Analytical expressions to estimate the effective piezoelectric tensor of a textured polycrystal for any crystal symmetry

Depósi o de In es igación de la Uni e sidad de Se illa h ps://idus.us.es/ This is an Accep ed Manusc ip o an a icle published by Else ie In Mechanics o Ma e ials, Vol. 151, on Decembe 2020, a ailable a : h ps://doi.o g/10.1016/j.mechma .2020.103604 Copy igh 2020 Else ie . En idUS Licencia C ea i e Commons CC BY-NC-ND Analy ical exp essions o es ima e he e ec i e piezoelec ic enso o a ex u ed polyc ys al o any c ys al symme y Julie a L. Bu oni Facul ad de Ciencias Exac as y Na u ales, Uni e sidad Nacional de Ma del Pla a, Deán Funes 3350, B7602AYL, Ma del Pla a, A gen ina Fede ico C. Bu oni E-mail: [email p o ec ed] Depa men o Mechanical Enginee ing and Manu ac u ing, Uni e sidad de Se illa, Camino de los Descub imien os s/n, 41092, Se ille, Spain Ad ián P. Cisilino INTEMA & Depa men o Mechanical Enginee ing, Uni e sidad Nacional de Ma del Pla a & CONICET, A . Juan B. Jus o 4302, B7608FDQ Ma del Pla a, A gen ina Rode ick Melnik MS2 Disco e y In e disciplina y Resea ch Ins i u e, Wil id Lau ie Uni e si y, 75 Uni e si y A e W, Wa e loo, On a io, Canada N2L 3C5 Luis Rod íquez-Tembleque and And és Sáez Depa men o Con inuum Mechanics and S uc u al Analysis, Uni e sidad de Se illa, Camino de los Descub imien os s/n, Se ille E-41092, Spain Abs ac . This wo k in oduces a se o analy ical exp essions ha simpli ies he p ocedu e o ob aining closed- o m o mulas o es ima e he e ec i e piezoelec ic p ope ies o polyc ys alline agg ega es o med by c ys als o all he 21 non- cen osymme ical classes and wi h a bi a y ex u es. The exp essions a e de i ed om o ien a ional a e ages o hi d-o de piezoelec ic enso s o he indi idual c ys als ha a e weigh ed by o ien a ional dis ibu ion unc ions. The a e aging is done by using gene alized sphe ical ha monic se ies expansions. Imp o emen s wi h espec o p e ious wo ks in he li e a u e a e wo old: all c ys al symme ies a e conside ed and no symme y es ic ions a e imposed o ex u e. The e sa ili y o he in oduced exp essions is demons a ed on an example o BaTiO3polyc ys als wi h uniaxial and biaxial ex u es cha ac e ized by single and double Gaussian dis ibu ions, espec i ely. The esul s o uniaxial ex u e a e in pe ec ag eemen wi h he esul s published in he li e a u e. The biaxial ex u e is discussed in de ail and analyzed o a se o limi ing cases. Manusc ip File Click he e o iew linked Re e ences A e aging piezoelec ic hi d-o de enso s 2 1. In oduc ion Piezoelec ic ce amics (Uchino, 2017) a e gene ally used in he o m o a collec ion o pe ec ly bonded piezoelec ic c ys als wi h a la ge numbe o dis inc pola iza ion o ien a ions. Mac oscopic piezoelec ic p ope ies o he ma e ial a e go e ned by his mic oscopic inhomogeneous c ys al mo phology. When he ma e ial is an agg ega e o andomly o ien ed piezoelec ic c ys als, no ne pola iza ion is ealized and so he ma e ial is no mac oscopically piezoelec ic. To achie e mac oscopic piezoelec ic p ope ies, he indi idual c ys als mus ha e a p e e en ial o ien a ion. Gene ally, piezoelec ic p ope ies exhibi la ge aniso opy, which is an a ac i e ea u e o ma e ial design. P e e ed o ien a ion o he c ys allog aphic domains can be accomplished by con olling g ow h pa ame e s du ing ab ica ion (Kim e al., 2006) o by subjec ing he ma e ial o a la ge elec ic ield a high empe a u e (Li e al., 2005). Op imiza ion me hods can be used o design ma e ials speci ic piezoelec ic p ope ies by ailo ing hei o ien a ion dis ibu ion ( ex u e). In he amewo k o he implemen a ion o an op imiza ion algo i hm, Fini e Elemen Analysis (FEA) is a e sa ile ool o compu e e ec i e piezoelec ic p ope ies ( he op imiza ion objec i e unc ion) by means o a nume ical homogeniza ion app oach, see o ins ance Jayachand an e al. (2011). Howe e , closed- o m analy ical exp essions o e ec i e piezoelec ic p ope ies a e always p e e able o e he FEA homogeniza ion app oach, as hey simpli y implemen a ions and speed up calcula ions. E ec i e piezoelec ic p ope ies can be es ima ed by a e aging he p ope ies o single c ys als in he agg ega e aking in o accoun mac oscopic ex u e. To calcula e he e ec i e p ope ies o a polyc ys al, he bes a e aging me hod is he sel -consis en , ex ended o piezoelec ic polyc ys als by Li (2000). Howe e his me hod equi es compu a ion o piezoelec ic Eshelby enso which in ol es, in gene al, nume ical in eg a ion. On he o he hand, he e a e he simple olume a e aging app oaches, like he Voig model (see Li and Dunn (2001)), which a e a ac i e because hey allow o deduce closed- o m exp essions o he e ec i e p ope ies. In pa icula , he Voig model assumes uni o m s ain and elec ic ields in he c ys alli es. This assump ion is alid o piezoelec ic polyc ys als wi h ibe ex u es unde ce ain condi ions (see Li e al. (1999)) o which exac es ima es o some o he elec oelas ic moduli esul . Wo king in he con ex o ibe ex u e, Li e al. (1999) and Li (2000) compu ed Voig -Reuss es ima ions o he elec oelas ic moduli o polyc ys als by app oaching he p oblem wi h gene alized sphe ical ha monics. Speci ically, Li e al. (1999) conside ed ibe ex u e o o ho hombic single c ys als belonging o class symme y 2mm, which includes as pa icula cases classes o e agonal 4mm and hexagonal 6mm symme ies (see Nye’s book (Nye, 1985)). An in e es ing inding by Li (2000) was ha agg ega es o 4mm c ys als o ba ium i ana e (BaTiO3) wi h hei o ien a ions a ound he pola iza ion di ec ion cha ac e ized by a Gaussian dis ibu ion p esen an ampli ica ion e ec o he piezoelec ic coupling wi h espec o he single c ys al p ope ies. Recen ly, some o he A e aging piezoelec ic hi d-o de enso s 3 au ho s o he p esen wo k ha e used he Li’s amewo k o modelling mode n lead- ee piezocomposi es o ind in e es ing beha iou s a ibu able o such ampli ica ion e ec o he polyc ys alline phase (K ishnaswamy e al., 2019a,b, 2020a,b,c). P e ious wo ks add ess only o ho hombic c ys al symme y wi h ans e sely iso opic ex u e. This wo k p esen s a comp ehensi e app oach o de i e closed- o m exp essions o he es ima ion o he e ec i e piezoelec ic s ess enso o c ys al agg ega es o any symme y and gene al ex u e. The p oposed app oach is based on he p e ious wo ks by Li e al. in he use o gene alized sphe ical ha monics expansion and ex ends he ideas by Bunge (1982) and Roe (1965) among o he s, om elas ici y o piezoelec ici y. The de i ed analy ical exp essions a e e i ied and hei e sa ili y demons a ed h ough a s udy p oblem consis ing in a polyc ys alline agg ega e o BaTiO3wi h a ious o ien a ional dis ibu ion unc ions. I is shown how he gene al exp essions in oduced in his wo k can be educed o hose esul s by Li (2000) o speci ic cases. The epea ed index con en ion o addi ion is used in his wo k. Explici sums a e indica ed in exp essions when deemed con enien o he sake o cla i y. 2. P elimina ies We conce n wi h he compu a ion o he aniso opic elec omechanical coupling o polyc ys al piezoce amics om he p ope ies o single c ys alli es and hei o ien a ions. A simple app oxima ion o compu e his coupling is h ough he olume a e age o he hi d-o de piezoelec ic s ess enso eijk, which sa is ies eijk =eikj (i, j, k =1...3). Since in his case eijk depends only on he c ys al o ien a ion g(xi)(i=1...3) a each poin x, he in eg a ion o e a ep esen a i e olume elemen Vcan be ca ied ou in wo s eps, as usual. We i s in eg a e o e all hose olume elemen s dVwi h o ien a ion g, and hen o e all o ien a ions g, he eby he a e age enso is compu ed by hei=8⇡2Ie(g)w(g)dg, (1) whe e he unc ion w:SO(3) !R0–called O ien a ion Dis ibu ion Func ion (ODF)– appea s as a weigh unc ion ha accoun s o he olume ic densi y o c ys alli es o ien ed in dg. This Haa measu e dgis —a e no maliza ion— gi en by dg=1/(8⇡2)sin✓d✓d dwhe e ( ,✓,)a e he Eule ’s angles. Then, he ODF is no malized such as 8⇡2Iw(g)dg=1.(2) The e o e, by exp essing he c ys al o ien a ion in e ms o ( ,⇠=cos✓,), he Voig a e age becomes hei=Z2⇡ 0Z2⇡ 0Z+1 1 e(⇠, ,)w(⇠, ,)d⇠d d,(3) A e aging piezoelec ic hi d-o de enso s 4 No e ha he no maliza ion ac o 8⇡2abo e esul s in a iso opic ODF, wiso =1/(8⇡2) (Bunge, 1982; Roe, 1965). Among he di e en al e na i es, we adop he ollowing con en ion o speci y o a ions h ough Eule ’s angles: one begins wi h he p incipal ma e ial coo dina e sys em x0 i(i=1,2,3) ixed o he c ys al and wi h he axes being pa allel o hose o he global coo dina e sys em xi(i=1,2,3); hen x0 i(i=1,2,3) is i s o a ed abou he x0 3-axis h ough he angle ; he second o a ion is abou he x0 2-axis (in i s new o ien a ion) h ough ✓and, inally, he hi d o a ion is again abou he x0 3-axis (in i s new o ien a ion) h ough he angle . All posi i e o a ions a e in coun e clockwise di ec ion. The piezoelec ic componen s o e0 ixed o a c ys al can be exp essed in he global coo dina e sys em xi(i=1,2,3) acco ding o he ans o ma ion law o hi d-o de enso s eijk(⇠, ,)=⌦ im⌦jn⌦koe0 mno =Timjnkoe0 mno,(4) whe e ⌦– he ma ix ep esen a ion o he o ien a ion gin he Eule ’s space– is he o hogonal ans o ma ion ma ix (SO(3)) gi en by ⌦=0 B @ cos(✓)cos( )cos()sin( )sin()cos(✓)cos( )sin()sin( )cos()sin(✓)cos( ) cos(✓)sin( )cos()+cos( )sin()cos( )cos()cos(✓)sin( )sin()sin(✓)sin( ) sin(✓)cos()sin(✓)sin()cos(✓) 1 C A.(5) The six h-o de ma ix Timjnko condenses he iple enso p oduc ⌦im⌦jn⌦ko; i possesses majo symme ies only (mino symme ies a e no p esen because ⌦is nonsymme ic) gi en by he pe mu a ions o he disjoin cycles o indices Cycles[(3 5)(4 6)],Cycles[(1 3)(2 4)],Cycles[(1 3 5)(2 4 6)].(6) These symme ies allow a conside able educ ion in he numbe o componen s o Timjnko o be compu ed. Thus, o he 729 componen s o Timjnko only 249 a e di e en . Finally, he a e aging o equa ion (4) esul s in heijk(⇠, ,)i=hTimjnkoe0 mnoi=hTimjnkoie0 mno,(7) which shows ha he es ima ion o he e ec i e piezoelec ic enso o he polyc ys alline agg ega e educes o inding he hTimjnkoi o he co esponding ODF. In wha ollows, he p ime symbol will be d opped om he piezoelec ic cons an s in he c ys al coo dina e sys em in o de o allow a cleane no a ion. 3. A e aging in e ms o gene alized sphe ical ha monic se ies expansions Following Gel’ and e al. (1963), gene alized sphe ical ha monics o m a comple e o hogonal basis o he Hilbe space L2(SO(3)), which is he se o all squa e in eg able eal- alued unc ions on SO(3) wi h inne p oduc de ined by (H1,H 2)=IH1(g)H⇤ 2(g)dg, H1,H 22L2(SO(3)),(8) A e aging piezoelec ic hi d-o de enso s 5 whe e ⇤deno es he complex conjuga e. Thus, a gene al eal unc ion H2L2(SO(3)) can be expanded in se ies o gene alized sphe ical ha monics. This is a use ul app oach o e alua e he a e age o a gene al eal unc ion hHi. To do his, we expand bo h he ODF w(⇠, ,)and H(⇠, ,)in o gene alized sphe ical ha monics se ies (Roe, 1965) as ollows: w(⇠, ,)= 1 X l=0 l X m=l l X n=l WlmnZlmn(⇠)eim ein,(9) and H(⇠, ,)= 1 X l=0 l X m=l l X n=l HlmnZlmn(⇠)eim ein,(10) whe e i=p1and Zlmn(⇠)is a gene aliza ion o he Legend e associa ed unc ion Pmn l ha can be exp essed as Zlmn(⇠)=i nm 2l+1 2Pmn l(⇠),(11) being Pmn l(⇠)= (1)lminm 2l(lm)! (lm)!(l+n)! (l+m)!(ln)!1 2 ⇥(1 ⇠) (nm) 2 (1 + ⇠)(n+m) 2 dln d⇠ln⇥(1 ⇠)lm(1 + ⇠)l+m⇤ .(12) The associa ed Legend e unc ion Pmn lcan be ei he eal o pu ely imagina y, acco ding o whe he m+nis e en o odd, espec i ely. Consequen ly, Zlmn(⇠)is always eal- alued. The coe icien s o he expansions in (9) and (10) a e (Bunge, 1982; Kocks e al., 2005) Wlmn =1 4⇡2Z2⇡ 0Z2⇡ 0Z+1 1 w(⇠, ,)Zlmn(⇠)eim eind⇠d d,(13) he so-called ex u e coe icien s, and Hlmn =1 4⇡2Z2⇡ 0Z2⇡ 0Z+1 1 H(⇠, ,)Zlmn(⇠)eim eind⇠d d.(14) No e ha all in o ma ion abou he ODF is con ained in he ex u e coe icien s, which a e complex quan i ies sa is ying Wlmn =(1)m+nW⇤ l¯m¯n,(15) due o he symme y p ope ies o Zlmn(⇠)(Roe, 1965). In equa ion (15), ¯m=m. F om he no maliza ion condi ion (2), i can be shown ha W000 =1 4p2⇡2.(16) In p ac ice, he no maliza ion o w(⇠, ,)is p e e ed by ensu ing (16) ins ead o (2). A e aging piezoelec ic hi d-o de enso s 6 Expansion coe icien s Hlmn a e complex and ypically in ol e in eg a ion o simple igonome ic unc ions (Hcon ains p oduc s o di ec ion cosines) and he gene alized sphe ical ha monics. Thei analy ical exp essions can be ound once His speci ied. Finally, he expansion o he a e age o a gene al eal unc ion hHi,is (Bunge, 1982; Kocks e al., 2005) hHi=4⇡2 R X l=0 l X m=l l X n=l HlmnWlmn,(17) whe e, om he unca ion heo em by Fe a i and Johnson (1988), R=3, he enso ank o eijk. In ac , all coe icien s Hlmn a e ze o o l>3. I is wo h o no e ha , unlike he wo ks by Li (2000); K ishnaswamy e al. (2019a); Li e al. (1999), he p esen analysis does no imposed any es ic ion on Wlmn. In he nex , we will use (17) o ob ain he exp essions o he a e age o a ion enso in (7), which, in u n, will be used o compu e he e ec i e piezolec ic enso . 4. E ec i e piezoelec ic enso s o all c ys al symme ies No e i s ha each o he componen s Timjnko a e eal unc ions Timjnko :SO(3) !R, simila o unc ion H. Then, de i a ion o he exp essions o he a e age o a ion ma ix in (7) can be pe o med ia exp ession (17). This me hodology in ol es he compu a ion o he coe icien s Hlmn o each o hse 249 di e en componen s o Timjnko acco ding o (14). This was done analy ically using Ma hema ica (2020). Due o space limi a ions, he ull exp essions a e epo ed in Supplemen a y Ma e ial 1. All 249 componen s a e necessa y in o de o conside a gene al case wi h any symme y. Two sou ces o symme y a e conside ed when compu ing he a e age piezoelec ic enso (7): he ones due o he c ys alli e s uc u es and he o he s due o he ODF. No symme y es ic ions a e imposed on he ODF –and hence o he ex u e coe icien s in (17)– in his wo k. In espec o he c ys alli e symme ies, all he non-cen osymme ical classes (Nye, 1985) a e conside ed. The exp essions o (7) o each o he 21 classes a e p esen ed in Supplemen a y Ma e ial 2. The heijki in Supplemen a y Ma e ial 2 a e unc ions o he ex u e coe icien s Wlmn, which a e necessa y o speci y o each case o analysis acco ding o he adop ed ODF. The p ope y (15) has been used o p esen he exp essions o heijkiin eal o m. No e ha om he 21 non-cen osymme ical classes only 20 a e piezoelec ic. Thus, he cubic class 432 esul s in all componen s o heijkiiden ically ze o, no ma e wha he Wlmn a e. Some o he piezoelec ic classes sha e he ma ix s uc u e educing he amoun o exp essions as i can be obse ed in Supplemen a y Ma e ial 2. Two supplemen a y ma e ials p o ide all he de ails o he main con ibu ion o his wo k. The nex sec ion analyzes he e agonal 4mm class in de ail. A e aging piezoelec ic hi d-o de enso s 7 5. S udy p oblem: Te agonal class 4mm The heo y in oduced abo e is applied o he analysis o polyc ys alline agg ega e o BaTiO3in o de o e i y and illus a e he de i ed esul s. Ba ium i ana e is bo h a lead- ee ma e ial (Sai o e al., 2004) and en i onmen ally iendly in i s p ocessing (Ibn-Mohammed e al., 2017). In e es ing no el applica ions (Wang e al. (2018) among o he s) ha e been p oposed while BaTiO3-based composi es can be ab ica ed in a scalable manne using eme ging addi i e manu ac u ing me hods (Kim e al., 2017, 2019; Pha ha apee anun e al., 2017). Li (2000), among o he s (see o example (Li e al., 1999; Ruglo sky e al., 2006)), has s udied he a e age piezoelec ic p ope ies o BaTiO3polyc ys alline agg ega es. The BaTiO3c ys al belongs o he poin g oup class 4mm, wi h hei piezoelec ic cons an s lis ed in Table 1 (Li, 2000). Thus, he exp essions o he a e age componen s heijmi o he e agonal symme y a e e ie ed om Supplemen a y Ma e ial 2 and summa ized below: he111i=4 105⇡2⇣p14(5eAeB)<⇣p3W310 p5W330⌘+ 14p3eB<(W110)⌘, (18) he122i=4 315⇡2⇣14p3<(W110)(2eAeB+3e C)+ p14(eB5eA)<⇣p3W310 +3 p5W330⌘⌘, (19) he133i=8⇡27<(W110)(2eAeB+3e C)+2 p14(5eAeB)<(W310) 105p3,(20) he123i=8⇡2(5eAeB)=(W320) 3p105 ,(21) he113i=2 315⇡2⇣7p6<(W100)(2(eA+e B)+3e C)+ 2p7(5eAeB)<⇣3p2W300 2p15W320⌘⌘, (22) he112i=4 315⇡2⇣7p3=(W110)(2(eA+e B)+3e C)+ p14(5eAeB)=⇣p3W310 3p5W330⌘⌘, (23) he211i=4 315⇡2⇣14p3=(W110)(2eAeB+3e C)+ p14(eB5eA)=⇣p3W310 3p5W330⌘⌘, (24) he222i=4 105⇡2⇣p14(5eAeB)=⇣p3W310 +p5W330⌘+ 14p3eB=(W110)⌘, (25) A e aging piezoelec ic hi d-o de enso s 8 he233i=8 105p3⇡2(7=(W110)(2eA+e B3eC)+ 2p14(eB5eA)=(W310)⌘, (26) he223i=2 315⇡2⇣7p6<(W100)(2(eA+e B)+3e C)+ 2p7(5eAeB)<⇣3p2W300 +2 p15W320⌘⌘, (27) he213i=8⇡2(5eAeB)=(W320) 3p105 ,(28) he212i=4 315⇡2⇣7p3<(W110)(2(eA+e B)+3e C)+ p14(5eAeB)<⇣p3W310 +3 p5W330⌘⌘, (29) he311i=4 315⇡2⇣7p6<(W100)(2eAeB+3e C)+ p7(eB5eA)<⇣3p2W300 2p15W320⌘⌘, (30) he322i=4 315⇡2⇣7p6<(W100)(2eAeB+3e C)+ p7(eB5eA)<⇣3p2W300 +2 p15W320⌘⌘, (31) he333i=4 105 ⇣2p14⇡2(eB5eA)<(W300)+7 p6⇡2eB<(W100)⌘,(32) he323i=4 105p3⇡2(7=(W110)(2(eA+e B)+3e C)+ 4p14(eB5eA)=(W310)⌘, (33) he313i=4 105p3⇡2(7<(W110)(2(eA+e B)+3e C)+ 4p14(eB5eA)<(W310)⌘, (34) and he312i=8⇡2(5eAeB)=(W320) 3p105 ,(35) whe e eA=2e113 +e311,eB=4e113 +2e311 +3e333 and eC=2e113 4e311. No e ha his se o analy ical exp essions educe o hose p esen ed by Li (2000) when ODF is such ha ex u e coe icien W320 anishes. Howe e , exp essions (18) - (35) a e mo e gene al as shown below. To e i y hese esul s, we i s conside he ODF w(⇠, ,)=(⇠1),(36) whe e (⇠1) is he Di ac del a unc ion a ⇠=1. This ODF desc ibes an agg ega e wi h all he c ys alli es o ien ed wi h ✓=0and andom alues o and , which A e aging piezoelec ic hi d-o de enso s 15 Figu e 7. E ec i e piezoelec ic moduli o polyc ys al BaTiO3as a unc ion o he Gaussian dis ibu ion pa ame e ✓ o µ✓=0,µ =⇡/3and  =1. Figu e 8. E ec i e piezoelec ic cons an he112io he BaTiO3polyc ys al as a unc ion o µ and  o µ✓=0and ✓=0.6. Table 2. Ma ix s uc u es o he e ec i e piezoelec ic coupling enso s esul ing om he wo Gaussian ODFs. ODF C ys al ma ix s uc u e Polyc ys al ma ix s uc u e wA0 @ 0000e223 0 000e223 00 e311 e311 e333 000 1 A0 @ 0000he223i0 000he223i00 he311ihe311ihe333i000 1 A wB0 @ 0000e223 0 000e223 00 e311 e311 e333 000 1 A0 @ he111ihe122ihe133ihe123ihe113ihe112i he211ihe222ihe233ihe223ihe123ihe212i he311ihe322ihe333ihe323ihe313ihe123i 1 A A e aging piezoelec ic hi d-o de enso s 16 6. Conclusions This wo k p o ides gene al analy ical exp essions o olume a e ages o piezoelec ic p ope ies o polyc ys alline agg ega es. The exp essions a e de i ed om o ien a ional a e ages o hi d-o de piezoelec ic enso s ha a e weigh ed by o ien a ional dis ibu ion unc ions ha accoun o ex u e. The exp essions a e de i ed by using sphe ical ha monic se ies expansions. Imp o emen s o his wo k wi h espec o he p e ious wo ks in he li e a u e a e wo old: all c ys al symme ies a e conside ed and no symme y es ic ions a e imposed o ex u e. Thus, his wo k p o ides he analy ical o mulas o compu e he a e age piezoelec ic enso o polyc ys alline agg ega es o med by c ys als o all he 21 non- cen osymme ical classes. These exp essions a e open o be specialized o any ex u e, which is speci ied in e ms o Eule ´s angles ( ,✓,). The e sa ili y o he in oduced exp essions is demons a ed o BaTiO3 polyc ys als wi h uniaxial ex u e gi en by a single Gaussian dispe sion o ✓, and wi h biaxial ex u e gi en by a double Gaussian dispe sion o ✓and . The esul s o he i s case, which consis ed in a Gaussian dis ibu ion cen e ed a ound ✓=0, showed o be in pe ec ag eemen wi h esul s published in he li e a u e. This case has he pa icula i y ha all he ex u e coe icien s and all he exp essions o he a e age piezoelec ic p ope ies a e eal. On he o he hand, no p e ious s udy was ound o e i y he esul s o he simple Gaussian dis ibu ions wi h mean alue ✓6=0 and o he case wi h biaxial ex u es. The biaxial ex u e leads o complex- alued ex u e coe icien s and complex o mula ions o he a e aged piezoelec ic p ope ies. Resul s o he BaTiO3polyc ys al wi h biaxial ex u e ha e been discussed in de ail and analyzed o a se o limi ing cases. In all cases, he esul s showed o be consis en . The main con ibu ion o his wo k is he se o analy ical exp essions ha simpli ies ob aining closed- o m o mulas o es ima e he e ec i e piezoelec ic p ope ies o polyc ys als wi h a bi a y symme ies and ex u es. These exp essions a e sui able o pe o m pa ame ic s udies like hose p esen ed in his wo k o BaTiO3polyc ys als, and –p o ided he ex u es cha ac e ized by smoo h ODFs– o implemen e icien analy ical op imiza ion me hods o ind con igu a ions a he mic os uc u al le el in o de o maximize piezoelec ici y. I is s aigh o wa d o adap he exp ession he ein de i ed o he piezoelec ic s ess enso e o o he piezoelec ic p ope ies, such as d,g,h(see IEEE S anda d on Piezoelec ici y (1988)) o he piezomagne ic moduli. Au ho ’s con ibu ions JLB: In es iga ion, So wa e, Valida ion, FCB: Concep ualiza ion, Fo mal Analysis, Me hodology, Visualiza ion, P ojec Adminis a ion, W i ing - O iginal d a p epa a ion, Re iew & Edi ing, APC: Supe ision, Resou ces, W i ing - O iginal d a p epa a ion, Re iew & Edi ing, RM: W i ing – Re iew & Edi ing, LRT: P ojec REFERENCES 17 Adminis a ion, W i ing - Re iew, AS: W i ing - Re iew Acknowledgmen s This wo k was suppo ed by he Minis e io de Economía y Compe i i idad o Spain unde p ojec DPI2017-89162-R, he Conseje ía de Economía, Conocimien o, Emp esas y Uni e sidad o he Jun a de Andalucía (Spain) unde p ojec P18-RT-3128 and he Uni e sidad Nacional de Ma del Pla a (A gen ina) unde p ojec 15/G511. R.M. is also acknowledging suppo o NSERC and CRC P og am. 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