Prediction of fretting fatigue crack initiation location

Tribology International 127 (2018) 245–254

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Tribology International journal homepage: www.elsevier.com/locate/triboint

Prediction of fretting fatigue crack initiation location and direction using cohesive zone model

T

K. Pereiraa, N. Bhattia, M. Abdel Wahabb,c,d,∗ a

Department of Electrical Energy, Systems and Automation, Faculty of Engineering and Architecture, Ghent University, Belgium Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam d Soete Laboratory, Faculty of Engineering and Architecture, Ghent University, Technologiepark Zwijnaarde 903, Zwijnaarde B, 9052, Belgium b

A R T I C LE I N FO

A B S T R A C T

Keywords: Fretting fatigue Finite element analysis Crack initiation Cohesive zone model

Contact stresses distributions may substantially reduce the fatigue life of components subject to fretting, leading to early unexpected failures. Accurately predicting the components lives is, therefore, an important topic to be addressed, in particular from a design point of view. This topic has received great attention in the past decades and several numerical tools that better estimate these components lives have been proposed. In this paper, the focus is in the crack initiation phase. At this stage, it is important to correctly predict the crack initiation location and orientation, which is often achieved by using critical plane approaches. Instead, the use of cohesive zone model (CZM) as an alternative approach to accurately estimate those parameters is investigated. Cohesive zone model as well as two of its common initiation criteria, namely quadratic traction-separation criterion and maximum nominal stress criterion, are used to study crack initiation location and orientation under fretting conditions. Our results are compared with the traditional critical plane approaches and with experimental data, suggesting that cohesive approaches can accurately be used in crack initiation prediction.

1. Introduction Numerical tools, especially finite element methods, that can predict the life of components under fretting conditions have been receiving considerable attention of the research community [1–4]. However, most models require assumptions and simplifications that may no longer be plausible for cases where there is considerable amount of fretting. For instance, many numerical methods available in the literature simulate the propagation phase assuming linear elastic fracture mechanics [5–8]. However, as there may be high stresses at contact interface, plasticity can play a significant role and linear elastic fracture mechanics theory may no longer be valid. Moreover, the life estimates often rely heavily on the empirical models for the crack growth law, which may not be adequate for non-proportional loading conditions, a common characteristic of fretting fatigue problems. A more robust alternative that does not rely on the assumptions above is to consider the failure using cohesive zone models (CZM). As discussed by Kuna and Roth [9], the CZM was developed to replicate the fracture process in front of a crack tip and its basic idea is to describe the entire fracture process in a thin cohesive region. The material behaviour inside this region follows a local law, based on the

∗

tractions and separations transferred across the cohesive zone. This constitutive law creates a more realistic description of the stress at crack tip, removing the stress singularity modelled in linear elastic fracture mechanics [10]. Fig. 1(a) shows a representation of the CZM, and it can be seen that damage starts once the traction reaches a strength parameter (Tmax) or separation reaches δ0. This defines two regimes characteristic of cohesive models: a reversible state from which there is no damage accumulated and a softening region, where the local material cohesive strength is reduced. Complete failure happens when cohesive strength reduces to zero or once separation reaches a critical value δf. Note that, as elucidated by Roth et al. [10], to model failure using CZM, there is no necessity to have an incipient crack in the model. Therefore, CZM allows a unique way to model crack initiation, propagation and final failure. As pointed out by Brocks et al. [11], the fact that cohesive zone model is a phenomenological model implies that the shape of the traction-separation constitutive law is independent on the material being analysed. In the literature, many different shapes have been proposed, for instance: polynomial function [12], exponential [13], bilinear function [14], among others. For simplicity, in this work, a bilinear traction-separation law is adopted and it is represented in

Corresponding author. Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam. E-mail addresses: [email protected], [email protected] (M. Abdel Wahab).

https://doi.org/10.1016/j.triboint.2018.05.038 Received 1 April 2018; Received in revised form 24 May 2018; Accepted 27 May 2018 Available online 02 June 2018 0301-679X/ © 2018 Elsevier Ltd. All rights reserved.


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Fig. 1. Cohesive zone model: (a) representation of its use for modelling fracture; (b) bilinear model.

strain obtained at the centroid of the XFEM elements, at each loading increment, is then used to compute a damage function. Once this damage function reaches a value of one (within some tolerance), a crack is introduced in the model, crossing one entire element. The cohesive tractions and separations at the crack faces are used to model the degradation and eventual failure of the enriched element. The initial crack location and direction can be directly formulated by the user, using a user defined initiation criterion, programmed by the UDMGINI subroutine in ABAQUS®. For further details, the reader is referred to the ABAQUS® documentations [18]. The crack initiation is assumed to happen at the start of the degradation of the cohesive response of the enriched element (Initiation point in Fig. 1(b)). This process of degradation starts when the stresses and strains in the material meet a specified initiation criterion. This criterion can be written as a normalized function of the stresses or strains with respect to the critical cohesive strength of the material (Tmax ), and it is here called damage initiation criterion. As it is a normalized function that describes the degradation of the element, it is likely that crack initiates when this damage function reaches a value of 1 with some tolerance. For mixed mode conditions, as in the fretting fatigue case, the critical cohesive strength of the material Tmax can be expressed as two material properties: the cohesive strength of the material under pure mode I condition (tn, c ) and the tangential cohesive strength of the material under pure mode II condition (ts, c ). Their values can be estimated based on laboratory tests using fracture specimens for each failure mode (I or II) individually correlating the fracture toughness (area under the graph Γ0 in Fig. 1(b)) and the critical separation δf . However, one may have a reasonable estimate of tn, c and ts, c based on the fracture mechanism. For instance, it is known that the fracture of brittle materials involves very little plasticity. Therefore, a reasonable assumption would be that tn, c is roughly one order of magnitude of the Young's Modulus of the material [19]. For ductile materials (such as the aluminium alloys studied in this paper), the fracture process involves large plasticity, with the nucleation, growth a coalescence of voids ahead of crack tip. For this case, tn, c may be approximated equal to or in the same order of magnitude of the ultimate strength of the material [17,20]. Regarding the tangential cohesive strength of the material

Fig. 1(b). The initial material response is presumed to be linear until a damage initiation criterion (initiation point) is fulfilled. After that, degradation of the material starts and damage follows a linear softening evolution law until complete failure. The use of cohesive zones to simulate fretting phenomenon has been restricted to only few papers in the literature [15–17]. The present paper attempts to extend the literature, investigating the use of a cohesive zone model (bilinear traction-separation law) to predict crack initiation location and orientation under fretting fatigue conditions, considering two experimental configurations (flat and cylindrical pads). Here, only the initiation point of the cohesive zone is considered, ignoring the accumulation of damage during loading/unloading in a fatigue cycle. This assumption simplifies the analysis, but at the expense of inhibiting the application of CZM for life predictions, which will require a cycle-by-cycle (with or without a time acceleration procedure) analysis in conjunction with a damage evolution law. Focusing on the use of CZM to estimate the crack initiation location and orientation, this paper is divided in the following way. Firstly, a brief description of the implementation and modelling of cohesive zone damage initiation is done in Section 2.1. In order to compare the accuracy of the results obtained using cohesive zone model, they are compared with traditional critical plane approaches. Two classical critical plane damage parameters have been used in this comparison (Findley (FP) and Fatemi-Socie (FS) parameters). Details of the implementation required to compute those parameters is presented in Section 2.2. Section 3 shows details of the finite element models used in this study. The results and discussions are presented in Section 4. Finally, some conclusions are drawn in Section 5. 2. Theoretical background 2.1. Cohesive zone models The cohesive zone model was incorporated in our simulations of fretting phenomenon by using ABAQUS® XFEM with cohesive segments module. The behaviour of this model can be summarized as presented in Fig. 2. Initially, a stress analysis of the undamaged material (subjected to some load and boundary conditions) is performed. The stress/

Fig. 2. Behaviour of XFEM with cohesive segments model in Abaqus®. 246


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Fig. 3. Scheme of the UDMGINI subroutine for the two implemented damage criteria for a single element.

The element, in which the global maximum value of damage parameters is obtained, defines the critical location for crack initiation. The angle θP , that corresponds to this maximum damage, is selected as the potential crack initiation orientation.

under pure mode II condition (ts, c ), for ductile materials, its value can be approximated considering the von Mises criterion, ts, c = 0.5tn, c , as described in Ref. [21]. Considering this reasoning, in this paper, tn, c was assumed equal to the ultimate tensile stress σu and ts, c = 0.5σu . As fretting fatigue is a mixed mode conditions, it is expected that a combination of normal and shear tractions may impact the crack initiation orientation and location. Thus, for this work, two stress-based damage initiation criteria commonly used for the cohesive model [22] are considered: the quadratic traction-separation and the maximum nominal stress criteria. Those criteria have been previously applied to plain fatigue and delamination problems [19,23,24]. As discussed by Mei et al. [19], under mixed mode conditions, an approach to account for the effect of mode mixity is to consider the initiation criterion as a quadratic combination of normal and shear tractions. The quadratic traction-separation criterion, herein defined by the damage variable f1, reflects a quadratic ratio between the maximum values in time of the normal tractions tn, θ and tangential tractions ts, θ acting in a potential crack plane oriented at an angle θ (see Fig. 3). This damage variable can be written as: 2

f1 =

2.2. Critical plane approaches The other parameters used in this paper are based on critical plane approaches. According to critical plane approaches, the cracks nucleate and grow on specific planes known as critical planes [25–29]. These planes are assumed to be maximum shear stress or strain planes, maximum tensile stress or strain planes or any combination of these entities using influence factors. For the present study, Findley (FP) and Fatemi-Socie (FS) parameters are employed to determine the crack initiation location and orientation. According to Findley parameter (FP), the crack initiates on a plane where the combination of maximum shear stress amplitude and maximum normal stress is maximum [30,31]. This plane, where the FP reaches its maximum value, is denominated as critical plane. Findley parameter is based on stresses and can be expressed as:

2

⎧ tn, θ ⎫ + ⎧ ts, θ ⎫ ⎨ ⎨ ⎩ tn, c ⎬ ⎭ ⎩ ts, c ⎬ ⎭

(1)

FP =

where, tn, c is the cohesive strength of the material under pure mode I condition, ts, c is the tangential cohesive strength of the material under pure mode II condition. Here, it is assumed that compressive tractions do not promote damage, therefore, tn, θ = tn, θ if tn, θ > 0 and tn, θ = 0 otherwise. In contrast, the maximum nominal stress criterion does not consider any interaction between normal and tangential tractions. Here, this damage initiation criterion is represented by the variable f2 and can be obtained by:

tn, θ ts, θ ⎫ , f2 = max ⎧ ⎨ ⎩ tn, c ts, c ⎬ ⎭

Δτmax + kσnmax 2 Δτmax 2

(3)

σnmax

and are the maximum shear stress amplitude and where, maximum normal stress on critical plane. The material constant k acts as an influence factor to the normal stress component. It can be determined using fatigue limits in tension and torsion of the material. For the present work, the value of k is determined as 0.16 using the experimental data from literature [32,33]. Fatemi and Socie performed multiaxial fatigue tests and proposed a parameter that is suitable for material and loading conditions, which produce shear mode failure [34]. They observed that, 90° phase difference induces additional cyclic hardening and can cause significant fatigue damage in LCF. This parameter (FS) incorporates non-proportional loading and mean stress effects. The FS parameter is based on shear strain and normal stress and can be written as:

(2)

where, tn, c and ts, c are the normal and tangential cohesive strength as described above. The implementation of the two criteria described above was done using the user defined subroutine UDMGINI in ABAQUS. This subroutine allows the user to define not only the damage criteria, but also the crack initiation orientation. Fig. 3 shows a scheme of the subroutine used in the present study. Firstly, for each loading increment, the normal traction tn, θ and tangential traction ts, θ as function of the potential crack orientation θ are obtained based on the stresses at the centroid of each element. These tractions are later used to compute the damage variables f1 and f2 for each element as function of the angle θ . This process is repeated for all loading increments in one fretting cycle.

FS =

Δγmax ⎛ σnmax ⎞ ⎜1 + k1 ⎟ 2 ⎝ σy ⎠ Δγmax

σnmax

(4)

where, and are the maximum shear strain amplitude and 2 maximum normal stress (in the plane of maximum shear strain range), respectively. The FP and FS damage parameters include both shear and normal effects for crack initiation. The first part of these models incorporates shear effects, while the second term includes normal effects with the application of influence factors. The influence factor determines the 247


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coefficient of friction between the cylindrical pad and specimen was kept constant and equal to 0.65 (as per experimental data from Ref. [36]) and for the case of flat indenter it was maintained equal to 0.8 (value measured in Ref. [33]). For both types of pads, the specimen is subject to an oscillatory axial stress σaxial and the pads are subjected to normal contact load F. The only difference between those two pad configurations, in terms of loading, is the presence of compliance springs (for the case of cylindrical pads) that generates an oscillatory tangential load Q, as discussed below. The cyclic loads are applied in three steps aiming to simulate a complete fretting fatigue cycle. The normal load F is applied at the top pad in the first step and this loading condition is kept constant until the end of the fretting cycle. Then, the cyclic axial load was applied to the sides of the specimen in two steps, i.e. the maximum axial load followed by the minimum one, as illustrated in Fig. 4. For the cylindrical pad, the pads are secured during experiment through compliance springs that generate an oscillatory tangential load Q at the interface between pad and specimen. Its effect was considered as a cyclic reaction stress σreaction [37]:

contribution of normal stress or strain in crack initiation and depends on material static and fatigue properties. The results of the contact problem are used to compute damage parameters. The stresses and strains are stored for each load increment of the complete loading cycle. To find the highest value of the damage parameter and orientation of the critical plane, the stresses and strains are retrieved for the nodes at the contact interface from -a to a, using a Matlab code. The code first reads the shear stresses, for the node located at the contact edge (x = –a), at maximum and minimum load during the cycle. The normal stress is taken at maximum loading condition, as it gives maximum normal stress during the cycle. By applying Mohr's circle transformation, the stresses are evaluated at the rotated planes from θ = −90°–90° (using the same reference frame as shown in Fig. 3) with increments of 1°. For each increment of plane, shear stress range and normal stress is calculated to give a value of FS and FP parameter. For this node, values of computed damage parameters are compared and the angle with the highest value of damage parameter is stored as local critical angle and local critical point. Using same procedure, the local damage parameter is computed for all other nodes at the contact interface. Then all local damage parameters are compared to find maximum value of damage parameter. The corresponding node and angle are stored as damage initiation location and critical plane. For detailed description on determination of stress or strain ranges and flow chart, readers are referred to [4,35].

σreaction = σaxial −

2Q bt

(5)

where b is the specimen width (10 mm) and t is its thickness (4 mm). The analyses were done considering a linear elastic material response. For the flat indenter, loading conditions and material properties (aluminium 7075-T6) are as the same as in the experiments conducted by Sabsabi et al. [33]. The tests were performed in a stress ratio of −1.0 for the axial load (σaxial). For completeness, the loading conditions of the tests analysed in this study are summarized in Table 1. For the cylindrical pad, they replicate tests from Hojjati-Talemi et al. [36] performed in samples from aluminium 2420-T3. During the experiments, a stress ratio of 0.1 for the axial load (σaxial) and a −1.0 for the tangential load Q were adopted. The loading and experimental conditions are replicated in Table 2. In both cases (cylindrical and flat pads), pads and specimen are made of the same material with same properties. Those material properties for both materials are listed in Table 3 for completeness. For the cohesive parameters, the cohesive strength of the material tn, c was assumed equal to the ultimate tensile stress σu, similarly to the work done by Kim et al. [17]. Regarding the mesh, a 2D quadrilateral, 4-node (bilinear), plane strain, full integration element (CPE4) was used to discretize the model.

3. Numerical models For this study, finite element models were created in ABAQUS® representing the laboratory test set-up for fretting fatigue. The models consisted of two parts: a pad and a specimen. Two pad configurations were considered, a flat indenter and a cylindrical one. Both models, with flat and cylindrical pads, replicate only half of the experimental set-up, due to symmetry of the test. This symmetry was modelled by the restriction of the vertical movement in the y direction and rotation around the z-axis at the bottom surface of the specimen. In order to avoid rigid body movement of the pads, the displacement in x direction at both sides of them were also restrained. Details of the boundary conditions are depicted in Fig. 4. The master-slave algorithm with surface to surface interaction and finite sliding was used for modelling the contact between pad and specimen. The tangential behaviour was modelled using a Lagrange multiplier formulation and the normal behaviour by a hard contact. The

Fig. 4. Model details for each pad configuration: dimensions, boundary conditions, loading history. 248


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be near the trailing edge of the contact [25,36,39–43]. Trailing edge is defined as, edge of the contact interface which is towards the cyclic bulk load. The numerical results of the present work also show that the most favourable site for damage initiation is the trailing edge. For both flat and cylindrical pad cases, the damage initiation variables (f1 and f2) computed via XFEM with cohesive zone models in junction with the UDMGINI subroutine showed its maximum values near the trailing edge. Fig. 6 shows the distribution of those damage parameters in the specimen, for the case of a flat indenter (test 1 of Sabsabi et al. [33]). Both parameters had their maximum value in a very local region at approximately the trailing edge of the contact, due to the high stress concentration present in this configuration. Although the maximum of both parameters is about at the same location, their distribution with respect to the depth of the specimen is quite different. For the damage initiation criterion f1, this distribution of damage suggests that the most probable direction of crack growth is oblique and inwards the contact region, which is not the case for f2. This difference can be justified by the influence of shear stresses in each of those parameters, as can be noted by Eqs. (1) and (2). For the damage variable f1, shear stresses play a significant role leading to a probable mixed mode condition for propagation which is also verified in experimental tests [25, 44, 45]. Based on that, one may conclude that this damage variable f1 captures better the behaviour of specimens under those fretting conditions than f2. Therefore, the quadratic traction-separation criterion seems to be an appropriate choice of initiation criteria for fretting conditions. Fig. 7 shows the distribution of the damage variables for a cylindrical pad configuration. Those parameters are more smoothly distributed below the contact region, while for flat contacts, the damage is highly concentrated at the edge of the contact. This can be explained by the fact that, though there is a stress concentration on the cylindrical pad configuration, it is not as significant as is in the case of flat contacts, due to the geometry of the problem. It is also important to notice that, also for the case of cylindrical pads, both parameters also have their maximum values at near the edge of the contact. Considering that the crack would initiate at the location of maximum damage, the results, for all tested cases with cylindrical pad (FF1 to FF9), showed initiation locations around 0.95 to 0.98 times the contact semi-width a, which is very close to the contact edge. Those predictions are in agreement with other numerical techniques, such as continuum damage mechanics [36] and also the experimental data mentioned above. Regarding the crack initiation orientation, the values of the damage variables obtained in one fretting cycle are also used. The element in the model with the highest damage parameters (f1 and f2) over the loading cycle is selected and the variation of those values of damage f1 and f2 with the potential angle of propagation are used for the selection of the probable crack initiation orientation direction. Fig. 9 shows the variation of f1 and f2 for the case of cylindrical pads, for test FF9. Similar profiles were observed for other tested cases (FF1 to FF8). The damage parameter f1 had its peak values at two shear planes, leading to two potential directions of crack growth: +30° ± 5° or −30° ± 5°. The results corroborate well with experimental results from literature, with those angles being measured with respect to the normal to the contact surface. Lykins et al. [40] have shown experimentally the initiation angles to be 40°,-45° and −39° whereas Namjoshi et al. [41] observed either at −45° or +45° with a variation of ± 15°. Hojjati-Talemi et al. [36] found the initiation angles between −35° to −45° for the aluminium alloy and Almajali [43] found the initiation angle for Titanium alloy at 41°. Those results are summarized in Fig. 8. For orientation of the critical plane researchers [40,41] have shown experimentally that initiation process was mainly controlled by maximum shear stress. These maximum shear stress planes exist on both sides of the principal plane. Therefore, both the planes have potential to initiate damage in these directions. Depending upon the microstructure of the material during experiment, the crack may take any preferred direction closer to maximum shear plane. Therefore, crack near the surface in first few

Table 1 Experimental conditions, from Sabsabi et al. [33]. Test

F [N]

σaxial [MPa]

1 2 3 13 14 15

2000 4000 8000 2000 4000 800

110 110 110 190 190 190

Table 2 Experimental conditions, from Hojjati-Talemi et al. [36]. Test

σaxial [MPa]

F [N]

Qmax [N]

FF1 FF2 FF3 FF4 FF5 FF6 FF7 FF8 FF9

100 115 135 135 160 190 205 220 220

543 543 543 543 543 543 543 543 543

155.165 186.25 223.7 195.55 193.7 330.15 322.1 267.15 317.845

Table 3 Material properties.

E ν σ0.2 σu

Modulus of Elasticity [GPa] Poisson's ratio Yield Strength [MPa] Ultimate tensile stress [MPa]

Al 2420-T3 [36]

Al 7075-T6 [33]

72.1 0.33 383 ± 5 506 ± 9

72 0.3 503 572

For analysis considering the cohesive zone behaviour, a region of the model near the trailing edge has been meshed with a structured mesh of CPE4 enriched elements (XFEM with cohesive segments modelling). In this XFEM formulation, the approximation for a displacement vector function u is given by the following: N

u=

∑ Ni (x )[ui + H (x ) ai] i=1

(6)

where Ni (x ) are the usual nodal shape functions for an isoparametric quadrilateral element, u i is the standard nodal displacement of the finite element solution, ai is the nodal enriched degree of freedom vector and the associated discontinuous Heaviside function is H (x ) across the crack surfaces. Note that, as cohesive segments have been adopted here and the crack has to propagate across an entire element at a time, there is no need to model the stress singularity and the enrichment functions for the near-tip asymptotic singularity are not required. For this study, the contact interactions between crack faces has been neglected. To guarantee that the model is able to correctly capture the stresses distributions at the contact between pad and specimen, a small element size at this interface (about 5 μm) was selected, based on previous convergence study detailed in Pereira et al. [38]. This element size was kept constant in the region modelled with enriched elements. The element size was then progressively increased far away from the contact and the XFEM region. Fig. 5 shows the mesh details for each of the models, with flat pad and cylindrical pad. 4. Results Generally, in fretting fatigue conditions, multiple cracks initiate at the contact interface, however only one crack propagates to cause the failure. Several researchers have found this location experimentally to 249


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Fig. 5. Mesh details.

Fig. 6. Damage initiation criteria distribution along the specimen, for test 1 of Sabsabi et al. [33]: (a) f1 and (b) f2 parameter.

Fig. 10 presents the variation of the damage variable f1 and f2 as function of the probable crack orientation angle, for the case of flat pads, for test 3. Similar to the case of cylindrical pads, the damage parameter f1 has its peak values at two shear planes. For flat pad configuration, the two possible orientations of crack initiation are slightly higher than those for cylindrical pads: +40° ± 5° or −40° ± 5°. In addition, the parameter f2 also predicts two potential orientations: +45° ± 5° or −45° ± 5°. The previous results are also compared with critical plane

grains can occur either inwards or outwards direction with reference to the contact edge. For the damage parameter f2, the peak values lead to one potential propagation direction, perpendicular to the contact surface and perpendicular to the axial loading. This expected result comes from the fact that the stress state is nearly uniaxial at the contact edge. Therefore, the shear stress has minor impact on the calculation of this parameter and fracture is mainly dominated for normal stresses. For fretting fatigue, these orientation predictions (+0° ± 5°) are not accurate.

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Fig. 7. Damage initiation variables distribution along the specimen, for test FF1 from Hojjati-Talemi et al. [36]: (a) f1 and (b) f2 parameter.

be seen within the stick zone. It is widely accepted that Stage I growth occurs due to reversal of shear stress to form a slip band. With the increased intensity of stress reversal, there are more chances to form a slip band [46]. Therefore, dislocations along persistent slip bands cause crack initiation along plane of maximum shear stress or close to it. Since there are (hypothetically) two possible shearing planes at ± 45° for each element, the crack may take any preferred orientation depending upon crystallographic orientation. Fig. 11 (b) and Fig. 12 (b) shows the variation of

approaches. Using those approaches, the initiation location and orientation can also be predicted. Fig. 11 (a) and Fig. 12 (a) show the variation of damage parameter at the contact interface using FP and FS parameters. The results are computed at the instant of maximum axial stress applied during the loading cycle. It is observed that regardless of the pad geometry for both the cases, highest value of the parameter is achieved at location x/a = 1. Hence, good correlation is observed with experimental observations. Although both parameters showed almost same initiation location, however some variation in damage profile can

Fig. 8. Summary of experimental data, exemplifying the crack initiation angle measured by different researchers. 251


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Fig. 9. Variation of damage parameters with angles (at a point of maximum damage during a cycle): (a) f1 and (b) f2, for cylindrical pad (test FF9 of Hojjati-Talemi et al. [36]).

Fig. 10. Variation of damage parameters with angles (at a point of maximum damage during a cycle): (a) f1 and (b) f2, for flat pad (test 3 of Sabsabi et al. [33]).

Fig. 11. Variation of FP and FS (a) at contact interface and (b) with angles at the location of maximum damage (x = a), for test 1 of Sabsabi et al. [33].

critical plane approaches also predict the critical angles for flat pad case to be slightly higher than cylindrical pad. Findley [30] also observed that the critical plane orientation varies with maximum stress and with combined stress state. They showed that for test with zero mean stress, critical plane varied from 45° to 21° for range of k from 0 to 1.1. Furthermore, its orientation was found few degrees from maximum shear

damage parameter with different orientation angles and reinforce this hypothesis. The maximum damage parameter at the initiation site showed two peaks near the shear planes for both the cases. The numerical results show that the orientation angle is either −45° ± 5° or +35° ± 5°. Analogous to the cohesive zone damage initiation criterion f1, the

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Fig. 12. (a) Variation of FP and FS (a) at contact interface and (b) with angles at the location of maximum damage (x = a), for test 1 of Hojjati-Talemi et al. [36].

plane for small values of k and few degrees from principal plane for large k . The change in material parameter k affects the initial crack orientation, since it influences the contribution of normal stress term the damage parameter.

[3]

[4]

5. Conclusions [5]

In this paper, the cohesive zone model in conjunction with the XFEM method has been used to predict crack initiation location and orientation in fretting fatigue using cylindrical and flat pads. The predictions were made through a user subroutine UDMGINI, where two common stress based initiation criteria, namely quadratic traction-separation and maximum nominal stress, were computed. The behaviour of each of those criteria and the accuracy of their predictions were compared with experimental data and traditional critical plane approaches. For the critical plane approaches, two damage parameters FS and FP were considered. Our results showed that cohesive zone model with a quadratic traction-separation criterion (f1 parameter) can accurately predict the crack initiation location and the orientation for both pad conditions. The results agreed with experimental data and also with traditional approaches (continuum damage mechanics and critical plane approaches). On the other hand, although the maximum nominal stress criterion (f2 parameter) predicted correct crack initiation locations, it was not possible to obtain satisfactory results for orientation angles, especially for the cylindrical pad cases. Therefore, care should be exercised while using this criterion under fretting conditions. It is important to note that the focus of this paper is on the crack initiation location and orientation predictions. For life predictions, it is necessary to adjust the cohesive model, in order to account for damage accumulation (during loading/unloading) in each fretting cycle. The authors are currently working on this point and intend to publish their results at a later stage.

[6] [7] [8]

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

[20]

[21] [22]

Acknowledgements The authors would like to acknowledge the financial support of the Research Foundation-Flanders (FWO), The Luxembourg National Research Fund (FNR) and Slovenian Research Agency (ARRS) in the framework of the FWO Lead Agency project G018916N ‘Multi-analysis of fretting fatigue using physical and virtual experiments’. The second author acknowledges the research scholarship supported by Pakistan government.

[23] [24]

[25] [26] [27]

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