Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 2, pp. 271-288, Warsaw 2003 INTERACTION BETWEEN A STRATIFIED ELASTIC HALF-SPACE AND AN IRREGULAR BASE ALLOWING FOR THE INTERCONTACT GAS Ihor Machyshyn Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine e-mail: labmtd@iapmm.lviv.ua Wiesław Nagórko Department of Engineering and Environmental Sciences, Warsaw Agricultural University e-mail: nagorko@alpha.sggw.waw.pl Thepaper is devoted to the investigationof contact interaction of a lamina- ted half-space and a rigid bodywith a smooth cylindrical depression under conditions of plane deformation allowing for an intercontact ideal gas. To describe the homogenized model of the laminated body with microlocal parameters and to describe the behavior of the gas – equations of ideal gas state are used. Applying the method of complex potentials the problem is reduced to the singular integral equation for the height of intercontact gap and its solution is obtained in a closed form. To find the length of the gap, its volumeand the gaspressure a systemof three equations is derived.With the aid of this system the dependence of the external loading and amount of the gas in the gap on the contact pressure and geometrical parameters of the gap is analyzed. Keywords: contact problem, laminatedhalf-space, gas, complex potentials, singular integral equations 1. Introduction For the time being, the theory of contact problems, which deals with the Hertz contact (e.g. Johnson, 1985) when conjugated bodies touch at point or along the linebefore loading, is developed sufficientlywell.Demandsofmodern 272 I.Machyshyn, W.Nagórko technical engineering require also the development of much less investigated models to study a contact of bodies with conformable surfaces that initially coincide everywhere excluding zones that aremuch less than the contact area. Suchmodels take into account imperfections of surfaces related to their small deviations from a flat onto local parts (Martynyak, 1985). They are physically grounded since real surfaces of conjugatedbodies frequentlyhaveunevennesses caused by their manufacturing, defects of various kinds and technological im- perfections. Despite the fact that these surface factors are small, considerable perturbationsof stress and strain fieldsappear in their vicinity (Monastyrskyy, 1999; Shvets et al., 1996), therefore the considering of surface irregularities is necessary from the point of view of contact durability and fracture.As a result of the unevennesses of mated surfaces intercontact gaps between the bodies appear. Very often they are filled by a certain substance. Depending on the type of the compound and conditions of its exploitation it could be, for exam- ple, a gas or liquid. The filler of the intercontact gaps takes a considerable part in the contact interaction as far as along the zones of contact it transmits traction between the surfaces, and this in one turn changes the distribution of contact pressure and geometrical parameters of the gaps. The contact interaction of bodies allowing for an intercontact substance was considered in literature earlier. Kuznetsov (1985) considered contact of rough bodies in the presence of a fluid lubricant. Martynyak (1998) studied the influence of the ideal gas within the gap on the contact of half-spaces. An interaction of bodies taking into account ”gas-liquid” phase transition in the filler of the gapwas analyzed byMartynyak andMachyshyn (2000) in the case of surface depression possessing corner points at its ends providing constancy of the gap length under compression of the bodies. Fast development of building and mechanical engineering along with the improvement of technology demands creation of newmodernmaterials which must satisfy a priori givenmechanical characteristics. One of theways to solve this problem is to produce materials which are compounds of elements with known characteristics, namely composites. To describe composites of periodi- cally repeated structures on themacro level different methods of averaging of their properties (e.g. Bahvalov and Panasenko, 1984; Christensen, 1979) are used. This paper is devoted to the investigation of contact interaction between a layered composite half-space and an uneven rigid base in the presence of the ideal gas in the intercontact gap. As a tool of the averaging of characteristics of a stratified body the homogenization model with microlocal parameters established byWoźniak (1986, 1987) and developed in numerous publications Interaction between a stratified elastic half-space... 273 (e.g. Kaczyński and Matysiak, 1998; Matysiak and Nagórko, 1989; Nagórko, 1989) is proposed. In contrast to other methods of homogenization it allows one to derive mean values of stresses and strains as well as their local values, that is stresses and strains in each component of the stratified body. 2. Model of an elastic stratified body 2.1. Governing equations for a periodic n-layered half-space A stratified half-space, in which every periodically repeated lamina of the thickness δ consists of n elastic isotropic layers of the thickness δ1,δ2, ...,δn (δ = ∑n i=1δi), is considered (Fig.1). Perfect bonding between the layers and laminas takes place. TheCartesian system of coordinates Ox1x2x3 is introdu- ced so that the axis Ox3 is parallel to the planes of the layers and the axis Ox1 is situated on the boundary of the half-space. We do not define the loading of the system so far, but we demand that it ensures the conditions of plane deformation. This makes it possible to consider an arbitrary half-plane per- pendicular to the planes of the layers instead of the three-dimensional body. As such we choose the half-plane x1Ox2. According to the model of homoge- nization with microlocal parameters (Woźniak, 1986, 1987) the displacement in the considered half-plane can be written in the form U1(x1,x2)= u1(x1,x2)+h A(x2)V A 1 (x1,x2) (2.1) U2(x1,x2)= u2(x1,x2)+h A(x2)V A 2 (x1,x2) where the summation convention for the index A = 1, . . . ,n − 1 is accep- ted; u1(x1,x2), u2(x1,x2) are components of the macrodisplacement vector; VAi (x1,x2) – unknown functions of microlocal parameters, i =1,2; h A(x2) – a priori given δ-periodicpiece-wise linear functions (so called shape functions), which will be determined afterwards. To find 2n unknown functions, namely u1(·), u2(·), VA1 (·), VA2 (·) we have a system of 2n equations (Woźniak, 1987) 1 2 〈BαβγδhA,β〉(uγ,δ +uδ,γ)+ 〈BαβγδhA,δhB,γ〉VBβ =0 (2.2) 1 2 〈Bαβγδ〉(uγ,δβ +uδ,γβ)+ 〈BαβγδhA,δ〉V Aγ,β =0 274 I.Machyshyn, W.Nagórko Fig. 1. Scheme of n-layered periodic half-space where the arguments of the functions are omitted in order to simplify the pre- sentation of the expressions; commadenotespartial differentiation; α,β,γ,δ = 1,2; A,B =1, . . . ,n−1; hA,1 =0; the summation convention for all repeated indices are accepted, 〈Bαβγδ〉 are the components of the tensor of elastic con- stants, and the symbol 〈ϕ〉 denotes the averaging of the function ϕ along the thickness of the lamina 〈ϕ〉= 1 δ δ ∫ 0 ϕ(x2) dx2 (2.3) After eliminating 2n − 2 unknown functions VAi from system (2.2) we receive equations of equilibrium for the displacement components u1(·) and u2(·). The obtaining of coefficients of these equations in an explicit form and in the general case of n layers is related with very complicated analytical transformations. In the case of two layers the coefficients of Lamé equations were calculated (Kaczyński and Matysiak, 1987) and the next subsection of this paper is devoted to the case of three layers. 2.2. The equilibrium equation on displacements for a periodic three- layered body Let us consider described in 2.1 stratified half-space in the case of n =3, that is periodically repeated laminaof the thickness δ comprises three layers of the thickness δ1,δ2,δ3 (Fig.2). The Lamé coefficients λ,µ of such a body are Interaction between a stratified elastic half-space... 275 δ-periodic, piece-wise constant functions along the thickness of the half-space (in the direction of the Ox2 axis), and can be written in the form λ(x2)=        λ1 for 0¬ x2 ¬ δ1 λ2 for δ1 ¬ x2 ¬ δ1+ δ2 λ3 for δ1+ δ2 ¬ x2 ¬ δ (2.4) µ(x2)=        µ1 for 0¬ x2 ¬ δ1 µ2 for δ1 ¬ x2 ¬ δ1+ δ2 µ3 for δ1+ δ2 ¬ x2 ¬ δ where λi,µi for i =1,2,3 denote the Lamé coefficients of the ith layer. Fig. 2. Scheme of three-layered periodic half-space By virtue of isotropy of the layers, the components of the tensor of the elastic constants Bαβγξ can be expressed via Lamé coefficients Bαβγξ = λ(x2)δαβδγξ +µ(x2)(δαγδβξ + δαξδβγ) (2.5) where δαβ is the Kronecker symbol. 276 I.Machyshyn, W.Nagórko The functions h1(x2), h 2(x2), given in the form h1(x2)=        x2− 1 2 δ1 for 0¬ x2 ¬ δ1 − δ1 δ2+ δ3 x2− 1 2 δ1+ δ1δ δ2+ δ3 for δ1 < x2 ¬ δ (2.6) h2(x2)=        x2− 1 2 (δ1+ δ2) for 0¬ x2 ¬ δ1+ δ2 −δ1+ δ2 δ3 x2− 1 2 (δ1+ δ2)+ (δ1+ δ2)δ δ3 for δ1+ δ2 < x2 ¬ δ and system (2.2) taking into account relations (2.5), can be written in the form 〈λ+µ〉uα,αβ + 〈µ〉uα,ββ + 〈λhA,β〉VAβ,α+ 〈µhA,β〉V Aα,β + 〈µhA,α〉VAβ,β =0 (2.7) 〈λhA,α〉uβ,β + 〈µhA,β〉uα,β + 〈µhA,β〉uβ,α+ 〈λhA,αhB,β〉V Bβ + +〈µhA,βhB,β〉V Bα + 〈µhA,βhB,α〉V Bβ =0 With the use of relations (2.3), (2.6) we can average the functions 〈ϕ〉= 1 δ [δ1ϕ1+ δ2ϕ2+ δ3ϕ3] 〈ϕh1,2〉= δ1 δ [ ϕ1− δ2ϕ2+ δ3ϕ3 δ2+ δ3 ] 〈ϕh2,2〉= 1 δ [δ1ϕ1+ δ2ϕ2− (δ1+δ2)ϕ3] (2.8) 〈ϕ(h1,2)2〉= 1 δ [ δ1ϕ1+ δ21 (δ2+ δ3)2 (δ2ϕ2+δ3ϕ3) ] 〈ϕ(h2,2)2〉= 1 δ [ δ1ϕ1+ δ2ϕ2+ (δ1+δ2) 2 δ3 ϕ3 ] 〈ϕh1,2h2,2〉= δ1 δ [ ϕ1− δ2ϕ2− (δ1+ δ2)ϕ3 δ2+ δ3 ] where ϕ is arbitrary, δ – periodic function of x2, which equals a constant value ϕi, i =1,2,3 within layer i. After eliminating themicrolocal parameters from relations (2.7) we receive equations on the macrodisplacements A2u1,11+(B∗+C)u2,21+Cu1,22 =0 (2.9) A1u2,22+(B∗∗+C)u1,12+Cu2,11 =0 Interaction between a stratified elastic half-space... 277 where A2 = 〈λ+2µ〉+ G H 〈λh1,2〉+ J H 〈λh2,2〉 A1 = 〈λ+2µ〉+ F H 〈(λ+2µ)h1,2〉+ I H 〈(λ+2µ)h2,2〉 C = 〈µ〉+ L M 〈µh1,2〉+ N M 〈µh2,2〉 (2.10) B∗ = 〈λ〉+ F H 〈λh1,2〉+ I H 〈λh2,2〉 B∗∗ = 〈λ〉+ G H 〈(λ+2µ)h1,2〉+ J H 〈(λ+2µ)h2,2〉 and F = 〈(λ+2µ)h1,2〉〈(λ+2µ)(h2,2)2〉−〈(λ+2µ)h1,2h2,2〉〈(λ+2µ)h2,2〉 G = 〈λh1,2〉〈(λ+2µ)(h2,2)2〉−〈λh2,2〉〈(λ+2µ)h1,2h2,2〉 H = 〈(λ+2µ)h1,2h2,2〉2−〈(λ+2µ)(h1,2)2〉〈(λ+2µ)(h2,2)2〉 I = 〈(λ+2µ)h2,2〉〈(λ+2µ)(h1,2)2〉−〈(λ+2µ)h1,2〉〈(λ+2µ)h1,2h2,2〉 (2.11) J = 〈λh2,2〉〈(λ+2µ)(h1,2)2〉−〈λh1,2〉〈(λ+2µ)h1,2h2,2〉 L = 〈µh1,2h2,2〉〈µh2,2〉−〈µh1,2〉〈µ(h2,2)2〉 M = 〈µ(h1,2)2〉〈µ(h2,2)2〉−〈µh1,2h2,2〉2 N = 〈µh1,2h2,2〉〈µh1,2〉−〈µ(h1,2)2〉〈µh2,2〉 As the obtaining of coefficients (2.10) requires complicated calculations in order to solve this problem, we have involved a powerful instrument of analytical transformations, that is Mathematica 4.0 package. Here is the find result A1 = L1L2L3 η3L1L2+η2L1L3+η1L2L3 A2 = A1+ 4(µ2−µ3)(λ2−λ3+µ2−µ3)(µ2µ3L1+µ1µ3L2+µ1µ2L3) η3L1L2+η2L1L3+η1L2L3 (2.12) 278 I.Machyshyn, W.Nagórko B∗ = B∗∗ ≡ B = η1λ1L2L3+η2λ2L1L3+η3λ3L1L2 η3L1L2+η2L1L3+η1L2L3 C = µ1µ2µ3 η1µ2µ3+η2µ1µ3+η3µ1µ2 where ηi = δi δ Li = λi+2µi i =1,2,3 We can see from the last relations that B∗ = B∗∗ ≡ B. This makes it possible to overwrite equation (2.9) in the form A2u1,11+(B +C)u2,21+Cu1,22 =0 (2.13) A1u2,22+(B +C)u1,12+Cu2,11 =0 whose coefficients are defined by relations (2.12). Note, that allowing for two arbitrary neighbor layers within the lamina to be made of the same material (then we have a two-layered body) coeffi- cients (2.12) equal the corresponding coefficients obtained by Kaczyński and Matysiak (1987) for periodically two-layered body. 3. Model of the contact problem The problem of the contact interaction between the stratified half-plane described in 2.2 and a rigid base with a cylindrical depression taking into account the intercontact gas is formulated in this Section. The interaction is considered under conditions of plane deformation that allows us to replace the bodies by half-planes D2 and D1, which are cross-sections of the bodies perpendicular to the generators of the depression on the surface of the rigid body,Fig.3. TheCartesian systemof coordinates Ox1x2 is introduced so that theaxis Ox1 coincidewith theundeformedboundary D0 of thehalf-plane D2. The upper half-plane D2 is elastic stratified, and is composed of periodic three-layered laminas. The lower half-plane D1 is rigid, and possesses a bo- undary depression within the interval (−b,b), described by a function r(x1), which satisfies the conditions |r(x1)|≪ b |r′(x1)|≪ 1 r(±b)= 0 r′(±b)= 0 (3.1) It means that the depression is supposed to be shallow and smooth. Interaction between a stratified elastic half-space... 279 Fig. 3. Scheme of the contact The half-planes are compressed by the uniformly distributed pressure p∞ at infinity. The intercontact gap, locatedwithin the depression on the symme- tric region of the unknown length 2a is assumed to be filled by the ideal gas, whose state is described by the equation pV = m µ0 RT (3.2) where p, V , T , m, µ0 are pressure, volume, absolute temperature, mass and molar mass of the gas, respectively. R is the absolute gas constant. The contact is supposed to be frictionless. Along the gapnormal stresses in the elastic half-plane are equal to the gas pressure p, which, for the timebeing, is unknown and should be found during solving the problem using equation (3.2) of the gas state. Taking intoaccount that theheight of thedepression is small in comparison with its length (3.1), in the framework of the linear theory of elasticity it is possible to write down boundary conditions on an undeformed boundary of the stratified half-plane that is on the axis Ox1. They are as follows σ12(x1,0)= 0 x1 ∈ D0 σ22(x1,0)=−p |x1| ¬ a (3.3) v(x1,0)= { r(x1) a < |x1| < b 0 |x1| > a (3.4) σ∞22 =−p∞ σ∞12 = σ∞11 =0 √ x21+x 2 2 →∞ (3.5) 280 I.Machyshyn, W.Nagórko 4. Reduction of the problem to the Singular Integral Equation (SIE) Kaczyński and Matysiak (1987) have shown that stresses and displace- ments in a stratifiedbodycanbepresented through twoarbitraryholomorphic functions ϕ and ψ of generalized complex variables zm σ22 =2Re[Φ(z1)+Ψ(z2)] σ12 =2Im[s1Φ(z1)+s2Ψ(z2)] σ11j =2Re[c1jΦ(z1)+ c2jΨ(z2)] u =−2Re[q1ϕ(z1)+q2ψ(z2)] v =−2Re[p1ϕ(z1)+p2ψ(z2)] (4.1) where zm = x1+ismx2 sm = √ A1A2−2BC −B2+(−1)m √ D 2A1C D =(A1A2−2BC −B2)2−4A1A2C2 and we assume that D > 0 cmj = (A2+s 2 mB)Dj − (B +s2mA1)Ej A1A2−B2 Dj = λj λj +2µj A1 Ej = 4µj(λj +µj) λj +2µj + λj λj +µj B (4.2) pm = s2mA1+B A1A2−B2 qm = i s2mB +A2 sm(A1A2−B2) Φ(zm)= ϕ ′(zm) Ψ(zm)= ψ ′(zm) index j =1,2,3 denotes the corresponding layer. It is often convenient to use other presentations through only one complex potential defined not only at the area of body occupation, like the potentials ϕ and ψ in relations (4.1), but also in its complement to the whole plane. In the case of a isotropic half-plane, with the use of the technique of analytical continuation, it was done by Muskhelishvili (1953) and extended on the case of stratified bodies by Kryshtafovych and Matysiak (2001). Expanding the definition of the complex potential Φ(z) to the whole plane by the formula (Kryshtafovych andMatysiak, 2001) Φ(zm)=− s2+s1 s2−s1 Φ(zm)− 2s2 s2−s1 Ψ(zm) Interaction between a stratified elastic half-space... 281 we can overwrite the relations for stresses anddisplacements (4.1) through one piece-wise holomorphic function Φ(z) σ22 =2Re[Φ(z1)+ t1Φ(z2)+ t2Φ(z2)]− (p∞−p) σ12 =2Im[s1Φ(z1)+s2t1Φ(z2)+s2t2Φ(z2)] σ11j =2Re[c1jΦ(z1)+ c2jt1Φ(z2)+ c2jt2Φ(z2)] (4.3) u =2Re[q1Φ(z1)+q2t1Φ(z2)+q2t2Φ(z2)] v =2Re[p1Φ(z1)+p2t1Φ(z2)+p2t2Φ(z2)] where K = s1s2(s1+s2)A1 A1A2−B2 tm = 1 2 [ (−1)ms1 s2 −1 ] m =1,2 (4.4) Having satisfied all boundary conditions (3.3)-(3.5) except for the second condition in (3.3) we received the following expressions for complex potentials via the height h(x) of the intercontact gap Φ(zm)= s2K 2π(s2−s1) ∫ L h′(t)+r′(t) t−zm dt zm ∈ D2 Φ(zm)=− s2K 2π(s2−s1) ∫ L h′(t)+r′(t) t−zm dt zm ∈ D1 (4.5) Satisfying the lastboundarycondition,namely the secondone inconditions (3.3), we obtained the SIE on the derivative of the gap height a ∫ −a h′(t) t−x1 dt =− b ∫ −b r′(t) t−x1 dt+πK(p∞−p)≡ F(x1) x1 ∈ (−a,a) (4.6) If one of the layers occupies the whole lamina and the thickness of two other layers equals zero (η1 = 1, η2 = η3 = 0 or η2 = 1, η1 = η3 = 0 or η3 =1, η1 = η2 =0) then, instead of the stratified half-plane D2, wewill have an isotropic homogeneous one. From relations (4.2), it follows that D = 0, s1 = s2 =1. In this case, presentations (4.1) are nomore valid (Kaczyński and Matysiak, 1987), and in order to solve the problem correctly we should utilize well known presentations of stresses and displacements through the piece- wise holomorphic complex function for an isotropic half-plane (Muskhelishvili, 1953). With the use of such presentations, Martynyak (1985) have derived 282 I.Machyshyn, W.Nagórko the singular integral equation that differs from equation (4.6) only by the coefficient K, which in the isotropic case equals K = λ+2µ 2µ(λ+µ) λ,µ – Lamé coefficients of the isotropic half-plane. In the case of the whole lamina occupied by only one layer we will receive the same expression for the coefficient K if we put s1 = s2 =1 into relations (4.4). Thismeans that in the case of the isotropic half-plane we can also use integral equation (4.6), though we can not use complex representations (4.3). 5. Solution to the problem The elastic half-plane smoothly conjugates with the base at the points x =±a. Therefore, the function h′(x) is equal to zero at these points h′(±a)= 0 (5.1) The corresponding solution of SIE (4.6) is following h′(x1)=− √ a2−x21 π a ∫ −a F(t)√ a2− t2(t−x1) dt (5.2) and exists under the imposed condition (Muskhelishvili, 1953) on the func- tion F(t) a ∫ −a F(t)√ a2− t2 dt =0 (5.3) A depression of the specific form r(x1)=−H √ ( 1− x21 b2 )3 |x1| ¬ b (5.4) was considered. After analytical calculations with the use of relations (4.6) and (5.4) for- mulas (5.2) and (5.3) take the form h′(x1)=− 6H b3 x √ a2−x21 K(p ∞−p)= 3H 2b ( 1− a2 b2 ) (5.5) Interaction between a stratified elastic half-space... 283 The height of the gap at the points of contact must be equal to zero h(a)= h(−a)= 0 (5.6) Integrating relation (5.5)1 and using condition (5.6), we obtain the height of the gap h(x1)= H b3 √ (a2−x21)3 (5.7) Having the height of the gap (5.7) we can calculate its volume per unit along the Ox1 direction V = a ∫ −a h(t) dt = 3 8 Hπ a4 b3 (5.8) To find the unknown parameters of our problem, namely the length of the gap 2a, its volume V and the pressure of the intercontact gas p, we have to solve a system of nonlinear equations (3.2), (5.5)2 and (5.8) pV = m µ0 RT V = 3 8 Hπ a4 b3 (5.9) K(p∞−p)= 3H 2b ( 1− a 2 b2 ) With the aid of relations (4.3), (4.5), (5.4), (5.5)1 we can obtain a formula for the contact pressure pc(x1)= p ∞+ + 3H Kb3 [ x1 sgnx1 ( S−(|x1|−a) √ x21−a2−S−(|x1|− b) √ x21− b2 ) + a2− b2 2 ] where S−(y)= { 0 for y < 0 1 for y ­ 0 284 I.Machyshyn, W.Nagórko 6. Numerical analysis of the solution All the calculations were carried out for the dimensionless variables η1 = δ1 δ η2 = δ2 δ η3 =1−η1−η2 ω = µ1 µ2 γ = µ1 µ3 a = a b H = H b h = h b V = V b2 p = p µ1 p∞ = p∞ µ1 (6.1) K = µ1K = s1s2(s1+s2) A1 µ1 A1 µ1 A2 µ1 − B2 µ2 1 m = m RT µ0µ1b 2 (6.2) The ratio between the share modulus of the first and second layer was taken ω =2, and between the first and third layer – γ =4. Itmeans that the first layer is the stiffest one, while the third layer – the least stiff. Poisson’s ratios of the first, second and third layers were taken ν1 =0.3, ν2 =0.35 and ν3 =0.4, respectively. Themaximum height of the depression was H =0.01. The solid line inFig.4 corresponds to the casewhen themass of the gaswithin the gap equals m =10−6 and the dashed line – to the case of m =10−7. Fig. 4. Half-length of the gap versus the external pressure Interaction between a stratified elastic half-space... 285 We have considered three cases of relative layers thickness. Case I – The first layer occupies the whole elastic half-plane. The thickness of the second and third layer equals 0 (η1 =1, η2 =0). Case II –The thickness of the first layer is 0.5, of the second layer is 0.3 and of the third one 0.2 (η1 =0.5, η2 =0.3). Case III – The third layer occupies the whole elastic half-plane. The thick- ness of the first and second layers equals 0 (η1 =0, η2 =0). Fig. 5. Volume of the gap versus the external pressure Fig. 6. Contact pressure distribution 286 I.Machyshyn, W.Nagórko In Fig.4 the dependence of the gap half-length on the external pressure is presented. We can see that the length of the gap greatly depends on the relative thickness of the layers as well as on the mass of the gas. The stiffer combination of layers we have, the longer gap it is. Similarly, themore gas we put in thegap, thegreater gapweget.The samedependenceof the relative line location on the geometrical parameters of the layers and the mass of the gas can be seen inFig.5, where the volume of the gap versus the external pressure is presented. Nevertheless, the shape of the lines differs from the one shown in previous figure. The contact pressure distributions under the external load p∞ =0.018 is shown in Fig.6. At the ends of the initial depression we can see peaks of the contact pressurewhich are caused by the form of depression. The horizontal areas correspond to the pressure in the gap and their size, i.e. to the length of the gap. For the stiffest combination of the relative thickness of the layers the gas pressure is smallest and the peak values are biggest, while vice versa for the least stiff combination. 7. Conclusions Numerical analysis of the solution to the problem revealed that the relative thickness of the layers of a stratified body and the amount of the gas within the intercontact gap have considerable influence on both contact pressure di- stribution and geometrical characteristics of the gap. It was established that the increasing of the gas amount and stiffness of the lamina enlarges the gap. Acknowledgement I.Machyshyn isgrateful toJ.MianowskiFund for the supportof thisworkthrough the scientific grant. References 1. BahvalovN.S., PanasenkoG.P., 1984,AveragingMethods for Processes in Periodic Bodies, Nauka,Moscow (in Russian) 2. Christensen R.M., 1979, Mechanics of Composite Materials, Wiley Inter- science Publ. NewYork 3. Johnson K.L., 1985,Contact Mechanics, Cambridge University Press, Cam- bridge Interaction between a stratified elastic half-space... 287 4. 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Nagórko W., 1989, On modelling of thin microperiodic plates, Mechanica Teoretyczna i Stosowana, 27, 2, 293-302 16. ShvetsR.M.,MartynyakR.M.,KryshtafovychA.A., 1996,Discontinu- ous contact of an anisotropic half-plane and rigid base with disturbed surface, Int. J. Eng. Sci., 34, 184-200 17. Woźniak C., 1986, Nonstandard analysis in mechanics, Advances in Mecha- nics, 3-35 18. Woźniak C., 1987, A nonstandard method of modelling of thermoelastic pe- riodic composites, Int. J. Engng. Sci., 25, 483-499 288 I.Machyshyn, W.Nagórko Oddziaływanie sprężystej półprzestrzeni warstwowej na sztywną przeszkodę oddzieloną szczeliną wypełnioną gazem Streszczenie W pracy analizuje się oddziaływanie sprężystej półprzestrzeni warstwowo- niejednorodnej na ciało sztywne zwalcowąszczelinąwypełnionągazem.Półprzestrzeń sprężystą poddaje się homogenizacji mikrolokalnej, a do rozwiązania otrzymanych równań modelowych stosuje się metodę potencjałów zespolonych. Analizowany pro- blem sprowadza się do poszukiwania rozwiązania równania całkowego.W pracy uzy- skuje się rozwiązania w postaci zamkniętej oraz przeprowadza analizę numeryczną zależności rozwartości szczeliny oraz jej objętości od obciążenia zewnętrznego. Manuscript received October 7, 2002; accepted for print January 14, 2003