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The contact problem for a periodic cluster of microcontacts
Journal of Applied Mathematics and Mechanics 76 (2012) 604–610
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The contact problem for a periodic cluster of microcontacts夽 I.I. Argatov St Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 11 March 2012
a b s t r a c t The three-dimensional contact problem of the linear theory of elasticity is investigated for a system consisting of a large number of small punches, periodically arranged within a limited area on the boundary of an elastic semi-infinite body. The problem is investigated by a method which combines the averaging method and the method of matched asymptotic expansions. It is assumed that the ratios of the diameters of the actual contact areas to the distance between them are small, and each such ratio for neighbouring contact areas is in proportion to the ratio of the dimension of the periodic cell to the diameter of the nominal contact area. The asymptotic form of the doubly periodic contact problem for an elastic halfspace is constructed. An approximate solution, in explicit form, is constructed for circular and elliptic contact areas. In the first case, the results agree with the well-known solution in the literature, obtained by another method. © 2012 Elsevier Ltd. All rights reserved.
The three-dimensional contact problem for an elastic half-space and a large number of punches was investigated for the first time1,2 in the special case of circular punches. The basis of the approach, developed in Refs [1–3], is Galin’s formula4 for the contact pressure under a circular punch when a concentrated force acts on the boundary of a circular half-space (outside the punch). This problem has been called the Galin problem (see Ref. [5], Section 2.1.1). The contact problem in its general formulation has been investigated6 by the method of matched asymptotic expansions,7–9 and expansions have been obtained in explicit form for the contact pressure densities in the case of a system consisting of elliptic punches. There has recently been increased interest in multiple-contact problems in view of the problem of interaction in clusters of microcontacts10–12 and in its analytical description.13,14 The case of a periodic cluster enables important relations to be established in explicit form for the contact pressure on a single contact area. Details of the complete asymptotic analysis have been published in a not easily accessible publication.15 1. Formulation of the contact problem Suppose G is a bounded region in the k2 plane with a smooth boundary ∂G, and l is its characteristic dimension. Suppose is a plane region, bounded by a smooth contour ∂ and lying inside the square
We will put (1.1) i,j
where is a small positive parameter. We will denote by the union of all the areas , contained within the region G. We will assume that the region is obtained from a certain fixed region 1 ⊂ K1 by contraction by a factor of −1 , i.e., (1.2)
夽 Prikl. Mat. Mekh., Vol. 76, No. 5, pp. 725–733, 2012. E-mail address: [email protected] 0021-8928/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2012.11.004
I.I. Argatov / Journal of Applied Mathematics and Mechanics 76 (2012) 604–610
605
Then, from the property of harmonic capacity we have: c = cap(1 ), where cap(1 ) is the harmonic capacity of the set 1 = { : ∈ ¯ 1 , 3 = 0}. Following the Pankovich-Neiber representation (see, for example, Ref. [1]), the contact problem of the linear theory of elasticity on indentation without friction into an elastic half-space of a punch with a flat base, which occupies in plan a set , at a depth ␦0 , reduces to the following mixed boundary-value problem of the theory of harmonic functions
(1.3) The punch pressure on the semi-infinite elastic body is calculated from the formula (1.4) where E is Young’s modulus, is Poisson’s ratio and ∂x3 = ∂/∂x3 . We will denote by Y the set of indices (i, j), for which the point belongs to the region G. The number N of elements of the set Y depends on the value of the parameter , where
and |G| is the area of the region G. Earlier, the linear contact problem for a system of small punches densely packed in a limited area on the surface of an elastic half-space was investigated17 by a method which combined the use of the method of matched asymptotic expansions7,8 and the averaging method;16 it was assumed that the diameters of the contact areas and the distances between neighbouring areas were comparable with one another. The principal terms of the outer and inner expansions for the contact pressure densities were constructed, and the following asymptotic representation was obtained
(1.5) Here
xli
= il,
j x2
= jl are the coordinates of the centre of the contact area
i,j (),
Z1 () is the special non-energy solution of the doui,j
bly periodic mixed boundary-value problem of the theory of harmonic functions for a half-space, and ()(1 , 2 ) are the stretched coordinates, where (for simplicity in the symbol i,j the subscripts i and j are omitted) (1.6) Finally,
p0 (x1 ,
x2 ) is the average density of the contact pressures, defined as the solution of the integral equation
(1.7) The case of a punch with a so-called fine-grained boundary, when the diameters of the contact areas in the extended coordinates, i.e., the diameter of the region is small (of the order of ) compared with the side of the square K1 , was investigated in Refs 18 and 19. It was established that formula (1.5), which has a structure of the product of the average contact pressure density and the so-called20 local contact pressure intensity factor, remains true, but the average contact pressure density p0 (x1 , x2 ) is defined as the solution of the following integral equation, which differs from (1.7)
(1.8) To clarify the mechanical meaning of the factor c , which occurs in relation (1.8), we introduce into consideration the actual capacity ij i,j i,j c = cap( ()) of the contact area (), where, according to the property of the harmonic capacity, (1.9) The factor c can then be written as (1.10) We emphasise that, in view of formula (1.9), c is independent of the parameter . As follows from formula (1.10), the value of c is equal to the ratio of the translational (in the terminology used in Ref [21]) capacity of a single contact area to the fraction of the nominal contact area corresponding to it. Similar problems were investigated earlier by other analytic methods within the framework of the Marchenko–Khruslov theory,22 whence the term “fine-grained boundary” was also borrowed. The contact problems considered arise in a natural way in the mechanics of
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the elastic unconcentrated contact of rough bodies (see, in particular, Refs 23 and 24). Here, in turn, one must formulate and solve periodic contact problems for a semi-infinite elastic body. These problems have been investigated by different methods.25–27 Below we construct an asymptotic solution of the contact problem for an elastic semi-infinite cylinder with a square cross section and with a fixed contact area at the end. 2. The doubly periodic contact problem for an elastic half-space It was established in Ref. [17] that a unique solution of the following homogeneous problem exists
(2.1) which satisfies the periodicity conditions
(2.2) and allows of the following asymptotic behaviour at infinity (2.3) According to the maximum principle for harmonic functions, the following inequalities hold
(2.4) Using Green’s formula we establish the relation
(2.5) where |K1 | = 1 is the area of the square K1 . The constant C1 ( ) from the asymptotic formula, which refines (2.3), (2.6) called17
is the generalized capacity of the contact area. The monotonicity property of the functional C1 ( ) was established in Ref. [17], and the following representation was obtained
(2.7) We put
(2.8) where G() is Green’s function of the doubly periodic problem for a half-space,
(2.9) It can be shown (see Ref. [15]), that for any fixed value of > 0 the double functional series in (2.9) converges absolutely and uniformly in ¯ = K¯ 1 × [0, ]. the parallelepiped The following expansion can be established by the method of separation of variables for the harmonic function G() (2.10) Then the following limit relation holds (2.11)
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We have denoted by s’n the sum (the prime on the summation sign denotes that the indices k and j do not vanish simultaneously)
From formula (2.11) we obtain C0 = 0.620. . . From asymptotic formula (2.10) for the function (2.8) as 3 → ∞ we obtain
(2.12) By obtaining coincidence of the principal terms of the asymptotic form in formulae (2.12) and (2.6), we derive the normalization condition
(2.13) Correspondingly, comparing formulae (2.6) and (2.12), taking condition (2.13) into account, we obtain (2.14) Note that (see Ref. [15])
Since the kernel of the integral in (2.8) has a singularity, defined by formula (2.9), the following equality holds
(2.15) Note that by substituting expression (2.15) into (2.5), we obtain equality (2.13). We will now write the integral equation for determining the density p1 (1 , 2 ), (1 , 2 ) ∈ . Assuming 3 = 0 in expression (2.8) and bearing in mind the homogeneous boundary condition Z1 ( , 0 +) =0 when ∈ , we obtain (2.16) Here
(2.17) Finally, note that normalization condition (2.13) serves to determine the constant A1 . 3. A circular contact area In the case of a circular area of radius a with centre at the origin of coordinates, following the well-known approach,28 we reduce Eq. (2.16) to a Fredholm integral equation of the second kind. Using Galin’s solution of the problem1 (see also Ref. [5], Section 2.1.1 and Section 2.1.3), we represent the solution of Eq. (2.16) in the form
(3.1)
(3.2)
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Hence, using relations (3.1) and (3.2), Eq. (2.16) can be converted to the form
(3.3) It is obvious that, by symmetry, the function p1 (1 , 2 ) is even with respect to both variables. In the case of a small contact area (i.e., when a → 0) from Eq. (3.3) we determine the principal term of the asymptotic
(3.4) and, from normalization condition (2.13), we obtain (3.5) We emphasise that the quantity a , like the coordinates (1 , 2 , 3 ), are dimensionless. In the case when a = A, where A is independent of the parameter , by formulae (2.14) and (3.5) for the generalized capacity of the contact area when → 0, we obtain the asymptotic C1 ( ) ∼ (4A)−1 . To refine asymptotic (3.5) we will apply the method of successive approximations to Eq. (3.3), choosing expression (3.4) as the initial approximation, i.e.,
(3.6) To simplify expression (3.6), assuming
we will expand the kernel K (1 , 2 ; ) in powers of the parameter and, confining ourselves to terms O(2 ), we substitute the result obtained into formula (3.6). After evaluating the integrals we will have
(3.7) We have introduced the notation ((s) is the Riemann zeta function)
(3.8) Finally, applying normalization condition (2.13) to function (3.7), in the limits of accuracy with which we derived formula (3.7), we obtain
(3.9) Correspondingly, using formula (2.14) we also determine the asymptotic of C1 ( ). The solution agrees with that obtained in Ref. [25]. 4. An elliptic contact area Suppose the region is bounded by an ellipse e with semimajor axis a = A, where the value of A is independent of the parameter . Using Aleksandrov’s method29,30 (see also Ref. [5], Sections 2.2.1 and 2.2.3) for the case of small values of the parameter , we construct an approximate solution of integral equation (2.16). For simplicity we will assume that the ellipse ∂ is referred to the principal axes. Using the fact that, within the region , the variables 1 , 2 and 1 , 2 are quantities of O() in modulus, we expand the kernel of the integral operator b in powers of these variables. We substitute the expansion obtained into Eq. (2.16) and we simplify the result using the evenness of the density p1 (1 , 2 ). Finally, confining ourselves to the first non-trivial term, we obtain the following so-called31 “combined” integral equation
(4.1) Here c1 is the constant (3.8).
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609
We emphasise that Eq. (4.1) is approximate. Taking normalization condition (2.13) into account, we rewrite it in the form
(4.2) By Galin’s
theorem1
the solution of Eq. (4.2) can be represented in the form
(4.3) According to the well-known solution of the contact problem for an elliptic of the coefficients of the right-hand side of Eq. (4.2) using the formulae
punch,32,33
the coefficients of (4.3) can be expressed in terms
(4.4) We have introduced the following notation (K(e) and E(e) are the complete elliptic integrals of the first and second kinds)
Direct calculations give (4.5) Here normalization condition (2.13) takes the form
(4.6) Substituting expression (4.5) into Eq. (4.4) we can express the coefficient H0 in terms of A1 , which then, in turn, is found from Eq. (4.6). Thus, up to terms O(3 ) we will have
(4.7) It can be seen that, in the case of a circular region (i.e., when e = 0) the principal term of asymptotic (4.7) is identical with (3.5). We emphasise that formulae (4.1), (4.3) and (4.7) also remain true when the elliptic region is rotated around the coordinate axes. Then, of course, (1 , 2 ) are local coordinates connected with the principal axes of the region . References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Galin LA. Contact Problems of the Theory of Elasticity and Viscoelasticity. Moscow: Nauka; 1980. Gladwell GML, Fabrikant VI. The interaction between a system of circular punches on an elastic half space. Trans ASME J Appl Mech 1982;V9(2):341–4. Goryacheva IG, Dobychin MN. Contact Problems in Tribology. Moscow: Mashinostroyeyeniye; 1988. Galin LA. Contact Problems of the Theory of Elasticity. Moscow: Gostekhizdat; 1953. Argatov II, Dmitriyev NN. Principles of the Theory of Discrete Elastic Contact. Moscow–St Petersburg: Politekhnika; 2003. Van Dyke MD. Perturbation Methods in Fluid Mechanics. New York: Acad. Press; 1964. Il’in AM. Matching of Asymptotic Expansions of the Solutions of Boundary–Value Problems. Moscow: Nauka; 1989. Argatov II, Nazarov SA. The method of matched expansions for problems with small contact areas. In: Contact Interaction Mechanics. Moscow: Fizmatlit; 2001. Argatov II. The interaction of punches in an elastic half-space. Izv Ross Akad Nauk MTT 1999;4:56–63. Boyer L. Contact resistance calculations: generalizations of Greenwood’s formula including interface films. IEEE Trans Compon Packag Technol 2001;24(1):50–8. Borovsky B, Krim J, Syed Asif SA, Wahl KJ. Measuring nanomechanical properties of a dynamic contact using an indenter probe and quartz crystal microbalance. J Appl Phys 2001;90(12):6391–6. 12. Ciavarella M, Greenwood JA, Paggi M. Inclusion of “interaction” in the Greenwood and Williamson contact theory. Wear 2008;265(5–6):729–34. 13. Agratov I, Sevostianov I. On relatios between geometries of microcontact clusters and their overall properties. Int J Eng Sci 2009;47:959–73. 14. Argatov I, Petrov Yu. Electrical contact resistance and dynamic contact stiffness for a cluster of microcontacts: Cross-property connection in low frequency range. Phil Mag 2012;92(14):1764–76.
610 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
I.I. Argatov / Journal of Applied Mathematics and Mechanics 76 (2012) 604–610 Argatov II. Asymptotic Models of Elastic Contact. St Petersburg: Nauka; 2005. Bakhvalov NS, Panasenko GP. Averaging of Processes in Periodic Media. Dordrecht: Kluwet; 1989. Argatov II, Mel’nyk TA. Homogenization of a contact problem for a system of densely situated punches. Europ J Mech A Solids 2001;20(1):91–8. Argatov II. The indentation of a punch with a fine-grained boundary into an elastic base. Zh Prikl Mekh Tekh Fiz 2004;45(5):176–86. Argatov II. Electrical contact resistance, thermal contact conductance and elastic incremental stiffness for a cluster of microcontacts: Asymptotic modelling. Quart J Mech Appl Math 2011;64(1):1–24. Fischer FD, Daves W, Werner EA. On the temperature in the wheel-rail rolling contact. Fatigue Fract Engng Mater Struct 2003;26(10):999–1006. Argatov II. The capacity characteristics of a punch with a flat smooth base. Izv Vuzov Stroitel’svo 2000;4:26–32. Marchenko VA. Khruslov yeYa. Boundary–Value Problems in Regions with a Fine-Grained Boundary. Kiev: Naukova Dumka; 1974. Bhushan B. Contact mechanics of rough surfaces in tribology: multiple asperity contact. Tribology Letters 1998;4(1):1–35. Goryacheva IG. The Mechanics of Functional Interaction. Moscow: Nauka; 2001. Panasyuk VV, Andreikiv AYe. A problem of the limit equilibrium for an unbounded brittle body with a crack. Prikl Mekh 1970;6(6):25–9. Johnson KL, Greenwood JA, Higginson JG. The contact of elastic regular wavy surfaces. Intem J Mech Sci 1987;27(6):383–96. Goryacheva IG. The periodic contact problem for an elastic half-space. Prikl Mat Mekh 1998;62(6):1036–44. Andreikiv AYe, Panasyuk VV. The indentation of a system of circular punches into an elastic half-space. Dop Akad Nauk UkrSSR Ser A 1971;6:534–6. Aleksandrov VM. Some contact problems for the elastic layer. J Appl Math Mekh 1963;27(4):1164–74. Aleksandrov VM, Shmatkova AA. The indentation of a parabolic punch into an elastic layer and of two parabolic punches into an elastic half–space. Izv Ross Akad Nauk MTT 1998;4:149–55. Argatov II. The asymptotic solution of the contact problem for a three-dimensional elastic solid of finite dimensions. J Appl Math Mekh 1999;63(6):937–42. Lurie AI. Three-Dimensional Problems of the Theory of Elasticity. New York: Interscience; 1964. Vorovich II, Aleksandrov VM, Babeshko VA. Non-classical Mixed Problems of Elasticity Theory. Moscow: Nauka; 1974.
Translated by R.C.G.
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The contact problem for a periodic cluster of microcontacts夽 I.I. Argatov St Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 11 March 2012
a b s t r a c t The three-dimensional contact problem of the linear theory of elasticity is investigated for a system consisting of a large number of small punches, periodically arranged within a limited area on the boundary of an elastic semi-infinite body. The problem is investigated by a method which combines the averaging method and the method of matched asymptotic expansions. It is assumed that the ratios of the diameters of the actual contact areas to the distance between them are small, and each such ratio for neighbouring contact areas is in proportion to the ratio of the dimension of the periodic cell to the diameter of the nominal contact area. The asymptotic form of the doubly periodic contact problem for an elastic halfspace is constructed. An approximate solution, in explicit form, is constructed for circular and elliptic contact areas. In the first case, the results agree with the well-known solution in the literature, obtained by another method. © 2012 Elsevier Ltd. All rights reserved.
The three-dimensional contact problem for an elastic half-space and a large number of punches was investigated for the first time1,2 in the special case of circular punches. The basis of the approach, developed in Refs [1–3], is Galin’s formula4 for the contact pressure under a circular punch when a concentrated force acts on the boundary of a circular half-space (outside the punch). This problem has been called the Galin problem (see Ref. [5], Section 2.1.1). The contact problem in its general formulation has been investigated6 by the method of matched asymptotic expansions,7–9 and expansions have been obtained in explicit form for the contact pressure densities in the case of a system consisting of elliptic punches. There has recently been increased interest in multiple-contact problems in view of the problem of interaction in clusters of microcontacts10–12 and in its analytical description.13,14 The case of a periodic cluster enables important relations to be established in explicit form for the contact pressure on a single contact area. Details of the complete asymptotic analysis have been published in a not easily accessible publication.15 1. Formulation of the contact problem Suppose G is a bounded region in the k2 plane with a smooth boundary ∂G, and l is its characteristic dimension. Suppose is a plane region, bounded by a smooth contour ∂ and lying inside the square
We will put (1.1) i,j
where is a small positive parameter. We will denote by the union of all the areas , contained within the region G. We will assume that the region is obtained from a certain fixed region 1 ⊂ K1 by contraction by a factor of −1 , i.e., (1.2)
夽 Prikl. Mat. Mekh., Vol. 76, No. 5, pp. 725–733, 2012. E-mail address: [email protected] 0021-8928/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2012.11.004
I.I. Argatov / Journal of Applied Mathematics and Mechanics 76 (2012) 604–610
605
Then, from the property of harmonic capacity we have: c = cap(1 ), where cap(1 ) is the harmonic capacity of the set 1 = { : ∈ ¯ 1 , 3 = 0}. Following the Pankovich-Neiber representation (see, for example, Ref. [1]), the contact problem of the linear theory of elasticity on indentation without friction into an elastic half-space of a punch with a flat base, which occupies in plan a set , at a depth ␦0 , reduces to the following mixed boundary-value problem of the theory of harmonic functions
(1.3) The punch pressure on the semi-infinite elastic body is calculated from the formula (1.4) where E is Young’s modulus, is Poisson’s ratio and ∂x3 = ∂/∂x3 . We will denote by Y the set of indices (i, j), for which the point belongs to the region G. The number N of elements of the set Y depends on the value of the parameter , where
and |G| is the area of the region G. Earlier, the linear contact problem for a system of small punches densely packed in a limited area on the surface of an elastic half-space was investigated17 by a method which combined the use of the method of matched asymptotic expansions7,8 and the averaging method;16 it was assumed that the diameters of the contact areas and the distances between neighbouring areas were comparable with one another. The principal terms of the outer and inner expansions for the contact pressure densities were constructed, and the following asymptotic representation was obtained
(1.5) Here
xli
= il,
j x2
= jl are the coordinates of the centre of the contact area
i,j (),
Z1 () is the special non-energy solution of the doui,j
bly periodic mixed boundary-value problem of the theory of harmonic functions for a half-space, and ()(1 , 2 ) are the stretched coordinates, where (for simplicity in the symbol i,j the subscripts i and j are omitted) (1.6) Finally,
p0 (x1 ,
x2 ) is the average density of the contact pressures, defined as the solution of the integral equation
(1.7) The case of a punch with a so-called fine-grained boundary, when the diameters of the contact areas in the extended coordinates, i.e., the diameter of the region is small (of the order of ) compared with the side of the square K1 , was investigated in Refs 18 and 19. It was established that formula (1.5), which has a structure of the product of the average contact pressure density and the so-called20 local contact pressure intensity factor, remains true, but the average contact pressure density p0 (x1 , x2 ) is defined as the solution of the following integral equation, which differs from (1.7)
(1.8) To clarify the mechanical meaning of the factor c , which occurs in relation (1.8), we introduce into consideration the actual capacity ij i,j i,j c = cap( ()) of the contact area (), where, according to the property of the harmonic capacity, (1.9) The factor c can then be written as (1.10) We emphasise that, in view of formula (1.9), c is independent of the parameter . As follows from formula (1.10), the value of c is equal to the ratio of the translational (in the terminology used in Ref [21]) capacity of a single contact area to the fraction of the nominal contact area corresponding to it. Similar problems were investigated earlier by other analytic methods within the framework of the Marchenko–Khruslov theory,22 whence the term “fine-grained boundary” was also borrowed. The contact problems considered arise in a natural way in the mechanics of
606
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the elastic unconcentrated contact of rough bodies (see, in particular, Refs 23 and 24). Here, in turn, one must formulate and solve periodic contact problems for a semi-infinite elastic body. These problems have been investigated by different methods.25–27 Below we construct an asymptotic solution of the contact problem for an elastic semi-infinite cylinder with a square cross section and with a fixed contact area at the end. 2. The doubly periodic contact problem for an elastic half-space It was established in Ref. [17] that a unique solution of the following homogeneous problem exists
(2.1) which satisfies the periodicity conditions
(2.2) and allows of the following asymptotic behaviour at infinity (2.3) According to the maximum principle for harmonic functions, the following inequalities hold
(2.4) Using Green’s formula we establish the relation
(2.5) where |K1 | = 1 is the area of the square K1 . The constant C1 ( ) from the asymptotic formula, which refines (2.3), (2.6) called17
is the generalized capacity of the contact area. The monotonicity property of the functional C1 ( ) was established in Ref. [17], and the following representation was obtained
(2.7) We put
(2.8) where G() is Green’s function of the doubly periodic problem for a half-space,
(2.9) It can be shown (see Ref. [15]), that for any fixed value of > 0 the double functional series in (2.9) converges absolutely and uniformly in ¯ = K¯ 1 × [0, ]. the parallelepiped The following expansion can be established by the method of separation of variables for the harmonic function G() (2.10) Then the following limit relation holds (2.11)
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607
We have denoted by s’n the sum (the prime on the summation sign denotes that the indices k and j do not vanish simultaneously)
From formula (2.11) we obtain C0 = 0.620. . . From asymptotic formula (2.10) for the function (2.8) as 3 → ∞ we obtain
(2.12) By obtaining coincidence of the principal terms of the asymptotic form in formulae (2.12) and (2.6), we derive the normalization condition
(2.13) Correspondingly, comparing formulae (2.6) and (2.12), taking condition (2.13) into account, we obtain (2.14) Note that (see Ref. [15])
Since the kernel of the integral in (2.8) has a singularity, defined by formula (2.9), the following equality holds
(2.15) Note that by substituting expression (2.15) into (2.5), we obtain equality (2.13). We will now write the integral equation for determining the density p1 (1 , 2 ), (1 , 2 ) ∈ . Assuming 3 = 0 in expression (2.8) and bearing in mind the homogeneous boundary condition Z1 ( , 0 +) =0 when ∈ , we obtain (2.16) Here
(2.17) Finally, note that normalization condition (2.13) serves to determine the constant A1 . 3. A circular contact area In the case of a circular area of radius a with centre at the origin of coordinates, following the well-known approach,28 we reduce Eq. (2.16) to a Fredholm integral equation of the second kind. Using Galin’s solution of the problem1 (see also Ref. [5], Section 2.1.1 and Section 2.1.3), we represent the solution of Eq. (2.16) in the form
(3.1)
(3.2)
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Hence, using relations (3.1) and (3.2), Eq. (2.16) can be converted to the form
(3.3) It is obvious that, by symmetry, the function p1 (1 , 2 ) is even with respect to both variables. In the case of a small contact area (i.e., when a → 0) from Eq. (3.3) we determine the principal term of the asymptotic
(3.4) and, from normalization condition (2.13), we obtain (3.5) We emphasise that the quantity a , like the coordinates (1 , 2 , 3 ), are dimensionless. In the case when a = A, where A is independent of the parameter , by formulae (2.14) and (3.5) for the generalized capacity of the contact area when → 0, we obtain the asymptotic C1 ( ) ∼ (4A)−1 . To refine asymptotic (3.5) we will apply the method of successive approximations to Eq. (3.3), choosing expression (3.4) as the initial approximation, i.e.,
(3.6) To simplify expression (3.6), assuming
we will expand the kernel K (1 , 2 ; ) in powers of the parameter and, confining ourselves to terms O(2 ), we substitute the result obtained into formula (3.6). After evaluating the integrals we will have
(3.7) We have introduced the notation ((s) is the Riemann zeta function)
(3.8) Finally, applying normalization condition (2.13) to function (3.7), in the limits of accuracy with which we derived formula (3.7), we obtain
(3.9) Correspondingly, using formula (2.14) we also determine the asymptotic of C1 ( ). The solution agrees with that obtained in Ref. [25]. 4. An elliptic contact area Suppose the region is bounded by an ellipse e with semimajor axis a = A, where the value of A is independent of the parameter . Using Aleksandrov’s method29,30 (see also Ref. [5], Sections 2.2.1 and 2.2.3) for the case of small values of the parameter , we construct an approximate solution of integral equation (2.16). For simplicity we will assume that the ellipse ∂ is referred to the principal axes. Using the fact that, within the region , the variables 1 , 2 and 1 , 2 are quantities of O() in modulus, we expand the kernel of the integral operator b in powers of these variables. We substitute the expansion obtained into Eq. (2.16) and we simplify the result using the evenness of the density p1 (1 , 2 ). Finally, confining ourselves to the first non-trivial term, we obtain the following so-called31 “combined” integral equation
(4.1) Here c1 is the constant (3.8).
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609
We emphasise that Eq. (4.1) is approximate. Taking normalization condition (2.13) into account, we rewrite it in the form
(4.2) By Galin’s
theorem1
the solution of Eq. (4.2) can be represented in the form
(4.3) According to the well-known solution of the contact problem for an elliptic of the coefficients of the right-hand side of Eq. (4.2) using the formulae
punch,32,33
the coefficients of (4.3) can be expressed in terms
(4.4) We have introduced the following notation (K(e) and E(e) are the complete elliptic integrals of the first and second kinds)
Direct calculations give (4.5) Here normalization condition (2.13) takes the form
(4.6) Substituting expression (4.5) into Eq. (4.4) we can express the coefficient H0 in terms of A1 , which then, in turn, is found from Eq. (4.6). Thus, up to terms O(3 ) we will have
(4.7) It can be seen that, in the case of a circular region (i.e., when e = 0) the principal term of asymptotic (4.7) is identical with (3.5). We emphasise that formulae (4.1), (4.3) and (4.7) also remain true when the elliptic region is rotated around the coordinate axes. Then, of course, (1 , 2 ) are local coordinates connected with the principal axes of the region . References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
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Translated by R.C.G.