Commun. Theor. Phys. 56 774 778 Vol. 56, No. 4, October 5, A Piezoelectric Screw Dislocation Interacting with an Elliptical Piezoelectric Inhomogeneity Containing a Confocal Elliptical Rigid Core JIANG Chun-Zhi ã,,,3, XIE Chao,, and LIU You-Wen Õ, State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 48, China College of Mechanical and Vehicle Engineering, Hunan University, Changsha 48, China 3 Department of Physics and Electronic Information Engineering, Xiangnan University, Chenzhou 43, China Received September 5, ; revised manuscript received November 9, Abstract The electro-elastic interaction between a piezoelectric screw dislocation and an elliptical piezoelectric inhomogeneity, which contains an electrically conductive confocal elliptical rigid core under remote anti-plane shear stresses and in-plane electrical load is dealt with. The analytical solutions to the elastic field and the electric field, the interfacial stress fields of inhomogeneity and matrix under longitudinal shear and the image force acting on the dislocation are derived by means of complex method. The effect of material properties and geometric configurations of the rigid core on interfacial stresses generated by a remote uniform load, rigid core and material electroelastic properties on the image force is discussed. PACS numbers: 6.7.Lk, 6..-x, 6.7.Bb Key words: complex variable method, piezoelectric screw dislocation, elliptical inhomogeneity, elliptical rigid core Introduction Piezoelectric materials are widely used in modern technology such as sensors, micropositioner, electromechanical actuator, and high power sonar transducers as a result of the intrinsic coupling behavior. However, the presence of various defects, such as dislocations, cracks, and inclusions, can greatly influence their characteristics and coupling behavior. So it is important to investigate the electro-elastic fields as a result of the presence of defects and inhomogeneities in these quasi-brittle solids. A great deal of work has been conducted on electroelastic coupling characteristics of piezoelectric composite materials. 8] The rigid inclusion can be formed inside the reinforcement due to chemical composition segregation during the crystallizing process in the piezoelectric materials. Wu and Du 9] have discussed the elastic field and electric field of a rigid line in a confocal elliptic piezoelectric inhomogeneity embedded in an infinite piezoelectric medium under the remote anti-plane shear and inplane electric field, and analyzed the characteristics of the elastic field and electric field singularities at the rigid line tip. In the present work, the electro-elastic coupling interaction between a piezoelectric screw dislocation and an elliptical piezoelectric inhomogeneity containing a confocal elliptical rigid core embedded in an infinite piezoelectric medium is investigated using the complex variable method. The matrix is subjected to the remote antiplane shear and inplane electric field. The image force acting on the piezoelectric screw dislocation is calculated by using the generalized Peach Koehler formula. Problem Description Consider an electrically conductive elliptical rigid core in a confocal elliptical piezoelectric inhomogeneity embeded in an infinite piezoelectric matrix. The inhomogeneity and the matrix are assumed to be perfectly bonded along the interface, having different material properties with electro-elasticity modulus M and electro-elasticity modulus M respectively. The matrix is subjected to a remote uniform load, in-plane electric load. A piezoelectric screw dislocation b = b z b ϕ T is located at arbitrary point in the inhomogeneity. Referring to the work of Ref. ] the mapping function is shown as follows z = ω= c R R, R = z c ], c z where = ξ iη, c = a b = a b, R = a b/a b, r = a b /a b, z = r e iθ, = r e iϕ, a and b are the major and minor diameters of the elliptical inhomogeneity, a and b are the major and minor diameters of the elliptical rigid core. Using the mapping function, the two elliptical curves in the z- Supported by the Science Fund of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body under Grant No. 6875 and the National Natural Science Foundation of China under Grant No. 8765 Corresponding author, E-mail: jiangchunzhi@6.com c Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn
No. 4 Communications in Theoretical Physics 775 plane are mapped onto the concentric circles L, L in the -plane with radius, r/r, respectively. For the problem described above, out-of-plane displacement w, strains γ xz and γ yz, stresses τ xz and τ yz, electrical field components E x and E y, electric potential ϕ, electric displacement components D x ans D y, all this components are only functions of x and y. Introducing the vector of generalized displacement w τxz U = ϕ, the generalized stresses Σ x = and Σ y = τyz D y D x can be written with an analytical function vector fz = f w z, f ϕ z T U = Refz], Σx i Σy = Mf z. With the mapping function, Eq. can be written in the -plane as follows U = f f], Σx i Σy = M f ω, 3 C44 e where M = 5 e 5 d ], C 44, e 5, and d are the elastic, piezoelectric, and dielectric constants respectively. 3 Solution of Problem The resultant force and the resultant normal component of electric displacement along any arc AB can be determined by formulate B T = Σx dy Σy dx = i Mf f]. 4 A According to the Schwarz symmetry principle, the following new analytical function vectors are introduced in the corresponding region f = f, < < R r, 5 f = f, <. 6 The hypothesis of the perfect bonding between the medium S and the medium S implies that f t f t] = f t f t], t =, 7 M f t M f t] = M f t M f t], t =, 8 where the subscripts and represent the regions inhomogeneity S and matrix S. The superscripts and denote the boundary values of the physical quantity as z approaches the interface from S and matrix S, respectively. The analytical function vector can be chosen as f z = B lnz z f z, z S, 9 f z = Γz B lnz f z, z S, where B = b/πi, Γ = M ] τ xz D x iτ yz id y ], f z and f z are holomorphic in the region S and S, respectively. Transforming into -plane, we derive f = B ln ln ] R f, f = a k k b k k, a k and b k are complex constant vectors r/r < < f = B ln R crγ f, >. Using Cauchy integrals ] and the Laurent expansion, we obtain M M a k k M M b k k = M crγ M M B k k k] k R, 3 M M ā k k M M b k k = M cr Γ M M B k k k R k ]. 4 Noting that U is a constant on the boundary L, and using Eqs., 5,, 3, and 4, yields where a = Ω M M Br R ] Ω M cr 3 Γ, 5 a k = k Λ M M Br k k R k ], k, 6 b = Ω M M B r M M B r r R ] r M cr Γ, 7 b k = Λ M M B k k M M B r k k r k k rk R k k Ω = M M R M M r, Λ = M M R k M M r k. ], 8 Having determined the complex constant vectors, the analytical function vectors can be determined to be
776 Communications in Theoretical Physics Vol. 56 f = B ln ln ] R Λ M M B k k Λ M M B r k k R k ] k k r k ] M M B r k k r R k] k Ω M crr Γ r Γ, 9 f = B ln ln ] B R k r k k k ] k Λ M M B k r k k M M B k Rk r k k crγ cr Γ r k k rk R k k Ω M cr Γr R. 4 Interfacial Stresses Distribution under Longitudinal Shear The interfacial generalized stresses generated by a remote uniform load at the interface Γ can be defined as f Σx = = Re M ] ω =, f Σy = = Im M ] ω =. 5 Image Force on Dislocation Once the complex potentials in both the matrix and the inclusion are determined, the image force acting on the dislocation can be obtained by using the generalized Peach Koehler formula F x if y = ib T Σ x i Σ y = F imag F Γ, where Σ x iσ y denotes the perturbation generalized stresses field. Referring to the work of Ref. ] F imag denotes the image force and F Γ denotes the force acting on dislocation generated by remote uniform load. Σ x i Σ y = M B r R 3 Λ k M M B R k k Ω M crr Γ r Γ Λ r k k M M B k 3 r k r M M B R 6 Numerical Results and Discussion k ] k 3 R cr c. 3 ] The interfacial stresses of the inhomogeneity and the matrix generated by a remote uniform load can be normalized as τxz = τ xz /τyz and τ yz = τ yz /τyz. We take the piezoelectric screw dislocation vector b =. 9 m T and the piezoelectric matrix material is PZT-5H with the electroelastic properties: C 44 =.56 N/m, e 5 =.7 C/m, d =.646 8 C/Vm. The inhomogeneity is another piezoelectric material. The variations of interfacial stresses with angel ϕ under different shear modulus ratios u = C 44 /C 44 and different piezoelectric coefficients ratios v = e 5 /e 5, =, a =.5, a = 3, are shown in Fig. to d /d Fig. 4. The variations of interfacial stresses with angel ϕ under different rigid core geometric ratio h = b /a are shown in Figs. 5 and 6. Fig. The variations of interfacial stress τ xz with angel ϕ under different u v =. At a given value of ϕ, the harder the medium S relative to S, the smaller the value of the interfacial stresses.
No. 4 Communications in Theoretical Physics 777 This is simply because that the interfacial stresses can be further intensified or diminished by the adjacent material having a higher or lower stiffness. The conclusion is in agreement with the results in Refs. 3] and 4]. The influence of the piezoelectric coefficients is smaller than that of the shear modulus. In addition, the interfacial stresses concentration increase with the increment of the rigid core geometric ratio h = b /a. Fig. 5 The variations of interfacial stress τ xz with angel ϕ under different h u =. Fig. The variations of interfacial stress τ xz with angel ϕ under different v u = Fig. 6 The variations of interfacial stress τ yz with angel ϕ under different h u =. Fig. 3 The variations of interfacial stress τ yz with angel ϕ under different u v =. Fig. 4 The variations of interfacial stress τ yz with angel ϕ under different v u = Let us consider the image force acting on the screw dislocation. If the dislocation lies on x-axis z = x, defining F x = πf x /C b 44 z. Figure 7 shows the normalized image force F x versus the location of the dislocation x with different u as v =. It is found that the magnitude of repulsion force on dislocation will be a large value when dislocation approaches the rigid core, no matter the shear modulus ratios. The image force is always positive when u <, this is because the stiff rigid core repels the dislocation, while the soft matrix attracts the dislocation. There is a stable equilibrium position where the image force equals to zero when u >, for both the stiff inhomogeneity and the matrix repel the dislocation. The normalized force F x versus the piezoelectric coefficients ratios v under different location x and u = is depicted in Fig. 8. It is seen that the closer to the rigid core the screw dislocation is located, the less the effect of the piezoelectric coefficients ratios v on the image force. The image force reaches minimum as the magnitude of the piezoelectric coefficients ratios v approximates zero, provided the location of the dislocation is certain.
778 Communications in Theoretical Physics Vol. 56 Fig. 7 The variations of normalized image force F x with location x under different u v =. Fig. 8 The variations of normalized image force F x with v u = under different location x. 7 Conclusions The technical of conformal mapping and the method of analytic continuation are applied to investigate the interaction between a piezoelectric screw dislocation and an elliptical piezoelectric inhomogeneity, which contains a confocal elliptical rigid core. The analytical solution is obtained by using the complex potential method. The interfacial stresses fields for the interface between inhomogeneity and matrix generated by a remote uniform load and the image force acting on the dislocation are also given. The numerical curves of image force and interfacial stresses are shown in this paper. The results indicate that the interfacial stresses can be further intensified or diminished by the adjacent material having a higher or lower stiffness. In addition, the stress concentration becomes more evident when the rigid core geometric ratio h = b /a tends to be bigger. Moreover, elliptical rigid core and material electroelastic properties play an important role in the interaction dislocation force. References ] W. Deng and S.A. Meguid, Int. J. Solids. Struct. 36 999 449. ] L.H. He and C.W. Lim, Composite Part B: Engineering 34 3 373. 3] Z.M. Xiao, J. Yan, and B.J. Chen, Acta Mech. 7 4 37. 4] X. Wang and E. Pan, Phys. Status Solid b 44 7 94. 5] B. Jin and Q.H. Fang, Arch. Appl. Mech. 78 8 5. 6] Q.H. Fang, Y.W. Liu, and P.H. Wen, Int. J. Mech. Sci. 5 8 683. 7] X. Wang and E. Pan, Int. J. Solids. Struct. 45 8 45. 8] Q. Li and Y.H. Chen, Acta Mech. Sin. 5 9 9. 9] L.Z. Wu and S.Y. Du, Int. J. Solids. Struct. 37 453. ] N.L. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoff, Leyden 975. ] C. Xie, Y.W. Liu, Q.H. Fang, and M. Deng, Theor. Appl. Fract. Mec. 5 9 39. ] S. Lee, Eng. Fract. Mech. 7 987 539. 3] Y.W. Liu, C. Xie, C.Z. Jiang, and Q.H. Fang, Appl. Math. and Mech. 3 5. 4] C.K. Chao, L.M. Lu, C.K. Chen, and F.M. Chen, Int. J. Solids. Struct. 46 9 959.