Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 1, pp. 89-105, Warsaw 2003 ADHESIVE CONTACT PROBLEM FOR TRANSVERSELY ISOTROPIC ELASTIC HALF-SPACE Marcin Pawlik Bogdan Rogowski Department of Mechanics of Materials, Technical University of Łódź e-mail: marcinp@kmm-lx.p.lodz.pl; brogowsk@ck-sg.p.lodz.pl The problem of adhesive contact for a transversely isotropic elastic half- space is considered.Theproblemis reduced to the solutionof twocoupled integral equations, and these are solved exactly. Explicit expressions are found for the contact compliance and for coefficients which characteri- se the singularities of contact stresses at the boundary of the contact region. The numerical results presented for some anisotropic materials show that the influence of anisotropy on the analysedmechanical quan- tities is significant. Key words: adhesive contact, anisotropy, integral equations, compliance, stress concentration factors 1. Introduction The problem of adhesive contact can be solved by the use of the Hankel transformsand the subsequentuse of theWeiner-Hopf technique.Theproblem was first solved byMossakovskǐı (1954) and thenwas considered byAbramian et al. (1956) and Spence (1968a,b). The solutions of many adhesive contact problems can be found in Gladwell’s book (1980). These solutions are related to isotropic materials. The adhesive contact problem in the context of a transversely isotropic elastic stratum is considered in this paper. Many of fiber-reinforced, platelet and laminated systems, some soils and, of course, a number of crystalic and other real materials have transversely isotropic mechanical properties. The present paper clarifies the effect of anisotropy on the mechanical quantities under consideration in the adhesive contact problem. 90 M.Pawlik, B.Rogowski 2. Basic elasticity equations and their solutions The axially symmetric problem of elasticity can be analysed by means of displacement functions, which are governed by differential equations ∇2iϕi(r,siz)= 0 i =1,2 (2.1) where ∇2i is Laplace’s operator referred to the cylindrical polar co-ordinate system (r,θ,zi) with zi = siz, where s1 and s2 are the parameters of a transversely isotropic medium. The displacement and stress can be uniquely expressed in terms of these displacement functions (Rogowski, 1975). The solution to equations (2.1) may be presented as a Hankel’s (in terms of r) representation of the harmonic functions ϕi(r,siz) in the domain (r,siz), as follows ϕi(r,siz)= ϑi ∞∫ 0 ξ−1Hi(ξsiz)J0(ξr) dξ (2.2) where ϑ1 =− s2 Gz(k+1)(s1−s2) ϑ2 = s1 Gz(k+1)(s1−s2) (2.3) Hi(ξsiz)= Ai(ξ)e −ξsiz and where Gz is the shear modulus along the axis of elastic symmetry of the material (z-axis) that has five components of the elastic stiffness cij or three equivalent parameters s1, s2 and k. The corresponding displacement and stress components take the form ur(r,z) = 1 Gz(k+1)(s1−s2) ∞∫ 0 [ ks2A1(ξ)e −ξs1z −s1A2(ξ)e−ξs2z ] J1(ξr) dξ (2.4) uz(r,z) = s1s2 Gz(k+1)(s1−s2) ∞∫ 0 [ A1(ξ)e −ξs1z −kA2(ξ)e−ξs2z ] J0(ξr) dξ σzz(r,z) =− 1 s1−s2 ∞∫ 0 ξ [ s2A1(ξ)e −ξs1z −s1A2(ξ)e−ξs2z ] J0(ξr) dξ (2.5) σrz(r,z) =− s1s2 s1−s2 ∞∫ 0 ξ [ A1(ξ)e −ξs1z −A2(ξ)e−ξs2z ] J1(ξr) dξ Adhesive contact problem for... 91 where Ai(ξ) (i =1,2) are arbitrary constants and Jν(ξr), (ν =0,1) are the Bessel functions. By using the substitutions A1(ξ)= s1t̂(ξ)− p̂(ξ) (2.6) A2(ξ)= s2t̂(ξ)− p̂(ξ) we can transform these equations to give the displacements and stress on the plane z =0, and the resulting equations are then ur(r,0)= 1 GzC [∞∫ 0 t̂(ξ)J1(ξr) dξ −ϑ0 ∞∫ 0 p̂(ξ)J1(ξr) dξ ] (2.7) uz(r,0)= 1 GzC [ −ϑ0s1s2 ∞∫ 0 t̂(ξ)J0(ξr) dξ+ ∞∫ 0 p̂(ξ)J0(ξr) dξ ] σzz(r,0)=− ∞∫ 0 ξp̂(ξ)J0(ξr) dξ (2.8) σzr(r,0)=−s1s2 ∞∫ 0 ξt̂(ξ)J1(ξr) dξ It is seen that p̂(ξ) and t̂(ξ) are the Hankel transforms of order zero and one, respectively, of the contact stress σzz(r,0)=−p(r) and σzr(r,0)=−s1s2t(r). In equations (2.7) the material constants C and ϑ0 are defined by equations C = (k+1)(s1−s2) (k−1)s1s2 =2 Gr Gz 1 (1−νrθ)s1s2(s1+s2) (2.9) ϑ0 = ks2−s1 (k−1)s1s2 = GzC√ c11c33+ c13 where Gr and νrθ are the shear modulus and Poissons ratio, respectively, in the isotropic plane.The constant C is real since s1 and s2 are real or complex conjugate; in consequence the parameter ϑ0 is also real. 92 M.Pawlik, B.Rogowski 3. Boundary conditions and integral equations Consider a rigid circular indenter loaded by the force P on a transversely isotropic half-space (Fig.1). Assume that the friction at the interface is suf- ficient to prevent any slip between the indenter and the edge of the stratum. This states that the contact region (r ¬ a) has a constant displacement δ in the z-direction and zeroth displacement in the r-direction. The remainder of the plane z =0 is stress-free. Thus uz(r,0)= δ 0¬ r ¬ a ur(r,0)= 0 0¬ r ¬ a σzz(r,0)= 0 r > a σzr(r,0)= 0 r > a (3.1) Fig. 1. Translation of a rigid indenter on a half-space The boundary conditions (3.1) will be satisfied provided that −ϑ0s1s2 ∞∫ 0 t̂(ξ)J0(ξr) dξ+ ∞∫ 0 p̂(ξ)J0(ξr) dξ = GzCδ 0¬ r ¬ a ∞∫ 0 t̂(ξ)J1(ξr) dξ−ϑ0 ∞∫ 0 p̂(ξ)J1(ξr) dξ =0 0¬ r ¬ a (3.2) ∞∫ 0 ξp̂(ξ)J0(ξr) dξ =0 r > a ∞∫ 0 ξt̂(ξ)J1(ξr) dξ =0 r > a (3.3) Adhesive contact problem for... 93 Introducing the auxiliary functions ϕ(t) and ψ(t), on the assumption that ψ(0)= 0, such that p̂(ξ)= a∫ 0 ϕ(t)cos(ξt) dt t̂(ξ)= a∫ 0 ψ(t)sin(ξt) dt (3.4) we obtain from equations (2.8) the contact stresses p(r)=− 1 r d dr a∫ r tϕ(t)√ t2−r2 dt 0¬ r < a t(r)=−s1s2 d dr a∫ r ψ(t)√ t2−r2 dt 0¬ r < a (3.5) and p(r) = 0= t(r) for r > a, where the results (A.1)-(A.4) have been used (see Appendix). The equilibrium equation of the punch gives P =2π a∫ 0 rp(r)dr (3.6) Substituting equation (3.5)1 into (3.6) and integrating, we obtain P =2π a∫ 0 ϕ(t) dt (3.7) Now substitute expressions (3.4) into equations (3.2), and use (A.5) and (A.6) −ϑ0s1s2 ∞∫ 0 t(ξ)J0(ξr) dξ + r∫ 0 ϕ(t)√ r2− t2 dt = GzCδ 0¬ r ¬ a 1 r r∫ 0 tψ(t)√ r2− t2 dt−ϑ0 ∞∫ 0 p(ξ)J1(ξr) dξ =0 0¬ r ¬ a (3.8) These equations are of Abel’s type. Applying the inverse Abel’s operator we obtain ϕ(t)− ϑ0s1s2 π d dt a∫ 0 ψ(x)ln t+x |t−x| dx = 2 π GzCδ (3.9) ψ(t)− ϑ0 π 1 t d dt a∫ 0 ϕ(x) ( 2t−x ln t+x |t−x| ) dx =0 94 M.Pawlik, B.Rogowski where results (A.7)-(A.10) have been used. Multiplying both sides of these equations by dt and tdt, respectively, and integrating with respect to t from 0 to a and using result (3.7) we obtain the following equations ϑ0s1s2 a∫ 0 ψ(x)ln a+x a−x dx = 1 2 P −2GzCδa (3.10) π a∫ 0 xψ(x) dx+ϑ0 a∫ 0 xϕ(x)ln a+x a−x dx = Paϑ0 π The problem is reduced to the solution of integral equations (3.10). 4. Solution to integral equations By a suitable change of the variables x′ = x a Θ(x′)= 1 2 ln 1+x′ 1−x′ tanhΘ(x′)=x′ 0¬ x′ < 1 0¬ Θ(x′) < ∞ (4.1) equations (3.10) become 2ϑ0s1s2 1∫ 0 ψ(x′)Θ(x′) dx′ = =2ϑ0s1s2 ∞∫ 0 ψ(tanhΘ)Θsech2Θ dΘ = P 2a −2GzCδ (4.2) π 1∫ 0 x′ψ(x′) dx′+2ϑ0 1∫ 0 x′ϕ(x′)Θ(x′) dx′ = = π ∞∫ 0 tanhΘψ(tanhΘ)sech2Θ dΘ+ +2ϑ0 ∞∫ 0 tanhΘϕ(tanhΘ)Θsech2Θ dΘ = Pϑ0 πa Adhesive contact problem for... 95 Using the fact that ψ(x′) is an odd function and ϕ(x′) is an even function, we assume the solution to these equations having the forms ψ(x′)=B(λ)sin(λΘ) (4.3) ϕ(x′)=A(λ)cos(λΘ) where A(λ) and B(λ) are constants and λ plays the role of an eigenvalue. Substituting equations (4.3) into equations (4.2), and using integrals (A.11), (A.12) and (A.13), see Appendix, we obtain the following algebraic equations B(λ) [ 1− π 2 λcoth (π 2 λ )] cosech (π 2 λ ) = 2GzCδ πϑ0s1s2 − P 2πϑ0s1s2a (4.4) B(λ)λ2cosech (π 2 λ ) + 4λϑ0 π A(λ) [ 1− π 4 λcoth (π 2 λ )] cosech (π 2 λ ) = 4Pϑ0 π3a The third equation is obtained from condition (3.7), which gives A(λ)cosech (π 2 λ ) = P π2aλ (4.5) where the integral (A.14) is used (see Appendix). Eliminating A(λ) from equations (4.4)2 and (4.5), we obtain B(λ)cosech (π 2 λ ) = Pϑ0 π2aλ coth (π 2 λ ) (4.6) Substituting (4.6) into (4.4)1, we have Pϑ0 π2aλ coth (π 2 λ )[ 1− π 2 λcoth (π 2 λ )] = 2GzCδ πϑ0s1s2 − P 2πϑ0s1s2a (4.7) If we define the eigenvalue λ by equation tanh (π 2 λ ) = ϑ0 (4.8) which has the solution λ = 1 π ln 1+ϑ0 1−ϑ0 (4.9) then δ = Ps1s2 2GzCa [ϑ0 πλ − 1 2 ( 1− 1 s1s2 )] (4.10) 96 M.Pawlik, B.Rogowski This solution determines the compliance of a transversely isotropic half-space in the adhesive contact problem. For real materials the quantity of ϑ0 is real, positive and 0 ¬ ϑ0 < 1. For example, ϑ0 takes the values: 0.1833; 0.2474; 0.4020 for cadmium, laminated composite consisting of alternating layers of two isotropic materials with µ/µ = 0.5, h/h = 0.5, µ = 104MPa (shear modulus) and for E-glass-epoxy composite, respectively. The constants A(λ) and B(λ) are equal, and are given by equation A(λ) =B(λ)= 2 π GzCδ 1 √ 1−ϑ20 (4.11) and the functions ψ(x) and ϕ(x) defined by equations (4.3) are as follows ψ(x) = 2 π GzCδ 1 √ 1−ϑ20 sin(λΘ) (4.12) ϕ(x) = 2 π GzCδ 1 √ 1−ϑ20 cos(λΘ) Θ = 1 2 ln a+x a−x For an isotropic material the following hold ϑ0 = 1−2ν 2(1−ν) C = 1 1−ν s1 = s2 =1 (4.13) and δ = P 4Gza 1−2ν ln(3−4ν) (4.14) Result (4.14) agrees with Spence’s solution (Spence, 1968a). For an incom- pressible material (ν =1/2 for isotropy or ϑ0 =0 for transverse isotropy) we have the limiting values lim ν→1 2 1−2ν ln(3−4ν) = 1 2 or lim ϑ0→0 ϑ0 ln 1+ϑ0 1−ϑ0 = 1 2 (4.15) so that δ = P 8Gza or δ = P 4Gza s1+s2 s1s2 for ks2 = s1 (4.16) and ψ(x)= 0 ϕ(x)= 2 π GzCδ for ϑ0 =0 λ =0 (4.17) Adhesive contact problem for... 97 Equations (3.9) show that for ϑ0 = 0 the solutions are given by (4.17). This is a confirmation of the correctness of the obtained results and the proper definition of the eigenvalue λ by equation (4.9). Equations (4.16) agree with the result of the frictionless contact related to the incompressible material of a half-space. 5. The stress in the contact region and displacements outside of one The contact stresses are given by equations (3.5) or, alternatively, by the following integrals p(r)= 1 ϑ0 1 r d dr r∫ 0 xψ(x)√ r2−x2 dx = 1 ϑ0 r∫ 0 dψ(x) dx dx√ r2−x2 (5.1) t(r)=− s1s2 ϑ0 d dr r∫ 0 ϕ(x)√ r2−x2 dx =− s1s2 ϑ0 1 r r∫ 0 dϕ(x) dx xdx√ r2−x2 where the second representations are obtained with the use of formula (A.17). Note, that the equivalence in equation (5.1)1 holds since ψ(x) is an odd func- tion, while in equation (5.1)2 it does since ϕ(x) is an even function. The displacements outside the contact region are defined by equations (2.7), in which the functions t̂(ξ) and p̂(ξ) are given by integrals (3.4). The substitution with the use of equations (A.3) and (A.4) (for displacements) yields p(ρ)= P π2a2λ ϑ0√ 1−ϑ20 S1(ρ) 0¬ ρ < 1 t(ρ)= Ps1s2 π2a2λ ϑ0√ 1−ϑ20 S2(ρ) 0¬ ρ < 1 ur(ρ)= δ ρ [2 π 1 √ 1−ϑ20 U2(ρ)−λ ] ρ ­ 1 uz(ρ)= 2 π δ 1 √ 1−ϑ20 U1(ρ) ρ ­ 1 (5.2) 98 M.Pawlik, B.Rogowski where S1(ρ) = − 1 ρ d dρ 1∫ ρ xcos(λΘ) √ x2−ρ2 dx = 1 ϑ0 1 ρ d dρ ρ∫ 0 xsin(λΘ) √ ρ2−x2 dx = = 1 ϑ0 1 ρ2 ρ∫ 0 d[xsin(λΘ)] dx xdx √ ρ2−x2 (5.3) S2(ρ) = − d dρ 1∫ ρ sin(λΘ) √ x2−ρ2 dx =− 1 ϑ0 d dρ ρ∫ 0 cos(λΘ) √ ρ2−x2 dx = = − 1 ϑ0 1 ρ ρ∫ 0 d[cos(λΘ)] dx xdx √ ρ2−x2 U1(ρ)= 1∫ 0 cos(λΘ) √ ρ2−x2 dx U1(1)= π 2 √ 1−ϑ20 U2(ρ)= 1∫ 0 xsin(λΘ) √ ρ2−x2 dx U2(1)= π 2 λ √ 1−ϑ20 (5.4) Θ = 1 2 ln 1+x 1−x 0¬ x < 1 In deriving the third representations in (5.3) the integral (A.17) is used. The stress distribution on the contact surface can be characterised by the load-transfer factor, P(ρ), which is defined as P(ρ)= 2πa2 1∫ ρ ρp(ρ) dρ (5.5) From equations (5.3)1 and (5.5), we have P(ρ)= 2 π P 1 λ ϑ0√ 1−ϑ20 1∫ ρ xcos(λΘ) √ x2−ρ2 dx = = P − 2 π P 1 √ 1−ϑ20 ρ∫ 0 √ ρ2−x2cos(λΘ) 1−x2 dx (5.6) P(0)= P P(1)= 0 Adhesive contact problem for... 99 In deriving P(0) and P(1) the integrals (A.14) and (A.15) were employed. Applying the differentiation rule of the integrand (equation (A.16), see Appendix), we derive the following relations from equations (5.3) S1(ρ)= cos(λΘ) √ 1−ρ2 +λ 1∫ ρ sin(λΘ) (1−x2) √ x2−ρ2 dx (5.7) S2(ρ)= ρ [ sin(λΘ) √ 1−ρ2 + 1∫ ρ sin(λΘ) x2 √ x2−ρ2 dx−λ 1∫ ρ cos(λΘ) x(1−x2) √ x2−ρ2 dx ] The integrals S1(ρ) and S2(ρ) show singularities at the boundary of the con- tact region, i.e. as ρ → 1, which results in the relevant stresses singularities, too. Such behaviour is well known in the analysis of contact and interface crack problems (Ting, 1990; Ni and Nemat Nasser, 1991, 1992). In the case of an incompressible material, i.e. when λ =0, we obtain the square root of the singularity for the normal stress, while the shear stress vanishes in this case. This corresponds to the frictionless contact problem of an incompressible half-space. The oscillations occur in the regions defined by 1+ρ 1−ρ > eπ/λ or ε0 < 2 1+eπ/λ ρ =1−ε0 (5.8) 1+ρ 1−ρ > e2π/λ or ε1 < 2 1+e2π/λ ρ =1−ε1 for the normal and shear stresses, respectively. For example, for an isotropic and extreme case when ν = 0 we have ϑ0 = 0.5, λ = 0.3497, ε0 = 0.00025. The calculation of the local extremum of the first term on the right hand side of equations (5.7) results in the following relationships, respectively tan[λΘ(ρ0)]= ρ0 λ extrS1(ρ0)= λ √ λ2+ρ20 √ 1−ρ20 (5.9) tan[λΘ(ρ1)]=−λρ1 extrS2(ρ1)= λρ21√ 1+λ2ρ21 √ 1−ρ21 There are many roots of ρ0 and ρ1 in the small intervals (1− ε0,1) and (1−ε1,1) which can be obtained from the foregoing equation. The one which 100 M.Pawlik, B.Rogowski yields thefirst extremumof the contact stresses at the end (ρ0 or ρ1) is chosen for numerical computation. The values of ρ0 and ρ1 for differentmaterials are presented in Table 1. The stress concentration factors defined by equations Kz = √ 2a(1−ρ0)p(ρ0) (5.10) Kzr = √ 2a(1−ρ1)t(ρ1) are obtained as follows Kz = P √ 2 π2a √ a ϑ0√ λ2+ρ20 √ 1+ρ0 (5.11) Kzr = P √ 2 π2a √ a ϑ0s1s2ρ 2 1√ 1+λ2ρ21 √ 1+ρ1 For ϑ0/λ → π/2 the obtained results are reduced to the following formulae p(ρ)= P 2πa2 1 √ 1−ρ2 0¬ ρ < 1 t(ρ)= 0 0¬ ρ < 1 ur(ρ)=    0 − 2 π δϑ0 1 ρ for the incompressible half−space, ρ ­ 1 for the frictionless contact, ρ ­ 1 uz(ρ)= 2 π δarcsin 1 ρ ρ ­ 1 δ = P 4GzCa (5.12) Equations (5.12) are well known, and therefore tend to confirm the present analysis. 6. Numerical results Table 1 shows the values of compliance (δaµ/P), the stress concentration factors andtheparameters ε0,ε1,ρ0,ρ1 obtained fromequations (4.10), (5.11), Adhesive contact problem for... 101 (5.8), respectively, for six different materials such as cadmium (denoted sym- bolically asC),magnesium(M) cristals, E-glass-epoxy (EG-E), graphite epoxy (G-E) composite materials and comparative layered (L) and isotropic (ISO) materials. For the layered material it is assumed µ/µ = 0.5 and h/h = 0.5; µ =104MPa. Table 1. Compliance, stress concentration factors, parameters ε0, ε1, ρ0, ρ1 for different materials ISO C M EG-E G-E L (ν =0.3) δaµ P 0.1216 0.09639 0.1894 0.1455 0.1605 0.1701 Kzπ 2a √ a P √ 2 0.1287 0.2114 0.2743 0.3647 0.1727 0.1986 Krzπ 2 a √ a P √ 2 0.1972 0.2080 0.1558 0.1121 0.1797 0.1986 ε0 0.5532 ·10 −11 0.3141 ·10−6 0.1865 ·10−4 0.7531 ·10−3 0.6536 ·10−8 0.1020 ·10−6 ε1 0.1530 ·10 −22 0.4932 ·10−13 0.1738 ·10−9 0.2838 ·10−6 0.2136 ·10−16 0.5205 ·10−14 1−ρ0 0.4050 ·10 −10 0.2261 ·10−5 0.1315 ·10−3 0.5085 ·10−2 0.4748 ·10−7 0.7369 ·10−6 1−ρ1 0.1120 ·10 −21 0.3550 ·10−12 0.1225 ·10−8 0.1904 ·10−5 0.1552 ·10−15 0.3759 ·10−13 Figure 2 shows the load-transfer curves obtained from equation (5.6) for different materials. Fig. 2. Load transfer characteristics P(ρ)/P (equation (5.6)) for different materials shown in Table 1 The distributions of contact stresses: normal p = p(ρ)a2/P and tangential t = t(ρ)a2/P for cadmium are shown in Fig.3 (see equations (5.2)1,2). 102 M.Pawlik, B.Rogowski Fig. 3. Contact stresses p = p(ρ)a2/P and t = t(ρ)a2/P , ρ = r/a for cadmium (equations (5.2)1,2) 7. Conclusions The equations derived in the papermake it possible to completely describe the compliance of the elastic transversely isotropic half-space loaded by a rigid indenter in the adhesive contact problem. As it could be expected, this compliance appears to be strongly dependent onmechanical properties of the presented materials. The contact pressures (normal and tangential), regadless of their closed mathematical structures, contain integrals which can only be determined nu- merically. Those integrals exhibit however singular behaviour, which results in the oscillations of contact stresses near the contact region edge.Theoscillation regions are characterised by two parameters: width (ε0 or ε1) and location of the first extremum (ρ0 or ρ1). These parameters are defined by the closed form equations. They are closely related to the material anisotropy. The contact stress distribution is illustrated by the load-transfer curves, and it is visible that these curves are almost material independent. A. Appendix A.1. Integrals involving Bessel functions The following relations are used 1 r d dr [rJ1(ξr)] = ξJ0(ξr) (A.1) Adhesive contact problem for... 103 d dr [J0(ξr)]=−ξJ1(ξr) (A.2) The following integrals are used ∞∫ 0 J1(ξr)cos(ξt) dξ =    1 r 0 < t < r 1 r [ 1− t√ t2−r2 ] t > r (A.3) ∞∫ 0 J0(ξr)sin(ξt) dξ =    0 0 < t < r 1√ t2−r2 t > r (A.4) ∞∫ 0 J0(ξr)cos(ξt) dξ =    1√ r2− t2 0 < t < r 0 t > r (A.5) ∞∫ 0 J1(ξr)sin(ξt) dξ =    t r √ r2− t2 0 < t < r 0 t > r (A.6) t∫ 0 rJ0(ξr)√ t2−r2 dr = sinξt ξ (A.7) t∫ 0 r2J1(ξr)√ t2−r2 dr = t ξ (sinξt ξt −cosξt ) =− d dξ sinξt ξ (A.8) A.2. Integrals involving trigonometric and hyperbolic functions ∞∫ 0 sinξtsinξx ξ dξ = 1 2 ln t+x |t−x| (A.9) ∞∫ 0 cosξx d dξ (sinξt ξ ) dξ =−t+ x 2 ln t+x |t−x| (A.10) ∞∫ 0 sin(λΘ)Θsech2Θ dΘ =−π 2 [ 1− π 2 λcoth (π 2 λ )] cosech (π 2 λ ) (A.11) ∞∫ 0 sin(λΘ)tanhΘsech2Θ dΘ = π 4 λ2cosech (π 2 λ ) (A.12) 104 M.Pawlik, B.Rogowski ∞∫ 0 Θcos(λΘ)tanhΘsech2Θ dΘ = (A.13) = π 2 λ [ 1− π 4 λcoth (π 2 λ )] cosech (π 2 λ ) ∞∫ 0 cos(λΘ)sech2Θ dΘ = π 2 λcosech (π 2 λ ) (A.14) ∞∫ 0 cos(λΘ)sechΘ dΘ = π 2 sech (π 2 λ ) (A.15) Results (A.12) and (A.13) have been deducted from the results given by Er- delyi (page 30 and 88 of Vol. I book by Erdelyi (1954)). The following rule of differentiation of the integrand was employed in deriving equations (5.7) and (5.3) d dr a∫ r h(t)dt√ t2−r2 =− rh(a) a √ a2−r2 +r a∫ r d dt (h(t) t ) dt√ t2−r2 (A.16) r d dr r∫ 0 f(t)dt√ r2− t2 = r∫ 0 df(t) dt tdt√ r2− t2 (A.17) References 1. 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Spence D.A., 1968a, A Weiner-Hopf equation arising in elastic contact pro- blem,Proc. R. Soc.,A305, 521 9. Spence D.A., 1968b, Self similar solutions to adhesive contact problems with incremental loading,Proc. R. Soc.,A305, 55 10. Ting T.C.T., 1990, Interface cracks in anisotropic media, Journal of the Me- chanics and Physics of Solids, 38, 505-513 Zagadnienie kontaktowe z adhezją dla poprzecznie izotropowej sprężystej półprzestrzeni Streszczenie Rozpatrzono adhezyjne zagadnienie kontaktowedla poprzecznie izotropowej sprę- żystej półprzestrzeni. Zagadnienie zredukowano do rozwiązania dwóch sprzężonych równań całkowych, które rozwiązano dokładnie. Znaleziono w postaci jawnej wzo- ry na podatność oraz współczynniki określające osobliwości naprężeń kontaktowych na brzegu obszaru kontaktu.Wyniki liczbowe przedstawione dla różnychmateriałów pokazują wpływ anizotropii na analizowanewielkości mechaniczne. Manuscript received July 2, 2002; accepted for print October 9, 2002