Chapter 3 Stress analysis

Transcription

1 Chat Stss analysis Th stss-tnso (. th dfinition of stss- Consid a body subjctd to a systm of tnal focs F, F, at statically quilibium. Stss T v lim A d F d A

2 (. th stss at ctangula coodinat systm u Ty Tyi Tyyj Tyzk T,lt y y τ, Tyy yy Basic stss: u T i τ y j τ zk u Ty τ yi yyj τ yzk u Tz τ zi τ zyj zz k τ τ uu τ τ y z y yy zy τ z τ yz zz τ i: sufac diction, j: foc diction v v i v j v k v (. Th stss at any sction: j

3 Fom th quilibium of foc: u ( vi g τ y( vgj τ z( vk g Siv g u u F τ y( v i yy( v j τ zy( v k S j v g g g g u uuu τ z( vi τ yz( v j zz g g ( vk g Skv g u τ y τ z l Si g S τ τ m u S j S g y yy zy y τ z τ yz zz n u Sz Sk g v S i j (4 Pincial stss and Pincial diction W can contol v ν S u that mans τ li li ( δ li δ τ τ y z τ y yy τ zy τ z τ yz zz, I I I I y z I τ τ τ ( y yz z y y z z y I τττ τ τ τ ( y z y yz z yz y z z y and Ss Tv Sn u, s S T v n v S : Sha stss, Not: I, I, I a invaiant S : nomal stss n

4 (5Equilibium quation of stss Fom Foc quilibium (6 Maimum and Octahdal sha stss A. Maimum sha stss Fj i τ τ ji Lt us tak th coodinat as in th incial diction. And any v v i v j v k v sction diction is j 4

5 v S and Fom i j v S, v S, v S S S S uu s S v s n uv Sv Sv g v v v n And u S v v v ( v v v s S S v v v v Sv S v S v n v v v with ( W sk th valus of v, v, v such that S s is a maimum. Diffntiating quation ( obtain, v[( v ( v ( ] v [( v ( v ( ] v v v 註 : Fom Lagang multili mthod Lt Fvv v s v v v (,,, λ λ( s,,, v v v λ 5

6 得 : v[( v ( v ( ] v [( v ( v ( ] v v v Fo v, v, ( (, v v ± Th solutions will b Th maimum sha stss is: Th nomal stss is: τ ± ( n ± ( τ ± ( ± ( τ ± ( ± ( 6

7 B. th Octahdal sha stss: If th lan ABC is also ointd that OAOBOC, thn th nomal on will tak qual angls with all th as, and v v v ±, Fom, S v v v ( v v v S ( ( 9 [( ( ( ] 9 7

8 sn v v v ( m ( ss ( ( τ ( ( ( oct m m m s s s oct m τ τ τ τ oct ( y ( y z ( z 6( y yz z J 結論 : 在 Octahdal 面上之垂直應力為靜水壓力, 其對降伏無影 响, 故降伏, 由 τ oct (7 Stss dviato Tnso: 稱 Von-mss s yild cition It is convnint in lasticity thoy to slit th stss tnso into two ats. On calld th shical stss tnso and th oth th stss dviato tnso. Th shical stss tnso is m δ m m m m ( ( y z I Th dviato stss tnso is 8

9 s m τy τ z Ρ τ τ τ z τ yz z m y y m yz s τ τ y z τ y sy τ yz τ τ s z zy z Fo incial stss, m m m s s s Lt si i m Th invaiant of th stss dviato tnso will b ssd by st S m fo to obtain: S S J SJ J J J ( I I [( y ( y z ( z 6( τy τz τ yz ] 6 J ( I 9 II 7 I 7 O in incial stss, J J [( ( ( ] 6 J ( ( ( m m m (8 Pu sha stss: If at a oint in a body, a st of coodinat as can b found such that, thn th oint is said to b in a stat of u sha. y z It can ally b show that a ncssay and sufficint condition fo a 9

10 stat of u sha stss to ist is ii y z (9In conclusion it should b mhasizd that all th quations dvlod and sntd in this chat a indndnt of th matial otis and thfo qually valid fo bodis which bhav lastically o lastically.

11 . Stain tnso ( Th dfinition of stain-th chang atio of th lativ osition of any two oints in a continuous body btwn nt quilibium and initial quilibium. Consid an infinitsimal lin in undfomd and dfamd body in incmnt thoy t. An infinitsimal lin can divid into th stags: tanslation, otation, stain to obtain th stain of an infinitsimal lin, th st as: A. to dlt th tain of an infinitsimal lin

12 and uuu uu uu L AB OB OA uuu uu uuuu ' OB U ' OB BB uuu uu uuuu ' OA U ' OA AA uuuu uuu uuu L ' ' ' ' A B OB OA uuuu uuu uuu uu uuu uu L L ( ( ' ' AB ' OB ' A B OB OA OA u U uuuu U uuuu U u u U, Tl lim ( 表示除去後之線變形 l l ' ' BB AA ( 一 Th dfinition of lin dfomation as: u u U Tl lim l l l -lin diction B. th dfinition of lin at ctangula coodinat systm lt u U Ui Uy j Uzk u y y y y U Ui Uy j Uzk u U U i U j U k z z z z y z Fom lin dfomation dfinition u T l u U lim Vl l

13 u uu U U U y U z U U y U uu z T lim lim ( i j k ( i j k Ti Ty j Tz k u y y y y uu U U U y U uuu z Ty lim ( i j k Tyi T yy j Tzzk y y y y y u z z z uu U U z U y U z Tz lim i j k Tzi Tzy j Tzz k z z z z z Th basic lin dfomation y z U U U y z T Ty T z y z Uy Uy U y T Ty Tyy Tzy y z Tz Tyz T zz y z Uz Uz U z y z and u u u du T d L C. To dlt th otation stain l If lin otation is han u u u u u u u u u u u u u u ' ' dl dl dl dl ( dl du ( dl du dldl dldu dudu u u dl du u u u dl T dl l

14 u u y u z y z d u u u y z dz u y uz u z y z y y z [ d dy dz] dy u u u y u u u z y u y uz uz d ( ddy ( ddz dy ( dydz dz y z y z y z u u y uz y z u u y u u u z y uz,, y z z y IF only otation u u uu T w l l l u u y uz u u u ( y u ( uz u u ( y u ( uz y z y z u u y u z u uy uy ( u z u ( z u uy u ( y u ( z y y y y y z y y z y u u y uz u y ( u u z u ( z u z u y z z z z z y z ( u u z u ( z z z y T Ty T z y z wy w z Ty Tyy Tzy y yy zy wy wzy Tz Tyz T zz z yz zz wz wyz D. Th stain at any diction if know th basic stain u uu u Fom du T d L l u and T l is basic lin dfomation tnso 4

15 And u du u uu u T v T B T i T y j l T z k dl Tv T i vj Th som mthod fo stain v i vj (Th incial stain Whn w contol v, and mak uu // v v v i i ( δ v i y z y yy zy z yz zz vj (Maimum and octahdal sha stains A. Maimum sha stains Fom vi vj, if tak th coodinat as in th incils diction 5

16 v, v, v v v v And uu uu v v v v v v uu n v v v v v...( s v n v v v ( v v v...( By th sam way as fo th stsss Th maimum sha stain as: ( ma± ( ( ma± ( ( ma± ( and v, v ±, v ± v ±, v, v ± v ±, v ±, v B. Octahdal sha stains If w consid th octahdal lans fo v v v ±, fom (,( n ( ( ( [( 9 9 ( ( s 6

17 oct s ( ( ( (. stain dviato tnso ( ( ( n n n ( ( ( 6( y y z z y yz z ' J ' J ( ( ' J Th stain tnso can b saatd into two ats: a shical at q and dviato at Th shical at is q m δ m m m ( and m Th dviato stain is m y z m y yz z m m y y m yz m Not: If w consid a ctangula aalllid with sids qual to a, b, and c, so that its initial volum is abc, thn aft staining its volum will b: a( b( c( abc abc( v ( v Th shical stain tnso is thus ootional to th volum chang. Th dviato stain tnso thn snts a u distotion stain 7

18 (a Comatibility of stain Fom th stain dfinition: u v w, y, z y z u v v w u w y (, yz (, z ( y z y z It is obvious that if th dislacmnts a scifid as continuous functions of, y, z thn th uniquly stain can b comut, but fo th invs oblm, to calculat th dislacmnts u, v, w in a body givn th stain comonnts. W hav 6-quation to solv -unknow, that mans too many solutions can b obtaind. (Assum th body is dividd into infinitsimal cubs, Lt ths nti cubs b saatd fom ach oth and lt ach of th cubs b subjctd to som abitatd stains. It is obvious that if w now ty to ut all th cubs togth th way thy w bfo, to oduc a continuous body This shows that must b som lationshis btwn th stains at th diffnt oint of a body in od that th body main continuous aft staining. Ths lationshis a calld th comatibility lation. 8

19 y y y y y z yz z y y z z z z z yz z y ( y z y z z y yz y ( y y z z y zy z z ( z z y y 4. Elastic stss-stain lation (Equations of Elasticity Fo an isotoic matial, th stss-stain lation as: G δ λi and ' G : sha modulus E : lastic modulus ' µ : oisson s atio I y z ' I y z o G δ λi ' µ m m man stss-stain lation E s dviato stss-stain lation G fo ij µ ii ( I g E µ y z ( y z ( y z G E 9

20 µ ( ( y z G E µ µ ( ( y z E E µ ( y z E µ m m man stss-stain lation E s dviato stss-stain lation G Fo octahdal sufac µ m m E oct τ oct 4G oct, τ s s s and s G oct J, τ J ' J J 4G oct ' oct (Elastic stain ngy functions Th lastic stain ngy is : ' U and G δλ I U G δ λi G λi ' ' Fo incial stss-stain U G( λ( yy zz y z and ( y z ( y z ( y z yz I J [ J -( ] ' ' ' y z y z U G( I J λi ' ' (G λi GJ ' ' '

21 Th fist tm is th ngy involvd in th changing of th volum; th scond tm is th ngy of distotion dsignatd as, ' Distotion ngy U GJ G 4G τ d oct oct 註 : 若只考慮塑性變形, 則第一項可忽略不計 5. Citia fo yilding ( What is th maning about yild cition? In this cas th stss is un-aial and this oint can adily b dtmind. But what if th a sval stss acting at a oint in diffnt diction? Th citia fo dciding which combination of multi-aial stss will caus yilding a calld citia. (. Thoy of yild cition- (A Tsa cition Yilding will occu whn th maimum sha stss achs th valus of th maimum sha stss occuing und siml tnsion. Th maimum sha stss in multi-aial stss th maimum sha stss in siml tnsion ma,, 換言之, 最大剪應力為 材料就降伏

22 Fo u sha ( k k k k 又 k Fo u sha ( k stat, th yilding is hand if k (Th von-miss yild cition Yilding bgin whn th octahdal sha stss achs th octahdal sha stss at yild in siml tnsion. τ oct τ oct, o 又

23 ( ( ( 6( oct y y z z y yz z τ τ τ τ τ oct, o 註 : τ oct, o 為八面體上之剪應力 故由 τoct τoct, o 得 ( z ( y ( z 6( τy τ yz τz 換言之, 八面體之剪應力為 材料就降伏 Fo th incial stss ( ( ( 又 [( ( ( 6( y y z z y z yz ] J τ τ τ 6 得 J o Fo th cas of u sha

24 k, 6 k k 註 : k 為純剪應力之大小故 von-miss yild 可簡化為 : z J k Discussion: Fo Tsca cition Fo Von-Miss Yild cition kt.5 V.577 k k > k V T (Yild sufac and Haigh-wstgaad stss sac Fom th yilding cition, th sha condition in multi-aial stss th sha condition in siml tnsion F( K( k < > : Th stss stat k : obtain fom siml tnsion 4

25 ( Yild sufac And Haigh-Wstgaad stss sac Fom th fom of yilding cition. That is Th sha condition in multiaial stss Th sha condition in siml tnsion F( K( k ( obtain fom siml tnsion Th stss stat (A. Rsnts a hy sufac in th si-dimnsional stss sac, any oint on this sufac snts a oints a oint at which yilding can bgin and function ( is calld th yilding function. Th sufac in th stss sac is calld th yild sufac. Sinc th otating th as dos not affct th yilding stat, w can choos th incial as fo th coodinats. F(,, K Futhmo, sinc it is always assumd that hydostatic tnsion o comssion dos not influnc yilding, w can assum that only th stss dviatos nt into th yilding function. f ( s, s, s K 5

26 and s, s, s can b witn in tms of th invaiants J, J, J J s s s Wh J ( s s s J ( s s s f ( J, J K( k Fo von-miss citial τoct s s s o ( 和 m無關 J o J o k, and k is th yild in u sha. Fo Tsca cition: ( m ( m ss o ( 和 m無關 4J 7J 6k J 96k J 64k 4 6 B. Haigh-Wst-gaad stss sac. > yilding cition can b ssd as function of (,, 6

27 故可以,, 為座標軸, 可得函數圖. Th incial (,, coodinat systm snts a stss sac calld th Haigh-wWst-gaad stss sac. uuu Consid a lin ON which assing though th oigin, and having qual angls With th coodinats as, thn vy oint on this lin is m uuu Th lan ndicula to ON, will b uuu uuu ON OPg uuu ρ and ρ : is th distanc fom oigin to th lan ( ON > ρ Whn, th lan is calld π-lan. And this is th u sha condition. uuu u uuu ON A A OP m ON And uuu 7

28 u u u B P A ( i ( j ( k m m m B ( ( ( m m m s s s ( Q J ( s s s ss B J # and J u B J > th comonnts of B a thfo th stss > diatos s, s, s uuu u u u u ON B P A B m uuu J u (at Haigh - Wst - gaad stss sac ON B Sinc it is assumd that yilding is dtmind by th dviatoic stat of stss only, it follows that if on of th oints on th lin uuu though aalll to ON lis on th yild sufac > thy must all li on th yild sufac, sinc thy all hav th sam dviatoic stss comonnts. Hnc th yild sufac must b comosd of lins uuu aalll to ON ; i., it must b a cylind with gnatos aalll to uuu ON. Not: (a Th intsction of this yild cylind with any lan ndicula to it will oduc a cuv calld th yild locus. Sinc this cuv will b th sam fo all lans ndicula to th cylind. > Fo this uos w choos th π-lan which m. 8

29 ( 達到 yild 是 isotoy (b If, as usual, isotoy is assumd so that otating th as dos not affct th yilding. That mans a lin ndicula to, ;, ;, ; a thfo lins of symmty and w now hav si symmtic sctos. (,-,,-,,- (II Th yild sufac must b symmtic in th incial stss sinc it ctainly dos not matt. > Hnc, w hav dividd th yild locus into symmtic sctos, ach of o. and w nd only consid th stss stats lying in on of ths sctos. 三, Th stss in π-lan 9

30 a cos cos ( / b sin sin ( 6 β γ a b ( ( ( ( m ( m ( m J ( b 6 tan tan tan a ( ( > > tan ( if Fo von Mis yild cition o o Th yild locus is thfo a cicl of adius o

31 NoT: (a fo, Fom tan ( and m ( ( at π-lan, That mans is u sha stat m (b fo, fom tan > uniaial stss (c If th yild locus is assumd to b conv, th bounds on yild loci will b btwn C A B and C A B. a yilding cuv blow CAB Which ass th C,A,B oint will not b conv is call low bound. a yilding cuv outsid C C A B B which ass th C,A,B oint

32 will not b conv also is call u bound. (4Subsqunt yild sufacs, Loading and unloading 前面所討論的是 initial yild 之問題, 但若 yilding 後, 其 yilding sufac 會如何呢? 即 stain hadns 之問題 Fo a yild Function F( k ( 加工硬化方程式 And k is a valu wich dfind a yild sufac. and stain-hadn function F( is loading function. Aft yilding has occud, k tak on a nw valu, dnding on th stain-hadning otis of th matial. Aft yilding has occud, k taks on a nw valu, dnding on th stain-hadning otis of th atial. 一. Loading and unloading fo a stain-hadning matial Th cass fo a stain-hadning matial: (a (b (c Loading lastic flow is occuing. F k, df d > Nutal loading stss stat moving on yild sufac. unloading F k, df d F k, df d < uu uu df d j ( j ( di j j not: 若取 (,, 為座標軸, j

33 u uu F d i j u F : 表示垂直 yild sufac 之向量 d i uu : 任意增量向量 j 故 df d j < j 表示曾增加向量為 unloading 二.Subsqunt yild loci ( 一 Isotoic hadning If ' >, thn th nw yild locus is a cicl of adius ' fo von-miss cition, which is lag than, but concntic with, th oigin yild cicl. th matial is calld stain hadn isotoically. ( 二 Bausching ffct- Pag Kinmatic modl Assum :(ath igid fam having th sha of th yild sufac (bth fam is assumd to b constaind against otation and to b fctly smooth, so that only focs nomal to th fam can b tansmittd to it. (cth stat of stss and th stat of stain a sntd in th modl in diffnt ways, Fo aml, fo a igid stain-hadning matial, th dislacmnt of cnt of th fam lativ to th oigin is ootional to th total stain, and th stat of stss is sntd by th osition of th in lativ to th oigin.

34 Not: th isotoic hadning assumtion is still gnally usd. Fo small lastic stain it obably givs answs that a sufficintly accuat. 6,Plastic stss-stain lations- 在彈性區之應力和應變之關係已經討論過現在, 則更進一步討論塑性區之 stss-stain. ( Gnal divation of lastic stss-stain lations. (not: 所有的運算皆為向量法則 Fo obtaining gnal stss-stain lation, two dfinition and two assumtion a ndd. Dfinition: (apositiv wok is don by tnal agncy duing th alication of th st of stss. dd > (bth nt wok fomd by it ov th cycl of alication and moval is zo o osition. dd (dlct lastic stain nggy Assum:(aA loading function ists. At ach stag of th lastic dfomation th ists a function ( so that futh lastic dfomation taks lac only fo ( > k 4

35 Both and k may dnd on th isting stat of stss and on th stain histoy. (bth lation btwn infinitsimals of stss and lastic stain is lina: G df d c d kl kl Not: (a c kl may b functions of stss, stain, and histoy of loading that imlis thy a indndnt of th d (bfom assumtion, it follows that fo lastic may b alid to th stss and stain incmnts. If '' d and d a two incmnts oducing lastic stain ' incmnts, '' d and d.and ' '' d Bl : incmntal stss ndicula to f ' dbl : incmntal stss tangnt to f( d ' B l oduc no lastic flow( 延著硬化方程式, 繞著走依然是 彈性範圍 ' '' '' Fom assumtion. d c ( d d c d kl kl kl kl kl ' '' Fom assumtion. df dkl ( dkl dkl > kl kl ' ' '' Q dkl dkl df dkl > kl kl kl 5

36 and Fom '' d kl a df a > kl kl kl df a > kl kl '' df d ckldkl ckla ckl ( F F k l kl df ckl ( df kl kl G df ( k d GdF and G Ck ( ( k k is calld otntial function Fom dfinition. d d ( d d d kl ' '' ' d d Bcaus oduc no W can chos any constc. kl ositiv o ngativ, ' '' ' will oduc th sam ( d d d < dd lastic incmnt d ' d G df Q df > d d ' G ' d GdF ( ( G is a constant which may k G G Ck b function of stss, stain ( ( and histoy of loading k k 6

37 f d G df GdF dλ Hadning valu ^ 垂直 yild locus之方向梯度 (d λ: is a nonngativ constant which may vay though out th loading histoy > Th lastic stain incmnt vcto must b nomal to th yild sufac. ( '' ( Not : Fom d G df df d kl G d kl '' kl kl df 之增加由 d 控制, 而 df 則為 yild locus 半徑增加量 '' kl 一 Th flow uls associatd with von-miss and Tsca ( 一 Fo th von-miss yild function F( J ( ( ( s s 6 o s s Fom d GdF GdF ( s m s GdFs dλs Paadtl - Russ quation 7

38 ( 二 Fo Tsca yild function Assuming it is known which is th maimum incial stss and minimum incial stss, > > F (,, o d dλ d d dλ (( a known as th flow uls associatd with th von-miss and Tsca citia. 二.Pfctly Plastic Matial Fo idal lasticity it is also assumd that F ( ists and is function of stss only, and that lastic flow tak lac without limit whn F ( k, and th matial bhavs lastically whn F ( Fo lastic flow. df d d // d dλ dd <k. 8

39 d // d dλ wh dλ is a scala. 三.Dtmination of th function G.. Effctiv stss and stain ( 一 Effctiv stss Fom yild cition F( K( o f ( Fom uni-aial tnsil tst. 即左式為多軸壓力狀態, 可相對某一值, 若該值等於 則降 伏 稱之 ffctiv stss. Dfinition F( K( J Fo Von-Miss d sd d d ' s kl kl, ss kl kl kl kl Not: 當降伏時, 即應力之降伏相當於單軸應力之降伏. ( 二 Effctiv lastic stain d Fom th dfinition of lastic wokeffctiv lastic wok d s d (If lastic flow is hand s s d d d and ss d d Gdλs s Gdλ 9

40 d d d d d d d dd (Fo Von-miss cition ( ( ( ( ( ( d dy dz dy dyz dz Fo uniaial tns it tst d y d, dz d d d 即降伏之塑性應變當於單軸應力之降伏 即 sd d d 塑性功能不變法則 達到塑性狀態所所需之功和過程無關 ( 三 Dtmination of th Function G. Fom d GdF and d dd d GdF dfg d d d d ' ' ( d kl kl kl kl d Th slo of th uniaial stss-lastic stain cuv at th cunt valu of 4

41 Fo Von-Miss cition kl skl, s s d d ( Q ss skl, s s s kl s s d d ' kl s d d ( Q ss s s kl kl ' s s d d (Th flow ul associatd with von miss yild cition kl o s s v v i j vi, ( j j s s ' and s If th vlocity fild is know v v i j vi, ( j j 4

42 s will b know, If in lastic stat. (? Incmntal and dfomation thois s s Fo d d d d a calld incmntal stss-stain lations bcaus thy lat th incmnts of lastic stain to th stss. Fo th cas of ootion o adial loading i. if all th stss a incasing in atio (stssd disk o cylinds, th incmntal thoy ducs of th dfomation thoy. s K K IF s s d d monotonically incasing function of K and K is Thn s Ks K s s d d d d s Th lastic stain is a function only of th cunt of stss and is indndnt of th loading ath. (?Convity of yild sufac. Singula oints. 一. Convity of yild sufac Lt som tnal agncy add stsss along som abitay ath insid th sufac until a stat of stss d is achd which is on th 4

43 yild sufac. Now suos th tnal agncy to add a vy small outwad ointd stss incmnt d which oducs small lastic stain incmnts d, as coll as lastic incmnts. Th wok don by th tnal agncy ov th cycl is ( Elastolastic oblms of shs and cylinds 一 Shical coodinats Th oblms of shs φ φ (a Th quilibium quation ( d( d dφ( dsinφd dφ sinφd sinφdddφ volum d d d d d d ( F and F body foc unit (b Th stain-dislacmnt o comatibility lation 4

44 44 > < > < d d d d d du Fom u d du φ φ (, ~~~~~, (c Th stss-stain lation E E E φ µ µ µ µ ] [( ] ( [ ( Ⅰ. Von-Mis yild function ] ( ( [( 6, S S J J and, ( Ⅱ.Pandfl-uss Equation Fom S d d d d d d sgn( d d 二 Pola coodinats-fo cylinds oblms ( Th quation of quilibium of stss F d d ( Th stain-dislacmnt lations o comatibility quation u d du, d d z

45 45 ( Th stss-stain lation ( ] ( [ ] ( [ ] ( [ z z z z E E E µ µ µ NOTE: fo th cas of lan stss, z,and fo th cas of lan stain z o. z const fo gnalizd stain. In both cass th sta stsss and stains a zo. 三 Thick Hollow sh with intnal ssu Consid a sh with inn adius a and out adius b, subjctd to an intnal ssu P. It is obvious that comlt symmty about th cnt will ist so that th adial and any two tangntial diction will b incial diction. ( Elastic solution ] [( ( E E Fom µ µ µ and d d quabhuiums d d quations comatituibility E d d d d E ( ( ] [( µ µ µ

46 46 d d d d µ µ µ µ ( c d d c c d d c d d Fom lt D d d d d dz d d d dz d dz d z z ϑ ϑ, 原式 ( c ϑ c c y z h ( ( ( ( c c c c y Λ ϑ ϑ ϑ ϑ c c c c y y y h Fom bounday condition ( ( c b c b c a c P a (, a b b Pa c a b Pa c ( ( a b Pa a b a b Pa a b Pa a b b a Fo convninc th following dimnsionlss quantitis a now intoducd:,, a a b ρ β,, S S P P ( ( β ρ β ρ β ρ β ρ P P S S ( dw d d d and : som stat of stss insid

47 th loading sufac π π Q dd d d cos (acut angl π π Q ( d cosψ (acut angl Fo conv sufac No vcto can ass outsid th sufac intscting th sufac twic. Th sufac must thfo b conv. π π ( at ψ condition 47

48 Fo sufac is not conv If th sufac is not conv, th ist, som oints, such that th vcto fom an obtus angl. 二. Singula oint- Th yild sufac has vtics o cons wh th gadint is not dfind (Tsca hagon. Such oint can b tatd by intoducing an auiliay aamt. 7.Alication To solv any lasticity oblm, fou sts of lations must b satisfid as: (a Th quation of quilibium of stss f j j (b Th stain-dislacmnt o comatibility lations: u u ( i j j i ii fo lasticity 48

49 (c Th stss-stain lations I Von-Miss yild function J, J s s II Pandtl-Russ Equations d d s and (d Th bounday conditions stss-bounday l i τ j dislacmnt-bounday uj U ss Tst by tnsidl-tt d dd J 6 49

50 If w know, thn any oint stss could b know. m If cuvs a now dawn in th y lan such that at vy oint of ach cuv th tangnt coincids with on of maimum sha diction, Th two familis of cuvs calld sha lins, o sli lins. α lin β lin Not:<a> α, β a mly aamts o cuvilina coodinats usd to dsignat th oint und considation, just as and y dsignat th oint. <b>tak a cuv lmnt, Fom mon s cycls 5

51 Fom d du v y v v y, y, y ( dt dt y y v cos vsin αβ, vy sin vβ cos m K c along α lin m K c along β lin If w choos th α, β cuv lina coodinat systm,, α y β m K α α m K c along α-cuv m m K c along β-cuv K β β Hncky quation, Fom bounday condition, w obtain c, c If w know,,, τ m y y 三. Giing Vlocitis quation Fom Pandtl-Russ quation, and incomssibility condition y d d dy y y d s d τ y τ y y y dii d dy y u u y u u y and, y, y ( y y d du v v y v v y, y, y ( dt dt y y 5

52 v v y ( ( y y v v y τ y ( y v v y ( y Sinc th incial as of stss and stss and of lastic stain incmnt coincid, it follows that th maimum sha stss lins and maimum sha vlocity lins coincids, o th stss sli lins a th sam as th vlocity sli lin. th stain ats nomal to th α and β diction a qual to th man stain at, that man. d α d β ( y Th a no tnsion, only dt dt shaing flows in th sli diction. ( 在靜水壓, 方向無應變率 Now consid th vlocitis in th sli diction And v vα cos vβ sin vy vα sin vβ cos Tak α, β cuv-lin coodinat and v vα α ( vβ α α vy v β β ( vα y β β dvα vβd along a α lin dvβ vαd along a β lin Giing vlocitis quation 四.Gomty of th sli-lin fild 5

53 Hncky s fist law th angl btwn two sli lins of on family at th oints wh thy a cut by a sli lin of th oth family is constant along thi lngths. Fom Hncky s quation m K c along α lin m K c along β lin along AD-α-lin md K D along DC-β-lin md mc 4K K K K K 4K D ma 4K 4K 4K 4K D B C A D B C A A B D C A C C A B along AD-α -lin ma KA md K D along DC-β -lin md KD mc KC 4K K K mc ma D A Th sam mthod along AB and BC D C K K 4K mc D B C A D B C A A B D C ma C A 4K K K K K 4K A C C A 4K 4K 4K 4K B B 5

54 (Maimum And Octahdal sha stss 一. Maimum sha stss Lt us tak th coodinat as in th incial diction. And any sction diction is v l i nj mk v j Fom vi Tvj v T, v T, v T v v v uuuu Tv Tvg Tv v v v 54