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УДК 539.612
Предельная форма, обусловленная фреттинговым износом адгезивного контакта, в приближении Дагдейла
Y.S. Chai1, В.Л. Попов234
1 School of Mechanical Engineering, Yeungnam University, Gyongsan, 712-749, South Korea 2 Berlin University of Technology, Berlin, 10623, Germany
3 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия
4 Национальный исследовательский Томский политехнический университет, Томск, 634050, Россия
Рассмотрен фреттинговый износ адгезивного контакта, обусловленный тангенциальными колебаниями малой амплитуды. В то время как существуют подробные теории как износа в контактах в отсутствие адгезии, так и адгезивных контактов без износа, до сих пор не было предпринято попытки объединить оба подхода. В настоящей статье проблема износа в адгезивном контакте изучается в упрощающем предположении, что адгезивное (притягивающее) напряжение между поверхностями постоянно до некоторого критического расстояния h и обращается в нуль на больших расстояниях (приближение Дагдейла). В этом приближении задача о нормальном адгезивном контакте может быть решена аналитически. В серии предшествующих работ было показано, что форма контактных поверхностей в результате прогрессирующего фреттингового износа стремится к предельной форме, определяемой исключительно решением нормальной контактной задачи. В настоящей статье эти идеи использованы для определения предельной формы изношенного контакта при наличии адгезии.
Ключевые слова: фреттинговый износ, адгезия, частичное проскальзывание, трение
Limiting shape due to fretting wear in an adhesive contact in the Dugdale approximation
Y.S. Chai1 and V.L. Popov2-3-4
1 School of Mechanical Engineering, Yeungnam University, Gyongsan, 712-749, South Korea 2 Berlin University of Technology, Berlin, 10623, Germany 3 National Research Tomsk State University, Tomsk, 634050, Russia 4 National Research Tomsk Polytechnic University, Tomsk, 634050, Russia
We consider fretting wear in an adhesive contact due to tangential oscillations of small amplitude. While both wear in non-adhesive contacts and adhesive contacts without wear have been studied in detail, there still have been no attempts to combine both approaches. In the present paper, we study the problem of wear in adhesive contacts under the simplified assumption, that the adhesive (attractive) stress is constant up to some critical distance h and vanishes beyond this range (the Dugdale approximation). In this approximation, the normal adhesive contact problem can be solved to a great extent analytically. In a series of previous works, it was shown that the worn shape due to fretting wear tends to some limiting shape which is determined solely by the solution of the normal contact problem. In the present paper, we exploit these ideas to derive the limiting shape of the worn body in an adhesive contact.
Keywords: fretting wear, adhesion, partial slip, friction
1. Introduction
Fretting wear occurs in frictional joints under action of tangential oscillations with small amplitude. Due to its technical importance, fretting wear has been intensively stud-
ied in a number of works with respect to various applications [1-3]. The physical reason for fretting wear is a stress
concentration at the boundary of a tangentially loaded contact, which leads to partial slip and wear in an annular region in the vicinity of the contact boundary [4, 5]. In [6], it
was suggested for the first time that the process of fretting wear might lead to some final shape. Indeed, according to the usual wear criteria like those of Khrushchov [7] and Archard [8], wear only occurs in the regions where there is relative tangential motion of contacting surfaces. Initially, slip occurs in the contact area in those regions where pressure is not large enough to withstand sliding, while in the stick region pressure is sufficiently large. Due to slip and wear, the stress in the slip region will decrease and be re-
© Chai Y.S., Попов В.Л., 2016
distributed to the stick region. Thus, the initial stick region remains in sticking state in the course of wear process while the initial slip region remains slipping up to the moment when the contact is lost due to progressive wear. In the final state, the nonworn area coincides with the initial stick region.
A simple method for the analytical calculation of the worn shape in this final state, based on the method of dimensionality reduction (MDR) [9], was suggested in [10]. In [11], it was shown that the approach developed in [10] can be generalized to multimode fretting, and it was argued that the limiting shape is determined solely by the solution of the normal contact problem of a "cylinder" with the contacting region coinciding with the nonworn shape in the region of permanent stick. These ideas are also applicable to the adhesive contact; in this case the pressure distribution should now be calculated while taking adhesive pressure into account.
Solving the adhesive contact for the case of a general adhesive potential is extremely complicated and can only be carried out numerically. However, for a very simple model interaction with a constant attractive stress up to some critical distance h and zero above this distance [12]
|CTo for f (r) - u2 3d (r) < h,
Padh(r) = 1n , ,, , (1)
I0 f°r f(r)-Mz,3D(r)>h,
the adhesive problem can be solved analytically to a great extent. This was first done for normal contact by Maugis [13] and recently generalized to tangential contact by Popov and Dimaki [14]. The latter paper is based on the method of dimensionality reduction [9], which is the simplest and most elegant technique to treat this complicated problem. As for wear, we will assume that only one of the bodies is worn out. In the solution procedure this will mean that we will consider an nondeformable body in contact with an elastic half-space, but assume that only the nondeformable body is worn.
2. Formulation of the adhesive contact problem in the framework of method of dimensionality reduction
To find the final profile we use the method of dimensionality reduction [9]. This method allows solving the normal and tangential contact problems for axisymmetric bodies by mapping them to a one-dimensional contact of a properly defined elastic foundation. As the limiting fretting shape is determined solely by the solution of the normal contact problem, we restrict ourselves to the formulation of the method of dimensionality reduction related to the normal contact problem. The main steps of the method of dimensionality reduction are the following. Given a three-dimensional profile z = f (r), we first determine the equivalent one-dimensional profile
" f '(r)
The back transformation is given by the integral
g (") = I" I
л/"2-
г dr.
(2)
/ (r )=21
g ( ") VT2-
rdx.
"ov r~ - "
The profile (2) is pressed to a given indentation depth d into an elastic foundation consisting of independent springs with spacing Ax, whose normal stiffness is given by
kz = E*A", (4)
where E is the effective elastic modulus:
l
E*
1 -v2 1-v2
(5)
E e2
E1 and E2 are the Young's moduli of the contacting bodies, and v1 and v2 are their Poisson ratios. Note that throughout this paper, we assume that the contacting materials satisfy the condition of "elastic similarity" guaranteeing the decoupling of the normal and tangential contact problems [15]. In particular, this condition is always satisfied in the important case of the contact between a rigid body and an incompressible elastomer.
The resulting vertical displacements of the springs are given by
Uz(") = d - g (" (6)
and the linear force density of the springs is
q(") = E*Uz(") = E*(d - g(")), (7)
which is valid in the whole contact region " < a.
According to MDR rules, the distribution of normal pressure p in the initial three-dimensional problem can be calculated using the following integral transformation [9]:
P(r) = -- J-
q( X)
П r si"2 - r The inverse transformation is
d"=E- ]-ßML
2 П rV"2^
rdx
q( ") = 2]-fM= dr. Wr - "
(8)
(9)
These transformations are valid for all contributions to the stress: both elastic and adhesive interactions. Thus, application of the transformation (9) to the adhesion stress (1) gives the equivalent linear adhesive force density in the one-dimensional MDR model:
qadh,z ( x) =
21
rP adh (r )
П2 2 sir -"
dr = 2I
ran
2
sir -j
rdr = 2a,
for " < h, (10)
0 for " > h,
where b is the outer radius of the region of adhesive interaction (that is of the region where the distance between both surfaces is smaller than h, see Fig. 1). The radius b depends on the surface deformation and should be determined as a part of solution of the contact problem.
Fig. 1. Adhesive contact according to Dugdale-Maugis model: In the subplot (a), the real contact of a rigid profile with an elastic halfspace is shown; subplot (b) shows the contact of a modified profile with a one-dimensional elastic foundation. In the three-dimensional case, constant adhesive stress g0 is acting between the surfaces at the points where the distance between the surfaces is smaller than h. At larger distances, there is no interaction any more. The radius of the region of action of adhesive stress b is larger than the contact radius a. In the MDR representation, the adhesive stress is replaced by one-dimensional adhesive linear force density qadh(x) given by Eq. (10)
Let us denote the initial, nonworn three-dimensional profile as f0 ( r ) and the corresponding one-dimensional image as g0(x) and the limiting shakedown shapes as (r) and g( x ) correspondingly.
The force AFcont z (x) between the spring at the point x and the effective profile g(x) is composed of the elastic spring force and the adhesion force:
AFcont,z (x) = Ax(-qz (x) + qadh,z (x)) =
= Ax(E(d -g ( x )) + #adh,z (x)). (11)
In the limiting state, this force should vanish in all points
starting with the radius c of permanent stick (for detailed
*
explanation see [11]): E (d - g(x)) + qadh z (x) = 0. For the limiting form of the one-dimensional MDR-image we thus get
g0(x) for 0 < x < c,
(12)
э( x) =
d + 2
E
for c < x < a.
This shape is schematically shown in Fig. 2.
In the following, we restrict ourselves to the case when the action range of adhesive forces h is smaller than any other characteristic length of the problem. This limiting case corresponds to the classical theory of Johnson, Kendall and Roberts [16, 17]. In this limit, the difference of the radii a and b is given by the equation [14] nhE*
b - a «-. (13)
4Gn V ;
Fig. 2. One-dimensional MDR-image of the final "shakedown" profile
For small enough h, the radii a and b can be considered as approximately equal for the sake of the profile calculation and the three-dimensional limiting shape can be calculated from (12) by the back transformation (3):
fc (r):
f0(r) for 0 < x< c,
2 c g0(x) . 2
_J_
п 0л/;1—
dx +—d J
1
-dx +
4 G0 rryja2 - x2
J
n E* c4r2 —
n cJT—x2 (14)
dx for c < x < a.
Integration gives the shape in the worn region c < x < a :
fc ( r ) - - J
g0( x)
п 0 л/;2—
rdx + <
1 —arcsin n
/ л c
4 Gn
+--* a
n E
r \ 1, r
- E
\
c r
where
E ( x, k) = J
Vi - k212
vrt
d t
(15)
(16)
o V1 -1
is the incomplete elliptic integral of the second kind.
As an example let us consider an indenter with parabolic initial profile fo (r) = r2/(2R) and the corresponding one-dimensional MDR-image go(r) = x2/R. According to (15), the limiting three-dimensional profile has the form
fc ( r ) - d — П
2 ^ d - i-2R
J
arcsin
r J
r
nR
xi E
■2' c ^
^ 2
4G
+ -a x П E
a '
V a J
- E
c r r ' a
(17)
Introducing dimensionless variables
~ f r x c a ~ d
f = —, r =—, x =—, c =—, a =—, d = — = 1, (18)
d aл aл a aл d
Fig. 3. Adhesive contribution to the fretting wear according to Eq. (21). Parameter c lineary increases from 0.1 (1) to 0.9 (9)
where a0 =^J~Rd is the contact radius of the initial profile, we can rewrite these equations in the form
/> ( r ) » 1 —
П
x a
(19)
Additional wear contribution due to adhesion is determined by the last term in Eq. (17):
/( r )adh
4 fVa * j
n E r
Vr -;
rdx =
rdx
_ 4 Goa r Vl - x
or in dimensionless form
4 go a r л/l-î
(20)
/ ( r )
adh
r.
n Ed r-J,
rdX _
4 ^a < e
n Ed
Í il, r ] — E с ~ r ~Z i ~ л r
r ) 1r a ;
(21)
This contribution (21) is shown in Fig. 3 for a representative set of parameters.
The additional wear contribution due to adhesion is governed by the dimensionless parameter g0a/(E*d) which has the physical meaning of the ratio of adhesive stress to a typical value of elastic stress in the contact area.
Up to now, we considered the radius c of the permanent stick as a given quantity parametrizing the limiting worn shape. For determination of the radius c as function of the oscillation amplitude, an additional consideration of tangential contact should be carried out. The corresponding procedures are described in [14]. The radius of the stick region is
*
determined by the condition that the normal force [E (d --g( c)) + qadh z (c)]A" between a spring with the coordinate
c of the one-dimensional MDR-model and the substrate multiplied with the coefficient of friction is equal to the tangential force G*u ^ Ax, where
1 G*
2-v1 + 2—v2
4G1 4G2
(22)
G1 and G2 are the shear moduli of the indenter and the half-space respectively, and uf is the amplitude of tangential oscillation (for details see [10, 14]). In explicit form, this condition reads:
E*(d - g0(c)) + 2a0Va2 - c2 = G*u{0. (23)
Solution of this equation with respect to c gives the radius of the permanent stick region. The complete sliding (gross slip) starts when the stick radius vanishes: c = 0, which gives the maximum amplitude of tangential oscillation, at which the fretting will occur:
u (o) = E*d + 2aoa
ux max * • (24)
G
This corresponds exactly to the findings of the paper [14].
3. Conclusion
We studied fretting wear in an adhesive contact. There are a number of explicit and implicit assumptions which we used in this study. First, we assumed the simplified Dugdale potential for the adhesive interaction. This assumption is not crucial for the study and was done only to simplify calculations. The most important physical assumptions are implicit assumptions that the wear proceeds "continuously" (without breaking out of larger wear particles) and that the wear particles can be transported out of the contact region without destroying the adhesive contact. These assumptions could be valid in micro- and nanoscopic contacts, especially of polymers and elastomers. Under the assumptions made, we come to the conclusion that small tangential oscillations lead to the appearance of a circular slip region with progressive wear, which finally leads to the formation of a stationary profile with no further wear. The shape of this limiting profile can be determined analytically. The profile occurs to be a superposition of the worn profile without adhesion and an adhesive contribution. The relative value of the adhesive contribution is determined by the dimensionless factor G0a j(E*d) having the order of magnitude of the ratio of adhesive stress to the average elastic stress in the contact.
Acknowledgments
This research is supported by Yeungnam University (Y.S. Chai) and by D.I. Mendeleev Foundation, project number 8.2.19.2015 (V.L. Popov).
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Поступила в редакцию 05.09.2016 г.
Сведения об авторах
Young S. Chai, Prof. Dr., Yeungnam University, South Korea, yschai@yu.ac.kr Valentin L. Popov, Prof. Dr., Berlin University of Technology, v.popov@tu-berlin.de
