Apparent Versus Local Friction Coefficients

Scratch Damage of Uncoated Material

## Introduction

In contact mechanics analysis, converting the stiffness calculated from load versus depth (in the case of indentation) or load contact versus contact radius (in the case of scratching) behaviors into a stress strain relationship is a major problem. Generally, when directly recording the load from the load cell, the true contact radius and the true depth depend on a model which depends on the kind of behavior (elastic, elastic-plastic, plastic).This is not a real problem for materials such as steel and other materials that quickly yield during contact. However, no models are available to consider the viscoelastic and/or viscoplastic behavior of the material or an elastic behavior at large strain, such as polymeric materials.

New apparatus was developed to study this problem. It controls the tip velocity over a large range, at a wide range of temperatures and is equipped with an “optical microscope” to perform in-situ control and measurement of the contact area and of the groove left on the surface. The prototype of this apparatus built under the name “Micro Visio Scratch” ^{[1]} at the Charles Sadron Institute is now available as an option for all types of Anton Paar Indentation and Scratch Testers.

** Figure 1. ** Anton Paar Nanoscratch Nanoindentation tester including the in-situ vision set-up

Figure 1 shows the prototype Mirau interferometer integrated with an Anton Paar Open Platform, which is mounted in a vacuum chamber. This optical device may be shifted under the Nanoscratch Module or under the anoindentation Tester Module. This article discusses the recent results acquired with this setup. The indenter tip used has a spherical geometry for two reasons: the stress is uniformly distributed around such an indenter and the in-situ vision allows controlling the mean contact strain proportional to the ratio a/R (where a is the contact radius and R the tip radius).

## Scratch Analysis

In-situ observation of the contact area improves the analysis of the scratch behavior. No model is required to predict the surface behavior as in the case of a blind test. Also, key information on the bulk material properties can be directly obtained from in-situ observations. For instance, for a polymer exhibiting a viscoplastic behavior, a logarithmic increase of the scratching speed causes a linear decrease of the true contact area without altering the shape of the contact area. Hence, only the contact radius decreases. During the scratching process, the local strain rate may be calculated as the tip speed divided by the contact radius. Here, all experimental measurements of the mean contact pressure may be plotted as a function of this strain rate. Therefore, the scratch hardness (mean contact pressure in case of plastic contact) appears to be a time and temperature activated processes property like other mechanical properties for this type of material ^{[1]}.

Figure 2 shows a typical image of the contact area and shape of the groove left on the surface. As can be seen in the figure, the contact area has a diameter larger than the width of the groove because of the elastic unloading just after contact. During the contact phase, the rear contact angle looks like a good parameter for characterizing the rate of yielding around and under the scratching tip. For an elastic contact, this angle is equal to π/2, and for a fully plastic contact it decreases up to 0.2 to 0.3 rad.

** Figure 2. ** Schematics of the constant load scratch on a cross-sectioned sample

Studies performed on a variety of polymeric surfaces have demonstrated that the imposed elastic-plastic strain controls the scratch tests in terms of maximum value and size, but also its localization through depth beneath the moving tip. Since the strain dependency can now be dissociated from the time and temperature dependencies, some basic finite element modeling (FEM) can now be used to explain the surface behavior, in contrast to the general use of FEM which aims to predict the In-situ observation of the scratch/indentation true contact area to provide analysis without being model-dependent scratch behavior ^{[3]}. The strain dependency during a scratch made at increased normal load, applied on a spherical tip, is illustrated in Figure 3.

** Figure 3. ** Mechanical transitions from elastic sliding to fully plastic scratching observed during experiments with increasing the applied normal load for PMMA (local friction coefficient was estimated at 0.2)

As a first approximation, there is an increase in the average plastic strain (ε_{p}) with the local friction for a given ratio a/R and for a constant local friction with the ratio a/R. FEM results explain experimental results: the average plastic strain ε_{p} and then the relative plastic strain field are more sensitive to the local friction coefficient for intermediate values of a/R, corresponding to elastic–plastic contacts on polymeric surfaces ^{[4,5]}. The combined effect of the friction coefficient and of the ratio a/R is shown in Figure 5.

** Figure 4. ** Equivalent plastic strain maps as computed by simulation for a/R=0.2 with µ_{loc}=0 and µ_{loc}=0.4 for PMMA.

** Figure 5. ** Experimental in situ observation of the contact geometry and the residual groove during scratch tests (top view) as a function of the mean geometrical strain a/R and the local friction coefficient.

## Apparent Versus Local Friction Coefficients

When a rigid sphere is allowed to slide on a soft polymer, the tip penetrates into the surface of the material and produces a “ploughing wave” wherein a large amount of energy is dissipated. This effect is added to the apparent friction coefficient µ_{app}, the ratio of the measured tangential force over the applied normal load. In order to consider this phenomenon and agree to a more “local” friction coefficient µ_{loc} expressed as the ratio between the local shear stress and the local contact pressure, a model was created ^{[6]} and applied to study the relationship between the local friction coefficient and the mean contact pressure, the yielding of the surface or the structural recovery of the surface ^{[7,8]}.

Therefore, the true contact area is the sum of a front area (half disc) and a rear area (part of the rear half disc). The challenge is to consider this rear contact to relate the true and ploughing frictions to the measured apparent friction. The input data required by the model is the shape of the rigid sliding tip and the true contact area (contact radius and rear contact angle). A relationship was easily established between the true friction coefficient and the apparent friction coefficient, using four integrals A, B, C, and D, which are the elementary action integrals of the local pressure and shear, on the normal axis, z and on the sliding axis, x (Figure 6) ^{[6]}.

** Figure 6. ** Rigid sphere sliding on a viscoelastic polymer showing the ploughing effect and the corresponding contact area.

F_{n} and F_{t} can be written as:

using the notations in Figure 6 A, B, C and D can be expressed as follows:

The model leads to and therefore,

This model allows an analysis of the interfacial rheological behavior of the polymer against the mean contact pressure. Also, the interface at the local scale with the control of tip roughness can be experimentally investigated. For example, the local friction coefficient (µ) is plotted against the mean contact pressure (P_{mean}) in Figure 7. From the results, the local friction coefficient follows a master curve for smooth indenters (Δ) below yield stress ( P_{mean}<100 MPa). Chemical etching (fluorhydric acid) was used to realize some roughened indenters in order to monitor the nanoroughnesses parameters. The friction coefficient (µ) gradually falls to a plastic-like constant value between 5.5 nm Rms and 140 nm Rms: nano-roughness monitors friction. The behavior of confined polymer layers ^{[9]} can now be studied due to the in-situ information of the contact area.

** Figure 7. ** Local friction coefficient µ versus mean contact pressure P_{mean} during sliding and scratching tests with roughened spherical lenses on a rejuvenated PC surface (v = 30 µm s-1, T = 30 °C). (Δ) R_{rms} = 0.7 nm (reference), (+) R_{rms} = 5.5 nm, (Δ) R_{rms} = 8.5 nm, (Δ) R_{rms} = 140 nm. Snapshots during sliding in the elastic, elasto-plastic and plastic regimes with the measured contact area in diagonal grey hatching.

## Scratch Damage of Uncoated Material

The scratch behavior of a thermoset polymer that exhibited a brittle behavior was studied using the experimental observations in combination with a finite element modeling. The 3D crack pattern, formed at the rear part of the contact and in the residual groove (due to the high level of tensile stress during scratch tests) has been analyzed using fluorescence confocal microscopy (Figure 8). FEM has been used to compute the normal contact pressure and the interfacial shear stresses in the contact area at the sample/indenter interface. By applying the resulting values as input data, the 3D crack network analysis has been performed using combined 3D localized multigrid and X-FEM/level set techniques. The fracture process that caused the formation of the crack pattern was identified as a complex 3D unloading/reloading process, primarily driven by mode I. Also, a failure strength around 90 MPa was calculated. It is the value attained by the computed tensile stresses at the rear edge of the contact area for a distance between the last crack and its rear, equal to the measured distance between two consecutive actual cracks ^{[10]}.

** Figure 8. ** In-situ photograph of CR39 damage. Cracking appears near the rear edge of the contact area (left) 3D crack pattern reconstruction from confocal microscopy. The surface fracture is roughly a part of a cylinder (right).

## Scratch Damage of Coating

It is a generally accepted fact that the critical load, causing the first damage in a scratch test, is representative of the behavior of a coating. Since the polymers have time and temperature dependent properties, the overall mechanical behavior cannot be described by a single value of the critical load. Also, cracking was observed to occur in the contact area and not always at the rear edge. For thin solid films, the ratio of the contact radius to the radius of the grooving tip was shown to be an important parameter to estimate the damage caused and did not rely on the scratching velocity or temperature.

The ratio of the coating thickness to the tip roughness is another important parameter: the coating prevents the roughness of the diamond tip from forming micro-scratches at the macro-groove surface. Hence, since the lack of micro-scratches is a condition for relaxation of the macro-groove, the coating thickness must be higher than the tip roughness, as illustrated in Figure 9 ^{[11]}. The interfacial adhesion of coatings with the substrate is the second important feature of their durability submitted to scratching.

In-situ observation is a key tool to investigate the damage mechanisms such as spalling and blistering. The fracturing of some thin coatings applied on different substrates was examined under different conditions of scratching speed and temperature. Based on these two variables, different kinds of fracture mechanisms were identified ^{[12]}. The adhesion of the system can be estimated using a global energy balance model ^{[13]} of the stable blistering process (Figure 10) obtained for some experimental conditions. To estimate adhesion, the delaminated area (quantified by image analysis) is followed as a function of the scratching distance during blistering. The adhesion relative to different substrate/thin film systems was estimated using FEM and the global energy balance model. The occurrence of some confinement effect was shown in a study of similar films with different thicknesses, thus leading to an analysis of the results with respect to the probe characteristics: roughness and size.

** Figure 9. ** Effect of the thickness of the coating on the true contact area (angle increases if no microscratches along the macro groove); R = 110 µm; Tip roughness R_{t} = 2.5 µm; a/R roughly constant.

** Figure 10. ** Full sequence of the crescent blister growth during scratching of the material with 10 µm/s scratching speed and a temperature of 90 °C. The indexes 1 to 7 chronologically number the shots. The dashed lines indicate the delamination area.

## Conclusion

In-situ observation is key to investigate the damage mechanisms which occur during scratching. Similar information cannot be obtained from post mortem observation owing to the fact that once unloaded, the shearing and compressive fields disappear after the contact time.

In-situ observation has shown the transition from scratching to sliding during the contact of a moving tip with a polymeric surface as a function of the sliding speed. It would be impossible to analyze such a transformation with a blind test as no residual scratch track remains on the surface (for elastic materials) ^{[2]}.

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** Acknowledgements**

Anton Paar would like to thank the Institut Charles Sadron (CNRS Strasbourg) and Prof Christian Gauthier for the results published in this Application Bulletin.

This information has been sourced, reviewed and adapted from materials provided by Anton Paar TriTec SA.

For more information on this source, please visit Anton Paar TriTec SA.

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