CC BY
68
yflK 539.612
Experimental investigation of the adhesive contact of an elastomer
L.B. Voll1 and V.L. Popov123
1 Technische Universität Berlin, Berlin, D-10623, Germany 2 National Research Tomsk State University, Tomsk, 634050, Russia 3 National Research Tomsk Polytechnic University, Tomsk, 634050, Russia
The present work is an experimental study of adhesion between an elastomer and a rigid cylindrical indenter. The experimental characterization was carried out using a specially developed apparatus. Adhesive force was measured as function of contact geometry, pull-off velocity, normal force, temperature, waiting time, and material properties. Experimental results are compared with existing theoretical models for adhesion of elastic and viscoelastic bodies. Our study shows that the adhesive force between the studied elastomer and a steel cylinder is determined by completely different mechanisms than assumed in the Kendall's theory. In particular, it does not depend on the surface energy and is almost entirely dominated by the viscosity of the elastomer.
Keywords: adhesion, elastomer, elastic contact, viscoelastic contact, temperature dependence, velocity dependence, seals, valves
1. Introduction
In various applications involving elastomers, adhesion forces may occur, a phenomenon which might be intended or undesirable. The particular application that motivated this study was to ensure and possibly improve the opening and closing behaviour of valves operating in a wide temperature range. For the case of simplicity of the theoretical analysis, we confine ourselves in the present work by consideration of contacts between an elastomer and cylindrical indenters (Fig. 1).
In cooperation with an industrial partner, we developed a measuring apparatus to investigate the adhesive force in such a contact. Our measuring apparatus and experimental investigations were application-oriented and hence were not meant to explain the cause of adhesion. Rather, we attempted to characterize the adhesion force empirically. The results obtained, are, however, of general interest.
Fig. 1. Contact between an elastomer and cylindrical indenter
2. Theoretical models for adhesion of elastic and viscoelastic bodies
The adhesion force of a cylindrical rigid indenter and an elastic half space was first calculated in 1971 by K. Kendall [1]:
Fa =yl 8 nE E y12a3, (1)
*
where E is the effective elastic modulus
E is the elastic modulus, v is Poisson's ratio, a is the contact radius, and y12 is the work of separation of contacting bodies per unit area. For elastomers, contrary to purely elastic bodies, different microscopic loss-of-contact criteria can be formulated due to the dependence of the stress on strain and strain rate [2]. In our investigation we used the so-called deformation criterion, which is described in detail in the publications [2, 3]. When using this criterion, the detachment of the indenter from a viscoelastic half space occurs when the critical separation is achieved determined by
«a = 21/2 nbc l"1/2 a12, (3)
2aE
where a is the radius of the cylinder and b, bc are parameters with the order of magnitude of a molecular diameter. The value (3) of the critical separation is motivated by a microscopic consideration of the stress concentration at the border of the adhesive contact, but for our purposes it can just be
© Voll L.B., Popov V.L., 2014
considered as an empirical parameter. Note that Eq. (3) is valid regardless of the concrete rheology of the elastomer (provided the above mentioned deformation criterion is applicable).
Let us now consider an elastomer which is characterised by the time dependent shear modulus G(t) [4]. Following Radok, the dependence of normal force on displacement of a rigid cylindrical indenter and an elastomer can be shown to be [3, 5]
/s/sss/sss/
t An
Fn(t) = 8a J G(t-t')™dtf .
(4)
du , . -. » (t »=
(5)
Let us assume that the rigid indenter does not move until time t = 0 and afterwards moves with a constant velocity
v0:
)V = 0, t = 0,
v = v0, t > 0.
According to Eq. (4), the normal force reaches its maximum value at the time of detachment: t = ua/ v0. This maximum value can be interpreted as the adhesive force Fa :
Fa = 8aV0 Uj\° G©di;. (6)
0
To write Eq. (6) in explicit form, information about the rheology of elastomer is needed. Let us assume that the elastomer is described by the simplest viscoelastic material model: the Kelvin body. The time dependent shear modulus can then be written [3] as
G (t ) = Gi + G2eT. (7)
Using Eqs. (3) and (7) the adhesive force for a cylindrical indenter is determined to be
Fa = 27/2 Gnbcb-V2
' 21/2 rcfeb-/2a"2^
^32 +
+ 8a v0 tG
1 - exp
vn t
(8)
y 5
This equation describes dependence of the adhesion force on three essential parameters: contact radius, pull-off velocity and, implicitly, temperature (G(t, T) ). Under the condition that
1 <<
212 nhb-V2 a12
<<
vn t
G
(9)
the equation can be simplified to
Fa = 8 av on, (10)
where we introduced the dynamical viscosity of the elastomer
n = ^2X. (11)
The influence of the temperature is included by the introduction of the temperature-dependent dynamic viscosity by means of the Andrade's equation
n(T) = AeBT, (12)
where A and B are empirical constants which are determined experimentally [6].
Weights
Indenter
Elastomer Force sensor
I
Fig. 2. Sketch of the apparatus for the measurement of adhesive force (temperature regulation not shown)
Equation (10) predicts a completely different dependencies of the force on adhesion on geometric and loading parameters as compared to the Kendlall's equation (1). In the region of validity of this model, the force of adhesion is directly proportional to the contact radius and to pull-off velocity. Further, the force of adhesion is proportional to the viscosity, and therefore has to show a strong (exponential) dependence on temperature. For cylindrical indenters, the adhesive force does not depend on the contact time because of the time-independence of the contact area. These theoretical predictions can be easily checked by experiments.
3. Experimental apparatus for adhesive force measurements
For the measurement of adhesion force an apparatus was developed which enables measurements in a temperature range of approximation from -80 to 80 °C. Figure 2 shows the main components and their functions.
The normal force is produced with fixed weights, which are made with an accuracy of 0.1 g. The maximum normal
Fig. 3. Adhesive force dependence on the cylindrical indenter radius (a = 1, 2, 3, 4 and 5 mm) and on the temperature (T = 20, 0, -20, -40 °C). The pull-off velocity, the contact time and normal force are constant. Measurement results are presented with absolute error. Every data point was averaged from 10 single measurements
Fig. 4. Dependence of the viscosity logarithm on the reciprocal absolute temperature
Fig. 6. Dependence of the adhesive force on the normal force. Contact time 10 s, pull-off velocity 2.5 mm/s, cylindrical indenter (a = 5 mm), test temperature 20 °C. Averaging was taken from 6 single measurements each
force is limited by the nominal load of the force sensor, at 50 N. The lower limit of 1.5 N is due to the weight of the indenters and the fixture. The steel indenter is moved with a stepping motor which drives a thread spindle. A special lever system guarantees that the indenter and its holder can only be moved vertically. The lever system also makes sure that the only applied force is in the normal direction. The structural implementation of the motor drive takes the resonance frequency of the force sensor into account. The stepping motor can be operated at variable speed and acceleration and allows controlling the displacement precisely. Cooling is provided by gaseous nitrogen flowing through a conduit system around the sample holder. Preliminary tests have shown that there is no difference between direct and indirect cooling with nitrogen. For the attachment of the material sample a gimbal bearing was designed, which makes sure that the sample is held horizontally. The measurement procedure starts with the insertion and alignment of the material sample and sealing of the measuring apparatus. Subsequently the normal force is adjusted by attaching weights. Next, the desired temperature is adjusted. When the material sample has reached the test temperature the
0.61-
I 0.2-<
0.0 n-i-1-1-1-1
0 2 4 6 8 10 Pull-off velocity, mm/s
Fig. 5. Dependence of the adhesive force on the pull-off velocity. Contact time 10 s, normal force 20 N, cylindrical indenter ( a = = 5 mm), test temperature 20 °C. Measurement results are presented with absolute error. Every data point was averaged from 6 single measurements
indenter is set down on the elastomer in a controlled manner. After a predefined contact time the indenter is pulled off by means of the lifting unit. The force necessary to detach the indenter from the elastomer is measured.
4. Results of the experimental investigation
The influence of temperature, contact time (holding time), pull-off velocity, normal force and indenter radius on the separation behaviour of the indenter in contact with the elastomer was examined. Figure 3 shows the dependence of the adhesive force on the contact radius and the temperature.
The adhesive force is with good accuracy directly proportional to the indenter radius, which substantiates the validity of Eq. (10) in contrast to the Kendall equation (1). The temperature influence is also consistent with the model described by Eq. (10): The adhesive force falls with rising temperature. In order to further examine the temperature dependence, the viscosity was determined from the measurement (Fig. 3). The following diagram shows (Fig. 4) the viscosity dependence on the absolute temperature.
The examined elastomer shows the temperature-dependence predicted by Eq. (12). The obtained viscosity ranges from 6 • 103 to 1.21 • 105 Pa • s. The next diagram (Fig. 5) shows the dependence of the adhesive force on the pull-off velocity.
101 102 103 Contact time, s
Fig. 7. Dependence of the adhesive force on the contact time. Normal force 20 N, pull-off velocity 7 mm/s, cylindrical indenter (a = 5 mm), test temperature 20 °C. Averaging was taken from 6 single measurements each
A linear increase of the adhesive force is apparent when varying the pull-off velocity between 0.25 and 10 mm/s. This linearity, again, confirms the model described by Eq. (10). The next diagram (Fig. 6) shows the dependence of the adhesive force on the normal force.
The adhesive force is found to be independent of the normal force in the examined range. Unfortunately the measuring apparatus is limited with regard to the minimum normal force. The next diagram (Fig. 7) shows the influence of the contact time on the adhesive force.
The contact time was varied from 1 to 600 s. As can be seen from Fig. 7, the adhesive force seems to be independent of the contact time, which is consistent with the theoretical prediction.
5. Summary
Our study shows that the adhesive force between some elastomers and a steel cylinder is determined by completely different mechanisms than assumed in the Kendall's theory of elastic contacts. In our experiments, the dependence of the adhesive force on system parameters could be completely represented by Eq. (10). The surface energy does not appear in this equation. The adhesive force is instead almost entirely dominated by the elastomer viscosity: the studied elastomer shows the same adhesive properties as a liquid with a corresponding viscosity. Our measurements show the linear dependence of the adhesive force on the contact
radius of cylinder-shaped indenters (Fig. 3), as well as direct proportionality between adhesive force and pull-off velocity (Fig. 5) in the examined velocity interval. The description of the temperature-dependent viscosity (12) with the Andrade's equation has turned out to be suitable (Fig. 4) too. The independence of the adhesive force from the normal force (Fig. 6) and the contact time (Fig. 7) underpins the validity of the theoretical description. According to our experimental results, the maximum adhesion with the studied elastomer would be obtained if the bodies were brought into contact at higher temperature (so that the surface roughness can be filled by the soft elastomer) and then cooled down.
References
1. Kendall K. The adhesion and surface energy of elastic solids // J. Phys.
D: Appl. Phys. - 1971. - V 4. - P. 1186-1195.
2. Popov V.L. Basic ideas and applications of the method of reduction of dimensionality in contact mechanics // Phys. Mesomech. - 2012. -V. 15. - No. 5-6. - P. 254-263.
3. Popov V.L., Heß M. Methode der Dimensionsreduktion in Kontaktmechanik und Reibung. Eine Berechnungsmethode im Mikro- und Makrobereich. - Berlin: Springer, 2013. - 267 s.
4. Popov V.L. Contact Mechanics and Friction: Physical Principles and Applications. - Heidelberg: Springer, 2010. - 362 p.
5. Radok J.R.M. Viscoelastic put under stress analysis // Q. Appl. Math. -
1957. - V. 15. - P. 198-202.
6. da C. Andrade E.N. A theory of the viscosity of liquids. Part I // London Edinb. Dub. Philos. May. J. Sci. - 1934. - V. 17(112). - P. 497-511.
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Voll Lars Bastian, Dipl.-Ing., Technische Universität Berlin, Germany, lars.voll@tu-berlin.de
Popov Valentin, Prof., Technische Universität Berlin, Tomsk State University, Tomsk Polytechnic University, v.popov@tu-berlin.de
