Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 4, pp. 753-771, Warsaw 2007 STRESSES OF HARD COATING UNDER SLIDING CONTACT Roman Kulchytsky-Zhyhailo Gabriel Rogowski Bialystok University of Technology, Faculty of Mechanical Engineering, Białystok, Poland e-mail: ksh@pb.edu.pl The two-dimensional contact problem of elasticity connected with an indentation of a rigid cylinder in an elastic semi-space covered by an elastic layer is considered.The detailed analysis of the stress distribution produced by contact pressures and tangential forces is presented. The obtained results for stresses are comparedwith ones obtainedwithin the framework of theory of elasticity for a half-space loaded by the Hertz pressure. Key words: hard coatings, contact problem, friction 1. Introduction The two-dimensional contact problem concerned with the pressing of a rigid cylinder in an elastic half-space which was coated with a layer of a different elasticmaterial, was consideredbyGupta et al. (1973) andGuptaandWalowit (1974). The authors conducted detailed analysis of the influence of parameters h= H/a (H is the thickness of the layer, a is the half-width of the contact zone) and E0/E1 (E0,E1 are Young’s moduli of the layer and the substrate, respectively) on basic contact characteristics (the contact width and distribu- tion of the contact pressure). Wide employments of hard top layers for the improvement of tribological properties are observed in last years, and they re- stored the great interest to investigations of the field of stresses responsible for destruction of layers, and namely, for fracture of coatings (cohesive failure) or delamination and spalling (adhesive failure) at the coating/substrate interface. It is shown that fundamental factors responsible for destruction of layers are: maximum tensile stress, which was considered by Gupta et al. (1973), Diao 754 R. Kulchytsky-Zhyhailo, G. Rogowski et al. (1994), Anderson and Collins (1995), Houmid Bennani and Takadoum (1999), Schwarzer (2000), Kato (2000), Bargallini et al. (2003), Bhomwick et al. (2003) and Xie and Tong (2005); the maximum shear stress, conside- red by Kouitat Nijwa and von Stebut (1999) and Bhomwick et al. (2003) (or Huber–Mises reduced stress – the approach used by Anderson and Col- lins (1995), Kouitat Nijwa et al. (1998), Diao (1999), Elsharkawy (1999) and Schwarzer (2000)) and themaximum shear stress at the coating/substrate in- terface, considered among others by Gupta et al. (1973) and Xie and Tong (2005). Therefore, the analysis of such stresses is very important. In majority of studies, when analysis of stresses in a layer due to contact loads is considered, two hypotheses are accepted: (1) tangential forcesworking in the area of contact as a result of friction have negligible influence on the di- stribution of contact pressure, (2) the actual pressuredistribution are replaced byHertz’s distribution. In thecase,when thefirst theorydoesnot raisedoubts, which was given by Johnson (1985) andElsharkawy andHamrock (1993), the second one seems undisputed in the case of very ”thin” (h< 0.2) or ”thick” (h> 0.7) coatings. As it is known fromGupta andWalowit (1974), the distri- bution of contact pressuremay considerably differ due toHertz’s distribution. The largest differences among these distributions occur when 0.3 5, the maximum value of the contact pressure is not in the centre of the contact area but in some distance from the centre. In the present work, we will focus on detailed analysis of the stress distri- bution produced by contact pressure described byGupta andWalowit (1974) and related tangential forces. We make a comparison between stress fields created by the distribution given by Gupta and Walowit (1974) and Hertz’s distribution. It permits one to define conditions in which the replacement of the contact stress distribution presented by Gupta and Walowit (1974) with Hertz’s distribution provides results satisfying for engineering practice. We propose a simplified algorithm for solving the contact problem together with the classical algorithm.This algorithm allows one to obtain a formula for pres- sure calculationwhich simplifies theanalysis of stress fieldsbydirectnumerical methods (finite element method or boundary element method). 2. Problem formulation Theproblemof an elastic half-space covered byan elastic layer inwhich a rigid cylinder is pressured, is considered.We assume that the cylinder is in the state Stresses of hard coating under sliding contact 755 of limit equilibrium(problemA,Fig.1a) or it slides along theboundarysurface of the half-space (problemB, Fig.1b). Fig. 1. A scheme of problem I (a) and of problem II (b) From the mathematical point of view, boundary conditions, typical for the problem of a cylinder sliding along a boundary surface of an elastic half- space with a constant velocity, are the same as the boundary conditions ty- pical for the problem of limit equilibrium. The field of displacements satisfies quasi-static equations of elasticity in movable coordinates. In these coordi- nates, the z-axis changes its position with constant velocity (the case of a two-dimensional problem was presented by Galin, 1980). On the basis of the solution to the problems connected with loading of the half-space surface by movable pressures (see by Eason, 1965), we can infer that when the velocity of the loading is much lower than the velocity of transverse waves in the con- sidered elastic media, quasi-static equations can replaced by equations of the static theory of elasticity. In engineering calculations, we can obtain that the solution to the contact problem concerned with the limit equilibrium is also the solution to the sliding problem. We take into account that the height of the cylinder aswell as its radius R are much larger than the width of the area of contact 2a. This approach permits one to consider this problem within the plane state of strain and to approximate the surface of the cylinder by a paraboloid z(x)= ax2 2R (2.1) where x, z are dimensionless Cartesian coordinates referred to the half-width of the contact. Let us consider that the tangential forces coincident with fric- 756 R. Kulchytsky-Zhyhailo, G. Rogowski tion in the contact region are connected with contact pressure by Amonton’s law q(x)= fp(x) (2.2) where f is the coefficient of friction. Simultaneously, we assume that the tangential forces exert a negligible influence on the size of the contact area and on the distribution of contact pressuredescribedbyJohnson (1985). The idealmechanical contact conditions between the layer and the half-space are assumed. All state functions denoted by the upper index 0 describe the state of displacement and stresses in the layer; the state functions for the elastic half- space are denoted by the index 1 (E is Young’smodulus, ν –Poisson’s ratio). We seek the final solution as a sum of two solutions. The first one is the solution to the contact problem (problem I), while the second one stands for the solution to the problemof elasticity for the half-space loaded by tangential forces on the boundary (problem II). In these problems, the equations of the elasticity theory should be satisfied (1−2νi)∆u(i)x + ∂θi ∂x =0 (1−2νi)∆u(i)z + ∂θi ∂z =0 i=0,1 (2.3) and boundary conditions u (0) x =u (1) x u (0) z =u (1) z σ (0) xz =σ (1) xz σ (0) zz =σ (1) zz z=0 (2.4) — problem I σ(0)xz =0 σ (0) zz =−p(x)H(1−x2) z=h (2.5) — problem II σ(0)xz = fp(x)H(1−x2) σ(0)zz =0 z=h (2.6) and σ (1) ij → 0 x 2+z2 →∞ (2.7) where u is the vector of the non-dimensional displacement referred to the half- width of contact, σ is the stress tensor, p is an unknown contact pressure, H(x) is theHeavisideunit step function.Theunknowncontact pressure should Stresses of hard coating under sliding contact 757 satisfy the boundary conditions connected with the shape of the pressured cylinder ∂u (0) z ∂x = ax R −1 0 we obtain a parti- tion of the interval [0,1], which condenses in the vicinity of the right end of the interval. Substituting Eq. (3.8) into integral (3.6) and comparing the left and right hand sides in points xj = (aj +aj−1)/2, j = 1,2, . . . ,m, we obta- in m linear algebraic equations containing (m+1) unknown parameters pi, i=1,2, . . . ,m and a0 m∑ i=1 Ajipi =−2(1−ν0)a20xj (3.10) where Aji = π 2 ai ∞∫ 0 a6(s)sin(sxj)J1(sai) ds (3.11) Themissing (m+1)-th equation π 4 m∑ i=1 pia 2 i =1 (3.12) is found by satisfying equilibrium condition (2.9). The solution to the system of equations is obtained bymaking use of a numericalmethod.The results are compatible with those given by Gupta andWalowit (1974). 4. A simplificed algorithm for solving the contact problem The distribution of contact pressure is sought in the form p∗(x)= [ p̂+ 16 π ( 1− π 4 p̂ ) x2 ]√ 1−x2 (4.1) 760 R. Kulchytsky-Zhyhailo, G. Rogowski which satisfies equilibriumcondition (2.9). It is necessary to point out that the parameter p̂ is the ratio of the contact pressure in thecentre of the contact area to the mean contact pressure. Contact boundary condition (2.8) is replaced by the following conditions u(0)z (1,h)−u(0)z (0,h) = a 2R 1∫ 0 [u(0)z (x,h)−u(0)z (0,h)] dx= a 6R (4.2) Substituting Eq. (4.1) into Eqs. (4.2), we obtain two equations for two unk- nown parameters (1−ν0)a20+Aip̂=Bi i=1,2 (4.3) The forms Ai and Bi, i=1,2 are presented in Appendix B. Figures 2 and 3 show that the solution to the system of equations (4.3) is well approximated by integral equation (3.6). The largest differences among the solutions appear in the case of ”thin” layers. They increase together with an enlargement of the ratio of the layer andYoung’smoduli of substrates. The error of estimation of the half-width of contact for h = 0.1 and E0/E1 = 8 is equal to 3.1%. Accuracy of contact pressure distributions is satisfactory for h> 0.2. The largest differences between contact pressure distributions for our contact problemandHertz’s pressure occurswhen h∈ [0.3,0.6]. In this range, we can use an approximate solution given in Eq. (4.1). Fig. 2. (a) Non-dimensional half-width of contact. (b) Pressure in the centre of the contact area as functions of the parameter E0/E1 (solution to integral equation (3.6) – gray curves, approximate solution – black curves, Hertz solution – dashed curve) Stresses of hard coating under sliding contact 761 Fig. 3. Distribution of contact pressure (solution of integral equation (3.6) – gray curves, approximate solution – black curves, Hertz solution – dashed curve) 5. The calculation algorithm for stress tensor components Stress tensor components in interior points of the non-homogeneous half-space can be obtained by numerical calculation of the integrals σ (i) jk (x,z) = 1√ 2π ∞∫ −∞ σ̃ (i) jk (s,z)exp(ixs) ds jk=xx,zz,xz i=0,1 (5.1) where the Fourier transforms σ̃ (i) jk (s,z) are described by Eqs (3.4) and (3.5). The accuracy of calculation of integrals (5.1) is supported by the continuity of stresses for z→h. The stresses σ(0)xz (x,h) and σ(0)zz (x,h) on the surface of the non-homogeneous half-space are known from the boundary conditions. The formula for calculating the stress σ (0) xx(x,h) is obtained taking into account the sixth equation of the system of (A1): σ(0)xx (x,h) = p(x)+ 2 √ 2d0√ π ∞∫ 0 [a4(s)cos(sx)+fâ4(s)sin(sx)]p̃(s) ds (5.2) where the relationship fâ4(s)p̃(s)=µ0a4(s)isgn(s) occurs in problem II. Asymptotic analysis of the solution to the system of equations (A1) for s→∞ shows that a4(s)=− 1 d0 +a∗4(s) â4(s)=− 1 d0 + â∗4(s) (5.3) lim s→∞ a∗4(s)= lim s→∞ â∗4(s)= 0 762 R. Kulchytsky-Zhyhailo, G. Rogowski Taking into consideration Eq. (5.3), we can write expression (5.2) in the fol- lowing form σ(0)xx (x,h) 1 p0 =−p∗(x)− 2 √ 2f√ π ∞∫ 0 p̃∗(s)sin(sx) ds+ (5.4) + 2 √ 2d0√ π ∞∫ 0 [a∗4(s)cos(sx)+fâ ∗ 4(s)sin(sx)]p̃ ∗(s) ds We can calculate the first integral from Eq. (5.4) analytically and the second one numerically. 6. Numerical results and discussion Upon the analysis of the relationship found for calculation of the dimension- less stresses σ (0) ij /p0, we conclude that they depend on five non-dimensional parameters: the ratio of the thickness of the top layer and the half-width of contact, the ratio of the layer andYoung’s moduli of substrates the coefficient of friction and the layer and the substrate Poisson ratios. In the contact problem for the homogeneous half-space, tensile stresses oc- cur at the trailing edge of theunloadedhalf-space surface,whichwasdescribed by Smith and Liu (1953). The maximum value of σ (0) xx/p0 occurs at the edge area of contact (x=−1,z=h) and is equal to 8f/π. In the case of the non- homogeneous half-space, tensile stresses in the mentioned zone also appeare (Fig.4). In that case, mechanical properties of the layer differ insignificantly from the substrate properties, themaximum tensile stress occur at the edge of contact. An increment of the parameter E0/E1 causes that themaximum ten- sile stresses occur in a certain distance from the edge of the contact area. This is more typical for ”thick” layers. When the thickness of the layer increases it leads to a decreasing level of dimensionless tensile stresses in the described area. Thesecondarea, inwhich tensile stresses canoccur is the coating/substrate interface (Fig.4b’,c’). In the case of ”thin” layers, the tensile stresses in this area do not occur at all or their level is much lower with respect to tensile stresses appearing at the surface. Together with the increasing thickness of the layer or E0/E1 parameter, tensile stresses at the interface also increase. For a certain layer thickness, the principal stress σ1 (we make an assump- tion that σ3 <σ2 <σ1) achieves the largest value at the interface. Figure 5 Stresses of hard coating under sliding contact 763 Fig. 4. Distribution of σ xx /p0 on the non-homogeneous half-space surface (a, b, c) and at the coating/substrate interface (a’, b’, c’): distribution of the contact pressure on the basis of integral equation (3.6) – black curves, distribution of actual contact pressure replaced by the Hertz solution – gray curve 764 R. Kulchytsky-Zhyhailo, G. Rogowski shows that every curve (except for gray curve 2) consist of two sections. The first section (adequate for smaller values of parameter h) is related with pa- rameters for which the maximum tensile stress occurs on the surface of the non-homogeneous half-space. The second section describes cases in which the maximum tensile stress at the interface is observed. Gray curve 2 shows that an increment in the coefficient of friction can cause that the dominant stress will be stress on the surface of the non-homogeneous half-space. Fig. 5. Relationship between the maximum tensile stresses and non-dimensional layer thickness (E0/E1 =4 – gray curves, E0/E1 =8 – black curves) The replacement of theactual contact pressurebyHertz’s distributiondoes not lead to large errors during calculation of themaximumvalue of the tensile stress on the surface. For the coefficient of friction f = 0.25, the calculation error is less than 5%, and for f = 0.5 is less than 9% (Fig.6). In the case of ”thin” layers, the error is smaller, and for the thickness h=0.2 is up to 4%. It is necessary to pay attention to the difference between results described in the present work and results published by Diao et al. (1994). The non- dimensional stress distributions σ (0) xx/pmax on the surface of the half-space calculated on the base of the Hertz contact pressure fully agree with the re- sults published by Diao et al. (1994). The difference is included in the para- meter pmax. In the present paper, the parameter pmax =4p0/π is calculated from the condition of equilibrium of the cylinder. However, in the work by Diao et al. (1994), it was assumed that pmax = px=0, where px=0 was the actual contact pressure in the centre of the contact area. The value px=0 was taken from the paper by Gupta andWalowit (1974). The maximum values of tensile stresses at the interface calculated of the basis of the Hertz contact pressure and the distribution of actual pressure considerably differ. Stresses of hard coating under sliding contact 765 Fig. 6. Error of calculation of the maximum tensile stresses (distribution of pressure replaced by the Hertz solution – gray curves, contact pressure approximated by Eq. (4.1) – black curves) In the casewhen h> 0.2, the distribution of contact pressuredescribed by Eq. (4.1) permits one to calculate the maximum value of tensile stresses with satisfactory precision both on the surface and at the interface (black curves in Fig.6). On the basis of the stress distribution σxx, it is easy to predict the maximum shear stress distribution in the plane of strain τ1 = = √ (σxx−σzz)2+4σ2xz/2 and the maximum shear stresses τ =(σ1−σ3)/2. Calculations confirmed the conclusion of the work by Kouitat Nijwa and von Stebut (1999) that the maximum shear stress τ1 in the plane of strain assu- med the minimal value in the central part of the top layer. The stress τ1 has the maximal value on the surface of the non-homogeneous half-space (for the determinated level of friction forces or in the case of ”thin” layers) or at the coating/substrate interface (for a small coefficient of friction for layers whose thickness exceeds the described value). Moreover, it is necessary to note that the maximum shear stress τ = (σ1 −σ3)/2 is frequently above the maximal value of the stress τ1. Typical distributions of maximum shear stresses on the surface of the non-homogeneous half-space are shown in Fig.7. Figure 7 pre- sents that the largest level of the stress τ is achieved in the central part of the contact area (0 0.7) lay- 766 R. Kulchytsky-Zhyhailo, G. Rogowski Fig. 7. Distribution of the maximum shear stresses τ/p0 (τ =(σ1−σ3)/2) on the surface of the non-homogeneous half-space: contact pressure distribution found from integral equation (3.6) – black curves, distribution of actual contact pressure replaced by the Hertz solution – gray curves ers. The largest deviation is observed when h ≈ 0.4. Describing the contact pressure by Eq. (4.1), it permits one to calculate the largest value of the ma- ximum shear stresses with an error up to 3% when h> 0.3 (black curves in Fig.9). The distribution of shear stresses σxz at the coating/substrate interface in the case of ”thin” layers was described in the paper by Houmid Bennani and Takadoum (1999). A typical distribution, which occurs in the case h > 0.3, is shown in Fig.10. Replacing of the actual contact pressure by the Hertz pressure does not cause considerable changes in the σxz stress distribution at the coating/substrate interface. Stresses of hard coating under sliding contact 767 Fig. 8. Relationship between τmax/σmax 1 and the non-dimensional layer thickness (E0/E1 =4 – gray curves, E0/E1 =8 – black curves) Fig. 9. Calculation error of the maximum τ1 stresses (distribution of pressure replaced by the Hertz solution – gray curves, contact pressure approximated by Eq. (4.1) – black curves) 7. Conclusions The analysis of stress distribution in the hard top layer proved that this layer can be divided into three sections: ”thin” layers (h < 0.2), ”thick” layers (h> 0.7) and those in between. In the case of ”thin” layers, the maximum tensile stress occurs at the trailing edge of theunloadedhalf-space surface.The largest value of the tensile stresses can bemostly observed at the edge of the area of contact (except for very large values of the ratio E0/E1). The maximum shear stress can turn 768 R. Kulchytsky-Zhyhailo, G. Rogowski Fig. 10. Typical distribution of σ xz /p0 stresses at the coating/substrate interface: contact pressure distribution found from integral equation (3.6) – black curves, distribution of actual contact pressure replaced by the Hertz solution – gray curves out to be much larger (more than twice) from the largest value of the tensile stress. The parameter τmax/σmax1 increases with growing of ratio E0/E1 and decreases with growing coefficient of friction. In engineering calculations, we can replace the distribution of actual pressure by the Hertz distribution. In the case of ”thick” layers, the predominant stresses are mostly tensile stresses at the coating/substrate interface (except for very large coefficients of friction). In engineering calculations, we can also apply Hertz distribution. In thecasewhen 0.2