homework_08 - HW#8 l PM%A Due Date In...

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Unformatted text preview: HW#8 l. PM]: %A. Due Date: November 26, 2007 In transformation—toughened ceramics, the phase transformation is mainly controlled by the hydrostatic stress. For simplicity, when the mean stress 0m equal to a critical value 0', , phase transformation takes place. The phase transformation criterion is defined as am = akk / 3 = 0', Assume that the linear elastic asymptotic crack-tip fields is true for determine the phase transformation size and shape. Determine the transformation radius r, as a function of 6 for a given K 1 and 0', near a crack tip under mode I loading conditions. Also, plot the transformation zones in the normalized coordinates 7c and y under plane strain and plane stress conditions. The Possion’s ratio of the ceramics can be assumed to be 0.2. The normalized coordinates are defined as x and y: y ’—_7 —_2 L12: Lfl 27: a, 27: 0', Consider a crack under mode III loading conditions. Under mode III loading conditions, only the out-of—plane shear stresses 0',Z and O'yz exist. The elastic ' asymptotic crack-tip stress axz and ayz are a = K’” [—sing] and a = K’” cosg H 42%?" 2 yz V270“ 2 where K,” represents the mode 111 stress intensity factor. Estimate the plastic zone radius ry as function of 6 for a given K I” and the tensile yield stress 0'0 based on the Mises yield criterion. Sketch the shape of the estimated plastic zone in the normalized coordinates f and y. The normalized coordinates 7c and y are defined as x and y: y “‘_—’_2 _“'—2 i 5n L Km 2x 00 27: 0'0 ><l ll MEflZ fill/#09 PA" Berk ‘7. lg” 06/ 5 m I x4 mnrfGth/Zn — foug/Lerzeaz ceramfcs, 7kg fiase fi’mrgrmmzf/Z m criterion, P / (j; : El; : a; 3 mafitfcaf vague. Assume, the [mew 6W2. 4:740 tot/2 crack ~29} fie/62’s, 677%”; KI, 07:) Mag/21 AMP/sz 31:42,. £8f6/‘M/ha fig Wsflrmaf/Bn mat/(213 f7: 4: a flaman O79 51, I 4’ P/ozf sz trxz/flsyémnaéép/q 20 r423 57 rM momna/fzep/ 9? émd’zr p/w/nx Sat/“€55 4449/ f/W WW2. (andft/an x a 52=LKHL 3'7: 4K1; 475‘“? _ 02 La) Ham. Smgs PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN .L ';(I+cos(9.) :9 '61“ 1 K $(S7n5’ 4— (@552 97/462) ’ Norma/éaf (2)7. = %(€098 4103161) :~ 1 % (Ex/«(9+ €03§~97fi<9> “‘fi/ PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN <5) aim/77 Cong/757w; 5 fee-=0 ) (Ea—750. _ 5‘ A l (9 <9 GEE: yx/ngzcos: 5’ #4562 71:44757‘977774'67fl/7 Offer/an. .L __ Kr: 2 .19 . A Am“ X :3— (I—HJ) (’09: 1 10M U; = O} *9 Phase TWGLHOM :> ' —, 2 1.? K1 _ C03; P 9. ga'I—V); CcDSar-ig1 x 3—7:: :— L ~ I Y3: = i X%CI+V)1(I+COS<9) £750: ———————————fl 0 800% 7/: s7émnaéfbn n61 fl 9?.” I: Y‘Cosél P _ DC '76— 3‘ N’me» ‘ “>99 ' SW9 PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN 0 Nor/27747224 {7%}? 7: = —;;(/+u)L (cosél-I-(bslél) = 729M494 (“351+‘oSLflD _ l A F. : '7'C/+V>l(§m6’—+d;m9m5‘¢9~) = £KAI/$4 (é/MflF/“SY/MQKOS‘CQ) 9‘ Misc/9 ai MM g/m was: md/ 19% Sim/71. PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN Radius, rvs. 6 under mode I loading a...» m-.. M.w..m.d.r.. . »»w.,..»u.v..,_smw.WWw, W. -vm ...................................................................................................................... ., r (plane stress) — — r lane strain) / ’ I‘ \ (p / r 0.6 \ a” .6 S '0 OJ .5 a E O C —180 -150 -120 -90 -60 ~30 0 30 60 90 120 150 9 (degree) Shape of the phase transformation area under mode | loading MWMWWOIJ5 WMWWWWWWWWmmmmtmmm plane stress — — plane strain normalized y Him fill/#09 PM; W I a 2735/6/77 2‘ @ngHg/J a, C/flaé under omfl ll mflfibfl. 7/42 e/asfia gym/Dfofi'c crack “5/? stress 5% = K1 (mg) JR]: I _ K1 4: L 03112 " W C031— nyen, K12 ) To ) Mm M3613 owx‘few/‘o/L, Estiywe 7:43 flats-67c 20m Mal/2L: r64 4s a 7/mZA/gn a7Q 62 81682505, "the whine a]? {fig p/asz‘i 2cm /2 fée mama/726’ Coofléfiwges I? d/AJ 34: , *2 — "X M - a ‘- _/_ KM"; ‘ 7 : ‘ (UV I 3% (€39 m Maw my Cw/“fer/En. .1 T _'__I,F . l \‘L L ‘ Ly wk LOX—0;.) +C$~®f+ (CE—Um +éCDE+O§+$H1 3: 3;: «4%.: U; B modem W96“ 622: G3 032 1‘— ‘:E; =>AII mews =0. .2 I L __2 L61 *. < GEE—L GEE") : lg S/flg-FCOSZ) = 033. -Lx K71):— =~Q¥ 3J6 C55 3 -i EL I; rg‘~3.75(0; \fl PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN ° NOV/ma ("7'2th 70/1. V )2 = 3603(9- :387/2c9 ——_—7f .4 ~_—,I % Cth'K’a/Vlt. 3.7L x Ga / 2 f a tug shaft? 07D fILa Mast/“c eone is“ a drag WITCIIL 15M Center is locateoI a} we crack +4}. ——> PM Hot '5 attaCde! at 7éf'709J7fifl— Pflfien PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN Shape of the plastic zone under mode “I loading E “WWW, WWW. _,MMMWMWMWWMWMVWVWMW .mmm > 85550: normalized x ’- .' A cf -"’"€-‘ , 2/ 5/- . s\,' _,,_/'//,—\"4.—. g f Ali 2 , ~ 1 0053(9 ' 7" {— [VI k- , fl / 1 6 ‘ V Jrfi :— ,*,-’— Km, ’0‘»? a _ _' ‘ I." ;_T’c-r" 4" , g ,Lkéwm 9:104 c— g‘ V‘G.r;‘; r H ’ t" \ .\ Cogfi/ /+ grfigzfi + (-9: A. J'- [email protected] ’— 1... . : ‘ t 71/? /\ '3 I”: {I} . '— F, . ‘- '— -— 'r‘ C011..— 4, + l i. / 1 2‘ 5‘ - (D§>’1ér~a§ ; ,. :_ If/ {CG-Ea” 3‘- f 4—,, \z. W #7:; I / /(z ‘2‘ J— . .I ( ) O”W+5_M+UH I T. 5 GM’ =:(/*L!) Lfiwkfi€):§:‘———-/t (05E0(’+U3 Z J 5 VTLV 4 , \3 K2 AIL #m’u.’ 1: C?= frag—é/Hm " 1’ Vt 1— .. _ 2 M." [an ’ .37’PL (05719 (/"D>” / ""5: L . )— L )2 7» V035) 7'51 Leora; (/¢U)LC<J$‘9 : ’1’, = ._ —- , :._ ; /[1.>1— i7£2>- "" ’f / “’- _ ! [£13,1— M 02/ m 07 31/ ,Egmmogérogrfi ,f . :9 Q. (; (.41‘L5051—C [2’40 :7 In pfcm. —-§1‘l'r~€_S§ / x / . 0.4} fl \\\ _--’7 ) O m: — d» a In. / //—fl\[\ \~\// / l V17 d1 {.2 -'\ ...
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homework_08 - HW#8 l PM%A Due Date In...

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