Fracture analysis of cracked thermopiezoelectric materials by BEM

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Amice Gaines
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1 Q. H. Q / Eero Joura o ouar Eee Vo. No Fraure aa o rae heroeoeer aera E Q-Hua Q Deare o eha a Uver a 37 P.R. Cha E-a: Qh@u.eu. ra he ouar eee oruao or aa rae heroeoeer aera ue o hera a eeroea oa revewe h aer. wa o Gree' uo or eoeer o wh ee a varaoa re a ouar eee oe E or a -D heroeoeer o wh varou ee ue. he eho aae o ue ra roe oh e a e o. Fa a re aee o he ouar eee oruao ae oer oe uera eae or re a eer aee SED e aor a a aruar ra- a ra e o eoeer aera. Irouo heroeeroea aa o ue ra e a eoeer o o oerae orae he e o raure eha a he eoeer aera oe oa a era rora eore ue. aa aa o heroeoeer o wh ee however ver u ue o oe eoer a aheaa oruao. hu ee uera ehue uh a ouar eee eho reure o eveo. I 99 ee a Ja ] erve a ouar era euao o eoeer ea he eho o wehe reua a ao oae he uaea ouo or ae eoeer u he oue Fourer raor ehoo. u a ahreho ] reee a varaoa ouar era euao or he ae roe. D e a. 3] eveoe a ouar era oruao whh ee or aa ra roe eoeer aera. Raaae ] ue hree ouar eee eho re ouar eho re ouar eee eho a ou re-eer hare eho oue eeroea roe. Xu a Raaae ] eee he oruao o he ae o eoeer o wh varou ee ave uo ra e.. U Rao raor ehue H a Farr 6] eree a ouaoa ouo or hree-eoa eoeer aera a e era or whh a e evauae uera. Khuoraa e a. 7] roue a ouar eee oruao or e-eee roe o ear eoeer. Ree Q 8] a Q a u 9] rooe a ouar eee oruao or raure aa o heroeoeer ae o he oao eho a he oea varaoa re. u a Hu] reee a ouar era euao or aa h eoeer a oa. he oow eveoe are ae o Q8] Q a u9] a Q a a]. 83

2 Q. H. Q / Eero Joura o ouar Eee Vo. No a oruao or heroeoeer Coer a ear eoeer o whh a e are aue o ee o o -ae oorae a. oae o a or eher ou veor or are ee o wheher ower ae or uer ae ue. he SED veor Π he ea aee a eer oea EDEP veor u eeraure a hea u h he o uee o oa a e eree er o oe aa uo a oow]: ϑ h ϑ h ϑ u F F φ F F Π φ Π φ wh F a ] 3 τ where overar eoe he oe ouae a re reree ereao a oa veor o e eere ouar oo u{u u u 3 ϕ} Π {σ σ σ 3 D } ; are he oee o hea ouo u a ϕ are he ea aee a eer oea eeraure hea u re a eer aee ϑ he hea ow uo τ a are he hea a eeroea eevaue o he aera whoe aar ar are ove a are arrar uo wh oe arue a reeve a are we-ee he eraure ee] or eae. 3 E or heroeoeer roe h σ a D Coer a wo-eoa heroeeroea o e o whh here are a uer o ra. he uera aroah o uh a heroeeroea roe w vove wo e: ove a hea raer roe r o oa he ea-ae e; auae he SED aue he e he erve a ohera ouo o a he orreo ehaa a eer ouar oo a a ove he oe roe or he EDEP a SED e. he ea are a oow. 3. E or eeraure ou roe Coer a hera e reo Ω wh a uer o ra oue a how F. a. he hea raer roe o e oere a e ae a Ω 3 h h o h h o are 8

3 Ω Ω Q. H. Q / Eero Joura o ouar Eee Vo. No h o 6 where he ora o he ouar h a are he rere h vaue o hea ow a eeraure whh a o he ouare reeve. For we ee ˆ o a h where ˆ he eeraure ou he uo o a ra a are ee F.. Furher we e Ω e he oeear reo o Ω.e. he uo o Ω a Ω or he e reo Ω a ˆ a are ee F. he roe how F a a e eee o he e ae ee F.. Ω Ω a F. : Courao o eoeer ae or E aa. 3.. Poea varaoa re I a ar aer o ha o Y a Ehraher 3] he oa eerae oea eer or he hera roe ee aove a e ve a P ˆ Ω h ˆ. 7 Ω raor he reo era e 7 o a ouar era we have P ˆ ϑ ˆ h ˆ 8 whh he reao h ˆ ˆ ˆ ] 9 h ϑ ϑ a he eeraure ou aue o e ouou over a e ero a he e o are ue. oreover eeraure e 8 a e eree er o ˆ hrouh ue o Gree uo reee ]. hu he oea eer a e urher wre a P ˆ ϑ ˆ ˆ h ˆ. 8

4 Q. H. Q / Eero Joura o ouar Eee Vo. No ouar eee oruao aa reu or he u oea are o eera oe a hereore a uera roeure u e ue o ove he roe. he oveoa E he ouare a are ve o a ear ouar eee or whh he eeraure ou a e aroae he u o eeea eeraure oue: ˆ ˆ F where ˆ he eeraure ou a oe N a N he uer o ra. I hou e eoe ha aearae o N ue o he uer o oe e oe ore ha he uer o eee eah ra. Wh he ue o Gree uo reee ] a e he eeraure a hea-ow uo a o or whh ee he e a e ve a I a ] ˆ ϑ Re a ] ˆ 3 where a ha ere or or ere roeee e or ea. I aruar he eeraure a oe a e wre a ] I{ a ]} ˆ. where are he oorae a oe. For he oa oea eer hrouh he uuo o e a 3 o oe oa P ˆ Kˆ ˆ G ˆ where K he o-ae e ar a G he euvae oa hea ow veor wh he or K Re a ] Re a ] 6 G h F 7 where h whe h a h or oher ae. h 86

5 Q. H. Q / Eero Joura o ouar Eee Vo. No he ao o Pˆ e K ˆ G. 8 he a or o ear euao o e ove oae ee he arorae euao ro e a 8. Euao w e hoe or hoe oe a whh he eeraure rere a e 8 w e hoe or he rea oe. er he oa eeraure oue have ee auae he aee a re a a o he reo a e evauae hrouh ue o e. he are U u ˆ Π ˆ Π ˆ 9 where u a have ere or or ere roe ee e or ea. 3. E or aee a oea ou roe Coer aa he oa Ω whh he over euao a ouar oo are ere a oow: Π Ω θ Π o u U o u u θ u u U U o 3 where a u θ θ θ are he ouare o whh he rere vaue o re a aee u are oe. Sar he reae oea eer or he ea roe a e ve a P ϕ θ ] where he eeroea ouo o uo ϕ a U aear aer have ee ee Chaer o Re.] a are ao reee e C or he ae o reeree. I hou e eoe ha we ue he ereae eer ouar oo o ra ae eaue o uh er aheaa reae a he a ha he eer oa o a eoeer aera are uh arer ha ha o he evroe eera ewee a 3 e arer. reae eore he ouare a are ve o a ere o ouar eee or whh he EDEP ou a e aroae hrouh ear eroao a 87

6 Q. H. Q / Eero Joura o ouar Eee Vo. No F. Wh he aroao he EDEP a SED uo a e eree he or U I D ] ϕ I D ] 6 where D ha ere or or ere roe. he uo D ve e D or our a roe. I aruar he aee a oe ve ] I{ D. 7 U ]} Suu e a 6 o e we have P / 8 where I D ] I D ] 9 G F 3 θ θ a G whe oe oae a he ouar G or he oher oe. he ao o e 8 ea o a e o ear euao:. 3 Sar he a or o he ear euao o e ove oae ee he arorae euao ro e 7 a 3. Euao 7 w e hoe or hoe oe a whh he EDEP rere a e 3 w e hoe or he rea oe. Oe he EDEP ou ha ee ou he SED a a o a e eree a I D ] Π I D Π P ]. 3 hereore he urae rao-hare veor Π a oorae e oa o a aruar ra e a he h ra a e eree he or Π 33 Ω { Π Π o } where Ω ha ee ee e 3.7 o Re.]. U e 33 we a evauae he SED e aor he oow eo 88

7 Q. H. Q / Eero Joura o ouar Eee Vo. No K { K K K K } rπ r. 3 II I III D r ao o E o eere SED e aor I raa ouao oe a evauae he SED e aor evera wa uh a eraoao oruae rao oruae J-era oruae ] ea-uare eho 6] a oher ]. I our aa he eho o ea-uare ue e o he EDEP e oae ro he E reure he roeure. I hereore oe o reure uh ouer e or he K-aor auao. oreover ver ea o ee he eho o our E ouer rora. ha wh we ee he ea-uare eho raher ha aoher o auae SED e aor.. Reao ewee SED uo a SED e aor I orer o ae o aou he ra- uar o he SED e we hooe he a uo 7] 3 3 w ξ ξ where he oorae o he ra uer oerao. Rea ha he eera ereo or he EDEP e a SED uo o a ear heroeoeer o are 6 7] U Re ] ϕ Re ]. he EDEP a SED e ξ-ae a he e wre a 36 U Re ξ ξ ] Π Re ξ / ξ ξ / ξ ]. 37 Wh he uua eo he veor o SED e aor K evauae K ξ ξ Π Re ξ ξ ]. 38 ξ he uo a ear he ra a e aue a e ooa o ξ ξ ξ ξ r ξ 39 where r 3K are ow oe oa a are rea oa veor o e eere. O he ra urae whh rao-hare ree.e. ϕ he uuo o e 39 o e 36 e 89

8 Q. H. Q / Eero Joura o ouar Eee Vo. No ϕ Re ξ r ξ ]. No ha ξ ξ / ao he ra urae where he ae ro ra o he o oere we have R I I{ r}] or 3 K - I R Re{ r}] or 6K whh R Re I I. Suu e a o e 39 a he o e 37 a 38 e U K Q ξ Re I S ξ ξ ] R I R I r r ] 3 where Q ξ I ] ξ S ξ ξ I r ] ξ r ξ 3 K R I R Q ξ I ] ξ S ξ ξ Re r ] ξ r ξ K. I R I. Sua K E a ea-uare eho he ea-uare eho a e eveoe oer he reua veor or EDEP e a o K R Q ξ S ξ ξ ] U K 6 where U he EDEP veor a o oae ro he E ve he revou eo. he u or he u o he uare o he reua veor { } Q] Q]{ } { } Q] { U} { S} er whou { } 7 rove Q] Q]{ } Q] { U} { S} 8 where } { K { U { U U K } 9 { } { } { } S S S K S } U S S ξ ξ 9

9 Q. H. Q / Eero Joura o ouar Eee Vo. No Q Q K Q Q Q K Q Q] K K K K Q Q K Q Q Q ξ. Oe he uow veor {} ha ee oae ro e 8 he SED e aor K a e evauae ro e 3. I he auao a arorae uer a e e o oa he reure aura. Nuera reu a urao he rooe ouar eee oe ae o he oow wo uera eae whh a uo a a ra are vove. I a he auao he aera or he ar a he e uo are aue o e ao 3 a Cau Seee reeve. he aera oa or he wo aera are a oow: aera oa or ao 3 e κ 3 33 GPa C/ e 6 κ 6 κ K 66 GPa C/ e GPa 6 K C / N λ aera oa or Cau Seee 33. C/ 6 GPa.33 3 κ N/CK κ GPa Perv o ree ae 7.GPa. GPa GPa GPa 3. GPa γ.6 6 NK γ NK 3.9 CK e 3.6 C e C e.38 C κ 8.6 C N κ C N. where ea e a 33 are hera eao oa λ 3 a 3 are roeer oa e eoeer oa γ eohera oa. Se he vaue o he oee o hea ouo or ao 3 a Cau Seee ou o e ou he eraure he vaue 33 /. or ao 3 a 33 /.8 or Cau Seee 3 a W/K are aue. I our aa ae ra eorao aue a he ra e aue o e he - ae.e. D 3 u 3. hereore he re e aor veor K ow ha o hree ooe K I K II K D. I he ea-uare eho he SED e aor are aee he araeer 9

10 Q. H. Q / Eero Joura o ouar Eee Vo. No a a where he uer o er e 6 a he au ae ro ra o he -o a whh he reua veor are auae a he u ae. I our aa e o e... a F. : Geoer o he ra-uo e. Eae : Coer a ra o eh a a uo eee a e ae a how F.. he uor hea ow h ae o he ra ae o. I our aa he ra wa oee ear eee. ae how ha he uera reu or he oee o re e aor a o ee F. o var wh a whe he ra ae a reeve where are ee K I h γ 33 / K II h γ / 3 K D h χ 3 D /. ae he E reu or oee v a Eae a / D SIE

11 Q. H. Q / Eero Joura o ouar Eee Vo. No F. 3: he oee veru ra ae. For oaro he uar era euao eho SIE ve 67] wa ue o oa orreo reu. I a e ee ro ae ha he reu oae u a are oe oer o hoe oae SIE ha u a. or a.. h eaue ore aa a e ue o he ea-uare eho or a are a u oo are a a o reree he ra- roere a aue error. Fure 3 how he reu o oee a a uo o ra ae whe a a. I eve ro he ure ha a he oee are o ver eve o he ra ae u var h wh. I ao eve ha he wo uera oe E a SIE rove ao he ae reu. Eae : Coer a reauar heroeoeer ae oa a ra a a uo a how F.. I he auao eah e o he ouer ouar oee ear eee a he ra ve o ear eee a a are ue. I F. he oee o SED e aor a o ee F. are reee a a uo o ra oreao ae. However uera reu or uh a roe are o e avaae he eraure. For oaro he we-ow e eee eho 8] ue o oa orreo reu. I he auao a eh-oe uaraera eee oe ha ee ue. I ao he hree oe ao oe o he e o eah o he uaraera eee are oae a he ra a he wo ao -o are ove o he uarer ae 9] orer o roue /r / e o uar. I a e ee ro F. ha he vaue o are ore eve o ra oreao ha hoe Eae. he reah her ea o o vaue a 37 or or 93

12 Q. H. Q / Eero Joura o ouar Eee Vo. No h h h. a h 3.a h a F. : Courao o he ra-uo e Eae a.. Pree eho Fe eee eho. D 6 8 e. F. : SED e aor veru ra ae. o a or D reeve. I ao eve ro F. ha he au rea ewee he uera reu oae ro he wo oe e ha %. 9

13 Q. H. Q / Eero Joura o ouar Eee Vo. No Couo ao o ouar eho o heroeoeer aera wh ee are ue h aer. ue oruao or rae ha-ae aera hoe a uo roe o eoeer ha ee reee. he u ae ha he oruao aae o ue ra roe oh e a e o. Nuera reu oae ro he ree ouar eee oruao a e eee aroah are oare a he are oo areee u he orer wh e eree o reeo. Reeree. ee JS a Ja Z. ouar era oruao a D uaea ouo or eoeer ea. eh. Re. Co. 99;:7-.. u P a ahreho O. varaoa ouar eee oruao or eoeer. eh. Re. Co. 99;: D HJ Wa GP a Che WQ. ouar era oruao a D uaea ouo or eoeer ea. Co. eh.. eh. E. 998;8:6-8.. Raaae RK. ouar eee eho or eoeer o. Proee o SPIE ahea a Coro Sar Sruure 997;339:8-8.. Xu X a Raaae RK. ouar eee aa o eoeer o wh ee. Cooe Par : Eeer 998;9: H R a Farr N. hree eoa eoeer ouar eee. Proee o SPIE ahea a Coro Sar Sruure 997;339: Khuoraa N Soa H a Zu WH. roae Gree uo a a ouar eee eho or eeroea aa o ave aera. Cou. & Sru. 998;66: Q QH. heroeeroea aa o ra eoeer ha-ae E. Coua. eh. 999;3: Q QH a u. E or ra-uo roe o ae heroeoeer o. I. J. Nuer. eh. E. ;8: u YJ a Fa H. a o h eoeer o he ouar eee eho. Co. eh.. eh. E. ;9: Q QH. Fraure eha o Peoeer aera. Chaer 6 WI Pre Souhao.. Q QH a a YW. E or ra-hoe roe heroeoeer aera. Eeer Fraure eha ;69: Y HP a Ehraher. Varaoa aroah o aee ou eho a aao o ra roe. I. J. Fraure 993;63:3-3.. aa H a Rooe DP. Nuera Fraure eha. Couaoa eha Puao: Souhao 99.. Ju SH. Sua re e aor or aoro aera he eauare eho. I. J. Fra. 996;8:

14 Q. H. Q / Eero Joura o ouar Eee Vo. No Ju SH. Sua hree-eoa re e aor he ea-uare eho. I. J. Nuer. eh. E. 998;3: Kha S Su C a Kwa WC. ao o a hr e eee eho o eere re e aor ureoa ooe. I. J. Fra. 986;3: Oe J a Kee E. Fe eee oruao o eera eeroheroea roe. I. J. Nuer. eh. E. 97;3: ero J. Fraure eha: Fuaea a ao. CRC Pre: oo 99.. Q QH Geera ouo or heroeoeer wh varou hoe uer hera oa. ;37:6-78 e : Ereo or a or a he uo a ha ere or or ere roe. For eae a { } { } or aera roe a or a ae oa a e uo a ue ra oue he uo whh a a a a { } 3 { } or he ha-ae roe wh ue ra a a } { 96

15 } { or a ae oa ue ra a a hoe o varou hae where are ee ]. a I he oruao aove ee a o o 6 a reree he oorae a oe he ae ewee he eee he e o oe a he -a wh ee ar. I hou e oe ou ha he uerr or varae a ha ee oe e a orer o he wr. e : Foruao o u a he ereo o u are reee or oow our ae: a a ae oa a e uo a ue ra oue he uo 3 I u I P τ I{ I 3 τ P 3 I I 3 where Q. H. Q / Eero Joura o ouar Eee Vo. No

16 e. aera roe ]} { I u ]} { I 6 ]} { I ]} { I 7 ]} { I. ]} { I 8 ha-ae roe wh ue ra ] I u ] I 9 ] I ] I ] I ] I where ] ]. v a ae oa ue ra a a hoe o varou hae Q. H. Q / Eero Joura o ouar Eee Vo. No

17 ] I τ P u ] I τ P 3 ] I P ] I P ] I. ] I hu he SED a EDEP o a ouar ue eeraure ou are o he or ˆ ˆ θ θ Π u U. 6 I eera over he ouar o whh SED rere a over he ouar o whh EDEP rere. o aae he SED a EDEP o he orreo ouare we u ueroe a ouo o he orreo ohera roe wh a SED veor or a EDEP veor eua a ooe o hoe o e 6. he ea are ve he u-eo 3.. θ θ U u e C: Ereo o a U ϕ a ae oa a hoe a a oao oue he uo I U ] I ] I C I ] I ] I ϕ C where aδ δ δ 3 δ ]. ha-ae roe I U ] I ] I C3 I ] I ] I ϕ C Q. H. Q / Eero Joura o ouar Eee Vo. No

18 Q. H. Q / Eero Joura o ouar Eee Vo. No oaera roe For a aera ae uee o a e oao oae he uer ha-ae a he ouo ve ] U I{ } I{ } C ϕ I{ } I{ } C6 or aera > a U I{ } C7 ϕ I{ } C8 where I ] I C9 I C wh he urae eae ar. v uo roe For he ae o a ae oa a e uo a a oao oue he uo he ereo o ϕ a U oo oe a we o hoe ea or oee whh a e ou 9]. e D: Ereo o D a ae oa a e uo a ue ra oue he uo D I E E 3

19 Q. H. Q / Eero Joura o ouar Eee Vo. No I D whh a e eree er o he reao a - a o o D a. ha-ae roe wh ue ra D where I I. D3 a. aera roe D I. D I v a ae oa ue ra a a hoe o varou hae ee Seo. o Rre.] D I D. I 3

ON THE EXTENSION OF WEAK ARMENDARIZ RINGS RELATIVE TO A MONOID

ON THE EXTENSION OF WEAK ARMENDARIZ RINGS RELATIVE TO A MONOID wwweo/voue/vo9iue/ijas_9 9f ON THE EXTENSION OF WEAK AENDAIZ INGS ELATIVE TO A ONOID Eye A & Ayou Eoy Dee of e Nowe No Uvey Lzou 77 C Dee of e Uvey of Kou Ou Su E-: eye76@o; you975@yooo ABSTACT Fo oo we