FRACTURE OF PIEZOELECTRIC MATERIALS

Transcription

1 CF000OR FRACUR OF PZOCRC MARAS ong-yi Zhng* nd Mingho Zho eprtment of Mechnic ngineering Hong ong University of Science nd echnoogy Cer Wter By owoon Hong ong Chin * -mi: mezhngt@ust.hk ABSRAC he present work studies theoreticy nd experimenty the frcture behvior of piezoeectric cermics. For n eectricy insuting crck the rtio of α / β pys n importnt roe in the energy reese rte where α is the rtio of the minor semi-xis b to the mjor semi-xis of the eipse nd β is the rtio of the dieectric constnt of the cvity to the effective dieectric constnt of the mteri. here re three imiting vues in the energy reese rte respectivey corresponding to α / β 0 for n eectricy permebe crck α / β finite nonzero vue for gener cses nd α / β for n eectricy impermebe crck. For n eectricy conductive crck the ppied eectric fied pre to the crck drives the crck to propgte. When eectric yied occurs t crck tip the gob energy reese rte is the sme s tht derived from iner frcture mechnics whie the oc energy reese rte shows iner retionship between the frcture toughness nd the ppied eectric fied. he experiment resuts iustrte tht for PZ-8 nd PZ- cermics the degree of scttering of mesured dt ws considerby enhnced by n ppied eectric fied. ither positive or negtive eectric fied reduced the frcture toughness. he experiment resuts so confirmed tht there exist mechnic- nd eectric-frcture toughness for PZ- cermics nd both re mteri properties. he mechnicy- nd eectricy- critic energy reese rtes re respectivey 8.7(±.) nd.7(±5.5) N/m. YWORS piezoeectric medium crck energy reese rte frcture toughness poriztion sturtion NROUCON Aong with the ppiction of piezoeectric mteris in more nd more sophisticted structures their mechnic reibiity becomes incresingy importnt. hus mny efforts hve being mde to study the frcture behvior of those mteris [-]. One of the most interesting spects is bout the effect of eectric fied on the frcture behvior. Mny experiment resuts re inconsistent or even contrdicted which mkes the study of frcture of piezoeectric cermics more chenging. he present work briefy summrizes the work done t the Hong ong University of Science nd echnoogy on this topic. WO-MNSONA SOUON Figure shows schemticy crck of ength in n infinite piezoeectric medium under remotey

2 uniform eectric ods nd in-pne nd nti-pne mechnic ods. Using Stroh s formism [-] the gener two-dimension soution cn be written in the form u Af + Af ψ f + f Σ ψ Σ ψ u ( u u u ϕ ) Σ ( ) Σ ( ) () where u ϕ i ij nd re the dispcements eectric i potenti stresses nd eectric dispcements respectivey. he two mtrixes A nd re reted to the mteri properties with A nd denoting their ij ij eements repectivey. he nytic function vector f( z ) ( f ) ( z) f ( z ) f( z ) f ( z ) is determined by the remote ods nd the boundry conditions. or i ε or i or ε i i NSUANG CRAC Zhng et. [] obtined the soutions for n insuting eiptic cvity using the exct boundry conditions trction-free surfce chrge-free nd continuity of eectric potenti ong the cvity surfce. he resuts depend grety on the rtio ofα / β where α is the rtio of the minor semi-xis b to the mjor semi-xis of the eipse nd β is the rtio of the dieectric constnt of the cvity to the effective dieectric constnt of the piezoeectric mteri. When the cvity shrinks into sit crck there re three imiting cses respectivey corresponding to α / β 0 for n eectricy permebe crck α / β finite nonzero vue for gener cses nd α / β for n eectricy impermebe crck. Ony the two cses of eectricy permebe nd impermebe crcks re discussed in the present pper. From these soutions the energy reese rte for the two cses cn be expressed in terms of the intensity fctors t the crck tip ( ( B ) or Fig. A crck in n infinite piezoeectric medium under combined remote odings. + B J () where ) with the components being the stress intensity fctors of the three modes nd nd the eectric dispcement intensity fctor respectivey. B is Hermitin mtrix B B B ia () B B he upper eft-hnd bock B is mtrix nd the ower right-hnd eement B is scr. For stbe mteris B hs the foowing properties [ ] ectricy permebe crck < B B B 0. () he eectricy permebe boundry conditions correspond to α / β 0 nd ed to the intensity fctors

3 π π B [( B + B ) + ( B + B ) + ( B + B ) ] π π (5) qution (5) shows tht the eectric dispcement intensity fctor is competey induced by the piezoeectric effect rther thn by the ppied eectric ods. hen substituting q. (5) into q. () we see tht the energy reese rte is independent of the ppied eectric fied either perpendicur or pre to the crck. n other words the energy reese rte hs nothing to do with the ppied eectric fieds. ectricy impermebe crck An eectricy impermebe crck ignores the eectric fied within the crck. hus the intensity fctors re given by ( ) ( π π π ) π. (6) qutions () nd (6) show tht the remote mechnic ods nd s we s the eectric od determine the energy reese rte whie the ppied eectric fied pre to the crck contributes nothing to the energy reese rte. Furthermore we cn esiy see tht the ppied eectric fied perpendicur to the crck impedes the crck propgtion becuse is negtive s given in q. (). B CONUCNG CRAC f the crck is eectricy conducting the boundry conditions ong crck fces re trction-free nd constnt eectric potenti. For simpicity it is ssumed tht there re no net eectric chrges on the crck fces nd the constnt eectric potenti on the crck fces is zero. hen the intensity fctors tke the form of π π. (7) π π he energy reese rte is expressed in terms of the intensity fctors where i nd J ( ) [ ] i PQ PQ (8) A A A A A A A A Q P. (9) A A A A A A A A qution (8) shows tht ony the eectric fied pre to the crck ffects the crck propgtion. We expicity demonstrte this resut with mode conductive crck. When the poing direction of trnsversey isotropic piezoeectric medium is perpendicur to the crck the nti-pne deformtion decouped from the in-pne deformtion nd the intensity fctors for mode conductive crck re given by

4 π π. (0) Consequenty the energy reese rte for the mode conductive crck is ( ) ( ) c e c J κ π. () qution () shows cery tht the ppied eectric fied pre to the crck drives the crck to propgte. POARAZAON SAURAON MO For n eectricy impermebe crck Go et. [] proposed strip poriztion mode to exmine the eectric yieding effect on the frcture behvior of eectricy insuting crcks. n this mode piezoeectric cermics re treted s mechnicy britte nd eectricy ductie mteris. he eectric sturtion is nogous to the cssic ugde mode. o emphsize the physic insight Go et. [] considered simpified piezoeectric mteri to mke the derivtion process strightforwrd. n the present work the compete governing equtions re used. he oc intensity fctors re the sum of the intensity fctors induced by the ppied fied nd the sturtion zone d + () where the ppied intensity fctor vector is given in q. (6) nd the intensity fctor vector cused by the poriztion zone tkes the form of d ( ) 0. () As resut the oc energy reese rte is given by + J ) ( B B. () Becuse the poriztion zone shieds the crck tip eectricy nd competey the oc eectric dispcement intensity fctor is zero. he ppied eectric fied contributes to the oc energy reese rte vi the piezoeectric effect. he oc energy reese rte s fiure criterion yieds iner retionship between the ppied mechnic od nd eectric fied. But the gob energy reese rte is the sme s tht from iner frcture mechnics which predicts tht either positive or negtive eectric fied impedes the crck propgtion. XPRMNA RSUS Frcture test Compct ension (C) tests nd ndenttion Frcture (F) tests were crried out on PZ-8 cermics under n ppied eectric fied. he mteri is poed PZ-8 with n verge grin size of -5 µm. he composition of PZ-8 is ner but in the tetrgon side of the morphotropic phse boundry. n the C tests the notch on ech smpe ws cut using dimond sw with 0. mm-thick bde nd the notch tip

5 ws further shrpened with wire sw of 0.05 mm in dimeter. he poing direction ws perpendicur to the notch. About 0 smpes were tested t ech eve of the eectric fieds except tht smpes were tested t the eectric fied of 5 kv/cm to study the distribution of frcture toughness. he energy reese rte ws ccuted by the finite eement method nd then converted into the mode stress intensity fctor. n the F tests the od ws 9.0 N nd the eectric fied of kv/cm ws ppied either pre or ntipre to the poing direction of the smpe. Under ech eve of the eectric fieds bout 0 indenttions were performed. Ony the frcture behvior of the crcks perpendicur to the poing direction is reported in this pper. Figures () nd (b) show the vrition of C with the ppied eectric fied obtined from the C nd F tests respectivey. Under purey mechnic oding the verged C s re. ± 0.05MP m nd.0 ± 0.06 MP m respectivey from the two methods. he men vue of C is reduced by either positive or negtive ppied eectric fied in both types of testing. n the C tests the negtive fied of 7.5 kv/cm reduces the verged C by 0.5 MP m whie the sme strength positive fied reduces the verged C by 0.0 MP m ectric fied (kv/cm) Simiry in the F tests reduction of 0. MP m or 0.0 MP m resuts from the ppiction of negtive or positive fied of kv/cm. hese fcts indicte tht negtive fied hs stronger infuence on the verged C thn positive fied does. Appying positive eectric fied of 5 kv/cm reduces further the verged C to 0.9 ± 0. MP m in the C tests resuting in retive reduction of 8%. Conducting crck C smpes used in this study were mde from poed PZ- piezoeectric cermics nd the poing direction ws pre to the notch. A 0.5 mm wide notch in every smpe ws cut nd its tip ws shrpened further by wire sw of rdius 0. mm. Siver pint ws fied into the notch to mke it function s n eectrode. hus the crck becomes eectricy conducting. wo oding types were ppied in this study. One is purey mechnic oding nd the other is purey eectric oding. n the eectric oding tests sttic votge ws ppied to smpe t the eectrodes nd incresed unti the smpe ws fied. A tests were crried out t room temperture nd 0 smpes were tested for ech oding type. t ws observed tht eectric brekdown ws usuy ccompnied by frcture. he frcture surfces re ft for smpes frctured under mechnic ods. A critic votge cuses dieectric brekdown tunne nd rough frcture surfces. he finite eement method ws used to ccute the energy reese rte for the given smpes. he critic energy reese rte under mechnic oding ws ccuted from the frcture od nd the corresponding igment s shown in Fig. (). he men of the mechnic critic M energy reese rte is G 8.7±0. N/m. t is cery seen tht the C iner regression of the pot is very cose to horizont ine indicting tht the critic energy reese rte is mteri constnt independent on the smpe igment. Simiry the critic energy C (MP m / ) C (MP m / ) (b) F test men.0.8 () C test ectric fied (kv/cm) Fig. ffect of n eectric fied on the frcture toughness. () igment (mm) reese rte under eectric oding ws ccuted from the critic votge nd the igment. Figure (b) presents the eectric critic energy reese rte versus smpe igment. he iner regression of the pot is so most horizont ine. his fct in nogy with the mechnic oding sitution mens tht the (N/m) G C M (N/m) G C xp. Regr. Men xp. Regr. Men (b) 00 (b) igment (mm) Fig. he energy reese rte versus the igment () mechnic oding nd (b) eectric oding.

6 eectric critic energy reese rte is mteri property with the vue G.7±7.0 N/m for PZ- cermics. he significnce of the existence of G C is tht it enbes frcture mechnics concepts to be used in understnding dieectric fiure nd it provides usefu mteri property for designers of eectronic nd eectromechnic devices. C CONCUNG RMARS he iner theoretic resuts show tht the eectric fied hs different effects on the energy reese rte for different types of crcks. he energy reese rte is independent of the ppied eectric fied for eectricy permebe crcks. he ppied eectric fied perpendicur to n impermebe crck impedes the crck to propgte whie the ppied eectric fied pre to conducting crck drives the crck to propgte. However the strip poriztion sturtion mode when eectric yied occurs in front of the crck tip predicts tht the propgtion of n impermebe crck is driven by perpendicur positive eectric fied nd impeded by perpendicur negtive eectric fied. he experiment resuts of both compct tension tests nd indenttion frcture tests show tht either positive or negtive eectric fied perpendicur to the eectricy insuting crck wys ssists the ppied mechnic ods to frcture the smpe. For eectricy conductive crcks there is critic energy reese rte under purey eectric oding just ike the cse under purey mechnic oding. n terms of the energy reese rte however the eectric frcture toughness is bout 5 times higher thn the mechnic frcture toughness. ACNOWGMNS: his work is supported by grnt from the Reserch Grnt Counci of the Hong ong Speci Administrtive Region Chin. RFRNCS. Brnett. M. nd othe J. (975) Phys. Stt. So. (b) Suo Z. uo C. M. Brnett. M. nd Wiis J. R. (99) J. Mech. Phys. Soids Zhng.-Y. Qin C.-F. nd ong P. (998) nt. J. Soids Structures 5.. Go H. Zhng.-Y. nd ong P. (997) J. Mech. Phys. Soids Co H. nd vns A. G. (99) J. Am. Cerm. Soc Cherepnov G. P. (979). Mechnics of Britte Frcture McGrw-Hi New York. 7. eeg W.F.J. (980). he nysis of disoction crck nd incusion probems in piezoeectric soids. Ph hesis Stnford University. 8. Fu R. nd Zhng.-Y. (998). J. Am. Cerm. Soc Fu R. Qin C.-F. nd Zhng.-Y.(000) App. Phys. etters Zhng.-Y. nd Hck J.. (99) J. App. Phys Zhng.-Y.(99) nt. J. Frcture 68 R.. Zhng.-Y. nd ong P. (996) nt. J. Soids Structures.. ynch C.S. (996) Act Mter. 7.. McMeeking R. M. (989) J. App. Mth. Phys Meht. nd Virkr A.V. (990) J. Am. Cerm. Soc Pk Y.. (990) J. App. Mech Sos H. nd Pk Y.. (990) nt. J. Soids Structures Sos H. (99) nt. J. Soids Structures Suo Z. (99) J. Mech. Phys. Soids Zhu. nd Yng W.(999) J. Mech. Phys. Soids Qin Q. H. Mi Y. W. nd Yu S. W. (999) nt. J. Soids Structures ogn. Hui C. Y. nd Mokov V. (996) nt. J. Soids Struct Zhng.-Y. Zho M. H. nd ong P. (00) Frcture of Piezoeectric Cermics Advnces in Appied Mechnics in press.