Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 4, pp. 1181-1192, Warsaw 2017 DOI: 10.15632/jtam-pl.55.4.1181 FRICTIONAL CONTACT OF TWO SOLIDS WITH A PERIODICALLY GROOVED SURFACE IN THE PRESENCE OF AN IDEAL GAS IN INTERFACE GAPS Nataliya Malanchuk, Bogdan Slobodyan, Rostyslav Martynyak Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU, Lviv, Ukraine e-mail: labmtd@iapmm.lviv.ua The frictional contact between two solids, one of which having a periodically grooved sur- face, under the action of normal and shear load is investigated. The interface between the contacting solids consists of a periodic arrayof gas-filled gaps anda periodic arrayof contact regions,where stick-slip contact occurs.The correspondingplane contact problem is reduced to a set of two singular integral equations. A solution to the contact problem is obtained for a certain shape of the grooves.The analysis of dependences of contact parameters of the solids on the applied load and gas pressure is carried out. Keywords: stick-slip contact, periodically grooved surface, gas pressure 1. Introduction The surfacemicrotexturing, which consists in forming regularly (periodically) arranged grooves of the same shape on a surface of a solid, is one way of improving performance of joints. A regular surface texture may be generated by several methods (Etsion, 2004; Greco et al., 2009; Nakano et al., 2007; Schreck and Zum Gahr, 2005; Stepien, 2011): laser texturing, precise dia- mond turning, rolling, embossing, etching, vibrorolling, abrasive jet machining, micro electrical dischargemachining, grinding.When periodically grooved surfaces are placed in contact, perio- dically arranged intercontact gaps occur at the interface. These gaps are usually filledwith some substance (a natural substance (gas, liquid), a substance used for functional purposes (grease, coolant) or a biological fluid (synovia)). The effect of the interstitial medium filling the gaps of different nature on mechanical behavior of bodies was investigated by Kit et al. (2009), Ma- chyshyn and Nagórko (2003), Martynyak and Slobodyan (2009), Evtushenko and Sulim (1981), Kaczyński andMonastyrskyy (2004), Monastyrskyy and Kaczyński (2007). The contact betweenmicrotextured surfaces is usually realized not only under normal load, but also under shear load. This shear load may cause partial slip of surfaces, and the filler of the gaps may have some effect on propagation of the slip zones. However, the existing studies of stick-slip contact (Ciavarella, 1998a,b; Chang et al., 1984; Block and Keer, 2008; Hills et al., 2016; Goryacheva andMartynyak, 2014) do not consider this effect. When contacting solids are subjected to heating, thermal deformations can also cause partial slip of surfaces. The thermally induced partial slip of a rigid flat-ended punch and an elastic half-spacewithdifferent temperatureswas investigated in (Pauk, 2007). The thermoelastic stick- slip contact problem for two semi-infinite solids in the presence of a single thermoinsulated gap at the interface was studied in (Malanchuk et al., 2011). The effect of thermal conductivity of a medium filling the interface gaps on partial slip between a textured half-space and a flat half-space, which was caused by an imposed heat flow, was studied in (Chumak et al., 2014). The goal of this research is to investigate partial slip between a half-spacewith a periodically grooved surface and a flat half-space, which is caused by the applied shear load, taking into 1182 N.Malanchuk et al. account pressure of an ideal gas in the interface gaps. The stick-slip contact between a surface with a single groove and a flat surface in the presence of an ideal gas in the interface gap was previously studied in (Slobodyan et al., 2014). Themodel of partial slip between a periodically grooved surface and a flat one was presented in (Slobodyan et al., 2016). However, that model did not take into account the effect of the interstitial medium. 2. Statement of the problem Consider the contact between two isotropic elastic half-spaces made of an identical material under plane strain conditions. The upper solid D2 has a flat surface. The surface of the lower solid D1 has an array of grooves of width 2b spaced with the period d (d > 2b). The shape of each groove is describedby a smooth function r(x) (r(x)≪ b, r′(x)≪ 1, r(±b)= 0, r′(±b)=0). The solids are successively loaded by normal and shear loads. At the first stage of loading, the solids are pressed together by amonotonically increasing nominal pressure pn =P/d applied at infinity, where P denotes the normal force per one period. The compressive load is assumed to vary quasistatically. Due to the regular surface texture of the lower solid, the interface consists of a periodic array of gaps and a periodic array of contacts. The width 2a (a< b) and height h(x) of the interface gaps are unknown and decrease monotonically with an increase of the nominal pressure. The gaps are filled with an equal amount of the ideal gas, whose pressure P1 changes during the loading. The relation between the gas pressurePl and gas volume V = l ∫a −ah(x) dx is described by the ideal gas law P1V = m1 µ RT (2.1) where m1 is mass of the gas, µ is molar mass of the gas, T is gas temperature, R is the ideal gas constant, R=8.3145JK−1mol−1 and l=1m. The shear displacements of the contacting surfaces due to the normal load are the same because of identity of the materials. The shear stresses do not, therefore, arise at the interface and the slip of the solids does not occur at the first loading phase. At the second loading phase, the normal load is held fixed and the bodies are subjected to the action of the nominal shear loading sn = S/d applied at infinity (Fig. 1). Here, S denotes the shear force per one period. According to theCoulomb-Amontons law, the surfaces of thebodies are in stick until the contact shear stress τxy is less than the contact pressure |σy| multiplied by the friction coefficient f (τxy < f|σy|). The applied shear force leads to the frictional slip of the contacting surfaces in the regions (−c+kd,−a+kd), (a+kd,c+kd), which, due to symmetry of the problem with respect to y-axis, are located symmetrically relative to the origin of each gap, k=0,±1,±2, . . .. In the slip zones, τxy = f|σy|. The direction of slip is indicated by arrows in Fig. 1. Denote Iik = [−i+ kd,i+ kd], Jik = [−d/2+ kd,−i+ kd]∪ [i+ kd,d/2+ kd], i = a,b,c, Y c,a k = [−c+kd,−a+kd]∪ [a+kd,c+kd], hereinafter k=0,±1,±2, . . .. The boundary conditions are: — at the gaps (x∈ Iak) σ−y (x,0)=σ + y (x,0) σ − y (x,0)=−P1 τ−xy(x,0)= τ + xy(x,0) τ − xy(x,0)= 0 (2.2) — in the slip zones (x∈Y c,ak ) σ−y (x,0)=σ + y (x,0) τ − xy(x,0)= τ + xy(x,0) v −(x,0)−v+(x,0)=−r(x) τ−xy(x,0)=−fσ−y (x,0) (2.3) Frictional contact of two solids with a periodically grooved surface... 1183 Fig. 1. Contact of the solids (within one period) — in the stick zones (x∈ Jck) σ−y (x,0)=σ + y (x,0) τ − xy(x,0)= τ + xy(x,0) v −(x,0)−v+(x,0)=−r(x) (2.4) — at infinity σy(x,±∞)=−pn σx(x,±∞)= 0 τxy(x,±∞)= sn (2.5) Here, r(x) = 0 when x∈ Jbk, σy(x,y), σx(x,y), τxy(x,y) are stress components, u(x,y), v(x,y) are displacement components, superscripts+and−denote the boundaryvalues of the functions on the x-axis in the upper and lower solid, respectively. Note thatbecause of identity of the contactingmaterials, the frictional slip does not influence the normal contact stress and width as well as and height of the gaps. 3. Solution to the problem Let us represent the stresses and displacements in both solids throughout three functions: height of the grooves r(x), height of the gaps h(x), and relative tangential shift of the solids surfaces U(x)=u−(x,0)−u+(x,0) (Slobodyan et al., 2014; Muskhelishvili, 1953) σx+σy =4Re[Φj(z)]−pn σy − iτxy =Φj(z)−Φj(z)+(z−z)Φ′j(z)−pn− isn 2G(u′+iv′)= (3−4ν)Φj(z)+Φj(z)− (z−z)Φ′j(z)+νpn+isn Φ1(z)=−Φ2(z) = (−1)j+1G 4π(1−ν) ∞ ∑ k=−∞ ( i c+kd ∫ −c+kd U ′(t) dt t−z + a+kd ∫ −a+kd h′(t) dt t−z + b+kd ∫ −b+kd r′(t) dt t−z ) z∈Dj j=1,2 (3.1) whereU(x)= 0when x∈ Jck, z=x+iy, i = √ −1, ν is Poisson’s ratio,G is the shearmodulus. 1184 N.Malanchuk et al. Representations (3.1) have been constructed so that they satisfy boundary conditions (2.3)1, (2.4) and (2.5). Taking into account the periodicity of the functionsU(x),h(x) and r(x) (Schmu- eser and Comninou, 1979), the complex functionsΦ1(z), Φ2(z) can be rewritten as Φ1(z)= (−1)j+1G 4d(1−ν) ( i c ∫ −c U ′(t)cot π(t−z) d dt+ a ∫ −a h′(t)cot π(t−z) d dt + b ∫ −b r′(t)cot π(t−z) d dt ) Φ2(z)=−Φ1(z) z∈Dj j=1,2 (3.2) The normal and shear stresses at the interface y=0 calculated from expressions (3.1), are N(x)=σy(x,0)= G 2d(1−ν) ( b ∫ −b r′(t)cot π(t−z) d dt+ a ∫ −a h′(t)cot π(t−z) d dt ) −pn S(x)= τxy(x,0)=− G 2d(1−ν) c ∫ −c U ′(t)cot π(t−z) d dt+sn (3.3) In order to satisfy boundary condition (2.2)1, we substitute (3.3)1 into (2.2)1, which leads to a singular integral equation 1 d a ∫ −a h′(t)cot π(t−x) d dt=−1 d b ∫ −b r′(t)cot π(t−x) d dt+ 2(1−ν) G (pn−P1) |x| ¬ a (3.4) From boundary conditions (2.2)2 and (2.3)2, and expression (3.3)2, we obtain a singular integral equation for the functionU ′(x) 1 d c ∫ −c U ′(t)cot π(t−x) d dt= 2(1−ν)sn G +                0 x∈ Iak f ( 1 d a ∫ −a h′(t)cot π(t−x) d dt+ 1 d b ∫ −b r′(t)cot π(t−x) d dt ) − 2(1−ν)f G (pn−P1) x∈Y c,ak (3.5) As a result, we get a set of two singular integral equations with the Hilbert kernel for the functions h′(x) andU ′(x). It is obvious from Eq. (3.5) that the gas pressure P1 influences the relative tangential shift U(x) of the contacting surfaces. The functions h(x) andU(x) satisfy the conditions h(±a)= 0 h′(±a)= 0 U(±c)= 0 U ′(±c)=0 (3.6) The first condition in (3.6)1 means that the gaps between the solids vanish in the contact regions. The second condition in (3.6)1 represents smooth closure of the gaps at x = ±a and Frictional contact of two solids with a periodically grooved surface... 1185 ensures that the normal contact stress is boundedatx=±a. Thefirst condition in (3.6)2 follows from the continuity of shear displacements of the contacting surfaces. The second condition in (3.6)2 ensures that the shear stress is bounded at the edges of the slip zones x=±c. By changing variables ξ = tan(πx/d), η = tan(πt/d), α = tan(πa/d), β = tan(πb/d), γ=tan(πc/d), we reduce set (3.4), (3.5) to a set of singular integral equations with the Cauchy kernel α ∫ −α h′(η) dη η− ξ =− β ∫ −β r′(η) dη η− ξ + 2d(1−ν) G(1+ ξ2) (pn−P1) |ξ| ¬α γ ∫ −γ U ′(η) dη η− ξ = 2d(1−ν)sn G(1+ ξ2) +                  0 |ξ| ¬α f ( α ∫ −α h′(η) dη η− ξ + β ∫ −β r′(η) dη η− ξ ) − 2d(1−ν)f G(1+ ξ2) (pn−P1) α¬ |ξ| ¬ γ (3.7) In the new variables, conditions (3.6) have the form h(±α) = 0 h′(±α) =0 U(±γ)= 0 U ′(±γ)= 0 (3.8) To solve set (3.7), the function r(x), which describes the shape of the grooves, should be specified. We preset it as follows: r(x) = −r0 ( 1− tan2(πx/d)/tan2(πb/d) )3/2 , x ∈ Ibk, where r0 is themaximum depth of the grooves, and 0β Since conditions (3.8)2 must be met, we find a solution to Eq. (3.17) that is bounded at ξ=±γ. The solution to singular integral equation (3.17) is (Muskhelishvili, 1953) U ′(ξ)= 2d(1−ν)snξ √ γ2−ξ2 πG √ 1+γ2(1+ ξ2) − √ γ2− ξ2 π2 γ ∫ −γ L(η) dη √ γ2−η2(η− ξ) |ξ| ¬ γ (3.18) Using a piecewise constant approximation of the functionL(ξ) for evaluation of the integral in the right-hand side of Eq. (3.18), we obtain U ′(ξ)= 2d(1−ν)snξ √ γ2−ξ2 πG(1+ξ2) √ 1+γ2 + 1 2π2 m−1 ∑ j=0 Lj ( Γ(γ,ξ,ζj+1)−Γ(γ,ξ,ζj) ) |ξ| ¬ γ (3.19) where ζj =−γ+2jγ/m, j=1,2, . . . ,m are nodes of the approximation;Lj are the nodal values of the functionL(ξ), that is Lj =L(ζj), j=1,2, . . . ,m; and Γ(γ,ξ,ζ)= ln γ2− ξζ+ √ (γ2− ξ2)(γ2− ζ2) γ2− ξζ− √ (γ2− ξ2)(γ2− ζ2) Integration of Eq. (3.19) from −γ to ξ in view of the first condition in Eq. (3.8)2 gives the relative tangential shift U(ξ)= d(1−ν)sn πG ( ln ∣ ∣ ∣ ∣ ∣ √ 1+γ2− √ γ2− ξ2 √ 1+γ2+ √ γ2− ξ2 ∣ ∣ ∣ ∣ ∣ + 2 √ γ2− ξ2 √ 1+γ2 ) + 1 π2 m−1 ∑ j=0 Lj { (ξ− ζj+1)Γ(γ,ξ,ζj+1)− (ξ− ζj)Γ(γ,ξ,ζj) +2 ( √ γ2− ζ2j+1− √ γ2− ζ2j )[ arcsin (ξ γ ) + π 2 ]} |ξ| ¬ γ (3.20) By setting ξ= γ in Eq. (3.20) and in view of the first condition in Eq. (3.8)2, we obtain an equation for width γ of the slip zones 2d(1−ν)sn G √ 1+γ2 + 1 π2 m−1 ∑ j=0 Lj ( √ γ2− ζ2j+1− √ γ2− ζ2j ) =0 (3.21) This equation is solved numerically. Substituting Eq. (3.19) into Eq. (3.3)2 and performing some integrations, we find shear contact stresses in the stick zones (|ξ| ­ γ) S(ξ)= sn √ 1+γ2 ( ξ2+1−|ξ| √ ξ2−γ2 ) + G 2d(1−ν)π (ξ2+1) · m−1 ∑ j=0 Lj [ arcsin (ζj+1 γ ) −arcsin (ζj γ ) −arcsin ( ξζj+1−γ2 γ(ξ− ζj+1) ) +arcsin ( ξζj −γ2 γ(ξ− ζj) )] (3.22) 1188 N.Malanchuk et al. Asd→∞, we obtain results for the single groove (Slobodyan et al., 2014). Byputting inEqs. (3.10)-(3.12) and (3.15)-(3.22) m1 = 0, we obtain results for the frictional contact interaction between two solids, one ofwhichhaving a regular surface texture in formof periodically arranged grooves, in the case when the gaps do not contain a filler (Slobodyan et al., 2016). 4. Numerical results and discussion The obtained results are illustrated in Figs. 2-6, where the dimensionless parameters x= x/d, a = a/d, b = b/d, c = c/d, r = r/d, h = h/d, U = U/d, σy = 4σy(1−ν)G−1, τxy = 4τxy(1− ν)G−1, pn =4pn(1−ν)G−1, sn =4sn(1−ν)G−1,P1 =4P1(1−ν)G−1,m1 =m1RTµ−1d−1 are introduced. The maximum depth of the grooves r0 is taken to be 10 −3, the half-width of the grooves b=0.3, and the friction coefficient f =0.1. The nonlinear dependence of the half-width a of the gaps on the applied pressure pn is given in Fig. 2. The half-width a of the gaps decreases with an increase in the applied pressure pn. Fig. 2. Dependence of the half-width a of the gaps on the applied pressure p n Figure 3a showsheight of the gaps for different values of theappliedpressurepn.Theheighth of the gaps decreases with the increasing nominal pressure and has its maximum value in the center of the gap. The height h of the gaps for different values of the gas mass m1 is given in Fig. 3b, where pn =0.003. The curve form1 =0 corresponds to the case of gaps without a filler. As seen in the figure, h is the largest in the case of filled gaps and increases with increasingm1. Figure 4a shows the dependence of the half-width c of the slip zones on the shear load sn for different values of the applied pressure pn. If the edge of the slip zone is located within the groove (a < c < b), then this dependence is nonlinear. If the edge of the slip zone exceeds the bounds of the groove (c > b), then this dependence is almost linear. The width c of the slip zones monotonically increases with the increasing shear load sn. The rate of the increase of c with a change of sn is greater for smaller values of pn. The curves c= c(sn) have specific kinks at the points c= b=0.3 (the right/left end of the right/left slip zone reaches the right/left end of the groove). If c > b, then the slope of curves c= c(sn) increases significantly. This effect can be explained by the fact that the normal contact stress has its maximum value at the edges of the grooves and rapidly decreases in the region beyond the grooves (see Fig. 6a). The dependence of the half-width c of the slip zones on the gas mass m1 is given in Fig. 4b, where pn = 0.003. As seen in the figure, for a fixed sn, c is the smallest in the case of gaps without a filler (the curve that corresponds to m1 = 0) and increases with increasingm1 (curves that correspond tom1 =8 ·10−7,m1 =9 ·10−7,m1 =10−6). Frictional contact of two solids with a periodically grooved surface... 1189 Fig. 3. The height h of the gaps Fig. 4. Dependence of the half-width c of the slip zones on the shear load s n Figure 5a shows the relative tangential shiftU of the surfaces within one period for different values of the applied pressure pn. The relative tangential shiftU of the surfaces increases with the increasing applied pressurepn andhas itsmaximumvalue in the center of the grooves.At the gaps, the relative tangential shiftU is larger than in the slip zones.The relative tangential shiftU of the surfaces within one period for four values of the gas massm1 (0; 8 ·10−7; 9 ·10−7; 10−6) is given in Fig. 5b, where pn = 0.001. The relative tangential shift U is smallest in the case of gaps without the filler (the curve that corresponds tom1 =0) and increases with increasingm1 (curves that correspond tom1 =8 ·10−7,m1 =9 ·10−7,m1 =10−6). Figure 6a shows the distribution of the contact normal stress σy for different values of the nominal pressure pn. The magnitude of the contact normal stress |σy| increases with the increasing nominal pressure pn and has its maximum value at the edges of the grooves (x= b= ±0.3). As required by boundary condition (2.2)1, σ±y =−P1 at the surfaces of the gaps. Beyond the grooves, the contact normal stress monotonically decreases. 1190 N.Malanchuk et al. Fig. 5. The relative tangential shiftU of the surfaces Fig. 6. Distribution of the stresses The distributions of the shear contact stress τxy (solid curves) and the normal contact stress f|σy| multiplied by the friction coefficient (dashed curve) are given in Fig. 6b. The solid curves are plotted for two values of the shear load sn. For sn = 1.3 · 10−4; 2.6 · 10−4, the correspondingvalues of care 0.28; 0.35.The shear contact stress τxy increaseswith the increasing shear load sn. The stress τxy has the maximum value at x = ±c (the ends of the slip zones) if the applied shear load sn is such that slip occurs within the grooves (c < b) (the curve that corresponds to sn = 1.3 ·10−4), and at x = ±b (the ends of the grooves) if the applied shear load sn is such that slip extends outside the grooves (c ­ b) (the curve that corresponds to sn =2.6 ·10−4). At the surfaces of the gaps, τxy =0 and f|σy|= fP1. As required by boundary condition (2.3)2, the solid curves coincide with the dashed curve in the slip zones. Outside the slip zones, τxy is less than f|σy| (the solid curves lie below the dashed curve). Frictional contact of two solids with a periodically grooved surface... 1191 5. Conclusions The contact problem for two half-spaces of the samematerial under sequential application of the nominal pressure and nominal shear stress has been considered. One of the contacting surfaces is textured in form of periodically arranged grooves, and the other is flat. Partial slip of the surfaces of the solids is induced by shear load, and the first points to slip are the ends of each gap. The contact problem has been solved for the case when the interface gaps are filled with the ideal gas. Analysis of contact parameters of the contact pair on the applied load has been carried out. It has been shown that the half-width and height of the gaps decrease when the nominal pressure increases. The height of the gaps is the largest in the case of filled gaps and increases with the increasing mass of the gas, and has the maximum value in the center of the gaps. The normal contact stress increases with the increasing nominal pressure and has the maximum value at the edges of grooves. The width of the slip zones, relative tangential shift of the surfaces and shear contact stress increase with the increasing shear load. For a fixed load, the relative tangential shift andwidth of the slip zones is the smallest in the case of gapswithout the filler and increases with the increasing gas mass. The shear contact stress has themaximum value at the ends of the slip zones if the applied shear load is such that slip occurs within the grooves and at the ends of the grooves if the applied shear load is such that slip extends outside the grooves. References 1. Block J.M., Keer L.M., 2008, Periodic contact problems in plane elasticity, Journal of Mecha- nics of Materials and Structures, 3, 7, 1207-1237 2. Chang F.-K., Comninou M., Sheppard S., Barber J.R., 1984, The subsurface crack under conditions of slip and stick caused by a surface normal force,ASME Journal of AppliedMechanics, 51, 311-316 3. 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