JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 4, pp. 745-761, Warsaw 2005 ON THE CONTACT PROBLEM FOR A SMOOTH PUNCH IN PIEZOELECTROELASTICITY Ewa Dyka Institute of Power Engineering, Technical University of Lodz e-mail: edyka@p.lodz.pl Bogdan Rogowski Chair of Mechanics of Materials, Technical University of Lodz e-mail: brogowsk@p.lodz.pl The problemof electroelasticity for piezoelectricmaterials is considered. For axially symmetric states, three potentials are introduced, which determi- ne displacements, electric potential, stresses, components of the electric field vector and electric displacements in the piezoelectric body.These fundamen- tal solutions are utilized to solve a smooth contact problem. Exact solutions are obtained for elastic and electric fields in the contact problem. The nu- merical results are presented graphically to show the influence of applied mechanical and electrical loading on the analyzed quantities and to clarify the effect of anisotropy of piezoelectric materials. It is also shown that the influence of anisotropy of the materials on these fields is significant. Key words: piezoelectric medium, transverse isotropy, potential theory method, circular punch 1. Introduction Mechanical durability and reliability of piezoelectric materials offer impor- tant considerations in the design of ”smart” structures and devices. Actually, over a hundred piezoelectric materials or composites are known. Piezoelec- tric materials, particularly piezoelectric ceramics, have been widely used for applications such as sensors, filters, ultrasonic generators and actuators. The piezoelectric composite materials have also been used for hydrophone appli- cation and transducers for medical imaging. The readers interested in this problem are referred to the state of the art survey by Rao and Sunar (1994). 746 E.Dyka, B.Rogowski The body of literature concerning the mechanics of piezoelectric materials is enormous. We referred to a few fundamental works (Cady, 1946; Berlincourt et al., 1964; Tiersten, 1969; Parton andKudryatvsev, 1988). In particular, the contact problem of electroelasticity is very interesting from the point of view of application, since the contact is the direct way of transmission of loading from one element to another. Fan et al. (1996) considered the two-dimensional contact problem of a piezoelectric half-plane. These authors bymeans of Stroh’s formalism formulated the nonslip and slip conditions of contact on thehalf-plane.The three-dimensional contact problem for piezoelectric materials was solved by Chen (2000), who used Fabrikant’s potentials (1989) and the solution whichwas found byDing et al. (1996). The solutions related to elliptical contact problems andpiezo-electro andmagneto- electro elastic bodies have been recently obtained in papers by Ding et al. (1999) and Hou et al. (2003). In this paper, three potential functions are introduced to simplify the basic equations for piezoelectric materials with transversely isotropic electrical and mechanical properties. Using the operator theory, we derive a general solu- tion that is expressed in terms of the three potentials. These functions satisfy differential equations of the second order and are quasi-harmonic functions. Making use of these fundamental solutions, the punchproblem is investigated. The integral equations are derived from the corresponding mixed boundary- value problems of a half space. The exact solutions are obtained.The formulae in a closed form, describing elastic and electric fields in piezoelectricmaterials, are obtained. Also relationships between the force, electric charge, indentation depth of the punch and the potential on the boundary are derived. These relationships are presented graphically. 2. Basic equations and their fundamental solution As our point of departure, we take partial differential equations of equili- brium of linear elasticity for a transversely isotropic piezoelectric material c11B1ur+ c44D2ur+(c13+ c44)D ∂uz ∂r +(e31+e15) ∂φ ∂r =0 c44B0uz + c33D2uz +(c13+ c44)D ∂[rur] r∂r +e15B0φ+e33D2φ=0 (2.1) (e31+e15)D ∂[rur] r∂r +e15B0uz +e33D2uz −ε11B0φ−ε33D2φ=0 On the contact problem for a smooth punch... 747 where the following differential operators have been introduced Bk = ∂2 ∂r2 + 1 r ∂ ∂r − k r2 k=0,1 D= ∂ ∂z (2.2) In above equations: ur and uz are components of the displacement vector in the radial and axial directions of the cylindrical coordinate system (r,θ,z), φ is the electric potential and cij, ekl, εkl stand for the elastic, piezoelectric and dielectric constants, respectively. The problem considered is axially symmetric in which uθ ≡ 0 and physical quantities are independent on the θ-coordinate. We applyHankel’s transforms of the first order for equation (2.1)1 and the zero order for equations (2.1)2,3, namely ûr(ξ,z) =H1[ur(r,z);r→ ξ]≡ ∞∫ 0 ur(r,z)rJ1(rξ) dr (2.3) { ûz(ξ,z), φ̂(ξ,z) } =H0[uz(r,z),φ(r,z);r → ξ]≡ ≡ ∞∫ 0 { uz(r,z),φ(r,z) } rJ0(rξ) dξ where J1(ξ) and J0(rξ) are theBessel functions of the first kind and order one or zero, respectively, and ξ is the transform parameter.We use the properties of Hankel’s transforms Hν[Bνf(r,z);r→ ξ] =−ξ2f̂ν(ξ,z) H1 [∂f(r,z) ∂r ;r→ ξ ] =−ξf̂0(ξ,z) (2.4) H0 [∂[rf(r,z)] r∂r ;r→ ξ ] = ξf̂1(ξ,z) where the index ν = 0 or ν = 1 denotes the transforms of the zero or first order, respectively. From partial differential equations of equilibrium (2.1) three-coupled or- dinary differential equations are then obtained, which may be written in the form D   ûr ûz φ̂  =   0 0 0   (2.5) where D is the following operator matrix D=   −c11ξ2+ c44D2 −ξ(c13+ c44)D −ξ(e31+e15)D ξ(c13+ c44)D −c44ξ2+ c33D2 −e15ξ2+e33D2 ξ(e31+e15)D −e15ξ2+e33D2 ε11ξ2−ε33D2   (2.6) 748 E.Dyka, B.Rogowski We have |D|=−a0(D2−λ21ξ2)(D2−λ22ξ2)(D2−λ23ξ2) (2.7) where λ2i (i = 1,2,3) are the roots of the following cubic algebraic equation in λ2i a0λ 6+ b0λ 4+ c0λ 2+d0 =0 (2.8) with the coefficients defined by equations a0 = c44(c33ε33+e 2 33) b0 =(e31+e15)[2c13e33− c33(e31+e15)]+2c44e33e31− c11e233+ − ε11c33c44−ε33c2 c0 =2e15[c11e33− c13(e31+e15)]+ c44e231+ε33c11c44+ε11c2 (2.9) d0 =−c11(c44ε11+e215) c2 = c11c33− c13(c13+2c44) By virtue of the operator theory, we obtain the following general solution to equations (2.5) ûr(ξ,z) =Ai1F̂(ξ,z) ûz(ξ,z)=Ai2F̂(ξ,z) (2.10) φ̂(ξ,z) =Ai3F̂(ξ,z) where Aij are the algebraic cominors of thematrix operator and F̂(ξ,z) is the zero order Hankel’s transform of the general solution F(r,z), which satisfies the equations, respectively |D|F̂(ξ,z) = 0 (2.11) (D2+λ21∆)(D 2+λ22∆)(D 2+λ23∆)F(r,z) = 0 Here, ∆= ∂2/∂r2+r−1∂/∂r is the Laplacian and D2 = ∂2/∂z2. Taking i=3 andwriting down the expressions for A3j, we obtain ûr(ξ,z) = (a1D 2+ b1ξ 2)ξDF̂(ξ,z) ûz(ξ,z)=−(a2D4+ b2ξ2D2+c2ξ4)F̂(ξ,z) (2.12) φ̂(ξ,z) = (a3D 4+ b3ξ 2D2+ c3ξ 4)F̂(ξ,z) where a1 = c33(e31+e15)− (c13+ c44)e33 b1 = c13e15− c44e31 a2 = c44e33 b2 =(c13+ c44)e31+ c13e15− c11e33 c2 = c11e15 a3 = c44c33 b3 = c 2 13+2c13c44− c11c33 c3 = c11c44 (2.13) On the contact problem for a smooth punch... 749 Note that in equations (2.12), ûr(ξ,z) is the first orderHankel’s transform of the displacement ur(r,z), while ûz(ξ,z) and φ̂(ξ,z) are the zero order Hankel’s transformsof thedisplacement uz(r,z) andelectric potential ,φ(r,z), as well as F̂(ξ,z) and F(r,z). Applying the inverse Hankel’s transforms to equations (2.12), the oryginal solution for the displacements and electric potential are obtained as follows ur(r,z) =−(a1D2− b1∆) ∂2F(r,z) ∂r∂z uz(r,z) =−(a2D4− b2∆D2+ c2∆2)F(r,z) (2.14) φ(r,z) = (a3D 4− b3∆D2+ c3∆2)F(r,z) Using the generalized Almansi’s theorem, the function F(r,z), which sa- tisfies equation (2.11)2, can be expressed in terms of three quasi-harmonic functions F =    F1+F2+F3 for distinct λi F1+F2+zF3 for λ1 6=λ2 =λ3 F1+zF2+z 2F3 for λ1 =λ2 =λ3 (2.15) where Fi(r,z) satisfies, respectively ( ∆+ 1 λ2i D2 ) Fi(r,z) = 0 i=1,2,3 (2.16) For the sake of simplicity, we proceed to consider the case of distinct roots here and after. On the other hand, the special case of multiple roots can be obtained from the general solution by appropriately limited calculation. Using ∆Fi =− 1 λ2i D2Fi (2.17) and summing in equations (2.14), we obtain ur(r,z) =− 3∑ i=1 αi1 ∂4Fi ∂r∂z3 uz(r,z) =− 3∑ i=1 αi2 ∂4Fi ∂z4 (2.18) φ(r,z) = 3∑ i=1 αi3 ∂4Fi ∂z4 The coefficients αij are αij = aj + bj λ2i + cj λ4i (2.19) where aj, bj and cj are defined by equations (2.13) and c1 ≡ 0. 750 E.Dyka, B.Rogowski It is now assumed that αi2 ∂3Fi(r,z) ∂z3 =− 1 λi ϕi(r,z) (2.20) then equations (2.18) can be further simplified to ur(r,z) = 3∑ i=1 ai1λi ∂ϕi ∂r uz(r,z) = 3∑ i=1 1 λi ∂ϕi ∂z (2.21) φ(r,z) =− 3∑ i=1 ai3 λi ∂ϕi ∂z where ai1 = αi1 αi2 1 λ2i = a1λ 2 i + b1 a2λ 4 i + b2λ 2 i + c2 ai3 = αi3 αi2 = a3λ 4 i + b3λ 2 i + c3 a2λ 4 i + b2λ 2 i + c2 (2.22) and ( ∆+ 1 λ2i ∂2 ∂z2 ) ϕi(r,z) = 0 (2.23) It can be verified that ai3 = c13+ c44 e31+e15 − c11− c44λ2i e31+e15 ai1 = −e15+e33λ2i ε11−ε33λ2i − (e31+e15)λ 2 i ε11−ε33λ2i ai1 (2.24) The relationships between stress, displacement and electric potential for a transversely isotropic piezoelectric medium, in the case of axial symmetry, are σrr = c11 ∂ur ∂r + c12 ur r + c13 ∂uz ∂z +e31 ∂φ ∂z σθθ = c12 ∂ur ∂r + c11 ur r + c13 ∂uz ∂z +e31 ∂φ ∂z (2.25) σzz = c13 ∂ur ∂r + c13 ur r + c33 ∂uz ∂z +e33 ∂φ ∂z σzr = c44 (∂ur ∂z + ∂uz ∂r ) +e15 ∂φ ∂r Substituting equations (2.21) into equations (2.25), we obtain σrr =− 3∑ i=1 ai4 λi ∂2ϕi ∂z2 − (c11− c12) ur r σzz = 3∑ i=1 ai4 λ3i ∂2ϕi ∂z2 σθθ =− 3∑ i=1 ai4 λ2i ∂2ϕi ∂z2 − (c11− c12) ∂ur ∂r σzr = 3∑ i=1 ai4 λi ∂2ϕi ∂r∂z (2.26) On the contact problem for a smooth punch... 751 where ai4 = e31c44λ 2 i +e15c11 e31+e15 ai1+ c44e31− c13e15 e31+e15 (2.27) The components of the electric field vector Er and Ez are obtained from relations Er =− ∂φ ∂r = 3∑ i=1 ai3 λi ∂2ϕi ∂r∂z Ez =− ∂φ ∂z = 3∑ i=1 ai3 λi ∂2ϕi ∂z2 (2.28) The electric displacements are defined by equations Dr = e15 (∂ur ∂z + ∂uz ∂r ) +ε11Er (2.29) Dz = e31 (∂ur ∂r + ur r ) +e33 ∂uz ∂z +ε33Ez and presented by potentials as follows Dr = 3∑ i=1 ai5λi ∂2ϕi ∂r∂z Dz = 3∑ i=1 ai5 λi ∂2ϕi ∂z2 (2.30) where ai5 = e33ε11−e15ε33 ε11−ε33λ2i − e31ε11−e15ε33λ 2 i ε11−ε33λ2i ai1 (2.31) The form of the solution is very simple. It can be used to solve various kinds ofmixed boundary - value problems of electroelasticity of a piezoelectric material, such as crack and punch problems. It can be easily verified that: Gauss’law (Parton andKudryatvsev, 1988) ∂Dr ∂r + Dr r + ∂Dz ∂z =0 (2.32) and equilibrium equations for stresses (Nowacki, 1973) ∂σrr ∂r + ∂σrz ∂z + σrr−σθθ r =0 (2.33) ∂σzr ∂r + ∂σzz ∂z + σzr r =0 are satisfied. 752 E.Dyka, B.Rogowski In the vacuum, constitutive equations (2.29) and governing equations (2.32) become Dr = ε0Er Dz = ε0Ez (2.34) ∂2φ ∂r2 + 1 r ∂φ ∂r + ∂2φ ∂z2 =0 where ε0 is the electric permittivity of the vacuum. For axially symmetric problems, the very useful is Hankel transform me- thod. Assumethe solutions todifferential equations (2.23) in the formofHankel’s integrals as follows ϕi(r,z) = ∞∫ 0 Ai(ξ)exp(−λiξz)J0(rξ) dξ (2.35) where Ai(ξ) (i=1,2,3) are arbitrary functions of the transformparameter ξ, which is to be determined from the boundary conditions and λi are the roots of equations (2.8), which have positive real parts to ensure the regularity conditions at z → ∞. Then we can easily obtain mechanical and electrical quantities (2.21), (2.26), (2.28) and (2.30). We have: ur(r,z) =− 3∑ i=1 ai1λiIi1(r,z) uz(r,z) =− 3∑ i=1 Ii0(r,z) φ(r,z) = 3∑ i=1 ai3Ii0(r,z) σrr(r,z) =− 3∑ i=1 ai4λiJi0(r,z)− (c11− c12) ur r σθθ(r,z) =− 3∑ i=1 ai4λiJi0(r,z)− (c11− c12) ∂ur ∂r σzz(r,z) = 3∑ i=1 ai4 λi Ji0(r,z) σzr(r,z) = 3∑ i=1 ai4Ji1(r,z) Er(r,z) = 3∑ i=1 ai3Ji1(r,z) Ez(r,z) = 3∑ i=1 ai3λiJi0(r,z) Dr(r,z) = 3∑ i=1 ai5λ 2 iJi1(r,z) Dz(r,z) = 3∑ i=1 ai5λiJi0(r,z) (2.36) On the contact problem for a smooth punch... 753 where Iiν = ∞∫ 0 ξAi(ξ)exp(−λiξz)Jν(rξ) dξ (2.37) Jiν = ∞∫ 0 ξ2Ai(ξ)exp(−λiξz)Jν(rξ) dξ ν =0,1 As an application of the obtained fundamental solution, the punch problem will be considered in the next Section. 3. Punch problem We assume that the circular punch is flat ended, maintained at a con- stants electric potential and loaded centrally by a concentrated force. On the- se assumptions, it is known that both the electric potential φ and the punch penetration δ are constants inside the contact region (Fig.1). Fig. 1. Circular punch problem In this case, we have the boundary conditions (a) uz(r,0) = δ 0¬ r¬ a (b) φ(r,0)=φ0 0¬ r¬ a (c) σrz(r,0) =0 r­ 0 (d) σzz(r,0)=0 r>a (3.1) As usual (Fan et al., 1996), the displacement and electric potential are prescribed in the contact region as δ and ϕ0, respectively. For the sake of practical convenience, the punch can be grounded and the electric potential 754 E.Dyka, B.Rogowski will be zero. Introducing two new unknown functions D1(ξ) and D2(ξ) for simplicity of the formulae for uz and φ and using boundary condition (3.1c), we may obtain the following system of equations A1(ξ)+A2(ξ)+A3(ξ)=D1(ξ) a13A1(ξ)+a23A2(ξ)+a33A3(ξ)=D2(ξ) (3.2) a14A1(ξ)+a24A2(ξ)+a34A3(ξ)= 0 The solution to this system of algebraic equations is m2Ai(ξ)= diD1(ξ)+ liD2(ξ) (3.3) where l1 =−a24+a34 l2 =−a34+a14 l3 =−a14+a24 d1 = a24a33−a34a23 d2 = a13a34−a14a33 d3 = a14a23−a13a24 m2 = 3∑ i=1 di (3.4) Boundary conditions (3.1a), (3.1b) and (3.1d) yield − ∞∫ 0 ξD1(ξ)J0(rξ) dξ= δ 0¬ r¬ a ∞∫ 0 ξD2(ξ)J0(rξ) dξ=φ0 0¬ r¬ a (3.5) m m2 ∞∫ 0 ξ2D1(ξ)J0(rξ) dξ+ m6 m2 ∞∫ 0 ξ2D2(ξ)J0(rξ) dξ=0 r >a where m and m6 are defined by equations (3.14) and m2 by equations (3.4). Dual integral equations (3.5) are converted to the Abel integral equation bymeans of the following integral representation for ξDi(ξ) (Sneddon, 1972) ξDi(ξ)= √ 2 π a∫ 0 Φi(x)cos(ξx) dx i=1,2 (3.6) Using the integrals involving Bessel and trigonometric functions, we may verify that equation (3.5)3 is satisfied identically, while equations (3.5)1,2 give − √ 2 π r∫ 0 Φ1(x)√ r2−x2 dx= δ √ 2 π r∫ 0 Φ2(x)√ r2−x2 dx=φ0 (3.7) On the contact problem for a smooth punch... 755 These equations are of Abel’s type and have the following solutions Φ1(x)=− √ 2 π δ Φ2(x)= √ 2 π φ0 (3.8) Substituting (3.8) into equation (3.6) and integrating, we obtain ξD1(ξ)=− 2 π δ sinξa ξ ξD2(ξ)= 2 π φ0 sinξa ξ (3.9) The stress σzz and electric displacement Dz on the crack plane z=0 are obtained as (r