International Journal of Solids and Structures

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1 Iteratial Jural f Slids ad Structures 48 (211) Ctets lists available at ScieceDirect Iteratial Jural f Slids ad Structures jural hmepage: Micrmechaical mdelig f smart cmpsites csiderig debdig f reifrcemets Yasser M. Shabaa a,, Matti Ristimaa b a Mechaical Desig Departmet, Faculty f Egieerig, El-Mataria, Helwa Uiversity, P.O. Bx 11718, Cair, Egypt b Divisi f Slid Mechaics, Lud Uiversity, P.O. Bx 118, 221 Lud, Swede article if abstract Article histry: Received 24 March 211 Received i revised frm 14 Jue 211 Available lie 24 July 211 Keywrds: Micrmechaical mdelig Smart cmpsites Debdig damage Prsity Macrscpic ad micrscpic behavirs Electr-maget-therm-mechaical prperties Usig the ifrmati f the micrstructure, this paper presets the develpmet f a icremetal cstitutive law gverig the respse f a electr-maget-therm-mechaical smart cmpsite. I this develpmet, differet shapes f reifrcemets that have maget-electr-therm-elastic prperties that differ frm the matrix material are csidered. Shapes such as ellipsidal (spherical, prlate ad blate) particles, elliptical ad circular cylidrical fibers, disk ad ribb ca be treated prvided that the crrespdig Eshelby tesr is used. The debdig f the reifrcemets frm the matrix is als a part f the micrscpic prcess csidered. The develped icremetal cstitutive law t ly predicts the macrscpic ad micrscpic electr-maget-therm-mechaical-elastic behavir f cmpsites while csiderig the debdig prcess, but it als characterizes their differet macrscpic effective prperties such as permittivity, permeability, stiffess mduli, pyrelectricity, pyrmagitivity ad thermal expasi cefficiet i differet directis. Mrever, the develped cstitutive law is applicable t prus materials ad cmpsites with multiple reifrcemets ad prsities. I the tw examples csidered belw, particular atteti is devted t assessig the effects f bth the shape ad the ccetrati f the iclusi ad/r prsity ad the damage evluti the multiphysical micrscpic ad macrscpic behavirs ad the effective prperties. The first example sheds light btaiig the macrscpic effective prperties, takig it accut the piezelectric BaTiO ctiuus fibers embedded i the piezmagetic CFe 2 O 4 matrix. While i the secd example, mechaical ladig is csidered, epxy is take as the matrix material ad the respse f the cmpsite is preseted while the evluti f damage i terms f debdig is takig place. Ó 211 Elsevier Ltd. All rights reserved. 1. Itrducti Cmpsite materials csistig f piezelectric ad piezmagetic phases have draw a sigificat iterest due t the rapid develpmet i adaptive material systems. Hwever, the icreased cmplexity f the micrstructure i these materials cmplicates their aalysis. Oe pssible rute t slve such cmplexity is t achieve useful mdels that calculate the effective material prperties. Therefre, may studies were ccered with the predicti f the effective r verall prperties usig varius techiques. Fr istace, Abudi (21) emplyed a hmgeizati micrmechaical methd t predict the effective mduli f electrmaget-therm-elastic cmpsites. Results fr fibrus ad peridically bilamiated cmpsites were cmpared with the geeralized methd f cells ad the Mri Taaka predictis. Li ad Du (1998b) develped a micrmechaical apprach t aalyze the average fields ad effective prperties f hetergeeus Crrespdig authr. Tel.: +2 (2) ; fax: +2 (2) addresses: yasser.shabaa@gmail.cm (Y.M. Shabaa), matti.ristimaa@ slid.lth.se (M. Ristimaa). media that exhibit full cuplig betwee statiary elastic, electric ad magetic fields. Usig the slutis btaied fr iclusi ad ihmgeeity prblems i a ifiite maget-electr-elastic medium (Li ad Du, 1998a), they established exact relatis fr the iteral field distributi iside a hetergeeus magetelectr-elastic slid. I additi, they btaied clsed-frm expressis fr the effective mduli f fibrus ad lamiated cmpsites as well as the exact cectis betwee the effective thermal mduli ad the effective maget-electr-elastic mduli f tw-phase cmpsites. Li (2) studied the average magetelectr-elastic field i a multi-iclusi r a ihmgeeity embedded i a ifiite matrix ad develped a umerical algrithm t evaluate the maget-electr-elastic Eshelby tesrs fr the geeral material symmetry ad ellipsidal iclusi. Based the framewrk f the Variatial Asympttic Methd fr Uit Cell Hmgeizati (VAMUCH), Tag ad Yu (29) develped a micrmechaics apprach t predict the effective prperties as well as the lcal fields f peridic smart materials respsive t fully cupled electric, magetic, thermal ad mechaical fields. Vids exist i smart materials ad sme iclusis may be partially r fully debded frm the matrix. These ca be attributed t 2-768/$ - see frt matter Ó 211 Elsevier Ltd. All rights reserved. di:1.116/j.ijslstr

2 21 Y.M. Shabaa, M. Ristimaa / Iteratial Jural f Slids ad Structures 48 (211) either the maufacturig prcess r the ladig cditis durig the usage f a smart material. Therefre, the effects f defects ad iclusis the prperties f smart materials are imprtat t be csidered. Zhg ad Meguid (1997) develped a geeralized ad mathematically rigrus mdel t treat the partially debded circular ihmgeeity prblem i piezelectric materials uder ati-plae shear ad i-plae electric field usig the cmplex variable methd. This eabled the explicit determiati f the cmplex ptetials iside the ihmgeeity ad the matrix. Deg ad Meguid (1999) dealt with the prblem f a partially debded piezelectric circular iclusi i a piezelectric matrix ad derived the clsed frm cmplex ptetials bth iside ad utside the iclusi. They explicitly derived the frmulae fr the electr-elastic field itesity factrs f the iterfacial crack. Chug ad Tig (1996) studied the tw dimesial prblems f a elliptic hle r iclusi i a slid f a istrpic piezelectric material. Fag et al. (21) preseted a theretical methd t study the multiple scatterig f electr-elastic waves resultig frm tw subsurface hles i a fuctially graded piezelectric material layer bded t a hmgeeus piezelectric material, ad als preseted the dyamic stress arud the hles. Based the abve itrducti, it is evidet that predictig the respse f a smart material while csiderig its micrstructure is bth theretically challegig ad practically imprtat. The eed fr estimatig the electr-maget-therm-mechaical elastic behavir f smart cmpsite structures while csiderig damage evluti ad differet gemetries f the reifrcemets has aturally arise as smart cmpsites are icreasigly used i differet egieerig applicatis. Accrdigly, the aim f this wrk is t preset a micrmechaically-based mdel ad its crrespdig icremetal cstitutive law fr a smart hetergeeus cmpsite while csiderig the damage evluti. The damage evluti is preseted by the debdig f the iclusis frm the matrix. This mdel icrprates the full cuplig f electric, magetic, thermal ad elastic mechaical behavirs. I additi, the resultig icremetal cstitutive law is able t calculate the lcal micrscpic fields i the matrix ad i the iclusi as well as the macrscpic field. Mrever, it ca predict the effective prperties icludig the effective elastic, piezelectric, piezmagetic, dielectric, magetic permeability ad maget-electric cuplig cefficiets as well as the thermal stress cefficiets ad the assciated pyrelectric ad pyrmagetic cstats. Takig the debdig prcess it csiderati, this cstitutive law is applicable t prus materials ad cmpsites with multiple reifrcemets ad prsities icludig a wide rage f iclusi ad/r pre gemetries ragig frm elliptical cylider t thi disk t sphere t ribb prvided that the prper Eshelby tesr is used. T validate the predictis f the prpsed mdel, the results are cmpared t thse btaied frm the fllwig fur appraches: the hmgeizati thery, the geeralized methd f cells ad the Mri Taaka apprach all preseted by Abudi (21) ad the results based the VAMUCH f Tag ad Yu (29). Agreemets betwee these five appraches have bee shw t be very gd. Here r ij ad e ij are the stress tesr ad strai tesr, respectively; D i ad E i are the electric displacemet ad electric field; B i ad H i are the magetic flux ad field. C ijkl, k il, ad l il are the elastic stiffess, the dielectric, ad magetic permeability tesrs, respectively. Stress is cupled with the electric ad magetic fields thrugh the piezelectric, e ijl, ad piezmagetic, q ijl, cefficiets, respectively, while electric ad magetic fields are cupled thrugh the maget-electric cefficiet, a il. Fially, the stress, electric displacemet, ad magetic flux are cupled with temperature chages dt thrugh the thermal stress tesr k ij (te that k ij = C ijkl a kl with a kl as the thermal expasi strai tesr), pyrelectric cefficiet p i, ad pyrmagetic cefficiet m i. I rder fr the differet multiphysical prperties f the smart materials t be readily cmputed, the theretical estimates are develped usig a matrix frmulati. Hece, the cstitutive equati (1) is writte i matrix frm as fllws: r C e q e k D 5 ¼ e T k a 54 E 5 4 p 5dT B q T a T l H m where usig the symmetry f the stress ad strai tesrs, C is a 6 6 submatrix fr elastic cstats, e is a 6 submatrix fr piezelectric cefficiets, q is a 6 submatrix fr piezmagetic cefficiets. Mrever, k is a submatrix fr dielectric cefficiets, a is a submatrix fr electrmagetic cefficiets ad l is a submatrix fr magetic permeability. I the fllwig aalysis, it is cveiet t treat the elastic, electric, ad magetic fields equally. T this ed, the tati itrduced by Barett ad Lthe (1975) fr piezelectric aalysis ad the geeralized t icrprate magetic cuplig by Alshits et al. (1992) is utilized. Here, ly the matrix frmats are csidered, but mre details ca be fud i Li ad Du (1998b). The geeralized stress R is itrduced as a 12 1 clum matrix ctaiig the glbal stress, glbal electric displacemet ad glbal magetic flux as R ¼ ½r 11 r 22 r r 2 r 1 r 12 D 1 D 2 D B 1 B 2 B Š T the superscript T detes the traspse. The geeralized strai Z is defied as a 12 1 clum matrix ctaiig the glbal strais ad the electric ad magetic fields as Z ¼ ½e 11 e 22 e c 2 c 1 c 12 E 1 E 2 E H 1 H 2 H Š T ð4þ Takig advatage f the geeralized frmats defied abve, allws the cstitutive Eq. (2) t be writte as R ¼ LZ WdT ¼ LðZ PdTÞ ð2þ ðþ ð5þ 2. Prperties f the cstituet materials Electr-maget-therm-elastic media that exhibit liear, static, aistrpic cuplig betwee the magetic, electric, thermal ad elastic fields are csidered. I this case, the cstitutive equatis fr bth the matrix ad reifrcemets ca be expressed as fllw: r ij ¼ C ijkl e kl þ e ijl ð E l Þþq ijl ð H l Þ k ij dt D i ¼ e ikl e kl k il ð E l Þ a il ð H l Þ p i dt B i ¼ q ikl e kl a il ð E l Þ l il ð H l Þ m i dt ð1þ where L is a effective material matrix ctaiig the effective multiphysics material prperties ad W is a 12 1 matrix ctaiig the secd-rder thermal stress tesr k, the vectr f pyrelectric p ad the vectr f pyrmagetic m. I additi, P equals L 1 W ad the superscript -1 is used t dete iversi. Fr a trasversely istrpic piezelectric-piezmagetic cmpsite with axis f symmetry rieted i the -directi, the material matrix L takes the fllwig frm where the cefficiets are labeled accrdig t Abudi (21) t facilitate a easy cmparis with results f ther studies i the literature.

3 Y.M. Shabaa, M. Ristimaa / Iteratial Jural f Slids ad Structures 48 (211) C 11 C 12 C 1 e 1 q 1 C 11 C 1 e 1 q 1 C e q C 44 e 15 q 15 C 44 e 15 q 15 C 66 L ¼ k 11 a 11 k 11 a 11 k a l l l I Fig. 1, the piezelectrmagetic cmpsite f iterest is shw. The states befre ad after a icremetal defrmati f the cmpsite ctaiig piezelectrmagetic elemets, i the damage prcess are depicted. The far field applied mechaical, electrical ad magetic ladig cditis are deted as R ad the respse field as Z at the start f a icremet where at the ed f the icremet the values are give by R + dr ad Z + dz. Mrever, the state befre the icremetal defrmati, shw i Fig. 1(a), is described i terms f the vlume fractis f the itact ad damaged particles f p ad f d. If the vlume fracti f the particles that are debded durig the icremetal defrmati is deted by df p, the the state after defrmati, shw i Fig. 1(b), ca be described i terms f the vlume fractis f the itact ad debded particles f p df p ad f d + df p. T btai the icremetal macrscpic ad micrscpic cstitutive respses, the aalyses will be based the frmulati prvided by Thg ad Chu (1996) ad Shabaa (2) fr the debdig case. Fllwig Eshelby equivalece priciple cmbied with Mri Taaka mea field ccept, the icremetal excitati field i the itact particle, dr p, is give by dr p ¼ dr þ der þ dr pt 1 ¼ L 1 dz P 1 dt þ dz e þ dz pt 1 ¼ L dz P dt þ dz e þ dz pt 1 dz 1 ð6þ ad W is give as W ¼½k 11 k 22 k k 2 k 1 k 12 p 1 p 2 p m 1 m 2 m Š T Frm here, R ad Z will be referred t as excitati field ad respse field, respectively.. Icremetal cstitutive equati The icremetal frmulati is itrduced s that the resultig cstitutive equati is valid fr cmpsites with differet prperties f the cstituets while csiderig the debdig prcess f the reifrcemets..1. Frmulati Itact particle Σ, Z Σ + dσ, Z + dz where the subscript refers t the matrix ad 1 refers t the reifrcemets. Sice air has a sigificatly smaller dielectric cstat relative t that f a piezelectric material (Ssa, 1992), it is assumed that the excitati field (i.e. the stress, electric displacemet ad magetic flux) iside a vid vaishes. Hece, the Eshelby equivalece priciple fr the debded reifrcemet ca be writte i the fllwig frm: ¼ dr þ der þ dr pt 2 ¼ L dz P dt þ dz e þ dz pt 2 dz 2 ð7þ Furthermre, sice i the debdig prcess the curret reifrcemet excitati field shuld be released i the ext icremetal defrmati, the fllwig equati is btaied: R p ¼ dr þ der þ R pt ¼ L 1 dz P 1 dt þ dz e þ Z pt ¼ L dz P dt þ d e Z þ Z pt Z I the abve equatis, dr ad der are the icremetal excitati field ad the icremetal average excitati field based Mri ad Taaka mea field ccept, ad they are related t dz ad d e Z by dr ¼ L ðdz P dtþ; der ¼ L d e Z ð9þ I Eqs. (6) (8), dr pt 1 ; drpt 2 ; Rpt ad dzpt 1 ; dzpt 2 ; Zpt represet the perturbed parts f the excitati ad the respse fields i the itact ad debded reifrcemets ad the reifrcemet t be debded, respectively. Mrever, dz 1 ; dz 2 ad Z are the Eshelby equivalet trasfrmati respse fields. The perturbed respse fields are related t the trasfrmati respse fields by ð8þ dz pt 1 ¼ SdZ 1 ; dzpt 2 ¼ SdZ 2 ; Zpt ¼ SZ ð1þ Debded particle (vid) Itact particle vlume fracti: f p Debded particle vlume fracti: f d Matrix vlume fracti: 1 - f p - f d (a) Befre icremetal defrmati Matrix Debded particle i the icremetal defrmati Itact particle: f p - df p Debded particle: f d + df p Matrix: 1 - f p -f d (b) After icremetal defrmati Fig. 1. Debdig f ihmgeeities i the defrmati prcess. where dr pt 1 ; drpt 2 ad Rpt are described by dr pt 1 ¼ L ðs IÞdZ 1 ; drpt 2 ¼ L ðs IÞdZ 2 ; Rpt ¼ L ðs IÞZ ð11þ The matrix I is a idetity matrix ad the matrix S detes the maget-electr-elastic Eshelby tesr f the particles, which is the key igrediet ecessary fr determiig the magetelectric cuplig f piezelectric-piezmagetic cmpsites (Wu ad Huag, 2). Fr ellipsidal iclusis, S is expressed as a fucti f the shape f the ihmgeeity ad the electr-maget-elastic mduli f the matrix. Explicit expressis fr the electr-magetelastic Eshelby tesrs are give i Li ad Du (1998a) ad Huag et al. (1998). Sice the icremetal verall excitati field dr is determied by the average excitati field f the cmpsite, it fllws that, dr ¼ðf p df p ÞdR p df p R p þð1 f p f d ÞðdR þ derþ ð12þ

4 212 Y.M. Shabaa, M. Ristimaa / Iteratial Jural f Slids ad Structures 48 (211) where the icremetal average excitati field der is give by der ¼ ðf p df p ÞdR pt 1 þ f ddr pt 2 þ df pr pt ð1þ Substitutig Eqs. (9) ad (11) it (1), the icremetal average respse field is expressed by dz e ¼ ðs IÞ ðf p df p ÞdZ 1 þ f ddz 2 þ df pz ð14þ Usig Eqs. (6) (8) ad takig Eqs. (1) ad (14) it csiderati, the fllwig relatis are btaied after sme mathematical maipulatis. h i L 1 Rp ¼ ðl L 1 Þ 1 ðl 1 P 1 L P ÞþP dt h i þ S ðl L 1 Þ 1 L dz 1 ðs IÞZ ð15þ L 1 Rp ¼ðS IÞdZ 2 ðs IÞZ ð16þ L 1 Rp ¼ dz P dt ðs IÞ ðf p df p ÞdZ 1 þ f ddz 2 þ df pz þðs IÞZ ð17þ The Eshelby equivalet trasfrmati respse fields, dz 1 ; dz 2 ad Z, are btaied by slvig the simultaeus Eqs. (15) (17). I the geeral case, there are three phases fr the reifrcemets i the cmpsite, ad there are twelve cmpets f the trasfrmati respse field fr each phase. The sluti ca be writte as dz 1 ¼ A 1 1 B 1 þ A 1 1 B 1ðS IÞH 1 df p ½I ðs IÞD 1 Š L 1 dr þ P dt þ A 1 1 B 1ðS IÞH 1 L 1 A 1 1 M 1 þ A 1 1 B 1ðS IÞH 1 df p ½P ðs IÞD 2 Š df pr p dt dz 2 ¼ A 1 2 B 2 þ A 1 2 B 2ðS IÞH 1 df p ½I ðs IÞD 1 Š L 1 dr þ P dt A 1 2 B 2ðS IÞH 1 L 1 df pr p þ A 1 2 M 2 A 1 2 B 2ðS IÞH 1 df p ½P ðs IÞD 2 Š dt Z ¼ H 1 ½ðS IÞD 1 IŠ L 1 dr þ P dt H 1 L 1 r p where þ H 1 ½P ðs IÞD 2 ŠdT D 1 ¼ðf p df p ÞA 1 1 B 1 þ f d A 1 2 B 2 D 2 ¼ðf p df p ÞA 1 1 M 1 f d A 1 2 M 2 A 1 ¼ðL 1 L Þ 1 R 1 f d R 1 R 2 B 1 ¼ f d R 1 L I M 1 ¼ f d R 1 L P ðl 1 L Þ 1 ðl 1 P 1 L P Þ A 2 ¼ R 1 1 ðl 1 L Þf d R 1 2 R ði SÞ B 2 ¼ R 1 2 L R 1 1 ðl 1 L Þ M 2 ¼ R 1 1 ðl 1P 1 L P ÞþR 1 2 L P H ¼ðS IÞðf p df p ÞA 1 1 B 1ðS IÞdf p þðs IÞf d A 1 2 B 2ðS IÞdf p þðs IÞð1 df p Þ R 1 ¼ L þðl 1 L ÞS þðl 1 L ÞðI SÞðf p df p Þ R 2 ¼ L ði SÞðf p df p Þ R ¼ L ð1 f d Þ ð18þ ð19þ The icremetal verall respse field, dz, f the cmpsite is expressed by the average respse field as fllws: dz ¼ðf p df p Þ dz þ dz e þ dz pt 1 þ f d þ df p dz þ dz e þ Z pt dz þ dz e þ dz pt 2 þð1 f p f d Þ dz þ d e Z Csiderig Eqs. (1) ad (14), the abve relati becmes dz ¼ dz þðf p df p ÞdZ 1 þ f ddz 2 þ df pz ð2þ ð21þ Substitutig frm Eq. (18), the verall icremetal respse field-, dz, icremetal excitati field, dr, relati f the cmpsite is btaied as dz ¼ L 1 dr þ Pdf p þ PdT where L 1 ¼ ði þ D 1 Þ ½D 1 ðs IÞ IŠH 1 df p ½ðS IÞD 1 IŠ P ¼½D 1 ðs IÞ IŠH 1 L 1 Rp ( P ¼ D ) 2 þ½d 1 ðs IÞ IŠH 1 df p ½P ðs IÞD 2 Š P ðði þ D 1 Þ ½D 1 ðs IÞ IŠH 1 ½ðS IÞD 1 IŠdf p Þ L 1 ð22þ ð2þ Eq. (22) shws that the icremetal macrscpic respse field f the cmpsite csists f three mai parts: the respse field icremet due t the excitati field icremet, the debdig damage ad the temperature chage. The effective electr-magetmechaical prperties f the cmpsite such as elastic, piezelectric, piezmagetic, dielectric, magetic permeability ad maget-electric cuplig cefficiets ca be extracted frm the first part. O the ther had, the thermal stress tesr k ad the assciated pyrelectric vectr, p, ad pyrmagetic vectr, m, that are icluded i the thermal matrix P f the cmpsite ca be extracted frm the third part. First, it is ccluded that W ¼ LP ð24þ ad fially the thermal expasi cefficiets i differet directis ca be evaluated frm a ¼ C 1 k ð25þ The apprach als allws fr the assessmet f the micrscpic cstitutive behavir, i.e. the resultig lcal excitati field ad thereby als allws fr the pssibility t desig failure-safe electr-maget-therm-mechaical smart cmpsites. Therefre, the micrscpic excitati ad respse fields f the matrix ad reifrcemets are itrduced here. The icremetal average excitati field f the matrix, dr m ¼ dr þ der, is give by h i dr m ¼ L dz P dt ðs IÞ ðf p df p ÞdZ 1 þ f ddz 2 þ df pz ð26þ Usig Eq. (18), the explicit expressi f the matrix excitati field ca be expressed as (! ði þd dr m 1 Þ ½D 1 ðs IÞ IŠH 1 ¼ L ði SÞ L 1 x½ðs IÞD 1 IŠdf p þði SÞ 1 S dr þ½d 1 ðs IÞ IŠH 1 df p L 1 Rp!) D 2 þ½d 1 ðs IÞ IŠH 1 df p ½P ðs IÞD 2 Š þp ðði þd 1 Þ ½D 1 ðs IÞ IŠH 1 ½ðS IÞD 1 IŠdf p IÞ dt ð27þ Mrever, the icremetal average excitati field f the itact reifrcemets is dr p ¼ dr þ der þ dr pt 1 dr p ¼ dr m þ L ðs IÞdZ 1 ð28þ

5 Y.M. Shabaa, M. Ristimaa / Iteratial Jural f Slids ad Structures 48 (211) Furthermre, the icremetal average respse fields f the matrix ad bth the itact ad damaged reifrcemets ca be evaluated frm dz m ¼ L 1 drm dz p ¼ L 1 1 drp dz d ¼ dz þ d e Z þ SdZ 2 ð29þ With Eqs. (27) (29) ad takig advatage f Eq. (18), it is the pssible t calculate the icremetal micrscpic excitati ad respse fields fr the differet cstituets..2. Vlume fracti f the particles i the debdig prcess Based the critical eergy criteri f iterfacial debdig (Che et al., 2), the threshld bd stregth r cr betwee the particle ad matrix materials depeds the particle size ad may be writte as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cð1 f r p ÞE cr ¼ c 1 ½f p ð1 2m Þþð1þm Þ=2Š ðþ where c is the specific iterface adhesive eergy; E is the iitial mdulus f the matrix; m is the Piss rati f the matrix; c 1 is the particle radius ad f p is the iitial particle vlume fracti. Suppse the prbability f debdig at the iterface ca be described by the Weibull s distributi fucti: P ¼ 1 exp r r cr S m ; ðr P r cr Þ ð1þ where S ad m are material parameters ad r is the average rmal stress at the iterface that defied i Che et al. (2). The, fr r > r cr, the cumulative vlume fracti f the damaged reifrcemets is represeted by f p P, ad the vlume fracti f the reifrcemets t be debded i the icremetal defrmati, df p, is give by df p ¼ f p dp dr dr ð2þ T clse the thery, the abve relatis are the used i the previusly-derived cstitutive equatis f the cmpsite. 4. Numerical results ad discussis I this secti, tw examples are csidered. The iteti f the first example is t demstrate the applicability f the prpsed mdel by applyig it t a tw-phase electr-maget-elastic cmpsite csistig f the piezelectric BaTiO ctiuus fibers embedded i piezmagetic CFe 2 O 4 matrix. While i the secd example, the matrix is chaged t be epxy istead f CFe 2 O 4 t evaluate the micrscpic ad macrscpic fields ad the damage evluti f the smart cmpsite uder mechaical ladig cditis. The electr-maget-therm elastic mduli f the cstituet materials are take frm Tag ad Yu (29) ad are preseted i Table 1. The csidered materials f the cstituets, BaTiO ad CFe 2 O 4, are trasversely istrpic with the axis f symmetry rieted i the -directi while epxy is istrpic. Table 1 Material prperties f the cmpsite cstituets (BaTiO, CFe 2 O 4 ad Epxy) (Tag ad Yu, 29). BaTiO CFe 2 O 4 Epxy C 11 (GPa) C 12 (GPa) C 1 (GPa) C (GPa) C 44 (GPa) e 15 (C m 2 ) 11.6 e 1 (C m 2 ) 4.4 e (C m 2 ) 18.6 k 11 (1 9 CV 1 m 1 ) k (1 9 CV 1 m 1 ) q 15 (N A 1 m 1 ) 55 q 1 (N A 1 m 1 ) 58. q (N A 1 m 1 ) l 11 (1 6 Ns 2 C 2 ) l (1 6 Ns 2 C 2 ) a 11 (1 6 K 1 ) a 22 (1 6 K 1 ) a (1 6 K 1 ) Fig. 2. Effective elastic mduli C 11 ad C 12 f a fibrus cmpsite agaist the vlume fracti f BaTiO fr differet values f the aspect rati (b =.1,.5, 1, 5, ad 1). cylidrical fibers (b = 1). While C 12 is hardly affected by the aspect rati, C 11 decreases with its icrease. This is maily because the dimesi i the 1-directi f the elliptic crss secti f the fiber decreases relative its dimesi i the 2-directi with the icrease f the aspect rati ad csequetly the stiffess decreases i the 1-directi ad icreases i the 2-directi. Beyd b = 5, the effect f the aspect rati C 11 is egligible. The slight deviati f C 12 with the aspect rati may refer t the secdary effect f the electrmagetic prperties f the cstituets. Figs. 5 reveal the Cefficiets f Thermal Expasi (CTEs), pyrelectric p ad pyrmagetic m prperties as fuctis f the 4.1. Electr-maget-therm elastic prperties Fig. 2 shws the variatis f the elastic mduli C 11 ad C 12 agaist the vlume fracti f the BaTiO elliptical cylidrical fibers fr differet values f the aspect rati b. It ca be see that bth mduli recver the mduli f the cstituet phases at the tw vlume fracti limits. Als, the btaied results are csistet with thse preseted i Abudi (21) fr the circular Fig.. Effective thermal expasi cefficiets a 11 ad a f a fibrus cmpsite agaist the damage evluti fr differet vlume fractis f BaTiO (f p =.5,.1,.2 ad.).

6 214 Y.M. Shabaa, M. Ristimaa / Iteratial Jural f Slids ad Structures 48 (211) Fig. 4. Effective pyrelectric cefficiet p f a fibrus cmpsite agaist the damage evluti fr differet vlume fractis f BaTiO (f p =.5,.1,.2 ad.). applicatis, eve if it is abset i each cstituet. The variati f the effective electrmagetic cuplig cefficiet a f the fibrus cmpsite agaist the damage evluti fr differet vlume fractis f the fibers is illustrated i Fig. 6. It ca be ticed that a decreases with the damage evluti ad its decreasig rate becmes higher fr the higher fiber vlume fracti. Hwever, fr the perfect cmpsite a is icreasig with the fiber vlume fracti ad its values are csistet with the results i Abudi (21). Fig. 7 depicts the effective magetic permeability cefficiets l 11 ad l f the smart material agaist the damage evluti fr differet vlume fractis f the fibers. It ca be see that l 11 icreases with the icrease f the iitial fiber vlume fracti, which is ppsite t l. Bth cefficiets are almst idepedet the damage evluti. It ca be bserved frm Fig. 8 that the effective axial dielectric permittivity k icreases sigificatly with the iitial fiber vlume fracti, while the effective trasverse dielectric permittivity k 11 is almst idepedet the iitial fiber vlume fracti. Althugh k 11 remais almst ivariat with the chage f the damage evluti, k decreases almst liearly with the damage evluti. The variatis f the effective piezelectric cefficiets e, e 1 ad e 15 are shw i Fig. 9. It is bvius that e ad e 1 are sigificat while e 15 is almst egligible. Hwever e ad e 1 have ppsite behavirs with the icrease f the iitial fiber vlume fracti ad the damage evluti; they apprach zer whe all fibers tur it vids. Hwever the piezmagetic cefficiet q 15 decreases with the icrease f the iitial fiber vlume fracti as shw i Fig. 1; it is almst idepedet the damage evluti. It is fud frm the umerical results that the ther Fig. 5. Effective pyrmagetic cefficiet m f a fibrus cmpsite agaist the damage evluti fr differet vlume fractis f BaTiO (f p =.5,.1,.2 ad.). damage evluti f d /f p due t debdig damage fr differet values f the circular cylidrical fiber vlume fracti. The values f CTEs, p ad m cicide with Tag ad Yu (29) whe f d =. Sice a 11 f the fiber is higher tha bth a ad that f the matrix, the icrease f the fiber vlume fracti results i a icrease i a 11 f the cmpsite ad a decrease i its a as shw i Fig.. This behavir ccurs because the lg fibers prevet the matrix frm expadig i the -directi, ad as a result the matrix is frced t expad mre tha usual i the trasverse directis. This results i a lwer CTE i the -directi tha that i the trasverse directis. It ca be see that a icreases whe the fibers gradually tur it vids due t the reducti f the fiber vlume fracti. This csequetly results i mre freedm that the matrix has t expad i the -directi. Whe all fibers tur it vids, a appraches the matrix limit. O the ther had, a 11 has a ppsite behavir as it decreases whe fibers gradually tur it vids. Althugh the pyrelectric ad pyrmagetic cefficiets are abset i each f the idividual cstituets, the pyrelectric effect is iduced i the cmpsite due t the iteracti betwee the piezelectric effect ad the thermal expasi. Mrever, the pyrmagetic effect is iduced due t the iteracti betwee the piezmagetic effect ad the thermal expasi. As depicted i Figs. 4 ad 5, the cefficiets p ad m are i geeral decreasig with the icrease f the fiber vlume fracti fr the perfect cmpsite ( debdig ccurs) ad these results are csistet with Tag ad Yu (29). Whe all fibers tur it vids, p appraches zer fr the differet values f the fiber vlume fracti. O the ther had, m decreases liearly with the damage evluti as shw i Fig. 5. The electrmagetic cefficiet is ather material prperty that is triggered i the cmpsite ad ca be utilized i practical Fig. 6. Effective magetelectric cefficiet a f a fibrus cmpsite agaist the damage evluti fr differet vlume fractis f BaTiO (f p =.5,.1,.2 ad.). Fig. 7. Effective magetic permeability cefficiet l 11 ad l f a fibrus cmpsite agaist the damage evluti fr differet vlume fractis f BaTiO (f p =.5,.1,.2 ad.).

7 Y.M. Shabaa, M. Ristimaa / Iteratial Jural f Slids ad Structures 48 (211) Fig. 8. Effective dielectric cefficiets k ad k 11 f a fibrus cmpsite agaist the damage evluti fr differet vlume fractis f BaTiO (f p =.5,.1,.2 ad.). Fig. 11. Effective thermal expasi cefficiets a 11 ad a f a fibrus cmpsite agaist the damage evluti fr differet gemetries f the reifrcemets (f p =.). Fig. 9. Effective piezelectric cefficiets e, e 1 ad e 15 f a fibrus cmpsite agaist the damage evluti fr differet vlume fractis f BaTiO (f p =.5,.1,.2 ad.). Fig. 12. Effective pyrelectric ad pyrmagetic cefficiets p ad m f a fibrus cmpsite agaist the damage evluti fr differet gemetries f the reifrcemets (f p =.). prperties p ad m are shw i Fig. 12. The ribb ad thi disk reifrcemets have the highest levels fr p ad m respectively. While disk reifrcemet results i egligible p, cylider ad ribb reifrcemets result i a egligible m relative t that f the disk reifrcemet. Mrever, bth prperties apprach zer whe all reifrcemets tur it vids. It is emphasized agai that the cmpsite exhibits these tw prperties eve if either cstituet exhibits them. It ca be see frm the previus results that the differet prperties have remarkable differeces accrdig t the csidered reifrcemet aspect rati, vlume fracti ad gemetry Effect f damage evluti durig ctiuus ladig Fig. 1. Effective piezmagetic cefficiets q 15 f a fibrus cmpsite agaist the damage evluti fr differet vlume fractis f BaTiO (f p =.5,.1,.2 ad.). piezmagetic cefficiets q 1 ad q have the same rder f the magitude f q 15 ad als have the same behavir f q 15. As a fial ivestigati, three differet gemetries f the reifrcemets are csidered i rder t ivestigate their effects the prperties f the smart cmpsite. These gemetries are circular cylidrical fiber, thi disk ad ribb with iitial vlume fracti equals.. It ca be see frm Fig. 11 that the trasverse CTE a 11 is higher tha the axial e a fr all gemetries. I additi, the cylidrical fiber has the highest level fr a 11 ad a, while the disk reifrcemet has the lwest e. Mrever, the disk is sigificatly affected by the damage evluti relative t the ther gemetries. The variati f the pyrelectric ad pyrmagetic As a example f the applicati f the preset cstitutive equati, a ctiuus ladig situati is csidered. The purpse f this is t highlight the usefuless f the frmulati (e.g. as a cstitutive driver i a -liear fiite elemet settig). The mdel is evidetly very geeral, but fr simplicity the respse uder uiaxial tesi i the axial directi is aalyzed. The specific micrstructure is described as a particle reifrced smart cmpsite takig it accut the debdig damage f the reifrcemets. The matrix material is take as epxy with the piezelectric BaTiO iclusis. The iitial reifrcemet vlume fracti, the reifrcemet diameter ad the specific iterface adhesive eergy are take t be.15, 1 lm ad.1 J/m 2, respectively. The variatis f the axial stresses i the cmpsite, reifrcemet ad matrix r z ; r p z ad r m z as well as the damaged reifrcemets vlume fracti f d are shw i Fig. 1 as fuctis f the cmpsite axial strai e z. It ca be see that bth f the

8 216 Y.M. Shabaa, M. Ristimaa / Iteratial Jural f Slids ad Structures 48 (211) effect f the cstituet materials, the prpsed cstitutive equati ca be used after remvig the magetic effect parts; thus, the stiffess ad Eshelby matrices will be 9 9 matrices i this case. If the electric parts are als remved, these matrices will be 6 6 matrices ad the cstitutive equati will deal with the thermmechaical behavirs f cmpsites. Mrever, the thermal part ca defiitely be mitted t leadig t predictig the mechaical field behavirs f traditial cmpsites. Evetually, differet cmbiatis f the fur differet fields f differet cmpsites ca be treated by the prpsed cstitutive equati. I shrt, the preseted cstitutive law ca be used t study the micrstructure-prperty-perfrmace relatiship f materials ad t guide the desig ad ptimizati f the smart structures. Fig. 1. Micrscpic ad macrscpic stress strai relatis f the cmpsites with debdig damage. b =1,f p =.15. micrscpic stresses are liear ad the reifrcemets exhibit the highest liear stress strai relati while the matrix exhibits the lwest liear e. O the ther had, bth the macrscpic stress strai relati ad the damage evluti are liear. Nt surprisigly, the vlume fracti f the damaged reifrcemets icreases with the axial strai. It ca be ticed that the stress i the reifrcemets is e rder f magitude higher tha bth the stress i the matrix ad the macrscpic stress f the cmpsite. Therefre, desigig smart structures based the macrscpic aalysis may give misleadig results ad may lead t uexpected failure f differet structures. Evetually, the micrscpic aalysis ad the damage evluti f the reifrcemets shuld be csidered whe desigig differet smart structures. Fr this ladig situati, it turs ut that the electric field E, which is iduced due t the piezelectric effect, icreases mtically with the axial strai as shw i the figure. 5. Cclusis A micrmechaical apprach, which results i a icremetal cstitutive equati, t aalyze the lcal multiphysical micrscpic fields i the matrix ad i the iclusi as well as the macrscpic multiphysical field f hetergeeus smart cmpsites, which are sesitive t thermmechaical, electric ad magetic fields, is preseted. Full cuplig betwee elastic, electric, magetic ad thermal fields ad differet gemetries f the reifrcemets as well as the debdig prcess f the reifrcemets due t high ladig cditis are csidered. The effective multiphysical prperties icludig the effective elastic, piezelectric, piezmagetic, dielectric, magetic permeability ad maget-electric cuplig cefficiets as well as the thermal stress cefficiets ad the assciated pyrelectric ad pyrmagetic cstats are extracted frm the macrscpic multiphysical field. The prpsed cstitutive equati is valid fr predictig fur differet field behavirs f smart cmpsites uder fur differet types f ladig cditis either idividually r cmbied. These are electric, magetic, thermal, ad mechaical ladig cditis. Als, it is valid fr cmpsites that exhibit e r mre f these field behavirs ad ladig cditis. Fr example, if the cmpsite uder csiderati is a piezelectric e with magetic Ackwledgemets This research wrk is fiacially supprted by the Eurpea Ui thrugh a Erasmus Mudus Iteratial Visitig Academic Staff Fellwship awarded t the first authr. This supprt is gratefully ackwledged. The wrk was cmpleted while the first authr was visitig the Divisi f Slid Mechaics, Lud Uiversity, Swede. Refereces Abudi, J., 21. Micrmechaical aalysis f fully cupled electr-magettherm-elastic multiphase cmpsites. Smart Mater. Struct. 1, Alshits, V.I., Dariskii, A.N., Lthe, J., O the existece f surface waves i halfifiite aistrpic elastic media with piezelectric ad piezmagetic prperties. 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