Hiroyuki Nagahama Hiroyuki Nagahama. Geophysics 84 4 : WAWA Article history received:. ABSTRACT The pressure dependence of elastic deformation at multiple contacts in thin cracks has been investigated through the variation of elastic wave velocity. You do not currently have access to this article. You could not be signed in. Librarian Administrator Sign In.
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Tony Diaz. Our new model involves adhesive elastic contact between a smooth, rigid paraboloid tip and a rough, semi-infinite, deformable solid substrate [see Fig. There are two major types of models used for studying contact between rough surfaces. The first type is based on the non-interacting asperity contact model pioneered by Greenwood and Williamson 22 , which is widely used for studying the effect of roughness on adhesion 23 , 24 , particle adhesion 25 , elasto-plastic contact 26 , and friction The second type is related to the self-affine fractal contact model put forward by Persson Ours is a non-interacting asperity type contact model, in which we assume that each substrate asperity interacts with the tip as though it were the only one interacting with it.
These large-scale instabilities correspond to the initial sudden drop in the contact force [ i. It was observed in the experiments 6 that each of these large-scale instabilities always occurred only once in a contact cycle. We assume that those surface asperities will come into contact through small-scale surface mechanical instabilities, as done in refs 6 , We assume that the detachment of the depth-dependent asperities takes place through the occurrence of small-scale instabilities, too.
This paper is organized as follows. Finally, we conclude by discussing the limitations of our model. The adhesive interactions are introduced using the Dugdale cohesive zone model As per this model, a surface material point experiences a traction only if its distance from the other solid in the direction normal to the surface is less than Z 0. Thus Z 0 denotes the range of the inter-body adhesive forces, which are thought to arise from van der Waals-type interactions between the surfaces.
In terms of these non-dimensional parameters, the magnitude of the contact force P and the indentation depth h at equilibrium are related as. The parameter c is defined such that all surface points whose radial coordinate in the undeformed configuration r is less than or equal to c experience a non-zero traction force [see Fig. The coordinate system corresponding to r is defined in Fig. The separation [ u 3 ] is defined in Fig. For this reason we refer to c as the contact radius.
The JKR theory applies to compliant materials having a large work of adhesion, while the DMT theory applies to stiff materials having a small work of adhesion. The schematic of a typical equilibrium P — h curve predicted by the MD theory is shown in Fig. In a displacement controlled experiment, the measured P — h curve will be the envelope of the equilibrium P — h curve. It is denoted as the shaded area in Fig. The limits of integration r i and r o in eq. However, we were able to obtain closed form expressions in three special cases. By analyzing the numerical data shown in Fig. Consider a region in the mean plane having an area of unit magnitude.
We say that this unit region contains an asperity whose apex has the coordinates x 1 , x 2 , z x 1 , x 2 , if it contains the point x 1 , x 2 , 0. The unit region will, in general, contain a large number of asperities. Motivated by those results, we model the variation of the different geometric characteristics of the asperities belonging to the unit region using the probability density functions PDFs given by Nayak That is, the PDFs characterizing the different geometric features of the asperities do not depend on the location or the orientation of the unit region.
For this special case, Nayak 21 gives the joint PDF of the heights and curvatures of the asperities belonging to the unit region to be. The constants C 1 , C 2 in eq. These moments can be computed from the equation. See Supplementary Material for its complete definition. We assume in our model that the contact between the tip and the substrate takes place only at the asperities.
Consequently, in our model the real contact region is smaller than the nominal contact region. We define the nominal contact region to be a circular region in the mean plane that contains all the contacting asperities [Fig. The nominal contact region is also referred to as the apparent contact region in the literature, since at the large-scale it is the region over which the solids appear to be in contact. The nominal contact region grows and recedes during the loading and unloading phases of a contact cycle, respectively.
The evolution of the real contact region is much more complicated. The definition of the nominal contact region, by itself, does not imply that all asperities contained within it are in contact with the tip. Indeed, it is possible that many asperities never make contact despite belonging to the nominal contact region during some instance of the contact cycle. However, as part of our model, we assume that all asperities within a nominal contact region make and break contact with the tip as that region forms and unforms.
The asperity density is the total number of asperities contained in a nominal contact region of unit area. Nayak 21 gives the total number of asperities contained in a region of the mean plane of unit area to be.
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Recall that the nominal contact region is part of the mean plane. In many contact experiments the nominal contact region is measured as part of the experiment e. In such cases, the total number of depth-dependent asperities can be estimated by using the measured nominal contact area values in conjunction with eqs 9 and In other situations, where such measurements are unavailable we believe that the best alternative is to estimate the nominal contact region using a classical adhesive elastic contact theory.
For example, in the next section, we estimate the nominal contact region in the experiments of Kesari et al. That energy loss is not constant between the asperities, but varies between them depending on their curvature. Using eq. Recall that the nominal contact region belongs to the mean plane. Therefore, the PDF eq. Thus, the mean energy loss per depth-dependent asperity can be computed as. Equation 14a follows from noting that in eq.
The eq. Multiplying the mean energy loss per depth-dependent asperity with the total number of those asperities we get. In this section, we use eq.
Depth-dependent hysteresis in adhesive elastic contacts at large surface roughness
The experiments involved contact between a spherical glass bead and PDMS substrates. The geometry of the contacting solids in the experiments is shown in the insets of Fig. In the experiments, both the substrate and the tip are rough [see Fig. However, in our model only the substrate is rough. This makes the quantitative comparison of our model with the experiments challenging. Despite its crude nature, we hope that some knowledge can yet be gained about the utility of our model from this comparison. The RMS roughness of this bead is To calculate the first two of these quantities we need to know the C iso function.
Given a substrate surface topography function, we are able to numerically compute C iso q , where q is the wavenumber magnitude, using the method presented in ref. We take the surface topography function in our model to be a scaled version of the surface topography function of the substrate in the experiments. For the purposes of the current comparison, we make the same assumption. The RMS roughness of those topographies ranged between 0.
Mechanics of elastic contacts - CERN Document Server
Since our model applies to the large surface roughness regime, we only consider the experiments on the substrate with the largest roughness, namely 1. We use that topography data to construct the surface topography function for the substrate in the experiments.
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The values of C iso corresponding to this surface topography function are shown in Fig. As can be seen, the values of C iso are approximately constant at small wavenumber magnitudes, and fall off rapidly at large wavenumber magnitudes. This behavior is similar to that of a power-law PSD function. To be specific, consider the PSD function. The value for q 0 was chosen independently before performing the minimization. To account for the roughness of the tip in the experiments in our model, in which the tip is always smooth, we take the surface topography function of the substrate in our model to be a scaled version of that of the substrate in the experiments.
We perform our comparison for a range of k values, which—as we shall discuss shortly—we selected by taking into consideration the measured roughness of the tip-in-the-experiments. Knowing the C iso function in our model, we numerically evaluate the integrals in eq. Substituting the values for the spectral moments given in 18 in eq.
In eq. However, we could not find a clear way to identify Z 0 in the experiments. Therefore, we treat Z 0 as a fitting parameter in our comparison. Unfortunately, neither Kesari et al. The JKR theory is the most widely used model for adhesive elastic contact, which only applies to contact between smooth surfaces.
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Kesari and Lew 16 presented a generalization of the JKR theory that applies to contact between rough surfaces and gives a prediction for the nominal contact area. However, as we discussed in the introduction, that model only applies in the regime of small surface roughness. As the final step for making the comparison, we need to choose appropriate values for k. Recall that we introduced the scaling parameter k to account for the roughness of the tip in the experiments in our model, in which the tip is smooth.
The tip in the experiments was a spherical glass bead. As can be noted from the table, the RMS roughness of these beads ranges from 2. While making these estimates we used Z 0 as a fitting parameter. As can be seen in Fig.
In fact, we found that the estimates for other cases in which k resp. In each of those cases we again used Z 0 as a fitting parameter. Recall that the parameter Z 0 is a measure of the distance of the inter-body cohesive forces and that we were unable to clearly identify it in the experiments of Kesari et al.
Thus, the best fit values for Z 0 in our comparision lie well within the experimentally reasonable range. The error bars indicate the standard deviation of the measurements of hysteretic energy loss taken at five locations on the PDMS substrates in the experiments. It is essentially a constant equal to 0. We computed these values using the data shown in Fig. Also can be seen in Fig.
We are unaware of any experimental data that can be used to check the validity of this theoretical prediction of our model. However, this trend is consistent with the numerical results reported by Song et al. We generated predictions from our model in the context of the experiments reported by Kesari et al.
In general, however, it is challenging to determine a priori whether or not it is reasonable to apply our model to a particular contact scenario. The reason is that we assumed in our model that the contact region is multiply connected and that there is no interaction between neighboring asperities. These are reasonable assumptions only if the size of the contact region formed at each asperity is much smaller than the separation between neighboring asperities.
Therefore, a general theory of the type developed by Johnson 36 that yields information about the topology of the contact region would form a valuable supplement to our model. Hua et al. However, it has been argued that in soft-lithography it can be challenging to replicate features due to surface stress flattening out features having high curvature 38 , 39 , Some preliminary insight into addressing this question can be obtained by considering the model presented by Style et al.
We found that the average distance between each asperity and its nearest neighbor on the Si mold surface shown in Fig. We conclude by noting that our model bears some similarities with the models presented in refs 44 ,