Chapter 7 Yielding criteria

Transcription

1 Chat 7 Yiling itia. Citia fo yiling What is th maning about yil ition? In this as th stss is un-axial an this oint an aily b tmin. But what if th a sval stss ating at a oint in iffnt ition? Th itia fo iing whih ombination of multi-axial stss will aus yiling a all itia.. Thoy of yil ition- A Tsa ition Yiling will ou whn th maximum sha stss ahs th valus of th maximum sha stss ouing un siml tnsion. Th maximum sha stss in multi-axial stss th maximum sha stss in siml tnsion max,, 換言之, 最大剪應力為材料就降伏

2 Fo u sha k k k 又 k k Fo u sha k stat, th yiling is han if k Th von-miss yil ition Yiling bgin whn th otahal sha stss ahs th otahal sha stss at yil in siml tnsion. τ ot τ ot, o 又

3 6 ot x y y z x z xy yz zx τ τ τ τ τ ot, o 註 : τ ot, o 為八面體上之剪應力故由 τot τot, o 得 x z x y x z 6 τxy τ yz τzx 換言之, 八面體之剪應力為材料就降伏 Fo th inial stss 又 [ 6 x y y z x z xy xz yz ] J τ τ τ 6 得 J o Fo th as of u sha

4 k, 6 k k 註 : k 為純剪應力之大小故 von-miss yil 可簡化為 : z J k Disussion: Fo Tsa ition Fo Von-Miss Yil ition kt.5 kv.577 k > k V T Yil sufa an Haigh-wstgaa stss sa Fom th yiling ition, th sha onition in multi-axial stss th sha onition in siml tnsion F K k < > : Th stss stat k : obtain fom siml tnsion 4

5 Yil sufa An Haigh-Wstgaa stss sa Fom th fom of yiling ition. That is Th sha onition in multiaxial stss Th sha onition in siml tnsion F K k obtain fom siml tnsion Th stss stat A. Rsnts a hy sufa in th six-imnsional stss sa, any oint on this sufa snts a oints a oint at whih yiling an bgin an funtion is all th yiling funtion. Th sufa in th stss sa is all th yil sufa. Sin th otating th axs os not afft th yiling stat, w an hoos th inial axs fo th ooinats. F,, K Futhmo, sin it is always assum that hyostati tnsion o omssion os not influn yiling, w an assum that only th stss viatos nt into th yiling funtion. f s, s, s K 5

6 an s, s, s an 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 itial τot s s s o 和 m無關 J o J o k, an k is th yil in u sha. Fo Tsa ition: m m ss o 和 m無關 4J 7J 6k J 96k J 64k 4 6 B. Haigh-Wst-gaa stss sa. > yiling ition an b xss as funtion of,, 6

7 故可以,, 為座標軸, 可得函數圖. Th inial,, ooinat systm snts a stss sa all th Haigh-wWst-gaa stss sa. uuu Consi a lin ON whih assing though th oigin, an having qual angls With th ooinat s axs, thn vy oint on this lin is m 即該線之點皆為靜水壓應力 uuu Th lan niula toon, its quation will b ρ Wh uuu uuu ON OPg uuu ρ an ρ : is th istan fom oigin to th lan ON 7

8 an, is th lan all π-lan. An this is th u sha stss onition. uuu u uuu ON A A OP m ON An uuu 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 # an J von Miss u B J > th omonnts of B a thfo th stss > iatos s, s, s uuu u u u u ON B P A B m uuu J u at Haigh - Wst - gaa stss sa ON B Sin it is assum that yiling is tmin by th iatoi stat of stss only, it follows that if on of th oints on th lin uuu though aalll to ON lis on th yil sufa > thy must all li on th yil sufa, sin thy all hav th sam iatoi stss omonnts. Hn th yil sufa must b omos of lins uuu aalll to ON ; i., it must b a ylin with gnatos aalll to uuu ON. 8

9 Not: a Th intstion of this yil ylin with any lan niula to it will ou a uv all th yil lous. Sin this uv will b th sam fo all lans niula to th ylin. > Fo this uos w hoos th π-lan whih m. b If, as usual, isotoy is assum so that otating th axs os not afft th yiling. That mans a lin niula to, ;, ;, ; a thfo lins of symmty an w now hav six symmti stos.,-,,-,,- Th yil sufa must b symmti in th inial stss sin it tainly os not matt. 9

10 > Hn, w hav ivi th yil lous into symmti stos, ah of o. an w n only onsi th stss stats lying in on of ths stos. C. Th stss in π-lan a os os / b sin sin 6 a b m m m J b 6 tan tan tan a > > tan if Fo von Mis yil ition

11 J o o Th yil lous is thfo a il of aius o NoT: a.fo tan an m at π-lan, That mans is u sha stat m b fo, fom tan > uniaxial stss If th yil lous is assum to b onvx, th bouns on yil loi will b btwn C' A B' an C A B.

12 A yiling uv blow CAB Whih ass th C, A, B oint will not b onvx is all low boun. a yiling uv outsi C C' A B' B whih ass th C,A,B oint will not b onvx also is all u boun. 4 Subsqunt yil sufas, Loaing an unloaing 前面所討論的是 "initial yil" 之問題, 但若 yiling 後, 其 yiling sufa 會如何呢? 即 "stain hans" 之問題 Fo a yil Funtion F k 加工硬化方程式 An k is a valu wih fin a yil sufa. an stain-han funtion F is loaing funtion. Aft yiling has ou, k tak on a nw valu, ning on th stain-haning otis of th matial. Aft yiling has ou, k taks on a nw valu, ning on th stain-haning otis of th matial.

13 A. Loaing an unloaing fo a stain-haning matial Th ass fo a stain-haning matial: A B C Loaing lasti flow is ouing. F k, F > Nutal loaing stss stat moving on yil sufa. F k, F unloaing F k, F < Not: 若取,, 為座標軸, F j j i j j u uu F i j uu u F : 表示垂直 yil sufa 之向量 i j j uu uu : 任意增量向量故 F j < j 表示曾增加向量為 unloaing B. Subsqunt yil loi A Isotoi haning If ' >, thn th nw yil lous is a il of aius ' fo von-misss ition, whih is lag than, but onnti with, th oigin yil il. Th matial is all stain han isotoially.

14 B Baushing fft 註 : 可發現當剪應力走不同方向時, 橢圓不在! Pag Kinmati mol Assum : a Th igi fam having th sha of th yil sufa bth fam is assum to b onstain against otation an to b ftly smooth, so that only fos nomal to th fam an b tansmitt to it. Th stat of stss an th stat of stain a snt in th mol in iffnt ways, Fo xaml, fo a igi stain-haning 4

15 matial, th islamnt of nt of th fam lativ to th oigin is ootional to th total stain, an th stat of stss is snt by th osition of th in lativ to th oigin. Not: th isotoi haning assumtion is still gnally us. Fo small lasti stain it obably givs answs that a suffiintly auat. 5

16 6,Plasti 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 ivation of lasti stss-stain lations. not: 所有的運算皆為向量法則 Fo obtaining gnal stss-stain lation, two finition an two assumtion a n. Dfinition: apositiv wok is on by xtnal agny uing th aliation of th st of stss. > bth nt wok fom by it ov th yl of aliation an moval is zo o osition. lt lasti stain nggy Assum:aA loaing funtion xists. At ah stag of th lasti fomation th xists a funtion so that futh lasti fomation taks la only fo > k an on th stain histoy. Both an k may n on th xisting stat of stss 6

17 lina: bth lation btwn infinitsimals of stss an lasti stain is G F Not: a may b funtions of stss, stain, an histoy of loaing that imlis thy a innnt of th bfom assumtion, it follows that fo lasti may b ali to th stss an stain inmnts. If '' an a two inmnts ouing lasti stain ' inmnts, '' an.an ' '' Bl : inmntal stss niula to f ' Bl : inmntal stss tangnt to f ' B l ou no lasti flow 延著硬化方程式, 繞著走依然是彈性範圍 ' '' '' Fom assumtion. ' '' Fom assumtion. F > ' ' '' Q F > '' an a F a > 7

18 Fom F a > '' F a F F k l F F G F G F an G C k is all otntial funtion k Fom finition. ' '' k k ' Baus ou no W an hos any onst. ositiv o ngativ, ' '' ' will ou th sam < lasti inmnt ' G F Q F > ' G ' GF G is a onstant whih may k G G Ck b funtion of stss, stain an histoy of loaing k k 8

19 f G F GF λ Haning valu ^ 垂直 yil lous之方向梯度 λ: is a nonngativ onstant whih may vay though out th loaing histoy > Th lasti stain inmnt vto must b nomal to th yil sufa. '' Not : Fom G F F G '' F 之增加由控制, 而 F 則為 yil lous 半徑增加量 '' 一 Th flow uls assoiat with von-miss an Tsa 一 Fo th von-miss yil funtion F J s s 6 o s s Fom GF GF s m s GFs λs Paatl - Russ quation 9

20 二 Fo Tsa yil funtion Assuming it is known whih is th maximum inial stss an minimum inial stss, > > F,, o λ λ a known as th flow uls assoiat with th von-miss an Tsa itia. 二.Pftly Plasti Matial Fo ial lastiity it is also assum that F xists an is funtion of stss only, an that lasti flow tak la without limit whn F k, an th matial bhavs lastially whn F Fo lasti flow. F // λ <k.

21 // λ wh λ is a sala. 三.Dtmination of th funtion G.. Efftiv stss an stain 一 Efftiv stss Fom yil ition F K o f Fom uni-axial 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 fftiv stss. Dfinition F K J Fo Von-Miss s ' s, ss Not: 當降伏時, 即應力之降伏相當於單軸應力之降伏. 二 Efftiv lasti stain Fom th finition of lasti wokefftiv lasti wok s If lasti flow is han s s an ss Gλs s Gλ

22 Fo Von-miss ition x y z xy yz zx Fo uniaxial tns it tst y x, z x x 即降伏之塑性應變當於單軸應力之降伏即 s x 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 Funtion G. Fom GF an GF FG ' ' Th slo of th uniaxial stss-lasti stain uv at th unt valu of

23 Fo Von-Miss ition s, s s Q ss s, s s s s s ' s Q ss s s ' s s Th flow ul assoiat with von miss yil ition o s s v v i j vi, x x j j s s ' an s If th vloity fil is know v v i j vi, x x j j

24 s will b know, If in lasti stat.? Inmntal an fomation thois s s Fo a all inmntal stss-stain lations baus thy lat th inmnts of lasti stain to th stss. Fo th as of ootion o aial loaing i. if all th stss a inasing in atio stss isk o ylins, th inmntal thoy us of th fomation thoy. s K K IF s s monotonially inasing funtion of K an K is Thn s Ks K s s s Th lasti stain is a funtion only of th unt of stss an is innnt of th loaing ath.?convxity of yil sufa. Singula oints. 一. Convxity of yil sufa Lt som xtnal agny a stsss along som abitay ath insi th sufa until a stat of stss is ah whih is on th 4

25 yil sufa. Now suos th xtnal agny to a a vy small outwa oint stss inmnt whih ous small lasti stain inmnts, as oll as lasti inmnts. Th wok on by th xtnal agny ov th yl is Elastolasti oblms of shs an ylins 一 Shial ooinats Th oblms of shs φ φ a Th quilibium quation φ sinφ φ sinφ sinφφ volum F an F boy fo unit b Th stain-islamnt o omatibility lation 5

26 6 > < > < u Fom u u φ φ, ~~~~~, Th stss-stain lation E E E φ µ µ µ µ ] [ ] [ Ⅰ. Von-Mis yil funtion ] [ 6, S S J J an, Ⅱ.Panfl-uss Equation Fom S sgn 二 Pola ooinats-fo ylins oblms Th quation of quilibium of stss F Th stain-islamnt lations o omatibility quation u u, z

27 7 Th stss-stain lation ] [ ] [ ] [ z z z z E E E µ µ µ NOTE: fo th as of lan stss, z,an fo th as of lan stain z o. z onst fo gnaliz stain. In both ass th sta stsss an stains a zo. 三 Thik Hollow sh with intnal ssu Consi a sh with inn aius a an out aius b, subjt to an intnal ssu P. It is obvious that omlt symmty about th nt will xist so that th aial an any two tangntial ition will b inial ition. Elasti solution ] [ E E Fom µ µ µ an quabhuiums quations omatituibility E E ] [ µ µ µ

28 8 µ µ µ µ Fom lt D z z z z z ϑ ϑ, 原式 ϑ y z h y Λ ϑ ϑ ϑ ϑ y y y h Fom bounay onition b b a P a, a b b Pa a b Pa a b Pa a b a b Pa a b Pa a b b a Fo onvnin th following imnsionlss quantitis a now intou:,, a a b ρ β,, S S P P β ρ β ρ β ρ β ρ P P S S w an : som stat of stss insi

29 th loaing sufa π π Q os aut angl π π Q osψ aut angl Fo onvx sufa No vto an ass outsi th sufa intsting th sufa twi. Th sufa must thfo b onvx. π π at ψ onition 9

30 Fo sufa is not onvx If th sufa is not onvx, th xist, som oints, suh that th vto fom an obtus angl. 二. Singula oint- Th yil sufa has vtis o ons wh th gaint is not fin Tsa hxagon. Suh oint an b tat by intouing an auxiliay aamt. 7.Aliation To solv any lastiity oblm, fou sts of lations must b satisfi as: a Th quation of quilibium of stss f j x j b Th stain-islamnt o omatibility lations: u u i j x j xi ii fo lastiity

31 Th stss-stain lations I Von-Miss yil funtion J, J s s II Pantl-Russ Equations s an Th bounay onitions stss-bounay l i τ j islamnt-bounay uj U ss Tst by tnsil-tt J 6

32 If w know, thn any oint stss oul b know. m If uvs a now awn in th xy lan suh that at vy oint of ah uv th tangnt oinis with on of maximum sha ition, Th two familis of uvs all sha lins, o sli lins. α lin β lin Not:<a> α, β a mly aamts o uvilina ooinats us to signat th oint un onsiation, just as x an y signat th oint. <b>tak a uv lmnt, Fom mon s yls

33 Fom x u v x y v v x y x, y, xy t x t y y x v os vsin αβ, vy sin vβ os m K along α lin m K along β lin If w hoos th α, β uv lina ooinat systm,, x α y β m K α α m K along α-uv m m K along β-uv K β β Hnky quation, Fom bounay onition, w obtain, If w know,,, τ m x y xy 三. Giing Vloitis quation Fom Pantl-Russ quation, an inomssibility onition x y x y x y x y s τ xy τ xy xy xy ii x y x y u u x y u u x y an x, y, xy x y y x x ux v v x y v v x y x, y, xy t x t x y y x

34 v v x y x y x y v v x y τ xy y x v v x y x y Sin th inial axs of stss an stss an of lasti stain inmnt oini, it follows that th maximum sha stss lins an maximum sha vloity lins oinis, o th stss sli lins a th sam as th vloity sli lin. th stain ats nomal to th α an β ition a qual to th man stain at, that man. α β x y Th a no xtnsion, only t t shaing flows in th sli ition. 在靜水壓, 方向無應變率 Now onsi th vloitis in th sli ition An vx vα os vβ sin vy vα sin vβ os Tak α, β uv-lin ooinat an vx vα α vβ x α α vy v β β vα y β β vα vβ along a α lin vβ vα along a β lin Giing vloitis quation 四.Gomty of th sli-lin fil 4

35 Hnky s fist law th angl btwn two sli lins of on family at th oints wh thy a ut by a sli lin of th oth family is onstant along thi lngths. Fom Hnky s quation m K along α lin m K 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 mtho along AB an 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

36 Maximum An Otahal sha stss 一. Maximum sha stss Lt us tak th ooinat axs in th inial ition. An any stion ition 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 6