Open Access. Xingli Ren, Hongli Hou and Yaoling Xu * ix 2. , p. , m , L 1 , R 1

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1 48 The Oen Materials Science Journal Oen ccess Study on Electro-Magneto-Elastic Fiber omosites with Periodically Distributed Reinorced Phases Under ntilane Deormation Mode and its lications Xingli Ren Hongli Hou and Yaoling Xu School o ivil Engineering and Mechanics Yanshan University Qinhuangdao 664 hina bstract: The resonse o electro-magneto-elastic long iber comosites containing eriodically distributed reinorced hases under antilane shear load couled with inlane electromagnetic load which lead to the antilane deormation mode is dealt with. There are three dierent electro-magneto-elastic materials in the unit cell (3-hase model) a rigorous analytical method is develoed by using the generalized eigenstrains concet integrated with the doubly quasieriodic Riemann boundary value roblem theory. The exressions o electro-magneto-elastic ields in each hase o comosites are obtained. s an imortant alication o the resented solution the eective electro-magneto-elastic moduli are redicted by using the average ield theorem and comared with the results derived rom the generalized sel-consistent method. Keywords: Electro-magneto-elastic comosites 3-hase model eriodically distributed reinorced hases eective roerty. 1. INTRODUTION The electro-magneto-elastic comosites which ossess excellent magneto-electric couling eect have recently attracted much attention due to the extensive alications or broadband magnetic ield robes electronic acaging medical ultrasonic imaging sensors and actuators [1-]. Boyd et al. [3] resented a method or using arrays o microelectro-mechanical systems electrodes and electromagnets to achieve micro scale ositioning o iezoelectric and iezomagnetic articles in liquid olymers which would then be solidiied to mae a olymer matrix magneto-electric comosite. This ability would reduce concentrations o stress electric ield and magnetic ield thereby increasing eective threshold roerties such as strength electric breadown ield and magnetic saturation ield. Recently by using inite element analysis and volumetric averaging technique a study on the eective electro-magneto-elastic roerties o comosites containing a doubly eriodic arrangement o iezoelectric and iezomagnetic hases was inished by Lee et al. [4]. In their analysis comlicated eriodic boundary conditions o the dislacement electric otential and magnetic otential were subjected to the boundaries o the unit cell. However the research results have been reorted rarely or electro-magneto-elastic comosites with eriodic microstructures. The resent aer deals with the electro-magneto-elastic comosites containing doubly eriodic arallelogram arrangement o reinorced hases under antilane shear load couled with in lane electromagnetic load which lead to the antilane deormation mode. analytical method is ddress corresondence to this author at the School o ivil Engineering and Mechanics Yanshan University Qinhuangdao 664 hina; Tel: ; Fax: ; s: xylysu@163.com xylysu@sina.com X/11 develoed the eective electro-magneto-elastic moduli are redicted by using the average ield theorem and comared with the results derived rom the generalized sel-consistent method (GSM).. STTEMENT OF THE PROBLEM s shown in Fig. (1a) the arallelogram P denotes an unit cell with the boundaries = which contains a circular cross-section iber with an annulus interhase. The unit cell is eriodically arranged in the comlex lane z = x 1 + ix which denote the cross-section o an electro-magneto-elastic comosite (see Fig. 1b). In the unit cell let m denote the regions occuied by the iber bounded by the contour L interhase bounded by the contours L L 1 and matrix resectively. m reresent the union o m and its eriodic congruent regions and L L 1 reresent the union o L L 1 and its eriodic congruent contours resectively. R R 1 denote the internal and external radius o the interhase annulus. 1 and denote two undamental eriods the vertices o the unit cell are and 1 resectively. The electro-magneto-elastic comosite is subjected to the ar-ield antilane shear stresses 13 3 couled with inlane electrical dislacements D 1 D and magnetic inductions B 1 B which lead to the antilane deormation mode o comosites and are not deicted in the Fig. (1b). 3. NLYSIS ND SOLUTION For an electro-magneto-elastic comosite under antilane shear couled with in lane electric ield and magnetic ield 11 Bentham Oen

2 Study on Electro-Magneto-Elastic Fiber omosites The Oen Materials Science Journal 11 Volume 5 49 (a) a unit cell P (b) the cross-section o comosites Fig. (1). unit cell and the cross-section o comosites. there are only the non-trivial antilane dislacement w strain comonents 13 and 3 stress comonents 13 and 3 electric otential electric ield comonents E 1 and E electric dislacement comonents D 1 and D magnetic otential magnetic ield comonents H 1 and H magnetic induction comonents B 1 and B with all ield quantities being only the unctions o coordinates x 1 and x. w and can be exressed by three analytical otential unctions F and resectively. W = Re U = 1 U + U (1) { } T is the generalized { } T. where W = w dislacement U = F The constitutive equations o the roblem can be exressed as [5] 1 i = M(Z 1 iz ) = M du () dz { } T is the generalized stress { } T is the generalized strain =1. where = 3 D B Z = 3 E H M is the roerty matrix o the electro-magneto-elastic materials 44 e 15 q 15 M = e d 11 q 15 d 11 μ 11 where μ 11 denote the elastic dielectric and magnetic ermeability moduli e 15 q 15 d 11 denote the iezoelectric iezomagnetic and magneto electric moduli resectively. (3) For the ollowing analysis we introduce B T = ( B dx 1 1 dx ) = Im[ MU ] To ormulate the roblem the concets o the generalized eigenstrain and equivalent medium are introduced. onsider two equivalent electro-magneto-elastic cases: ase 1: n ininite homogeneous electro-magneto-elastic solid with material roerties M m (the same as the matrix) is subjected to uniorm ar-ield generalized stresses rom here and aterward i not mentioned seciically =1. arently in the entire lane the generalized stresses are uniorm i.e. =. From Eq. () the generalized strains Z = M 1 m are also uniorm. For an actual electromagneto-elastic iber comosite the resence o the doubly eriodic regions and with roerty matrix M and M disturbs the uniorm generalized strains and stresses. The ' ' disturbance ield quantities being denoted by Z in ' ' Z in ' ' m Z m generalized stresses and strains are ' + ' + Z ' + Z m resectively. in ' + m (4) in m then total Z ' + Z Z ' + Z m ase : n ininite homogeneous electro-magneto-elastic solid with roerty matrix M m which is the same as the matrix is subjected to uniorm ar-ield generalized stress. Instead o dealing with the resence o actual doubly eriodic ibers and interhases with dierent roerty matrix M and M we introduce suitable generalized eigenstrain ields Z in and Z in such that the equivalent homogeneous solid has the same electro-magneto-elastic couling ields. The introduction o the generalized eigenstrains disturbs the uniorm electro-magneto-elastic ields with the disturbance ield quantities being denoted by in in

3 5 The Oen Materials Science Journal 11 Volume 5 Ren et al. '' '' Z '' '' in Z '' '' in m Z m total generalized stresses and strains are '' + Z '' + Z m Let Z '' and Z m Z '' + Z '' + Z + Z in m resectively. '' + Z ' = Z m ' = Z in Z in m then " Z in " + m '' + Z ' = Z in in in m as the generalized eigenstrains and Z are not related to the generalized stresses the conditions o the equivalence or case 1 and case require (M M m )[(Z '' 1 + Z 1 ) i(z '' + Z )] + M (Z 1 iz ) = in (5) (M M m )[(Z '' 1 + Z 1 ) i(z '' + Z )] + M (Z 1 iz ) = in (6) ccording to Eqs. (5) and (6) we need to determine our '' '' '' '' unnown generalized strain vectors Z 1 Z Z 1 Z induced by the generalized eigenstrain Z 1 Z Z 1 Z. { } T and Let W '' = w '' '' '' U '' = { F '' '' '' } T denote the generalized dislacement vector and otential unction vector o the electro-magneto-elastic ields induced by the generalized eigenstrain Z and Z resectively. Deine otential unction vector T { } U = F corresonding to the generalized eigenstrain Z exand it into Taylor series in U = F = B z z =1 and From Eq. () the generalized eigenstrain can be exressed as (Z 1 iz ) = du = dz B z 1 z =1 Noting Eq. (1) the generalized eigen-dislacement vector W (t) can be derived on L (7) (8) w (t) W (t) = (t) = 1 U (t) + U (t) (t) t L = 1 B t 1 + B R t =1 (9) Deine otential unction vector T { } U = F corresonding to the generalized eigenstrain Z which can be exanded into Laurent series in F U = = B z + B z z (1) =1 (Z 1 The generalized eigenstrain can be exressed as iz ) = du = B dz z 1 B z (+1) z (11) =1 The generalized eigen-dislacement vector can be derived on L and L 1 W W 1 B + 1 B R t (t) = 1 t L (1) =1 + R B + B 1 t B + 1 B R 1 (t) = 1 t L 1 (13) =1 t + R 1 B + B 1 t

4 Study on Electro-Magneto-Elastic Fiber omosites The Oen Materials Science Journal 11 Volume 5 51 The jum conditions o W '' on L and L 1 can be written as [W '' (t)] + [W '' (t)] = W P (t) W (t) t L (14) [W '' (t)] + [W '' (t)] = W P1 (t) t L 1 (15) where the suerscrits + and - signiy the corresonding boundary values o the hysical quantities as aroached rom the interior and the exterior regions o the contour L or L 1. From Eqs. (1) and Eqs. (14) (15) we have ([U '' (t)] + + [U '' (t)] + ) ([U '' (t)] + [U '' (t)] )= [W (t) W (t)] t L (16) ([U '' (t)] + + [U '' (t)] + ) ([U '' (t)] + [U '' (t)] )= W 1 (t) t L 1 (17) The continuity conditions o the resultant generalized stress on L and L 1 can be written as T + (t) = T (t) t L and L 1 (18) The substitution o Eq. (4) into Eq. (18) yields ([U '' (t)] + [U '' (t)] + )= ([U '' (t)] + [U '' (t)] ) t L and L 1 (19) From Eqs. (16) (17) and (19) it is seen that [U '' (t)] + [U '' (t)] = W (t) W (t) t L () [U '' (t)] + [U '' (t)] = W 1 (t) t L 1 (1) ccording to the results or the doubly quasi-eriodic Riemann boundary roblem [6] the general solutions o Eqs. () and (1) in unit cell P can be exressed as F '' U '' = '' = z + 1 W (t) W (t) (t z)dt L '' i + W P1 (t) (t z)dt () L1 where ( ) is Weierstrass Zeta unction = { 1 3 } T is comlex constant vector to be determined. Substituting Eqs.(9) (1) and (13) into Eq.() U '' can be derived z + U 1 + U P1 + U P3 U '' = z + U + U P + U P3 z + U + U P + U P4 z z z m (3) where U 1 = 1 B z ( 1) + B ( 1)! R [ 1 z ](1) (4) =1 U = 1 ( 1) B ( 1)! R (1) (5) =1 B + 1 B R U 1 = 1 R B (6) =1 ( z 1) + [ 1 ( 1)! z ](1) + B U = 1 ( 1) R B ( 1)! + B (1) (7) =1 B + 1 B R z 1 U 3 = 1 R 1 B (8) =1 ( 1) + [ 1 ( 1)! z ](1) + B

5 5 The Oen Materials Science Journal 11 Volume 5 Ren et al. U 4 = 1 ( 1) R 1 B ( 1)! + B (1) (9) =1 where the suerscrit (-1) denotes the (-1) order derivative with resect to z. For the generalized eigenstrain roblem no generalized stress at ininity is alied. ccording to the eriodicity the resultant generalized stress vanishes on each boundary (=1 34) o the unit cell P. From Eq.(4) it ollows that [U '' U '' ] = (3) = onstant vector can be determined by Eq. (3) 1 3 = R S + S (R 1 R )( 1 B B B B 1 1 ) (31) Noting Eq.() and comleting the irst derivative o U '' '' we can obtain the disturbed generalized strains Z in '' Z in '' Z m in m resectively. The comlex constants B B B (=1 ) can be uniquely determined by Eqs. (5) and (6). Then the total generalized stress and strain in case1 and case can be obtained they are equivalent. 4. ELETRO-MGNETO-ELSTI EFFETIVE MODULI onsider a multihase electro-magneto-elastic comosite subjected to homogeneous generalized stress boundary conditions. The average stress theorem o elasticity can be generalized to show N = i i + m m = (3) i=1 where the bracet denote the volume average which can be calculated in the undamental cell P or the doubly eriodic roblem under consideration. The subscrit m denotes the matrix and i(i=1 N) is used or numbering the disersed hases. i denotes i-th hase volume raction. The eective roerties o the electro-magneto-elastic comosite are deined as relations between the volume average o the generalized stress and the generalized strain. Noting Eq. (3) we have = = M Z (33) where M is the electro-magneto-elastic eective moduli. Z also can be written as the volume-weighted average o Z over each hase N Z = i Z i + m Z m (34) Z m i=1 Noting 1 = M m m Z i 1 = M i i (35) rom Eqs. (3)-(34) it is obtained that N (M 1 M 1 m ) = i (M 1 1 i M m ) i (36) i=1 5. EXMPLES ND NLYSIS Electro-magneto-elastic eective moduli lay an imortant role in the design o electro-magneto-elastic comosites. Once the averaged generalized stress is determined in each disersed hase by the resent method electro-magneto-elastic eective moduli M can be determined by Eq. (36). s an imortant alication o the resented solution the eective electro-magneto-elastic moduli are redicted and comared with the results derived rom the GSM. Material roerties are listed in Table 1. Examle 1: onsider two arrays o reinorced hases i.e. the square and hexagonal arrays as shown in Fig. (). Table 1. Material Proerties BaTiO 3 (Piezoelectric) ofe O 4 (Piezomagnetic) Eoxy (Pure Elastic) 44 (GPa) e 15 ( / m ) 11.6 q 15 (N / m) (1 9 / Nm ) d 11 (1 9 Ns / V) μ 11 (1 6 Ns / )

6 Study on Electro-Magneto-Elastic Fiber omosites The Oen Materials Science Journal 11 Volume 5 53 The total volume raction o iezoelectric hase and iezomagnetic hase is.6. The change o eective electromagneto-elastic moduli with volume raction o iezomagnetic hase or the square and hexagonal arrays are shown in Fig. (3). The dilute sel-consistent Mori- Tanaa dierential and generalized sel-consistent methods have been extensively used or redicting the eective roerties o comosites. Generally the results redicted by the generalized sel-consistent method (GSM) are in good agreement with the exeriment data. s a comarison with the resented method the results redicted by GSM are also deicted in Fig. (3). (a) Square (b) Hexagonal (Fig. 3) contd.. (c) Eective iezomagnetic modulus (d) Eective dielectric modulus Fig. (). Fiber square and hexagonal arrays and unit cell (threehase model). (a) Eective elastic modulus (e) Eective magneto-electric modulus (b) Eective iezoelectric modulus () Eective magnetic ermeability modulus Fig. (3). Results and comarisons o eective electro-magnetoelastic moduli.

7 54 The Oen Materials Science Journal 11 Volume 5 Ren et al. From Fig. (3) it is seen that the results are dierent between square array and hexagonal array some reasons or this behavior has been discussed by Pettermann and Suresh [7]. It is also seen that the results redicted by the GSM are almost in exact agreement with those or the hexagonal array redicted by the resent method. It aears that the GSM relects the idealized even distribution o inclusions and in the doubly eriodic arrays the hexagonal array aroaches most the idealized even distribution. When = and =.6 the reinorced hase is iezoelectric hase (BaTiO 3 ) or iezomagnetic hase (ofe O 4 ) there in no magneto-electric couling eect eective magneto-electric modulus is zero. When < <.6 the three-hase electro-magneto-elastic comosite contain a non-zero magneto-electric modulus. When is about.5 the magneto-electric couling eect is maximum. Examle : The hexagonal array o ibers is adoted in this examle. Tae the iber and matrix as the iezoelectric hase BaTiO 3 and the iezomagnetic hase ofe O 4 resectively. ure elastic coating Eoxy is between the iber and matrix as shown in Fig. (4). The dimensionless coating thicness = (R 1 R )/a =.1. The coating elastic modulus 44 = 1.8GPa (dierent value o denotes dierent coating elastic modulus). The variations o the electro-magneto-elastic eective moduli with or dierent iber volume raction are deicted in Fig. (5). (b) Eective iezoelectric modulus (c) Eective magneto-electric modulus (Fig. 5) contd.. Fig. (4). n unit cell with Eoxy coating. (a) Eective elastic modulus (d) Eective magnetic ermeability modulus

8 Study on Electro-Magneto-Elastic Fiber omosites The Oen Materials Science Journal 11 Volume 5 55 (Fig. 5) contd.. (e) Eective iezomagnetic modulus () Eective dielectric modulus Fig. (5). Eect o the coating elastic moduli on the magnetoelectric eective moduli. From Fig. (5) the eective elastic increase monotonously with the increase o the coating stiness. The eective iezoelectric and magneto-electric moduli increase no monotonously with the increase o the coating stiness which can be interreted by the act that a sti or a sot coating is disadvantageous to transer strain rom matrix (iezomagnetic hase ofe O 4 ) to ibers (iezoelectric hase BaTiO 3 ). When coating is too sti or too sot the strain in ibers is very little and the eective iezoelectric and magneto-electric moduli are close zero. Numerical results also indicate that the coating stiness has very little inluence on the eective magnetic ermeability iezomagnetic and dielectric moduli. 5. ONLUSIONS For the electro-magneto-elastic comosites with a doubly eriodic arallelogram array o reinorced hases under arield antilane shear couled with inlane electromagnetic load a rigorous analytical method is develoed by introducing the concets o generalized eigenstrain integrated with the doubly quasi-eriodic Riemann boundary roblem. Numerical results show the dierence between square array and hexagonal array o reinorced hases. comarison o the resent solution with results rom GSM demonstrates the eiciency and accuracy o the resent method. KNOWLEDGMENTS The ommittee o IMSE 11 acnowledges the suort rom Bentham or ublishing the acceted aer in her OPEN Journals the wor is suorted by the National Natural Science Foundation o hina under Grant NNSF and the Natural Science Foundation o Hebei Province under Grant E REFERENES [1] Nan W. Magnetoelastic eect in comosites o iezoelectric and iezoelectric hases. Phys Rev B 1994; 5: [] boudi J. Micromechanical anslysis o ully couled electromagneto-thermo-elastic multihase comosites. Smart Mater Struct 1; 1: [3] Boyd JG Lagoudas D Seo S. rrays o micro-electrodes and electromagnets or rocessing o electro-magneto-elastic multiunctional comosite materials. SPIE 1th nnual International Symosium on Smart Structures San Diego US 3. [4] Lee J Boyd JGIV Lagoudas D. Eective roerties o threehase electro-magneto-elastic comosites. Int J Eng Sci 5; 43: [5] Fang QH Liu YW Jiang P. On the interaction between a generalized screw dislocation and circular-arc interacial rigid lines in magnetoelectroelastic solids. Int J Eng Sci 5; 43: [6] Lu JK. Boundary Value Problems or nalytic Function Singaare: World Scientiic [7] Pettermann HE Suresh S. comrehensive unit cell model: a study o couled eects in iezoelectric 1-3 comosites. Int J Solids Struct ; 37: Received: ugust 1 Revised: October 5 1 cceted: February Ren et al.; Licensee Bentham Oen. This is an oen access article licensed under the terms o the reative ommons ttribution Non-ommercial License (htt://creativecommons.org/licenses/ bync/3./) which ermits unrestricted non-commercial use distribution and reroduction in any medium rovided the wor is roerly cited.