Predicting effective magnetostriction and moduli of magnetostrictive composites by using the double-inclusion method

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1 Mchanics of Matrials 35 (003) Prdicting ffctiv magntostriction and moduli of magntostrictiv composits by using th doubl-inclusion mthod Xu Fng a, Daining Fang a, *, Ai-kah Soh b, Kh-Chih Hwang a a Dpartmnt of Enginring Mchanics, Tsinghua Univrsity, Bijing 00084, China b Dpartmnt of Mchanical Enginring, Th Univrsity of Hong Kong, Hong Kong Rcivd Fbruary 00; rcivd in rvisd form 5 July 00 Abstract Th most important proprtis of magntostrictiv composits ar th ffctiv magntostriction and ffctiv lastic moduli. In this papr, an attmpt was mad to prdict th ffctiv proprtis of magntostrictiv composits consisting of magntostrictiv particls disprsd in a non-magntostrictiv matrix. This was don by xtnding th doubl-inclusion modl to dal with magntolastic problms. It has bn shown that th magntostriction of magntostrictiv composits dpnds not only on th volum fraction and matrial proprtis, but also on th shap and orintation of th llipsoidal inhomognitis and th common magntostriction constants of th cubic microcrystallit. Th analytical rsults hav bn compard with th xisting xprimntal rsults to dmonstrat th applicability of th proposd doubl-inclusion modl. Ó 00 Elsvir Scinc Ltd. All rights rsrvd. Kywords: Magntostrictiv composit; Doubl-inclusion modl; Effctiv magntostriction; Magntolastic moduli. Introduction Magntostriction is a consqunc of th rotation of atomic momnts insid a spcial crystallin. Th rar-arth iron alloys attract xtnsiv intrst bcaus of thir comparativly larg magntostriction. Clark (980) has discussd in dtails th magntostrictiv bhavior and magntomchanical proprtis of rar-arth iron compounds. * Corrsponding author. Tl.: ; fax: addrss: fangdn@mail.tsinghua.du.cn (D. Fang). Howvr, th high cost and wak mchanical proprtis of ths matrials limit xtnsiv us of such matrials in nginring applications. Th composits consisting of magntostrictiv particls and a non-magntostrictiv matrix, such as Al, which is suscptibl to lastic dformations, constitut a vry intrsting group of magntostrictiv matrials. This is bcaus such matrials do tak th advantag of ach constitunt and consquntly posssss suprior magntolastic proprtis as compard with thos of th convntional magntostrictiv matrials (Pinkrton t al., 997). Du to th uniqu proprty in trms of th magntolastic coupling ffcts arising from /03/$ - s front mattr Ó 00 Elsvir Scinc Ltd. All rights rsrvd. doi:0.06/s (0)0077-6

2 64 X. Fng t al. / Mchanics of Matrials 35 (003) th magnto-mchanical intraction, th magntostrictiv composits hav attractd grat attntions of th practicing nginrs for th dsign of intllignt or activ structurs. A rmarkably wid varity of applications, which includ th dvlopmnt of magnto-mchanical transducrs, magntic snsors, strain gags, ultrasonic gnrators, and strss snsors (Karl t al., 000; Shim t al., 998), rquir sufficintly larg magntostriction. In rcnt yars, som modls hav bn dvlopd by diffrnt rsarchrs to prdict th magntostriction of magntostrictiv composits. Hrbst t al. (997) proposd a simpl modl for prdicting th ffctiv magntostriction of SmF / Al and SmF /F composits. This modl tratd magntostriction as an lastic problm in modling an lastically isotropic magntostrictiv sphr mbddd in an infinit matrix. This was just an approximat modl bcaus it was insnsitiv to matrials constants. Chn t al. (999) masurd th dpndnc of saturatd magntostriction for Trfnol-D composits with diffrnt matrics. Guo t al. (00) producd th poxy and glass matrix Trfnol-D composits and masurd th saturatd magntostriction and ovrall modulus of th composit. Nan (998, 999) proposd a mor rigorous modl basd on GrnÕs function. Howvr, his modl was dvlopd basd on a spcial avraging tchniqu for approximation whil th intraction btwn th magntic fild and strss was uncoupld by introducing two indpndnt GrnÕs functions. It is worth noting that th ffctiv magntostriction ratio k prdictd using NanÕs modl for th SmF s /F composit was smallr than that for th SmF s /Al composit, which was inconsistnt with th xprimntal data availabl in th opn litratur. For lastic two-phas composits, many thoris hav bn proposd for prdicting th ffctiv proprtis of th composits in trms of th proprtis of th constitunts and thir microstructur. Dtaild discussions hav bn providd by Nmat-Nassr and Hori (993) and, thrfor, will not b ritratd hr. In this papr, th doubl-inclusion modl dvlopd by Hori and Nmat-Nassr (993, 994) is xtndd to prdict th ffctiv modulus and magntostriction of th magntostrictiv composits. This modl accounts for th intraction btwn th particls (inclusions) and matrix, and btwn th particls (inclusions) and particls (inclusions). Th ffctiv proprtis of th doubl-inclusion ar th approximatd valus of thos of th composits, and th avrag filds in th doubl-inclusion can b obtaind analytically. Thrfor, it is asy to obtain th avrag strain filds inducd by magntostriction in th doubl-inclusion whn th ign-strains inducd by magntostriction ar givn. Morovr, th doubl-inclusion modl can b unifid with othr modls by propr slction of th shap and orintation of th inclusion and matrix, such as th Mori Tanaka mthod, th slfconsistnt schm, th Castanda WillisÕ modl, and th Kustr Toksoz modls (Hu and Wng, 000). In rcnt yars, th doubl-inclusion modl has bn mployd to analyz thrmolastic and magnto-lctro-lastic problms (Li, 999, 000a,b).. Prliminaris Th constitutiv quations for a matrial xhibiting magnto-mchanical coupling ffcts can b xprssd as (Nan and Wng, 999): r ij ¼ E ijkl kl q kij H k v ijkl H k H l ðþ B i ¼ q ijk jk þ l ij H j þ v ijkl jk H l ðþ whr r ij, ij, B i, and H i ar th strsss, total strains, magntic inductions, and magntic filds, rspctivly; E ijkl and l ij ar th lastic moduli and prmability, rspctivly; q ijk and v ijkl ar th pizomagntic and magntostrictiv cofficints, rspctivly. From th abov quations, th magntomchanical ffcts can b dividd into two catgoris: pizomagntic and magntostrictiv ffcts. Th pizomagntic ffct is linar, whras, th magntostrictiv ffct is non-linar. In gnral, most frromagntic matrials ar composd of cubic crystals with giant magntostriction, and ths matrials do not xhibit any pizomagntic ffct. By ignoring th pizomagntic ffct, Eqs. () and () can b rwrittn as

3 X. Fng t al. / Mchanics of Matrials 35 (003) r ij ¼ E ijkl kl E ijkl ms kl B i ¼ l ij ð kl ; H k ÞH j ð3þ ð4þ whr ms ij ar strains inducd by magntostriction, which is th function of magntic fild, H i. Th boundary conditions of th composit can b xprssd as u i ðsþ ¼ 0 ij x j; /ðsþ ¼H 0 i x i ð5þ whr u i, / and S ar th lastic displacmnts, magntic potntial and xtrnal surfac of th composit, rspctivly. Th ffctiv magntostrictiv bhaviour of an inhomognous composit is dfind in trms of th avragd filds (Nan and Wng, 999) as follows: hr ij i¼e ijkl h klie ijkl ms kl hb i i¼l ij hh ji ð6þ ð7þ whr Eijkl, ms, and l ij ar th ffctiv lastic moduli, magntostrictiv strains and prmability of th composits, rspctivly. Th most important proprtis of th magntostrictiv composits ar th saturatd magntostriction and ffctiv moduli of th composits. Thrfor, th focus of th prsnt study is on ths two factors. In fact, whn th magntostriction bcoms saturatd in th magntostrictiv phas, th magntostriction is indpndnt on th magntic fild and only dpndnt on th magntostriction constants of th cubic microcrystallit (Nan and Wng, 999). With rfrnc to Eq. (6), which prsnts th magntostrictiv bhavior of th htrognous matrials, it is convnint to divid th lastic fild into two parts: on is du to th applid linar displacmnt, i.., fild I, which is purly lastic; and th othr is du to th magntostriction inducd by th magntic fild, i.., fild II, which dpnds on th xtrnal magntic fild. In accordanc with this classification mthod and th avrag strain thorm (Nmat-Nassr and Hori, 993), Eq. (6) can b rwrittn as and hr H ij i¼e ijkl ms kl ð9þ whr hr S iji dnots th pur lastic strsss arising from th applid linar displacmnts, and hr H ij i dnots th strsss inducd by th magntostrictiv strains, which dpnd on th xtrnal magntic fild. 3. Th doubl-inclusion modl Th ffctiv magntostriction of th magntostrictiv composits can b dtrmind by first introducing th doubl-inclusion mthod and, hnc, xtnd it to handl th magntolastic problms. A doubl-cll X is dsignatd for ach inclusion, as shown in Fig.. Th doubl-cll consists of th matrix rgion X X and th inhomognity X. Th doubl-cll X is thn mbddd in an infinit matrix D of lastic moduli E 0 and its boundary is subjctd to a far fild (rfr to Fig. ). For convninc, X and X X ar dnotd by C and C, rspctivly, and th corrsponding volum fractions ar f and f. Not that f þ f ¼. E and E ar th lastic moduli of th inhomognity C and th matrix C, rspctivly. 3.. Th ffctiv lastic moduli For dtrmining th ffctiv lastic moduli, only th fild I du to th applid displacmnt is considrd. Whn a uniform fild ij is applid at th boundary of th infinit matrix, a disturbanc hr S ij i¼e ijkl 0 kl ð8þ Fig.. Schmatic of th doubl-inclusion modl.

4 66 X. Fng t al. / Mchanics of Matrials 35 (003) fild d ij will b gnratd du to th prsnc of th inhomognity and th ign-strain. In accordanc with th quivalnt inclusion mthod, th consistncy rlationships can b xprssd as E r ijkl d kl j r ¼ E 0 ijkl d kl j r kl j r r ¼ ; ð0þ whr ij j r ar th quivalnt ign-strains introducd to convrt th inhomognity to an inclusion. Th abov consistncy quation (0) can b rarrangd as E ijkl þ S klmn þðs klmn S klmn Þ ¼ E 0 ijkl kl þ S klmn I klmn þðs klmn S klmn Þ ðþ and E ijkl ¼ E 0 ijkl þs klmn þf ðs klmn f S klmn Þ þ S klmn I klmn þ f ðs klmn f S klmn Þ ðþ whr Sijkl and S ijkl ar th Eshlby tnsors of X and X, rspctivly, and I klmn is th fourth-ordr unit tnsor. Th strain concntration tnsors A ijkl and A ijkl can b dfind as follows: ij j ¼ A ijkl kl ð3aþ ij j ¼ A ijkl kl ð3bþ Thus, th ffctiv lastic moduli can b obtaind from th avrag fild in th doubl-inclusion as follows: h E ijkl ¼ E0 ijrs I rspq þ S rsmn h I pqkl þ S pqcd f i A cdkl þ f A cdkl i I rsmn f A mnpq þ f A mnpq ð4þ If X and X ar similar and coaxial, it rsults in Sijkl ¼ S ijkl ¼ S ijkl. Hnc, th strain concntration tnsors can b xprssd as E A r ijkl ¼ E0 ijmn E r 0 ijmn mnkl S ijkl r ¼ ; ð5þ 3.. Th ffctiv magntostriction For dtrmining th ffctiv magntostriction, th fild II causd by th xtrnal magntic fild is considrd. In fact, th strain inducd by magntostriction can b considrd as ign-strain. Th ign-strains in C and C ar j and j, rspctivly. Th consistncy quation (0) can thn b rarrangd as (): E ijkl þ S klmn þðs klmn S klmn Þ kl j ¼ E 0 ijkl kl þ S klmn I klmn and E ijkl þðs klmn S klmn Þ þ S klmn ¼ E 0 ijkl þ S klmn f þ ðs klmn f S klmn Þ þ f ðs klmn f S klmn Þ kl j I klmn ð6aþ ð6bþ If X and X ar similar and coaxial, it rsults in Sijkl ¼ S ijkl ¼ S ijkl. Thus, th quivalnt ign-strain j r ðr ¼ ; Þ ar obtaind as follows: ij j r ¼ E0 ijkl E r ijkl Sijkl kl þ E 0 ijkl E r ijkl Sijkl ðs klmn I klmn Þ r r ¼ ; ð7þ It is asy to dtrmin S ijkl whn th inhomognity is of a spcial shap, such as sphr, cylindr or pnny. Th avrag fild in th doubl-inclusion can thn b xprssd as h H ij i¼ ij þ S ijkl f kl j kl j þ f kl j kl j ð8þ

5 X. Fng t al. / Mchanics of Matrials 35 (003) Th far fild can b obtaind basd on th lastic boundary conditions. If no traction is applid on th boundary, from th avraging thorm, th avrag strss in th composit causd by th xtrnal magntic fild should b zro. Thus, ij can b xprssd as ij ¼ I ijpq X r¼ f r þ S ijrs S pqrs þ I rskl kl j r I ijrs Arspq I pqrs A r rsmnð S mnkl I mnkl Þ ð9þ For pur magntostriction, th fr boundary conditions ar usd. Th rinforcmnt is mad up of many crystallits of cubic symmtry. Thrfor, th local magntostrictiv strains along th crystallographic axs of a microcrystallit can b xprssd as (Clark, 980; Nan and Wng, 999) ms ij ¼ ka þ k 3 00 a 3i 3 ; i ¼ j 3 a ð0þ 3ia 3j k ; i 6¼ j whr k 00, k, and k a ar th common magntostriction constants of a cubic microcrystallit. ða ij Þ is th tnsor for transforming th local crystallographic axs Xj 0 to th matrial axs X i. As mntiond abov, th magntostrictiv inducd strain can b considrd as ign-strain. Thus, th proprtis of th composit and th avrag fild in it can b approximatd by mans of doubl inhomognity using th doubl-inclusion mthod (rfr to Fig. ). Thrfor, from Eq. (8), th avrag fild of th composit can b obtaind by rplacing th ign-strain ij with ms ij. Sinc th matrix is non-magntostrictiv, ij j is zro. From Eq. (8), th ffctiv magntostrictiv strain fild can b xprssd as ms ¼h H i ¼ f fs½a ðs IÞþIŠðI þ f SA Þ ½ðS IÞA þ IŠ ðs IÞ ½A ðs IÞþIŠg ms ðþ Basd on th dfinition of magntostriction, th ffctiv magntostriction can b xprssd as k s ¼ 3 ðms k ms? Þ ðþ whr ms k and ms? ar macroscopically magntostrictiv strains paralll and prpndicular to th applid magntic fild H 3, rspctivly. Whn th matrix in isotropic or transvrsly isotropic and th rinforcd particls ar of spcial shaps such as lliptical shap, th Eshlby tnsors can b obtaind analytically. Othrwis, th Eshlby tnsors can only b dtrmind using numrical computations. It can b obsrvd from Eq. () that th ffctiv magntostriction dpnds on th proprtis of th matrix, th shap of th rinforcd particls as wll as th common magntostriction constants of th cubic microcrystallit. In this cas, whn th matrix is ithr isotropic or transvrsly isotropic, ms k ms k and ms? can b xprssd as ¼ ms 33 ; ms? ¼ ms ¼ ms ð3þ Th ffctiv normalizd magntostriction k can b asily obtaind as follows: k ¼ k s ðf Þ=k s ðf ¼ Þ 4. Numrical rsults and discussion ð4þ Th analytical prdictions of th ffctiv magntostriction of SmF /Al, SmF /F and Trfnol-D/glass composits ar mad basd on th abov proposd modl. Th analytical rsults ar thn vrifid by comparing with th xprimntal rsults obtaind by Pinkrton t al. (997) and Guo t al. (00). In th prsnt study, th magntostriction composits ar assumd to consist of randomly orintd sphrical magntostrictiv particls and a non-magntostrictiv matrix. 4.. Prdictions of ffctiv magntolastic proprtis for SmF /Al and SmF /F composits Th prdictions of th ffctiv magntostriction and lastic proprtis for SmF /Al and SmF /F composits ar obtaind for comparison with th xprimntal rsults obtaind by Pinkrton t al. (997). Tabl prsnts th proprtis of th matrix and th rinforcd particls of ths two composits. If th matrial is lastically isotropic,

6 68 X. Fng t al. / Mchanics of Matrials 35 (003) Tabl Proprtis of SmF inclusions and th mtal matrix Al, F (Pinkrton t al., 997; Nan, 998) YoungÕs modulus (GPa) PoissonÕs ratio E (GPa) E (GPa) E 44 (GPa) k (ppm) k 00 (ppm) SmF 8 66 )00 )700 Al F th lastic moduli E ijkl can b dtrmind onc th YoungÕs modulus and PoissonÕs ratio hav bn obtaind. Sinc th matrix is lastically isotropic, Fig.. Th prdictd composit lastic constants vrsus volum fraction f of SmF inclusions distributd in F matrix. E ijkl of th matrix ar obtaind from its YoungÕs modulus and PoissonÕs ratio. Figs. and 3 show th prdictd lastic constants vrsus th volum fraction of th rinforcd SmF magntostrictiv particls mbddd in th F matrix and th Al matrix, rspctivly. Whn th matrix of th magntostrictiv composit is F, th ovrall lastic proprtis E, E and E 44 dcras with th incras of th volum fraction of th rinforcd particls. Howvr, whn th matrix of th magntostrictiv composit is Al, th ovrall lastic constant E incrass, instad of dcrass lik E or rmains mor or lss constant lik E 44, with th incras of th volum fraction. This is bcaus th lastic proprtis of th SmF ar not isotropic and is much closr to thos of th Al matrix than thos of th F matrix. Figs. 4 and 5 show th ffctiv saturatd magntostriction ratio k of th SmF /F and SmF /Al magntostriction composits, rspc- Fig. 3. Th prdictd composit lastic constants vrsus volum fraction f of SmF inclusions distributd in Al matrix. Fig. 4. Comparison of th prdictd ffctiv saturatd magntostriction ratio k of th SmF /F magntostrictiv composits obtaind using th proposd doubl-inclusion modl with th xisting xprimntal rsults and thos obtaind by othr rsarchrsõ modls.

7 X. Fng t al. / Mchanics of Matrials 35 (003) Fig. 5. Comparison of th ffctiv saturatd magntostriction ratio k prdictd using thr diffrnt modls for th SmF /Al magntostrictiv composits with th xprimntal rsults. Fig. 6. Th prdictd composit lastic constants vrsus volum fraction f for th magntostrictiv composit consists of Trfnol-D inclusions and glass matrix. tivly, vrsus th volum fraction f of th SmF magntostrictiv particls. Th k valus prdictd for th SmF /Al and SmF /F by using th proposd doubl-inclusion modl (DI modl) ar in bttr agrmnt with th xprimntal data obtaind by Hrbst t al. (997), as compard with thos of NanÕs modl and HrbstÕs modl (HCP modl). 4.. Prdiction of ffctiv proprtis for Trfnol- D/glass composits Th proprtis of th matrix and rinforcd particls of Trfnol-D/glass composits ar prsntd in Tabl. In th xprimntal invstigation conductd by Guo t al. (00), both th matrix and th rinforcd Trfnol-D particls wr assumd to b lastically isotropic vn though th Trfnol-D particls wr not. Fig. 6 shows th prdictd lastic constants vrsus th volum fraction of th rinforcd magntostrictiv particls whn th matrix of th magntostrictiv composit is mad of glass. All Fig. 7. Comparison of th YoungÕs modulus calculatd for th Trfnol-D/glass magntostrictiv composit using th proposd modl with th xisting xprimntal rsults. th lastic proprtis E, E and E 44 incras with th incras of th volum fraction. A comparison of th prdictd YoungÕs moduli obtaind using th proposd modl and th corrsponding xprimntal rsults (Guo t al., 00) ar mad in Tabl Proprtis of Trfnol-D inclusions and th glass matrix (Guo t al., 00; Nan and Wng, 999) YoungÕs PoissonÕs ratio E (GPa) E (GPa) E 44 (GPa) k (ppm) k 00 (ppm) modulus (GPa) Trfnol-D Glass

8 630 X. Fng t al. / Mchanics of Matrials 35 (003) Fig. 8. Comparison of th magntostriction ratio k calculatd for th Trfnol-D/glass magntostrictiv composit using th proposd modl with th xisting xprimntal rsults. Fig. 7. Th ffctiv YoungÕs modulus is approximatly proportional to th volum fraction f. It is obvious that th analytical prdictions ar in good agrmnt with th xprimntal rsults. Fig. 8 shows th ffctiv saturation magntostriction ratio k of th Trfnol-D/glass magntostriction composit vrsus th volum fraction f of th Trfnol-D magntostrictiv particls. Th k valu prdictd for th Trfnol-D/glass composit using th proposd modl incrass with th incras of th volum fraction f, which has th sam trnd as that of th xprimnt and HCP modl. Not that th prdictd k valus ar always lowr than th corrsponding xprimntal rsults (Guo t al., 00). This is bcaus th actual Trfnol-D is not lastically isotropic and th ffctiv magntostriction should tak into account not only th orintation of th microcrystallit but also som othr microstructural faturs of th microcrystallit, such as th angls btwn th local crystallographic axs and th xtrnal magntic fild. 5. Concluding rmarks Th most important proprtis of th magntostrictiv composits ar th ffctiv magntostriction and th ovrall lastic moduli. Th doubl-inclusion mthod has bn succssfully xtndd to dvlop a modl that is abl to prdict th ffctiv magntostriction and ovrall lastic moduli of th magntostrictiv composit, which consists of magntostrictiv particls and a nonmagntostrictiv matrix. Sinc most frromagntic matrials possss cubic crystals with giant magntostriction and such matrials do not xhibit any pizomagntic ffct du to cubic symmtry, th magntostrictiv inducd strain can b considrd as ign-strain. Thus, th proprtis of th composit and th avrag fild in it can b approximatd by mans of doubl inhomognity using th doubl-inclusion mthod. Th solution is obtaind by sparating th magntolastic filds into two parts: on is du to th applid linar displacmnt, which is purly lastic, and th othr is du to th magntostriction inducd by th magntic fild, which dpnds on th xtrnal magntic fild. Thus, th gnral rlations for prdicting th ffctiv proprtis of th composits ar obtaind. It has bn shown that th magntostriction of magntostrictiv composits dpnds not only on th volum fraction and matrial proprtis, but also on th shap and orintation of th llipsoidal inhomognitis and th common magntostriction constants of th cubic microcrystallit. Comparison of th rsults obtaind using th proposd doubl inclusion modl with th xisting xprimntal data and thos of othr rsarchrsõ modls dmonstrats that th proposd modl is in bttr agrmnt with th xprimntal data than othr modls. Acknowldgmnts Th authors ar gratful for th support by National Natural Scinc Foundation of China. Rfrncs Chn, Y. t al., 999. Effct of th lastic modulus of th matrix on magntostrictiv strain in composits. Appl. Phys. Ltt. 74, 59. Clark, A.E., 980. In: Wohlfarth, E.P. (Ed.), Frromagntic Matrials, vol.. North-Holland, Amstrdam, p. 53. Guo, Z.J., Busbridg, S.C., t al., 00. Effctiv magntostriction and magntomchanical coupling of Trfnol-D composits. Appl. Phys. Ltt 78, Hrbst, J.F., Caphart, T.W., Pinkrton, F.E., 997. Estimating th ffctiv magntostriction of a composit: a simpl modl. Appl. Phys. Ltt 70, 304.

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