Modeling of a creep process between rough surfaces under tangential loading

Grzemba B. / Физическая мезомеханика 17 3 (2014) 39-42

УДК 539.62

Modeling of a creep process between rough surfaces under tangential loading

B. Grzemba

Technische Universität Berlin, Berlin, D-10623, Germany

If two bodies with rough surfaces are brought into contact and an increasing tangential load is applied, there will be some relative tangential displacement of the bodies due to the contact stiffness and partial slip in the contact area. The sliding areas will increase with increasing load until the condition of complete sliding is fulfilled and macroscopic sliding starts. In this paper it is suggested that an essential part of the slow creep observed in many contact problems is due to this combined effect of elasticity and partial slip. To prove this hypothesis, tangential creep in contact of rough surfaces is simulated using the assumption of a constant coefficient of friction. The results of the simulation are compared with high resolution experimental measurements of creep between steel samples. In the most cases, the surface parameters can be adjusted to achieve a good description of the experimental data.

Creep in contacts is a universal phenomenon, which is observed in contacts of all scales, from nanoindentation [1] to earthquake dynamics [2, 3], both in the normal [4] and tangential direction. In the literature, one can find different usage of the notion "creep". The standard meaning of creep is a slow deformation or displacement occurring under conditions of constant load. However, in rolling contact mechanics, the slow relative movement of bodies due to their elastic deformation and partial slip in the contact area is also called "creep" [5]. In a pure tangential contact under a constant load, elastic deformation and partial slip do not lead to progressive slow motion. However, if the load is increasing, it is not possible to differentiate between the "true" creep due to plastic deformation and due to elastic deformation and partial slip. In the present paper we suggest that an essential part of the "creep" in tangential contacts under unsteady loading conditions may be due to the mentioned effects of contact stiffness and partial slip.

To prove this we consider a contact of rough surfaces under constant normal and increasing tangential load. This is a contact setup often used for experimental investigations especially in the field of earthquake dynamics. Such systems tend to a stick-slip movement, where creep occurs in the sticking phase prior to sudden slip events.

For moderate normal loads, a rough surface will usually not come into complete contact. When a tangential load is applied, some regions of contact will be still sticking whereas others will be already in sliding state. Because of the increase of tangential load, there will be an increasing relative displacement of bodies, which macroscopically looks like slow "creep". In the following, we will investigate this relative displacement and call it "creep".

The maximum displacement due to this creep before macroscopic sliding is a characteristic length parameter of the contact configuration. It depends on the surface and material properties and the normal force [6].

Instable frictional processes as the considered ones are often described using rate-and-state friction laws [7-9]. This class of friction laws represents microscopic creep processes using three or four empiric parameters in the global friction law. In contrast, in this paper the microscopic effects are represented by a resolution of the surface roughness and a local Coulomb friction law.

In the area of rock friction the considered surfaces are fractal and self-affine in their roughness [10]. This concept for roughness can also be used in other engineering areas [11]. The surface topography is then characterized by two to three parameters: the Hurst exponent, rms roughness and cutoff wavelength. The normal contact problem for such

© Grzemba B., 2014

Grzemba B. / u3unecKan Me30MexanuKa 17 3 (2014) 39-42

fractal surfaces has been studied extensively over the last years [12, 13].

For fractal rough surfaces, the accurate representation of normal contact stiffness by the method of dimensionality reduction (MDR) has been proven in [14, 15]. The validity of the method of dimensionality reduction for tan-gentially loaded contact has been shown for rotational symmetric bodies [16] and the corresponding proof for fractal surfaces is in progress.

Consider a 3D contact of two bodies with rough self-affine, fractal surfaces which can be characterized by rms roughness hrms and the Hurst exponent H. In the present model, it is assumed, that the surface roughness has no cutoff wavelength, which means the longest present wavelength is the size of the system itself. In a series of recent papers, it has been shown that the contact problems of three-dimensional rough bodies can be described equivalently by one-dimensional contacts with elastic foundations (method of dimensionality reduction) [14-18]. In the frame of the method of dimensionality reduction this problem is replaced by a problem of contact of a rough line generated accordingly to [17] with an elastic foundation having the following normal and tangential stiffnesses [18]:

cn =-E . A x=F*A x, ct =-2Gr A x = G * A x, (1)

n 2(1 -v2) t 2-v2

where E and G are Young&s and shear modulus, respectively, v is Poisson&s ratio and Ax represents the distance between two springs.

An example of such fractal line is given in Fig. 1. The phasing of all created fractal lines is random. Congruously, all numerical calculations where performed for hundred sample lines and the results are given by the mean values. The simulation proceeds in the following steps. First, the rough line is brought into normal contact with the elastic plane until the desired normal force is reached. Then step by step a tangential displacement is forced upon the system and the according tangential force is deduced as the sum of all spring forces. The Coulomb friction law

Ff ^oF (2)

is applied for each single contact point i at the end of each spring. The tangential spring forces ft* are then given by

)ctUt for Ctut <Mufn,i >

Fig. 1. Schematic of the contact configuration in the method of dimensionality reduction. In the numerical realization, the distance between two springs Ax is the same as the resolution of the finest roughness

IV«,* for ctut ^Mf •

Here, m0 is the friction coefficient and ut is the global tangential displacement which is the same for all points, f * is the normal force of spring i.

This is performed as a static process, equilibrium is assumed at all times. Since each single state of the system is unique, fixing the displacement or fixing the force is equivalent. The static approach is valid if the considered creep processes proceed slowly.

A plot of the global tangential displacement ut on the global tangential force Ft for different surfaces shows the influence of the Hurst exponent on the creep process (Fig. 2). Here, normalization is applied: The maximum tangential displacement is given by the displacement where the last spring goes into sliding state. That is the spring with maximum normal deflection dmax located beneath the highest asperity; dmax is also equivalent to the macroscopic indentation depth. The complete sliding state is reached when

CtU tmax = M0fnmax ^

M f c (4)

u = ^0fnmax =M cn d utmax M&O d maxCt Ct

The maximum tangential force is given by the macroscopic friction force

Ftmax =M0Fn& (5)

with Fn being the global normal force.

For lower Hurst exponents the frictional joint breaks loose more sudden. Surfaces with high Hurst exponents are characterized by one larger region of contact with varying normal stress whereas surfaces with low Hurst exponents will typically come in contact with several smaller regions with similar stresses. Further tests show that as long as the contacting regions are much smaller than the total size of the system the normal force has no apparent influence on the creep curve.

The above model is a static one: the displacement depends only on loading conditions but not on time. However, if the tangential force is linearly growing in time, then the displacement becomes time dependent too. The given dependence on Ft in Fig. 2 is then equivalent to a dependence on time except for a linear factor. If it is assumed that the force is applied via a linear spring with stiffness cs and a driving velocity v0 an equivalent virtual time t can be found:

Ft(%) = CsV01 ^ t =—(6)

This virtual time allows a comparison of simulated and experimental process.

Grzemba B. / 0u3unecKaH Me30Mexanum 17 3 (2014) 39-42

Fig. 2. Normalized tangential displacement on normalized tangential force for hrms = 2.86 ¡m and different Hurst exponents

The experimental setup is based on a steel specimen placed between two V-shaped steel base plates and pulled by a soft spring (Fig. 3, a). The sharp edges of specimen emerge a little so that it slides on four distinct points (Fig. 4, c).

The driving velocity of the spring base point is constant and can be adjusted to rates down to 0.05 mm/s. The position of the specimen is measured from the back using a laser vibrometer (Fig. 3, b). The specimen moves in a very regular stick-slip manner.

Due to the high resolution (8 nm) of the laser vibrometer it is possible to capture the creep that precedes a slip event. This creep is very regular and can even be used for a prediction of the upcoming slip event [19]. A typical event with the preceding creep is shown in Fig. 4.

The model was adjusted to simulate one of the four contacts between the specimen and base plate. The normal force is given by Fn =42mg/4 ~ 0.31 N. The friction coefficient of the setup is known from previous studies as ¡0 = = 0.21. It should be assured that the length of the rough line is large enough to avoid saturation, meaning the complete contact of all points. This was satisfied in all simulations using a line length of L = 20 ¡m.

In the model, the simulation runs up to the point where all regions in contact slide. From this point, no change in the tangential force is possible because the macroscopic friction force is reached. With this force applied, arbitrary displacements can be obtained in this model.

Compared to the experiment this is equivalent to the point in time where the slip event starts. Although there is no obvious definition of a criterion for this starting point, it can be assumed that the static approach will only meet the experimental curves for low velocities. A limiting velocity of vlim = 1 ¡im/s has proven useful to define the start of the predominantly dynamic region. The simulated data is shifted in such way that it ends at the time where the experimental

Fig. 3. Experimental setup: the arrangement of specimen, spring and driving linear motor (a), laser vibrometer position (b), and the steel specimen (c)

velocity reaches vlim. To have comparible time scales for the simulation and experimental data, the mentioned virtual time t from Eq. (6) was used for the numerical data.

For each experimental creep process the optimal values for hrms and H are found using a least square method. The ranges of the surface values are given by

hrms e (0.35 pm, 6 pm), H e (0,1). (7)

These values represent normal rough surfaces with typical qualities from usual finishing processes. For analyzing rock contacts, higher roughnesses should be expected.

Since the model is based on a static theory of equilibrium, only measurements with the lowest possible driving velocity of v0 = 0.05 mm/s are used. The fit was performed for 27 experimental data sets of creep processes. Some measurements were irregular and some curves were not well fitted. The fitting was satisfying for 63 % of the data sets. An example of a fitted simulated curve and the experimental data is shown in Fig. 5. Note that the simulated creep is shown depending on the virtual time t.

u.uuu , ,-,-,—

Fig. 4. Creep process. Specimen displacement with time

Grzemba B. / Физическая мезомеханика 17 3 (2014) 39-42

Fig. 5. The simulation and experimental data of creep process. The fitted optimal surface properties are hrms = 2.46 Mm and H=03

The optimal values for hrms and H differ for each data set. Although the contact point can be different for each measurement, the overall surface properties should be comparable. An average over all successful fits gives the mean values

<H> = 0.24, <hrms> = 1.87 pm. (8)

The found optimal values scatter relatively strong over the data sets. The standard deviation for both the mean roughness and the Hurst exponent lie in the same order of magnitude as the mean values.

A contact between two rough surfaces has been modeled to analyze the progress of tangential creep under a linearly growing tangential load. The influence of surface roughness parameters on the creep process has been examined. By comparison with high-resolution creep measurements the model has shown capable of reproducing creep processes at low creep rates well if the surface parameters are adjusted.

Compared to conventional friction models that allow creep, e.g. ones including a rate-and-state friction law, the introduced model has the advantage of using only well-defined surface characteristics as parameters. The Hurst exponent and the mean surface roughness are clearly defined and measurable quantities, whereas the empiric parameters of rate-and-state laws often are not or lack a clear physical interpretation.

For faster processes with higher influence of dynamic forces an extension of the introduced model is possible. Including the inertia forces to the model would lead to a simple differential equation that can be solved numerically as suggested in [20]. Furthermore, for future investigations the inclusion of a macroscopic surface curvature is suggested.

References

Experimental Studies of Rock Friction with Application to Earthquake Prediction, 1977. - V. 116. - P. 205-237.

Press, 1985. - 468 p.

Поступила в редакцию 17.02.2014 г.

Сведения об авторе

Grzemba Birthe, Dipl.-Ing., Research Assistant, Technische Universität Berlin, Germany, birthe.grzemba@tu-berlin.de