A theoretical justification of the slip index concept in fretting analysis

Fretting in the partial-slip and gross-slip regimes under a constant normal load is considered. The tangential force—displacement relations for the forward and backward motions are described based the generalized Cattaneo—Mindlin theory of tangential contact and Masing’s hypothesis on modelling the force—displacement hysteretic loop. Besides the critical force and displacement parameters (characterizing the triggering of sliding), the model includes one dimensionless fitting parameter that tunes the tangential contact stiffness of the friction—contact interface. Explicit expressions are derived for the main tribological parameters of the fretting loop, including the slip index and the signal index. The presented phenomenological modelling approach has been applied to the analysis of two sets of experimental data taken from the literature. It has been shown that the experimentally observed simple relation of a rational type between the slip index and the slip ratio corresponds to the gross-slip asymptotics of the corresponding model-based predicted relation. The known quantitative criteria for the transition from the partial slip regime to the gross slip regime are expressed in terms of the stiffness parameter, and a novel geometric transition criterion is formulated.


Introduction
The phenomenon of fretting has been observed for a long time [1], and it is still the subject of active tribological research, both experimental [2] and theoretical [3]. Whereas fretting wear, fretting fatigue, and fretting corrosion are frequently encountered in frictional contacts subjected to prolonged tangential oscillations of small amplitude, overall, the accumulated fretting damage may become a critically decisive factor for long-term functionality of the contact interfaces. An important example of serious consequences that may result from fretting wear is associated with flow induced vibrations in the pressurized water reactor system of nuclear power plants [4,5].
In addition to the stick regime of fretting, which is regarded as a non-dissipating regime [6], there are distinguished two main regimes of fretting wear: partial slip and gross slip, and their distinction from experimentally observed tribological characteristics of a contact interface, such as variations of the tangential force and the relative displacement, is rather to be regarded as one of poorly understood aspects of fretting process [7]. This is mainly because the regime transition occurs when a relative motion is undertaken over the entire contact interface, which is usually hidden from observation in the real engineering applications. Significant progress in the analysis of the fretting regime transition was achieved when Fouvry et al. [8] introduced certain transition criteria to quantify the boundary between the partial and gross slip. In particular, based on the classical Cattaneo-Mindlin theory [9,10] of tangential contact (in a ball on flat contact configuration), the effects of partial slip and dissipated energy were highlighted and, in particular, the so-called slip and energy ratios were suggested as dimensionless transition criteria.
Varenberg et al. [11] have made another important contribution to our understanding of the transition from fretting to reciprocal sliding by introducing a similarity criterion, termed as the slip index, as a result of their dimensional analysis of the mechanics of fretting contact. Further, Varenberg et al. [12] have established a simple empirical relation between the slip ratio and the modified slip index that incorporates the friction coefficient.
In response to the need of monitoring fretting characteristics in real-time, Kim et al. [13] introduced the so-called fretting signal index as a normalized phase difference between the friction force signal and the displacement signal, when one of them vanishes. However, to the best of the authors' knowledge, there are no published reports revealing the interdependence between the different transition criteria introduced so far.
In the present study, we develop a unified mathematical modelling framework, which incorporates as a special case the Cattaneo-Mindlin theory-based models used in Refs. [8,13]. To keep the analysis simple, we employ a one-free-parameter model for the tangential force loading of a contact-friction interface, which is based on Masing's hypothesis (see, e.g., [14]) for a hysteresis loop formed by the unloading and reloading curves in cyclic loading. In particular, the developed approach allows to theoretically justify the experimentally established relation [12] between the slip index and the slip ratio. As an important result of the presented analysis, we formulate a simple modelfree transition criterion, which is based on geometric properties of the slip ratio/slip index variation curve.

Masing hysteretic model for fretting contact
We consider a contact interface that may experience relative tangential motion under a constant normal load, N. Let F and x denote the tangential force and the corresponding relative displacement, respectively. Moreover, let * F and * x denote the critical force and displacement at incipient sliding starting from the position of rest, when the tangential load is gradually increasing. For quasi-static (non-accelerating) sliding, we assume * F F  and, thus, the tangential force F is balanced by the friction force * F N   , where  is the Coulomb coefficient of friction. The functional dependence ( ) F x of the tangential force F on the tangential displacement x in the case of initial loading is termed as the backbone curve (see Fig. 1). We note that the derivative d ( ) / d F x x is called the incremental tangential stiffness, and its value at the zero point is equal to 0 tan( )  , where 0  is the angle of inclination of a tangent line to the backbone curve at the origin (see Fig. 1). The slop c 0 tan( ) S   is usually called the (initial) interface contact stiffness. For frictional interfaces, c S determines the maximum incremental tangential stiffness.
The Masing model for reciprocating quasi-static sliding between the contacting surfaces with the force amplitude 0 F below the critical value, that is 0 gives the following analytical expressions for the forward, ( ) F x  , and backward, ( ) F x  , force-displacement curves (see Fig. 2), which are based on a chosen expression ( ) F x for the backbone curve, It is to emphasize that the Masing model exploits the symmetry condition ( )  Following the Ref. [15], we assume that We note that the case 3 / 2 m  corresponds to the Cattaneo-Mindlin theory [9,13] (see also Ref. [8]). In its general form, the Eq. (3) represents the model of tangential reaction of a lap-type joint (we refer to the Ref. [13] for details). In the present study, the exponent parameter 1 m  is used as a fitting constant, along with the parameters * x and * F . If the displacement amplitude is not large, that is 0 * x x  , then the substitution of Eq. (3) into Eqs. (1) and (2) yields where, in view of Eq. (3), the force and displacement amplitudes are related by If the displacement amplitude is large, that is 0 then the force amplitude 0 F equals * F , and thus, Eqs. (1) and (2) imply that and  Fig. 3. In the case of displacement-controlled loading, when the displacement variable x is specified, the value of the displacement amplitude 0 x may take an arbitrary positive value. In the force-controlled tangential loading, the force amplitude 0 F , of course, cannot exceed the critical value * .
can be obtained from the backbone curve ( ) F x by means of an affine transformation without shearing, that is C  , which is in complete agreement with Eq. (1).

Slip ratio
We recall [11] that the slip amplitude is defined as the amplitude of the relative displacement between two positions where the tangential force vanishes. In other words, we consider the equations ( The so-called slip (or sliding [8]) ratio is defined as

and s
A is the resulting slip amplitude. In When the backbone curve is defined by Eq. (3), it is readily seen that the root of Eq. (9) is independent of * F (this conclusion also follows from a simple dimensional analysis) and thus depends on the values of the model parameters * x and m as well as on the value of the imposed displacement amplitude 0 x . Hence, the slip ratio will be a function of the dimensionless parameter m and the relative displacement amplitude: In view of Eqs. (5) and (8), Eqs. (9) and (10) yield where we have introduced the auxiliary notation (  and the dimensionless stiffness parameter m that tunes the tangential contact stiffness of the friction-contact interface.

Slip index
We recall that the notion of the slip index,  , was introduced [11] based on the dimensional analysis of the hysteretic friction loop by the equation   d c / A S N , where c S is the tangential stiffness of the contact interface, and N is the normal load. In our notation, we have According to Ref. [11], the stiffness c S is defined as the slop of the forward force-displacement curve ( ) F x  at the beginning of the forward motion (see Fig. 2), that In view of Eqs. (1) and (7), we easily find that c , and therefore, Eq. (3) implies that Thus, the substitution of Eq. (13) into Eq. (12) yields where we have taken into account that  Thus, in the framework of the developed generalized Cattaneo-Mindlin theory-based model, the slip index is found to be proportional to the coefficient of friction  , the stiffness parameter m, and the relative displacement amplitude 0 * / x x that we denoted above as  .

Energy ratio
Following Ref. [8], we consider the ratio between the dissipated energy, d W , and the total energy, t W , and The total energy t W is defined as the energy input, i.e., whereas the dissipated energy (per cycle) is given by If the displacement amplitude is not large, that is 0 * x x  , then the substitution of Eqs. (4) and (5) into Eq. (17) yields where, in view of Eq. (3), we have In particular, when the imposed displacement amplitude coincides with the critical value of displacement, that is 0 It is of interest to note that * d 0 W  in the special case 1 m  , which corresponds the case of linearly elastic tangential response (without dissipation), when Eq. (3) simplifies as It can be simply verified that in the case 1 m  , Eq. (20) also yields d 0 W  for any 0 then the substitution of Eqs. (7) and (8) into Eq. (17) implies that where * d W is given by Eq. (21). Thus, the energy ratio  as a function of the relative displacement amplitude 0 * / x x can be evaluated by Eqs.  16) and (20) or (22) followed by simple algebra. However, when the derived formulas are applied for numerical calculations, it is much simpler to program the dissipated ratio d t / W W in three simple steps: first, we calculate the total energy t W , second, depending on the value of the relative displacement amplitude, we calculate the dissipated energy d W , using one of Eqs. (20) or (22), and then, we take the ratio of d W to t W .

Signal index
Regarding the periodic character of fretting oscillations and per se following Ref. [13], we introduce the phase of displacement signal  by assuming that It is clear (see Fig. 3) that, when the phase angle  (being measured in radians) increases from zero to π / 2 , the tangential displacement x decreases from the maximum value 0 x (displacement amplitude) to zero.
Let 1  denote the phase corresponding to the In view of Eq. (23), the following relations hold true:

Transition criteria
Following Ref. [8], we consider quantitative characteristics of the transition between a partial and a gross slip behaviour at the fretting contact interface that complies with the Masing hypothesis. According to the assumed backbone curve (3), the transition between the partial slip and gross slip regimes is represented by the condition  Fig. 4), such that they tend to unit as m tends to infinity.    Hence, by generalizing the observed results, we can formulate the following general transition criterion: the transition from partial slip to gross slip is identified with the inflection point of any of the curves s,  , and  versus the relative displacement amplitude  .

Gross-slip asymptotics
It is of interest to observe that the slip ratio s and the energy ratio  in the gross slip regime are given by simple rational expressions. Indeed, according to Eqs. (11) and (22) (29) and (30), it becomes clear that the signal index slower approaches to the limit value as  tends to infinity.

Variation of the slip ratio vs. the slip index
First, we note that, in view of Eqs. (10) and (14) where  is the slip index.
It is of interest to observe that, based on a large number of experimental results, Varenber et al. [12] empirically established the relation 1 , which is of the form of Eq. (32). By comparing Eqs. (32) with (29), it becomes evident that this experimental law corresponds to the gross-slip asymptotics (see Eq. (29)). Figure 6 shows the result of fitting of the experimental data for nano-and microscale fretting from Refs. [11,16], which were represented in Ref. [12] based on average friction coefficient values. It is readily seen that while the simple approximation from Ref. [12] fits well the experimental data in the gross Experimental data is according to Refs. [11,16].  11), is capable of fitting also the data in the partial slip regime.

Discussion
As it was already mentioned above, the tangential contact model (see Eqs. (1)-(3)) reduces to the Cattaneo-Mindlin contact model in the special case 3 / 2 m  , which corresponds to the Hertzian normal contact. Of course, the adopted phenomenological approach does not provide expressions for * F and * x similar to those furnished by the Cattaneo-Mindlin theory of elastic tangential contact. We recall that the latter theory, which assumes the Hertzian contact geometry and isotropy of the material properties, was generalized to arbitrary axisymmetric and non-axisymmetric geometries [17,18] and transverse isotropy [19]. In particular, if the initial gap between the contacting surfaces, which is measured in the undeformed state in the direction normal to the contact interface, is described by the monomial shape function ( ) Λ r r    , where r is the polar radius from the centre of circular contact area, then the following relation holds true: ( 1)/ m     . Moreover, the case 1   and, thus, 2 m  corresponds to the conical contact geometry, and m decreases to 1 as the shape parameter  increases to infinity and, thereby, the contacting surfaces become flatter (the limiting case 1 m  was noticed in Section 3.1). The effect of the tangential contact stiffness parameter m on the initial part of the normalized forcedisplacement relation of the backbone curve is shown in Fig. 7, where the effect of different friction coefficients for partial slip and sliding (static and kinetic coefficients of friction, s  and k  , according to the terminology [20]) is illustrated as well.
We underline that, compared to Eq. (3), the above formula introduces only one additional dimensionless parameter, namely, the friction coefficient ratio k s /   . Figure 8 shows the results of fitting the experimental data presented in Ref. [21] by using the model predictions that are based on the backbone curves given by Eqs. (3) and (33) (see curves 1 and 2, respectively). We note that the total energy in the slip regime was evaluated as t 0 * 4 W x F  , where according to Ref. [21], * F is the maximum tangential force associated with the displacement amplitude 0 x . Evidently, the refined model based on Eq. (33) allows a better fit of the energy ratio results, and, in particular,  x . Experimental data is according to Ref. [21]. Curve 1 is drawn based on the constant friction coefficient model; curve 2 takes into account a drop of the friction coefficient from 0.9 to 0.85 in the transition to sliding. this modified model accommodates the observed jump in the energy ratio upon the transition from partial slip to sliding. It should be emphasized that the ratio k s /   was not used as a fitting variable, and its value was evaluated based on the friction coefficients s 0.9

 
and k 0.85   taken from the data presented in Ref. [21].
Yet another point that deserves a comment is a pronounced discrepancy between the model predictions and the experimental data for larger displacement amplitudes (see Fig. 8). Apparently, this can be explained by the effect of the system stiffness, that is of the tangential accommodation of the testing device [8], since the dissipated energy is evaluated as d t e W W W   , where e W is the elastic energy. That is why, when the elastic energy is also stored in the system besides the contact interface, the share of the dissipated energy d W in the total energy t W decreases. At the same time, the models presented above implicitly take into consideration only contact deformations at the contact interface.
Another important result of the presented analysis is that the empirical relation , which was established in Ref. [12] between the slip ratio s and the slip index  , represents the so-called gross-slip asymptotics. This, in particular, means that the empirical relation, if applied for determining the transition between the partial and gross slip, may introduce a www.Springer.com/journal/40544 | Friction systematic error as a priori can be expected from any asymptote, provided insufficient experimental data is available for the analysis, and other additional tools of analysis (like the newly introduced convexity/concavity geometric criterion) are not applied.
We recall that differential calculations were applied in Ref. [8] for evaluating the transition point from the maximum of the second derivative of a certain contact parameter (tangential force or dissipated energy) with respect to the contact displacement. It is to note here that the second derivative of the dissipated energy is identically equal to zero in gross slip. However, as Fig. 8 shows, the tribological data is very noisy and, therefore, the application of the numerical differentiation tools is rather problematic. On the contrary, the convexity/concavity transition criterion can be easily implemented, and, moreover, in many cases the transition point may be approximately established by the direct inspection of the plotted data. It should be emphasized that a number of transition criteria have been introduced since the seminal paper by Fouvry et al. [8], but to the best of the authors' knowledge, the simple convexity/concavity transition criterion (without the necessity of specifying the backbone curve) has not been reported in the literature up to now.
It is to note here that the fretting signal index was introduced in Ref. [13] by assuming the harmonic variation of the tangential force, that is like the functional dependence 0 cos F F   , where  is the phase angle.
However, this approach effectively works only when 0 * F F  , when there is a one-to-one correspondence between the variables F and  in the base interval [0,π]   . It is also to note here that, though Eq. (23) adopts the first harmonic variation for the tangential displacement x, following the Ref. [13], the dependence of x on the phase angle  can be approximated by the equation 0 ( ) with an arbitrary (including sawlike modulation) function ( ) Friction 11(7): 1265-1275 (2023) | https://mc03.manuscriptcentral.com/friction so-called microscopic point of view, the transition from rest to gross slip involves microslips between asperities forming the rough surfaces. However, the related questions about the variation of the coefficient of friction at the contact interface lie outside of the scope of the present study, as they refer to the microscale modelling approach. On the contrary, in the present study we necessarily take a so-called macroscale point of view, as the slip index is measured from the macro-scale characteristics of fretting loops.
Finally, we observe that the presented above simple mathematical modelling framework can be further generalized to account for the effect of variation of the backbone curve due to wear. Recently, the phenomenon of non-monotonic behaviour of the dissipated energy in the partial-slip regime of fretting wear was highlighted in Refs. [22,23]. In this way, the stiffness parameter m (which is shown to be dependent on the contact geometry) is likely to become dependent on the number of fretting cycles due to the contact geometry adaptation. Another possible generalization concerns the application of artificial neural networks (ANNs) (see, e.g., Ref. [24]) for the purpose of realistic description of the backbone curve based on experimentally observed data for the tangential force-displacement relation in the fretting loop.

Conclusions
In the present study, a unified mathematical modelling approach for the analysis of the tangential forcedisplacement hysteretic loops in fretting has been developed based on Masing's hypothesis about the scaling of the forward and backward forcedisplacement curves from the backbone curve. By adopting a one-free-parameter generalized Cattaneo-Mindlin contact model of the frictional tangential contact loading, explicit relations for the main tribological parameters of the fretting loop (slip and energy ratios among others) have been derived. As a result, novel transition criterions, which are parameterized by the interface stiffness parameter, have been introduced, including the convexity/concavity geometric criterion that is shown to be a model-free transition criterion.