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УДК 539.3, 539.62
Применение метода редукции размерности для расчета силы трения между осесимметричным индентором и вязкоупругим полупространством
S. Kusche
Берлинский технический университет, Берлин, 10623, Германия
Метод редукции размерности применен для описания касательного контакта между эластомером и осесимметричным индентором. Найдены аналитические решения уравнений контактной задачи. Также обсуждается численная реализация метода. Установлена зависимость нормальной и касательной силы от глубины индентирования и скорости скольжения.
Ключевые слова: метод редукции размерности, вязкоупругое основание, сила трения, касательный контакт
Applying the method of dimensionality reduction to calculate the friction force between a rotationally symmetric indenter and a viscoelastic half-space
S. Kusche
Berlin University of Technology, Berlin, 10623, Germany
The method of dimensionality reduction is applied to the tangential contact between an elastomer and a rotationally symmetric indenter. The resulting equations are solved analytically, and numerical methods are discussed. The dependence of the normal and tangential force on the indentation depth and velocity is indicated.
Keywords: method of dimensionality reduction, viscoelastic foundation, friction force, tangential contact
1. Introduction
In many applications friction plays an important role [1, p. 231ff]. To achieve a high friction coefficient rubber and elastomers are commonly used. A standard problem is the rigid indenter which is pressed by constant force into a viscoelastic half space and moved tangentially.
Unfortunately the mathematical treatment is difficult as the material properties are time dependent and contact area is not known a priori. While in the case of pure normal contact a closed-form solution has been found [2], in the case of a sphere rolling on a viscoelastic half-space it is still missing. For this case numerical procedures have been developed [3]. One drawback is the missing generality of the obtained results, since the exact geometric shape of the indenter and the material law must be defined.
A widely used simplification is to neglect the interaction of the stress and deformation between different points of the contact area. The benefit is the simplified geometric determination of the contact area. It has been used for ex-
ample on the problem of rolling cylinders [4] and spheres [5]. A similar approach is the method of dimensionality reduction (MDR) [6, 7]. This procedure greatly simplifies three dimensional contact problems. The analogous model results from an integral transformation and consists of a one dimensional bedding including independent elements. This foundation replaces the half-space, while the contacting model is replaced by a transformed one-dimensional profile. This reproduction is exact for rotationally symmetric displacement and stress fields [8]. This requirement is not exactly true in the tangential contact. Nevertheless the mechanism can be studied and qualitative results can be obtained.
The method of dimensionality reduction has been used for viscoelastic tangential contacts before [9, 10]. This paper now proceeds at this point and introduces analytical solutions. The following problems are to be examined: A rigid indenter is pushed into an elastomer and simultaneously moved tangentially at a constant speed. The resis-
© Kusche S., 2015
tance it experiences is caused exclusively by the deformation of the viscoelastic material; the stress in the contact is assumed as pure pressure. This conceptual model corresponds to a lubricated contact, or to the case that both objects roll on each other without creep [11]. In reality, this effect is only rarely observed, although the influence of the creep can be completely neglected in case of appropriate geometry.
For the viscoelastic solid in the limiting cases of small and large tangential speed, the expected elastic solutions are reproduced exactly. In the transition, a qualitatively correct result is shown.
2. Model
2.1. Linear viscoelastic material
It is well-known that the temporal deformation behavior of elastomers [12, 13] can be described by a continuous spectrum. This can also be approximated through a small number of discrete frequencies, and it results in the Prony series. It also benefits from a mechanical substitute model consisting of spring and damping elements. In this way, all the time-dependent material constants can be represented as a series. It appears, however, that many elastomers are nearly incompressible and exhibit a Poisson's ratio of about 1/2. Thus it is sufficient to only adopt the shear modulus as a variable in order to fully describe the time-dependent material behavior:
/
t
G (t ):
G^+S G exp
i=1
(1)
A distinction must be made between viscoelastic solid and viscoelastic liquid [12]. The latter offer no resistance to very slow deformations, therefore G(t ^ = 0 is assumed.
2.2. Method of dimensionality reduction
The method of dimensionality reduction replaces the viscoelastic half space through a one dimensional bedding. It consists of mutually independent deformable elements that match the featured mechanical substitute model. As a time-dependent rigidness, the product of lattice spacing and effective elasticity modulus is used: k = hxE. The effective modulus Eq. (2) allows the deformation to be completely attributable to the bedding.
J_ G*
2-V 2-V
(2)
4G1 4G2
In the case of the anticipated incompressibility and that the indenter is modeled as a rigid body, the results from Eq. (2) are: G*(t) = 4G(t). The deformation history is now of importance to the acting force. The consequent law of force (q being a distributed load) is:
q(t) = 4 J G(t - 0 w( t')d t\
(3)
Ultimately, the reproduction of the indenter in contact is still missing. It must be reproduced to a one-dimensional
profile in accordance with the rule of Heß. This is described three-dimensionally by the contour z(r) = c3Drn. This is reproduced to the one-dimensional contour: z(x) = = c1D | x | n. The new constant is calculated as follows:
Vn nT(n/2)
c1D
= Knc
nc3D'
KM =-
2 r((n +1)/2)'
(4)
2.3. General procedure
The indenter is pulled at a constant velocity v over the elastomer, so that each point in contact with the bedding adapts to the indenter's contour. The coordinate x measures from the apex point to another point in the contact. It is valid for the length of the contact in the range -a1 < x < a2 (see Fig. 1 for geometry description). Being that t = 0 and x = -a1, then it follows that x = -a1 + vt. Since the elements are independent of each other, the deformation at the first point of contact is zero and the indentation depth follows the equation 8 = c1D a?. The deformation of the elastomer perpendicular to the surface (normal direction) and its speed is as follows:
w (x) = 8- z (x),
w (x) = -z (x)x(t) = -vz'(x).
The distributed load q is calculated from the deformation history, starting with the first contact with -a1 up to the current point x. The maximum of x goes until the end of the contact area at a2, the separation point. Due to x = = -a1 + vt the integration in time can instead be performed in place. With the substitution vdt = dx' t -1' = = (x - x ')/v and the Prony series for G(t) it follows:
(5)
m
q(x) = -4Gtc J z'dx-4S Gte VT- J
'Z d X.
(6)
i =1
The function q(x) is positive up to the separation point a2. Its implicit conditional equation is q(a2) = 0. If a1 and a2 are known, the normal force can then be determined by reintegration. As the indenter has a slope, the normal force also functions in tangential direction. Multiplied by the slope, and integrated, these results in the tangential force:
Fn = J q(x)dx, Ft =- J q(x)z'(x)dx = |wFn.
(7)
Fig. 1. Indenter (above) in contact with an elastomer (below). The indenter moves with velocity v to the left. Prior to the first contact, the deformation is zero, because the elements are independent. At the outlet there is a nonzero deformation in general
- a
a2/a\ —^ Lb
0.8 ■ '......"
0.4 n \\
0.0 '— Viscoelastic solid V-. ......Viscoelastic liquid
-6 -4 -2 0 lg(ax/m) with ax = (à/clD)l/n
Fig. 2. Coefficient of friction (a) and detachment point (b) for a cone (n = 1) and a paraboloid (n = 2) in contact with a viscoelastic solid (G^ = 0.5 GPa) and a viscoelastic liquid (G^ = 0) elastomer. The material is described by the series G(t) = 105Pa x x [exp(-t • 102/s) + exp(-t • 10Ys) + exp(-t • 10Ys)]
3. Solutions
3.1. Analytical treatment
The integrals can be calculated, in principle, in the sense that functions such as the incomplete gamma function (see Appendix) appear in the solutions. The first task is to solve for the detachment point a2. If this fails, the forces cannot be specified as a function of the indentation depth. This is in general not possible. For the standard model and the powers 1 and 2, however, this was successful, with the solutions indicated in Sect. 3.3. The second step is to calculate the forces. Here as well, difficulties of a numerical nature quickly result if the functions grow rapidly. Especially the generalized hypergeometric series 2 F2 exhibits these characteristics.
The analytical solutions can nevertheless be simplified. In the limiting cases of small and large indentation depths, the separating conditions are easier. A series expansion at expansion point a1 = 0 provides vanishing terms at the lowest order if a2 = a1 is selected. This is regardless of whether it is a viscoelastic liquid or solid. If a1 is very large and G^ ^ 0, the order of the second term decreases as a result of the exponential function. Thus, the term before Gis observed, and this provides again the condition a2 = a1. Finally, the difference in the gamma functions is crucial for G^ = 0. A series expansion at expansion point a1 =^>
shows that the gamma function with the positive argument does not exhibit exponential growth. Unlike the gamma function with the negative argument, which is why a2 = 0 is chosen here. Through these highly simplified detachment points, the asymptotes can be calculated from the solutions for the forces by corresponding series expansions. They are given in Table 1.
These limiting cases can also be physically interpreted. The elastomer follows the indenter contour completely for smaller indentation depths. The detachment point lies at the same distance from the vertex as the first point of contact. For larger indentation depths, a permanent deformation ensues for the non-crosslinked elastomer. It detaches in the limiting case at as early as the vertex.
The coefficient of friction is determined by Popov [6] as well as by the dissipation power at a certain harmonic stimulation. For this, a representative frequency is needed with which the system is stimulated. As a rough estimate, the length of the contact X = 2 a1 is used for the wavelength of the stimulation.
Thus, the dissipation power can be determined depending on the frequency-dependent loss modulus G"(q). This in turn is related to the tangential force and the velocity, and the following coefficient of friction ensues:
Table 1
Asymptotes of the solution for an arbitrary indenter in the form: z(x) = c1D | x |n, n > 1
8^0 G^ = 0 G^ 0
a2 « a1 0 a1
F « n+1 8n „ C1D a1 —"T G0 n +1 m C1Da", 41GTiv l =1 c an+1 8n G C1D a1 —T n +1
Ft « 4X n+1 16n 2 y G 1D 1 (1 + n)(1 + 2 n) S T v c2 a 2n-1 4n2 yG C1D a1 , Gi TiV 2n -1 i=1 c2 a 2n-1 8n2 yG t C1D a1 , Gi TiV 2n -11=1
^ « n 2 n m Gl c1D a1-y-l— 1D 1 1 + 2 nl=Î G0Tl v c an-1 n2 C1D a1 2n -1 n-2 n (n +1) m GlTlv c1D a1 2 —-- y l l 1D 1 2 n -1 ri G^
Table 2
Asymptotes for small and large indentation depths 8 in Eq. (8)
I GI
5 ^ 0
- £ G
n i=i G0x. v
-n £
i=1 G0 Tiv
= 0
n-1
^ 0
n £ Gi Ti v
-i=1 G^
_n-2 ^ wi v
i=1
m rnT.
G ( œ) = G^+£ G;—^— i=1 œTi -1
= G + iG '
In Eq. (8), the above limiting cases are also discussed. For a diminishing indentation depth respective to the contact radius, the frequency of the stimulation is infinite. Conversely, the frequency tends towards zero for larger indentation depths, since the contact length is very large. These limiting cases are summarized in Table 2. The same dependencies, which were also derived from the analytical solution, are revealed.
3.2. Numerical treatment
The analytical solutions are difficult to apply numerically. Firstly, the determination of the detachment point is possible only through an implicit equation. Second, the exact calculation of the generalized hypergeometric series, which is required for the tangential force, is nontrivial. Both problems can be circumvented by introducing a discrete lattice:
xt = ihx, hx = 2a1/n, i e [-n, -n + 1, ..., n], n e N+. (9)
For this, a2 < a1 must always apply. For each discrete point, the analytical solution for the distributed load can now be multiplied with the grid spacing in order to calculate the force in an element. The coordinate of the detachment point arises from the condition that there can only be positive forces in contact. The integrals for all normal and tangential force simplify themselves to sums.
j ^ q (x:) > 0 a q (x:+1) < 0, a2 = xf,
j j (10)
Fn = hx £ q(xiX Ft =-hx £ z'(xi)q(x).
1=-n 1=-n
To determine the distributed load, the gamma function for positive and negative arguments must be analyzed. The latter is also unlimited in the product with the exponential function with a negative argument.
The simplest case is a question of an indenter with integer power n. Then the incomplete gamma function with Eq. (11) [see 14, p. 178] can be defined in a finite sequence of power functions. It is very good in terms of accuracy and calculation speed. The example shown in Fig. 2 has been calculated in this way.
r( n, x )
r( n)
n-1 ~ j
£ ^, n eN+. j=0 j !
(11)
If this is not the case, then normalized gamma functions must be used. Furthermore, the exponential function can be combined together with the gamma function to a generalized hypergeometric series 1F1 (Kummer's confluent function) [14, p. 328]. For this, there are specialized algorithms [15, 16]:
ne~x (-1)n y (n, - x) = X 1F
1
1 + n
(12)
As an alternative to an analytical solution for the distributed load, a quadrature can be used. The midpoint rule provides, together with the Prony series, an iterative algorithm for integration. In Eq. (3) it is assumed that no deformation has taken place for t < 0 and a material with one relaxation frequency has been analyzed. The algorithm for one step reads as follows:
q (t) = 4G00) w(Od t' + 4 G11( t),
0
I (t )
t/ht
; ht £ exp
j=0
t - A
V
w (jht ),
(13)
I (t + ht ) « exp
A
I (t ) + htw (t + h ).
With a suitable choice of the time increment hx = vht the instants of time coincide directly with the locations of discrete elements, see Eq. (14). Furthermore, this method can also be used if the deformation history is not known a priori. This may be the case if it is not clear how a stationary state appeared, or when the transient process of immersion needs to be examined in the elastomer.
q( x = x) = q
/ x; + — ^
t = —-1 = (i + n) ht
(14)
Fig. 3. Dimensionless friction coefficient as a function of dimen-sionless distance from the first point of contact. For the indenter, this is a question of a cone (n = 1) and a parabola (n = 2). The upper curve in each case exhibits the Maxwell element (G = 0). Standard models follow beneath with G = {1,1/5, 1/20}
I!
oc
Table 3
Solution for the conical indenter. Material is the standard model: G (t) = G (G + exp(-t/t))
Conical indenter
Conditional equation a2 : 0 = G(a1 - a2 ) + 2 exp(-a2 ) -1 - exp(-a - a2 )
_ a2 -G+ r [G-1(2-e~a')exp(G-ä1)], G * 0,
I ln(2 - e"_>), G = 0
in
F„
cidG(to)'
• = 4(_i - _2) + 2GH_i2 - _2 + 2(_i - _2 + _i_2)]
i =
i
1 4dG(tV)2
= 4+ _ 2 - 2(1 - e _')J + 2G(_ - a 2)(_ 1 - a 2 - 2)
3.3. Special cases
The Maxwell element and the standard model are examined as special cases. The first model consists of the series connection of spring and damper elements. It is an example of a viscoelastic liquid. The second additionally consists of a spring connected in parallel. For ease of distinction, the ratio of shear moduli occurring is defined as the characteristic variable: GG1 = G^. Both models also only have one discrete relaxation time. This results in product with the velocity of movement in a characteristic length Tv. This can be used to normalize the system length. Resulting from the analytical solutions, the following dimen-sionless variables are offered:
(15)
Vn f = F^G(tu)" Ft = itCiDG(Tvr" ^ = pCiD(xv)"
_ = _TV = (8/£VD)v", Fn = FnCiD
2 ,, -TT- í^^n-l
Their significance is more common than it appears at first glance. Thus, the constant c1D can be replaced by its three-dimensional representation. When Master curves are generated by experiments or numerical simulations with these variables, parameters can be altered and the master curves retain their shape. For example, experiments with
different geometry, e.g. the radius of a rolling ball, can be compared.
For the indenter, the cone (n = 1) and the parabola (n = 2) are examined. The cone is defined by the opening angle n -26 and the parabola by the radius of curvature R at the vertex:
n = 1: C3D = tg6 = -
n = 2: c3D =
1
n 1
c1D,
(16)
c1D *
2 R 2
The results are shown in Tables 3 and 4. The master curves for the coefficient of friction can be found in Fig. 3.
4. Conclusions and outlook
The method of dimensionality reduction was applied to examine the tangential contact problem between elastomers and rigid bodies. The analytical solutions point to the principle dependencies. It remains to be examined to what extent the consequent dimensionless variables are quantitatively and qualitatively reproducible in the solutions of three-dimensional theory.
Table 4
Solution for the conical indenter. Material is the standard model: G (t) = G (G + exp(-t/t))
Parabolic indenter Conditional equation a2: 0 = G(a2 - a22) + 2 [1 - a2 - (1 + a1 )exp(-a1 - a2)]
a2 = = 1 + W[-(1 + ai)exp(-1 - a)], G = 0 TV
F = —= 4(ai2 - of)+4 G (a + Ü2) [2ai2 + (a - a2)(3 + a2) ] c1dG(tv) 3 L J
_ F 88 r 2 2 "1 _ Ft = 2 ' ,4 = Ja1(2a1 3) + a2(2a2 +3) + 2G(a1 a2 2)(2 + a1 + a2) 4,G (tv)4 3 L J
The function W is the Lambert W function, which solves the equation x = Wexp(W).
References
1. Popov V.L. Contact Mechanics and Friction: Physical Principles and Applications. - Berlin: Springer-Verlag, 2010. - P. 231-270.
2. Lee E.H., Radok J.R.M. The contact problem for viscoelastic bodies // J. Appl. Mech. - 1960. - V. 27. - No. 3. - P. 438-444.
3. Carbone G., Putignano C. A novel methodology to predict sliding and rolling friction of viscoelastic materials: Theory and experiments // J. Mech. Phys. Solids. - 2013. - V. 61. - No. 8. - P. 1822-1834.
4. May W.D., Morris E.L., Atack D. Rolling friction of a hard cylinder over a viscoelastic material // J. Appl. Phys. - 1959. - V. 30. - No. 11. -P. 1713-1724.
5. Flom D.G., Bueche A.M. Theory of rolling friction for spheres // J. Appl. Phys. - 1959. - V. 30. - No. 11. - P. 1725-1730.
6. Popov V.L., Hess M. Method of Dimensionality Reduction in Contact Mechanics and Friction. - Berlin: Springer-Verlag, 2013. - 267 p.
7. Popov V.L., Hess M. Method of dimensionality reduction in contact mechanics and friction: A users handbook. I. Axially-symmetric contacts // Facta Universitatis. Ser. Mech. Eng. - 2014. - V. 12. - No. 1.-P. 1-14.
8. Heß M. On the reduction method of dimensionality: The exact mapping of axisymmetric contact problems with and without adhesion // Phys. Mesomech. - 2012. - V. 15. - No. 5-6. - P. 264-269.
9. Li Q., Popov M., Dimaki A., Filippov A.E., Kürschner S., Popov V.L. Friction between a viscoelastic body and a rigid surface with random self-affine roughness // Phys. Rev. Lett. - 2013. - V. 111. - P. 034301.
10. Popov V.L., Voll L., Li Q., Young S.C., Popov M. Generalized law of friction between elastomers and differently shaped rough bodies // Sci. Rep. - 2014. - V. 4. - P. 3750.
11. Grosch K.A. The relation between the friction and visco-elastic properties of rubber // Proc. Roy. Soc. Lond. A. Math. Phys. Eng. Sci. -1963. - V. 274. - P. 21-39.
12. Ferry J.D. Viscoelastic Properties of Polymers. - New York: John Wiley & Sons, 1980. - P. 15-17.
13. Tschoegl N.W. The Phenomological Theory of Viscoelastic Behavior. - Berlin: Springer-Verlag, 1989. - P. 158-170.
14. Olver F.W.J., Lozier R.F., Boisvert R.F., Clark C.W. NIST Handbook of Mathematical Functions. - New York: Cambridge University Press, 2010.
15. Muller K.E. Computing the hypergeometric function, M(a, b, x) // Numer. Math. - 2001. - V. 90. - No. 1. - P. 179-196.
16. Pearson J.W., Sheehan O., Porter M.A. Numerical Methods for the Computation of the Confluent and Gauss Hypergeometric Functions, 2014, arXiv:1407.7786v1 [math.NA].
Appendix
The solution is given in terms of the incompleted gamma function [6, p. 174]:
J tn-1e- dt + J tn-1e~t dt = J tn-1e~' dt
0 X 0
(A1)
Y (n, x)
r( n, x)
r( n)
and the generalized hypergeometric series (see [6, p. 404] and for Pochhamer's symbol [6, p. 136]):
F p q
1' ***' ap
Y
; z
(a)k =
h'-' bq
r(a + k)
= E
(fli)k '••(ap )k
k=o(bi)k■••(bq)k k!
(A2)
r(a)
The distributed load follows from Eq. (6)
q (x < 0)
~4C
= G
1D
,[< - (-x)n ]-
+ n£ (T v)nG-e
i=1
a1 n,—— x
-Y n,--
t- v i Ti v i
q (x > 0)
4c
= G
1D
.[an -xn]-
(A3)
+ n£ (Ti v) nGte i=1
( a1 Y n,— Ti v
f
/ 1 \-n
- (-1) Y
Ti
The normal and tangential force follows from Eq. (7): Fn = g a2(an - an) + naj1 (a + a2) +
4c1D n +1
+ E G-Tiv(an - a2n) + nE Gi (Ti v)n+1 e T-v x i=1 i=1
(-1)-n Y
n, —
a,,
Ti v
V /
n n\2 . 2
( a Y n,——
Ti v i
(A4)
-Ft = 1G^a - a2" )2 + n2 E G (Ti v)2n Y
4c1D 2
( „ Y
i=1
n,-
Ti v i
(
(-1)-n Y
Y
Ti v
( „ Y"
n,-
Ti v i
- 2 E Gi
2 i=1
2n
- a2 2F2
2n a1 2 F2
( 1,2 n
( 1,2 n
a
;1
1 + n,1 + 2n Tiv
a
Y
1 + n,1 + 2n Ti v
(A5)
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Stephan Kusche, Dipl.-Ing., Berlin University of Technology, Germany, [email protected]
