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Dimaki A.V., Popov V.L. / Физическая мезомеханика 20 4 (2017) 5-10
5
УДК 531.44, 53.091
Dynamics of the coefficient of friction between a rigid conical indenter and a viscoelastic foundation under step-wise change of sliding velocity
A.V. Dimaki12, V.L. Popov2 3 4
1 Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634055, Russia
2 National Research Tomsk State University, Tomsk, 634050, Russia
3 Berlin University of Technology, Berlin, 10623, Germany 4 National Research Tomsk Polytechnic University, Tomsk, 634050, Russia
We numerically calculated the coefficient of friction between a rigid cone and a viscoelastic Kelvin body under step-wise change of the velocity of sliding. The time dependence of the coefficient of friction has been empirically approximated. We show that the transition process has different character for the cases of increasing and decreasing of the sliding velocity.
Keywords: sliding friction, elastomers, Kelvin material, sliding velocity, step-wise change
Динамика коэффициента трения между жестким коническим индентором и вязкоупругим основанием при скачкообразном изменении скорости скольжения
А.В. Димаки1-2, В.Л. Попов2-3-4
1 Институт физики прочности и материаловедения СО РАН, Томск, 634055, Россия 2 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия
3 Берлинский технический университет, Берлин, 10623, Германия
4 Национальный исследовательский Томский политехнический университет, Томск, 634050, Россия
Проведен численный расчет коэффициента трения между жестким коническим индентором и основанием Кельвина при скачкообразном изменении скорости скольжения. Построена эмпирическая оценка, аппроксимирующая зависимость коэффициента трения от времени. Показан различный характер протекания переходного процесса в случае увеличения и уменьшения скорости скольжения.
Ключевые слова: трение скольжения, эластомеры, материал Кельвина, скорость скольжения, скачкообразное изменение
1. Introduction
Friction of elastomers is an important topic for many industrial applications [1]. Greenwood and Tabor [2] have
shown as early as 1958 that the friction of elastomers can be attributed to deformation losses in the volume of the material [3]. In the following years, the role of rheology
[4] and of surface roughness [5, 6] in friction of elastomers has been studied in detail for a stationary regime of sliding. However, till now there were practically no studies of the kinetics of the coefficient of friction under nonstationary loading conditions. It is well established that there exist different contributions to elastomer friction, which may prevail under different conditions. The main three contributions are the static hysteresis which is equivalent to re-
maining plastic deformation after loading [7, 8], the adhesion contribution [7] and the hysteresis losses due to vis-coelasticity. For the plastic contribution, the heterogeneous deformation controlling the frictional forces has been analyzed in [9]. In the present paper, we consider solely the viscoelastic contribution.
Since Coulomb [10], it has been known that nonstationary regime of sliding may result in a nonlocal response of the system in time. A time dependency of static friction was reported already by Coulomb, and was later investigated in detail by Dieterich [11]. Based on his experimental investigations, Dieterich formulated a law of nonstationary sliding for geological materials [12], introducing a single internal "parameter of state". Ruina suggested a formulation
© Dimaki A.V., Popov V.L., 2017
of the found dependencies in form of differential equations [13, 14]. Multiparametrical rate-and-state laws have been analysed in detail by Gu [15]. At the present time, the rate-and-state laws are used in wide range of applications including the analysis of earthquake generation and seismic cycle [16], simulation of friction of lubricated surfaces [17], fracture propagation [18] etc. The physical background of rate-and-state laws has been analysed in [19].
The role of viscosity of a material on the time-dependent mechanical response of contacting bodies in nonstationary sliding has not yet been sufficiently studied. Despite a number of experimental works in this field (see, for example [20-22]), there is no theory that predicts kinetics of coefficient of friction for materials with complex rheology, e.g. elastomers.
Recently, Li et al. have carried out a numerical simulation of nonstationary sliding of a rigid body with fractal roughness on a viscoelastic foundation, and proposed an empirical relation that approximates the dynamics of the coefficient of friction under a step-wise change of sliding velocity [23]. In the present paper, we solve a simpler but more fundamental problem in the framework of a "single-asperity" approach assuming that contact patches are situated far enough from each other that prevents their interaction. In this approximation it becomes possible to obtain analytical expressions for time-dependent forces, shape of the contact and, thus, for a coefficient of friction.
In order to achieve a basic understanding of the influencing factors, we consider the simplest model: (a) the elastomer is modeled as a simple incompressible Kelvin body, which is completely characterized by its static shear modulus and viscosity, (b) the surface of the elastomer is assumed to be initially plane and frictionless, (c) we consider only one single contact ("single-asperity model") in the form of a cone, that is absolutely rigid, (d) no adhesion or capillarity effects are taken into account, (e) we consider a one-dimensional model. These simple assumptions still result in nontrivial and complicated behavior in nonstationary regime of sliding.
For simulation of the system under consideration we use the method of dimensionality reduction (MDR) [24, 25]. In the case of simulation of rheological contribution to the
force of friction between a rigid indenter and an elastomer, the MDR is not exact but provides a good approximation. For the case of stationary sliding, this was illustrated by comparison of results obtained using MDR [26] and those of direct three-dimensional simulations with boundary element method [27]. A detailed analysis of the methodology of applying the MDR to frictional contacts of elastomers and its accuracy can be found in [28].
Recently, Popov and Hess [25] have obtained a complete analytic solution of the described problem for the case of stationary sliding. In the present paper we generalize these results for the case of nonstationary sliding.
2. Friction between a rigid cone and a viscoelastic medium under constant indentation depth
We first recapitulate the solution of the stationary problem as we will need these results for reference in the following consideration. Let us consider a rigid conical in-denter z = f (r) = r tg 0, where z is the coordinate normal to the contact plane, and r is the in-plane polar radius (Fig. 1, a). In the framework of the MDR, the original profile is replaced by a plane MDR transformed profile; for the case of a conical indenter, the effective MDR profile is given by [21, 24]
Z = g(x) = f| x | tg 0 = c | x |. (1)
In the second step, viscoelastic half-space is replaced by a row of independent elements with a small spacing Ax, each element consisting of a spring with normal stiffness Akz = 4G ax (2)
and a dashpot having the damping constant
Ad = 4nAx, (3)
where G is the shear modulus and n is the viscosity of the elastomer [25] (Fig. 1, b).
The profile (1) is now pressed into the viscoelastic foundation defined according to rules (2) and (3) to a constant depth d and is moved tangentially with the velocity v (Fig. 1, a), so that the profile shape at a given time t is given by
Z = g (x + vt) = g (x), (4)
where we introduced the coordinate x = x + vt in the frame of reference that moves with the rigid indenter.
Fig. 1. Contact between an elastomer and rigid conical indenter, which is moved tangentially with the velocity v (a); rheological model for a viscoelastic medium (b)
The coordinates of the boundary of the contact area x = -aj and x = a2 (Fig. 1, a) are then stationary. The left boundary of the contact area is determined by the condition uz (-aj) = 0 giving
aj = d/c (5)
and the right boundary by the condition of fN (a2) = 0, where fN ( x) = 0 is the normal force of a spring at the position x. To calculate the force, consider first the vertical displacements
uz (x, t) = d - g(x + vt) = d - g(x). (6)
The vertical velocities are the calculated as
duz ( x, t ) = dg( x + vt )
• = -vg'{x)
(7)
dt dt and the force acting on a single element is equal to
fN (X) = AkzUz + AdUz =
= 4[ G (d - g (X))-nvg( x )]Ax. (8)
Equating the force to zero, we get the coordinate of the right boundary of the contact region:
aj = d/c, a2 = d/c - vt, (9)
where we have introduced the relaxation time
T = n/ G. (10)
We can consider two velocity domains:
I: v < d/(ct), (11)
II: v > d/(ct). (12)
In the first domain, the right contact point lies to the right of the tip of the indenter. In the second, it coincides with the tip of the indenter.
In the velocity domain II the coefficient of friction remains constant. This will obviously be valid even for nonsta-tionary sliding velocity. Thus, in the following we consider only velocities in the domain I, in particular, we assume that any changes of the sliding velocity occur inside this domain.
Variation of the tangential velocity leads to variation of the detachment coordinate a2 = a2(t). With this coordinate, the normal force becomes function of time:
a2(t)
Fn(t) = 4 J [G(d-g(X))-nvg'(x)]dx =
= 4G
J (d + cx + Tcv(t))dx+
a2(t)
+ J (d - cx -tcv(t))dx
0
= 4G
d (aj + a2(t )) + tcv (t ) x
X (aj - a2 (t)) + 2 (aj - a2 (t))
(13)
In the limiting case of stationary sliding, the solution was provided by Popov and Hess [25]:
Fn (v) =
4G
d 2 + 2(cvt)2
(14)
The time dependence of the total tangential force can be written as follows:
a2(t)
Fx(t) = -4g'(x) J [G(d-g(X))-nvg (X)]dX =
= 4Gc
J (d + cx - tcv(t))dx -
- J (d + cx - tcv(t))dx
0
= 4cG
d(aj - a2 (t)) - 2 (aj2 - a2 (t)) +
+ tcv(t )(aj + a2 (t ))
(15)
The corresponding value of tangential force for a stationary sliding was obtained in [25]:
Fx (v) = 4Gc
2d (vt) - 2 (vt)2
(16)
From the Eqs. (13) and (15) it is evident that the time dependence of the coefficient of friction is completely determined by the time dependence of the detachment coordinate a2(t ):
F (t )
^(t) = -Ftt = c\d( aj - a2(t)) + Fn (t) L
+ 0.5c(aj2 + a\ (t )) + tcv(t )(aj + a2 (t ))] x
j + a2 (t)) + 0.5c(aj2 - a\ (t)) +
+ tcv(t)(aj - a2 (t)) ] j.
(17)
In the case of stationary sliding with the tangential velocity v the coefficient of friction is
2(cvr/ d ) - 0.5(cvr/ d )2
Fx Fn
j + 0.5(cvr/ d )2
(18)
which we will need for further reference.
3. Transition process after a step-wise change of sliding velocity
A step-wise change of sliding velocity is a simplest kind of nonstationary sliding, that can result in a nontrivial kinetics of frictional forces, and correspondingly of the coefficient of friction. Let us suggest that at a time t0 = 0 the value of sliding velocity changes from v0(t0) = v to vj (t0 + dt ) = v + dv.
1. If dv > 0, the detachment coordinate a2 changes immediately to a new value
a2 = d/c - vjt. (19)
2. If dv < 0, a transition process from a2,t0 to a2t0+Atm starts. Let us first estimate the characteristic time attrans of this process. From purely geometric considerations, we can write the displacement of an element in the point of detachment:
uz (a2, J = ca2, t0 - d = tcv (20)
-a
-a
-a
During the time interval AttI
the detachment coordi-
nate becomes equal to the "stationary" value (19). At the same time, the indenter is displaced by a/trans = attrans Vj. In order to stay in contact, the element must move up to the position
uz (a2,t0 +Attrans ) = Ca2,t0+At,m„ - d = C VJAttrans ' (21)
Since an element of Kelvin material has the exponential law of relaxation, we can write:
uz K t0+Attrans ) = Uz K t0)eXP(-AttranJt)- (22)
From (22) it is easy to obtain the equation for Attrans:
(
Attians = -t ln
e z (a
2,t0 +Att„
= -xln
e z (a2,t0)
\
(t + Attrans ) v
tV
(23)
This equation demonstrates that: (i) the transition time depends only on the material constant t and the values of sliding velocity before and after the jump; (ii) for a particular case vj = 0 (i.e. for a "start-stop" test) the transition time tends to infinity.
The transition process takes place within a time interval [t0; t0 + Attrans] while a distance of detachment changes from a21 to a2,to +Attr and the displacement of the foundation in the point of detachment changes from (20) to (21). At the moment of the jump of sliding velocity we have the following "initial profile" of a foundation:
uz (x) = g(x) - d = cx - d, x < a2,to,
uz (x) = uz (a2. ) exp
( x - a2. ^
Vt
(24)
=cvtexp
x - d/c + vt
vt
, x > a2,t0
After the jump of sliding velocity we can rewrite the equation (24) in the following way:
uz(x, t) = g(x, t) - d = c(x - Vjt) - d, x < a2(t), x - d/c + vt |
uz (x, t ) = c vt exp I -
VT
- Iexp(- t/T), (25)
X > «2(0-
The latter gives the equation
c(a2 (t ) - v1t ) - d =
a2(t ) - dc + ^ exp (-/ t), (26)
= cvtexp
vt
which can be easily transformed into a nonlinear equation with respect to a2(t):
a2(t) - d/c
vt
= exp
a2(t) - d/c -1(v1 - v)
-1
vt
(27)
Note the equation (27) is valid only inside the time interval
[t0; t0 + attrans ]. After the time t exceeds the value t = t0 + +Attrans the value of a2(t) becomes constant and equal to
(19). In order to provide shorter notations we introduce notations
a2(t) - d/c
y(t ) = -
vt
p(t ) = ill - v
t1 v
(28)
and rewrite equation (27) in the form:
y(t) = eY(t )-1. (29)
In the second step, let us estimate the values of force jumps at the step-wise change of the sliding velocity. Let us suggest that in the moment t = t0 of the velocity jump the detachment coordinate still remains the same as before a2(t = t0) = d/c - vt, but the sliding proceeds with new velocity v1. After simple transformations of the Eq. (13) we receive the maximum value of normal force immediately after the velocity change: Fjmv = Fn (to, v = vi) = 4G'
d2 + (cvt)2 j ^ -1
(30)
Applying the same approach to tangential force (see the equation ) we get
FT* = Fx (to, v = vj) =
' 2
= 4Gc
2d (v1t) - c(vt)2
(31)
4. Numerical model
The described model can be easily implemented numerically. Each ith spring (Fig. 1), having been displaced to ui = = u( xi ), moves up to an equilibrium position in accordance to an exponential law
ut(t) = u(xi, t + at) = u(xt, t)e"At/T. (32)
If an element comes into a contact with indenter, it produces a local normal force
fz ;i (t) = -4Gax(u. (t)-Tiit (t)) (33)
and a local tangential force proportional to a local gradient of the indenter surface:
fx,i (t ) = fz ,i (t ) g '( x )- (34)
The total normal force and the total tangential force are given by the sums over all elements in contact:
(35)
(36)
FN = E fz i,
Ncont
= Y f ..
x J x.
Ncon,
The coefficient of friction, obviously, is
^ = Ff/ FN. (37)
In the framework of the developed model, the coefficient of friction is determined only by geometry of a contact, namely, by the coordinate of detachment. At the end of a time step at = ax/v each element of the viscoelastic foundation moves by ax, and the described procedure is repeated.
1.5
w
5
1.0
0.5 -1
FN(t»to)
1
tlx
jump FN
Fig. 2. Normal (a) and tangential (b) forces in the contact versus time
5. Kinetics of the coefficient of friction
We consider a "start-stop" test, when the initial value of the velocity v corresponds to a border value between velocity domains I and II (see Eqs. (11) and (12)), and the value of the velocity after the step-wise change is close to zero, v1 = v • 10-4. In the Fig. 2 the time dependencies of normal and tangential forces, acting on the indenter, are shown. It is seen that a step-wise change of the sliding velocity leads to an immediate jump of a force, followed by a "relaxation" to a new stable value. It is interesting to note that the time dependence of the coefficient of friction does not have the same peak (Fig. 3, a).
The evolution of the detachment coordinate a2(t) is shown in the Fig. 3, b. In the conditions of the "start-stop" test, a2(t ) goes from zero (that corresponds to the tip of the cone) to its maximum value, determined by the indentation depth of the indenter. Since v1 > 0, the transition time Attrans has a finite value.
6. Conclusion
We studied the kinetics of the frictional force and of the coefficient of friction in the simplest one-asperity contact with a viscoelastic Kelvin body after a velocity jump. Both analytical and numerical results show that the behavior is different for positive and negative jumps (accelerating and decelerating of slip). Thus, in the case of a positive veloc-
ity jump, the force of friction and the coefficient of friction have also a jump but there is no relaxation process. In the case of a negative velocity jump, on the contrary, the jump in the frictional force is followed by a relaxation to a new stationary value. Depending on the initial and final velocity, the relaxation time can change essentially. We have found explicit analytical equation for this time as a function of the relaxation time of the viscoelastic medium and the starting and finishing velocities. We would like to stress that these results are valid only for the pure hysteretic vis-coelastic contribution to the force of friction. Two other important contributions—that of the static hysteresis (plasticity) and adhesion will be considered in the future. For the further research, it would also be interesting to find a formulation of these regulations in form of differential equations similar to the state-and-rate law by Dieterich.
The authors thank the German Academic Exchange Service for financial support. The implementation of the numerical model and numerical simulations of the kinetics of the coefficient of friction have been carried out under the financial support of the Russian Science Foundation (Project 14-19-00718).
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Fig. 3. Coefficient of friction (a) and the coordinate of detachment (b) in the contact versus time. The vertical dashed line indicates the end of the transition process
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Dimaki Andrey Victorovich, PhD, researcher, ISPMS SB RAS, senior researcher, TSU, [email protected] Popov Valentin Leonidovich, Prof., Berlin University of Technology, Prof., TSU, Prof., TPU, [email protected]
