Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 3, 39, 2001 ON 3D PUNCH PROBLEMS FOR A PERIODIC TWO-LAYERED ELASTIC HALF-SPACE Andrzej Kaczyński Faculty of Mathematics and Information Science, Warsaw University of Technology e-mail: akacz@alpha.mini.pw.edu.pl Stanisław J. Matysiak Faculty of Geology, University of Warsaw e-mail: matysiak@geo.uw.edu.pl Within the framework of the linear elasticity with microlocal parameters three-dimensional contact problems for a half-space region consisting of al- ternating layers of two homogeneous, isotropic and linear-elastic materials are examined. Effective results can be obtained on the basis of similarity in governing equations for the homogenized model of the laminated medium and transversely isotropic elastic solid. Key words: periodic two-layered half-space, rigid punch, integral equation 1. Introduction Considerable progress has been made with the modelling and analysis of contact problems.Extensive accounts can be found in the books byShtaerman (1949), Galin (1953, 1980), Rvachev and Protsenko (1977), Gladwell (1980), Johnson (1985), Mossakovskii et al. (1985), Goryacheva andDobykhin (1988) and in the recent proceedings by Raous et al. (1995). Willis (1966) and Hwu and Fan (1998) made significant contributions to the development of research for the contact of anisotropic bodies. Complete and new solutions to seve- ral three-dimensional contact problems were presented in twomonographs by Fabrikant (1989, 1991). Thispaper is devoted to theanalysis of three-dimensional contact problems for a periodic two-layered elastic half-space. It is a sequel of our earlier investi- gations in the two-dimensional case (seeKaczyński andMatysiak, 1988, 1993). 524 A.Kaczyński, S.J.Matysiak The study is based on the use of the homogenized model of the micro- periodic two-layered composite, proposed by Woźniak (1987), Matysiak and Woźniak (1988). In Section 2 we review briefly the governing equations of this model in the three-dimensional case of the linear elasticity with micro- local parameters. Due to close similarity to the fundamental equations for a transversely isotropic elastic solid, the general solutions in terms of harmonic potentials, well suited to contact problems, are constructed. Section 3 presents a general elastostatic contact problem of indentation of the two-layered half- space by a frictionless rigid smooth punch treated within the homogenized model. According to the analysis, the well-known governing integral equation of the elastic contact problem similar to that in the case of contact on a transversely isotropic half-space is obtained. This factmay be utilised to yield closed-form solutions following directly from those given, for example, by Fa- brikant (1989). The complete solution in the case of a flat centrally loaded circular punch is presented in Section 4. 2. Governing equations We consider a three-dimensional static contact problem of a two-layered microperiodic elastic half-space with a rigid smooth punch (see Fig.1). Let λl, µl be Lamé’s constants and δl be the thicknesses of the subsequent layers; in the following, all the quantities (material constants, stresses, etc.) with the index l or (l) are related to the layers denoted by l = 1 or l = 2. The Cartesian coordinate system (x1,x2,x3) is devisedwith the x3-axis normal to the layering and the x1x2-plane of boundary.Referring to this system, denote at the point x=(x1,x2,x3) the displacement vector by u= [u1,u2,u3] and the stresses by σ11, σ12, σ22, σ13, σ23, σ33. To analyse the problem of a punch penetrating this layered body we ta- ke into consideration the specific homogenization procedure called microlocal modelling, proposed byWoźniak (1987) and then developed byMatysiak and Woźniak (1988), applicable to a certainmacro-homogeneousmodel of the tre- ated body.We present only a brief outline of its governing equations. The homogenized model of the layered body under study is characterised by the shape function with the period δ= δ1+ δ2, defined as h(x3)=    x3− δ1 2 for x3 ∈ 〈0,δ1〉 δ1−ηx3 1−η − δ1 2 for x3 ∈ 〈δ1,δ〉 η= δ1 δ (2.1) On 3D punch problems for a periodic... 525 Fig. 1. Periodic two-layered half-space indented by a frictionless rigid punch Note that the values of this function are small whereas the values of its derivative h′ =    1 if x belongs to the 1st layer − η 1−η if x belongs to the 2nd layer are not small even for very thin layers. The following representations and approximations are postulated within the elasticity with microlocal parameters1 ui =wi+h(x3)di ≈wi ui,α ≈wi,α u (l) i,3 ≈wi,3+h ′di σ (l) αβ ≈µl(wα,β +wβ,α)+ δαβλl(wi,i+h′d3) (2.2) 1Indices i,j run over 1,2,3 while α,β,γ run over 1,2. They are related to the Cartesian coordinates. The summation convention holds for both kinds of the afore- mentioned indices. 526 A.Kaczyński, S.J.Matysiak σ (l) α3 ≈µl(wα,3+w3,α+h ′dα) σ (l) 33 ≈ (λl+2µl)(w3,3+h ′d3)+λlwγ,γ Here δαβ is theKronecker delta, wi and di areunknown functions interpreted as the macro-displacements andmicrolocal parameters, respectively. Following the special homogenization procedure (cf Woźniak, 1987), we arrive at the governing equations and constitutive relations of a certainmacro- homogeneous medium (the homogenized model), given (after eliminating the microlocal parameters and in the absence of the body forces) in terms of the macro-displacements wi as follows (see Kaczyński, 1993) 1 2 (c11+ c12)wγ,γα+ 1 2 (c11− c12)wα,γγ + c44wα,33+(c13+ c44)w3,3α =0 (c13+ c44)wγ,γ3+ c44w3,γγ + c33w3,33 =0 σα3 = c44(wα,3+w3,α) σ33 = c13wα,α+ c33w3,3 (2.3) σ (l) 12 =µl(w1,2+w2,1) σ (l) 11 = d (l) 11w1,1+d (l) 12w2,2+d (l) 13w3,3 σ (l) 22 = d (l) 12w1,1+d (l) 11w2,2+d (l) 13w3,3 Positive coefficients appearing in the above equations are given in the Ap- pendix. They depend on the material and geometrical characteristics of the subsequent layers. It is noteworthy that the condition of perfect bonding be- tween the layers (the continuity of the stress vector at the interfaces) is sa- tisfied. We also observe that setting λ1 = λ2 ≡ λ, µ1 = µ2 ≡ µ entails c11 = c33 = λ+2µ, c12 = c13 = λ, c44 = µ and the well-known equations of the elasticity for a homogeneous isotropic body with Lamé’s constants λ, µ are recovered. The general solutions to governing equations (2.3) in terms of three har- monic poten- tials have becomepossible due to close similarity to the displace- ment and stress-displacement relations for a transversely isotropic solid (see, for example, Kassir and Sih, 1975). According to the results obtained by Ka- czyński (1993), the form of the potential representations is dependent on the material constants of the layers and is given below in two cases2. 2The constants ti,mα are defined in Appendix. On 3D punch problems for a periodic... 527 Case 1: µ1 6=µ2 The displacement field can be expressed through the three potentials ϕ̂i(x1,x2,zi), in which zi = tix3, such that ∇2ϕ̂i ≡ ( ∂2 ∂x21 + ∂2 ∂x22 + ∂2 ∂z2i ) ϕ̂i ∀i∈{1,2,3} as follows w1 =(ϕ̂1+ ϕ̂2),1− ϕ̂3,2 w2 =(ϕ̂1+ ϕ̂2),2+ ϕ̂3,1 (2.4) w3 =m1t1 ∂ϕ̂1 ∂z1 +m2t2 ∂ϕ̂2 ∂z2 From stress-displacement relations (2.3), the stresses σ3i can be expres- sed as σ31 = c44 [ (1+m1)t1 ∂ϕ̂1 ∂z1 +(1+m2)t2 ∂ϕ̂2 ∂z2 ] ,1 − t3 ∂2ϕ̂3 ∂z3∂x2 σ32 = c44 [ (1+m1)t1 ∂ϕ̂1 ∂z1 +(1+m2)t2 ∂ϕ̂2 ∂z2 ] ,2 + t3 ∂2ϕ̂3 ∂z3∂x1 (2.5) σ33 = c44 [ (1+m1) ∂2ϕ̂1 ∂z21 +(1+m2) ∂2ϕ̂2 ∂z22 ] Formulas for the remaining stresses σ (l) αβ are not of immediate interest and have been omitted. Case 2: µ1 =µ2 ≡µ, λ1 6=λ2 Here, the displacement equations take the classical form (B+µ)wj,ji+µwi,jj =0 (2.6) provided B= λ1λ2+2µ[ηλ1+(1−η)λ2] (1−η)λ1+ηλ2+2µ (2.7) and the representation in terms of the three harmonic functions ϕi(x1,x2,x3) (satisfying ∇2ϕi ≡ϕi,jj =0 ∀ i∈{1,2,3}) is given as follows 528 A.Kaczyński, S.J.Matysiak w1 =(ϕ1+x3ϕ2),1−ϕ3,2 w2 =(ϕ1+x3ϕ2),2+ϕ3,1 w3 =ϕ1,3+x3ϕ2,3− B+3µ B+µ ϕ2 σ31 =2µ [ ϕ1,3− µ B+µ ϕ2+x3ϕ2,3 ] ,1 −µϕ3,23 (2.8) σ32 =2µ [ ϕ1,3− µ B+µ ϕ2+x3ϕ2,3 ] ,2 +µϕ3,13 σ33 =2µ [ ϕ1,33− B+2µ B+µ ϕ2,3+x3ϕ2,33 ] Putting in the above case λ1 = λ2 ≡ λ implies B = λ, passing to the known representation for the homogeneous isotropic bodywith the Lamé con- stants λ, µ. 3. Frictionless contact problem Consider the general problem of indentation of the two-layered periodic half-space by a frictionless smooth rigid punch. Let S is the known contact area (see Fig.1). Within the framework of the homogenized model presented in Section 2 we can formulate the following mixed conditions on the entire plane x3 =0, denoted by Z w3(x1,x2,0)=ω(x1,x2) ∀(x1,x2)∈S σ33(x1,x2,0)= 0 ∀(x1,x2)∈Z−S σ31(x1,x2,0)=σ32(x1,x2,0)= 0 ∀(x1,x2)∈Z (3.1) where ω(x1,x2) is a known function that describes the profile of the punch. Wenowproceed to construct thepotential functionswell suited to the abo- ve mixed boundary conditions. This will be done by using the same potential representation as in Kaczyński (1993), for the corresponding crack problem with symmetric loading. Case 1: µ1 6=µ2 The potentials are expressed in terms of a single harmonic function f̂(x1,x2,x3) (i.e. ∇2f̂ ≡ f̂,jj =0) as follows On 3D punch problems for a periodic... 529 ϕ̂1(x1,x2,z1)=− 1 t1(1+m1) f̂(x1,x2,z1) ϕ̂2(x1,x2,z2)= 1 t2(1+m2) f̂(x1,x2,z2) (3.2) ϕ̂3 ≡ 0 SubstitutionEqs (3.2) intoEqs (2.4) andmakinguse ofEqs (2.3) yields the following representations of the displacements wi and stresses σ3i in terms of the potential f̂ wα(x1,x2,x3)= 2∑ k=1 (−1)k tk(1+mk) f̂,α(x1,x2,zk) w3(x1,x2,x3)= m2 1+m2 ∂ ∂z2 f̂(x1,x2,z2)− m1 1+m1 ∂ ∂z1 f̂(x1,x2,z1) (3.3) σ3α(x1,x2,x3)= c44 [ ∂ ∂z2 f̂(x1,x2,z2)− ∂ ∂z1 f̂(x1,x2,z1) ] σ33(x1,x2,x3)= c44 [ 1 t2 ∂2 ∂z22 f̂(x1,x2,z2)− 1 t1 ∂2 ∂z21 f̂(x1,x2,z1) ] The remaining stresses (discontinuous on the interfaces) are found to be σ (l) 11(x1,x2,x3)= 2∑ k=1 (−1)k tk(1+mk) · · [ d (l) 11 f̂,11(x1,x2,zk)+d (l) 22f̂,22(x1,x2,zk)+d (l) 13mkf̂,33(x1,x2,zk) ] σ (l) 22(x1,x2,x3)= 2∑ k=1 (−1)k tk(1+mk) · (3.4) · [ d (l) 12 f̂,11(x1,x2,zk)+d (l) 11f̂,22(x1,x2,zk)+d (l) 13mkf̂,33(x1,x2,zk) ] σ (l) 12(x1,x2,x3)= 2µl 2∑ k=1 (−1)k tk(1+mk) f̂,12(x1,x2,zk) It easily follows from Eqs (3.3) that on the boundary x3 = 0 (then z1 = z2 =0, ∂f̂(x1,x2,zα)/∂zα = ∂f̂(x1,x2,x3)/∂x3) the third condition in (3.1) σ3α = 0 is satisfied. In addition, in view of Eqs (3.3), the components of the 530 A.Kaczyński, S.J.Matysiak displacement and stress that act along the x3-axis on the surface x3 = 0 + take the form w3(x1,x2,0)= ( m2 1+m2 − m1 1+m1 )[ f̂,3(x1,x2,x3) ] x3=0 (3.5) σ33(x1,x2,0)= c44 ( 1 t2 − 1 t1 )[ f̂,33(x1,x2,x3) ] x3=0 The above relations reduce the contact problem given by Eqs (3.1) to the classical mixed problem (cf Sneddon, 1966) for finding the harmonic func- tion f̂ in the half-space x3 ­ 0, which vanishes at infinity and satisfies the boundary conditions [ f̂,3(x1,x2,x3) ] x3=0 = (1+m1)(1+m2) m2−m1 ω(x1,x2) ∀(x1,x2)∈S [ f̂,33(x1,x2,x3) ] x3=0 =0 ∀(x1,x2)∈Z−S (3.6) Case 2: µ1 =µ2 ≡µ, λ1 6=λ2 The solution to Eqs (2.6) in terms of one harmonic function f with the assumption that the boundary x3 = 0 is free from tangential stresses is achieved by taking in Eqs (2.8) ϕ1 = µ B+µ f ϕ2 = f,3 ϕ3 =0 (3.7) Then it follows from Eqs (2.8) that the displacement and stress components are wα = µ B+µ f,α+x3f,3α w3 =− B+2µ B+µ f,3+x3f,33 σ3α =2µx3f,α33 σ33 =2µ(−f,33+x3f,333) σ (l) 11 =2µ(D (l) 11f,11+D (l) 12f,22+x3f,113) (3.8) σ (l) 22 =2µ(D (l) 12f,11+D (l) 11f,22+x3f,223) σ (l) 12 =2µl ( µ B+µ f,12+x3f,123 ) On 3D punch problems for a periodic... 531 where D (l) 11 =1+ 2µ(λ1−B) (B+µ)(λ1+2µ) D (l) 12 = λ1 B+µ ( 1+ λ1−B λ1+2µ ) (3.9) The following expressions are found on the plane x3 =0 w3 =− B+2µ B+µ f,3 σ33 =−2µf,33 (3.10) Application of conditions (3.1) yields a similar problem to that appearing in Eqs (3.6) in finding the harmonic function f [ f̂,3(x1,x2,x3) ] x3=0 =−B+2µ B+µ ω(x1,x2) ∀(x1,x2)∈S [ f̂,33(x1,x2,x3) ] x3=0 =0 ∀(x1,x2)∈Z−S (3.11) Themixed boundary-value problems for the harmonic functions f̂ in Ca- se 1 and f in Case 2 can be reduced to integral equations by using the re- presentations of their first x3-derivatives through the potentials of the simple layer, namely f̂,3(x1,x2,x3)= ∫∫ S σ̂0(x,y) dxdy√ (x1−x)2+(x2−y)2+x23 (3.12) f,3(x1,x2,x3)= ∫∫ S σ0(x,y) dxdy√ (x1−x)2+(x2−y)2+x23 where the unknown functions σ̂0 and σ0 will be determined from the well- known properties of these potentials f̂,33 ∣∣∣ x3=0 = { −2πσ̂0(x1,x2) ∀(x1,x2)∈S 0 ∀(x1,x2)∈Z−S (3.13) f,33 ∣∣∣ x3=0 = { −2πσ0(x1,x2) ∀(x1,x2)∈S 0 ∀(x1,x2)∈Z−S Notice that the second condition in Eqs (3.6) and (3.11) is satisfied and the first one gives, in view of Eqs (3.5) and (3.10), the equations for σ̂0 and σ0 t1t2 c44(t1− t2) σ33(x1,x2,0)=−2πσ̂0(x1,x2) Case 1 − 1 2µ σ33(x1,x2,0)=−2πσ0(x1,x2) Case 2 (3.14) 532 A.Kaczyński, S.J.Matysiak Hence, denoting the normal contact traction σ33(x1,x2,0)≡ p3(x1,x2) on S, we obtain (see Appendix) σ̂0(x1,x2)= t1t2 2πt−c44 p3(x1,x2) Case 1 σ0(x1,x2)= 1 4πµ p3(x1,x2) Case 2 (3.15) Substitution of Eqs (3.15) in Eqs (3.12) leads to the potentials expressed by p3(x1,x2) (unknown function) as follows f̂,3(x1,x2,x3)= t1t2 2πt−c44 ∫∫ S p3(x,y) dxdy√ (x1−x)2+(x2−y)2+x23 Case 1 f,3(x1,x2,x3)= 1 4πµ ∫∫ S p3(x,y) dxdy√ (x1−x)2+(x2−y)2+x23 Case 2 (3.16) Finally, satisfaction of the first conditions inEqs (3.6) and (3.11) yields the governing integral equationof the considered contact problemfor a two-layered periodic half-space −H ∫∫ S p3(x,y) dxdy√ (x1−x)2+(x2−y)2 =ω(x1,x2) (3.17) where H is the same constant as used byFabrikant (1989) in study of contact on a transversely isotropic half-space, here taking on the values H =    t1t2 2πc44t− m1−m2 (1+m1)(1+m2) = t+ √ c11c33 2π(c11c33−c213) Case 1 1 4πµ B+2µ B+µ Case 2 (3.18) Once the contact stresses p3(x1,x2) are known from the solution of the above integral equation, the complete displacement and stress fields can be written down using Eqs (3.3), (3.4) in Case 1 and Eqs (3.8) in Case 2 with themain potentials f̂ and f, determined fromEqs (3.16) by integrating with respect to x3 f̂(x1,x2,x3)= = t1t2 2πt−c44 ∫ ∫ S ln [√ (x1−x)2+(x2−y)2+x23+x3 ] p3(x,y) dxdy On 3D punch problems for a periodic... 533 f(x1,x2,x3)= (3.19) = 1 4πµ ∫ ∫ S ln [√ (x1−x)2+(x2−y)2+x23+x3 ] p3(x,y) dxdy Integral equation (3.17) has been widely known, but its solution presents considerable difficulties. However, marked progress has been made by Fabri- kant (1989, 1991) in obtaining exact and complete solutions to various contact problems (in elementary functions) for a circular punch of anypolynomial pro- file. Owing to the same governing equation (3.17), these solutions will be used for solving the corresponding problems of contact on a periodic two-layered half-space within the framework of the elasticity with microlocal parameters, presented in Section 2. For the sake of simplicity, the results will be presented for the simplest case of indentation by a flat punch in the next section. 4. Example: flat centrally loaded circular punch Consider the casewhenaflat rigid circular punchof the radius a is pressed against a two-layered periodic elastic half-space x3 ­ 0 by the centrally applied normal force P. This problem is characterised by mixed boundary conditions (3.1) with the contact area S = {(x1,x2) : ρ2 ≡ x21 +x22 ¬ a2} and the punch settlement ω(x1,x2)= const ≡ω0 > 0. Several methods of solving axisymmetric punch problems were reported in the literature (see for example a review by Barber, 1992). A wide range of new investigations in the field of contact problems related to a transversely isotropic body and directed to obtain complete solutions has been carried out by Fabrikant (1989, 1991). At present, by making use of his results, we present the exact solution of the posed problem within the framework of the homogenized model. The solution to governing integral equation (3.17) gives the contact stresses p3(x1,x2)=σ3(x1,x2,0)=− ω0 π2H √ a2−x21−x22 (4.1) The total force P is related with the punch settlement ω0 by the rela- tionship P =− ∫∫ S p3(x1,x2) dx1dx2 = 2ω0a πH (4.2) 534 A.Kaczyński, S.J.Matysiak Nowwe substituteEq. (4.1) intoEqs (3.16) in order tofindthemainpoten- tial functions giving a complete solution. Themethod of Fabrikant yields the results in elementary functions as follows (for Case 1 andCase 2, respectively) f̂(x1,x2,x3)=− t1t2ω0 π2Ht−c44 [ x3arcsin a l2 − √ a2− l21 +a ln(l2+ √ l22−ρ2) ] f(x1,x2,x3)=− ω0 2π2Hµ [ x3arcsin a l2 − √ a2− l21+a ln(l2+ √ l22 −ρ2) ] where in his notation l1 ≡ l1(a,ρ,x3)= 1 2 [√ (ρ+a)2+x23− √ (ρ−a)2+x23 ] (4.3) l2 ≡ l2(a,ρ,x3)= 1 2 [√ (ρ+a)2+x23+ √ (ρ−a)2+x23 ] Appropriate differentiation of the above potentials (see Appendix 5 in the book by Fabrikant, 1991) and thenmaking use of Eqs (3.3) in Case 1 andEqs (3.8) inCase 2 give the complete displacement and stress field in the following concise form: Case 1 wα = 2ω0axα πρ2 2∑ k=1 1 tk(mk−1) [ 1− √ a2− l21k a ] w3 = 2ω0 π 2∑ k=1 mk mk−1 arcsin a l2k (4.4) σ3α = ω0t1t2xα π2Ht−ρ 2∑ k=1 (−1)2 l1k √ a2− l21k l2k(l 2 2k − l21k) σ33 = ω0t1t2 π2Ht− 2∑ k=1 (−1)2 √ a2− l21k tk(l 2 2k− l21k) Here the notations l1k and l2k for k = 1,2 are understood as l1(a,ρ,zk) and l2(a,ρ,zk), respectively. The evaluation of σ (l) αβ is not given because of the complexity. Case 2 wα = 2ω0xα πρ B+µ B+2µ [ − µ B+µ a− √ a2− l21 ρ + x3l1 √ l22−a2 l2(l 2 2 − l21) ] On 3D punch problems for a periodic... 535 w3 = 2ω0 π [ arcsin a l2 + B+µ B+2µ x3 √ a2− l21 l2(l 2 2 − l21) ] (4.5) σ3α =− ω0µxαx3(B+µ) π(B+2µ) √ a2− l21(3l22 + l21 −4a2) (l22 − l21)3 σ33 = ω0µ(B+µ) π(B+2µ) { − √ a2− l21 l22 − l21 + x23[l 4 1 +a 2(ρ2−2a2−2x23)]√ a2− l21(l22 − l21)3 It is of interest to record the normal displacement and stress distribution of the boundary x3 =0. Taking into account that l1 ∣∣∣ x3=0 = l1k ∣∣∣ x3=0 =min(a,ρ) l2 ∣∣∣ x3=0 = l2k ∣∣∣ x3=0 =max(a,ρ) one obtains w3(x1,x2,0)=    ω0 if x 2 1+x 2 2 ¬ a2 2ω0 π arcsin a √ x21+x 2 2 if x21+x 2 2 >a 2 (4.6) σ33(x1,x2,0)=    − ω0 π2H √ a2−x21−x22 if x21+x 2 2 a 2 with H definedbyEqs (3.18). Assuming λ1 =λ2 ≡λ inCase 2we obtain the well-known solution of the contact problem under study for a homogeneous isotropic elastic half-space with Lame’s constants λ and µ. 5. Conclusion The three-dimensional contact problem for a periodic two-layered half- space has been investigated within the homogenized model with microlocal parameters. The governing integral equation of this problem turns out to ha- ve the classical form well known from the consideration of the corresponding 536 A.Kaczyński, S.J.Matysiak problem of an arbitrary frictionless rigid punch pressed against a transversely isotropic elastic half-space. Hence, complete solutions to several punch pro- blems, which were included in Fabrikant (1989, 1991), may be extended and adopted in the case of contact on the laminar half-space under study. A. Appendix • Denoting by η = δ1/δ, bl = λl +2µl (l = 1,2), b = (1− η)b1 + ηb2, the positive coefficients in governing equations (2.3) are given by the following formulae c11 = c33+ 4η(1−η)(µ1−µ2)(λ1−λ2+µ1−µ2) b c13 = (1−η)λ2b1+ηλ1b2 b c33 = b1b2 b c12 = λ1λ2+2[ηµ2+(1−η)µ1][ηλ1+(1−η)λ2] b c44 = µ1µ2 ηµ2+(1−η)µ1 d (l) 13 = λlc33 bl d (l) 11 = 4µl(λl+µl)+λlc13 bl d (l) 12 = 2µlλl+λlc13 bl • The constants appearing in Eqs (2.4) and (2.5) are given as follows t1 = 1 2 (t+− t−) t2 = 1 2 (t++ t−) t3 = √ ηµ1+(1−η)µ2 c44 mα = c11t −2 α −c44 c13+ c44 ∀α∈{1,2} where t± = √ (A±±2c44)A∓ c33c44 A± = √ c11c33±c13 Note that t1t2 = √ c11/c33, m1m2 =1. On 3D punch problems for a periodic... 537 References 1. Barber J.R., 1992, Elasticity, Kluwer Academic Publishers, Dordrecht- Boston-London 2. Fabrikant V.I., 1989,Applications of Potential Theory in Mechanics. A Se- lection ofNewResults,KluwerAcademicPublishers,Dordrecht-Boston-London 3. Fabrikant V.I., 1991, Mixed Boundary Value Problem of Potential Theory and their Applications inEngineering,KluwerAcademicPublishers,Dordrecht- Boston-London 4. Galin L.A., 1953,Contact Problems in the Theory of Elasticity, (in Russian), Gostekhteorizdat,Moscov 5. Galin L.A., 1980,Contact Problems of the Theory of Elasticity and Viscoela- sticity, (in Russian), Izd. Science, Moscov 6. GladwellG.M.L., 1980,Contact Problems in the Classical Theory of Elasti- city, Sijthoff and Noordhoff, Amsterdam 7. Goryacheva I.G., Dobykhin M.N., 1988, Contact Problems in Tribology, (in Russian), Machine-Construction,Moscow 8. Hwu C., Fan C.W., 1998, Contact Problems of Two Dissimilar Anisotropic Elastic Bodies,ASME Journal of Applied Mechanics, 65, 580-587 9. Johnson K.L., 1985,Contact Mechanics, Cambridge University Press, Cam- bridge 10. Kaczyński A., 1993, On the 3-D Interface Crack Problems in Periodic Two- Layered Composites, Int. J. Fracture, 62, 283-306 11. Kaczyński A., Matysiak S.J., 1988, Plane Contact Problems for a Periodic Two-Layered Elastic Composite, Ingenieur-Archiv, 58, 137-147 12. KaczyńskiA.,Matysiak S.J., 1993,Rigid SlidingPunchon aPeriodicTwo- Layered Elastic Half-Space, J. Theor. Appl. Mech., 31, 2, 295-306 13. KassirM.K., SihG.C., 1975,Three-DimensionalCrackProblems,Mechanics of Fracture, 2, Noordhoff Int. Publ., Leyden 14. Matysiak S.J.,WoźniakC., 1988,On theMicrolocalModelling of Thermo- elastic Periodic Composites, J. Tech. Phys., 29, 85-97 15. Mossakovskii V.I., Kachalovskaia N.E., Golikova S.S., 1985,Contact Problems of the Mathematical Theory of Elasticity, (in Russian), Izd. Naukova Dumka, Kiev 16. RaousM., JeanM.,Moreau J.J., 1995,Contact Mechanics, PlenumPress, NewYork 538 A.Kaczyński, S.J.Matysiak 17. Rvachev V.L., Protsenko V.S., 1977, Contact Problems of the Theory of Elasticity for Non-Classical Domains, (in Russian), Izd. NaukovaDumka, Kiev 18. ShtaermanA.I., 1949,Contact Problems of the Theory of Elasticity, (inRus- sian), Gostekhteorizdat,Moscow 19. Sneddon I.N., 1966, Mixed Boundary Value Problems in Potential Theory, North-Holland Publ. Co., Amsterdam 20. Willis J.R., 1966, Hertzian Contact of Anisotropic Bodies, J. Mech. Phys. Solids, 14, 163-176 21. Woźniak C., 1987, A Nonstandard Method of Modelling of Thermoelastic Periodic Composites, Int. J. Engng Sci., 25, 483-499 O trójwymiarowych zagadnieniach kontaktowych dla periodycznej dwuwarstwowej półprzestrzeni sprężystej Streszczenie Wramachliniowej teorii sprężystości zparametramimikrolokalnymizbadanokon- taktowe zagadnienia przestrzenne dotyczące wciskania stempla w periodycznie dwu- warstwowąpółprzestrzeń sprężystą. Efektywnewyniki uzyskano dzięki podobieństwu rządzącychrównańmodelu zhomogenizowanegopółprzestrzeni z równaniamidla ciała sprężystego z poprzeczną izotropią. Manuscript received December 15, 2000; accepted for print January 19, 2001