crack driving force. tip This is not the case if an inhomogeneous and strongly deformed material surrounds the crack tip. Then the procedure calculating the crack growth rate must be modified. As long the magnitude of the crack tip velocity, | v tip |, is direction independent (e.g. no material influence) the crack grows into the negative direction of the configurational crack driving force at the crack tip, f . tip The energy dissipated per unit crack extension is equal to J tip e ( f tip ).
(9)
e is the unit vector in crack growth direction. This is identical to Rice’s J-integral notated for a finite
strain setting and expresses the crack driving force at the crack tip.
After decomposition of Jtip into two perpendicular directions in a local coordinate system at the crack tip see, [21], [29], the crack growth angle is calculated using Equation 10: J tip 2 , J tip 1
(10)
arctan
where Jtip1 and Jtip2 are the crack driving force components oriented in the direction of x, respectively y axis in the coordinate system of the crack, see Fig. 4.
The magnitude of the crack driving force is computed by Equation 11: 2 2 J tipM J tip 1 J tip 2 .
(11)
The cyclic crack driving force at the crack tip J tipM is calculated according to [39] as J tipM J tipM max J tipM min 2 J tipM max J tipM min ,
where the J tip
M
max
(12)
is a maximum value calculated during the wheel passes the crack, and the J tip
M
min
is a
minimal value when the wheel passes the crack, see Figs. 5-8. Then the stress intensity factor range at the crack tip K tipM for plane strain conditions is calculated using the relation K tipM
J tipM E
1 2
,
where E is Young´s modulus, is a Poisson number.
(13)
The crack growth rate is calculated using the Walker equation according to [40] as: m
K tipM , da C 1 dN 1 R
(14)
where C , m , are the material parameters taken from [40], see Table 2, and R is the load ratio. The load ratio R is calculated as R
K tipM min . K tipM max
(15)
The crack growth rate computed by Equation 14 takes into account the load ratio.
Figure 4: Local coordinate system at the crack tip. The yellow line denotes the embedded inclined crack. Jtip is divided into a component parallel (Jtip1) and perpendicular (Jtip2) to the surface
Table 2: Material parameters used in equation (14) [40]
C [mm]
4.01e-9
m
3.1319
0.8246
3 RESULTS
The results for the calculated crack driving forces are presented in three parts with a different focus. Firstly, the dependency of the crack driving forces on the value is shown. The four chosen slip values of 0% up 1.5% represent the rolling-sliding behaviour of a wheel from free rolling up to full slip for the chosen wheel load of 150 kN. Secondly, the dependencies on the contact pressure and the slip for three different crack depths are shown. Finally, the crack growth rate is computed for three crack depths.
As the wheel moves from left to right over the modelled rolling distance, it passes the crack at a specific rolling time. During the 10th pass, the wheel center is located just over the crack at 47.5 seconds. During one pass the wheel position and wheel load determines the number, locations, and heights of the computed crack driving force peaks, see Figs. 5-8. The computed crack driving force is shown for a constant normal force of 150 kN and various slip values. A free rolling case (slip 0%) produces two peaks, see Figure 5a. The first peak develops at a position when the wheel just passes the crack. The crack faces are pressed together and slide on each other. Hence, the computed crack growth tendency is showing upwards. The crack driving force vector is sketched at this wheel position, see sketch 1 in Fig. 5b. The second peak is calculated when the wheel has passed the crack. The calculated crack growth tendency shows downwards, see sketch 2 in Fig. 5b.
If a slip of 0.5 % is applied, only one peak is computed, see Fig. 6a. This wheel load represents a partial slip case. The computed peak is calculated when the wheel has passed the crack position, see Fig. 6a,b. The computed magnitude of the crack driving force is much smaller than in the free rolling case, compare Figures 5a and 6a. The crack will likely grow parallel to the crack growth direction, see Fig. 6b. The small crack driving force is a result of the – in this case in the depth of the crack tip - developed residual compression stress around the crack tip during the preceding nine wheel passes. It must be remarked that in the rail in track the plastification due to the rolling-sliding contact will change continuously over millions of cycles. The crack has, in reality, a long time to grow to the investigated length. However, our simulations have shown that nine cycles are necessary to reach an almost steady state of the crack tip plastic zone. Thus no big influence on the driving force from the sudden introduction of the crack can be remarked in the following cycles. This is at least valid for the material parameters given in Table 1.
At higher slips of 1 and 1.5 % the calculated peaks are located in both cases at the same wheel position, see Figs. 7 and 8. The first peak develops before the wheel approaches the crack; the acceleration traction opens the crack. Just at the moment, the wheel passes the crack; the crack faces are pressed together and slide on each other. The crack growth tendency shows downwards when the wheel is in front of the crack. The upward crack growth tendency is calculated when the wheel passes the crack position. A higher crack driving force is calculated at higher slip. This effect results due to the higher traction. The change between the up and downward crack growth tendency is sketched in Figs. 7b and 8b.
a)
b)
Figure 5: a) The computed crack driving force in direction of its unity vector em (m denotes the direction of maximum dissipation) and corresponding crack growth angle over the relative rolling time in a free rolling case (slip 0 %). b) The wheel position and the computed magnitude and direction of the crack driving force vector are shown. The crack tip is located in 0.5 mm depth, and the applied normal force amounts to 150 kN.
a)
b)
Figure 6: a) The computed crack driving force in direction of its unity vector em (m denotes the direction of maximum dissipation) and corresponding crack growth angle over the relative rolling time in a 0.5% slip case (partial slip). b) The wheel position and the computed magnitude and direction of the crack driving force vector are shown. The crack tip is located in 0.5 mm depth, and the applied normal force amounts to 150 kN.
a)
b)
Figure 7: a) The computed crack driving force in direction of its unity vector em (m denotes the direction of maximum dissipation) and corresponding crack growth angle over the relative rolling time in a near full slip case (slip 1 %). b) The wheel position and the computed magnitude and direction of the crack driving force vector are shown. The crack tip is located in 0.5 mm depth, and the applied normal force amounts to 150 kN.
a)
b)
Figure 8: a) The computed crack driving force in direction of its unity vector em (m denotes the direction of maximum dissipation) and corresponding crack growth angle over the relative rolling time in a full slip case (slip 1.5 %). b) The wheel position and the computed magnitude and direction of the crack driving force vector are shown. The crack tip is located in 0.5 mm depth, and the applied normal force amounts to 150 kN.
The influence of the contact pressure and amount of slip on the crack growth tendency is summarized in Figs. 9a-c. Table 3 shows the computed maximum contact pressures in the 10th rolling-sliding loading cycle as a function of the applied normal load. Three crack depths are investigated: 0.25, 0.50 and 0.75 mm, see Figs. 9a,b,c, respectively. It is assumed that only the maximum of the computed crack driving forces during one wheel pass is responsible for crack growth and crack direction. Only the value of crack driving forces with vectors showing downwards are plotted in Figs. 9a,b,c. Within the hatched area the vectors are showing upwards. It can further be assumed, according to [35], that no crack growth for such pearlitic rail steels will occur below a crack driving force value of 500 J/m2 which corresponds to a ΔK threshold value of about 11 MPaˑm1/2 at R=0.11.
The computed crack growth tendencies differ between a crack tip depth of 0.25, 0.50 and 0.75 mm, compare Figs. 9a,b,c. For the case crack depth 0.25 mm, the upward crack growth tendency is computed for a maximum contact pressure higher than 900 MPa and all regarded slip values.
The full picture of the computed crack driving force values is plotted in Figure 10a,b,c, independently if the crack will grow up- or downwards. The maximum value of the computed crack driving force is shown. The computed crack driving forces rise with crack depth, see Figure 10a,b,c. For a slip value of 0.5 %, a compressive residual stress state evolves just in the depth of the crack tip, and lower crack driving forces are computed. The absolute maximum of the computed crack driving force is calculated for the highest investigated maximum contact pressure of approximately 1200 MPa and the highest investigated slip of 1.5 %.
From contact mechanics together with the chosen Coulomb friction law, the traction force at the surface cannot be increased after reaching full slip between wheel and rail applying a higher slip value. Therefore, the crack driving forces will reach nearly their maximum near a value of 1.5% slip.
The smallest crack growth rate is calculated for slip 0.5 % for every investigated crack depth, but the area with the smallest crack growth rate nearly disappears at a crack depth of 0.75 mm and a slip of 0.5% in Fig. 11c. A smaller crack growth rate is calculated for a crack depth of 0.25 mm compared to deeper cracks. The highest crack growth rates are calculated for high maximum contact pressures 1000 – 1200 MPa and slips of 1 and 1.5 %, see Figs 11a,b,c.
a)
b)
c) Figure 9: The computed crack growth tendencies for a) 0.25 mm, b) 0.50 mm and c) 0.75 mm crack depth. The hatched area represents upward crack growth tendency, the colored area downward crack growth.
a)
b)
c)
Figure 10: The computed crack driving forces for a) 0.25 mm, b) 0.50 mm and c) 0.75 mm crack depth.
a)
b)
c)
Figure 11: The computed crack growth rates for a) 0.25 mm, b) 0.50 mm and c) 0.75 mm crack depth.
Table 3: Conversion table between applied normal force and computed maximum contact pressure in the elastic-plastic 10th rolling-sliding cycle
Normal force
Max. contact
[kN]
pressure [MPa]
50
639
100
892
150
1060
200
1206
4 MODEL VALIDATION
The computed crack growth rate of the rail steel R260 is compared with the test rig experiment, see [15]. The average crack length after 100000 wheel passes under dry conditions is 6.98 mm and crack growth angle is 34.4°, see Table 3 in [15]. The cracks were produced on the “head checks affected area” with calculated slips between 0.47 and 1.53 %, see Table 6 in [14]. The normal force used in the test is slightly higher than the maximum normal force used in the crack growth rate simulations (230 kN compared to 200 kN). Simulations with 230 kN using the 2D model resulted in numerical instabilities. Besides that, the lower forces in the simulation can be justified as it is described in [6] that the contact pressures at these high loads are reduced in the test rig after some wheel cycles. Therefore a reduction of the maximum contact pressure from about 1400 MPa to 1200 MPa which corresponds to 200 kN in the 2D model seems reasonable.
The mean computed crack growth depicted from Fig. 11 is shown in Table 4 along with the geometry of the crack. Assuming that the cracks grow from a length of 0.5 to 1 mm, and from 1 to 1.5 mm with the computed rate, see Table 4 and after that approximately with the constant rate of 4.375*10 -5, we get a crack growth length of 4.83 mm after 100000 cycles which is approximately 30 % less than the crack length measured in the test rig experiment, see Table 3 in [15].
As contact conditions are changing during the test rig experiments due to the plastic deformation and wear resulting from wheel-rail contact some deviations in our comparison are expected. The verification of computed crack growth rates under contact conditions needs to our opinion an extensive experimental program at the test rigs as described in the conclusions. However, the presented computed crack growth rate calculation regards the contact loading conditions, non-proportional cyclic materials loading and the plastic deformation near and around the crack.
Table 4: The summarized crack geometry and computed mean crack growth rate (taken from Fig. 11) for maximum contact pressure 1200 MPa (200 kN) and slip 1 %.
Crack
Crack
Crack
Mean crack
depth
inclination
length
growth rate
[mm]
[°]
[mm]
[mm/cycle]
0.5
3.075*10-5
1
6.5625*10-5
1.5
4.375*10-5
0.25 0.5 0.75
30
5 DISCUSSION
The influence of wheel load and slip on the crack driving force is investigated using a two-dimensional finite element model. Two-dimensional modelling of the real three-dimensional (3-D) situation is a simplification; however, it is necessary to detect the main crack driving parameters. The transfer from the 3-D to the 2-D situation can be done taking a cut through the plane of the contact patch and the maximum contact pressure as it is outlined in [7] using the equations in [42]. Making this transfer carefully the errors by the simplification can be reduced to the unavoidable contact mechanical fact, that a 2-D plane strain contact description always describes a line contact, while the real situation is often close to point contact.
The crack driving force is calculated using a configurational force concept. This concept allows cyclic analyses using an incremental elastic-plastic material description. The use of an elastic-plastic material description adds several numerical problems to the contact calculations but changes the stress fields near the crack and its tip severely. The plastic zone around the crack tip can be modelled as well as the entire plastic zone caused by the wheel-rail contact. The interaction of the open crack faces with the rolling wheel is included in our model.
As we introduce cracks into an unloaded material, we have to account for this. Calculations have shown that for the used material parameters about 10 cycles are necessary to establish a constant crack driving force for all investigated contact conditions. This effect happens even though the crack is situated in a steadily plastically deforming field. However, the calculated crack driving forces are strongly influenced by the developing residual stresses near the crack tip after 10 cycles. That means that our calculations include the assumption that wheels contacting at the same rail position always produce
similar loadings, and after some wheel passages, an almost stable characteristic residual stress field develops. The development of the plastic zone is depicted in Figs 12 a,b,c,d in the 1 st, 2nd, 9th, and 10th cycle, respectively. There is a remarkable change from the first to the following cycles. The equivalent accumulated plastic strain (PEEQ) is strongly increasing due to the rolling sliding contact at the crack tip and in the whole wheel/rail contact region. However, the shape of the plastic zone remains the same after a few cycles. The development of the crack driving force is for the case 30°, 150 kN (1060 MPa maximum contact pressure) and 1,5 % slip already stable after two cycles. Other loading cases and crack geometries sometimes need more cycles. In Fig.13, for a similar loading case but 1 mm deep straight crack, the crack driving force (CDF) and the longitudinal residual stresses (S11) around the crack tip are stable after about 7 cycles.
a)
b)
c)
d) st
nd
th
Figure 12: Development of the plastic zone around a crack tip in the a) 1 , b) 2 , c) 9 , and d) 10th cycle. The accumulated equivalent plastic strain (PEEQ) increases, but the shape of the plastic zone remains the same.
Figure 13: Development of the maximum crack driving force J tip during 10 loading cycles for a 1 mm deep straight crack and the horizontal residual stress in the elements surrounding the crack tip after each cycle.
Depending on the normal and tangential wheel load, one, in most cases two and sometimes more crack driving force peaks are calculated while the wheel is passing the crack. As each peak shows a distinct direction of the configurational force vector, it must be assumed that the relative height of the peaks to each other decides the crack growth direction. The peaks will be calculated during Mode 1 or 2 crack loading, and there are only very few validated crack growth rates available related to Mode 2. From experiments, it is not clear which Mode will dominantly influence the crack growth rate. Therefore, it is assumed that the calculated crack driving force direction at the highest peak will decide about the crack direction. This assumption may not be valid as the calculated crack driving forces show in most cases not in the direction of usually observed cracks in rails with angles between 20-40 degrees. This finding supports that both peaks should be regarded for the calculation of crack growth or that cracks in the investigated depths are usually located in the deformed zone where the assumption that the crack velocity is direction independent is not valid. In this case, the crack velocity must be adapted using findings in [25] where for deformed pearlite a strong direction dependent change of these velocities is measured.
Taking the simple highest peak assumption, the results show a crack growth tendency downwards for high normal loads and low slip values. High slip values produce mainly crack growth directions
upwards. The highest growth rates are calculated for high slip values and high loads, however, such cracks show a tendency to grow to the surface. High normal loads and small slip (<0.5 %) may cause crack growth downwards. There is no quantitative validation of our results yet available. The results at hand must be evaluated by practical experience. However, a comparison of calculated to measured crack growth for similar loading conditions and crack geometries delivered a satisfying agreement in growth rates.
Experiments for future verification must be discussed as this is not a straightforward procedure. Surface cracks are typically neighboured by other cracks, and they will surely shield each other. It can be assumed that the calculated crack driving forces will be reduced in most loading cases when crack systems are present.
A possible quantitative verification can be done by full-scale wheel-rail test rig experiments where the contact loading can be calculated at each point of the rail-head in the lateral direction after reaching some steady state. However, the determination of the cycles to crack initiation and afterward measuring the crack length is a challenging task. As test rig experiments are highly reproducible, the test can be stopped at different amount of cycles, and the crack growth rate and direction between these intervals can be determined after metallographic investigations and compared to simulations at each lateral rail position. A similar procedure can be tried in track at specific curves with continuous crack production but this is of course very costly, and the results depend on boundary conditions that are known only to a certain extent, and that cannot be controlled.
6 CONCLUSIONS
The paper offers a new method to evaluate the influence of contact pressure and slip on the crack driving force, crack growth rate and growth direction. The paper is intended to help to understand which contact loading will cause high or low crack growth rates and which loading eventually decides the crack growth direction.
The paper explains how the configurational force method can be used to investigate cracks loaded by rolling-sliding contact.
It is shown how stress intensity values can be calculated from the configurational force results.
It is discussed how crack directions can be derived if multiple crack tip loadings are present during one wheel pass.
Crack growth rates are presented for an inclined crack with different depth in dependence of contact pressure and slip.
All crack driving force calculations include the plastic deformation in the rail and the crack tip plasticity after some cycles thus regarding a characteristic residual stress field due to a given wheel load.
Comparison to literature experiments qualitatively and half-quantitatively verifies the crack growth results.
7 Acknowledgment
The authors gratefully acknowledge the financial support under the scope of the COMET program within the K2 Center “Integrated Computational Material, Process and Product Engineering (ICMPPE)” (Project No 859480). This program is supported by the Austrian Federal Ministries for Transport, Innovation and Technology (BMVIT) and for Digital and Economic Affairs (BMDW), represented by the Austrian research funding association (FFG), and the federal states of Styria, Upper Austria and Tyrol.
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Highlights
Shear cracks are investigated under rolling sliding contact and cyclic loading regarding plasticity and residual stresses
Amount and direction of crack driving forces are calculated for full or partial slip
A method is proposed how to relate crack driving forces to crack growth rates for rolling contact
Similar crack growth rates as measured in tests are calculated