An Interface Anticrack in a Periodic Two Layer Piezoelectric Space under Vertically Uniform Heat Flow

Transcription

1 Mechanics and Mechanical Engineering Vol., No c Lodz University of Technology An Interface Anticrack in a Periodic Two Layer Piezoelectric Space under Vertically Uniform Heat Flow Andrzej Kaczyński Faculty of Engineering and Economics State Higher Vocational School in Ciechanow Narutowicz 9, Ciechanow, Poland akacz@mini.pw.edu.pl Bartosz Kaczyński Department of Medical Informatics and Telemedicine Medical University of Warsaw Banach 1a, Warsaw, Poland bartosz.kaczynski@wum.edu.pl Received 16 May 018 Revised 17 July 018 Accepted 17 October 018 This paper aims to investigate 3D static thermoelectroelastic problem of a uniform heat flow in a bi-material periodically layered space disturbed by a thermally and electricallyinsulated rigid sheet-like inclusion so-called anticrack situated at one of the interfaces. An approximate analysis of the considered laminated composite is given in the framework of the homogenized model with microlocal parameters. Accurate results are obtained by constructing suitable potential solutions and reducing to the corresponding homogeneous thermoelectromechanical or thermomechanical anticrack problems. The governing boundary integral equation for a planar interface anticrack of arbitrary shape is derived in terms of a normal stress discontinuity. As an illustration, a complete solution for a rigid circular inclusion is obtained in terms of elementary functions and interpreted from the failure perspective. Unlike existing solutions for defects at the interface of materials, the solution obtained displays no oscillatory behavior. Keywords: periodic two-layered composite, homogenized model, interface anticrack, thermo-electroelasticity, heat flow, thermal stress singularity. 1. Introduction Due to the inherent coupling between mechanical and electrical properties, multilayered piezoelectric electronic structures e.g., filters, radiators and converters are in the focus of special attention in the manufacture of composite materials Rao and Sunar [1]. In many cases, various defects such as cracks and inclusions may appear at the interface between the layers. These defects cause high thermal, stress

2 668 Kaczyński, A. and Kaczyński, B. and electric field concentrations, which may lead to failure, fracture and dielectric breakdown. Some industrial applications involve temperature changes producing the pyroelectric effect. In this context, there is tremendous interest in studying the failure behaviors of interface defects in thermopiezoelectric materials under thermal loadings. A comprehensive overview in the field of interface cracks in piezoelectric materials was presented by Govorukha et al. []. In addition to cracks, rigid lamellate inclusions also called anticracks, for brevity are objects around which stress concentrations occur and further affect the behavior of material. In comparison with thermoelastic problems of cracks in homogeneous materials or bi-materials, the study of rigid inclusion problems are rather limited. Most investigations was conducted on two-dimensional problems. Due to mathematical complexity, the more practical and realistic three-dimensional problems involving anticracks subjected to thermal actions seem to remain inadequately treated. Certain progress in homogeneous media has been achieved lately by Kaczyński [3]; see also extensive references therein. The present contribution is a sequel to some earlier investigations Kaczyński and Matysiak [4-5], Kaczyński and Monastyrskyy [6] dealing with interface crack or rigid inclusion problems in a bimaterial periodically layered space subjected to thermal loading. It is devoted to examine the piezoelectric effect in analyzing the obstruction of a uniform steady heat flow in a two-layer periodic stratified medium by an interface rigid sheet-like inclusion anticrack that is isothermal and electrically impermeable. The paper is organized as follows. In Section, the description of the problem under study and the use of the homogenized model of the considered composite are demonstrated. Section 3 presents the resulting boundary-value problem and and its solution method within the framework of this homogenized model that is almost identical to that for the corresponding homogeneous thermoelectromechanical or thermomechanical anticrack problem considered by Kaczyński and Kaczyński [7]. As an application of the theory, a closed form solution in terms of elementary functions is given and discussed from the point of view of material failure for a circular rigid disc-inclusion in Section 4. Finally, some conclusions are made in Section 5.. Problem description and governing equations of the homogenized model The composite being considered is a periodic stratified space consisting of a repeated thin fundamental layer of thickness δ which is composed of two homogeneous bonded sublayers with different mechanical and thermopiezoelectric properties, denoted by 1 and, with thicknesses δ 1 and δ so δ = δ 1 + δ as shown in Fig. 1. In the following, all quantities material constants, stresses, etc. pertinent to these sublayers will be denoted with the superscript r taking the values 1 and, respectively. Assume that the piezoelectric materials filling sublayers are chosen with 6 mm hexagonal symmetry Nye [8], characterized by the following 33, cr 44 elastic stiffnesses, er 31, er 33, er 15 dielectric permittivities, kr 1, kr 3 thermal con- thermal coefficients of linear expansion. Besides, system of constants: piezoelectric constants, ε r 11, εr 33 ductivity coefficients, α r c r 11, cr 1, cr 13, cr 1, αr 3

3 An Interface Anticrack in a Periodic Two Layer Piezoelectric β r 1 = c r 11 + cr 1 α r 1 + c r 13 αr 3, βr 3 = c r 13 αr 1 + c r 33 αr 3 are the thermomechanical moduli and p r 3 = e r 31 αr 1 + e r 33 αr 3 are the pyroelectric constants. Figure 1 A bi-material periodically layered space with an interface anticrack Referring to the Cartesian coordinate system O x 1, x, x 3 with the x 3 -axis normal to the layering and being the axis of symmetry and polarization, denote at the typical point x = x 1, x, x 3 the temperature strictly, a deviation of the temperature from the reference state by θ, the electric potential by Φ and the components of the displacement vector, stress tensor and electric induction vector by u i, σ i j, D i i, j {1,, 3}, respectively. Suppose that a rigid sheet-like inclusion anticrack serving as a mechanical defect in this periodically layered composite occupies a domain S with smooth boundary at the interface x 3 = 0, and there is a constant heat flow q = [0, 0, q 0 ], q 0 > 0 in the direction of the negativex 3 -axis Fig. 1. The anticrack S is assumed to be thermally insulated and electrically impermeable. The perfect mechanical, thermal and electrical contacts between the layers excluding the inclusion region S are required. Moreover, the stress and electric displacement field should decay to zero at infinity. To analyze the thermal stress and electric displacement field disturbed by this defect, a direct analytical approach becomes intricate because of the complicated

4 670 Kaczyński, A. and Kaczyński, B. geometry and complex boundary conditions. Therefore, a specific homogenization procedure called microlocal modelling Woźniak [9] will be employed in order to seek an approximate solution within certain homogenized model of the considered composite. We utilize this approach for periodically layered piezoelectric composites presented in Evtushenko et al. [10] and attempt to derive the governing equations of this model by using the homogenized process based on the uncoupled thermoelectroelasticity. However, we omit the presentation of mathematical assumptions and detailed calculations; see the papers cited above for details. In the subsequent considerations the following notation will be used: Latin subscripts always assume values 1,, 3 and the Greek ones 1,. The Einstein summation convention holds and a comma followed by an index denotes the partial differentiation with respect to the corresponding coordinate variable. The following representations and microlocal approximations for the temperature θ r, fluxes q r i, the electric potential Φ r, displacements u r, stresses σ r and electric displacements D r i are postulated within the stationary thermo-electroelasticity with microlocal parameters see Evtushenko et al. [10], Kaczyński [11], Chen [1]: i i j θ r = ϑ + h Γ = ϑ, θ r, α = ϑ, α, θ r, 3 = ϑ, 3 + h Γ 1 q α r = k r ϑ, α, q r 3 = k r ϑ, 3 + h Γ, 3 Φ r = φ + h H = φ, Φ r, α = φ, α, Φ r, 3 = φ, 3 + h H 3 u r i = w i + h d i = wi, u r i, α = w i, α, u r i, 3 = w i, 3 + h d 3 4 σ r 3α = c r 44 w α, 3 + w 3, α + h d 3 + e r 15 φ, α 5 σ r 33 = c r 13 w γ, γ + c r 33 w 3, 3 + h d 3 + e r σ r 1 = 0, 5 c r 11 cr 1 33 φ, 3 + h H β r ϑ 6 w 1, + w, 1 7 σ r 11 = c r 1γ w γ, γ + c r 13 w 3, 3 + h d 3 + e r 31 φ, 3 + h H β r ϑ 8 σ r = c r 1 3 γ w γ, γ + c r 13 w 3, 3 + h d 3 + e r 31 φ, 3 + h H β r ϑ 9 D r α = e r 15 w α, 3 + w 3, α + h d 3 ε r 11 φ, α 10 D r 3 = e r 31 w γ, γ + e r 33 w 3, 3 + h d 3 ε r 33 φ, 3 + h H + p 3 ϑ 11 In the above, ϑ, φ, w i and Γ, H, d i are unknown functions interpreted as the macro-temperature, the macro-electric potential, macro-displacements and the microlocal thermal, electric and kinematic parameters, respectively. Moreover, the postulated a priori function h, called the shape function, characterizes the special approximate model of the layered composite. As for the treated stratified body, this function being sectional linear, δ- periodic with its derivative is chosen as follows: h x 3 = { x3 0, 5 δ 1, x 3 0, δ 1 δ 1 η x 3 / 1 η 0, 5 δ 1, x 3 δ 1, δ η = δ 1 /δ 1

5 An Interface Anticrack in a Periodic Two Layer Piezoelectric h = { 1 for r = 1 η/ 1 η for r = According to the results of Evtushenko et al. [10], the asymptotic approach to the macro-modelling of the considered layered composite leads to the governing relations of certain macro-homogeneous medium the homogenized model, given in terms of the macroscopic temperature ϑ, the macroscopic electric potential φ and macroscopic displacements w i after eliminating all microlocal parameters Γ, H, d i and in the absence of heat sources, body forces and electric charges as follows: The governing equation of heat conduction for the macro-temperature ϑ: 13 ϑ, ϑ, + K 0 ϑ, 33 = 0 ϑ + K 0 ϑ, 3 3 = 0 14 The governing equations for macro-displacements w i and the macro-electric potential φ: C 1 j w j, 1j +C 66 w 1, + w, 1 +C 44 w 1, 33 + w 3, 13 +E φ, 31 = K 1 ϑ, 1 15 C j w j, j +C 66 w 1, 1 + w, 11 +C 44 w, 33 + w 3, 3 +E φ, 3 = K 1 ϑ, 16 C 3 j w j, 3j +C 44 w 1, 13 + w, 3 + w 3, γ γ +E 15 φ, γ γ +E 33 φ, 33 = K 3 ϑ, 3 17 E w 1,13 + w,3 + E 15 w 3,γγ + E 33 w 3,33 Ĕ11φ,γγ Ĕ33φ,33 = P 3 ϑ,3 18 The constitutive relations for fluxes q r i, stresses σ r i j and electric displacements D r i : q r α = k r ϑ, α, q 3 = K ϑ, 3 19 σ α3 = C 44 w α, 3 + w 3, α + E 15 φ, α 0 σ 33 = C 13 w 1, 1 + w, + C 33 w 3, 3 + E 33 φ, 3 K 3 ϑ 1 σ r 1 = cr 66 w 1, + w, 1 σ r 11 = dr 11 w 1, 1 + d r 1 w, + d r 13 w 3, 3 + E r 31 φ, 3 K r 1 ϑ 3 σ r = dr 1 w 1, 1 + d r 11 w, + d r 13 w 3, 3 + E r 31 φ, 3 K r 1 ϑ 4 D α = E 15 w α, 3 + w 3, α Ĕ15 φ, α 5 D 3 = E 31 w 1, 1 + w, + E 33 w 3, 3 Ĕ33 φ, 3 + P 3 ϑ 6 The coefficients appearing in Eqs are given in Appendix A. They depend in a complicated way on material and geometrical properties of the composite constituents. It is noteworthy to point out the close relation of these equations to fundamental equations of piezothermoelasticity in stationary case for an elastic homogeneous body with transverse isotropy specified by five elastic stiffnesses C 11, C 1, C 13, C 33, C 44, three piezoelectric constants E 31, E 33, E 15, two dielectric permittivities Ĕ11, Ĕ 33, one pyroelectric constant P 3 and two thermal conductivity coefficients k 1, K. The difference manifests itself in the fact that fluxes q α r and stresses σ r suffer α β

6 67 Kaczyński, A. and Kaczyński, B. a discontinuity on the interfaces. Notice that the condition of ideal thermal and electromechanical contact between the layers is satisfied. Finally, assuming that the two sublayers have the same thermo-electromechanical properties, we pass directly to the well-known equations of classical theory of uncoupled stationary thermoelectroelasticity for a homogeneous transversely isotropic solid Chen [1]. 3. Formulation of the anticrack problem and its potential solution Within the scope of the presented homogenized model we are concerned with the following boundary-value problem: find fields ϑ and φ, w i, σ i j, D i suitable smooth on R 3 \S such that Eqs hold subject to: Thermal conditions: q 3 = K ϑ,3 = 0 x 1, x, x 3 = 0 ± S 7 heat-insulated anticrack q 3 = K ϑ, 3 q 0 as x 1 + x + x 3 8 vertical flow of uniform heat Mechanical conditions on S: w 1 = w = 0, w 3 = ε, x 1, x, x 3 = 0 ± S 9 rigid inclusion with a vertical unknown small translation ε The condition of electrically-insulated impermeable rigid inclusion S: D 3 = 0, x 1, x, x 3 = 0 ± S 30 Mechanical and electrical conditions at infinity stress and electric-free state: σ i j = 0, D i = 0 31 It is noteworthy here that a similar boundary-value problem is formulated and solved in Kaczyński and Kaczyński [7]. In what follows, proceeding as in this paper, only main results will be presented in order to solve the above anticrack problem within the homogenized model. To satisfy the global boundary conditions 8 31, superposition is applied to separate the problem into two parts: the first one attached by 0 relating to a basic state of the homogenized space with no inclusion and the second, corrective part with tilde describing the perturbations caused by the existence of the rigid inclusion. Thus, the solution is written as: ϑ = 0 ϑ + ϑ, w i = 0 w i + w i, Φ = 0 Φ + Φ, σ i j = 0 σ i j + σ i j, D i = 0 D i + D i 3

7 An Interface Anticrack in a Periodic Two Layer Piezoelectric The results for the first 0-problem involving the solution to the basic equations with conditions 8 and 31 are computed as: 0 ϑ x 1, x, x 3 = q0 K x 3, 0 Φ = q 0 K n 3x 0 3, σ i j = 0, 0 w α = q0 0 D i = 0 k 3 n γ x γ δ α γ x 3, 0 [ w 3 = q0 K n x 3 n 1 x 1 + x] where the constants n i are found from the linear system of equations C 11 + C 1 C 13 E 31 C 13 C 33 E 33 n 1 n = K 1 K 3 34 E 31 E 33 Ĕ 33 n 3 P 3 Attention will be concentrated next on the non-trivial solution of the perturbed problem. The disturbing thermal field ϑ, which is odd in x 3 and vanishes at infinity, is determined by solving Eq. 14 in the half-space x 3 0 with the following boundary conditions: 33 ϑ + K0 ϑ, 33 = 0, x 1, x, x 3, x 3 0 ϑ, 3 S + = q 0 K, ϑ R S + = 0, ϑ 0 as x 1 + x + x 3 35 Making use of the potential theory Kellogg [13], a solution is written via the thermal potential ω x 1, x, z 0 such that ϑ x 1, x, x 3 = ω x 1, x, z 0 z0, z 0 = K 0 x 3 0, + z0 ω = 0 36 Assuming that ω x 1, x, z 0 = S ln x 1 ξ 1 + x ξ + z0 + z 0 γ ξ 1, ξ dξ 1 dξ 37 we obtain from 35 the following integro-differential equation of Newton s potential type for the unknown density γ x 1, x : γ ξ 1, ξ dξ 1 d ξ = q 0 38 x 1 ξ 1 + x ξ K K 0 S Note that this equation is similar to that which arises in mode I crack problem for the case of constant loads Fabrikant [14]. Moreover, the desired macrotemperature has a jump on the surface S: ϑ x 1, x, x 3 = 0 + ϑ x 1, x, x 3 = 0 = 4πγ x 1, x, x 1, x S 39 We proceed now to the associated problem of electroelasticity that is governed by Eqs and 0 6 with the unknowns marked by the tilde. Because of the anti-symmetry of the temperature and stress system, and bearing in mind Eqs. 3, 33, 9, 30 and the resulting conditions for the displacements

8 674 Kaczyński, A. and Kaczyński, B. i.e., w 1, w, D3 are odd in x 3, w 3 is even in x 3, the anticrack perturbed problem may be formulated as a mixed problem over a half-space x 3 0 with the following boundary conditions: w α x1, x, x 3 = 0 + = 0, x 1, x R α = 1, 40 w 3 x1, x, x 3 = 0 + = q 0 n 1 K x 1 + x + ε, x1, x S 41 σ 3 3 x1, x, x 3 = 0 + = 0, x 1, x R S, x 1, x R 4 D 3 x1, x, x 3 = 0 + = 0, x 1, x R 43 1 w i = O as x = x 1 x + x + x 3 44 For solving this boundary-value problem we use the potential function approach with some modifications well suited to the above boundary conditions, derived in Kaczyński and Kaczyński [7]. It is based on constructing a special representation of governing equations expressed by a single harmonic function f x 1, x, x 3 as follows: [ ] w α = b i f x1, x, z i + A i [ ω x 1, x, z i ], α,α +c 1 [ ω x 1, x, z 0 ],α 45 [ ] w 3 = m i s i b i f x1, x, z i + m i s i A i [ ω x 1, x, z i ],zi,z i c K 0 [ ω x 1, x, z 0 ],z0 46 [ ] Φ = l i s i b i f x1, x, z i + l i s i A i [ ω x 1, x, z i ], zi, z i c 3 K 0 [ ω x 1, x, z 0 ],z0 47 [ ] σ 3 α = a i s i b i f x1, x, z i + a is i A i [ ω x 1, x, z i ], zi α, z i α +δ 1 [ ω x 1, x, z 0 ],z0 α 48 [ ] σ 3 3 = a i b i f x1, x, z i + a i A i [ ω x 1, x, z i ], zi z, z i z i i δ 3 [ ω x 1, x, z 0 ], z0 z 0 49 [ ] D α = d i s i b i f x1, x, z i + d is i A i [ ω x 1, x, z i ], zi α, z i α +τ 1 [ ω x 1, x, z 0 ],z0 α 50 [ ] D 3 = d i b i f x1, x, z i + d i A i [ ω x 1, x, z i ], zi z, z i z i i τ 3 [ ω x 1, x, z 0 ], z0 z 0 51 here, z 0 = K 0 x 3, z i = s i x 3 and the material constants s i, m i, l i, c i, a i, b i, d i, δ 1, δ 3, τ 1, τ 3, A i are defined in Appendix B. It is then shown that the anticrack perturbed problem described by Eqs is reduced to the classical mixed potential problem cf. Sneddon [15] of finding

9 An Interface Anticrack in a Periodic Two Layer Piezoelectric the harmonic function f in the half-space x 3 0 with the boundary conditions: [ m i s i b f ] x 1, x, x 3 i = r 3 x 1, x, x 1, x S 5 x 3 x 3=0 + with: [ ] f x 1, x, x 3 x = 0, 3 x 3 =0 + x 1, x R S 53 [ ] r 3 x 1, x = β ω x1, x, z 0 z 0 β = c K 0 m i s i A i z 0 =0 + q 0n 1 K x 1 + x + ε A well-known solution to this classical boundary problem in potential theory Kellogg [13] may be written in the following form: 54 f x 1, x, x 3 = 1 π a i b i S 55 σ 33 ξ1, ξ, 0 + ln x 1 ξ 1 + x ξ + x 3 + x 3 dξ 1 dξ Then, enforcing the displacement boundary condition 5, we arrive at the governing two-dimensional singular integral equation of Newtonian potential type to determine the unknown normal stress σ + 33 x 1, x σ 33 x 1, x, 0 +, x 1, x S on the upper side of S: m j s j b j π a i b i S σ + 33 ξ 1, ξ dξ 1 dξ x 1 ξ 1 + x ξ = r 3 x 1, x, x 1, x S 56 The above equation is analogous to the well-known governing equation of the frictionless contact problem of an isotropic half-space under the action of a rigid punch see, for instance, Fabrikant [14]. Having found the stress σ 33 + S from the solution to this equation, the parameter εwill be obtained from the equilibrium condition having a form: σ 33 + x 1, x dx 1 dx = 0 57 S Moreover, the main potential fis found from Eq. 55 and the whole perturbed electroelastic fields can be obtained from relations It is worth mentioning that for a rigid inclusion with an arbitrary shape S, the governing equations 38 and 56 can be generally solved by numerical methods.

10 676 Kaczyński, A. and Kaczyński, B. 4. Example: interface circular anticrack in a uniform heat flow For illustration, making use of the results of Kaczyński and Kaczyński [7], a solution expressed in elementary functions will be presented for a rigid circularly shaped inclusion, i.e.: S = {x 1 = r cos θ, x = r sin θ, x 3 = 0 : 0 r = } x 1 + x a 0 θ π Accordingly, the axially-symmetric solution to the thermal perturbed problem is given by: q 0 γ x 1, x = γ r = a r π k1, 0 r a 58 K [ q ω r, z 0 = 0 z 0 a π k1 + K 3 z 0 r sin 1 a l [l 0 + ]] l0 59 r a l 10 5r 10 3 a l l 10 ϑ r, z 0 = ω z 0 = q 0 π k1 K 3 a3 ln z 0 sin 1 a l 0 a l10 where Fabrikant s [14] notation has been used: L 1 L 1 a, r, x 3 = 1 ] [ r + a + x 3 r a + x 3, l 10 = L 1 a, r, z 0 61 L L a, r, x 3 = 1 ] [ r + a + x 3 + r a + x 3, l 0 = L a, r, z 0 along with the following properties: [L 1 ] = [l x3=0 10] = min a, r, [L z0=0 ] = [l x3=0 0] z0=0 = max a, r 6 where: In turn, an analytical solution to the governing equation 56 is: σ e β 3 r = q 0 a 3r π a r, 0 r < a 63 β e 3 = a ib i β 0 ; β 0 = n 1 3m j s j b j K β k1 K Moreover, the vertical rigid displacement is found from 57 as: ε = a q 0 β 3 k1 K + n 1 K The main harmonic potential for the electroelastic perturbed problem is obtained by calculating integral 55 with the use of 63. As a result, we find that for x 3 0: β e 3 q f x 1, x, x 3 = π a i b i [x 3 sin 1 a L a 3 r + x3 + + ] 66 a L 1 5r a L 11 6 L 1

11 An Interface Anticrack in a Periodic Two Layer Piezoelectric The full-space piezothermoelastic field can be obtained from formulas The derivation is omitted here to save the space of the paper. To investigate the singular behavior of the thermal-electric-stress field near the disc edge, however, some relevant interfacial quantities in the plane of the anticrack are presented below: ϑ x 1, x, 0 ± = { ± q 0 π K K a 0 r, 0 r a 0, r > a 67 q 3 r, 0 ± = K ϑ,3 r, 0 ± = σ 33 r, 0 ± = { 0, 0 r < a sin 1 a r a r q a 0, r > a { q 0 π e β 3 ± 0 a 3r π a r 0 r < a 0, r > a σ 3 r r, 0 ± = { σ 31 r, 0 ± cos θ + σ 3 r, 0 ± sin θ = βe q 0 r, 0 r < a = q 0 βe π r sin 1 a r e β r a3 β e a r a r, r > a r r a D r r, 0 ± { = D 1 r, 0 ± cos θ + D r, 0 ± sin θ = βed q 0 r, 0 r < a = q 0 βed π r sin 1 a r β ed a r a r, r > a ed β r a 3 r r a with the following constants: β e = β ed = e 3 β 3 a js j b j 4 a i b i a ks k A k +δ 1 k1 K 3 β e 3 djsjbj 4 a i b i d ks k A k +τ 1, βe r = a is i b i 3 m j s j b j β 0 k1, βed r = d is i b i K 3 m j s j b j β 0 7 Now, it can be emphasized that the singularity of the thermal stresses and electric displacements close to the edge of the anticrack has the order r 1/, contrary to the oscillatory type observed in the elastic fields relating to bimaterial interfaces Li and Fan [16]. Analyzing the above expressions, we reveal that: The anticrack S obstructs locally the heat flow, producing the jump of temperature and the drastic change of its gradient on the surface near its front. The sign of normal stress σ 33 S changes at r = /3 a. Moreover, this stress suffers the jump across S and exhibits the inverse square root singularity at r = a. One would expect a separation detachment of the surrounding matrix material from the anticrack surface described by the stress singularity coefficient: S ± I = lim r a π a r σ33 r, 0 ± = β e 3 q 0 a a π 73

12 678 Kaczyński, A. and Kaczyński, B. Another mechanism controlling the material cracking around the anticrack front is Mode II edge-sliding described by the thermal and electric stress intensity factors: K e II = lim r a + π r a σ3 r r, 0 = K ed II = lim r a + π r a Dr r, 0 = e β r q 0 a a 74 π ed β r q 0 a a 75 π The above-mentioned parameters may be used in the appropriate criterion for initiating fractures at the edge of the inclusion. Finally, the obtained solution for a stratified medium containing of a large number of alternating plane-parallel layers of two different piezoelectric materials can be converted to that considered in Kaczyński and Kaczyński [7] for homogeneous case. 5. Conclusions The three-dimensional thermal stress problem for an interface insulated rigid inclusion obstructing a uniform heat flux in a two-layer microperiodic piezoelectric space has been investigated within the homogenized model with microlocal parameters. Using the potential function method, the problem involving the inclusion of arbitrary shape has been reduced to classical boundary problems of potential theory. Specifically, with the knowledge of the steady-state temperature distribution, the governing equation similar in form to that reported in the literature on contact problems in elasticity was derived. In particular, for a circularly-shaped inclusion the complete solution was obtained in terms of elementary functions. Explicit expressions for the thermo-piezoelastic fields at the plane of inclusion surface were given and interpreted from the point of view of linear fracture theory. References [1] Rao, S.S. and Sunar, M.: Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey, Appl. Mech. Rev., 47, , [] Govorukha, V., Kamlah, M., Loboda, V. and Lapusta, Y.: Interface cracks in piezoelectric materials, Smart Mater. Struct., 5, 03001, 016. [3] Kaczyński, A.: On 3D problems of thermoelastostatics for transversely isotropic solids with anticracks, Eur. J. Mech. A / Solids, 58, , 016. [4] Kaczyński, A. and Matysiak, S.J.: Thermal stresses in a bimaterial periodically layered composites due to the presence of interface cracks or rigid inclusions, J. Theor. Appl. Mech., 36,, , [5] Kaczyński, A. and Matysiak, S.J.: On the three-dimensional problem of an interface crack under uniform heat flow in a bimaterial periodically-layered space, Int. J. Fract., 13, , 003. [6] Kaczyński, A. and Monastyrskyy, B.: Thermal stresses in a periodic two-layer space with an interface rigid inclusion under uniform heat flow, Acta Mech., 03, , 009.

13 An Interface Anticrack in a Periodic Two Layer Piezoelectric [7] Kaczyński, A. and Kaczyński, B.: On 3D problem of an anticrack under vertically uniform heat flow in a transversely isotropic electro-thermo-elastic space, Eur. J. Mech. A / Solids, 66, 15-5, 017. [8] Nye, J.F.: Physical properties of crystals, Clarendon Press, Oxford, [9] Woźniak, Cz.: A nonstandard method of modelling of thermoelastic periodic composites, Int. J. Eng. Sci., 5, 5, , [10] Evtushenko, O.O., Matysiak, S.I. and Pauk, V.I.: On homogenized model of periodically layered piezoelectric composites, Mat. Sci., 3, 3, , [11] Kaczyński, A.: Three-dimensional thermoelastic problems of interface cracks in periodic two-layered composites, Eng. Fract. Mech., 48, 6, , [1] Chen, W.Q.: On the general solution for piezothermoelasticity for transversely isotropy with application, J. Appl. Mech., 67, , 000. [13] Kellogg, O.D.: Foundation of potential theory, Dover, New York, [14] Fabrikant, V.I.: Mixed boundary value problems of potential theory and their applications in engineering, Kluwer Academic Publishers, Dordrecht, [15] Sneddon, I.N.: Mixed boundary value problems in potential theory, North-Holland Publ. Co., Amsterdam, [16] Li, X.F. and Fan, T.Y.: The asymptotic stress field for a rigid circular inclusion at the interface of two bonded dissimilar elastic half-space materials, Int. J. Solids Struct., 38, , 001. Appendix A: The effective constants in the governing equations 14 6 of the homogenized model are derived by using the following averaging operators on the material function M taking constant values M r in the r-sublayer: M = ηm η M, [M] = η M 1 M ˆM = ηm 1 + They are given as follows: η 1 η M, η = δ 1 /δ A.1 K 0 = k1 /K K = k 3 [k 3] 1/ k1 = η k η k 1 ˆk 3 = k 1 3 k 3 1 η k η k 3 C 11 = C = c 11 11, C 1 = C 1 = c 1 11, C 13 = C 31 = C 3 = C 3 = c = [c 1 3] ˆε 33 [c 1 3 ] + ê 31 [e 31 ] + [e 31 ] ê 31 [c 1 3 ] ĉ 33 [e 31 ] ĉ 33ˆε 33 + ê 31 C 33 = c = [c 33] ˆε 33 [c 3 3 ] + ê 31 [e 33 ] + [e 31 ] ê 31 [c 3 3 ] ĉ 33 [e 33 ] ĉ 33 ˆε 33 + ê 31 A. A.3 A.4

14 680 Kaczyński, A. and Kaczyński, B. C 44 = c 44 [c 44] ĉ 44 = c 1 44 c 44 1 η c η, C 66 = c 11 c 1 / A.5 c 44 E = ẽ 31 δ 31 + ẽ 15 δ 15, E 15 = ẽ 15 δ 15, E 33 = ẽ 33 δ 33 δ 31 = [c 1 3] ˆε 33 [e 31 ] + ê 31 [ε 33 ] + [e 31 ] ê 31 [e 31 ] ĉ 33 [ε 33 ] ĉ 33ˆε 33 + ê A.6 31 δ 15 = [c 44] [e 15 ], δ 33 = [c 33] ˆε 33 [e 31 ] + ê 31 [ε 33 ] + [e 33 ] ê 31 [e 3 1 ] ĉ 33 [ε 33 ] ĉ 44 ĉ 33ˆε 33 + ê 31 Ĕ 11 = ε 11 Ĕ 33 = ε 33 [e 33] ˆε 33 [e 3 1 ] + ê 31 [ε 33 ] + [ε 33 ] ê 31 [e 3 1 ] ĉ 33 [ε 33 ] ĉ 33 ˆε 33 + ê 31 K 1 = β 1 [c 1 3] ˆε 33 [β 3 ] ê 31 [p 3 ] + [e 31 ] ê 31 [β 3 ] ĉ 33 [p 3 ] ĉ 33 ˆε 33 + ê 31 K 3 = β 3 [c 33] ˆε 33 [c 3 3 ] + ê 31 [e 33 ] + [e 31 ] ê 31 [c 3 3 ] ĉ 33 [e 33 ] ĉ 33 ˆε 33 + ê 31 P 3 = p 3 [c 33] ˆε 33 [β 3 ] ê 31 [p 3 ] + [e 31 ] ê 31 [β 3 ] + ĉ 33 [p 3 ] ĉ 33 ˆε 33 + ê 31 d r 11 = cr 11 ˆε cr 13 h 33 [c 13 ] + ê 31 [e 13 ] ĉ 33ˆε 33 + ê e r 31 h ê31 [c 13 ] ĉ 33 [e 13 ] 31 ĉ 33ˆε 33 + ê 31 d r 1 = cr 1 ˆε cr 13 h 33 [c 13 ] + ê 31 [e 13 ] ĉ 33ˆε 33 + ê e r 31 h ê31 [c 13 ] ĉ 33 [e 13 ] 31 ĉ 33ˆε 33 + ê 31 d r 13 = cr 13 ˆε cr 13 h 33 [c 33 ] + ê 31 [e 33 ] ĉ 33ˆε 33 + ê e r 31 h ê31 [c 33 ] ĉ 33 [e 33 ] 31 ĉ 33ˆε 33 + ê 31 E r 31 = er 31 ˆε er 31 h 33 [e 31 ] + ê 31 [ε 33 ] ĉ 33ˆε 33 + ê e r 31 h ê31 [c 13 ] ĉ 33 [ε 33 ] 31 ĉ 33ˆε 33 + ê 31 K r 1 = β r 1 c r ˆε 13 h 33 [β 3 ] + ê 31 [p 3 ] ĉ 33ˆε 33 + ê e r ê 31 h 31 [β 3 ] ĉ 33 [p 3 ] 31 ĉ 33ˆε 33 + ê 31 A.7 A.8 A.9 A.10 A.11 A.1 A.13 A.14 Appendix B: Here, the constants involved in the solution of anticrack perturbed problem developed by Kaczyński and Kaczyński [7] are presented within the homogenized model. The positive, real and distinct s i are three roots of the material characteristic equation W s a 0 s 6 b 0 s 4 + c 0 s d 0 = 0 where the coefficients are: B.1

15 An Interface Anticrack in a Periodic Two Layer Piezoelectric a 0 = C 44 C 33 Ĕ 33 + E33 b 0 = C 33 C 44 Ĕ 11 + E + Ĕ33C + E 33 C 44 E 15 + C 11 E 33 CE c 0 = C 44 C 11 Ĕ 33 + E + Ĕ11C B. + E 15 C 11 E 33 + C 44 E 15 CE d 0 = C 11 C 44 Ĕ 11 + E15 provided: C = C 11 C 33 C 13 C 13 + C 44, C = C 13 +C 44, E = E 15 +E 31 B.3 The corresponding constants m i and l i are expressed as: m i = C 44 E 33 s 4 i +C 11E 33 +C 44 E 15 C E s i C 11 E 15 s i[c E 33 E C 33s i C E15 E C44] l i = C44 C33 s4 i C C44 s B.4 i +C11C44 s i[ce 33 EC 33s i C E15 E C44] The constants c 1, c, c 3 are the solution of the following linear system: C 11 C 44 K0 C K0 E K0 C C 33 K0 C 44 E 33 K0 E 15 c 1 c = K 1 K 3 E E 33 k0 E 15 Ĕ 11 Ĕ33K0 c 3 P 3 B.5 The remaining material coefficients occurring in the representation are as follows: a i = C m i + E 15 l i, d i = E m i Ĕ11l i b 1 = d 3 d, b = d 1 d 3, b 3 = d d 1 δ 1 = K 0 [C [ 44 c 1 c E 15 c 3 ], ] δ 3 = C 13 c 1 + K0 C 33 c + E 33 c 3 K 3 τ 1 = K 0 E 15 c 1 c + Ĕ11 c 3, τ 3 = E 31 c 1 + K0 and the constants A 1, A, A 3 are the solution of the following linear system: d 1 d d 3 a 1 a a 3 A 1 A A 3 = c 1 τ 3 δ 3 E 33 c Ĕ33c 3 + P 3 B.6 B.7

16