TO PREVENT STATIC A NEW. regated. tle material. method. element. applied

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1 Bluchr Mchanical Eninrin Procdins May 04, vol., num. A NEW TRE BAED TOPOLOGY OPTIMIZATION METHOD TO PREVENT TATIC AND DYNAMIC FAILURE OF DUCTILE OR BRITTLE B MATERIAL. H. Jon, D. H. Choi, G. H. Yoon 3 Dpartmnt of Mchanical Eninrin, Hanyan Univrsity Dpartmnt of Mchanical Eninrin, Hanyan Univrsity 3 Dpartmnt of Mchanical Eninrin, Hanyan Univrsity (corrspondin-author, hy@hanyan.ac.r) Abstract. A nw strss-basd topoloy optimization mthod (TOM) is dvlopd in ordr to considr various static failur critria such as th imum shar s strsss (M) thory, th brittl Coulomb-Mohr (DCM)) thory, and th modifid Mohr (MM) thory. Bcaus of non- topoloy optimization (TO) mthod considrin th static failur has not bn proposd yt. diffrntiability of failur critria of ths static failur thoris, it sms that a succssful In ordr to solv TO problm which minimiz th usa of matrial subjct to th t nonby usin diffrntiabl static failur critria, w formulat th diffrntiabl failur critria imum and minimum oprators. For a stabl TO procss, th t p-norm strss masur apth s- proximatin th imum valu of th strss norms and th adjustmnt paramtr in ratd dsin domain ar implmntd. Furthrmor, a prliminary rsarch considrin th dynamic fatiu failur in th framwor of th modifid Goodman thory is prsntd. Th validity and usfulnss of th prsnt TOM ar dmonstratd by solvin typical TO bnchmar problms. Kywords: trss-basd topoloy optimization, static failur thoris, ductil matrial, brit- tl matrial.. INTRODUCTION This rsarch papr prsnts a rliabl strss-basd topoloy t optimization mthod (TOM) considrin th static failur critria for th brittl and ductil matrials. inc th topoloy optimization (TO) for continuum stiff structurs wass introducdd in th lat 980s, numrous wors hav invstiatd its thortical and practical applications. Thus,, svral topoloy optimization mthods, such as th homonization basd b mthod [, ], th t solid isotropic matrial with pnalization (IMP) mthod [3, 4], th lvl l st mthod [5, 6],, and th lmnt connctivity paramtrization (ECP) mthod [7, 8], hav bn dvlopd and applid to a varity of ninrin problms. Howvr, fw studis iv nouh invstiationn into th

2 topoloy optimization of lmnt-wis strss constraints as wll as th various static and dynamic failur critria, which ar mathmatically not diffrntiabl with rspct to both th TO dsin variabls and th principal strsss. Consquntly, this papr prsnts a nw TO framwor that utilizs th TOM for th static failurs by introducin diffrntiabl formulations of ths static critria usin diffrntiabl imum and minimum oprators.. TRE BAED TOPOLOGY OPTIMIZATION FOR NON DIFFERENTIABLE FAILURE CRITERIA.. Topoloy optimization formulation with static failur critria By followin [9-], th topoloy optimization to minimiz matrial usa subjct to failur critria can b statd as follows. Minimiz V( γ) = v ( γ : Filtrd dnsity) γ NE subjct to RN. () γ ( γ) with th dnstiy filtr ( ) ( ) whn and th th lmnt xists. () whr th imum valu of strss constraints in th th rion is dnotd by and th numbr of subdividd rions for th strss constraint valuation is RN. In (), th imum oprator is rplacd by th p-norm approximation multiplid by a corrction factor at th itrth optimization itration as follows from [9-]: itr c PN ( ). (3) / p ( ) p PN ( ). (4) c itr itr, itr ( ) c 0< < itr. (5) PN itr itr whr c,, and ar th corrction factor at th itrth optimization itration, th ral imum valu of th constraint functions in th th rion, and th dampin factor, rspctivly. Th p valu is st to 4 in this rsarch by followin [9]. Th dampin factor, usd in quation (5), is fixd to a valu of 0.5 by followin th wor in [9].

3 .. Formulations of static failur critria with phnomnoloical failur thoris In this subsction, a short xplanation and associatd formulations of th static failur thoris ar prsntd. A thorouh dscription of ths failur thoris can b found in []. In th cas of ductil matrials, thr ar thr failur thoris: th DE thory, th M thory, and th DCM thory. Th critria of ths thoris ar ivn in (6-8). In th cas of brittl matrials, thr ar two failur thoris: th MM thory and th BCM thory. Th critria of ths thoris ar ivn in (9-3). Not that, th othr critria xcpt for th critrion basd on th DE thory ar not diffrntiabl with rspct to principal strss du to imum, minimum, and th loical if oprator. Thrfor th diffrntiabl imum and minimum oprators amon thr principal strsss ar dvlopd []. Th Distortion Enry (DE) Thory for Ductil Matrials DE / x x y y 3 xy. (6) y Th Maximum har trss (M) Thory for Ductil Matrials M,, min,, 3 3. (7) y Th Ductil Coulomb-Mohr (DCM) Thory for Ductil Matrials DCM,, min,, (8) t c t c Th Brittl Coulomb-Mohr (BCM) Thory for Brittl Matrials BCM,, min,, 3 3. (9) ut uc Th Modifid Mohr (MM) Thory for Brittl Matrials if 0 (0) ut ut if 0 and () uc ut uc ut uc if 0 and () if 0 (3) uc

4 whr, t, c c, ut, and uc rprsnt th tnsil yild strnth, th comprssiv yild strnth, th ultimat tnsil strnth, andd th ultimat comprssiv strnth, rspctivly. 3. NUMERICAL EXAMPLE For th numrical xampls, w considrr an L-shapd bam and a cantilvr bam problms. Not that, th mthod of movin asymptots [5] is usd as an optimizr. o Exampl : L-bract considrin static failur thiros For th first numrical xampl, w considr an L-shapd bam as shown in i fiur (a). This xampl has bn widly usd by many prvious TOM rsarch [9,3,4] bcaus it has a rntrant cornr whr strss concntration occurs. To tst th M and DE thoris, th matrial proprtis of th carbon stl 08, a ductil mtal, wr usd. Also, to tst th BCM and MM thoris, th matrial proprtis of th ATM A48 A ray iron 40, a brittl mtal, wr mployd [ 9,3,4]. Bcaus th BCM and th DCM thoris shar th sam mathmatical formulation, only th BCM thory was tstd hr. Th dtaild omtry, boundary condition, loadin condition, and matrial proprtis ar ivn in fiur. Th numbr of subdomains RN was fixd to iht for this xampl Fiur shows th optimal topoloy layouts for ach failur f critrion. At th cornr, th smooth boundary appars to avoidd strss concntrationn in all layouts, as xpctd. Bcaus th M thory is mor consrvativ than th DE thory, th masss usa by th t M failur critrion was larr than t that by th DE failur critrion. Not that bcaus th considrd matrial from Fiur (c) hass a larr ultimat strnth valu in comprssion than that in tnsion, th thicr mmbr is appard at th uppr riht l. inc th BCM thory t is mor consrvativ than th MMM thory, th BCM modl usd mor m mass than th MMM modl. (a)

5 y y y E 0 (GPa) y DE thory M thory 358 (MPa) uc ut ut E 4 (GPa) ut uc MM thory BCM thory 93 (MPa) 970 (MPa) y (b) (c) Fiur. An L-shapd bam structur: (a) th omtry (Th xtrnal load, 500 N,, was distributd on th six nods of th tip off th riht d, 0.33 ) (b) th matrial proprtis and th failur nvlops for th DE andd M thoris for carbon stl 08, and (c) and th matrial proprtis and th failur nvlops for th BCM and MM thoris for ATM A48 ray iron 40. uc (a) (b) (c) (d)) Fiur. Th optimizd layouts of th L-shapd bam with th t DE thory and th M thory (RN=8, n =3 and n =0.5, s A and : unsortd principal strsss). (a) Th layout usin B th DE thory (V / V0 0.93), (b) th layout usin th M thory ( V / V0 0.33), (c) th layout usin th MM thory ( V / V ), and (d) th layout usin th BCM thory (VV / V ). 0 Exampl : Cantilvr bam considrin dynamic fatiu failur constraints (Modifid Goodman thory) Utilizin th thoris dvlopd abov, w found that it is possibl to considr th dynamic fatiu failur in th topoloy optimization problm minimizin th matrial usa subjct to dynamic and static failur critria; th dtaild thory and optimization rsults will b rportd in soon. To considr th dynamic fatiu ffct, th static analysis and th dynamic analysiss ar considrd in (4). In this rsarch, L in quation (4) is now th dynamic failur critrion basd on th modifid Goodman thory; th ffcts of th othr dynamic critria will b rportd. Furthrmor, in this problmm is basd on th DE thory (symmtric condition). To tst our formulations, th numrical xampl of Fiur 4 is considrd. It is intrstin that t th anti-symmtric layout of o fiur 4 is obtaind as th dynamic fatiu constraint L inors th comprssiv strss ffct on th fatiu strnth. It is also obsrvd that th thic mmbr is appardd in th uppr part bcaus th tnsil strss is mainly applid to ths aras.

6 Minimiz V ( γ ) = v ( γ : Filtrd dnsity) γ subjct to p L L, (,,, RN) / p * p. /, (,,, RN) γ ( γ) with th dnstiy filtr Ku NE F K M u D K u / p F D D D (4) (a) (b) Fiur 4. Th omtry of a cantilvr bam structur and th optimal o layout considrin th modifd Goodman thory. (Th load was distributd on th fiv nods of th tip of th riht * 3 d, E 0 GPa, 0.3, u 500 MPa, 79 MPa, and 780 / m, rad/s, tatic load: F 3000 N and Fully rvrsd dynamic load: F 500 N ). D 4. CONCLUION This rsarch prsntss a nw topoloy optimization framwor that considrin th various failur critria, includin th DE thory, th M thory, th DCMM thory, th BCM thory, and th MMM thory, by introducin diffrntiabl imum and minimum oprators as wll as a diffrntiabl loical oprator. Ths oprators wr w ssntial for drivin th snsitivity analysiss of ths failur f critria with rspct to both th topoloy optimization dsin variabls and th principal strss valus. Also, th prliminary rsarch considrin dynamic fatiu failur ar rprsntd. By solvin svral numrical xampls, th validity and usfulnss of th prsnt mthod ar vrifid. Acnowldmnts This wor was supportd by th National l Rsarch Foundation of Kora (NRF) rant fundd by th Kora ovrnmnt (MET) (No ). Also, this wor was supportd by rants from Dvlopmnt of th Prototypin Ball Barins for a Roct Turbopump projct of Ministry of Education cinc and Tchnoloy (MET). Th authors than to MET.

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