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The periodic contact problem of the plane theory of elasticity. Taking friction, wear and adhesion into account
Journal of Applied Mathematics and Mechanics 77 (2013) 245–255
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The periodic contact problem of the plane theory of elasticity. Taking friction, wear and adhesion into account夽 I.A. Soldatenkov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 29 May 2011
a b s t r a c t A solution of the plane problem of the contact interaction of a periodic system of convex punches with an elastic half-plane is given for two forms of boundary conditions: 1) sliding of the punches when there is friction and wear, and 2) the indentation of the punches when there is adhesion. The problem is reduced to a canonical singular integral equation on the arc of a circle in the complex plane. The solution of this equation is expressed in terms of simple algebraic functions of a complex variable, which considerably simplifies its analysis. Asymptotic expressions are obtained for the solution of the problem in the case when the size of the contact area is small compared with the distance between the punches. © 2013 Elsevier Ltd. All rights reserved.
A solution of the periodic contact problem of the theory of elasticity was given for the first time for a system of punches with plane bases, in contact with an elastic half-plane when there is no friction.1 The case of periodic contact of convex punches was considered later,2,3 and friction and sliding on the contact was also taken into account in Ref. 4. A solution of the periodic contact problem with adhesion for specified boundary displacements and a fixed contact area is also known.5 There is a fairly complete description of the different formulations of plane periodic contact problems of the theory of elasticity and methods of solving them in Ref. 6. In this paper we consider periodic contact problems for an elastic half-plane, which differ from those mentioned above in the fact that the adhesion component of the friction and wear of the half-plane are taken into account, as well as the increase in the contact area when punches are indented with adhesion.
1. Formulation of the problem and fundamental equations Consider an elastic half-plane, the boundary of which is in contact with an infinite system of similar absolutely rigid convex punches, spaced a distance L from one another (Fig. 1). We will assume that each punch is loaded in the same way by an external shear force P1 and an external normal force P2 . We will choose an arbitrary punch and connect with it a system of coordinates Oxy, the x axis of which is directed parallel to the undeformed boundary of the half-plane, while the y axis is perpendicular to it. We will assign the number 0 to the chosen punch, and we will assign the numbers n = −1, −2,... (n = 1, 2,...) to the punches situated to the left (to the right) of this punch. All the punches are of the same form, and we will describe the form by the equation y = g(x), where g(x) is a specified function and g(0) = 0. We will denote the shearing contact stress and the contact pressure by q1 = xy |y=0 and q2 = −y |y=0 ≡ p, and the displacements of the upper boundary of the half-plane along the x and y axes by u and v respectively. We will denote the contact areas of the punches with the half-plane by the sections [an , bn ] of the x axis (n = 0, ±1, ±2,. . .), where, to simplify the writing of the later calculations, we will additionally put a = a0 and b = b0 . By virtue of the above assumptions, the stress-strain state of the half-plane is periodic, so that
(1.1)
夽 Prikl. Mat. Mekh., Vol. 77, No. 2, pp. 337–351, 2013. E-mail address: [email protected] 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.07.017
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Figure 1.
where 2l = a + b is the length of the contact area, 2l < L. This property enables us henceforth to confine ourselves to the section [−a, b] to describe the properties of the quantities considered and the relations between them. We will consider two forms of boundary conditions, corresponding to Problem 1 on the uniform sliding of the punches in the direction of the x axis with velocity V when there is friction and wear of the half-plane, and Problem 2 on the gradual penetration of the punches into the half-plane when there is adhesion. In Problem 1 we assume a specified and constant load P2 , whereas in Problem 2 the loads P1 and P2 increase monotonically, being connected by a loading law of the form (1.2) Assuming that the friction of the punch along the half-plane is described by Coulomb’s law, and the increase in the wear W of the half-plane with time t obeys a linear wear law dW/dt = cW Vp, we will represent the boundary conditions in the form (1.3)
(1.4) Here is the friction coefficient, 0 is the adhesive component of the friction, cW is the wear-resistance parameter of the material of the half-plane and (x) is the tangential displacement within the contact area, where, by virtue of the choice of the system of coordinates, (0) = 0. In Problem 2 the monotonically increasing dimension a of the contact area is used to write the functions as a time argument, which takes into account the change in the deformation conditions of the half-plane as the punches penetrate. The fact that the function (x) is independent of a is explained by the fact that points of the boundary of the half-plane that come into contact with the punch do not undergo any displacements with respect to the punch – the condition of complete adhesion of the contacting bodies. Henceforth we will assume that, for each value of the dimension a of the contact area, the distributions of the contact stresses q1,2 (x) satisfy the Hölder condition, the derivative g’(x) is a smooth function, while the derivative ’(x) is a piecewise-smooth function with a possible discontinuity at the point x = 0:
(1.5) To determine the relation between the displacements of the boundary of the half-plane and the contact stresses we will use the relations9
(1.6) where E is Young’s modulus and is Poisson’s ratio. Following Shtayerman,3 we will bear in mind property (1.1) of the periodicity of the contact stresses q1 , 2 (x) and of the corresponding contact areas and we use the representation
(1.7)
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Figure 2.
The last equation is obtained by changing the order of summation and integration, in view of the uniform convergence of the corresponding functional series, and using the tabulated expression10 for this series. Representation (1.7) enables us, for an arbitrarily specified contact area [−a, b], to obtain from relation (1.6) a system of integral equations with a Hilbert kernel
(1.8) Equations (1.8) must be supplemented by the punch equilibrium conditions, which, assuming its shape to be mildly sloping, |g (x)| 1, have the form
(1.9) Equations (1.8) and (1.9), together with boundary conditions (1.3) or (1.4), form a system of equations for finding the contact stresses q1 , 2 and the dimensions a, b of the contact area in the problems formulated above. In Problem 2 we use in addition, loading law (1.2) and it is also required to obtain the tangential displacement (x) within the contact area. Note that, in Problem 1, the wear and shearing load P1 are calculated from the formulae
where the first formula is obtained by integrating the wear law dW/dt = cW Vp,7 while the second is obtained by integrating the first boundary condition of (1.3), taking equilibrium conditions (1.9) into account. 2. General solutions We will use the following quantities in further calculations (2.1) the first of which represents the relative size of the contact area, while the second represents its degree of asymmetry. To obtain a general solution of Eqs (1.8), we will change in them from integrals with a Hilbert kernel to Cauchy-type integrals, using the well-known method.11 Thus, we will introduce, instead of x and , new complex variables s0 and s using the formulae
(2.2) so that
Moreover, it can be shown that for replacement (2.2), the section [−a, b] of the real x axis changes into the contour , situated in the complex plane and representing the arc of a circle of radius 1, which intersects the real axis at the points s1 = −sin and s2 = sin, and intersects the imaginary axis at the point –i + y0 (Fig. 2). When changes from −a to b the corresponding point s is displaced along the contour from s1 to s2 . Henceforth we will assume that the contour does not include the ends s1 and s2 .
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When these results are taken into account, and also equilibrium conditions (1.9), replacement (2.2) enables us to give Eqs (1.8) the required form
(2.3) (2.4) Replacement of the variable (2.2) in equilibrium conditions (1.9) gives the equalities
(2.5) We will further put
(2.6) Problem 1. Taking into account the definition of the functions 1 ,2 (s0 ) and 1 ,2 (s0 ) given above, we conclude that replacement (2.2) gives boundary conditions (1.3) the form (2.7) The first equality of (2.7) provides the possibility of eliminating from consideration the function 1 (s0 ), corresponding to the shear contact stress q1 (x), and enables us to use only the second equation of (2.3) to solve the problem with sliding. We will write this equation in canonical form11
(2.8) the coefficients and right-hand side of which are found using equalities (2.7), so that
Taking conditions (1.5) into account, we will seek a solution of Eq. (2.8) in the class of bounded functions. This solution can be represented using the operator K∗ in the form11
(2.9) with the necessary and sufficient condition
(2.10) where
(2.11) We will henceforth use the following properties of the function Z(s)
(2.12) which can be established if we close the line by the section [s1 , s2 ] of the real axis (Fig. 2) and apply Cauchy’s integral theorem12 to the closed contour obtained, and also use well-known tabulated integrals.10
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The first integral of (2.12) enables us, when writing solution (2.9), to drop the additive term 0 = iP2 /L in the function (s) and to represent this solution in the form of two expressions
(2.13) while the second integral of (2.12) enables us to represent condition (2.10) in the form of two equations
(2.14) Moreover, for Problem 1 we have the second equilibrium condition (2.5), which relates the function 2 (s0 ) with the specified load P2 . However, this condition is equivalent to the second equality of (2.14), which can be shown if we substitute the first expression of (2.13) for 2 (s0 ) into the second equality of (2.5) and, in the repeated integral obtained, change the order of integration, using the Poincare -Beltran formula.11 Hence, satisfaction of condition (2.10) ensures the correctness of second equilibrium condition (2.5), and hence equalities (2.13) and (2.14) are sufficient to construct a solution of the problem with sliding. In fact, the quantities and r, present in Eqs (2.14), are defined in terms of the dimensions a and b of the contact region by relations (2.1). Consequently, these equations enable us to obtain the quantities a and b, after which, using one of expressions (2.13), we can calculate the contact pressure 2 (s0 ) = p(x). A similar algorithm was used when solving Problem 1 in the case of a single punch.7 Note that the adhesion component of the friction 0 and the load P2 only occur in Eqs (2.14), affecting the dimensions a and b of the contact area, whereas the diagram of the contact pressure 2 (s0 ) = p(x) itself, according to expressions (2.13), is independent of 0 and P2 . When there is no friction and wear ( = 0) one can obtain, using the second equation of (2.14), explicit relations between the dimension a = l of the contact area and the load P2 . For example, for a parabolic punch g(x) = kx2 one obtains the well-known formula13
(2.15) in which a0 is the Hertz dimension of the contact area for a single parabolic
punch.9,14
For a sinusoidal punch
(2.16) one obtains another well-known
formula3
(2.17) Problem 2.
Using replacement (2.2), we introduce into consideration the complex-valued function
Then, taking into account definition (2.4), boundary conditions (1.4) of the problem with adhesion can take the following form (2.18) method.14,15
To simplify the calculations, we will reduce system (2.3) to a single equation, following the well-known To do this we multiply the first equation of (2.3) by the imaginary unit and add it to the second equation of (2.3). As a result, bearing in mind Eq. (2.18), we obtain the equation
(2.19) with respect to the complex-valued contact stress (s,a) = (s,a) + i2 (s,a). Here
The bounded solution of Eq. (2.19) is determined using the operator K∗ of the form (2.9):11 (2.20) with the necessary and sufficient condition (2.10). According to definition (2.6) of the quantity the function Z(s, a) and the coefficients D1 , A* and B* have the previous expressions (2.11) with the exception of the fact that here
(2.21) The presence of the argument a in the function Z(s, a) is due to the fact that the position of the ends s1 and s2 of the line depends on the value of a in Problem 2.
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For the function Z(s, a) considered, properties (2.12) remain in force, which, as also for Problem 1, enable solution (2.20) and the corresponding condition (2.10) to be converted, by representing them in the form of equalities
(2.22)
(2.23) We must add to these equalities equilibrium condition (2.5), which, taking the definitions given above for (s0 , a) and P(a) into account, can be expressed using a single relation:
(2.24) In turn, if we substitute the first expression of (2.22) for the function (s0 , a) into the last equality and change the order of integration in the repeated integral obtained, using the Poincare–Beltran formula, equilibrium condition (2.24) can be written as follows:
(2.25) Eliminating the complex load P from Eq. (2.23) using condition (2.25), we obtain the equation (2.26) which will be used below instead of (2.23). Equations (2.25) and (2.26), being additional to loading law (1.2), enable us, for a specified dimension a of the contact area, to obtain its second dimension b and the tangential displacement (x) within the contact area. After this, we can calculate the contact stresses 1,2 (s0 ) = q1,2 (x) using one of the expressions in (2.22). A similar algorithm was used to solve Problem 2 in the case of a single punch in Ref. 8. The equalities obtained above, which give solutions of Problems 1 and 2, can be written in the original variables x and by substituting expressions (2.2) into them. In order to x represent the equalities converted in this way in a single form for both problems, we introduce the following function
(2.27) Substituting (2.2) into expressions (2.13) and (2.22) and taking definitions (2.6), (2.11) and (2.27) into account, we obtain
(2.28) Here
(2.29) The parameter is defined by relations (2.11) and (2.21). The integrals 0 , 1 (a) in Eqs (2.14), (2.25), and (2.26) are defined by expressions (2.14), (2.23) and (2.25) and, as a result of substituting (2.2), take the form
(2.30) Note that, for the problem with sliding, the argument a in Eqs (2.28)–(2.30) is used formally.
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3. Asymptotic analysis Below we will consider the asymptotic of the solutions of both problems for small values of . By virtue of definition (2.1) of , this case occurs when the dimension 2l of the contact area is small compared with the distance L between punches. Using Taylor’s formula for the elementary functions present in definition (2.29) of the function R(x, a), we can establish the existence of a quantity * such that when ∈ (0, * )
(3.1) and also
(3.2) where
The last terms in Eqs (3.1) and (3.2) are remainder terms of the expansions in the small parameter , where the functions 1 (x; ) and 2 (,x;) are continuous in , x ∈ [−a,b] × [−a,b] and bounded in the set
Substituting expressions (3.1) and (3.2) into formulae (2.28) and (2.30), with the condition 1, we obtain the following asymptotic expressions
(3.3)
(3.4) where
(3.5)
(3.6) Here the existence of the quantities wm and w’m is ensured by properties (1.5) of functions (x) and g(x). Problem 1. We will consider the case of parabolic punches, g(x) = kx2 , and we will assume that the condition 1 is satisfied due to the increase in the distance L between the punches for a specified load P2 (rarefied contact), i.e., L−1 = O(). For parabolic punches, by relations (2.27) and (3.6), we have
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Hence, we can assume that, when the load P2 on the punch remains unchanged and the distance L between the punches increases without limit, the quantities wm and w’m remain bounded, i.e.
Taking the above into account, and also the fact that, for parabolic punches, the integrals Jn , defined by formula (3.5), are taken in explicit form,10 we can substitute expressions (3.4) for 0 into the first two equalities of (2.14) and reduce them to algebraic equations
(3.7) with respect to the unknown l and r¯ ≡ r/l, which are uniquely related to the dimensions a and b of the contact area. Note that, in these equations, the quantity is expressed in terms of l by equality (2.1). The solution of Eqs (3.7) can be represented in the form
(3.8)
(3.9) The quantity a0 is defined by the second formula of (2.15) and is used as the load parameter, while ls and r¯ s denote the values of the quantities l and r¯ for the problem with a single punch when there is friction and wear:7
(3.10) The presence of the second terms in Eqs (3.8) and (3.9) is due to the mutual interaction of the punches, situated at a distance L from one another. If we introduce into consideration the quantity = l − ls – a correction to the value of ls , due to the mutual interaction of the punches, we can obtain the following expression from equality (3.8)
(3.11) which indicates that the correction takes negative values, i.e., the mutual interaction of the punches reduces the contact area. This conclusion agrees with the results of other investigations of multiple contact.16 The second term in Eq. (3.9) and, consequently, the shift of the contact area due to the mutual influence of the punches, can have different signs, depending on the parameters of the problem and the value of the load P2 . Remark 1. When there is no friction and wear ( = 0) for a system of parabolic punches or a sinusoidal punch, the asymptotic expressions for the dimension a of the contact area may be obtained directly from formulae (2.15) or (2.17). As might have been expected, when 1 formula (2.15) takes the form of equality (3.8), if we replace l and ls by a and a0 in it. However, the expansion of expression (2.17) in terms of has a form which differs from (3.8), namely,
(3.12) The positive value of the second term in this equation confirms a well-known result: the dimension a of the contact area for a sinusoidal punch exceeds the Hertz dimension a0 .14,17 Nevertheless, the correction = a–as for a sinusoidal punch, as previously, is negative and is calculated from formula (3.11). In order to show this, it is sufficient, in the well-known relation18
to take as the form g(x) a single bulge of sinusoid (2.16), expand the function sin(2x/L) in a power series and put a/L 1,a = as . As a result, we arrive at the equality
the subtraction of which from (3.12) leads to formula (3.11). Hence, and also in the case of a sinusoidal punch, the mutual influence of its bulges reduces the contact area in the same way as in the case of a system of parabolic punches.
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Remark 2.
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Letting P2 → 0 in formula (3.10), we obtain
In other words, for small values of the load P2 the contact area [−a, b] may be shifted as a whole into the region of negative values of the x coordinate as a result of the action of the adhesion component of the friction. The distribution of the contact pressure p(x) when « 1 is found using the general asymptotic expression (3.3). According to our assumptions, in case of sliding of parabolic punches, we have the equality
As a result, expression (3.3) takes the form
The expression obtained for p(x), together with equalities (3.8) and (3.9), enable us to calculate the deformation component Fdef of the sliding resistance force on each parabolic punch.19 The result of such calculations is the asymptotic formula
which indicates that, for a fixed load P2 , the interaction between the punches leads to an increase in the component Fdef . Problem 2.
We will consider the symmetrical case of the indentation with adhesion of parabolic punches into a half-plane
and we will investigate the initial stage of this process (a slightly loaded contact), assuming that the dimension a of the contact area is limited to a certain value a* L. Then
The condition P1 (a) ≡ 0 denotes that loading law (1.2) with the function N(P2 ) ≡ 0 is used, and this law is automatically satisfied. Henceforth, instead of we will use the parameter , defined by formulae (2.21). We will also introduce into consideration the function (x) = ’(x), which is even in view of the symmetry of the problem. Taking the above assumptions into account, we substitute expressions (3.4) for 0 (a) and 1 (a) into equalities (2.25) and (2.26) and, after reduction, we obtain
(3.13)
(3.14) where
Note that in the symmetrical case considered, equality (2.25) gives also the expression P1 (a) = 0 + O(4 ), which agrees with the above assumption that there is no shear load on the punches.
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In the case of definitions (2.27) and (3.6) the following estimates hold:
(3.15) 5 .
The first of these denotes that the residual terms in Eqs (3.13) and (3.14) are of the order of The existence of the quantity ’* is ensured by property (1.5) of the derivative ’(x) = (x), while to derive the first estimate of (3.15), the equality (0) = 0 is used, which is established by taking the limit as a → 0 in relation (3.14), taking into account the following properties of the integrals introduced above:
We will obtain an asymptotic expression for the function (x) when « 1, using Eq. (3.14). To do this we will use the following representation, obtained using Taylor’s formula:
(3.16) (4) x ∈ C[0,
imposing the constraint a* ], additional to (1.5), and bearing in mind the equality (0) = 0 established above. The coefficients An , present in Eq. (3.16), are related in a known way to the derivatives of the function (x) and remain to be determined. The remainder term in formula (3.16) has the form20
Taking this estimate into account, we substitute expression (3.16) for (x) into Eq. (3.14) and, after reduction, we obtain the equality
Equating terms with like powers of an here, we can obtain a system of recurrent equations in the unknown coefficients An . The solution of this system has the form
and, taking into account the property of evenness of the function (x), we can obtain, using (3.16), the required expression
(3.17) in which, as can be established, ␦1,3 > 0 when v ∈ [0, 1/2]. Note that expression (3.17) satisfies the smoothness conditions (1.5), and in the limiting case as L → ∞, as might have been expected, takes the form corresponding to the problem with a single punch.21 Substituting the expression obtained for (x) into (3.13) we obtain the asymptotic dependence of the load P2 on the dimension a of the contact area in the following form
(3.18) where
The positive value of the second term on the right-hand side of (3.18) denotes that the mutual influence of the punches leads to a reduction in the contact area, as also for Problem 1 considered above. The distributions q1 , 2 (x, a) of the contact stresses in the case considered of parabolic punches are found from the general asymptotic expression (3.3) using expression (3.17) for the function ’(x) and have the form
(3.19)
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Here the values of q1 , 2 (x, a) for negative x are obtained from the symmetry properties of the problem:
Note that expression (3.19) becomes the well-known solution for a single punch,22 when L → ∞ and, correspondingly, when A3 → 0. Acknowledgement This research was supported financially by the Russian Foundation for Basic Research (11-01-00650, 12-08-90434, 11-01-90419). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Sadowsky M. The two-dimensional problem of elasticity theory. ZAMM 1928;8(2):107–21. Westergaard HM. Bearing pressures and cracks. Trans ASME J Appl Mech 1939;6:49–53. Shtayerman IYa. The Contact Problem of Elasticity Theory. Moscow, Leningrad: Gostekhizdat; 1949. Galin LA. Mixed problems of the theory of elasticity for a half-plane. Dokl Akad Nauk SSSR 1943;39:88–93. Kuznetsov EA. Periodic fundamental mixed problem of elastic theory for a half-plane. Soviet Appl Mech 1976;12:942–8. Block JM, Keer LM. Periodic contact problem in plane elasticity. J Mech Material and struct 2008;3:1207–37. Soldatenkov IA. The wear of a half-plane by a disc counterbody. Izv Akad Nauk MTT 1989;6:107–10. Soldatenkov IA. Formulation and solution of the problem of the indentation with adhesion of a punch into an elastic half-plane. Izv Ross Akad Nauk MTT 2000;4:39–52. Galin LA. Contact Problems of the Theory of Elasticity and Viscoelasticity. Moscow: Nauka; 1980. Prudnikov AP, Brychkov YuA, Marichev OI. Integrals and Series. Elementary Functions. New York: Gordon and Breach; 1986. Muskhelishvili NI. Singular Integral Equations. Groningen: Noordhoff; 1953. Shabunin MI, Sidorov YuV. Theory of Functions of a Complex Variable. Moscow: Lab Baz Znanii etc; 2002. Kuznetsov YeA. The use of automorphic functions in the plane theory of elasticity. Izv Akad Nauk SSSR MTT 1978;6:35–44. Johnson KL. Contact Mechanics. New York: Cambridge Univ. Press.; 1982. Aleksandrov VM. On plane contact problems of the theory of elasticity theory in the presence of adhesion or friction. J Appl Math Mech 1970;34(2):228–32. Goryacheva IG. Frictional Interaction Mechanics. Moscow: Nauka; 2001. Mitrofanov BP. The effect of the shape and dimensions of contacting bodies on the value of the approach and area of actual contact. In: Theory of Friction and Wear. Moscow: Nauka; 1965. p. 112–4. Muskhelishvili NI. Some Basic Problems of the Mathematical Theory of Elasticity. Amsterdam: Kluwer; 1974. Soldatenkov IA. Calculation of the energy losses in a sliding elastic contact with friction and wear (the plane problem). J Appl Math Mech 2007;71:632–42. Il’in VA, Poznyak EG. Principles of Mathematical Analysis. Moscow: Nauka; 1971. Pt. 2. 1973. Soldatenkov IA. The indentation with adhesion of a symmetrical punch into an elastic half-space. J Appl Math Mech 1996;60(2):261–7. Mossakovskii VI, Fotiyeva NN. The indentation of a symmetrical punch into an elastic half-space when there is adhesion along the contact line. Izv Akad Nauk SSSR Mekhanika 1965;6:67–70.
Translated by R.C.G.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The periodic contact problem of the plane theory of elasticity. Taking friction, wear and adhesion into account夽 I.A. Soldatenkov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 29 May 2011
a b s t r a c t A solution of the plane problem of the contact interaction of a periodic system of convex punches with an elastic half-plane is given for two forms of boundary conditions: 1) sliding of the punches when there is friction and wear, and 2) the indentation of the punches when there is adhesion. The problem is reduced to a canonical singular integral equation on the arc of a circle in the complex plane. The solution of this equation is expressed in terms of simple algebraic functions of a complex variable, which considerably simplifies its analysis. Asymptotic expressions are obtained for the solution of the problem in the case when the size of the contact area is small compared with the distance between the punches. © 2013 Elsevier Ltd. All rights reserved.
A solution of the periodic contact problem of the theory of elasticity was given for the first time for a system of punches with plane bases, in contact with an elastic half-plane when there is no friction.1 The case of periodic contact of convex punches was considered later,2,3 and friction and sliding on the contact was also taken into account in Ref. 4. A solution of the periodic contact problem with adhesion for specified boundary displacements and a fixed contact area is also known.5 There is a fairly complete description of the different formulations of plane periodic contact problems of the theory of elasticity and methods of solving them in Ref. 6. In this paper we consider periodic contact problems for an elastic half-plane, which differ from those mentioned above in the fact that the adhesion component of the friction and wear of the half-plane are taken into account, as well as the increase in the contact area when punches are indented with adhesion.
1. Formulation of the problem and fundamental equations Consider an elastic half-plane, the boundary of which is in contact with an infinite system of similar absolutely rigid convex punches, spaced a distance L from one another (Fig. 1). We will assume that each punch is loaded in the same way by an external shear force P1 and an external normal force P2 . We will choose an arbitrary punch and connect with it a system of coordinates Oxy, the x axis of which is directed parallel to the undeformed boundary of the half-plane, while the y axis is perpendicular to it. We will assign the number 0 to the chosen punch, and we will assign the numbers n = −1, −2,... (n = 1, 2,...) to the punches situated to the left (to the right) of this punch. All the punches are of the same form, and we will describe the form by the equation y = g(x), where g(x) is a specified function and g(0) = 0. We will denote the shearing contact stress and the contact pressure by q1 = xy |y=0 and q2 = −y |y=0 ≡ p, and the displacements of the upper boundary of the half-plane along the x and y axes by u and v respectively. We will denote the contact areas of the punches with the half-plane by the sections [an , bn ] of the x axis (n = 0, ±1, ±2,. . .), where, to simplify the writing of the later calculations, we will additionally put a = a0 and b = b0 . By virtue of the above assumptions, the stress-strain state of the half-plane is periodic, so that
(1.1)
夽 Prikl. Mat. Mekh., Vol. 77, No. 2, pp. 337–351, 2013. E-mail address: [email protected] 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.07.017
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Figure 1.
where 2l = a + b is the length of the contact area, 2l < L. This property enables us henceforth to confine ourselves to the section [−a, b] to describe the properties of the quantities considered and the relations between them. We will consider two forms of boundary conditions, corresponding to Problem 1 on the uniform sliding of the punches in the direction of the x axis with velocity V when there is friction and wear of the half-plane, and Problem 2 on the gradual penetration of the punches into the half-plane when there is adhesion. In Problem 1 we assume a specified and constant load P2 , whereas in Problem 2 the loads P1 and P2 increase monotonically, being connected by a loading law of the form (1.2) Assuming that the friction of the punch along the half-plane is described by Coulomb’s law, and the increase in the wear W of the half-plane with time t obeys a linear wear law dW/dt = cW Vp, we will represent the boundary conditions in the form (1.3)
(1.4) Here is the friction coefficient, 0 is the adhesive component of the friction, cW is the wear-resistance parameter of the material of the half-plane and (x) is the tangential displacement within the contact area, where, by virtue of the choice of the system of coordinates, (0) = 0. In Problem 2 the monotonically increasing dimension a of the contact area is used to write the functions as a time argument, which takes into account the change in the deformation conditions of the half-plane as the punches penetrate. The fact that the function (x) is independent of a is explained by the fact that points of the boundary of the half-plane that come into contact with the punch do not undergo any displacements with respect to the punch – the condition of complete adhesion of the contacting bodies. Henceforth we will assume that, for each value of the dimension a of the contact area, the distributions of the contact stresses q1,2 (x) satisfy the Hölder condition, the derivative g’(x) is a smooth function, while the derivative ’(x) is a piecewise-smooth function with a possible discontinuity at the point x = 0:
(1.5) To determine the relation between the displacements of the boundary of the half-plane and the contact stresses we will use the relations9
(1.6) where E is Young’s modulus and is Poisson’s ratio. Following Shtayerman,3 we will bear in mind property (1.1) of the periodicity of the contact stresses q1 , 2 (x) and of the corresponding contact areas and we use the representation
(1.7)
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Figure 2.
The last equation is obtained by changing the order of summation and integration, in view of the uniform convergence of the corresponding functional series, and using the tabulated expression10 for this series. Representation (1.7) enables us, for an arbitrarily specified contact area [−a, b], to obtain from relation (1.6) a system of integral equations with a Hilbert kernel
(1.8) Equations (1.8) must be supplemented by the punch equilibrium conditions, which, assuming its shape to be mildly sloping, |g (x)| 1, have the form
(1.9) Equations (1.8) and (1.9), together with boundary conditions (1.3) or (1.4), form a system of equations for finding the contact stresses q1 , 2 and the dimensions a, b of the contact area in the problems formulated above. In Problem 2 we use in addition, loading law (1.2) and it is also required to obtain the tangential displacement (x) within the contact area. Note that, in Problem 1, the wear and shearing load P1 are calculated from the formulae
where the first formula is obtained by integrating the wear law dW/dt = cW Vp,7 while the second is obtained by integrating the first boundary condition of (1.3), taking equilibrium conditions (1.9) into account. 2. General solutions We will use the following quantities in further calculations (2.1) the first of which represents the relative size of the contact area, while the second represents its degree of asymmetry. To obtain a general solution of Eqs (1.8), we will change in them from integrals with a Hilbert kernel to Cauchy-type integrals, using the well-known method.11 Thus, we will introduce, instead of x and , new complex variables s0 and s using the formulae
(2.2) so that
Moreover, it can be shown that for replacement (2.2), the section [−a, b] of the real x axis changes into the contour , situated in the complex plane and representing the arc of a circle of radius 1, which intersects the real axis at the points s1 = −sin and s2 = sin, and intersects the imaginary axis at the point –i + y0 (Fig. 2). When changes from −a to b the corresponding point s is displaced along the contour from s1 to s2 . Henceforth we will assume that the contour does not include the ends s1 and s2 .
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When these results are taken into account, and also equilibrium conditions (1.9), replacement (2.2) enables us to give Eqs (1.8) the required form
(2.3) (2.4) Replacement of the variable (2.2) in equilibrium conditions (1.9) gives the equalities
(2.5) We will further put
(2.6) Problem 1. Taking into account the definition of the functions 1 ,2 (s0 ) and 1 ,2 (s0 ) given above, we conclude that replacement (2.2) gives boundary conditions (1.3) the form (2.7) The first equality of (2.7) provides the possibility of eliminating from consideration the function 1 (s0 ), corresponding to the shear contact stress q1 (x), and enables us to use only the second equation of (2.3) to solve the problem with sliding. We will write this equation in canonical form11
(2.8) the coefficients and right-hand side of which are found using equalities (2.7), so that
Taking conditions (1.5) into account, we will seek a solution of Eq. (2.8) in the class of bounded functions. This solution can be represented using the operator K∗ in the form11
(2.9) with the necessary and sufficient condition
(2.10) where
(2.11) We will henceforth use the following properties of the function Z(s)
(2.12) which can be established if we close the line by the section [s1 , s2 ] of the real axis (Fig. 2) and apply Cauchy’s integral theorem12 to the closed contour obtained, and also use well-known tabulated integrals.10
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The first integral of (2.12) enables us, when writing solution (2.9), to drop the additive term 0 = iP2 /L in the function (s) and to represent this solution in the form of two expressions
(2.13) while the second integral of (2.12) enables us to represent condition (2.10) in the form of two equations
(2.14) Moreover, for Problem 1 we have the second equilibrium condition (2.5), which relates the function 2 (s0 ) with the specified load P2 . However, this condition is equivalent to the second equality of (2.14), which can be shown if we substitute the first expression of (2.13) for 2 (s0 ) into the second equality of (2.5) and, in the repeated integral obtained, change the order of integration, using the Poincare -Beltran formula.11 Hence, satisfaction of condition (2.10) ensures the correctness of second equilibrium condition (2.5), and hence equalities (2.13) and (2.14) are sufficient to construct a solution of the problem with sliding. In fact, the quantities and r, present in Eqs (2.14), are defined in terms of the dimensions a and b of the contact region by relations (2.1). Consequently, these equations enable us to obtain the quantities a and b, after which, using one of expressions (2.13), we can calculate the contact pressure 2 (s0 ) = p(x). A similar algorithm was used when solving Problem 1 in the case of a single punch.7 Note that the adhesion component of the friction 0 and the load P2 only occur in Eqs (2.14), affecting the dimensions a and b of the contact area, whereas the diagram of the contact pressure 2 (s0 ) = p(x) itself, according to expressions (2.13), is independent of 0 and P2 . When there is no friction and wear ( = 0) one can obtain, using the second equation of (2.14), explicit relations between the dimension a = l of the contact area and the load P2 . For example, for a parabolic punch g(x) = kx2 one obtains the well-known formula13
(2.15) in which a0 is the Hertz dimension of the contact area for a single parabolic
punch.9,14
For a sinusoidal punch
(2.16) one obtains another well-known
formula3
(2.17) Problem 2.
Using replacement (2.2), we introduce into consideration the complex-valued function
Then, taking into account definition (2.4), boundary conditions (1.4) of the problem with adhesion can take the following form (2.18) method.14,15
To simplify the calculations, we will reduce system (2.3) to a single equation, following the well-known To do this we multiply the first equation of (2.3) by the imaginary unit and add it to the second equation of (2.3). As a result, bearing in mind Eq. (2.18), we obtain the equation
(2.19) with respect to the complex-valued contact stress (s,a) = (s,a) + i2 (s,a). Here
The bounded solution of Eq. (2.19) is determined using the operator K∗ of the form (2.9):11 (2.20) with the necessary and sufficient condition (2.10). According to definition (2.6) of the quantity the function Z(s, a) and the coefficients D1 , A* and B* have the previous expressions (2.11) with the exception of the fact that here
(2.21) The presence of the argument a in the function Z(s, a) is due to the fact that the position of the ends s1 and s2 of the line depends on the value of a in Problem 2.
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For the function Z(s, a) considered, properties (2.12) remain in force, which, as also for Problem 1, enable solution (2.20) and the corresponding condition (2.10) to be converted, by representing them in the form of equalities
(2.22)
(2.23) We must add to these equalities equilibrium condition (2.5), which, taking the definitions given above for (s0 , a) and P(a) into account, can be expressed using a single relation:
(2.24) In turn, if we substitute the first expression of (2.22) for the function (s0 , a) into the last equality and change the order of integration in the repeated integral obtained, using the Poincare–Beltran formula, equilibrium condition (2.24) can be written as follows:
(2.25) Eliminating the complex load P from Eq. (2.23) using condition (2.25), we obtain the equation (2.26) which will be used below instead of (2.23). Equations (2.25) and (2.26), being additional to loading law (1.2), enable us, for a specified dimension a of the contact area, to obtain its second dimension b and the tangential displacement (x) within the contact area. After this, we can calculate the contact stresses 1,2 (s0 ) = q1,2 (x) using one of the expressions in (2.22). A similar algorithm was used to solve Problem 2 in the case of a single punch in Ref. 8. The equalities obtained above, which give solutions of Problems 1 and 2, can be written in the original variables x and by substituting expressions (2.2) into them. In order to x represent the equalities converted in this way in a single form for both problems, we introduce the following function
(2.27) Substituting (2.2) into expressions (2.13) and (2.22) and taking definitions (2.6), (2.11) and (2.27) into account, we obtain
(2.28) Here
(2.29) The parameter is defined by relations (2.11) and (2.21). The integrals 0 , 1 (a) in Eqs (2.14), (2.25), and (2.26) are defined by expressions (2.14), (2.23) and (2.25) and, as a result of substituting (2.2), take the form
(2.30) Note that, for the problem with sliding, the argument a in Eqs (2.28)–(2.30) is used formally.
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3. Asymptotic analysis Below we will consider the asymptotic of the solutions of both problems for small values of . By virtue of definition (2.1) of , this case occurs when the dimension 2l of the contact area is small compared with the distance L between punches. Using Taylor’s formula for the elementary functions present in definition (2.29) of the function R(x, a), we can establish the existence of a quantity * such that when ∈ (0, * )
(3.1) and also
(3.2) where
The last terms in Eqs (3.1) and (3.2) are remainder terms of the expansions in the small parameter , where the functions 1 (x; ) and 2 (,x;) are continuous in , x ∈ [−a,b] × [−a,b] and bounded in the set
Substituting expressions (3.1) and (3.2) into formulae (2.28) and (2.30), with the condition 1, we obtain the following asymptotic expressions
(3.3)
(3.4) where
(3.5)
(3.6) Here the existence of the quantities wm and w’m is ensured by properties (1.5) of functions (x) and g(x). Problem 1. We will consider the case of parabolic punches, g(x) = kx2 , and we will assume that the condition 1 is satisfied due to the increase in the distance L between the punches for a specified load P2 (rarefied contact), i.e., L−1 = O(). For parabolic punches, by relations (2.27) and (3.6), we have
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Hence, we can assume that, when the load P2 on the punch remains unchanged and the distance L between the punches increases without limit, the quantities wm and w’m remain bounded, i.e.
Taking the above into account, and also the fact that, for parabolic punches, the integrals Jn , defined by formula (3.5), are taken in explicit form,10 we can substitute expressions (3.4) for 0 into the first two equalities of (2.14) and reduce them to algebraic equations
(3.7) with respect to the unknown l and r¯ ≡ r/l, which are uniquely related to the dimensions a and b of the contact area. Note that, in these equations, the quantity is expressed in terms of l by equality (2.1). The solution of Eqs (3.7) can be represented in the form
(3.8)
(3.9) The quantity a0 is defined by the second formula of (2.15) and is used as the load parameter, while ls and r¯ s denote the values of the quantities l and r¯ for the problem with a single punch when there is friction and wear:7
(3.10) The presence of the second terms in Eqs (3.8) and (3.9) is due to the mutual interaction of the punches, situated at a distance L from one another. If we introduce into consideration the quantity = l − ls – a correction to the value of ls , due to the mutual interaction of the punches, we can obtain the following expression from equality (3.8)
(3.11) which indicates that the correction takes negative values, i.e., the mutual interaction of the punches reduces the contact area. This conclusion agrees with the results of other investigations of multiple contact.16 The second term in Eq. (3.9) and, consequently, the shift of the contact area due to the mutual influence of the punches, can have different signs, depending on the parameters of the problem and the value of the load P2 . Remark 1. When there is no friction and wear ( = 0) for a system of parabolic punches or a sinusoidal punch, the asymptotic expressions for the dimension a of the contact area may be obtained directly from formulae (2.15) or (2.17). As might have been expected, when 1 formula (2.15) takes the form of equality (3.8), if we replace l and ls by a and a0 in it. However, the expansion of expression (2.17) in terms of has a form which differs from (3.8), namely,
(3.12) The positive value of the second term in this equation confirms a well-known result: the dimension a of the contact area for a sinusoidal punch exceeds the Hertz dimension a0 .14,17 Nevertheless, the correction = a–as for a sinusoidal punch, as previously, is negative and is calculated from formula (3.11). In order to show this, it is sufficient, in the well-known relation18
to take as the form g(x) a single bulge of sinusoid (2.16), expand the function sin(2x/L) in a power series and put a/L 1,a = as . As a result, we arrive at the equality
the subtraction of which from (3.12) leads to formula (3.11). Hence, and also in the case of a sinusoidal punch, the mutual influence of its bulges reduces the contact area in the same way as in the case of a system of parabolic punches.
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Remark 2.
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Letting P2 → 0 in formula (3.10), we obtain
In other words, for small values of the load P2 the contact area [−a, b] may be shifted as a whole into the region of negative values of the x coordinate as a result of the action of the adhesion component of the friction. The distribution of the contact pressure p(x) when « 1 is found using the general asymptotic expression (3.3). According to our assumptions, in case of sliding of parabolic punches, we have the equality
As a result, expression (3.3) takes the form
The expression obtained for p(x), together with equalities (3.8) and (3.9), enable us to calculate the deformation component Fdef of the sliding resistance force on each parabolic punch.19 The result of such calculations is the asymptotic formula
which indicates that, for a fixed load P2 , the interaction between the punches leads to an increase in the component Fdef . Problem 2.
We will consider the symmetrical case of the indentation with adhesion of parabolic punches into a half-plane
and we will investigate the initial stage of this process (a slightly loaded contact), assuming that the dimension a of the contact area is limited to a certain value a* L. Then
The condition P1 (a) ≡ 0 denotes that loading law (1.2) with the function N(P2 ) ≡ 0 is used, and this law is automatically satisfied. Henceforth, instead of we will use the parameter , defined by formulae (2.21). We will also introduce into consideration the function (x) = ’(x), which is even in view of the symmetry of the problem. Taking the above assumptions into account, we substitute expressions (3.4) for 0 (a) and 1 (a) into equalities (2.25) and (2.26) and, after reduction, we obtain
(3.13)
(3.14) where
Note that in the symmetrical case considered, equality (2.25) gives also the expression P1 (a) = 0 + O(4 ), which agrees with the above assumption that there is no shear load on the punches.
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In the case of definitions (2.27) and (3.6) the following estimates hold:
(3.15) 5 .
The first of these denotes that the residual terms in Eqs (3.13) and (3.14) are of the order of The existence of the quantity ’* is ensured by property (1.5) of the derivative ’(x) = (x), while to derive the first estimate of (3.15), the equality (0) = 0 is used, which is established by taking the limit as a → 0 in relation (3.14), taking into account the following properties of the integrals introduced above:
We will obtain an asymptotic expression for the function (x) when « 1, using Eq. (3.14). To do this we will use the following representation, obtained using Taylor’s formula:
(3.16) (4) x ∈ C[0,
imposing the constraint a* ], additional to (1.5), and bearing in mind the equality (0) = 0 established above. The coefficients An , present in Eq. (3.16), are related in a known way to the derivatives of the function (x) and remain to be determined. The remainder term in formula (3.16) has the form20
Taking this estimate into account, we substitute expression (3.16) for (x) into Eq. (3.14) and, after reduction, we obtain the equality
Equating terms with like powers of an here, we can obtain a system of recurrent equations in the unknown coefficients An . The solution of this system has the form
and, taking into account the property of evenness of the function (x), we can obtain, using (3.16), the required expression
(3.17) in which, as can be established, ␦1,3 > 0 when v ∈ [0, 1/2]. Note that expression (3.17) satisfies the smoothness conditions (1.5), and in the limiting case as L → ∞, as might have been expected, takes the form corresponding to the problem with a single punch.21 Substituting the expression obtained for (x) into (3.13) we obtain the asymptotic dependence of the load P2 on the dimension a of the contact area in the following form
(3.18) where
The positive value of the second term on the right-hand side of (3.18) denotes that the mutual influence of the punches leads to a reduction in the contact area, as also for Problem 1 considered above. The distributions q1 , 2 (x, a) of the contact stresses in the case considered of parabolic punches are found from the general asymptotic expression (3.3) using expression (3.17) for the function ’(x) and have the form
(3.19)
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Here the values of q1 , 2 (x, a) for negative x are obtained from the symmetry properties of the problem:
Note that expression (3.19) becomes the well-known solution for a single punch,22 when L → ∞ and, correspondingly, when A3 → 0. Acknowledgement This research was supported financially by the Russian Foundation for Basic Research (11-01-00650, 12-08-90434, 11-01-90419). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Sadowsky M. The two-dimensional problem of elasticity theory. ZAMM 1928;8(2):107–21. Westergaard HM. Bearing pressures and cracks. Trans ASME J Appl Mech 1939;6:49–53. Shtayerman IYa. The Contact Problem of Elasticity Theory. Moscow, Leningrad: Gostekhizdat; 1949. Galin LA. Mixed problems of the theory of elasticity for a half-plane. Dokl Akad Nauk SSSR 1943;39:88–93. Kuznetsov EA. Periodic fundamental mixed problem of elastic theory for a half-plane. Soviet Appl Mech 1976;12:942–8. Block JM, Keer LM. Periodic contact problem in plane elasticity. J Mech Material and struct 2008;3:1207–37. Soldatenkov IA. The wear of a half-plane by a disc counterbody. Izv Akad Nauk MTT 1989;6:107–10. Soldatenkov IA. Formulation and solution of the problem of the indentation with adhesion of a punch into an elastic half-plane. Izv Ross Akad Nauk MTT 2000;4:39–52. Galin LA. Contact Problems of the Theory of Elasticity and Viscoelasticity. Moscow: Nauka; 1980. Prudnikov AP, Brychkov YuA, Marichev OI. Integrals and Series. Elementary Functions. New York: Gordon and Breach; 1986. Muskhelishvili NI. Singular Integral Equations. Groningen: Noordhoff; 1953. Shabunin MI, Sidorov YuV. Theory of Functions of a Complex Variable. Moscow: Lab Baz Znanii etc; 2002. Kuznetsov YeA. The use of automorphic functions in the plane theory of elasticity. Izv Akad Nauk SSSR MTT 1978;6:35–44. Johnson KL. Contact Mechanics. New York: Cambridge Univ. Press.; 1982. Aleksandrov VM. On plane contact problems of the theory of elasticity theory in the presence of adhesion or friction. J Appl Math Mech 1970;34(2):228–32. Goryacheva IG. Frictional Interaction Mechanics. Moscow: Nauka; 2001. Mitrofanov BP. The effect of the shape and dimensions of contacting bodies on the value of the approach and area of actual contact. In: Theory of Friction and Wear. Moscow: Nauka; 1965. p. 112–4. Muskhelishvili NI. Some Basic Problems of the Mathematical Theory of Elasticity. Amsterdam: Kluwer; 1974. Soldatenkov IA. Calculation of the energy losses in a sliding elastic contact with friction and wear (the plane problem). J Appl Math Mech 2007;71:632–42. Il’in VA, Poznyak EG. Principles of Mathematical Analysis. Moscow: Nauka; 1971. Pt. 2. 1973. Soldatenkov IA. The indentation with adhesion of a symmetrical punch into an elastic half-space. J Appl Math Mech 1996;60(2):261–7. Mossakovskii VI, Fotiyeva NN. The indentation of a symmetrical punch into an elastic half-space when there is adhesion along the contact line. Izv Akad Nauk SSSR Mekhanika 1965;6:67–70.
Translated by R.C.G.