Anti-plane electro-mechanical behavior of piezoelectric composites with a nano-fiber considering couple stress at the interfaces

Anti-plane electro-mechanical behavior of piezoelectric composites with a nano-fiber considering couple stress at the interfaces Xue-Qian Fang, Xiang-Lin Liu, Jin-Xi Liu, and Guo-Quan Nie Citation: Journal of Applied Physics 114, 054310 (2013); doi: 10.1063/1.4817721 View online: http://dx.doi.org/10.1063/1.4817721 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/114/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effective shear modulus of piezoelectric film embedded with square nano-fibers under anti-plane shear waves J. Appl. Phys. 115, 154315 (2014); 10.1063/1.4871677 Electro-mechanical coupling properties of piezoelectric nanocomposites with coated elliptical nano-fibers under anti-plane shear J. Appl. Phys. 115, 064306 (2014); 10.1063/1.4863615 Effect of interface energy on effective dynamic properties of piezoelectric medium with randomly distributed piezoelectric nano-fibers J. Appl. 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Downloaded to ] IP: 139.184.30.132 On: Tue, 28 Oct 2014 16:19:11 http://scitation.aip.org/content/aip/journal/jap?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/879969015/x01/AIP-PT/JAP_ArticleDL_101514/aplmaterialsBIG_2.jpg/47344656396c504a5a37344142416b75?x http://scitation.aip.org/search?value1=Xue-Qian+Fang&option1=author http://scitation.aip.org/search?value1=Xiang-Lin+Liu&option1=author http://scitation.aip.org/search?value1=Jin-Xi+Liu&option1=author http://scitation.aip.org/search?value1=Guo-Quan+Nie&option1=author http://scitation.aip.org/content/aip/journal/jap?ver=pdfcov http://dx.doi.org/10.1063/1.4817721 http://scitation.aip.org/content/aip/journal/jap/114/5?ver=pdfcov http://scitation.aip.org/content/aip?ver=pdfcov http://scitation.aip.org/content/aip/journal/jap/115/15/10.1063/1.4871677?ver=pdfcov http://scitation.aip.org/content/aip/journal/jap/115/6/10.1063/1.4863615?ver=pdfcov http://scitation.aip.org/content/aip/journal/jap/115/6/10.1063/1.4863615?ver=pdfcov http://scitation.aip.org/content/aip/journal/jap/112/9/10.1063/1.4764869?ver=pdfcov http://scitation.aip.org/content/aip/journal/jap/112/9/10.1063/1.4764869?ver=pdfcov http://scitation.aip.org/content/aip/journal/jap/107/9/10.1063/1.3402937?ver=pdfcov http://scitation.aip.org/content/aip/journal/apl/85/12/10.1063/1.1790030?ver=pdfcov Anti-plane electro-mechanical behavior of piezoelectric composites with a nano-fiber considering couple stress at the interfaces Xue-Qian Fang,1,a) Xiang-Lin Liu,2 Jin-Xi Liu,1 and Guo-Quan Nie3 1Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, People’s Republic of China 2Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, People’s Republic of China 3Graduate School, Shijiazhuang Tiedao University, Shijiazhuang 050043, People’s Republic of China (Received 24 June 2013; accepted 12 July 2013; published online 6 August 2013) Many nanostructures are assumed to be characterized by couple stress effects. Based on the couple stress theory and surface/interface theory, the anti-plane electro-mechanical behavior of piezoelectric composites with a nano-fiber is studied, and the effect of stress and electric displacement at the interface on the size-dependent behavior is examined. By introducing a decoupling function, the governing equations in piezoelectric materials are simplified into two classical governing equations. The displacement and electric potential are expressed by Bessel functions. By satisfying the boundary conditions around the nano-fiber with surface/interface effect, the expanded coefficients are obtained. Excellent agreement with the previous results validates this model. It can be concluded that the interface around the nano-fiber shows greater effect on the stress than that on the electrical fields, and the interface effect increases greatly due to the existence of couple stress. The effect of material properties of nano-fiber on the stress and electric field under different coupling stresses is also examined. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4817721] I. INTRODUCTION To enable the prediction of the size effect experimen- tally observed in problems with a geometric length scale comparable to material’s microstructural length, couple stress theory was proposed by Mindlin and Tiersten.1 In the following years, there has been a significant amount of researches devoted to the effect of nano-inclusions on the structural behavior and related problems within the frame- work of couple stress theory. For very thin beams and wires, the bending and torsional strengths around them are higher. The physical rationale for the extension of the classical to micropolar or couple stress theory is that the classical theory is not able to predict the size effect experimentally observed in couple stress prob- lems. The theory of couple stress elasticity assumes that: (i) each particle has three degrees of freedom, (ii) a supplemen- tal form of the Euler-Cauchy principle with a non-vanishing couple traction, and (iii) the strain energy density depends on both the strain and the gradient of rotation.2 The research in couple stress of material response (both elastic and plastic) has been intensified during the last dec- ade, largely because of an increasing interest to describe the deformation mechanisms and manufacturing of micro- and nano-structured materials and devices.3,4 To date, the influ- ence of couple stress on stress concentration around the spherical and circular voids,5 an elliptical hole,6 a rigid cir- cular inclusion,7 a circular inhomogeneity,8–10 and a spheri- cal inhomogeneity11 was investigated. An extensive summation of micropolar and couple stress elasticity can be found in review articles by Dhaliwal and Singh,12 and Jasiuk and Ostoja-Starzewski.13 Piezoelectric nanocomposites are a new class of nano- composites, and attracting more and more attentions. They can harvest the vibration-based mechanical energy in the sur- roundings to offer electric power, and allow the electronic devices to exclude energy storage components. So, piezo- electric nano-structures hold great promises for directly drawing energy from ambient mechanical resources, power- ing small electronics, and achieving self-powered electronic devices.14,15 The piezoelectric nano-structures will suffer from many kinds of electro-elastic coupling loadings, and the precise prediction on the response of them is very important for the optimal design. In recent years, problems dealing with piezo- electric nano-inclusions embedded in an elastic medium and piezoelectric materials containing nano-defects have been of primary concern.16,17 In piezoelectric nanocomposites, the surface/interface around the nano-inhomogeneities shows significant effect on the behavior of nano-structures under static and dynamic loadings. In nanocomposites with piezoelectric nano-fibers, the surfaces/interfaces in nanocomposites are subjected to not only the normal and tangential forces but also the moments per unit area, and the couple stress may play an im- portant role in predicting the electro-elastic behavior. Recently, the effect of couple stresses on the anti-plane elec- tro-mechanical behaviour of piezoelectric media with a hole or inhomogeneity was investigated.18 However, in the piezo- electric nanostructures with couple stress, the surface energy density is not related to the surface couple stress, the surface a)Author to whom correspondence should be addressed. Electronic mail: [email protected] 0021-8979/2013/114(5)/054310/6/$30.00 VC 2013 AIP Publishing LLC114, 054310-1 JOURNAL OF APPLIED PHYSICS 114, 054310 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 139.184.30.132 On: Tue, 28 Oct 2014 16:19:11 http://dx.doi.org/10.1063/1.4817721 http://dx.doi.org/10.1063/1.4817721 http://dx.doi.org/10.1063/1.4817721 mailto:[email protected] http://crossmark.crossref.org/dialog/?doi=10.1063/1.4817721&domain=pdf&date_stamp=2013-08-06 stress, and the electric displacement. Most recently, the structure behavior of piezoelectric nanocomposites embed- ded with a nano-hole considering couple stress effect has been studied by Fang et al.19 In this paper, the work of Fang et al.19 is extended to the case of a nano-fiber embedded in piezoelectric nanocompo- sites. The solutions for anti-plane electric and mechanical fields in the piezoelectric medium are derived. Combing the couple stress and surface/interface theories, the boundary conditions around the nano-fiber are expressed, and the expanded coefficients are solved. The numerical solutions of electric field and stress around the nano-fiber are graphically illustrated. The effects of the couple stress at the surface/ interface and the material properties of surface/interface and nano-fiber on the electric field and stress in the matrix mate- rial are analyzed. II. PROBLEM FORMULATION A nano-fiber of radius a with infinite length embedded in a large piezoelectric matrix is considered, as depicted in Fig. 1. Both the nano-fiber and matrix phases are assumed to have the material orientation in that it has been poled along the z-direction exhibiting transversely isotropic behavior. Interface exits between the nano-fiber and the matrix, and the interface usually penetrates several atomic layers into the matrix. The electrical and mechanical properties in the vicin- ity of surfaces as well as the interplay between the electricity and elasticity therein are very complicated. The equilibrium and constitutive equations in the interface are the same as those in the couple stress theory, but the presence of surface stress, couple stress, and electric displacement gives rise to the nonclassical boundary condition. For convenience of notation, the matrix is denoted by “M,” the nano-fiber by “F,” and the surface/interface by “S.” The matrix is subjected to the uniform far field anti-plane loading ryz ¼ r0, and in-plane electrical field Ey ¼ E0. In couple stress theory, the rotation vector is not only in- dependent of the displacement vector and electric potential but also subjected to the constraint ui ¼ 1 2 eijkxjk; (1) where eijk is the skew-symmetric alternating tensor and xjk are the rectangular components of the infinitesimal rotation tensor. According to the couple stress theory, a surface element dS can transmit a force vector TidS. The surface forces Ti are in equilibrium with the non-symmetric Cauchy stress tij, the surface couples Mi are in equilibrium with the non- symmetric couple stress mij, and the applied surface charge q is in equilibrium with the electrical displacement, i.e., Ti ¼ tijnj; (2) Mi ¼ mijnj; (3) q ¼ �Dini; (4) where ni and nj ði; j ¼ 1; 2; 3Þ are the components of the unit vector orthogonal to the surface element under consideration. In the absence of body forces, body couples, and free charge, the differential equations of equilibrium are expressed as tji;i ¼ 0; (5) mji;i þ eijktjk ¼ 0; (6) Di;j ¼ 0; (7) where tji, mji, and Di ði; j; k ¼ 1; 2; 3Þ are the components of stress, couple stress, electrical displacement, respectively. eijk is the permutation tensor. The stress tensors, couple stresses, and electric displace- ments are expressed as rij ¼ Cijklekl � ekijEk þ Bijkljkl; (8) mij ¼ Kijkljkl � FkijEk þ Bklijekl; (9) Di ¼ cikEk þ eijkejk þ Fijkjjk; (10) where Cijkl and Kijkl are the components of elasticity tensors. ekij and cik are the components of the piezoelectric and dielectric tensors, respectively. The third rank tensor Fkij and fourth rank tensors Bijkl are related with the couple stresses. The components of stress tji are expressed as tji ¼ Cijklekl � ekijEk þ Bijkljkl � 1 2 �eijk½Klkrsjrs;l þ Ftlk/;tl þ Brsklers;l�; (11) where �eijk is Levi-civita signal. For the anti-plane strain problems, the in-plane and out- of-plane displacements (u, v, w) and the in-plane electric potential / are expressed as u ¼ v ¼ 0; w ¼ wðx; yÞ; / ¼ /ðx; yÞ: (12) Taking account of couple stress, the governing equations for anti-plane problem can be expressed as c44r2w þ e15r2/ � k66 4 r4w ¼ 0; (13) e15r2w � v11r2/ ¼ 0; (14) FIG. 1. Piezoelectric composites with a nano-fiber considering interface effect. 054310-2 Fang et al. J. Appl. Phys. 114, 054310 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 139.184.30.132 On: Tue, 28 Oct 2014 16:19:11 where r2 ¼ @2=@x2 þ @2=@y2 is the two-dimensional Laplace operator in the variables x and y. c44 is the elastic stiffness of piezoelectric materials measured in a constant electric field, v11 is the dielectric constant measured in con- stant strain, and e15 is the piezoelectric constant. The elastic stiffness, dielectric constant, and piezoelectric constant of the matrix are denoted as cM44, e M 11, and e M 15, respectively. To decouple Eqs. (13) and (14), another electro-elastic field w is introduced w ¼ / � ku; (15) where k ¼ e15=e11: Then, Eqs. (13) and (14) are rewritten as r2w � l2r4w ¼ 0; (16) r2w ¼ 0; (17) where l2 ¼ k66 4ðc44þe215=v11Þ . In the couple stress theory, l is called the characteristic length. It can be seen that it is related to the piezoelectric media, and contains not only the mechani- cal properties but also the electrical properties of the material. The general solution of Eq. (16) can be expanded, in po- lar coordinate, as w ¼ ½a0 þ b0 lnðrÞ þ c0I0ðr=lÞ þ d0K0ðr=lÞ�ðe0 þ hÞ þ X1 n¼1 anr n þ bnr�n þ cnInðr=lÞ þ dnK0ðr=lÞ½en cosðnhÞ þ fn sinðnÞ�; (18) where Inð•Þ and Knð•Þ ðn ¼ 0; 1; 2;…1Þ are the modified Bessel functions of the first and second kind of order m, respectively. The solution of / can be obtained from Eq. (15). III. DISPLACEMENT AND ELECTRIC POTENTIAL AROUND THE PIEZOELECTRIC NANO-FIBER According to the general solution of governing equation with couple stress effect, the displacements and electric potentials around the nano-fiber can be expressed as follows: (a) The displacements inside the piezoelectric nano-fiber and the matrix wðFÞ ¼ a1r sin h þ a2I1ðr=lÞsin h; (19) wðMÞ ¼ ða3r þ a4=rÞsin h þ a5K1ðr=lÞsin h: (20) (b) The electric potentials inside the piezoelectric nano- fiber and the matrix /ðFÞ ¼ b1r sin h þ eF15 eF11 a2I1ðR=lÞsin h; (21) /ðMÞ ¼ ðb3r þ b4=rÞsin h þ e ðmÞ 15 vðmÞ11 a3K1ðr=lÞsin h: (22) The electrical displacement, stresses, and couple stresses may be expressed, in polar coordinate system, as Dr ¼ e15 @w @r � v11 @/ @r ; (23) trz ¼ c44 @w @r � k66 4 @r2w @r þ e15 @/ @r ; (24) thz ¼ c44 r @w @h � k66 4 1 r @ @h ðr2wÞ þ e15 1 r @/ @h ; (25) mrr ¼ k11 � k12 2 @ @r 1 r @w @h � � ; (26) mrh ¼ � k66 þ k69 2 @2w @r þ k69 2 r2w: (27) Following the work of Mindlin and Tiersten,1 the boundary conditions at any point of the smooth boundary consist of the modified stress and couple stress tractions are written as �trz ¼ trz � 1 2r @mrr @h ; (28) �thz ¼ thz � 1 2r @mrh r@h : (29) IV. SURFACE/INTERFACE THEORY IN PIEZOELECTRIC NANOCOMPOSITES WITH COUPLE STRESS EFFECT Around the nano-fiber, the surface/interface usually pen- etrates several atomic layers into the matrix. The interface is usually modeled as a media with different material properties from the nano-fiber and matrix. In piezoelectric nanocompo- sites, the electrical and mechanical properties in the vicinity of surfaces as well as the interplay between the electricity and elasticity therein are more complicated. The equilibrium and constitutive equations in solids are the same as those in the couple stress theory, but the presence of surface stress and electric displacement gives rise to a nonclassical bound- ary condition. The surface/interface stress rSab, couple stress m S ab, and electric displacement DSa in solids are linked with the surface strain tensor eSab, surface curvature j S ab, and surface electric field ESa by the relations rSab ¼ s0dab þ @CðeSab; jSab;ESaÞ @eSab ; mSab ¼ m0dab þ @CðeSab; jSab;ESaÞ @jSab ; (30) DSa ¼ Da0 þ @CðeSab; jSab;ESaÞ @ESa ; a; b ¼ x; y; z; (31) where dab is the Kronecker delta function, and the surface energy density C around the nano-fiber depends on the strain at the surface, as well as the surface curvature and the 054310-3 Fang et al. J. Appl. Phys. 114, 054310 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 139.184.30.132 On: Tue, 28 Oct 2014 16:19:11 surface electric field. s0 is the residual surface stress. m0 is the residual surface stress. Da0 is the residual surface electric displacement. It is noted that the superscript S denotes the surface/interface. The constitutive relations at the surface/interface are written as rSab ¼ CSabkleSkl � eSkabESk þ BSabkljSkl; (32) mSab ¼ KSabkljSkl � FSijkESk þ BSabkleSkl; (33) DSa ¼ cSakESk þ eSabkeSk þ FSabkjSbk; (34) where CSabkl and K S abkl are the components of elasticity ten- sors of interface, respectively. eSkab and c S ak are, respectively, the components of the piezoelectric and dielectric tensors of interface. BSabkl and F S abk are the components of the axial ten- sors of interface. For this anti-plane problem with couple stress in this study, the constitutive relations can be simplified as rShz ¼ cS44eShz þ eS15ESh � kS66 4 jSh; (35) DSh ¼ Dh0 þ eS15eShz � vS11ESh; (36) where cS44, v S 11, e S 15, and k S 66 are the elastic stiffness, dielectric constant, piezoelectric constant, and coupling coefficient of interface. In experiment, extensive atomistic simulations can be used to determine the values of cS44, v S 11, e S 15, and k S 66. In the following numerical examples, it is assumed that the ma- terial coefficients are known. In the surface/interface theory with couple stress effect, it can be seen that the surface stress tensor rShz is charge and couple stress dependent, and the sur- face electric displacement DSh is deformation and couple stress dependent. For convenience, the residual surface stress and electric displacement are omitted, since the residual sur- face electric displacement Dh0 always gives rise to an addi- tional deformation field. V. BOUNDARY CONDITIONS AROUND THE NANO-FIBER According to the surface/interface theory, the surface/ interface around the piezoelectric nanofiber usually penetrates several atomic layers into the matrix. The material interface is usually regarded as a piezoelectric medium with material properties different from those of the fiber and matrix. In piezoelectric nanocomposites with couple stress effect, the surface/interface model requires a modification to reflect the effect of stress, couple stress, and electric dis- placement. The boundary conditions are quite different from those in conventional piezoelectric nanocomposites. Following the work of Gurtin and Murdoch,20 the dis- placement, electric potential, and couple stress are assumed to be continuous across the interface, i.e., wMjr¼a ¼ wFjr¼a; (37) /Mjr¼a ¼ /Fjr¼a; (38) mMrhjr¼a ¼ mFrhjr¼a: (39) Around the nano-fiber, the equilibrium equations are characterized by the following three additional constitutive laws: tFrzjr¼a � tMrz jr¼a ¼ � @tShz a@h ���� r¼a ; (40) �tFrzjr¼a � �tMrz jr¼a ¼ � @�tShz a@h ���� r¼a ; (41) DFr jr¼a � DMr jr¼a ¼ @DSh a@h ���� r¼a : (42) A coherent surface/interface is assumed in this study. Thus, the interfacial strain, surface curvature, and electric potential are equal to the associated tangential strain, surface curvature, and electric potential in the abutting bulk materi- als, i.e., eShz ¼ eMhz; jSh ¼ jMh ; ESh ¼ EMh : (43) According to the applied force and electric field at the infinity, the following conditions, as r ! 1, should be satisfied: ryz ¼ r0; Ey ¼ E0: (44) Substituting Eqs. (19)–(22) into Eqs. (38)–(44), the eight expanded coefficients can be determined. VI. NUMERICAL EXAMPLES AND ANALYSES In the following numerical analyses, the matrix is assumed to be PZT-5H ceramic, and the electroelastic prop- erties are: CM44 ¼ 3:53 � 1010 Nm�2, eM15 ¼ 17 Cm�2, and vM11 ¼ 1:51 � 10�8 CV�1m�1. To analyze the surface/inter- face effect around the nano-fiber on the distribution of elec- tric field and stress, the ratio of material properties of the nano-fiber to those of the matrix is introduced, i.e., C�44 ¼ CF44=CM44, e�15 ¼ eF15=eM15, v�11 ¼ vF11=vM11. The radius of the nano-fiber is denoted as a ¼ 1 nm. Since the objective of this study is to obtain an intuitive physical picture of the interface effects around the nano- fiber, a characteristic factor g is introduced to express the relation between the material properties of surface/interface and those of the matrix, i.e., CS44 ¼ gCM44, eS15 ¼ geM15, and vS11 ¼ gvM11. To validate this model, comparison with the existing so- lution is given in Fig. 2. When g ¼ 0, the surface/interface effect disappears. Fig. 2 shows the circumferential stress dis- tribution along x axis when the characteristic length is small. In this case, the couple stress effect is weaker. It can be seen that the numerical solution is in good agreement with the results in Shodja and Ghazisaeidi.18 It is also found that the circumferential stress increases greatly due to the existence of interface, especially at the positions near the nano-fiber. The greater the value of g becomes, the greater the increase 054310-4 Fang et al. J. Appl. Phys. 114, 054310 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 139.184.30.132 On: Tue, 28 Oct 2014 16:19:11 of the stress is. If the value of r=a is greater than 3.5, the sur- face/interface nearly shows no effect on the stress. To find the effect of the characteristic length on the dis- tribution of circumferential stress, Fig. 3 is given. It can be seen that the circumferential stress increases greatly due to the effect of couple stress. The interface effect also becomes greater if the value of g increases. In addition, the excellent agreement with the results in Shodja and Ghazisaeidi18 can be seen. Fig. 4 illustrates the radical stress distribution along x axis when the characteristic length is great. It can be seen that the radical stress shows great variation because of the effect of surface/interface. The direction of radical stress also changes when the value of g becomes great. By compar- ing with the results in Fig. 3, it is observed that the effect of interface on the circumferential stress is greater than that on radical stress. Fig. 5 illustrates the radical stress distribution along x axis when the nano-fiber is softer than the matrix. By com- paring with the results in Fig. 4, it can be seen that the softer nano-fiber results in the decrease of radical stress, and the interface effect around the softer nano-fiber also decreases greatly. Fig. 6 illuminates the radical electric field distribution along the x axis with couple stress effect. It can be seen that the electric field decreases due to the existence of the sur- face/interface, and the surface/interface effect increases with the increase of the value of g. When the ratio of r=a is greater than 4.5, the surface/interface effect vanishes. By comparing with the results in Figs. 2–6, it can be concluded that the interface effect on the stress around the nano-fiber is greater than that on the electric field, especially at the case of a stiffer nano-fiber. FIG. 2. Distribution of stress T�z along the x axis (C � 44 ¼ e�15 ¼ v�11 ¼ 5:0, a/l¼ 5.0). 1. g ¼ 0, obtained from Shodja and Ghazisaeidi;18 2. g ¼ 0, obtained from this paper; 3. g ¼ 3:0; and 4. g ¼ 8:0. FIG. 3. Distribution of stress T�z along the x axis (C � 44 ¼ e�15 ¼ v�11 ¼ 5:0, a/l¼ 2.0). 1. g ¼ 0; 2. g ¼ 3:0; and 3. g ¼ 8:0. FIG. 4. Distribution of radical stress Trz along the x axis (C�44 ¼ e�15 ¼ v�11 ¼ 5:0, a/l¼ 2.0). 1. g ¼ 0, obtained from Shodja and Ghazisaeidi;18 2. g ¼ 0, obtained from this paper; 3. g ¼ 3:0; and 4. g ¼ 8:0. FIG. 5. Distribution of radical stress Trz along the x axis (C�44 ¼ e�15 ¼ v�11 ¼ 0:2, a/l¼ 2.0). 1. g ¼ 0; 2. g ¼ 3:0; and 3. g ¼ 8:0. 054310-5 Fang et al. J. Appl. Phys. 114, 054310 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 139.184.30.132 On: Tue, 28 Oct 2014 16:19:11 VII. CONCLUSION By combining couple stress theory and surface/interface theory, the electro-mechanical behaviour of an infinite piezo- electric composite with a nano-fiber subjected to electro- elastic load is investigated. The piezoelectric surface/inter- face model with considering couple stress has been intro- duced to analyze the surface/interface effect. Comparison with the previous literatures validates this present model. It is found that the stress and electric field increase greatly due to the existence of interface, especially at the positions near the nano-fiber. The interface effect on the stress is greater than that on the electric field. Due to the effect of couple stress, the interface effect increases greatly. The softer nano-fiber can result in the decrease of stress. The interface effect around the softer nano-fiber also decreases greatly. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundations of China (Nos. 11172185; 11272222; and 11272221) and the National Key Basic Research Program of China (No. 2012CB723300). 1R. D. Mindlin and H. F. Tiersten, Arch. Ration. Mech. Anal. 11, 415 (1962). 2P. A. Gourgiotis and H. G. Georgiadis, Int. J. Solid. Struct. 45, 5521 (2008). 3N. A. Fleck and J. W. Hutchinson, Adv. Appl. Mech. 33, 295 (1997). 4Material Instabilities in Solids, edited by R. De Borst and E. Van der Giessen (John Wiley, Chichester, 1998). 5R. D. Mindlin, Arch. Ration. Mech. Anal. 16, 51 (1964). 6M. Majumdar, Indian J. Pure Appl. Math. 13, 1526 (1982). 7C. B. Banks and M. Sokolowski, Int. J. 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Lett. 93, 58 (2013). 20M. E. Gurtin and A. I. Murdoch, Arch. Ration. Mech. Anal. 57, 291 (1975). FIG. 6. Distribution of radical electrical field Er along the x axis (C�44 ¼ e�15 ¼ v�11 ¼ 5:0, a/l¼ 2.0). 1. g ¼ 0; 2. g ¼ 3:0; and 3. g ¼ 8:0. 054310-6 Fang et al. J. Appl. Phys. 114, 054310 (2013) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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