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Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading
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International Journal of Solids and Structures 0 0 0 (2016) 1–13
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Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading Olympia K. Panagouli∗, Konstantina Mastrodimou Laboratory of structural analysis and design, Department of Civil Engineering, University of Thessaly, Pedion Areos, 38334 Volos, Greece
a r t i c l e
i n f o
Article history: Received 15 October 2015 Revised 18 October 2016 Available online xxx Keywords: Friction coefficient Roughness Fractal interpolation functions Asperity resolution Normal pressure Cyclic loading
a b s t r a c t In this paper the friction mechanism that is developed between metallic rough interfaces with irregularities of different scales, submitted to cyclic loading is studied. The main purpose of the paper is to investigate how the resolution δ n , which takes into account asperities of different scales of the rough interfaces and the intensity of the excitation, affect the friction that develops between the two surfaces. Friction is assumed to be the result of gradual plastification of the interface asperities in a metallic body with elastic-plastic behaviour. The interface of the body is described by a fractal interpolation function f with iterative regular construction, which permits the analysis of interfaces with irregularities of different scales. In the sequel, the structures resulting from the different iterations (i.e. resolutions) of the rough interface are submitted to different acceleration histories that induce forces leading to plastification of the interface asperities. The apparent dynamic friction coefficient is studied macroscopically, as the ratio between the forces that develop parallel and normal to the interface. Finally, the influence of the applied normal pressure to the friction mechanism is investigated with respect to the fractal resolution and the acceleration history of the interfaces. © 2016 Published by Elsevier Ltd.
1. Introduction A good understanding of tribological process at different normal forces and length scales is essential for improving efficiency in many applications. There is a gamut of engineering materials like metals, ceramics, polymers and composite materials used in many applications where friction challenges both efficiency and reliability. The common method for calculating the static friction force (Coulomb friction law) was based on the assumption that the friction force is independent of the nominal contact area and it results by multiplication of the normal applied load by a constant, known as the static friction coefficient. Static friction coefficient was taken as a function of the contacting materials for at least 300 years (Dowson, 1979). However, the values of the static friction coefficient resulting from the above assumptions represent average coefficients of friction, because the friction coefficient is presently recognized as both material and system dependent (Persson and Tossati, 1996; Blau, 2001) and is definitely not an instinct property of two contacting materials. More specifically, in Blau (2001) indicated that the friction coefficient is a misunderstood quantity in the field of science and engineering, due to the fact that the fun-
∗
Corresponding author. Fax: +302310344824. E-mail address: [email protected] (O.K. Panagouli).
damental origins of sliding resistance are not so clear, and hence, it is very important to understand the tribological process. Tabor in his general work of friction understanding (Tabor, 1981), pointed out three basic elements that are involved in the friction of dry solids: • The true contact area between rough surfaces. • The type and strength of bond formed at the interfaces where contact occurs. • The way in which the material around the contacting areas is sheared and ruptured during sliding. Chang et al. (1988) presented a model (CEB friction model) for predicting static friction coefficient of rough metallic interfaces based on the three elements indicated by Tabor (1981). This model uses a statistical representation of surface roughness (Greenwood and Williamson, 1966) and calculates the static friction force that is required to fail all of the contacting asperities, taking into account the normal loading. This approach demonstrated that the classical Coulomb friction law is a limiting case of a more general behaviour where static friction coefficient decreases with increasing applied load. This friction model underestimates friction coefficient values in the case of contact of rough surfaces since it neglects the ability of elastic-plastic deformed asperity to resist additional loading before failure, as it was demonstrated in Kogut and Etsion (2003). More specifically, the approximate
http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010 0020-7683/© 2016 Published by Elsevier Ltd.
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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applied displacements
Fig. 1. The considered problem.
generated self affine curve f with an iterative regular construction which permits us to analyse pre-fractals of arbitrary generation and therefore of arbitrary resolution δ n . Every structure resulting from each resolution of the interface is submitted to a constant normal loading and to different acceleration histories in order to induce forces leading to the plastification of the interface asperities. The resulting contact problems are solved by using the F.E. code MSC-MARC and the apparent dynamic friction coefficient is studied macroscopically. 2. Self affinity for the modelling of roughness
Fig. 2. The applied horizontal acceleration.
models for the calculation of static friction coefficient presented in Chang et al. (1988) assume failure of contacting asperity at the time the first plastic point appears and hence, underestimate the friction coefficient. The model presented later in Kogut and Etsion (2004) with the assumption of elastic perfectly-plastic material behaviour, demonstrates a strong effect of the external force and nominal contact area on the static friction. It was also demonstrated that this assumption may be invalid in cases where the contact approaches fully plastic state where small static friction was found. The numerical model presented in Panagouli and Iodranidou (2013) relates the interface roughness both with the resolution and the fractal dimension of rough interfaces which are described by fractal geometry. The model considers strain hardening effects and the friction mechanism is assumed to be the result of the interlocking mechanism developed between the interfaces and the gradual plastification of the interface asperities. This model also demonstrates a strong effect of the external load and roughness on the apparent friction coefficient. In this paper the friction mechanism developed between metallic rough interfaces with irregularities of different scales, submitted to dynamic excitations is studied. The main purpose of the paper is to investigate how the resolution δ n , which takes into account asperities of different scales of the interfaces, the intensity of the excitation and the applied normal loading affect the friction that develops between the two interfaces of a body with elastic-plastic material behaviour. The rough interface of the body is a computer
Experimental studies have shown that the true contact area between rough surfaces depends strongly on the surface parameters used for the modelling of the geometry of interfaces, which in many cases is too complicated to be described by a few parameters. To overcome problems resulting from traditional approaches based on scale-dependent statistical surface parameters and to model real surfaces demonstrating multi-scale roughness, in contact mechanics studies of rough surfaces, fractal geometry is used in many cases in order to describe surface topography (Mandelbrot et al., 1984; Majumdar and Tien, 1990; Majumdar and Bhushan, 1990; Majumdar and Bhushan, 1991; Borodich and Mosolov, 1992; Warren and Krajcinovic, 1995; Panagouli, 1997; Borri-Brunetto et al., 1999; Borri-Brunetto et al., 2001; Persson et al., 2002; Chiaia, 20 02; Hyun et al., 20 04; Zavarise et al., 20 07; Panagouli and Mistakidis, 2011). The Hausdorff dimension (Falconer, 1985) which is used for the characterization of a set as a fractal one, has no application to the curves originated in mechanics. For that, other methods, based on experimental or numerical calculations, have been developed for the estimation of the fractal dimension of a curve, a parameter which is of great importance because of its scale-independent character. One of the most useful methods in this category, is the Richardson method, described in Mandelbrot (1982), which uses dividers with a lower cut-off length δ . Moving with these dividers along the curve so that each new step starts where the previous step leaves off, the number of steps N(δ ) is obtained. The curve is said to be of fractal nature if, by repeating this procedure for different values of δ , the relation
N (δ ) ∼ δ −D
(1)
δ∗
is obtained in some interval <δ< The power D denotes the fractal dimension of the profile, which is in the range 1 ≤ D < 2.
∗ .
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 3. F.E. Discretization of the structure which corresponds to the 7th resolution of the interface.
The relation between the fractal dimension D of this profile and the dimension Ds of the corresponding surface is Ds = D + 1. Typically, a rough profile can be measured by taking height data yi with respect to an arbitrary datum at N equidistant discrete points xi and following the procedure presented in Goerke and Willner (2008). Here, fractal interpolation functions are used for the passage from this discrete set of data {(xi , yi ), i = 0, 1, 2, ..., N} to a continuous model, where f(xi ) = yi , i = 0, 1, ..., N. It has been proved in Barnsley (1988) that there is a sequence of functions fn + 1 (x) = (Tfn )(x), where
fn+1 (x ) = (T fn )(x ) = ci li−1 (x ) + di fn li−1 (x ) + gi
(2)
for x ∈ [xi − 1 , xi ], i = 1, 2, ..., N. The operator T converges to a fractal curve f, as n → ∞. The transformation li transforms [x0 , xN ] to [xi − 1 , xi ] and it is defined by the relation
li ( x ) = ai x + bi .
(3)
The factors di are the hidden variables of the transformations and they have to satisfy 0 ≤ di < 1 in order for T: C0 → C0 to have a unique fixed point. Moreover, the remaining parameters are given by the following equations:
Fig. 4. Details of discretization in the neighbourhood of the interface.
ai = (xi − xi−1 )/(xN − x0 )
(4)
ci = (yi − yi−1 )/(xN − x0 ) − di (yN − y0 )/(xN − x0 )
(5)
bi = (xN xi−1 − x0 xi )/(xN − x0 )
(6)
gi = (xN yi−1 − x0 yi )/(xN − x0 ) − di (xN y0 − x0 yN )/(xN − x0 ).
(7)
The fractal interpolation functions give profiles which look quite attractive from the viewpoint of a graphic roughness simulation. In higher iterations these profiles appear rougher because irregularities of new scales are taken into account. It must be mentioned here that an important advantage of the fractal interpolation functions is that their fractal dimension can be N obtained numerically (Barnsley, 1988). More specifically if di > i=1
1, and the interpolation points do not lie on a single straight line, then the fractal dimension of the graph G of the fractal interpolation function is the unique real solution D of
δ −D ≈
N i=1
Fig. 5. Accelerations scaling parameter which leads to the plastification of asperities with respect to the resolution of roughness.
|di |aDi −1 δ −D ⇔
N
|di |aDi −1 = 1.
(8)
i=1
In the case where the interpolation points are equally spaced it results that xi = x0 + Ni (xN − x0 ) for i = 1, ..., N. Then from (4) it
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 6. Variations of the horizontal forces with respect to the time considering different resolutions, for normal pressure 0.4fy , 0.6fy and 0.8fy .
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 7. Details of the variations of the horizontal forces for normal pressure 0.4fy , 0.6fy and 0.8fy .
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 8. Variations of the horizontal forces with respect to the time considering different normal loading for the 4th and the 6th resolution.
follows that ai =
1 N
for i = 1, ..., N and if
N i=1
di > 1, then the fractal
dimension D of the interpolation functions obeys N
|di |(1/N )D−1 = (1/N )D−1
i=1
N
|di | = 1.
(9)
i=1
From the above relation it results that in this case the fractal dimension D is given:
Log( D=1+
N i=1
|di | )
Log(N )
.
(10)
3. Description of the considered problem The problem treated here is presented in Fig. 1. It consists of two bodies which are divided by a rough interface. Unilateral contact with friction conditions are assumed to hold at the interface,
which is assumed to be the graph of a fractal interpolation function f interpolating the points with coordinates (in mm): {(0.0, 0.0), (40.0, 5.0), (80.0, −5.0), (120.0, 0.0)}. The free parameters have been chosen to have the values d1 = d2 = d3 = 0.365 so that the dimension of the interface given by relation (10), have the value D = 1.0826 which is suitable for metallic rough surfaces considering asperities of the scales studied here (Kotowski, 2006). In the sequel, the iterations n = 1, 2, … of the fractal interpolation function f are obtained by using the iterative scheme presented in the previous section. It must be mentioned here that seven approximations are taken into account, in order to study the influence of the resolution of the fractal interface to the friction mechanism for the scales studied here. As it is shown in Fig. 1a, every structure is separated by the interface into two parts, which are made of steel with modulus of elasticity E = 210 GPa and Poisson ratio ν = 0.3. Inelastic material behaviour is considered. In this respect the von-Mises yield criterion is assumed to hold with yield stress fy = 235 MPa. The thickness of the two parts is taken equal to t = 10 mm and plane stress
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 9. Details of the variations of the horizontal forces considering different normal loading for the 4th and the 6th resolution.
Fig. 10. Variations of the friction coefficient with respect to the resolution of the interface considering different normal loading.
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 11. a–c Plastic strains for various resolutions of the interface and for different values of normal pressure.
conditions are assumed to hold. The vertical displacements at the boundary nodes of the lower part are fixed, whereas the applied horizontal acceleration history is given in Fig. 2, where the maximum value of the excitation is 1m/sec2 . This acceleration history is multiplied by a factor α so that the resulting acceleration history in conjunction with the normal loading Pv , lead to the plastification of the asperities of the interface. For Pv , which is applied on the upper part of the structure, six cases are considered by varying its magnitude so that it produces a normal stress equal to 0.3fy , 0.4fy , 0.5fy , 0.6fy , 0.7fy , and 0.8fy , respectively, where fy = 235 MPa. As it has already been mentioned in the previous section, fractal interpolation functions are adopted for the simulation of the interface geometry. The computer generated interfaces fn , n = 1,
2, ... are characterized by a precise value of the resolution δ n of the interface. The resolution δ n is related to the (n)-th iteration of the fractal interpolation function and represents the horizontal projection of the basic linear size of the interface. As it is shown in Table 1, where the characteristics of the second to seventh iterations of the fractal interfaces are presented, the resolution of the interface changes rapidly when higher order iterations are taken into account. Taking into account the scales studied here in conjunction with the work of Mandelbrot et al. (1984) where is pointed out that the roughness of surfaces in metals develops fractal characteristics for at least three orders of magnitude, seven approximations are used for the simulation of the roughness interface. It is important to mention here that the opposite sides
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 11. Continued
Table 1 Characteristics of the considered structures. Resolution
Horizontal projection of resolution δ n (mm)
Length(mm)
Number of finite elements
2nd 3rd 4th 5th 6th 7th
13.333 4.444 1.481 0.494 0.165 0.055
123.98 126.47 129.18 132.37 135.98 140.06
19,686 21,158 22,340 23,592 27,201 31,154
of the interface is assumed to be perfectly matching surfaces, so only one side of the interface was generated by using the described procedure.
At the interfaces, unilateral contact with friction conditions are assumed to hold. The inherent coefficient of static friction for the material is taken to be equal to μ = 0.1. At each scale, a contact problem is solved by using the F.E. code MSC-MARC. In order to have results which are not influenced by the discretization (Hu et al., 20 0 0), the F.E. density is exactly the same for all the structures. For that, the linear boundaries were first divided into equal parts. The boundaries simulating the fractal curves were divided into segments having more or less equal lengths, of the order of 0.07 mm. Notice that this length is about equal to the fundamental length δ 7 of the finest resolution. In Fig. 3 the F.E. discretization of the structure corresponding to the highest resolution of the interface is presented, whereas details of the discretized
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 11. Continued
structures in the neighbourhood of the interface are presented in Fig. 4. Concluding, the parameters considered here are the following: • The interface resolution δ n : seven (7) different cases are considered. • The value of the normal loading applied on the upper body, Pv : Six (6) cases are considered with values 0.3fy , 0.4fy , 0.5fy , 0.6fy , 0.7fy and 0.8fy . • The value of the horizontal excitation. The combination of the first two parameters leads to 42 different problems. Each of these problems is solved for different values of the horizontal excitation (different values of factor α which
multiplies the acceleration history of Fig. 2b) until the complete plastification of the asperities of the interface. From the numerical point of view, all the aforementioned problems were solved by means of time history dynamic analysis that took into account the nonlinearities of the problem i.e., plasticity, large displacements and strains and unilateral contact with friction. The unilateral contact was treated by means of a double “segment to segment” algorithm, i.e. the finite element sides of each interface boundary were checked against penetration with the finite element sides of the opposite interface boundary. In the parts of the interface that were found to be in contact, standard Coulomb friction boundary conditions are assumed to hold, with the inherent friction coefficient of the material mentioned earlier (μ = 0.1).
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 12. Variations of the apparent friction coefficient with respect to the normal pressure considering different resolutions of the interface.
The employed contact algorithm proved to be of high efficiency and accuracy. The dynamic problem was solved by means of the Newmark method. The time step considered (0.001 s) proved to be sufficiently small to ensure the stability of the method even with the strong nonlinearities involved. Finally, four-node isoparametric plain stress elements were used for the discretization. 4. Multiresolution analysis The aim here is the study of the apparent friction coefficient for each one of the problems corresponding to the combination of the considered parameters, as presented in Section 3. Initially, our interest is focused on the values of the horizontal acceleration which lead to the plastification of the asperities of the interface. For that the acceleration history of Fig. 2 is multiplied by a factor α . Fig. 5 depicts the change of the values of factor α for which the resulting acceleration history leads to the plastification of the asperities with respect to the resolution of the interface, for different values of the normal loading. It is noticed that the values of the factor α become bigger as we go to higher resolution structures. In all the cases it is noticed that for the higher resolutions δ 5 , δ 6 and δ 7 significantly higher horizontal accelerations appear leading to higher horizontal forces. Fig. 6 depicts the changes of the horizontal forces with respect to the time for different resolutions δ n of the interface and for different values of the normal loading (0.4fy , 0.6fy and 0.8fy respectively). It is noticed that as the resolution increases all the structures indicate a stronger behaviour. It is also noticed that for the higher resolutions δ 5 − δ 7 there is a significant increase of the horizontal forces in all the cycles of loading. Fig. 7 shows in magnification the change of the horizontal forces with respect to the time for the different resolutions of the interface and for the normal loading cases presented in Fig. 6. It is important to mention here that for the lowest resolution, the diagrams present a plateau when the excitation takes its larger values even in the cases the normal loading is rather small, whereas structures with higher interface resolutions present the same behaviour when the normal loading takes larger values. This means that at the lowest interface resolution (δ 2 ), a rather small interface sliding resistance appears, on the contrary to the higher interface resolutions which present a great interface sliding resistance as a result of the interlocking of their asperities in conjunction with the fact that the plastifica-
tion of these interfaces, even in cases where large values of normal loading are assumed, is restricted near the area of the asperities. From the above it is obvious that in the problem studied here sliding instabilities appear. The differences of the obtained responses are also presented in the diagrams of Figs. 8 and 9 (Fig. 9 presents in magnification the diagrams of Fig. 8), which compare the change of the horizontal forces with respect to the time for the different cases of the normal loading and for the fourth and the sixth resolutions of the interface. As it was expected horizontal forces increase with increasing normal loading, in all structures resulting from the different resolutions of the interface. It must be mentioned here that the increase of the horizontal forces becomes smaller as higher values of normal loading are taken into account, because the contact is more plastic. As the applied normal loading is constant for each case treated here, the apparent dynamic friction coefficient can be calculated by using the formula
μ=
max Fx , Pv
(11)
where max Fx is the maximum value of the sum of the horizontal forces applied on the lower part of the structure and Pv is the normal loading applied on the upper part of it. Here, it should be clarified that the above convention regarding the maximum calculated value of the total horizontal forces has to be done due to the fact that the horizontal forces are not constant in all the considered cases, but their values vary depending on the time. In order to examine the resolution effects on the friction coefficient, the variation of the apparent friction coefficient μ with respect to the characteristic linear size δ n (in logarithmic scale) for different values of normal loading, is given in Fig. 10. It is noticed that the apparent friction coefficient increases with the resolution, fact which is strongly connected to the number and the scales of the asperities of the interface. More specifically, as higher resolutions are taken into account, more asperities of different scales are considered, thus increasing the resistance developed by the interlocking between the two parts of the interface. Moreover, it is observed that the apparent dynamic friction coefficient decreases as the value of the normal loading increases, fact which is connected with the stronger plastification of the asperities. This observation is verified by Fig. 11 which shows the plastic strains
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 13. Variations of the horizontal forces with respect to the normal pressure considering different resolutions of the interface.
developed at the interface for three different values of normal pressure (0.4fy , 0.6fy , 0.8fy ), when the horizontal forces take their max values (time steps 2.96 s, 4.24 s, 5.52 s). As it is shown in this figure the contact is more plastic at lower resolutions of the interface even in cases where the normal loading takes small values. It must be mentioned here that the rate of increase of the apparent dynamic friction coefficient becomes maximum when we go from the fourth to the fifth resolution of the interface, in all cases of normal pressure. At higher resolutions this rate is reduced and there is convergence in the values of the friction coefficient when the seventh resolution of the interface is taken into account. The convergence becomes faster as the normal pressure takes larger values. Fig. 12 presents the variations of the apparent friction coefficient μ with respect to the normal pressure for the different resolutions of the interface. It is important to be mentioned here that the friction coefficient is strongly depended on the normal loading of the interface. It is obvious that this dependence becomes stronger as the interface becomes rougher. More specifically, in rougher interfaces (interfaces with resolution δ 5 − δ 7 ) the rate of decrease of the apparent friction coefficient by increasing normal pressure is higher when normal pressure takes smaller values. This is due to the fact that at these resolutions the interlocking between the two parts of the interface is more intense. As normal pressure takes larger values the contact becomes more plastic (see Fig. 11) and for that the rate of decrease of μ becomes higher. On the contrary, in smoother interfaces (e.g. second resolution) the rate of decrease of the friction coefficient is small in smaller values of the normal pressure, because the interlocking between the two parts of these interfaces is not intense and the only factor that influences the apparent friction coefficient is the plastification of the asperities, which is more intense in cases where the normal pressure takes larger values. These conclusions can be also drawn from the study of Fig. 13 where the maximum values of the horizontal forces with respect to the normal pressure, considering different resolutions of the interface, are presented. It can been noticed that at a given normal pressure the friction forces decrease with decreasing the resolution of the interface, because at lower resolutions the contact is more plastic and for that many asperities are unable to support any tangential load and hence, they do not contribute to the friction mechanism. Increasing the normal pressure at a given resolution of the interface increases the number of such interface asperities, but at the same time brings into contact more asperities that were initially noncontacting. It turns out that the latter effect is more dominant in cases where normal pressure
takes smaller values (0.3fy –0.6fy ), whereas the first effect appears to be dominant in cases where larger values of normal pressure are taken into account. 5. Conclusions The paper deals with the evolution of friction between two metallic interfaces with irregularities of different scales subjected to cyclic loading. The attention has been concentrated on the investigation of the effects of the resolution and the normal pressure of the interfaces on friction. The problem is formulated numerically within the context of the F.E. method, using an elastic perfectly plastic material law. Moreover, unilateral contact and friction phenomena are accurately taken into account. The following conclusions may be traced: • The intensity of the excitation which leads to the plastification of the asperities of the interface increases as we go to higher resolutions of the interface, fact which strongly connected with the interlocking phenomena. • The apparent dynamic friction coefficient μ increases with the resolution of the interfaces. This is due to the fact that as the resolution increases, the interfaces become rougher because asperities of different scales are taken into account and therefore the interface sliding resistance increases. The effect of the interlocking on the friction mechanism was found to be negligible at lower resolutions of the interface and large values of normal pressure, where the contact appears more plastic. In these cases small friction coefficient was found and an improved model that takes into account strain hardening effects may be required. • The development of the sliding between the two bodies depends on the geometry of the interface and the combination of the vertical and horizontal loading. For rougher surfaces the interlocking phenomena are stronger and the relative displacements that develop are rather small, whereas for relatively smoother surfaces the sliding is more pronounced. • The study of the effect of the normal loading shows that friction forces increase with increasing normal loading for all scales studied here. The rate of the increase of the friction forces becomes smaller when normal pressure takes large values because the contact is more plastic and for that many asperities are unable to support any tangential load. However, in contrast to the behaviour of the friction forces, increasing normal loading decreases the dynamic friction coefficient, a fact which is connected with the plastification of the asperities.
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The above results lead to the conclusion that both parameters, resolution and normal pressure have an important impact on the apparent friction coefficient in metallic interfaces under cyclic loading, for the scale ranges studied here. It must be mentioned here that in the approach presented here the contact area is strongly affected by the assumption that there is perfect matching at the interface and thus the apparent contact area is larger compared to the real contact area. Moreover, the friction model presented here treats the friction coefficient as a plastic yield failure mechanism strongly affected by the normal pressure, because as it takes larger values the roughness of the interface almost disappears. References Barnsley, M., 1988. Fractals Everywhere. Academic Press, Boston- New York. Blau, P.J., 2001. The significance and use of the friction coefficient. Tribol. Int. 34, 585–591. Borodich, F.M., Mosolov, A.B., 1992. Fractal roughness in contact problems. J. Appl. Math. Mech. 56, 681–690. Borri-Brunetto, M., Carpinteri, A., Chiaia, B., 1999. Scaling phenomena due to fractal contact in concrete and rock fractures. Int. J. Fract. 95, 221–238. Borri-Brunetto, M., Chiaia, B., Ciavarella, M., 2001. Incipient sliding of rough surfaces in contact: a multiscale numerical analysis. Comput. Methods Appl. Mech. Eng. 190, 6053–6073. Chang, W.R., Etsion, I., Bogy, D.B., 1988. Static friction coefficient model for metallic rough surfaces. ASME J. Tribol. 110, 57–63. Chiaia, B., 2002. On the sliding instabilities at rough surfaces. J. Mech. Phys. solids 50, 895–924. Dowson, D., 1979. History of Tribology. Longman Inc., New York. Falconer, K.J., 1985. The Geometry of Fractal Sets. Cambridge Press, Cambridge University. Goerke, D., Willner, K., 2008. Normal contact of fractal surfaces- experimental and numerical investigations. Wear 264, 589–598. Greenwood, J.A., Williamson, J.B., 1966. Contact of nominally flat surfaces. Proc. R. Soc. London Ser. A. 295, 300–319.
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Hu, G.D., Panagiotopoulos, P.D., Panagouli, O., Scher, O., Wriggers, P., 20 0 0. Adaptive finite element analysis of fractal interfaces in contact problems. Comp. Methods Appl. Mech. Eng. 182, 17–37. Hyun, S., Pei, L., Molinari, J.F., Robbins, M.O., 2004. Finite element analysis of contact between elastic self-affine surfaces. Phys. Rev. E70. Kogut, L., Etsion, I., 2003. A semi-analytical solution for the sliding inception of a spherical contact. ASME J. Tribol. 125, 499–506. Kogut, L., Etsion, I., 2004. A static friction model for elastic plastic contacting rough surfaces. ASME J. Tribol. 126, 34–40. Kotowski, P., 2006. Fractal dimension of metallic fracture surface. Int. J. Fract. 141, 269–286. Majumdar, A., Tien, C.L., 1990. Fractal characterization and simulation of rough surfaces. Wear 136, 313–327. Majumdar, A., Bhushan, B., 1990. Role of fractal geometry in roughness characterization and contact mechanics of surfaces. Trans. ASME J. Tribol. 112, 205–216. Majumdar, A., Bhushan, B., 1991. Fractal model of elastic-plastic contact between rough surfaces. J. Tribol. 113, 1–11. Mandelbrot, B., 1982. The Fractal Geometry of Nature. W.H. Freemann & Company, New York. Mandelbrot, B., Passoja, D., Paullay, A., 1984. Fractal character of fractured surfaces of metals. Nature 308, 721–723. Panagouli, O.K., 1997. On the fractal nature of problems in mechanics. Chaos Solit. Fract. 8 (2), 287–301. Panagouli, O.K., Mistakidis, E.S., 2011. Dependence of contact area on the resolution of fractal interfaces in elastic and inelastic problems. Eng. Comput. 28 (6), 717–746. Panagouli, O.K., Iordanidou, K., 2013. Dependence of friction coefficient on the resolution and fractal dimension of metallic fracture surfaces. Int. J. Solids Struct. 50, 3106–3118. Persson, B.N.J., Bucher, F., Chiaia, B., 2002. Elastic contact between randomly rough surfaces: comparison of theory with numerical results. Phys. Rev. B 65 (18), 1841061–1841067. Persson, B.N.J., Tosatti, E., 1996. Physics of Sliding Friction. Kluwer, Dordrecht. Tabor, D., 1981. Friction-the present state of our understanding. ASME J. Lubr. Technol. 103, 169–179. Warren, T.L., Krajcinovic, D., 1995. Fractal models of elastic-perfectly plastic contact of rough surfaces based on the Cantor set. Int. J. Solids Struct. 19, 2907–2922. Zavarise, G., Borri-Brunetto, M., Paggi, M., 2007. On the resolution dependence of micromechanical contact models. Wear 262, 42–54.
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Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading Olympia K. Panagouli∗, Konstantina Mastrodimou Laboratory of structural analysis and design, Department of Civil Engineering, University of Thessaly, Pedion Areos, 38334 Volos, Greece
a r t i c l e
i n f o
Article history: Received 15 October 2015 Revised 18 October 2016 Available online xxx Keywords: Friction coefficient Roughness Fractal interpolation functions Asperity resolution Normal pressure Cyclic loading
a b s t r a c t In this paper the friction mechanism that is developed between metallic rough interfaces with irregularities of different scales, submitted to cyclic loading is studied. The main purpose of the paper is to investigate how the resolution δ n , which takes into account asperities of different scales of the rough interfaces and the intensity of the excitation, affect the friction that develops between the two surfaces. Friction is assumed to be the result of gradual plastification of the interface asperities in a metallic body with elastic-plastic behaviour. The interface of the body is described by a fractal interpolation function f with iterative regular construction, which permits the analysis of interfaces with irregularities of different scales. In the sequel, the structures resulting from the different iterations (i.e. resolutions) of the rough interface are submitted to different acceleration histories that induce forces leading to plastification of the interface asperities. The apparent dynamic friction coefficient is studied macroscopically, as the ratio between the forces that develop parallel and normal to the interface. Finally, the influence of the applied normal pressure to the friction mechanism is investigated with respect to the fractal resolution and the acceleration history of the interfaces. © 2016 Published by Elsevier Ltd.
1. Introduction A good understanding of tribological process at different normal forces and length scales is essential for improving efficiency in many applications. There is a gamut of engineering materials like metals, ceramics, polymers and composite materials used in many applications where friction challenges both efficiency and reliability. The common method for calculating the static friction force (Coulomb friction law) was based on the assumption that the friction force is independent of the nominal contact area and it results by multiplication of the normal applied load by a constant, known as the static friction coefficient. Static friction coefficient was taken as a function of the contacting materials for at least 300 years (Dowson, 1979). However, the values of the static friction coefficient resulting from the above assumptions represent average coefficients of friction, because the friction coefficient is presently recognized as both material and system dependent (Persson and Tossati, 1996; Blau, 2001) and is definitely not an instinct property of two contacting materials. More specifically, in Blau (2001) indicated that the friction coefficient is a misunderstood quantity in the field of science and engineering, due to the fact that the fun-
∗
Corresponding author. Fax: +302310344824. E-mail address: [email protected] (O.K. Panagouli).
damental origins of sliding resistance are not so clear, and hence, it is very important to understand the tribological process. Tabor in his general work of friction understanding (Tabor, 1981), pointed out three basic elements that are involved in the friction of dry solids: • The true contact area between rough surfaces. • The type and strength of bond formed at the interfaces where contact occurs. • The way in which the material around the contacting areas is sheared and ruptured during sliding. Chang et al. (1988) presented a model (CEB friction model) for predicting static friction coefficient of rough metallic interfaces based on the three elements indicated by Tabor (1981). This model uses a statistical representation of surface roughness (Greenwood and Williamson, 1966) and calculates the static friction force that is required to fail all of the contacting asperities, taking into account the normal loading. This approach demonstrated that the classical Coulomb friction law is a limiting case of a more general behaviour where static friction coefficient decreases with increasing applied load. This friction model underestimates friction coefficient values in the case of contact of rough surfaces since it neglects the ability of elastic-plastic deformed asperity to resist additional loading before failure, as it was demonstrated in Kogut and Etsion (2003). More specifically, the approximate
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applied displacements
Fig. 1. The considered problem.
generated self affine curve f with an iterative regular construction which permits us to analyse pre-fractals of arbitrary generation and therefore of arbitrary resolution δ n . Every structure resulting from each resolution of the interface is submitted to a constant normal loading and to different acceleration histories in order to induce forces leading to the plastification of the interface asperities. The resulting contact problems are solved by using the F.E. code MSC-MARC and the apparent dynamic friction coefficient is studied macroscopically. 2. Self affinity for the modelling of roughness
Fig. 2. The applied horizontal acceleration.
models for the calculation of static friction coefficient presented in Chang et al. (1988) assume failure of contacting asperity at the time the first plastic point appears and hence, underestimate the friction coefficient. The model presented later in Kogut and Etsion (2004) with the assumption of elastic perfectly-plastic material behaviour, demonstrates a strong effect of the external force and nominal contact area on the static friction. It was also demonstrated that this assumption may be invalid in cases where the contact approaches fully plastic state where small static friction was found. The numerical model presented in Panagouli and Iodranidou (2013) relates the interface roughness both with the resolution and the fractal dimension of rough interfaces which are described by fractal geometry. The model considers strain hardening effects and the friction mechanism is assumed to be the result of the interlocking mechanism developed between the interfaces and the gradual plastification of the interface asperities. This model also demonstrates a strong effect of the external load and roughness on the apparent friction coefficient. In this paper the friction mechanism developed between metallic rough interfaces with irregularities of different scales, submitted to dynamic excitations is studied. The main purpose of the paper is to investigate how the resolution δ n , which takes into account asperities of different scales of the interfaces, the intensity of the excitation and the applied normal loading affect the friction that develops between the two interfaces of a body with elastic-plastic material behaviour. The rough interface of the body is a computer
Experimental studies have shown that the true contact area between rough surfaces depends strongly on the surface parameters used for the modelling of the geometry of interfaces, which in many cases is too complicated to be described by a few parameters. To overcome problems resulting from traditional approaches based on scale-dependent statistical surface parameters and to model real surfaces demonstrating multi-scale roughness, in contact mechanics studies of rough surfaces, fractal geometry is used in many cases in order to describe surface topography (Mandelbrot et al., 1984; Majumdar and Tien, 1990; Majumdar and Bhushan, 1990; Majumdar and Bhushan, 1991; Borodich and Mosolov, 1992; Warren and Krajcinovic, 1995; Panagouli, 1997; Borri-Brunetto et al., 1999; Borri-Brunetto et al., 2001; Persson et al., 2002; Chiaia, 20 02; Hyun et al., 20 04; Zavarise et al., 20 07; Panagouli and Mistakidis, 2011). The Hausdorff dimension (Falconer, 1985) which is used for the characterization of a set as a fractal one, has no application to the curves originated in mechanics. For that, other methods, based on experimental or numerical calculations, have been developed for the estimation of the fractal dimension of a curve, a parameter which is of great importance because of its scale-independent character. One of the most useful methods in this category, is the Richardson method, described in Mandelbrot (1982), which uses dividers with a lower cut-off length δ . Moving with these dividers along the curve so that each new step starts where the previous step leaves off, the number of steps N(δ ) is obtained. The curve is said to be of fractal nature if, by repeating this procedure for different values of δ , the relation
N (δ ) ∼ δ −D
(1)
δ∗
is obtained in some interval <δ< The power D denotes the fractal dimension of the profile, which is in the range 1 ≤ D < 2.
∗ .
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Fig. 3. F.E. Discretization of the structure which corresponds to the 7th resolution of the interface.
The relation between the fractal dimension D of this profile and the dimension Ds of the corresponding surface is Ds = D + 1. Typically, a rough profile can be measured by taking height data yi with respect to an arbitrary datum at N equidistant discrete points xi and following the procedure presented in Goerke and Willner (2008). Here, fractal interpolation functions are used for the passage from this discrete set of data {(xi , yi ), i = 0, 1, 2, ..., N} to a continuous model, where f(xi ) = yi , i = 0, 1, ..., N. It has been proved in Barnsley (1988) that there is a sequence of functions fn + 1 (x) = (Tfn )(x), where
fn+1 (x ) = (T fn )(x ) = ci li−1 (x ) + di fn li−1 (x ) + gi
(2)
for x ∈ [xi − 1 , xi ], i = 1, 2, ..., N. The operator T converges to a fractal curve f, as n → ∞. The transformation li transforms [x0 , xN ] to [xi − 1 , xi ] and it is defined by the relation
li ( x ) = ai x + bi .
(3)
The factors di are the hidden variables of the transformations and they have to satisfy 0 ≤ di < 1 in order for T: C0 → C0 to have a unique fixed point. Moreover, the remaining parameters are given by the following equations:
Fig. 4. Details of discretization in the neighbourhood of the interface.
ai = (xi − xi−1 )/(xN − x0 )
(4)
ci = (yi − yi−1 )/(xN − x0 ) − di (yN − y0 )/(xN − x0 )
(5)
bi = (xN xi−1 − x0 xi )/(xN − x0 )
(6)
gi = (xN yi−1 − x0 yi )/(xN − x0 ) − di (xN y0 − x0 yN )/(xN − x0 ).
(7)
The fractal interpolation functions give profiles which look quite attractive from the viewpoint of a graphic roughness simulation. In higher iterations these profiles appear rougher because irregularities of new scales are taken into account. It must be mentioned here that an important advantage of the fractal interpolation functions is that their fractal dimension can be N obtained numerically (Barnsley, 1988). More specifically if di > i=1
1, and the interpolation points do not lie on a single straight line, then the fractal dimension of the graph G of the fractal interpolation function is the unique real solution D of
δ −D ≈
N i=1
Fig. 5. Accelerations scaling parameter which leads to the plastification of asperities with respect to the resolution of roughness.
|di |aDi −1 δ −D ⇔
N
|di |aDi −1 = 1.
(8)
i=1
In the case where the interpolation points are equally spaced it results that xi = x0 + Ni (xN − x0 ) for i = 1, ..., N. Then from (4) it
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Fig. 6. Variations of the horizontal forces with respect to the time considering different resolutions, for normal pressure 0.4fy , 0.6fy and 0.8fy .
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Fig. 7. Details of the variations of the horizontal forces for normal pressure 0.4fy , 0.6fy and 0.8fy .
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Fig. 8. Variations of the horizontal forces with respect to the time considering different normal loading for the 4th and the 6th resolution.
follows that ai =
1 N
for i = 1, ..., N and if
N i=1
di > 1, then the fractal
dimension D of the interpolation functions obeys N
|di |(1/N )D−1 = (1/N )D−1
i=1
N
|di | = 1.
(9)
i=1
From the above relation it results that in this case the fractal dimension D is given:
Log( D=1+
N i=1
|di | )
Log(N )
.
(10)
3. Description of the considered problem The problem treated here is presented in Fig. 1. It consists of two bodies which are divided by a rough interface. Unilateral contact with friction conditions are assumed to hold at the interface,
which is assumed to be the graph of a fractal interpolation function f interpolating the points with coordinates (in mm): {(0.0, 0.0), (40.0, 5.0), (80.0, −5.0), (120.0, 0.0)}. The free parameters have been chosen to have the values d1 = d2 = d3 = 0.365 so that the dimension of the interface given by relation (10), have the value D = 1.0826 which is suitable for metallic rough surfaces considering asperities of the scales studied here (Kotowski, 2006). In the sequel, the iterations n = 1, 2, … of the fractal interpolation function f are obtained by using the iterative scheme presented in the previous section. It must be mentioned here that seven approximations are taken into account, in order to study the influence of the resolution of the fractal interface to the friction mechanism for the scales studied here. As it is shown in Fig. 1a, every structure is separated by the interface into two parts, which are made of steel with modulus of elasticity E = 210 GPa and Poisson ratio ν = 0.3. Inelastic material behaviour is considered. In this respect the von-Mises yield criterion is assumed to hold with yield stress fy = 235 MPa. The thickness of the two parts is taken equal to t = 10 mm and plane stress
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Fig. 9. Details of the variations of the horizontal forces considering different normal loading for the 4th and the 6th resolution.
Fig. 10. Variations of the friction coefficient with respect to the resolution of the interface considering different normal loading.
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Fig. 11. a–c Plastic strains for various resolutions of the interface and for different values of normal pressure.
conditions are assumed to hold. The vertical displacements at the boundary nodes of the lower part are fixed, whereas the applied horizontal acceleration history is given in Fig. 2, where the maximum value of the excitation is 1m/sec2 . This acceleration history is multiplied by a factor α so that the resulting acceleration history in conjunction with the normal loading Pv , lead to the plastification of the asperities of the interface. For Pv , which is applied on the upper part of the structure, six cases are considered by varying its magnitude so that it produces a normal stress equal to 0.3fy , 0.4fy , 0.5fy , 0.6fy , 0.7fy , and 0.8fy , respectively, where fy = 235 MPa. As it has already been mentioned in the previous section, fractal interpolation functions are adopted for the simulation of the interface geometry. The computer generated interfaces fn , n = 1,
2, ... are characterized by a precise value of the resolution δ n of the interface. The resolution δ n is related to the (n)-th iteration of the fractal interpolation function and represents the horizontal projection of the basic linear size of the interface. As it is shown in Table 1, where the characteristics of the second to seventh iterations of the fractal interfaces are presented, the resolution of the interface changes rapidly when higher order iterations are taken into account. Taking into account the scales studied here in conjunction with the work of Mandelbrot et al. (1984) where is pointed out that the roughness of surfaces in metals develops fractal characteristics for at least three orders of magnitude, seven approximations are used for the simulation of the roughness interface. It is important to mention here that the opposite sides
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Fig. 11. Continued
Table 1 Characteristics of the considered structures. Resolution
Horizontal projection of resolution δ n (mm)
Length(mm)
Number of finite elements
2nd 3rd 4th 5th 6th 7th
13.333 4.444 1.481 0.494 0.165 0.055
123.98 126.47 129.18 132.37 135.98 140.06
19,686 21,158 22,340 23,592 27,201 31,154
of the interface is assumed to be perfectly matching surfaces, so only one side of the interface was generated by using the described procedure.
At the interfaces, unilateral contact with friction conditions are assumed to hold. The inherent coefficient of static friction for the material is taken to be equal to μ = 0.1. At each scale, a contact problem is solved by using the F.E. code MSC-MARC. In order to have results which are not influenced by the discretization (Hu et al., 20 0 0), the F.E. density is exactly the same for all the structures. For that, the linear boundaries were first divided into equal parts. The boundaries simulating the fractal curves were divided into segments having more or less equal lengths, of the order of 0.07 mm. Notice that this length is about equal to the fundamental length δ 7 of the finest resolution. In Fig. 3 the F.E. discretization of the structure corresponding to the highest resolution of the interface is presented, whereas details of the discretized
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Fig. 11. Continued
structures in the neighbourhood of the interface are presented in Fig. 4. Concluding, the parameters considered here are the following: • The interface resolution δ n : seven (7) different cases are considered. • The value of the normal loading applied on the upper body, Pv : Six (6) cases are considered with values 0.3fy , 0.4fy , 0.5fy , 0.6fy , 0.7fy and 0.8fy . • The value of the horizontal excitation. The combination of the first two parameters leads to 42 different problems. Each of these problems is solved for different values of the horizontal excitation (different values of factor α which
multiplies the acceleration history of Fig. 2b) until the complete plastification of the asperities of the interface. From the numerical point of view, all the aforementioned problems were solved by means of time history dynamic analysis that took into account the nonlinearities of the problem i.e., plasticity, large displacements and strains and unilateral contact with friction. The unilateral contact was treated by means of a double “segment to segment” algorithm, i.e. the finite element sides of each interface boundary were checked against penetration with the finite element sides of the opposite interface boundary. In the parts of the interface that were found to be in contact, standard Coulomb friction boundary conditions are assumed to hold, with the inherent friction coefficient of the material mentioned earlier (μ = 0.1).
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Fig. 12. Variations of the apparent friction coefficient with respect to the normal pressure considering different resolutions of the interface.
The employed contact algorithm proved to be of high efficiency and accuracy. The dynamic problem was solved by means of the Newmark method. The time step considered (0.001 s) proved to be sufficiently small to ensure the stability of the method even with the strong nonlinearities involved. Finally, four-node isoparametric plain stress elements were used for the discretization. 4. Multiresolution analysis The aim here is the study of the apparent friction coefficient for each one of the problems corresponding to the combination of the considered parameters, as presented in Section 3. Initially, our interest is focused on the values of the horizontal acceleration which lead to the plastification of the asperities of the interface. For that the acceleration history of Fig. 2 is multiplied by a factor α . Fig. 5 depicts the change of the values of factor α for which the resulting acceleration history leads to the plastification of the asperities with respect to the resolution of the interface, for different values of the normal loading. It is noticed that the values of the factor α become bigger as we go to higher resolution structures. In all the cases it is noticed that for the higher resolutions δ 5 , δ 6 and δ 7 significantly higher horizontal accelerations appear leading to higher horizontal forces. Fig. 6 depicts the changes of the horizontal forces with respect to the time for different resolutions δ n of the interface and for different values of the normal loading (0.4fy , 0.6fy and 0.8fy respectively). It is noticed that as the resolution increases all the structures indicate a stronger behaviour. It is also noticed that for the higher resolutions δ 5 − δ 7 there is a significant increase of the horizontal forces in all the cycles of loading. Fig. 7 shows in magnification the change of the horizontal forces with respect to the time for the different resolutions of the interface and for the normal loading cases presented in Fig. 6. It is important to mention here that for the lowest resolution, the diagrams present a plateau when the excitation takes its larger values even in the cases the normal loading is rather small, whereas structures with higher interface resolutions present the same behaviour when the normal loading takes larger values. This means that at the lowest interface resolution (δ 2 ), a rather small interface sliding resistance appears, on the contrary to the higher interface resolutions which present a great interface sliding resistance as a result of the interlocking of their asperities in conjunction with the fact that the plastifica-
tion of these interfaces, even in cases where large values of normal loading are assumed, is restricted near the area of the asperities. From the above it is obvious that in the problem studied here sliding instabilities appear. The differences of the obtained responses are also presented in the diagrams of Figs. 8 and 9 (Fig. 9 presents in magnification the diagrams of Fig. 8), which compare the change of the horizontal forces with respect to the time for the different cases of the normal loading and for the fourth and the sixth resolutions of the interface. As it was expected horizontal forces increase with increasing normal loading, in all structures resulting from the different resolutions of the interface. It must be mentioned here that the increase of the horizontal forces becomes smaller as higher values of normal loading are taken into account, because the contact is more plastic. As the applied normal loading is constant for each case treated here, the apparent dynamic friction coefficient can be calculated by using the formula
μ=
max Fx , Pv
(11)
where max Fx is the maximum value of the sum of the horizontal forces applied on the lower part of the structure and Pv is the normal loading applied on the upper part of it. Here, it should be clarified that the above convention regarding the maximum calculated value of the total horizontal forces has to be done due to the fact that the horizontal forces are not constant in all the considered cases, but their values vary depending on the time. In order to examine the resolution effects on the friction coefficient, the variation of the apparent friction coefficient μ with respect to the characteristic linear size δ n (in logarithmic scale) for different values of normal loading, is given in Fig. 10. It is noticed that the apparent friction coefficient increases with the resolution, fact which is strongly connected to the number and the scales of the asperities of the interface. More specifically, as higher resolutions are taken into account, more asperities of different scales are considered, thus increasing the resistance developed by the interlocking between the two parts of the interface. Moreover, it is observed that the apparent dynamic friction coefficient decreases as the value of the normal loading increases, fact which is connected with the stronger plastification of the asperities. This observation is verified by Fig. 11 which shows the plastic strains
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Fig. 13. Variations of the horizontal forces with respect to the normal pressure considering different resolutions of the interface.
developed at the interface for three different values of normal pressure (0.4fy , 0.6fy , 0.8fy ), when the horizontal forces take their max values (time steps 2.96 s, 4.24 s, 5.52 s). As it is shown in this figure the contact is more plastic at lower resolutions of the interface even in cases where the normal loading takes small values. It must be mentioned here that the rate of increase of the apparent dynamic friction coefficient becomes maximum when we go from the fourth to the fifth resolution of the interface, in all cases of normal pressure. At higher resolutions this rate is reduced and there is convergence in the values of the friction coefficient when the seventh resolution of the interface is taken into account. The convergence becomes faster as the normal pressure takes larger values. Fig. 12 presents the variations of the apparent friction coefficient μ with respect to the normal pressure for the different resolutions of the interface. It is important to be mentioned here that the friction coefficient is strongly depended on the normal loading of the interface. It is obvious that this dependence becomes stronger as the interface becomes rougher. More specifically, in rougher interfaces (interfaces with resolution δ 5 − δ 7 ) the rate of decrease of the apparent friction coefficient by increasing normal pressure is higher when normal pressure takes smaller values. This is due to the fact that at these resolutions the interlocking between the two parts of the interface is more intense. As normal pressure takes larger values the contact becomes more plastic (see Fig. 11) and for that the rate of decrease of μ becomes higher. On the contrary, in smoother interfaces (e.g. second resolution) the rate of decrease of the friction coefficient is small in smaller values of the normal pressure, because the interlocking between the two parts of these interfaces is not intense and the only factor that influences the apparent friction coefficient is the plastification of the asperities, which is more intense in cases where the normal pressure takes larger values. These conclusions can be also drawn from the study of Fig. 13 where the maximum values of the horizontal forces with respect to the normal pressure, considering different resolutions of the interface, are presented. It can been noticed that at a given normal pressure the friction forces decrease with decreasing the resolution of the interface, because at lower resolutions the contact is more plastic and for that many asperities are unable to support any tangential load and hence, they do not contribute to the friction mechanism. Increasing the normal pressure at a given resolution of the interface increases the number of such interface asperities, but at the same time brings into contact more asperities that were initially noncontacting. It turns out that the latter effect is more dominant in cases where normal pressure
takes smaller values (0.3fy –0.6fy ), whereas the first effect appears to be dominant in cases where larger values of normal pressure are taken into account. 5. Conclusions The paper deals with the evolution of friction between two metallic interfaces with irregularities of different scales subjected to cyclic loading. The attention has been concentrated on the investigation of the effects of the resolution and the normal pressure of the interfaces on friction. The problem is formulated numerically within the context of the F.E. method, using an elastic perfectly plastic material law. Moreover, unilateral contact and friction phenomena are accurately taken into account. The following conclusions may be traced: • The intensity of the excitation which leads to the plastification of the asperities of the interface increases as we go to higher resolutions of the interface, fact which strongly connected with the interlocking phenomena. • The apparent dynamic friction coefficient μ increases with the resolution of the interfaces. This is due to the fact that as the resolution increases, the interfaces become rougher because asperities of different scales are taken into account and therefore the interface sliding resistance increases. The effect of the interlocking on the friction mechanism was found to be negligible at lower resolutions of the interface and large values of normal pressure, where the contact appears more plastic. In these cases small friction coefficient was found and an improved model that takes into account strain hardening effects may be required. • The development of the sliding between the two bodies depends on the geometry of the interface and the combination of the vertical and horizontal loading. For rougher surfaces the interlocking phenomena are stronger and the relative displacements that develop are rather small, whereas for relatively smoother surfaces the sliding is more pronounced. • The study of the effect of the normal loading shows that friction forces increase with increasing normal loading for all scales studied here. The rate of the increase of the friction forces becomes smaller when normal pressure takes large values because the contact is more plastic and for that many asperities are unable to support any tangential load. However, in contrast to the behaviour of the friction forces, increasing normal loading decreases the dynamic friction coefficient, a fact which is connected with the plastification of the asperities.
Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010
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Please cite this article as: O.K. Panagouli, K. Mastrodimou, Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading, International Journal of Solids and Structures (2016), http://dx.doi.org/10.1016/j.ijsolstr.2016.11.010