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УДК 539.612, 544.722.54
Mapping of two-dimensional contact problems on a problem with a one-dimensional parametrization
V.L. Popov123
1 Technische Universität Berlin, Berlin, 10623, Germany 2 National Research Tomsk Polytechnic University, Tomsk, 634050, Russia 3 National Research Tomsk State University, Tomsk, 634050, Russia
We discuss a possible generalization of the ideas of the method of dimensionality reduction (MDR) for the mapping of two-dimensional contact problems (line contacts). The conventional formulation of the MDR is based on the existence and uniqueness of a relation between indentation depth and contact radius. In two-dimensional contact problems, the indentation depth is not defined unambiguously, thus another parametrization is needed. We show here that the Mossakovskii-Jäger procedure of representing a contact as a series of incremental indentations by flat-ended indenters can be carried out in two-dimensions as well. The only available parameter of this process is, however, the normal load (instead of indentation depth as in the case of three-dimensional contacts). Using this idea, a complete solution is obtained for arbitrary symmetric two-dimensional contacts with a compact contact area. The solution includes both the relations of force and half-width of the contact and the stress distribution in the contact area. The procedure is generalized for adhesive contacts and is illustrated by solutions of a series of contact problems.
Keywords: line contact, two-dimensional contact, method of dimensionality reduction, Mossakovskii-Jäger superposition principle, adhesion
Отображение двумерных контактных задач на задачи с одномерной параметризацией
В.Л. Попов1-2-3
1 Берлинский технический университет, Берлин, 10623, Германия
2 Национальный исследовательский Томский политехнический университет, Томск, 634050, Россия
3 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия
Обсуждается обобщение идей метода редукции размерности на случай отображения двумерных контактных задач. Классическая формулировка метода редукции размерности основана на существовании однозначной зависимости между глубиной ин-дентирования и радиусом контакта. В двумерных контактных задачах глубина индентирования однозначно не определена и требует другой параметризации. Показано, что процедура представления контакта в виде суперпозиции индентирований плоскими штампами возрастающего радиуса, предложенная Mossakovskii и позже Jäger, может быть применена и в двумерном случае. Единственное изменение состоит в использовании нормальной силы как единственного однозначно определенного параметра состояния вместо глубины индентирования. Получено полное решение двумерной контактной задачи для произвольных симметричных профилей с односвязной областью контакта. Решение включает соотношения между силой и полушириной контакта и распределение давления в области контакта. Процедура обобщена для случая адгезивных контактов и проиллюстрирована решением ряда контактных задач.
Ключевые слова: двумерные задачи механики контактного взаимодействия, метод редукции размерности, принцип суперпозиции Mossakovskii-Jäger, адгезия '
1. Introduction
It is well known that axially symmetrical contacts with
compact contact area can be easily reduced to a contact of a modified profile with a one-dimensional elastic founda-
tion [1, 2]. This procedure is based on the solutions of the axially symmetric contact problems obtained by Galin [3] and later popularized by Sneddon [4]. In the last decade, this approach was generalized for treating a large variety
© Popov V.L., 2017
of problems in the framework of the method of dimensionality reduction (MDR) both for homogeneous media [5, 6] and power-law gradient materials [7, 8].
In the present paper we undertake an attempt to generalize the MDR to the case of two-dimensional contact problems (which are usually called "line contacts"). The basis of the traditional MDR-formulation is representation of the contact as a superposition of indentations by flat cylindrical punches as first considered by Mossakovskii [9] and later re-invented by Jäger [10]. This principle is generally also applicable to two-dimensional problems. However, in the presentations of Mossakovskii and Jäger, indentation depth was used as the "governing parameter" of the indentation process. But in two-dimensional contact problems the indentation depth is not uniquely defined [11]. Thus the question arises whether it is possible to apply the same procedure using another parametrization of the contact process. Jäger also undertook an analysis based on similar assumptions as in the present paper [12].
The basis for the application of the Mossakovskii-Jäger-procedure is the solution for indentation of a half-space by a flat-ended punch as schematically shown in Fig. 1. Let P be the force intensity per unit length distributed along the y-axis and a the half-width of the punch in the x-direction. Under assumption of frictionless contact, the stress distribution under the punch is known to be [11] P
P (x) -Ö17T • (1)
n(a
-x 2)'/2
The displacement of the surface outside the punch is equal to [11]
/ N ^ 2P I x uz ( x) = d--* ln \- +
nE I a which can be rewritten as
^ 2P | x d - Uz ( x) =—* ln\- + nE a
\
2 V/2^
-1
a ,
% -1 a ,
1/2-
where E is the reduced elastic modulus
*E E =-
1 -v2
(2)
(3)
(4)
with Young's modulus E and Poisson number v.
Note that while d and uz (x) in Eq. (3) are both defined relative to an arbitrary reference point, their difference is defined unambiguously. For an infinitesimal indentation
d ( d - Uz ( x) ) = —* ln \ - + nE I a
1/2
— 1
dP.
(5)
Now let us consider the indentation of a rigid body with a profile z = f(x). We follow the indentation process from the first contact to the final indentation depth and investigate the current values of the force intensity, indentation depth and contact half-width P, d and a. The entire process consists of a change of the contact half-width from a = 0 to a = a, and contact force intensity from P = 0 to P = P. Understanding this process as successive indentations by flat-ended punches with increasing half-width (as shown in Fig. 2), we can integrate Eq. (5) and set x = a: [d (a) - uz (a) ] - [d (0) - uz (0) ] =
2 h I a
= —* Iln\- +
nE 0 I a
2
a 1 ~ -1
Va y
(6)
Since [d(a) - uz (a)]-[d(0) - uz (0) ] = f (a), Eq. (6) can be rewritten as
f (a ) = 4* I ln +
nE 0 I a
Partial integration gives
2
a 1 - 1
va y
(7)
f (a ) = — nE
p( a ) ln \ a+
a
2
a 1 ~ -1
va y
\1/2
a=0
P(a )
2a a + —* J I
nE 0 av a2 - a2
da.
(8)
We assume that P(a) tends to zero when a ^ 0 more rapidly than ln |a|, which is e.g. valid for all power-law profiles f (x) ^ \x\n with n > 0 (see solution below for power-law profiles). Under this assumption, the first term in (8) vanishes and we come to the equation 2a % P(a)
f (a) =^a* I-
rv a -<
^d a.
(9)
0 aV a - a
Solution of this integral equation with respect to P(a) reads
[13]:
P (a) = E'a 2 J( f (a V a >' da.
J / 2 -2 0 V a - a
(10)
The pressure distribution can now be calculated by writing Eq. (1) in differential form
Fig. 1. Two-dimensional contact of a flat-ended punch with an elastic half-space
Fig. 2. Single flat-ended indenter (a) and a superposition of indentations by flat-ended indenters of different half-width (b)
Table 1
Coefficients Zn according Eq. (17) for several exponents n
1
n/ 2
3n/ 4
8/3
15n/16
10
315n/256
dp ( x) =
dP
n(52 -x2)12
and integrating it p( x) = J
dP
n(<% 2
(11)
(12)
-x2)12 da
Let us consider as an example a parabolic profile f (x) = = x2 /(2R ). Substitution into (10) gives
P (a) =
E a
2 a
-I-
da
nE a
(13)
,. e * a ada P(x) = ^7^
2 R ¿V a 2 - - 2 4 R Differentiating and inserting into (12) we find the following stress distribution
* _
E I-
— I—--T"T7T = — va -x , (14)
2 RX (%2 - x2)1/2 2 R ' V '
which coincides with the Hertz's solution in the limiting case of very elongated ellipsoid [14].
The same procedure can be applied to any arbitrary shape. Consider as a second example a general power-law profile:
f (x) = cnjxl". (15)
Substituting into (10) gives
P(a) = E\2jif^Mda -
o va2 - a2
* a % n - 2
= E a2cn (n - 1)I ,-= da =
' /""2 -2 ) Va -a
1 fcn-2
where
= E *Cn (n -1) an d^ = E*Cn Znan, (16)
oJl-42
Z n
_( n - 1)Vrcr(1/2 n -12 )
, • (17)
2r(l2n) V 7 These first few of these coefficients are tabulated in Table 1. In particular, for a wedge:
f (x) - |x| tan e, Zi -1. (18) For the force we get
P (a) - E*a tan e (19) and for the pressure distribution E *tan e7 1
P( x) =
I
n x (a2 -xJ)12
E *tan 0
da =
-ln
n
a — +
x
'a^ 2
-1
(20)
These results coincide of course with those obtained first by Truman et al. [15].
Fig. 3. A cylindrical profile with a small flat of half-width b in contact with an elastic half-space
As a less trivial example let us consider a cylinder lying on its side that has a small flat area ("worn cylinder") and is in contact with an elastic half-plane (Fig. 3). The shape of this profile is described as follows: f0 for 0 < x < b,
f ( x ) =
whence V ( x y
for b < x < a,
2 R
0 for 0 < x < b,
1 b
+
2R 2Rx 2
for b < x < a.
(21)
(22)
Substitution into (10) results in
P( «) = E^ I
2R
E a2
2
b
1 + " a 7
da
2R
n
— arcsin 2
a
1 s
2
a
+b-. h -
(23)
This dependence is plotted in Fig. 4 (solid line) together with the corresponding dependence for a parabolic profile without flat (dashed line).
Let us further consider the contact between a lying cylinder and an elastic half-space. Contrary to the parabolic
Fig. 4. Solid line shows the dependence (23) of the normalized force intensity 2rp(a)/(e*b2) on the normalized contact radius a/b. The dashed line shows the corresponding dependence for a parabolic profile
Z
The critical condition reads
Fig. 5. Exact solution (26) (solid line) and parabolic approximation (Hertz, dashed line) for the dependence of the force intensity on the contact half-width
approximation of Hertz, here the exact cylindrical form should be taken into account. The exact profile of a cylinder of radius R is given by the equation
2
f ( x) = R-4W-; whence
( f (x ) Y 1
R
n/R2^
Substitution into (10) and integration gives
P(a) = E* RI K R
where
n/2 :-
E(k) = J V1 - k2 sin2 8 d6
(24)
(25)
(26)
(27)
is the incomplete elliptic integral of the second kind and
n/2 de
K(k) -J (28)
o vi - k sin2 e
the complete elliptic integral of the first kind. The dependence (26) is shown in Fig. 5 (with solid line) in comparison with the solution based on the parabolic approximation of the cylinder (dashed line).
2. Adhesive contact
We consider adhesive contacts in the approximation used by Johnson, Kendall and Roberts [16]. This approximation assumes infinitely small range of action of adhesive forces. Adhesive contact in the JKR approximation can be thought of as a superposition of the normal contact without adhesion and a pulling contact of a flat-ended punch with a critical stress intensity factor. Denoting the pulling part of the force by - p, we get
P( x):
n(a2 - x2y/2
(29)
P
2Yi2 E
na V na
This corresponds to the critical value of the stress intensity factor at which the boundary of the contact is in equilibrium (for details see [1, Chapter 4] or [17]). For critical value of P1 it follows
P1 =7 2naY12 E *
(31)
For a power-law profile, the compressive part of the force is given by (16):
Po( a ) = E *Cn Z nan. (32)
Thus, the total force is equal to
P = EeCnZnan -V2naY12E*. (33)
In the case of a parabolic profile with n = 2 and Zn = = n/ 2, this force intensity assumes the minimum value
Pmin =- Padh = — (4nRE * Y122)1/3,
(34)
which of course coincides with the adhesive force obtained by Barquins [17]. For an arbitrary power-law profile, the force of adhesion is equal to
Padh =
2n -1
( 2n-1 n Yn2 E *n-1 Y1/(2 n-1)
cn Z n
(35)
(36)
2nn1(2 n-1) In the case of a wedge we have
p = 1 ^12 adh 2 tan 8'
In this case, the adhesion force does not depend on the elastic modulus.
3. Conclusion
Using the Mossakovskii-Jäger presentation of a contact between bodies with curved surfaces as a superposition of indentations with a series of flat-ended punches, we derived the general solution of a contact problem of two-dimensional profiles with compact contact area. This solution closely resembles the MDR solution for three-dimensional contacts but uses the indentation load instead of the indentation depth for parameterization of the indentation process. The generalization of the presented method for tangential contact problems should be straight forward, as the Cattaneo-Mindlin-Ciavarella-Jäger superposition principle (reducing the tangential contact to normal contact) is exactly applicable for line contacts [18]. Jäger indeed already have shown that this generalization is possible [12].
Acknowledgments
The author thanks M. Heß for valuable discussion and for pointing out to the paper of Jäger [12]. The present paper has been partially supported by the program of competitiveness of the Tomsk State University.
References
1. Popov V.L., Heß M. Method of Dimensionality Reduction in Contact Mechanics and Friction. - Berlin: Springer, 2015. - 265 p.
2. Popov V.L., Hess M. Method of dimensionality reduction in contact mechanics and friction: a user's handbook. I. Axially-symmetric contacts // Facta Univer. Mech. Eng. - 2014. - V. 12. - No. 1. - P. 1-14.
3. Galin L.A. Contact Problems in the Theory of Elasticity. - North Carolina State College, 1961. - 233 p.
4. Sneddon I.N. The relation between load and penetration in the axisym-metric Boussinesq problem for a punch of arbitrary profile // Int. J. Eng. Sci. - 1965. - V. 3. - No. 1. - P. 47-57.
5. Popov V.L. Contact Mechanics and Friction. Physical Principles and Applications. - Berlin: Springer, 2017. - 369 p.
6. WillertE., Heß M., Popov V.L. Application ofthe method of dimensionality reduction to contacts under normal and torsional loading // Facta Univer. Mech. Eng. - 2015. - V. 13. - No. 2. - P. 81-90.
7. Heß M. A simple method for solving adhesive and non-adhesive axi-symmetric contact problems of elastically graded materials // Int. J. Eng. Sci. - 2016. - V. 104. - P. 20-33.
8. Hess M., Popov V.L. Method of dimensionality reduction in contact mechanics and friction: a user's handbook. II. Power-law graded materials // Facta Univer. Mech. Eng. - 2016. - V. 14. - No. 3. - P. 251268.
9. Mossakovskii V.I. Compression of elastic bodies under conditions of adhesion (axisymmetric case) // J. Appl. Math. Mech. - 1963. - V 27. -No. 3. - P. 630-643.
10. Jaeger J. Axi-symmetric bodies of equal material in contact under torsion or shift // Arch. Appl. Mech. - 1995. - V. 65. - P. 478-487.
11. Johnson K.L. Contact Mechanics. - Cambridge: University Press, 1987. - 452 p.
12. Jäger J. Half-planes without coupling under contact loading // Arch. Appl. Mech. - 1997. - V. 67. - No. 4. - P. 247-259.
13. Bracewell R. The Fourier Transform and Its Applications. - New York: McGraw-Hill Book Company, 1965. - 381 p.
14. Landau L.D., Lifshitz E.M. Theory of Elasticity. Course of Theoretical Physics. V. 7. - London: Pergamon Press, 1959.
15. Truman C.E., Sackfield A., Hills D.A. Contact mechanics of wedge and cone indenters // Int. J. Mech. Sci. - 1995. - V. 37. - No. 3. -P. 261-275.
16. Johnson K.L., Kendall K., Roberts A.D. Surface energy and the contact of elastic solids // Proc. Roy. Soc. Lond. A. - 1971. - V. 324. - P. 301313.
17. Barquins M. Adherence and rolling kinetics of a rigid cylinder in contact with a natural rubber surface // J. Adhesion. - 1988. - V. 26. -P. 1-12.
18. Ciavarella M. The generalized Cattaneo partial slip plane contact problem. I. Theory // Int. J. Solid. Struct. - 1998. - V. 35. - No. 18. -P. 2349-2362.
Поступила в редакцию 16.04.2017 г.
Сведения об авторе
Valentin L. Popov, Prof., Technische Universität Berlin, Germany; Prof., TPU; Prof., TSU, [email protected]
