Effect of the model ’ s geometry in fretting fatigue life prediction

This paper analyses the influence that the type of geometry used to obtain the stress/strain fields in a cylindrical contact has on fretting fatigue life predictions. In addition, this work considers the effect that the fatigue crack shape assumed has on these fretting fatigue life predictions. The strain/stress fields are calculated using a series of finite elements models that consider the following three types of behaviour: plane stress, plane strain (2D geometries) and 3D. Each of these models gives a different crack initiation life and a different evolution of the stress intensity factor (SIF), which are calculated using the weight function method. These models therefore provide different fretting fatigue life predictions. Finally, the lives obtained using the numerical models are compared with experimental lives.


INTRODUCTION
retting is a superficial damage phenomenon that is found in certain mechanical joints when are subjected to fluctuating loads.Consequently, time variable contact stress distributions arise at the contacting surfaces, potentially leading to surface crack formation [1].If there is a bulk tension at the joint in addition to the contact stress field, surface cracks may cause mechanical failure at the joint [2].This phenomenon is referred to as fretting fatigue.Because the most common form of transmitting forces between solids is mechanical contact, fretting or fretting fatigue is present in a wide variety of mechanical components.Due to this commonality, fretting fatigue is the focus of many studies [3][4][5].With the research aim of predicting the failure of mechanical components, a series of fatigue life prediction models have been developed [6][7][8][9][10].In many circumstances, these models are applied to geometries and load systems whose behaviour can be assumed to be in plane stress or plane strain conditions, as observed in 2D models with actual 3D geometries.The focus of this work is to analyse the effect produced by the geometry of the model (2D or 3D) on fatigue life prediction.For this task, fatigue life predictions were completed over a series of experimental tests, considering both 2D and 3D models.Tests were performed using cylindrical contact in which a cylindrical element was pressed against a dogbone type fatigue test specimen while a static normal load, N, was applied.Both components were made of Al 7075-T651 alloy.The test specimen was then subjected to a cyclic axial stress, , and due to the assembly used and the friction between the contact pair, generates a tangential cyclic load Q (Fig. 1).More details about the experimental set up can be found in [11].

FATIGUE LIFE PREDICTION MODEL
he fretting fatigue life predictions model used in these analyses is based on a previous model proposed by the authors [7].This model combines the crack initiation and crack propagation phases by analysing each phase independently.Using fracture mechanics procedures, the crack propagation phase is calculated using a curve.This curve offers the crack propagation cycles, N p (a), that are required to grow a crack from an initial length, a, to the final fracture crack length at which the test specimens fails.This curve is obtained by integrating the crack propagation law between a and the final crack length.Moreover, the crack initiation phase offers the number of crack initiation cycles, N i (a), required to nucleate/generate a crack of length equal to a.The N i (a) curve is obtained by considering the stress and strain fields along the prospective crack path.The total fatigue life, N t , is obtained by adding the curves, Nt(a)=Ni(a)+Np(a).As a result, the total fatigue life, Nt(a), is a function of the crack length, a, which defines when the crack initiation phase ends and the crack propagation phase begins.Depending on the value assumed for the initial crack length, a, the total fatigue life estimated is different, and the curve Nta can be obtained by repeating the calculation process using diverse values of a.As shown in [7], close to the contact surface the fatigue life is controlled by the crack initiation process, and far from it by the crack propagation.Therefore, the value of a that produces the minimum of the curve Nt(a) is considered the nexus between the crack initiation and the crack propagation phases [8].For this reason, and because it is the crack initiation length that offers the most conservative prediction, the minimum value of N t is established as the fatigue life prediction.

Crack propagation phase
A fracture mechanics-based law reflecting a generic initial crack length, a, is used to calculate the crack propagation phase.Because the value of a can be measured on the order of microns, the proposed law also tries to model the short crack growth phase.For this reason, a modified threshold dependent on the actual crack length is introduced into the crack growth law [9].
In Eq. ( 1), is the threshold SIF range for long cracks, f is a parameter equal to 2.5 [10], l0 is the distance between the surface and the first microstructural barrier (a grain boundary), C and n are the Paris' crack growth parameters and a0 is the El Haddad's parameter, which is defined by in which FL   is the stress range at the fatigue limit.The factor that multiplies 1) is based on a theoretical approximation of the Kitagawa-Takahashi diagram that considers the threshold stress to be a function of the crack length.

T
Many different options were considered to model the short crack growth phase [9], but it was found that one offers the best results is that shown in Eq. ( 1).To study the effect exerted by the model geometry (2D or 3D) on the crack growth phase, SIF values were calculated according to the type of geometry used to model the contact pair that was used between the pad and the test specimen.In those models with a plane stress or a plane strain formulation, the SIF was obtained using the weight function proposed by Orynyak [12] and calculated as follows: where  represents the surface crack,  is the angle for a certain point along the crack front, σ(y) is the normal stress distribution along the prospective crack faces and g symbolises the crack geometry.In the case of a 2D model, this stress distribution only varies with the y coordinate.Conversely, when a 3D model is applied, the SIF is obtained by adapting Eq. ( 3) to the bidimensional stress field along the crack surface: where the normal stress σ(y,z) is the bidimensional stress distribution on the crack surface.
Obtaining SIF using Eq. ( 3) and ( 4) is only valid until the crack becomes a through crack.From that moment forward, the SIF is calculated using the unidimensional weight function proposed by Bueckner [13]: In Eq. ( 5), σ(y) is the normal stress distribution along the prospective crack path.In 3D cases, this stress distribution is averaged through the test specimen thickness.Both stress distributions, σ(y) (for Eqs. ( 3) and ( 5)) and σ(), have been obtained from a series of elastic-plastic finite element analyses, which simulate the mechanical behaviour of the contact pair between the pad and the test specimen.
The software used for the finite element models (FEM) was ANSYS ® 14.0.Fig. 2 shows the mesh and boundary conditions used in the 2D FEM.In these cases and due to the symmetrical conditions of the actual contact pair, only half of the test specimen has been introduced into the models.In the 3D models, these symmetrical conditions allow to consider only half of the pad and a quarter of the test specimen during modelling.The FEM mesh for 3D case modelling is shown in Fig. 3. Based on experimental observation in both models (2D and 3D), the cracks have been modelled as surface cracks that are semi-elliptical and centred along the test specimen thickness and emanating from the contact trailing edge (Fig. 1 and 4).It is further experimentally observed that the crack shape evolves as a crack grows; thus, for the SIF calculations, a variable crack aspect ratio, b/a, has been considered.The evolution of the crack aspect ratio can be obtained from the following equation: In this equation, it is implicitly assumed that the crack grows according to the same law at the surface point ( =π/2) and at the deepest point ( =0) of the crack front.According to Eq. ( 6), the procedure used to calculate the crack aspect evolution is as follows: 1. Using the present crack length, a, and aspect ratio, b/a, the SIF values at the surface and the deepest point of the crack front are calculated.2. A small crack length increment Δa is imposed; in the present case, Δa =1 µm.3. Assuming that the SIF values remain unchanged during the increment Δa, Δb is obtained from Eq. ( 6).Suspecting that the assumed initial crack aspect ratio could have a potential influence on the a/b evolution, a series of simulations were carried out while considering different initial crack aspect ratios.The result of one of these simulations is shown in Fig. 5, in which an initial crack length, a, of 1 µm has been considered for all cases.These simulations prove that the assumed initial crack aspect ratio only affects the aspect ratio at any length for cracks smaller than 20 µm.

Crack initiation phase
To analyse the crack initiation phase, the fatigue model here applied takes ideas from the work of McClung et al. [14].The first step is to obtain from smooth fatigue test specimens the fatigue curve N smooth (ε,a).This curve gives, as a function of the applied strain range, ε, the number of cycles needed to initiate a crack of length a in smooth test specimens.For each value of ε and a, the number of fatigue cycles, N smooth (ε,a), is calculated as follows: In Eq. ( 7), Nt  In the case where a multi-axial stress/strain state and a stress/strain gradient are present, as in fretting fatigue, the same procedure can be applied, but with some changes.First, a multi-axial fatigue parameter should be used.In this work, the Fatemi-Socie parameter (FS) has been used [15].Using the set of N smooth (ε,a) curves, a new set of N smooth (FS,a) curves are obtained.However, in a fretting fatigue situation where a stress/strain gradient with depth is present, the FS parameter also varies with depth.Therefore, the crack initiation life depends on where the parameter is evaluated.Considering the results from previous works [16], the mean value of the FS parameter between the surface and a depth equal to the prospective crack initiation length, a, is considered the most appropriate option.With this average value and the set of N smooth (FS,a) curves, the number of cycles required to initiate a crack with a length equal to a is obtained.Because a mean value is used, the calculation is approximate in nature.With this option, it is the hypothesis that zones with the same mean value of the parameter require the same number of cycles to initiate a crack.gives the total number of fatigue cycles needed as a function of the assumed initial crack length, a.The value of a that minimises the total number of cycles is considered the crack initiation length.The minimum value of N t is the fatigue life.This fatigue model presents the advantage that there is no need to determine a priori when the crack initiation phase finishes and, hence, when the crack propagation phase starts.The decision is made based on the value of a that minimises the total fatigue life N t .Fig. 7 shows the curves N i (a), N p (a) and N t (a) obtained in a simulation of the test type 3, using a 3D model.The parameters defining this type of test will be shown further in this document.In the following figure, it is observed that the crack initiation length, i.e., the value of a that minimises Nt, is approximately equal to 65 µm.

FATIGUE TESTS
ollowing the same procedure described in [11], a series of fretting fatigue tests were performed.The data for completed tests are shown in Tab. 1. Seven varying load combinations were applied to meet two main fretting fatigue test goals: completing tests with a wide range in fatigue lives and carrying out tests with varying predominate loads.

FATIGUE LIFE PREDICTIONS
he fretting fatigue life predictions were performed using the mechanical and fatigue properties for the Al 7075-T651, as shown in Tab. 2 [17].In this table, the σ f ´, b, ε f ´ and c are parameters defining the Δε-N curve, and the constants K´ and n´ define the stress-strain Ramberg-Osgood curve.The fretting fatigue predictions obtained for the 2D (plane stress or plane strain) and 3D models, are shown in Tab. 3. In addition, Tab. 3 indicates the experimental lives.
To consider the full picture of the behaviour offered by each model (2Ds or 3D), Fig. 8 presents the predictions, N pred , versus the experimental lives, N exp , for all cases.This plot illustrates the solid predictions obtained using both 2D models; furthermore, it illustrates that both models provide similar predictions.Regarding the 3D model, this plot also indicates good behaviour; however, for test type 2, the results were worse than those obtained using the 2D models.
To complete the prediction obtained using the models, Fig. 9 represents the percentage of N i and the crack initiation length as a function of the predicted life Npred.This plot shows that the crack initiation lengths are continuously longer than 20 m, justifying the low influence exerted on the fatigue predictions by the assumed initial crack aspect ratio.Similarly, this plot displays that the comparative relevance of both the crack initiation and the crack propagation phases can vary across test results.This fact shows that it is not possible to ignore either the crack initiation phase or the crack propagation phase when conducing these tests.Finally, Fig. 9 also indicates that the crack initiation length is always shorter than 70 µm; a model that considers the short crack growth phase is expedient.

CONCLUSIONS
n this work, the effect that the model's geometry has on the fretting fatigue life predictions has been studied.For the cylinder-plane contact pair considered, all of the models give similar results.These results are proven to be virtually identical in the 2D cases.These results additionally justify the use of a 2D FEM that provides the advantage of lower computational cost compared with 3D models.

Figure 4 :
Figure 4: Crack configuration in the models.

Figure 5 :
Figure 5: Crack aspect evolution for different initial crack aspect.
smooth (ε) represents the total number of fatigue cycles obtained in a smooth test specimen subjected to a strain range ε.N p smooth (ε,a) is the number of cycles required to propagate the crack from a to the final fracture crack length, a f , for a smooth test specimen subjected to ε.Finally, f(ε,a) represents the fatigue crack growth law Eq.(1), in which KI is calculated according to the applied ε and the geometry of the smooth fatigue test specimen.For the Al 7075-T651 alloy, the initiation curves N smooth (ε,a) for different initiation cracks lengths are represented in Fig. 6.The Nt smooth (ε) curve is also plotted in this figure.
When the curves Ni(a) and Np(a) are acquired, these values are added to obtain a new curve Nt(a)= Ni(a)+Np(a), which

Table 2 :
Mechanical and fatigue properties for the Al 7075-T651.