Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 1, pp. 107-124, Warsaw 2004 PLANE CONTACT PROBLEMS WITH PARTIAL SLIP FOR ROUGH HALF-SPACE Volodymyr Pauk Department of Road and Traffic Engineering, Technical University of Kielce e-mail: pauk@tu.kielce.pl Bernd W. Zastrau Institute of Mechanics and Applied Informatics, Technische Universität Dresden, Germany e-mail: Bernd.W.Zastrau@mailbox.tu-dresden.de Planecontactproblemswith thepartial slip in the contactareaare consi- dered in thepaper.Tomake theproblemsmore realistic, thedeformation of roughness of the contacting boundary is involved. The Shtayerman model of roughness is generalized on the case of tangential problems. The problems are treated by the boundary integral method. Examples of the contact of a flat rigid punch and a rigid cylinder with an elastic half-space involving boundary imperfections are studied. The effects of roughness parameters on the distribution of normal and shearing trac- tions as well as on the stick-slip transition are investigated. Key words: Cattaneo-Mindlin problem; boundary roughness; integral equation 1. Introduction When two elastic bodies are normally pressed against each other and, subsequently, shifted by amonotonically increasing shearing force in the tan- gential direction, slip zones develop in the mutual contact area. This kind of a contact problem is referred to as the Cattaneo-Mindlin problem, see Catta- neo (1938), Mindlin (1949). The practical importance of this problem is very great, its results are applied to the investigation of the fretting in the contact zone. The bibliography on the Cattaneo-Mindlin problem is wide. It can be 108 V.Pauk, B.W.Zastrau found in well-knownmonographs on contact mechanics, Johnson (1985), Hills et al. (1993). Some generalizations of this problemwere done by Jäger (1997), Ciavarella (1998). Known solutions to partial slip tangential contact problemswere obtained on the assumption that the contacting surfaces are ideally smooth. But real boundaries of real bodies are not perfectly smooth, they include roughness which has an influence on the contact. There are many approaches to the modelling of boundary roughness. Our approach, which is presented in Section 2, is based on the Shtayerman (1949) assumption, that theboundary roughness causes additional deformationunder the punch.We extend this assumption on that relevant to tangential contact problems. On the base of the model proposed, the normal and tangential contact problems are considered in the next Sections.Wewill studyuncoupled problems postulating that the normal traction has no effect on the tangential displacements and the shearing traction on the normal displacements. The problems are assumed to be plane and steady-state. One contacting body is considered as a rigid punch while for the second body the Hertz assumptions are applied, and it is considered as an elastic half-space. Two main contact geometries are studied: the flat punch and the cylinder approximated by a parabola. To solve the partial slip contact problem, Cattaneo’s superposition for the shearing traction is usedand integral equations for a corrective traction are derived. Contrary to the case of ideal contact boundaries, the solution to these integral equations can not be obtained analytically, and a numerical technique has to be applied. 2. Model of boundary roughness The boundary roughness acts like a thin compliant layer on the surface of a body, Johnson (1985). As a result, additional deformation takes place under the contact of rough bodies. This assumption was first used by Shtayerman (1949), who proposed amodel of boundary imperfections postulating that the normal displacement of the roughboundary subjected to the normal load p(x) consists of two parts v(x)= ve(x)+vr(x) (2.1) where ve(x) is the displacement due to the elastic deformation of the body and vr(x) areadditional local displacementsdueto the roughnessdeformation. Plane contact problems with partial slip... 109 The first part of the displacements can be found as a solution to the elasti- city equations. To describe the additional displacement, Shtayerman used the relation vr(x)=αp(x) (2.2) where the constant α is called the roughness parameter. The Shtayerman model of the boundary roughness states an analogy with thewell-knownWin- kler (1867) assumption, and can be successfully applied to the solution to normal contact problems for rough bodies. But this model is not useable in the tangential contact problems because it neglects the shearing traction and displacements. In the solution to the tangential contact problems, we will use a similar idea, presenting the normal displacements in form (2.1) and tangential ones as u(x)=ue(x)+ur(x) (2.3) where ue(x) is a displacement due to elastic deformation of the body and ur(x)=βq(x) (2.4) states the tangential displacement due to the deformation of the boundary ro- ughness subjected to the action of the shearing traction q(x). The constant β will be called the roughness parameter. Thus the proposedmodel is characte- rized by two roughnessparameters α and β. Equations (2.2), (2.4) present the simplest model of the boundary roughness. Another model of the roughness known as the Greenwood-Williamsonmodel (see Greenwood andWilliamson, 1966) has a broader application range, and is widely used for the investigation of the normal contact of rough bodies. However, it is not easy to generalize the Greenwood-Williamson model, i.e. extend it onto the case of tangential problems. Analysing formulae (2.2) and (2.4) it is easy to observe an analogy between the proposedmodel of the boundary roughness and the simplifiedmodel of the elastic foundationusedbyKalker (1973) for the investigation of rollingcontact. The proposed model treats the boundary roughness as a set of independent springs or as a ”wire brush”. The problem of determination of the roughness parameters is discussed in Appendix. Considering the body as an elastic half-space and taking the solutions ue(x) and ve(x) in the well-known forms, Johnson (1985), the total displace- 110 V.Pauk, B.W.Zastrau ments of the rough boundary of the half- space can be presented as v(x) = αp(x)+ 2(1−ν2) πE a∫ −a p(ξ)ln |ξ−x| dξ+ + (1−2ν)(1+ν) 2E a∫ −a q(ξ)sgn(x− ξ) dξ (2.5) u(x) = βq(x)+ 2(1−ν2) πE a∫ −a q(ξ)ln |ξ−x| dξ− − (1−2ν)(1+ν) 2E a∫ −a p(ξ)sgn(x−ξ) dξ where ν,E are Poisson’s ratio andYoung’s modulus of the half-space, respec- tively, a is the half-width of the contact area. As was stated in Introduction, in the further analysis we will consider an uncoupled problem in which the tangential traction has no effect on the nor- mal displacements and the normal pressure on the tangential displacements. This situation takes place if ν =0.5 (assumed here) or when the mechanical properties of contacting bodies are identical. It is important to notice that the effect of the coupling between the tangential and normal problems is not great also in the general case of material properties, see Johnson (1985), and thus can be neglected. 3. Normal contact problems Let us assume that the rigid punch is pressed symmetrically by the normal load P against the rough boundary of the elastic half-space. The punch geo- metry is described by the function h(x). Satisfying the boundary condition by making use of expression (2.5)1 v(x)= δy −h(x) x∈ (−a,a) (3.1) where δy = const describes the normal approach of the contacting bodies, we arrive at the following integral equation αp(x)+ 2(1−ν2) πE a∫ −a p(ξ)ln |ξ−x| dξ= δy −h(x) x∈ (−a,a) (3.2) Plane contact problems with partial slip... 111 This equation with the equilibrium condition a∫ −a p(x) dx=P (3.3) determines a system of integral equations of the normal contact problem.We will consider two types of the punch geometry. 3.1. Flat punch In this case. the function h(x) = 0 and after introducing dimensionless variables, i.e. contact pressure and parameters s= x a η= ξ a p∗(s)= ap(x) P α∗ = αE a(1−ν2) δ∗y = δyE 1−ν2 (3.4) the system of integral equations (3.2), (3.3) can be transformed to the form α∗p∗(s)+ 2 π 1∫ −1 p∗(η)ln |η−s| dη= δ∗y s∈ (−1,1) (3.5) 1∫ −1 p∗(s) ds=1 Equations (3.5) are then solved numerically for different values of the di- mensionless roughness parameter α∗. The effects of boundary roughness on the contact pressure distribution is presented in Fig.1a by dotted curves. For α∗ =0, we obtain the well-known solution for the smooth half-space, Johnson (1985) p∗(s)= 1 π √ 1−s2 (3.6) which is unbounded for s→±1. But if α∗ > 0, the contact pressure no longer tends to the infinity at the punch edges. This result is due to the boundary roughness and was first obtained by Shtayerman (1949). 112 V.Pauk, B.W.Zastrau Fig. 1. 3.2. Cylindrical punch Assuming the punch geometry in the form h(x)= x2 2R (3.7) and introducing dimensionless parameters s= x aH η= ξ aH p∗(s)= aHp(x) P a∗ = a aH α∗ = αE aH(1−ν2) δ∗y = δyE 1−ν2 (3.8) we obtain the dimensionless form of integral equations (3.2), (3.3) α∗p∗(s)+ 2 π a∗∫ −a∗ p∗(η)ln |η−s| dη= δ∗y − 2 π PH P s2 s∈ (−a∗,a∗) (3.9) a∗∫ −a∗ p∗(s) ds=1 Here, aH and PH are the contact size and normal load in the Hertz problem, respectively, see Johnson (1985) a2H = 4(1−ν2)RPH πE (3.10) Plane contact problems with partial slip... 113 In the numerical analysis, the normal load is equal to that in the Hertz problem, i.e. PH/P = 1, and the unknown contact size a ∗ is determined iteratively from the physical condition p(±a∗)= 0 (3.11) The distribution of the contact pressure in the present case is shown in Fig.1b by dotted curves for three values of the roughness parameter α∗. For α∗ =0 the classical solution, Johnson (1985) p∗(s)= 2 π √ 1−s2 (3.12) is obtained.Weobserve that thecontact area isbigger, andthemaximumvalue of the contact pressure is lower in the presence of boundary imperfections. 4. Complete stick contact problems Let us nowassume that the bodies are in contact aswas stated in Section 3 and, subsequently, the tangential load Q is applied.First,wewill consider fully adhesive contact described by the condition u(x)= δx x∈ (−a,a) (4.1) where δx = const is the tangential component of the rigid motion of contac- ting bodies. Satisfying this condition, using formula (2.5)2, we obtain an integral equ- ation for the shearing traction βq(x)+ 2(1−ν2) πE a∫ −a q(ξ)ln |ξ−x| dξ= δx x∈ (−a,a) (4.2) which has to be considered together with the equilibrium condition a∫ −a q(x) dx=Q (4.3) Integral equations (4.2), (4.3) have been solved numerically. The effect of the dimensionless roughness parameter β∗ =βE/[aH(1−ν2)] on the distribu- tion of the dimensionless tangential traction q∗(s)= aHq(x)/P is presented in 114 V.Pauk, B.W.Zastrau Fig.1 by solid curves in the cases of the flat punch and the parabolic cylinder, respectively. These results were obtained for Q∗ = Q/P = 0.2, and the con- tact area in the case of the cylindrical punch was equal to that in the normal contact problem for α∗ =0.5. The presented results needmore comments. Integral equations (4.2), (4.3) have a structure like equations (3.2), (3.3) of the normal contact of the rigid flat punch. Thus, the tangential traction q(x) is unbounded at the edge of the contact area if β∗ =0, and is limited when β∗ > 0. Note that this behaviour is independent of the punch geometry. Considering the ratio q∗(a∗)/p∗(a∗) in the case of the rigid cylinder (Fig.1b) we can state that this value is always equal to infinity. Itmeans that in order to satisfy the complete stick condition over the whole contact area, we must apply the infinite friction force at the contact zone edges, which is physically impossible. So, some slip under the punch near the points s=±a∗ is inevitable in the case of the parabolic geometry, and the partial slip contact problem has to be solved. Identical behaviour takes place for this geometry in the complete stick contact problem for the ideally smooth boundary, Johnson (1985). A significantly different situation is observed in the case of the flat punch. In the classical case, when the boundary is ideal (α = β = 0), the shearing traction is q∗(s)=Q∗p∗(s) s∈ (−1,1) (4.4) where the normal pressurehas the formof (3.7). Itmeans that the stick occurs everywhere when Q∗ ¬ f (f is the friction coefficient), and if Q∗ > f, the punch slides over the half-space. Thus, no partial slip solution exists on the classical assumptions. Let us note here, that Ciavarella et al. (1998) obtained the partial slip solution for the flat punch assuming that the punch edgeswere slightly rounded. To study possible slip near the flat punch edges, let us examine the ratio q∗(1)/p∗(1) for different values of the roughness parameters. Figure 2 presents this ratioversus thedimensionless load Q∗ for somevalues of theparameter α∗ (straight solid lines; β∗ = 0.5 is fixed) and for some values of β∗ (straight dotted lines; α∗ = 0.5 is fixed). Drawing a horizontal line f = const (for example f = 0.4), we can conclude that no partial slip solution exists if α∗ <β∗ =0.5. If α∗ >β∗ =0.5 there are regions inwhich the stick conditions q∗(1)β∗. This property can alsobeprovedusingananalogybetween thenormalandtangential problemfor identical elastic half-spaces discovered by Jäger (1997) and Ciavarella (1998). Thus, if the tangential load Q∗ is monotonically increasing from zero to the value Q0, the contact is fully adhesive; if Q0 f, the punch slides over the half-space. 5. Partial slip contact problems The previous section shows that, even if Q