C.Z. Jiang

Y.W. Liu

Y.X. Zhao

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 340 (2012) 661–667Contents lists available at SciVerse ScienceDirectComptes Rendus Mecaniquewww.sciencedirect.comA piezoelectric screw dislocation near an elliptical inhomogeneitycontaining a confocal rigid lineC.Z. Jiang a,b,c,∗, Y.X. Zhao a,b, Y.W. Liu a,ba State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, PR Chinab College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, PR Chinac Department of Physics and Electronic Information Engineering, Xiangnan University, Chenzhou 423000, PR Chinaa r t i c l e i n f o a b s t r a c tArticle history:Received 23 May 2012Accepted after revision 10 July 2012Available online 1 September 2012Keywords:Material engineeringPiezoelectric screw dislocationElliptical inhomogeneityRigid lineComplex variable methodThe interaction between a piezoelectric screw dislocation and an elliptical inhomogeneityin piezoelectric composite material which contains an electrically conductive confocal rigidline is studied, especially analyzing the shielding effect of a piezoelectric screw dislocationnear an elliptical inhomogeneity. By applying the complex variable method, the analyticalsolution to the elastic field and the electric field, the field intensity factors at the tip ofthe rigid line are derived. The image force acting on the piezoelectric screw dislocationis calculated by using the generalized Peach–Koehler formula. Accordingly, the locationand the orientation of the dislocation, the material properties upon the shielding or anti-shielding effect on the stress intensity factors, as well as the effects of the rigid line andthe electroelastic properties of the piezoelectric materials on the image force are discussed.© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionPiezoelectric materials are widely used in modern technology such as sensors, micropositioner, electro-mechanical actu-ator and high power sonar transducers as a result of the intrinsic coupling behavior [1–5]. However, the presence of variousdefects, such as dislocations, cracks and inclusions, can greatly influence their characteristics and coupling behavior. So itis important to investigate the electro-elastic fields as a result of the presence of defects and inhomogeneities in thesequasi-brittle solids. The intensity of the electro-elastic fields near the tip of the crack or anti-crack (rigid line) in piezo-electric materials could be characterized by a factor depending on geometric shapes, loading conditions and the materialproperties, etc. Above the factor are known as the field intensity factors, which may include the stress intensity factor (SIF)of the elastic fields and the electric displacement intensity factor of the electric fields. The field intensity factors can beused to be an indicator to if the crack will propagate or not. When the field intensity factors are greater than the criticalfield intensity factors, the crack will propagate. Here the critical field intensity factors are determined by the experiment,and relate to many parameters, such as temperature, plate thickness, strain velocity, etc. There are many articles [6–12] inwhich the electro-elastic coupling behavior of the inhomogeneity and dislocation in piezoelectric materials has been deeplyresearched in recent years. The schistose rigid inclusion can be formed inside reinforcement due to chemical compositionsegregation during the crystallizing process in the piezoelectric materials. Fang and Li [13] considered the problem for theelectro-elastic interaction between a piezoelectric moving dislocation and interfacial collinear rigid lines under combinedlongitudinal shear and in-plane electric field. Then Fang [14] investigated the interaction between a screw dislocation andan elliptical inclusion with interfacial rigid lines in piezoelectric solids. Wu and Du [15] discussed the elastic field andelectric field of a rigid line in a confocal elliptic piezoelectric inhomogeneity embedded in an infinite piezoelectric medium* Corresponding author at: Department of Physics and Electronic Information Engineering, Xiangnan University, Chenzhou 423000, PR China.E-mail address: jiangchunzhi@126.com (C.Z. Jiang).1631-0721/$ – see front matter © 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crme.2012.07.002662 C.Z. Jiang et al. / C. R. Mecanique 340 (2012) 661–667Fig. 1. The model in physical plane.under the remote anti-plane shear and in-plane electric field, and analyzed the characteristics of the elastic field and elec-tric field singularities at the rigid line tip. However, the interaction of a piezoelectric screw dislocation with a rigid linein a confocal elliptic inhomogeneity has not been studied due to the complexity of the calculation. In the current paper,the electro-elastic coupling interaction between a piezoelectric screw dislocation and an elliptical inhomogeneity, and theshielding effect of a piezoelectric screw dislocation on the rigid line are studied by using the complex variable method ofsingularity principal analysis, conformal mapping, analytic continuation and Cauchy singular integral.2. Statement of the problem and solutionConsider an infinite piezoelectric matrix containing an elliptical cylindrical inhomogeneity with an electrically conductiveconfocal rigid line which is infinitely long in the z-direction and free of force, under in-plane electrical loads. Both thematrix and the inhomogeneity are assumed to be transversely isotropic with an isotropic xoy-plane. A piezoelectric screwdislocation b = {bz bϕ}T is located at arbitrary point z0 in the matrix.For the present problem, out-of-plane displacement and in-plane electric field need to be considered, so that thereare only non-trivial displacement w , strains γxz and γyz , stresses τxz and τyz , electrical field components Ex and E y ,electric potential ϕ and electric displacement components Dx and D y in the Cartesian coordinates. All components are onlyfunctions of x and y. Introducing the vector of generalized displacement U ={wϕ}, the generalized strains Y x ={γxz−Ex},Y y ={γyz−E y}and generalized stresses Σx ={τxzDx}and Σ y ={τyzD y}can be written with an analytical function vector f (z) ={ f w(z) fϕ(z)}T , where z = x + iy is the complex variable.U = Re[ f (z)] Y x − iY y = f ′(z) Σ x − iΣ y = M f ′(z) (1)where M =[C44 e15e15 −d11], C44, e15 and d11 are the elastic, piezoelectric and dielectric constants, respectively.In order to facilitate the analysis, we introduce the following function vectorT =B∫A(Σ x dy − Σ y dx) = M Im[f (z)]BA (2)where [ ]BA represents the change in the bracketed function vector from point A to point B along any arc AB (not passingthrough interfaces of dissimilar phase).Introducing the following mapping function [16]z = ω(ζ ) = c2(Rζ + 1Rζ)Rζ = zc[1 +√1 −(cz)2 ](3)where ζ = ξ + iη, c = √a2 − b2, R =√a+ba−b , ζ = reiθ . Using the mapping function, the elliptical curve and the rigid line inthe z-plane (z = reiθ ) are mapped onto the concentric circles in the ζ -plane (ζ = ςeiϕ) with radius 1, 1R , respectively, seeFig. 1 and Fig. 2.With the mapping function (3), Eqs. (1) and (2) can be rewritten in the ζ -plane.The hypothesis of the perfect bonding between the medium S+ and the medium S− implies thatC.Z. Jiang et al. / C. R. Mecanique 340 (2012) 661–667 663Fig. 2. The model in transformed plane.U +1 (t) = U −2 (t) T +1 (t) = T −2 (t) along the interface |t| = 1 (4)where the subscripts 1 and 2 represent the regions inhomogeneity S+ and matrix S− . The superscripts + and − denote theboundary values of the physical quantity as z approaches the interface from S+ and S− , respectively.According to the Schwarz symmetry principle, the following new analytical function vectors are introduced in the corre-sponding regionf 1∗(ζ ) = − f̄ 1(1ζ)1 < |ζ | < R (5)f 2∗(ζ ) = − f̄ 2(1ζ)|ζ | < 1 (6)Then the boundary conditions express as[f 1(ζ ) + f 2∗(ζ )]+ = [ f 2(t) + f 1∗(t)]− |t| = 1 (7)[M1 f 1(t) − M2 f 2∗(t)]+ = [M2 f 2(t) − M1 f 1∗(t)]− |t| = 1 (8)The generalized analytical function vector in the matrix asf 2(z) = B ln(z − z0) + f 20(z) z ∈ S− (9)where B = b2π i , f 20(z) is holomorphic in the region S− .Transforming into ζ -plane, we obtainf 2(ζ ) = B ln(ζ − ζ0) + f 20(ζ ) |ζ | > 1 (10)where f 20(ζ ) is holomorphic in the region |ζ | > 1.f 1(ζ ) can be expanded into a Laurent series in the annular regionf 1(ζ ) =∞∑k=0akζk+1 +∞∑k=0bkζ−(k+1) ak and bk are complex constant vectors,1R< |ζ | < 1 (11)The boundary value problems of Eqs. (7) and (8) can be solved by using the Cauchy integrals, and then we can obtainthe generalized analytical function vectors.f 1(ζ ) = −2∞∑k=0Λ−1 M2 B(k + 1) ·[(R2ζζ0)k+1+(1ζ0ζ)k+1](12)f 2(ζ ) = B ln(ζ − ζ0) −∞∑k=0Bk + 1 (ζ0ζ )−k−1 + 2∞∑k=0Λ−1 M2 B(k + 1) ·[R2k+2 − 1](ζ0ζ )−k−1 (13)where Λ = [(M1 + M2)R2k+2 + (M1 − M2)].664 C.Z. Jiang et al. / C. R. Mecanique 340 (2012) 661–667If M1 = M2, the solution on the interaction between a piezoelectric screw dislocation and a finite-length rigid line inhomogeneous piezoelectric material can be calculatedf 1(ζ ) = −∞∑k=0B(k + 1)R2k+2 ·[(R2ζζ0)k+1+(1ζ0ζ)k+1](14)f 2(ζ ) = B ln(ζ − ζ0) −∞∑k=0B(k + 1)R2k+2 (ζ0ζ )−k−1 (15)If we take M =[C44 00 0], b → a, then c → 0 and R → ∞, the solutions are reduced to the interaction between a screwdislocation and a circular inhomogeneity in elastic material,f 1(ζ ) = −2∞∑k=0[M1 + M2]−1 M2 Bk + 1 ·(ζζ0)k+1(16)f 2(ζ ) = B ln(ζ − ζ0) −∞∑k=0Bk + 1 (ζ0ζ )−k−1 + 2∞∑k=0[M1 + M2]−1 M2 Bk + 1 (ζ0ζ )−k−1 (17)which are in agreement with the results in [17].3. Field intensity factorsThe field intensity factors are important parameters of piezoelectric composite material. According to [18], the fieldintensity factors at the right tip of the rigid line can be calculated asK =[k3kD]= √2π limζ→1/R√ω(ζ ) − ω(1/R) [τyz D y ]T = 1iR√πc∞∑k=0M1Λ−1 M2b ·[Rk+2ζ0k+1− Rk+2ζ k+10](18)where k3 is the stress intensity factor, kD is the electric displacement intensity factor.4. Perturbation stress and image force on the dislocationOne of the major interests in discussing the interaction problem of the piezoelectric dislocation with the inhomogeneityis the image forces on the piezoelectric screw dislocation. According to the generalized Peach–Koehler formula [19], theimage forces can be written asFx − i F y =[12π∞∑k=0bT M2bζ0−k−1ζ−k−20 −1π∞∑k=0bT M2Λ−1 M2b(R2K+2 − 1)ζ0−k−1ζ−k−20]· 2Rζ20cR2ζ 20 − c(19)5. Numerical examples and discussionIt is shown in Eq. (18), that the field intensity factors are directly proportional to b and relevant to the location of thedislocation ζ0 = r0eiθ when the size of the inhomogeneity and material constants are certain. The stress intensity factor andthe electric displacement intensity factor are all equal to zero as the dislocation lies on x-axis.As a practical example, we assume that the piezoelectric screw dislocation is b = {1.0 × 10−9 m 0}T , the piezoelectricmatrix is PZT-5H with the electroelastic properties: C (2)44 = 2.56 × 1010 N/m2, e(2)15 = 12.7 C/m2, d(2)11 = 0.646 × 10−8 C/V m.The inhomogeneity is another piezoelectric material. Let us introduce the shear modulus ratio u = C (2)44 /C (1)44 , piezoelectriccoefficients ratio v = e(2)15 /e(1)15 and dielectric constants ratio d(2)11 /d(1)11 = 1.The normalized stress intensity factor is k30 =√πcC (2)44 bzk3. The variation of k30 with respect to θ with different values of ufor piezoelectric material and elastic material when r0 = 1.5 are depicted in Fig. 1 and Fig. 2, respectively. Fig. 3 shows k30versus θ with different values of v , and Fig. 4 shows k30 versus r with different values of u when θ = π/3.It can be found from Figs. 3–6, when a positive screw dislocation is located in the upper half-plane, the stress intensityfactor will be positive (anti-shielding effect); while in the lower half-plane, the stress intensity factor will be negative(shielding effect). The absolute value of k30 increases with the decrement of u, namely, the shielding or anti-shielding effecton the stress intensity factor of the rigid line increases with the decrement of the shear modulus ratio. A comparisonof Fig. 3 with Fig. 4 indicates that the piezoelectric properties can enhance the shielding effect of the screw dislocation.It is also found that the shielding or anti-shielding effect will weaken gradually with the dislocation move away from theC.Z. Jiang et al. / C. R. Mecanique 340 (2012) 661–667 665Fig. 3. k30 versus θ with different u and v = 1.Fig. 4. k30 versus θ with different u for elastic material.Fig. 5. k30 versus θ with different v and u = 1.666 C.Z. Jiang et al. / C. R. Mecanique 340 (2012) 661–667Fig. 6. k30 versus r with different u and v = 1.Fig. 7. Fx0 versus x0/a with different u, v = 1.interface of the inhomogeneity, and the smaller the shear modulus ratio, the greater the rate of change of the shielding oranti-shielding effect.The influence of the different parameters to the image force can be analyzed by using Eq. (19).If the dislocation lies on the x-axis (z0 = x0), let us define Fx0 = 2π Fx/C (244b2z . Fig. 7 shows Fx0 versus x0/a with differ-ent u as v = 1. It can be found that the absolute value of Fx0 may be increased acutely from zero when the screw dislocationapproaches the interface of the inclusion. The image force is always positive as u  1, for which the rigid line and the stiffinhomogeneity repel the screw dislocation. There exists an unstable equilibrium position of the screw dislocation in thematrix and the image force equals zero at that point for the case as u > 1.The variation of Fx0 with respect to u is given in Fig. 8 with different x0/a. We observe that the unstable equilibriumposition of the screw dislocation only appear near the elliptical interface. The influence of the shear modulus ratio on theimage force will decrease acutely as the distance between the screw dislocation and the elliptical interface increases.6. ConclusionsThe problem of the electroelastic interaction between a piezoelectric screw dislocation and an elliptical inclusion con-taining a rigid line under in-plane electrical loads is dealt with by using of an efficient complex variable method. Theexplicit series solutions for the complex potentials of the matrix and the inclusion, as well as the field intensity factors atthe tip of the rigid line are calculated. The generalized Peach–Koehler force acting on the screw dislocation is also given. Theresults show that the electroelastic properties of the material, the location and the orientation of the dislocation have largeinfluence of the shielding or anti-shielding effect on the stress intensity factors of the rigid line. The shielding effect of thescrew dislocation in piezoelectric composite material is better than that in corresponding elastic material. The effect of therigid line on the equilibrium position of the dislocation near an elliptical inclusion is significant. An unstable equilibriumposition of the screw dislocation in the matrix can exist when the dislocation near the soft elliptical inclusion.C.Z. Jiang et al. / C. R. Mecanique 340 (2012) 661–667 667Fig. 8. Fx0 versus u with different x0/a, v = 1.AcknowledgementsThe authors gratefully acknowledge research support from the Science Fund of State Key Laboratory of Advanced Designand Manufacturing for Vehicle Body (No. 60870005), the NNSFC (11172094 and 11172095) and the NCET-11-0122. The workwas also supported by the Fundamental Research Funds for the Central Universities, Hunan University.References[1] F. dell’Isola, M. Porfiri, S. Vidoli, Piezo-electromechanical (PEM) structures: passive vibration control using distributed piezoelectric transducers, C. R.Mecanique 331 (2003) 69–76.[2] C. Falconi, G. Mantini, A. D’Amico, Z.L. Wang, Studying piezoelectric nanowires and nanowalls for energy harvesting, Sens. Actuators B 139 (2009)511–519.[3] A. Mahieddine, J. Pouget, M. Ouali, Modeling and analysis of delaminated beams with integrated piezoelectric actuators, C. R. Mecanique 338 (2010)283–289.[4] Y.D. Li, K.Y. Lee, N. 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A piezoelectric screw dislocation near an elliptical inhomogeneity containing a confocal rigid line

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A piezoelectric screw dislocation near an elliptical inhomogeneity containing a confocal rigid line

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