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Adhesive properties of contacts between elastic bodies with randomly rough self-affine surfaces: A simulation with the method of reduction of dimensionality
V.L. Popov, A.E. Filippov1
Berlin University of Technology, Berlin, 10623, Germany 1 Donetsk Institute for Physics and Engineering NASU, Donetsk, 83114, Ukraine
Adhesive properties (adhesion force and adhesion coefficient) of contacts between elastic bodies with rough surfaces have been investigated and adhesion maps constructed showing the dependence of adhesive properties on the roughness, rms slope of the surface, elastic modulus, surface energy and fractal dimension. Simulations have been carried out in the frame of the method of reduction of dimensionality.
Keywords: contact, adhesion, rough surfaces, self-affine surfaces, adhesion coefficient, pressure sensitive adhesion
1. Introduction
The roughness of surfaces has a great influence on many physical phenomena, such as friction, wear, sealing, adhesion, as well as electrical and thermal conductivity [1-4]. Bowden and Tabor [1] were the first to realize the importance of the surface roughness of bodies in contact. The main qualitative understanding of contact properties of rough surfaces — including self-affine surfaces — is due to the works by Archard [5]. Greenwood and Williamson published a theory of contacts with rough surfaces [6], which became the most influencial model for many years. The most important findings of Archard as well as Greenwood and Williamson were that the main effect of the roughness is due to the increase in the number of microcontacts with normal force while the average size and the loading condic-tions in individual microcontacts change only slowly. While the surfaces considered by Archard were partially multiscale self-affine surfaces, they have not been “randomy rough.” The surfaces considered by Greenwood and Williamson were on the contrary randomly rough but neither multiscale nor self-affine. Further generalization of the contact mechanics and its application to random self-affine surfaces is due to a number of researchers. Main contributions are the Persson theory [4] as well as numerical simulations by Hyun and Robbins [7] and Campana and Muser [8]. The main interest was focused over several years on the real contact
area. Recently, the contact stiffness was studied in detail [9-11]. In the present paper we will deal with adhesive contacts between elastic bodies with fractal surfaces. The physical foundations of the underlying theory have been established by Griffith [12] in his theory of adhesive crack. This theory has been microscopically supported by Prandtl
[13] and applied to contact problems between an elastic sphere and a halfspace by Johnson, Kendall and Roberts
[14]. Some generalizations of this theory can be found in
[15]. In the last decade, adhesion of structured and rough surfaces became a hot topic, due to the interest in the adhesion mechanism in living systems [16, 17]. Adhesion is further considered to be one of contributions to the friction force [18].
Inspite of theoretical and practical importance of the adhesive contact problem, no numerical simulations of contact between bodies with fractal surfaces were done until now, because of the complexity of this problem for numerical implementation. In the present paper we close this gap by direct numerical simulations of adhesion properties between stochastically rough, self-affine surfaces. For this sake, we use the method of reduction of dimensionality. This method was proposed in [19] and allows us to substitute a real three-dimensional contact with a contact with a onedimensional elastic foundation. This has been initially proposed for normal contact between cylindrical and parabolic
© Popov V.L., Filippov A.E., 2012
indenters and an elastic half-space [19] and then extended to randomly rough (but not fractal) surfaces [20, 21]. In [22] proof has been provided that the method of reduction of dimensionality gives exact results for contacts of arbitrary bodies of revolution, both with and without adhesion. Quite recently, a comparison of direct three-dimensional calculations and calculations with the method of reduction of dimensionality have shown that it is applicable to self-affine fractal surfaces as well, in the range of fractal dimensions from 2 to 3 [23]. In the meantime, the method has been succesfully applied to simulation of frictional force between a rigid rough surface and an elastomer [24]. However, it was never applied to adhesive contact of rough surfaces. In the present paper we fill this gap and simulate adhesive contacts between randomly rough fractal surfaces with the method of reduction of dimensionality.
2. The model
According to the method of reduction of dimensionality, the contact between a rigid body with a rough surface and an elastic half-space can be substituted by a one-dimensional contact problem with an elastic foundation, as illustrated in Fig. 1. To be equivaleent to the initial three-dimensional problem, the stiffness of each spring of the elastic foundation must be chosen according to the rule
Ak = E *A x, (1)
where Ax is the space between adjacent springs (discretization step) and * E
*E = 7-2- (2)
E being the Young modulus of the elastic half-space and v, its Poisson ratio.
Additionally, for adhesive contacts, the following rule of Hess [22, 25] must be applied. Consider a one-dimensional substitution system sketched in Fig. 1. If the upper body is first pressed onto underlying elastic foundation and then pulled off, then the most stressed outer springs will detach from the body when they achieve the critical length
ALax(a) =
2ащп
(3)
where 2a is the length of the contact. It has been shown in [22] that application of this rule provides exact results for
adhesive contact with an arbitrary body of revolution with respect to relations of the normal force Fn, the indentation depth d and the contact radius a.
One further rule is needed in the case of randomly rough surfaces. A randomly rough surface z(x) can by characterized by its spectral density 1
C2D(q) “ '
(2n)
rj(z(x)z(0)) e iqxd2X,
(4)
where (...) means ensemble averaging. In the case of self-affine fractal surfaces, the spectral density has the form [4]
\-2( H +1)
C2D = const
Kq0j
(5)
where H is the Hurst exponent. It is directly associated with
the fractal dimension Df of the surface:
Df = 3 - H. In
the three-dimensional physical space, Df can change between 2 and 3 so that the Hurst exponent takes values in the range of 0 < H < 1. Typical values for real physical surfaces are around Df = 2.3 and H- 0.7. Generally, the validity of the power law (5) is limited by some cut-off wave vectors q^m and qmax. In the present paper, we consider surfaces without cut-off at the lower limit of the wave vectors (or large wave lengths). The only natural cut-off is due to the size L of the system:
qmin =L - (6)
It was argued in [20] that the contact with a one-dimensional elastic foundation will be equivalent to the threedimensional problem if the spectral density C1D(q) of the one-dimensional equivalent “rough line” is defined as
C1D (q) = nqC2D (|q|)* (7)
The one-dimensional profile is generated according to the rule
qmax
h(x) = J dqB,D(q)cos(qx +q<q))- (8)
qmin
where ^(q) is a random phase and 12k
B1D(q) =y “LC1D( q) = B 1D(-q)- (9)
In Fig. 2, typical profiles of equivalent one-dimensional rigid lines for two values of the Hurst exponent are shown. Furthermore, the current position of the elastic (upper) sur-
Fig. 1. Schematic representation of the one-dimensional model used as well as of the “rule of Hess” for the adhesion criterion
f(x), y 4
Jl .J. A
I. I ..I. к il
-4
иг T||i4 г x
1| .......................I I"
0.0
0.2
0.4
0.6
0.8
Fig. 2. Typical profiles of equivalent one-dimensional rigid lines for two values of the Hurst exponent H = 0.25 and H = 0.75 (black curves on the subplots (a, c)). The current position of the elastic (upper) surface in the adhesive case are shown as gray lines. To the right of the surfaces, on the subplots (b, d), the height distribution histograms for the shown realizations are presented
face in the adhesive case, deformed with account of elasticity and adhesion, are shown as gray lines. To the right of the surfaces, the histograms of the height distributions are shown. The height distributions strongly differ from the Gaussian distribution and remains non-self-averaging at least up to a number of N = 220 - 106 discretization points.
Figure 3 shows a small fragment of the rigid surface z(x). It illustrates that it is smooth enough on the smallest scale.
3. Results
Numerical experiments have been carried out as follows. The rigid rough line was moved stepwise towards the elastic foundation. The surface of the elastic foundation, which was planar at the beginning, becomes deformed and contacts the rigid counter body in a set of connected fragments with lengths Sxk. These connected contact regions are considered to be microcontacts, and the rule of Hess (3) in the form
Л/,
max,&
8xk ^12
(10)
is independently applied to each microcontact. Figure 3 illustrates the definition of the mentioned fragments. In the course of vertical displacement of the rigid body, the length of the microcontacts changes. At every step of vertical dis-
placement, it is verified whether the immediate neighbors to the boundary point satisfy the condition (10). If this is the case, then the contact length and the condition (10) will be changed. The procedure is repeated until the system achieves equilibrium. Sometimes, the growing microcontacts coalesce, which enhances the tendency toward adhesion. The determination of the equilibrium at any given vertical position of the rigid body, as well as the corresponding force, is the main part of the calculation procedure. As the adhesion condition does depend on whether some points are already in contact or not, the contact regions are generally different on the approaching and pulling-away stages — as it is in the case even for the simplest forms, as for spheres in the JKR-theory [14]. Depending on the roughness and other parameters, three different regimes have been observed. For very smooth surfaces, the adhesion takes place at once over almost the entire surface and does not depend on the applied normal force. This case of ideal adhesion means that the roughness does not play an important role and the surfaces can be considered to be almost ideally smooth (Fig. 4(a)). If the roughness gets larger than at some critical value, the macroscopic adhesion force after the first contact of surfaces (without applying a normal force) vanishes almost completely. However, after application of a normal force, a finite adhesion force is observed (as shown in Fig. 4(b)). Fuller and Tabor [26] have shown that in this
0.100
0.101
0.102
Fig. 3. A small fragment of the contact surface illustrating the definition of the boundary points of microcontacts (bounding the contact region from the left and from the right). The boundary points are shown with white circles
Fig. 4. Three qualitatively different types of adhesive behavior realized in the studied system: almost pure adhesive contact with the adhesive force being independent of the applied normal force (a); “pressure sensitive” adhesive contact: the adhesion force appears only if a finite normal force is applied (b); almost no adhesion, independently of how strong the bodies are pressed against each other (c)
case the adhesion force is approximately proportional to the normal force. The coefficient of proportionality can be called the “adhesion coefficient” (see also [3]). For very large roughness, the adhesion coefficient vanishes (in numerical simulation, it becomes very small). This is the case of no adhesion (Fig. 4(c)).
According to a simple analytical estimation (Eq. (7.41) of [3]), the adhesive properties of rough bodies are governed by the dimensionless parameter
(11)
h( Vz)E *
V Y -
where Vz is the rms value of the surface gradient. Note that while the rms height h is a well-defined macroscopic quantity, which depends on the size of the system but does not depend on the cut-off wavelength on the smallest scale, the rms gradient Vz, defined as
. . qmax
(Vz2 = J
const ■
-2H-1 2
q d q =
qmin
qmax
= J const • q 2H+1d q =
qmin
const r 2-2H 2-2H -
= 2 - 2H Lqmax - qmin -- (12)
converges at the lower limit but diverges at the upper limit of integration. However, for Hurst exponents in the vicinity of H - 1, Eq. (12) takes the form
qmax
[Vz2j - J Sq_1dq = Sln(q,/q0)
for H - 1.
(13)
In this case, the integral (13) diverges weakly (logarithmically) at both limits. Therefore, all parts of the spectrum contribute to the surface gradient. Thus, the rms surface gradient is, strictly speaking, not a well-defined macroscopic
(h(Vz) EVy )12 2.0
(h(Vz) EVy)12 2.0
Fig. 5. Phase diagram of the Hurst exponent H vs. the dimensionless parameter combination -Jh(Vz) E /y. The diagram shows the region of existence of pressure insensitive adhesion (a) and pressure sensitive adhesion (b). The gray scale shows in (a), the averaged adhesion force Fa, and in (b), the adhesion coefficient defined as the ratio Fa/Fn. In both cases, the corresponding quantities are normalized by their maximum values. The horizontal dash line shows the boundary between pressure insensitive and pressure sensitive behavior. The dark region in the upper right corner of the diagram (b) corresponds to the case of the vanishing adhesion force, as classified in Fig. 4(c)
property. However, it is defined with logarithmic accuracy. This is valid only for Hurst exponents in the vicinity of 1. This is, however, exactly the range of Hurst exponents which are of the most practical importance.
In the following, we plot the applied normal force Fn and the adhesion force Fa as well as adhesion coefficient in the phase diagram in coordinates <^h(Vz)E*/y and H. These diagrams are shown in Fig. 5. The diagram Fig. 5(a) shows the force of adhesion Fa, which appears by a contact without finite normal force. The adhesion force is normalized by its maximum value
Fa,max =^12E*L - (I4)
which is achieved for an absolutely smooth surface; L is the apparent diameter of the macroscopic system. It is clearly seen that for small roughness, there is a finite adhesion force, while it disappears completely for large roughness. The boundary between the adhesive and non-adhesive surfaces can be roughly identified as
h{ Vz)E *
= 0.6.
Y
(15)
For larger values of the parameter (11), the adhesive force without preliminary application of normal force becomes zero. However, the adhesion force appears after the application of the normal force. In this region, the adhesion coefficient can be defined as the ratio Fj Fn. The adhesion coefficient is shown in the diagram Fig. 5(b). Theoretically, the adhesion coefficient at the boundary between pressure insensitive and pressure sensitive cases should be infinite. In the simulations, it is only very large and decreases with further increase of the parameter (11).
The horizontal dash line in Fig. 5 marks the boundary between the pressure sensitive and pressure insensitive cases. For the pressure insensitive case, the diagram in Fig. 5(a) should be used, and for the pressure sensitive case, the diagram in Fig. 5(b). A partial overlapping of these regions is due to the statistical nature of the system. The dark region in the upper right corner of the diagram in Fig. 5(b) corresponds to the case of vanishing adhesion force, as classified in Fig. 4(c). Figure 6 illustrates the quantitative behavior of the adhesion force and the adhesion coefficient by the cross-sections of the diagrams in Fig. 5. One can see that the pressure independent adhesion force is non-zero only to the left of the critical value of the parameter (11), while the adhesion coefficient, only to the right (with exception of the above mentioned small statistical overlap). From the insert of Fig. 6(b), one can see that the adhesion coefficient remains non-zero even at relatively large values of the parameter (11), but decreases exponentially when this parameter increases.
4. Discussion
With the method of reduction of dimensionality, we produced phase diagrams for adhesive contacts between elas-
h(vz)E*/y
Fig. 6. Fragments (a) and (b) are the cross-sections of the phase diagrams for the fixed value of the Hurst exponent H = 0.3 (along the vertical dot-dash lines in Fig. 5(a and b) correspondingly. In the inlay (c), the adhesion coefficient Fa/Fn is plotted on a logarithmic scale. One can see that after a maximum in the region of pressure sensitive adhesion, the adhesion coefficient exponentially decreases in the region of almost pure elastic contact
tic bodies with randomly rough, fractal, self-affine surfaces. We have found that the main parameters governing the adhesive behavior is the dimensionless combination yjh(Vz)E*/y and the Hurst exponent of self-affine surfaces. The most important is the first parameter, and the fractal dimension has only slight influence on the behavior. We have identified three different types of adhesive behavior: (i) pressure insensitive adhesion for very smooth surfaces, (ii) pressure sensitive adhesion for roughness in some intermediate range and (iii) surfaces without adhesion (more exactly, with exponentially small adhesion) for very rough surfaces.
We gratefully acknowledge valuable discussions with M. Hess and R. Pohrt. A.E. Filippov acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) and Deutscher Akademischer Austauschdientst (DAAD).
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Поступила в редакцию 10.06.2012 г.
Сведения об авторах
Попов Валентин Леонидович, д.ф.-м.н., проф. Берлинского технического университета, v.popov@tu-berlin.de Filippov Alexander E., Prof., Donetsk Institute for Physics and Engineering of NASU, Ukraine, filippov_ae@yahoo.com
