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(1 - s,) [ tr- s' ) (tr-L' ) : o

P - ( S r t t + S r ) t + S , S , t S , I 3 + J r l , ) t r - 5 1 5 ' J 3 = o ( 6 )

Jx *,x/

Y'*A

,t"*j l J'/, s)Jax s?

I : p-! v*

t -,^

.! *+ -tr

{ SJJ

-"y""Y %(4) o^ot Et(5),

I J , - J r - t s - t s3' f

iL:-F, tr- t s.s-r t srJ,)

t J3 :- sr !. 5; - i(si*

corfarig LXrH o"Yrd tX ( 6) ,

f J r

:'rj€- 3et (. 3:n*l;:i1-i*,.,+1,'+r'r.tzr,r+

r / ^ 1 , r l r r L \ / + a * n c . e r r v t=::= ta,t'J+ ":l

(- :y:,,1* r* Ji-

sii,tsi,) ( -f,+st*,j:\t#'H"i'** h3*- Ho{t

\ -{ ,, J" - s slSr=-J -fr J.(J, -"-,

)

!r,'r. 3er (ur.^Io- r-irrtkr :o /

I o =[t^r+Jyy+jsr;"= r^its; + \i*\ +lJxx i"r+rJry!**r-rrrr

f. --(J xx Syy t S.1r !*o* 5** 5^') + ( 5 f- t Sui * S"i I

= +(sk+ Ivi + Sti ) '' ( tYk {- s& tJ 'o I

I :

\ t ,.r, ,1.

\

",

Y\orm b+ ̂ vt 'tov \uarrtd

fn rYY JYL

II .J+, Sy 5l* I

1 \. J l

I

/I

(^' l ol/ttwL F*d"*'s^rt*-Q. ftvre*tt'n't

6

tl<t-

f-tVU- 0/rg thg S+eA rnUnUa,'nf,S 6i{) I'rn'rre- obla,,"i'A'd

Ir --36, Tr=-i(oi '61*c;,- i ,T' ') , I j=-duf(E)

J r : c , f L = | S J J j , , J 3 . : d ' e ' t t f ' ; )

yigtd c,rmdr'-fio-"<- ca'^ bL cuzi#t,rn. aa ?.lth'*rr'

{ c t , , r , , T 3 ) : o6{ t'

+ ( 6 , f " . , J r ) : o

*f*4, it w,'t M +ha'f 6 hat ntt tryfe 41""t o"'4t

ltellj-'U** (a'nd pla*2fut"^^t.* ''o geru^! "'e^\h 6nt2'r'\'et vnl'il^tu)'

Jo tfti- Tleld u'*ldt^ ( u'*Vu tzt'pAsam co'n bz un;lt<'r a't

f c L , r s ) : o

I^ thc moSf wida\ u5"4 '1ie(J con&'*-rw (l'-7\^^+afu\-

tt<r- l.zyead^n ^ I, i5 alto Jr"\y*l'

{ c f , ) - - L - t -o - l /o^ M;x t ' c ' t i }+ ' ;o " "

L* ot'urtlilnl, ,"nJ,!,(,,'t l// al ?u,ru (ii:n^dn*#xn )

tl TrV 6,V6) , 1"h-{'* 6t

k, ;l -th!, qrc^rteil+ s fu*t ttu/A '

- 63, - '1, i€' t'-fnr7^th*-r,,

r^ WA , (s,-'ii,)'- 't'k;lfrt;-s' ;'-v*;l[rs'-;;'-vt|]: Q

-) + (T,,Js) = +J -;'- tT Jr'- 3t ft;'T" + 16 W]' - 6+1 : o

1,,'lr* Irrn-^^h'$' 14 u'ull't- rn'v7t-'-^1'Y

Mdle d^'ffi a-Af h ,rt)w + f"h"pt{ Sfi+*t n'{tt- -L'*"^* J.,'o"I4

L^- pr^c,h'cL , fuctiu;$un* 6*hwp^ vrn /v13u a'nd Treacp'

c,/rfry"itu is Lp*l -(tlrrr; ts"{, utilhll tJl';c^t uyvo*J ?mqT

-f t\t_ s,w,7l* l-rr.r^ ,+ vm M\ tu.t c*i{e*'tw'. 7t5l<0a rf mui-f ,Do7''"1 wr'

3? Mohv/s attis

(',-F* &x p't' i.ivr'-r'vfo{ "&"-i' ' rp'}asvl'l tJ' fii' ftt'iz'\ C.r:,k;,;s'^n, i?-R* l,-e,^r+,"r1M& 2 )

ME 3+L F^^d xvw\a'^fu]- 4{L^h]ilt':ltl^,^' l t

V,Jo ** y{,{'J:e. tfu pnwmtttttv k t' +W 1'Yld s*az df h +tu"n|c- +?.!*'

L" +trrtvl,!- ktvf , fX^ 2 o, 6'tY:::: r^f,o* - o

6': * a,)_

Jxx :; rxx t syy =' ' -- j- 6^r,a

Jz = i(si" + ryits*l,- )= i +

k ':yie} pwt fz= * ui : k:'

6 y : J 3 k ,r C!-1K- ;T1i=

(V,'n nfitu )

( fu Tvusc., g'rt: (d gna J^'-*rt'a, kr : 3;

i: :r,.,f'r-ted 6 ,lrn* sh-e.o'" s'lrttA *7 G!,(6thuL si'lzrt

Y'

th-e/L f z: 6- y

4t onn* r{ VNtd, 5^/=T" I' : TJ =' k'

,.- T. : k- , i.e L riJ ths- cnlrc*!- .9h-p-ort sM'$ ^rtr"'L ( V,r', Miel)

A lrtittL Shuarr S*etYt iJ *Yb"eJ

C Fto',' Tru.* fl.\{ cn'r$hrsrr , l<r: T, ols' )

<-l[][-,-*.--> 6xY

W*.u^- 6v a^d ()i:r:,7, a^t th{- truug 1.,li.n-}*/ln s&v:l c ̂ rg'r^&nA/R'J

_L_3

L

"1" rTxT

V"n Um \J c^:nd^*i6r4

5nl6; + o,li, = k'

L

Mil

II

;

iw

k=F: ae]ffi+k, =t ' 'o /

q--i51.d y = . k f

N^vr-lYt

5xx

l2xtn'o&t=_ &ari tlre- Cw-uL *Y*fan*]

( t" tl^a- fvga6e, W(d cszt$-$7tn 611 +/tt'

\ fxx - gx/

1tl*'r* *infr' M,'fbL

& ? : - i * '- z l - - L6,r, =

T ov

ry tlt!- s^r' 7lx

6it

Tivxo'\

MF3+L Fnrla,wt*u17" q-^firy lD,4 l zTn ginQ L;l f ud On'fe,,,,r-nyr... br) EX pero, nt^e.,,t'5

Co,t*idp'- a +^lx- 9ulje"r"t',eA t, botl, PMirn f"r* tr

a r " d , f u I * T

Thr.ulrl"er^t SlwwL a,bw^(L wil( h- s"t\u^redh brtlL?tvr-LaQ st rtn Ex ,und S /rza,+' Sfu'v1 {y7.

f^urut";xg F*d T toqlrtlaL,Aht@f*y + a* oo**+,,^zf

a,UnTNS u^ b, ,'hupsa+." cfy-a a," d Qt W witl,

E : :-s>yvf t

MeaUnny4,w ";hZ{,,rr,-truns { 6<x, A* F WttwwS ?* ia/,%u6 ,,^l lp m^f tts j/htd Swr,(-r-e- o^rl u*Wv t-th flr$-phb! drh .

* Vrn ltth'sp,s ortfu'l"n Tp17-,,*- b "g+.n

t**rr wr+l+ %PUit^QlfE +he^ ftWraia Ou i--

'l

Yo

Trescd law Misss law

o CoPP',

X Aluminium

o Mrld Srcel

fFro. {. Experirnontal recults of Taylor and Qrriruney from combinod toraion snd

tension tsets, each motsl being work.hardonrxl rge tho samo ststn for all 0eet8.

Tho Miees law is ot*3rr : yt, whilo tho I'rcpce law ig orl4tr - Yr, whoro

o : ten-silo atreaa, r -- ahear atresS, .ll' ,: teneile yiold s11,r988.

Fto;,P. H;U, Tl',n-lttodttnc*hUtLr/-%{ltr,,rn*2xN (r;, Prt >7"/75 0P.2L

To/n, a^ol Qu;"Y'Y

pld(, Tr^"a kj5oc, A L3o,)23 ( l?3t )

.5

,4

.3

-2

' l

. 9'8,7,4'3,z. l r.0

ME 3+L r - t r A -iwn d o/r'\ti w*tNl fi t t a*t}rrt,S

t

C"" r j

tq Ca!4,,h-{ rl\rq- (Sirer4{-Jnra^4^) &e[ot#in4 in thr Pl@

5" ,^sdb , h.att wL cl-,nz,,.&.,.u ^ y$^lr\ pl"*tt. -aon^[ cno s1qin havLda,.'v1

TL*^ ;n tb pta,r+Z reyta {Uo;t')): J,-t: o, L rv^"'^ c.anrla4nr-

i .e. ! r : t , j , : o

dlau.twe4 tlrD "Ine'd naf rnz'a,n tha* a// s+ruv ctYr'p'*ta slalz cni"'"'*

I, - kt ^' ,1^'e[J snj"u-' ln tb s+t'+t tface-

7hz s+w ^rf"ffi ^ wU d+Lin +A* TVld tW

Rxro,!,(- 7, : 4$h + sy', -rS'-+* ) + t\in 1*i+l*i )

ir= f^r 5x" t l,1y iyy t Szti* t rEzls tr 5+"J*, t.l 5x/ty : sg'sr': o

-nM, k a, c.a,t*tral-,t on+Az stre* mte Sj

NhnL" $w sfre/ SW M +h!-

p{a*frv Strar'o *; , So th^t

g^f /e'rna; f of(n+iv^J

W'td 5v'n{aw,

ll;a^.y: ttf *

t" ,.tiWyt s-1

n^rfu?r {- w')it xndtryo

t, tfl E;iJ,) = tU €r1

tl,l*s-a ' *r-4 |

-,

: + zru e(

Tt +r\k ̂colq&

Il

!,

Lt becan'*

fu,twn''ngJ- r r l

Lnsteaa ,

Q-= Haro

^ )

h dPlu>rrv;fie- 8,!i'

+iy/\!- fu 5q2"Jtil?5 b' f,.d o'^/r th-s- c'uLnL^+ -ry h

( *lfr, -lnerJ ;nYnq sto"r'6 ) '

,] J

-?L )r L',.t*mp> dLwep'-k, t"€' ' ' f Cannot bg+n!6 .J is ,hdahrY' "-l-

. '

thL UeL!'v\f Sl^rai$ S-tatu ( ttt )'

rnwy(be-cl-p ' tw,n-. , ,^ ld,JACCUtrtwl"AnftSnLJltTlcz|-r l tpnf i5.

€:;

I^ a,Jhm, i? ln'"t fuf' f-^ai' tj'vt pta'>R' ry PIod^'so

*f g;bl.!- val^^rN- cj\tur\&L ' € iPiL:o, i:: - o

"tzufw'1o-n-c- pla,+ht. Str-alt )'til-o- -lc,rg- ^-o pl.a":lr''L }k--'u)'

h"*a

" ? L-J

piL$rt

UrS. P L.i j

ME3+L F*^d*^{^t^[ Eqwahbn'xr Gu t+

A,ts**yhrrt

*A*+h;t, ,,$L 2y nf := -i)' )

NoR. bd\" tlre ds'aatlni<' ela'+t.'z S+Ylin

r"ftffffi\ the- de4^u,'+lYui:c- s+e,et

z/ery : ,u(dff" e,f') =: .s,i + ls,r

T:ft,;,/

ov'ctittL'k 5+aud oA Ca*z'?'^uw ). ) , - :

' , ; t ' - ) , U ; - r -

' '

W =Jxx (xx *Syl 4y t Sz+ d** + ,. i Sf*ey+ + a J2* €*x +)ixy*,

:o f* Fr"le^H"U Plw*z r'tr'ne'xa"l

^'-Sii - zT -'l''- :- zT *) " )

Wftlrrr. ,'yVt4 co"1d4't56Y\ it 't'tzaclsd ) i{: Wfi il!. ch,e,,hw,f ri,

,rrr.{"-q- }" iJ a. pos;Nt- scoJ*{ {-,i",/

tF bq- rL{",tg:u'ra,i*rd

, a 0 .

i,.,u ?,1 : J,'-- ' J )

1 F= A J r ' iJ

2y e;'

?Ad, +lw plt/>'Fv slra,)*4-';f*lL

@ t )( €,51 t( J,-t ')

, '

6l = S,, €rJ'

. e I- - - I i i fet i +,J ' J

2/"

kecaft

W : 5 ; ( S , , j + ) , r l )

- T - c - C ^ ,Jz - - t ; j ' ) '

2 r ?

2 / N) / 1

r',-l!. tlo"; t, oklo-*'"n fi; (-**-+*.**

s,.f'- {r. s.; , J',)-- ,lt i( - 'y(e",-a!,!)" ' l z { J )

rr^t \tnstan s*ut +)ra,h *x:, f$' ; s''t' >*;rf, rxwa - - 3 K 8 , o = 3 k 6 / t + L r - - f u L , ,

T dl,l*an;nw d/-2- f-"tt-t X , ^/Q' n'a2/ to inttp duco'

<- Y*tt, ':f Na?k l-ovti-- ax)ocr^fteA ̂ ':' shy+ eV

. Dattf ''of

^l';'l- 1ot% It stnzt' elashL %u^3y

i'i,\) ' p^fi ,f *AnrJ' C^tt;y",ted ^4 t"k''f

(w,r.l,. J'r','!- a''vnaz-tqd wftLwLww c'Wry+ A

plt3*7, Furdo''rraatfxf h,,7' vr,x*ilsrui G*t t5

Sw^ n',*ut. Llta 1MAS , t*nd*,,, a. ryec,.t';ud clr{rQ ) Jkah, rch i

t/r-a- plw'u*L teJ ri,vrtt- !

r* d a*u"r'nt 9ru4<l Oi,' , b

r -l-.. L7 : J- 9..- *--> (J' :=:= J K €\-

3 (-f (

i'n

GtW,r,_ f', -' q J

i , 7

'''J -.--=-*

5 to Su-rtttran3 "f

Earttn R"ryq =

)ied c*trrd^f-s '

Ptq*'c Hft*.

J ^ r | / ' , / ) f-f;nd 6;i ( ifn^Aifte- ^ rn-al4-L f-ai,n ntJ' 'J

p rrien S

----__,\ ;

. )g , ' ) = S )+66 !

6,j

\ ' l n , ' t , /-"' J) : rJ Vj-yf tla'.

f f : 3 K t

Sii : zf e;iJ

Att Ffl"n'"L^'{

t l^ad^m+.rht- pcv/t t

I cl-il,r.ntuFur,tc- ?^n/= )

\

J, : I s,; s) -- kl'- [ V"n Miq'a )

oy - Js k- ( rlp'eld s*,tzf i'n *'nam)

ef : +

i t llo{:r,r,{a*mrr- e,\d,/'4'r sfa,,}^ t*fu )

; ?t W c-. ( p te*h-c S+rci^,' rc^rtc )eU Ztr -')' / ( d-uriu,.*o,r.ic ylr+44 rdx. )si : u^ ( e,l - if;"::;5 )G :3K E (l '\^4d*erldpt{L' S+re'' ta'fu)

Cu+lrt:" t 7;e Id icoryh'frvw|.

gr"yt":'n{ reTrw**un

/

r*e;e uafutsath-L rna*-en;t*L

f'vlls$^fre

{q,6,6t ) =,o

ai-.

r , 4r? tJ Sr^^tar&\ / t J

^/1EWZ fuftr"fl fbfrwdarw

I,n +Arb k t'|wt, ) t^)o gzW m,tnt dnsu''W-WS Ftr'r

( t/rn l\4|x/ wrtd Tvwc* ) , irC,ud,ty f 6-Qh

arn d U p e r' ft<-ot\t+t^l, vw f; to**icr|-,

4t Uratd !;4't^*Yy:!* li:::!{PWIf -t/L!- m^teno-Q. rJ riolryic , tlu_ thyee pricryJ

<:antzttM rtJi -tlAL in.fvr,ma.hst'L oefi^t ntuttltt-

Aaa rgaJ-e.d tl\e uyield 6md;4*rvt {L rwf

Tn V+,14trL waA4, th'y W1q.td cn1di+",fru iS (€a,d,ret trrL ^ zD SrY*,

/\€ tfu- ,feil.lrf*, in du- 3D syae syto.nnnd *,V 6 ) q, q+(6, , 6.. ft S : ()

rur,++** vnrrT€- t OYlu.;1<+nts

6: t@t*'6.* {j) [rlf no

f,,^ue -sbo,n4 tha* lvyd nS+^,tl- lfuA

,,rl*' irt ye ld

-* 6p:6"=d3 Th,b lrtewa^*! throf tl?-e- Ortd n'r'/rrz *Y

dl

cf ;t6i63:o

- 6p6y=67ur^L

-(,^r.,V ̂ l>rtgrn$c s ha-1,Y , wittL t/v- a,xet

lJ^J^!- 1 ut -(u-,ts "te-fin J b-t 61=6r:-6; ( l, ?- [ tttl d,ie o*")0

L, o-tl^!N wwtd,A, thL lrt*+'r[*'Ai:vt bafwe**

+hL "f(l'J s"f"* f Cr,, du, d1) : Q n'od-

hrt y:L'v,ru- Cuii"t I b,Y {rt6v*61 : c-

hu*t .\ c*rP I drs Sumu slt"ye-

Lrnaa.+n* lr;l *uk C".nrr," dn +h{ 6ttA,tdr : oU )

t/4e^ t',,t'L %,4il4- Vie ll suvJiro;;- h" L4 ti ke-

^ Q,'-'''a:f' .

I^ ofks,'\- uvr=e{ il,Ll u,e rr*d ra -h /'i /-S1eCI;f,/f- t/,t Shaiz+ 'f tLA c.{.t'ue, fr* yhvtd S,n lzr,-r-I t u"L,t rt-bl;i.. u**r"fu.J hV heTalr-h,ng +t Di Chtj,f- aI/^1 [t1J.

6ft6r+ 63= o 7lc', rre'

TAN+L h**fuc-L PA,4ffii Co;* f /

- i/ i/ :

,iI

!i

. - . - . / / ^ i0 t : j u : d r U I L A \| ) 'l

iIli

Movfe*rr'{. ;soffif"J reqwil \'- tlrL y'elJ

Syrrtmu+*' hJi+h *ff^f 1- f "f,

ro{v,*wvt

6+ trr, 6z- 4, f)-, S, ,

urfril au5 mtvvvt "r'fu*tl,* , s-g, 6 -61 , 61J t E

khli {aAitu)<JfitJ civ4 be inpn-d 6n +h!- yfitd t** sy":fb,

tl,L TetJ wW iJ w""tfi, nuMtuw-Q,d t, he c$nvrN, hrlsJn wLr,^l i I { dr'S r^rr> 1 qd% .

9z Vrn Mtre,.t Yietl h'd^'li.,.

Mofs- J ' z : 6 t - { , Sg= fS -F

s,^,y'oz* be

a"rawd +ha-

, t D

M

J,= +(s l+sJ+s; ) = : : L '

6: |Gtt6,+

ds) , .$,r : 6t -T ,

J, : i &" -T ft {6,-6i'-t (c,-6 )-_ * G'*oJ+ 6)'- te,1= tt6,-t6J+6t-

- (rr,6Lt c-{r

- *[8,-

6r)- t ( fr- ci',,,)' + ( 6v-

+ 6t6)l

q ,,J

Vn lrl^V-;ePldw Tfro Vrn l/t'7'u

^ r$"u&tt,

urvl' t0-w1.6l:6-= fs

$/Le-

V4Vtd sr-$^t rJ

witl,t A C4Tu"r!-s,/f

Th-L Frr( X- y)rnn ,h tht- Iruxf fW.

NIEH-L

[oo r-1

6,4r=(5Wt^rYag, il'ayTWn\*z&W', Cq.,4' 3

de sh^llv,PnT6 6t't,

[trt1 Io tol6L To be mo{L prcdu,,

€417P4, J in Wnu ot'

the- 'IlL-? lo'nQ,', i' L'

S , , S , , S , ,

tr+6i,f 6r=o N ' o & , . S r t j r t - t ) : o| [ . i i ]

L t o t J

frzi i l5l

Vnn f{LiSg5 yVl'J crrd"'*ibrrt

15 o\ c'Nd* of r adnvtA

^?rc : € trtln tho- pla,na of 5,, E l-

sr: *Cr

lVote +hosf S,, , Sz, 9j ana npl- ,"}rdl? erilr^ti

Ttt&orYrb 4 Sr, 5', S3 o/re not

I ;

Pr^t*U.,rf.-o"r h Qn'l^ "t'hl- i

iin$^dnn c-o q a&oL c*x*) ,) 7er7a"d^'c"{e- i

+ +{,.{- orYb of S.

- r r , ' ( , ) q - qs , - 5 ( r D - r ) " \ f r ) = j '

tli gco^ro-e'

t { ( , 0

t t ( 1 oLUe hd'+;e

ltr r 11* s .

- t ) + i r , e , ) l = t ( L 1 lr ) + i ( t 2 , ) | = l ( . r 2 )

| - - _ -i t lzTJ s = !(r -r -t).&Yryl

' g ' i

iz-- i(-r L -l)-(?)==FI\cJ-: ) ,

- L \ t ? -t ) - , - 1 - - R/

tvttS+z f j \ n ,(. t , i n^ r l

tttmltl,r'co!. lteyvwr**W_ !0,4'-_ 4-',' , f L ' t

C,,ri'lt"n{n i'^ rlfunn rfu'vt iVvn [Arv,s6 , { t : o

The inlo,rt"tat{6n rJ fuAn(a cryt:hdtu

ulrh fr=o flw rJ a/^ el.lyW

\/- Witb C^d+r^firn in thl, P|,*"

tytfrJ4, 7ht- t,yrei.i r."-fa,.u rn (,- !. yla,r*

<5t= 6{ 6v: o

rf , : Q 6r-*dY

tfr :dy 6r-=dy

's {= k G= -l t

J{

- R / 6

,/ _k,/

V .,---,/ .-o(

6, ' -66r+a,7:6k-J-- - q

SpjrYn*fr- q*N po{htd d-n-

fletd sv't,*

,f s,- 5.{,y dx"frr.nfiiln , Sr t !-,- -F $ -_ o

5- : : - i I ' t I '_)

!.: .(sl-t IJ+ tr" )

- - : ( t l * S l+(5,+ i " ) 'J

: 5i t '!'r' -t J', S "

6r^pt$*-A. ,9 - l_Cei 59 I Trv>co / re(J u^J,r'nvn.

5z

9g=r., I5.

- t

;.15u

Urn$ t_

x 3) r - L

, + +

{ l '

T'U

S '

:i,

I^l rl rll t

l rf J -

l i

_15z"5-

L

i

IL ko

TUCt "tidd annda?inn-

*u : i,"

sr

Tyurn ,ne1 cov,dn'L1:w irt(dt:o) = 16,- 4rl-- 6'r,

i ( a pol*t A otq- in Sc-'k o!' r J c l u l

eflLlt SZ car\nlt P an '.l"vY t"f - ' I a

l's Wr6."w r7 t;d4-W

ir1 t l'I-a- yta,rtz tj S, , J., 9t

F= o'r..7 *

c) q

o-6c7 Fv

J-'

g ,*,4 J2

== rE

== 2fu.

Truco 4f'eld cozrrhhM 15 \

twr;,,r* irts on kzd in tAL

oircb c-<Drelfwhrf h #L

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4/16/13 11:34 AM D:\My Documents\Courses\ME342-Inela...\plane_strain_uniaxial.m 1 of 3

% ME342 Theory and Applications of Inelasticity% Wei Cai, caiwei@stanford.edu%% Example: plane strain uniaxial loading % Material parametersmu = 100; % shear modulusnu = 0.3; % Poisson's ratiosig_Y = 1; % yield stress K = 2*mu*(1+nu)/3/(1-2*nu); % bulk modulusE = 2*mu*(1+nu); % Young's modulusk = sig_Y/sqrt(3); % condition at yieldsig_xx = sig_Y/sqrt(nu^2-nu+1); % see HW 1.1(a)sig_yy = 0;sig_zz = nu*sig_xx; % hydrostatic stresssig_bar = (sig_xx + sig_yy + sig_zz)/3; % deviatoric stresss_xx = sig_xx - sig_bar;s_yy = sig_yy - sig_bar;s_zz = sig_zz - sig_bar;J2 = (s_xx^2 + s_yy^2 + s_zz^2)/2; % J2 should equal k^2 at yield % strain (elastic at onset of yield)eps_xx = sig_bar/(3*K) + s_xx/(2*mu);eps_yy = sig_bar/(3*K) + s_yy/(2*mu);eps_zz = sig_bar/(3*K) + s_zz/(2*mu); % strain array (going beyond yield point)eps_xx_data = eps_xx + [0:120]*1e-4; % initialize arrays to store resultssig_xx_data = zeros(size(eps_xx_data));sig_zz_data = zeros(size(eps_xx_data));lambda_dt_over_2mu_data = zeros(size(eps_xx_data));s_xx_data = zeros(size(eps_xx_data));s_yy_data = zeros(size(eps_xx_data));s_zz_data = zeros(size(eps_xx_data));J2_data = zeros(size(eps_xx_data)); % the first data point is the yield point considered abovesig_xx_data(1) = sig_xx;sig_zz_data(1) = sig_zz; s_xx_data(1) = s_xx; s_yy_data(1) = s_yy; s_zz_data(1) = s_zz;

Plane Strain Tension 7

4/16/13 11:34 AM D:\My Documents\Courses\ME342-Inela...\plane_strain_uniaxial.m 2 of 3

J2_data(1) = J2; for i = 2:length(eps_xx_data), param = [mu, nu, k, sig_xx_data(i-1), sig_zz_data(i-1), ... eps_xx_data(i-1), eps_xx_data(i)]; %options = optimset('Display','iter','TolFun',1e-10); options = optimset('Display','off','TolFun',1e-10); trial = [sig_xx_data(i-1), sig_zz_data(i-1), 0]; sol = fsolve('eqns_plane_strain_uniaxial_plast', trial, options, param); sig_xx = sol(1); sig_zz = sol(2); lambda_dt_over_2mu = sol(3); % compute hydrostatic stress and deviatoric stress sig_bar = (sig_xx + sig_zz)/3; s_xx = sig_xx - sig_bar; s_yy = sig_yy - sig_bar; s_zz = sig_zz - sig_bar; J2 = (s_xx^2 + s_yy^2 + s_zz^2)/2; % save results to array sig_xx_data(i) = sig_xx; sig_zz_data(i) = sig_zz; lambda_dt_over_2mu_data(i) = lambda_dt_over_2mu; s_xx_data(i) = s_xx; s_yy_data(i) = s_yy; s_zz_data(i) = s_zz; J2_data(i) = J2;end % plot solution% plot stressfigure(1);subplot(3,1,1);plot(eps_xx_data, sig_xx_data, '.-');xlabel('\epsilon_{xx}');ylabel('\sigma_{xx}');subplot(3,1,2);plot([0 eps_xx_data], [0 sig_zz_data], '.-');xlabel('\epsilon_{xx}');ylabel('\sigma_{zz}');subplot(3,1,3);plot([0 eps_xx_data], [nu sig_zz_data./sig_xx_data], '.-');xlabel('\epsilon_{xx}');ylabel('\sigma_{zz} / \sigma_{xx}'); % plot deviatoric stressfigure(2);subplot(3,1,1);plot(eps_xx_data, s_xx_data, '.-');xlabel('\epsilon_{xx}');ylabel('s_{xx}');subplot(3,1,2);plot(eps_xx_data, s_yy_data, '.-');xlabel('\epsilon_{xx}');ylabel('s_{yy}');

Plane Strain Tension 8

4/16/13 11:34 AM D:\My Documents\Courses\ME342-Inela...\plane_strain_uniaxial.m 3 of 3

subplot(3,1,3);plot(eps_xx_data, s_zz_data, '.-');xlabel('\epsilon_{xx}');ylabel('s_{zz}'); % plot lambda_dt_over_2mu and J2 (should equal k^2)figure(3);subplot(2,1,1);plot(eps_xx_data, [NaN lambda_dt_over_2mu_data(2:end)], '.-');xlabel('\epsilon_{xx}');ylabel('\lambda \Delta t / (2\mu)');subplot(2,1,2);plot(eps_xx_data, J2_data, '.-');xlabel('\epsilon_{xx}');ylabel('J_2');ylim([0.9 1.1]*k^2);

Plane Strain Tension 9

4/15/13 12:42 AM D:\My Documents\Courses\...\eqns_plane_strain_uniaxial_plast.m 1 of 2

% ME342 Theory and Applications of Inelasticity% Wei Cai, caiwei@stanford.edu%% equations to be solved by plane_strain_uniaxial.m function F = eqns_plane_strain_uniaxial_plast(vars, param)% F(1): eps_xx_tot (specified)% F(2): eps_zz_tot = 0% F(3): yield condition sig_xx = vars(1);sig_zz = vars(2);lambda_dt_over_2mu = vars(3);sig_yy = 0; mu = param(1);nu = param(2);k = param(3);sig_xx_0 = param(4);sig_zz_0 = param(5);sig_yy_0 = 0;eps_xx_0 = param(6); % total strain at previous time stepeps_xx = param(7); % total strain at current time stepeps_zz_0 = 0;eps_zz = 0; K = 2*mu*(1+nu)/3/(1-2*nu); % bulk modulus % change of stressdsig_xx = sig_xx - sig_xx_0;dsig_yy = sig_yy - sig_yy_0;dsig_zz = sig_zz - sig_zz_0; dsig_bar = (dsig_xx + dsig_yy + dsig_zz)/3;ds_xx = dsig_xx - dsig_bar;ds_yy = dsig_yy - dsig_bar;ds_zz = dsig_zz - dsig_bar; % increment of elastic straindeps_elast_xx = dsig_bar/(3*K) + ds_xx/(2*mu);deps_elast_yy = dsig_bar/(3*K) + ds_yy/(2*mu);deps_elast_zz = dsig_bar/(3*K) + ds_zz/(2*mu); % hydrostaic stresssig_bar = (sig_xx + sig_yy + sig_zz)/3;sig_bar_0 = (sig_xx_0 + sig_yy_0 + sig_zz_0)/3; % deviatoric stresss_xx = sig_xx - sig_bar;s_yy = sig_yy - sig_bar;s_zz = sig_zz - sig_bar;

Plane Strain Tension 10

4/15/13 12:42 AM D:\My Documents\Courses\...\eqns_plane_strain_uniaxial_plast.m 2 of 2

s_xx_0 = sig_xx_0 - sig_bar_0;s_yy_0 = sig_yy_0 - sig_bar_0;s_zz_0 = sig_zz_0 - sig_bar_0; % average deviatoric stressave_s_xx = (s_xx + s_xx_0)/2;ave_s_yy = (s_yy + s_yy_0)/2;ave_s_zz = (s_zz + s_zz_0)/2; % increment of plastic straindeps_plast_xx = lambda_dt_over_2mu * ave_s_xx;deps_plast_yy = lambda_dt_over_2mu * ave_s_yy;deps_plast_zz = lambda_dt_over_2mu * ave_s_zz; % construct equationF = [0 0 0]'; % total strain in xxF(1) = (eps_xx_0 + deps_elast_xx + deps_plast_xx) - eps_xx; % total strain in zzF(2) = (eps_zz_0 + deps_elast_zz + deps_plast_zz) - eps_zz; % yield conditionJ2 = (s_xx^2 + s_yy^2 + s_zz^2)/2;F(3) = J2 - k^2;

Plane Strain Tension 11

2 4 6 8 10 12 14 16

x 10−3

1.14

1.16

1.18

εxx

σ xx

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0160

0.2

0.4

0.6

0.8

εxx

σ zz

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0160.3

0.35

0.4

0.45

0.5

εxx

σ zz /

σ xxPlane Strain Tension 12

2 4 6 8 10 12 14 16

x 10−3

0.58

0.6

0.62

0.64

εxx

s xx

2 4 6 8 10 12 14 16

x 10−3

−0.65

−0.6

−0.55

−0.5

−0.45

εxx

s yy

2 4 6 8 10 12 14 16

x 10−3

−0.2

−0.15

−0.1

−0.05

0

εxx

s zzPlane Strain Tension 13

2 4 6 8 10 12 14 16

x 10−3

1.55

1.6

1.65

1.7

1.75x 10

−4

εxx

λ Δ

t / (

2μ)

2 4 6 8 10 12 14 16

x 10−3

0.31

0.32

0.33

0.34

0.35

0.36

εxx

J 2Plane Strain Tension 14

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4/26/13 1:42 PM D:\My Documents\Courses\ME342-Inelas...\torsion_plast_example.m 1 of 8

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2D Cylinder Example: Torsion with plasticity using triangular elements% % Wei Cai caiwei@stanford.edu% William Kuykendall wpkuyken@stanford.edu%% adapted from FeCalc.m written by Peter Pinsky pinsky@stanford.edu% universal meshing from Michael Hunsweck%% First Adapted 01/03/2013%% Last Modified 04/22/2013%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Part 1: Set simulation parameters%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear;global nNodes nElements Coord u IEN K F Nxe Nye EBC ID g penalty_factornDoF = 1; % number of degrees of freedomnDim = 2; % number of dimensions% L: [l w pl pw]% l, w: domain length and width [m] % pl, pw: density of triangles along length and width % Circular cylinder [x0 y0 r]--[center coordinates, radius] [m m m]%shape = 'circle'; L = [0.8 0.8 10 10]; c = [0 0 0.2]; % Elliptical cylinder [x0 y0 a b]--[(x-x0)/a]^2+[(y-y0)/b]^2=1%shape = 'ellipse'; L = [1.3 1.3 10 10]; c = [0 0 0.6 0.2];% Rectangular rod [x0 y0 a b]-- [x_center, y_center, length, width]%shape = 'rectangle'; L = [0.4 0.4 5 5]; c = [0 0 0.4 0.4];shape = 'rectangle'; L = [0.6 0.4 5 5]; c = [0 0 0.6 0.4];E = 200e9;nu = 0.32;mu = E/(2*(1+nu));% shear modulus [ N/m^2 ] tau_max = 4.0e6; % maximum shear stress%beta = 3.5e-4; % twist per unit length [1/m]beta = 4e-4; % twist per unit length [1/m] % options for solving plasticity problem by iterationNiter = 2000; dt = 5e6; plotfreq = 100; penalty_factor = 2e-14; % set axis limits for plottingif strcmp(shape,'circle') clim = c(3);elseif strcmp(shape,'ellipse') clim = max(c(3:4));elseif strcmp(shape,'rectangle') clim = max(c(3:4))/2;else fprintf('Unrecognized shape for plotting.\n'); clim = 1;

4/26/13 1:42 PM D:\My Documents\Courses\ME342-Inelas...\torsion_plast_example.m 2 of 8

end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Part 2: Generate Mesh, Initialize Arrays, Set Up Boundary Conditions%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Step 1: Generate solution mesh[Coord,IEN,c,shape] = CreateMesh(L,c,shape); % Generates the meshIEN = IEN';nNodes = size(Coord,1); % Number of nodesnElements = size(IEN,2); % Number of elementsnNodesElement = size(IEN,1); % Number of nodes per elementnEdgesElement = 3; % Number of edges per element % Step 2: Set Tolerance for detecting boundary nodesboundary_tol = 1e-8;onSurface = logical(zeros(1,nNodes)); % Step 3: Allocate ArraysC = spalloc(nElements,nDoF,nElements);f = spalloc(nElements,nDoF,nElements);g = spalloc(nNodes,nDoF,round(nNodes/4));EBC = spalloc(nNodes,nDoF,round(nNodes/4));% Step 4: Set Essential Boundary Condition % Step 4.1: Get coordinates of all nodes X = Coord(:,1); % x-position of all nodes Y = Coord(:,2); % y-position of all nodes % Step 4.2: Define essential boundary condition value(s) T = [ 0 ]; % Traction % Step 4.3: Set g, EBC for all nodes on the surface % Step 4.3a: Find all nodes on external surface if strcmp(shape,'circle') % distance squared of node from circle center D2 = (X-c(1)).^2 + (Y-c(2)).^2; onSurface = (D2 >= c(3)^2-boundary_tol); elseif strcmp(shape,'ellipse') % distance squared of node from ellipse center D2 = ((X-c(1))/c(3)).^2 + ((Y-c(2))/c(4)).^2; onSurface = (D2 >= 1-boundary_tol); elseif strcmp(shape,'rectangle') for ii = 1:nNodes onSurface(ii) = abs(X(ii)-c(1))>(c(3)/2-boundary_tol) ... || abs(Y(ii)-c(2))>(c(4)/2-boundary_tol) ; end else error('Code should not get here. Shape should have been set.\n'); end % Adjust the interior mesh coords[newCoord] = AdjustMesh(Coord, IEN, onSurface, 200, 0e-1, 200); Coord = newCoord;

4/26/13 1:42 PM D:\My Documents\Courses\ME342-Inelas...\torsion_plast_example.m 3 of 8

% Step 4.3b: Set g, EBC for node on surface EBC(onSurface) = 1; g(onSurface) = T; % Step 5: Define the Natural Boundary Condition % Step 5.1: Allocate h array h = spalloc(nEdgesElement,nElements,round(nElements/4)); % In this problem there is not any natural boundary condition applied% Step 6: Define the Load Value (often RHS of equation you are solving) % Step 1: Set f for all elements f(:) = 0.1; f(:) = 10; f(:) = 1; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Part 3: Create Destination Array (ID) and Location Matrix (LM)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Step 1: Create Data Processing Arrays % Step 1.1: Allocate arrays ID = zeros(nNodes,nDoF)'; % Destination Array: ID(A) = P if A is not on essential boundary % 0 if A is on the essential boundary % index is global node number (A) % result is global equation number (P) LM = zeros(nNodesElement*nDoF,nElements); % Location Matrix: P = LM(a,e) % row is degree of freedom (a) % column is element number (e) % result is global equation number (P) % Step 1.2: Populate ID Array I = full(EBC' == 0); % Creates logical indexing array nEquations = sum(sum(I)); % Count number of DoF's ID(I) = 1:nEquations; % Assign Equation numbers ID = ID'; % Step 1.3: Populate LM Array P = ID(IEN,:)'; LM(:) = P(:); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Part 4: Assembly of K and F Matrices%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Step 1: Allocate K and FK = spalloc(nEquations,nEquations,20*nDoF*nEquations);F = spalloc(nEquations,1,nEquations);% Step 2: Assemble K and F % Step 2.1: Compute element contributiontions for e = 1:nElements % Step 2.2: Get coordinates of element nodes x = Coord(IEN(:,e),1); y = Coord(IEN(:,e),2); % Step 2.3: Define Shorthand

4/26/13 1:42 PM D:\My Documents\Courses\ME342-Inelas...\torsion_plast_example.m 4 of 8

x12 = x(1) - x(2); x23 = x(2) - x(3); x31 = x(3) - x(1); y12 = y(1) - y(2); y23 = y(2) - y(3); y31 = y(3) - y(1); % Step 2.4: Compute element Jacobian J_e = [ -x31, x23; -y31 y23 ]; % Step 2.5: Compute element size elementSize = 0.5*det(J_e); Ae = elementSize; %================================================================== % Subsection 1: Computation of k_e %================================================================== % For isotropic case. k_e = 1/(4*2*mu*beta*Ae)* ... [y23^2+x23^2, y23*y31+x23*x31, y23*y12+x23*x12; y23*y31+x23*x31, y31^2+x31^2, y31*y12+x31*x12; y23*y12+x23*x12, y31*y12+x31*x12, y12^2+x12^2]; %================================================================== % Subsection 2: Computation of f_e %================================================================== ff = f(IEN(1,e)); f_e = ff*Ae/3*[1;1;1]; %================================================================== % Subsection 3: Computation of f_g %================================================================== g_e = g(IEN(:,e)); f_g = -k_e*g_e; %================================================================== % Subsection 4: Computation of f_h %================================================================== % Step 4.1a: Edge load intensity (homogeneous boundary) f_h = h(:,e); % Step 4.1b: Edge load intensity (non-homogeneous boundary) l12 = sqrt(x12^2 + y12^2); l23 = sqrt(x23^2 + y23^2); l31 = sqrt(x31^2 + y31^2); h_e = h(:,e); f_h = h_e(1)*l12/2*[1;1;0] + h_e(2)*l23/2*[0;1;1] + h_e(3)*l31/2*[1;0;1]; %================================================================== % End of Subsections %================================================================== % Step 2.6: Get Global equation numbers P = LM(:,e); % Step 2.7: Eliminate Essential DOFs I = (P ~= 0); P = P(I); % Step 2.8: Insert k_e, f_e, f_g, f_h K(P,P) = K(P,P) + k_e(I,I); F(P) = F(P) + f_e(I) + f_g(I) + f_h(I); end % Global matrices for computing spatial gradients and area of elementsNxe = zeros(nElements,nNodes); Nye = zeros(nElements,nNodes); Ae = zeros(nElements,1);

4/26/13 1:42 PM D:\My Documents\Courses\ME342-Inelas...\torsion_plast_example.m 5 of 8

for e = 1:nElements % Get element coordinates xe = Coord(IEN(:,e),1); ye = Coord(IEN(:,e),2); % Define Shorthand x12 = xe(1) - xe(2); x23 = xe(2) - xe(3); x31 = xe(3) - xe(1); y12 = ye(1) - ye(2); y23 = ye(2) - ye(3); y31 = ye(3) - ye(1); % Area of a triangle: Ae(e) = (xe(1)*(ye(2)-ye(3))+xe(2)*(ye(3)-ye(1))+xe(3)*(ye(1)-ye(2)))/2; % Compute element jacobian J_e = [ -x31, x23; -y31 y23 ]; % Calculate natural derivatives of N1,N2,N3 N1es = 0; N2es = 1; N3es = -1; N1er = 1; N2er = 0; N3er = -1; d_Nrs = [N1er N1es; N2er N2es; N3er N3es]; d_Nxy = d_Nrs/J_e; Nxe(e,IEN(:,e)) = d_Nxy(:,1)'; Nye(e,IEN(:,e)) = d_Nxy(:,2)';end % compute volume contribution from each nodefor ii = 1:nNodes [row,col] = find(IEN == ii); V_ave(ii)= sum(Ae(col))/3; % 3 for triangular elements end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Part 5: Compute Finite Element Solution (including plasticity)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Step 1: Solve elasticity problemd_slash = K\F; % Step 2: Solve plasticity problem by% minimize (0.5*d'*K*d - F'*d)_in_elast_region + (penalty)_in_plast_region% using steepest descent algorithm% Plast_region is defined for nodes on elements whose slope exceeds tau_max% Elast_region is defined for remaining nodes% For nodes in Plast_region, the (elastic) gradient % from (0.5*d'*K*d - F'*d) term is set to zero, and the gradient from% penalty function = (tau^2-tau_max^2)*penalty_factor is added. % initialize d from elasticity solution (scaled)if ~exist('d') d = d_slash*0.25;endI = (EBC == 0); u = zeros(size(g)); tau_max2 = tau_max^2;for iter = 1:Niter, % grad is the residual of Poisson's equation grad = K*d - F; u(I) = d(ID(I)); u(~I) = g(~I); dxe = Nxe*u; dye = Nye*u;

4/26/13 1:42 PM D:\My Documents\Courses\ME342-Inelas...\torsion_plast_example.m 6 of 8

% total torque Torque = 2*dot(V_ave, u); % find elements whose slope exceeds tau_max tau2 = (dxe).^2+(dye).^2; element_factor = (tau2 >= tau_max2); ind_e = find(element_factor); id_in_elements = reshape(ID(IEN(:,ind_e)),length(ind_e)*3,1); non_zero_entries = find(id_in_elements>0); % for these elements, set the residual term from Poisson's equation to zero grad(id_in_elements(non_zero_entries)) = 0; % for elements whose slope exceeds tau_max, add penalty function penalty = sum( (tau2-tau_max2).*element_factor ) * penalty_factor; dpenalty_du = zeros(size(u)); dpenalty = zeros(size(d)); for e = 1:nElements, if element_factor(e) % derivative of slope wrt nodal value dpenalty_du = dpenalty_du + ... (2*(Nxe(e,:)*u)*Nxe(e,:)' + 2*(Nye(e,:)*u)*Nye(e,:)'); end end dpenalty_du = dpenalty_du * penalty_factor; dpenalty(ID(I)) = dpenalty_du(I); grad = grad + dpenalty; % move d in the steepest descent direction d = d - grad*dt; % plot intermediate results during minimization if (mod(iter,plotfreq)==0) disp(sprintf('iter = %d/%d Torque = %e sum(d) = %e norm(grad) = %e', ... iter,Niter,Torque,sum(d),norm(grad))); figure(1) x = Coord; T = trisurf(IEN',x(:,1),x(:,2),u); xlim([-clim clim]); ylim([-clim clim]); figure(3) for ii = 1:nNodes [row,col] = find(IEN == ii); du_ave(ii,1)= mean(dxe(col,:),1); du_ave(ii,2)= mean(dye(col,:),1); end; T = trisurf(IEN',x(:,1),x(:,2),sqrt(du_ave(:,2).^2+du_ave(:,1).^2)); xlim([-clim clim]); ylim([-clim clim]); drawnow endend % Step 3: Arrange results (d) into solution vector (u)

4/26/13 1:42 PM D:\My Documents\Courses\ME342-Inelas...\torsion_plast_example.m 7 of 8

u = zeros(nNodes,nDoF);I = (EBC == 0);u(I) = d(ID(I));u(~I) = g(~I);Torque = 2*dot(V_ave, u); % Step 4: Create coordinate arrayx = Coord; % Step 5: compute average slope on nodesfor ii = 1:nNodes [row,col] = find(IEN == ii); du_ave(ii,1)= mean(dxe(col,:),1); du_ave(ii,2)= mean(dye(col,:),1); end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Part 6: Plotting%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Step 1: Plot Prandtl's stress function (u, i.e. phi)figure(1)T = trisurf(IEN',x(:,1),x(:,2),u);title('Torsion: Prandtl Stress Function',... 'Interpreter','Latex','FontSize',12);xlabel('$x$','Interpreter','Latex','FontSize',12);ylabel('$y$','Interpreter','Latex','FontSize',12);zlabel('$\phi$','Interpreter','Latex','FontSize',12);set(gca,'FontSize',8);axis([-clim clim -clim clim 0 max(u*1.1)])co = colorbar;set(co,'FontSize',8);colormap jetshading interpset(T,'EdgeColor','k'); % Step 2: Plot sigma_xz and sigma_yz (gradient of u)figure(2)subplot(2,1,1)T = trisurf(IEN',x(:,1),x(:,2),du_ave(:,2));axis equalaxis([-clim clim -clim clim])view([0 90])title('FE Solution for $\sigma_{xz}$',... 'Interpreter','Latex','FontSize',12);xlabel('$x$','Interpreter','Latex','FontSize',12);ylabel('$y$','Interpreter','Latex','FontSize',12);zlabel('$\sigma_{xz}$','Interpreter','Latex','FontSize',12);co = colorbar; set(co,'FontSize',8);colormap jet; shading interpset(T,'EdgeColor','k');subplot(2,1,2)

4/26/13 1:42 PM D:\My Documents\Courses\ME342-Inelas...\torsion_plast_example.m 8 of 8

T = trisurf(IEN',x(:,1),x(:,2),-du_ave(:,1));axis equalaxis([-clim clim -clim clim])view([0 90])title('FE Solution for $\sigma_{yz}$',... 'Interpreter','Latex','FontSize',12);xlabel('$x$','Interpreter','Latex','FontSize',12);ylabel('$y$','Interpreter','Latex','FontSize',12);zlabel('$\sigma_{yz}$','Interpreter','Latex','FontSize',12);co = colorbar; set(co,'FontSize',8);colormap jet; shading interpset(T,'EdgeColor','k'); % Step 3: Plot maximum shear stress (magnitude of gradient of u)figure(3)T = trisurf(IEN',x(:,1),x(:,2),sqrt(du_ave(:,2).^2+du_ave(:,1).^2));xlim([-clim clim]); ylim([-clim clim]);title('Torsion: Maximum Shear Stress',... 'Interpreter','Latex','FontSize',12);xlabel('$x$','Interpreter','Latex','FontSize',12);ylabel('$y$','Interpreter','Latex','FontSize',12);zlabel('$\sigma_{xz}$','Interpreter','Latex','FontSize',12);cbar = colorbar; set(cbar,'FontSize',8);colormap jet; shading interpset(T,'EdgeColor','k'); % Step 4: Plot contour of stress function with elast-plast boundaryfigure(4);lxi = linspace(-1,1,300);lyi = linspace(-1,1,300);[xi,yi] = meshgrid(lxi,lyi);ui = griddata(x(:,1),x(:,2),u,xi,yi,'cubic');dui = griddata(x(:,1),x(:,2),sqrt(du_ave(:,2).^2+du_ave(:,1).^2),xi,yi,'cubic');[c h]=contour(xi,yi,ui,20);hold onc2=contourc(lxi,lyi,dui,[0.95 0.95]*tau_max,'b--');plot(c2(1,2:end),c2(2,2:end),'r.');hold offtitle('Torsion: Prandtl Stress Function',... 'Interpreter','Latex','FontSize',12);axis equal; axis([-clim clim -clim clim])figure(5);xi = linspace(-1,1,300);yi = linspace(-1,1,300);[xi,yi] = meshgrid(xi,yi);contour(xi,yi,dui,20);title('Torsion: Maximum Shear Stress',... 'Interpreter','Latex','FontSize',12);axis equal; axis([-clim clim -clim clim])

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σθθnum

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σθθanl

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−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

r / a

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(2k)

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1 1.2 1.4 1.6 1.8 20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r / a

σ θθ /

(2k)

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1 1.2 1.4 1.6 1.8 2−0.2

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−0.1

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0

0.05

0.1

0.15

0.2

0.25

0.3

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13

1.0

2.0

1.0

2.0

1.0

2.0

4/29/13 10:04 AM D:\My Documents\Courses\ME342-Inelastic...\cylind_tube_elast.m 1 of 2

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2D Example: solve strain field in pressurized cylindrical tube % elastic deformation only% % Wei Cai caiwei@stanford.edu% Created 03/24/2013% Last Modified 04/29/2013%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mu = 100;nu = 0.3;lambda = 2*mu*nu/(1-2*nu); a = 1; b = 2;dr = 0.05;r = [a:dr:b];p0 = 1; eps_rr = zeros(size(r));eps_qq = zeros(size(r)); Niter = 10000;plotfreq = 1000; % 50; param = [a b dr mu nu p0]; F0 = eqns_cylind_tube_elast([eps_rr, eps_qq],param);%options = optimset('Display','iter','TolFun',1e-8);options = optimset('Display','off','TolFun',1e-8);sol = fsolve('eqns_cylind_tube_elast', [eps_rr, eps_qq], options, param);eps_rr = sol(1:end/2); eps_qq = sol(end/2+1:end);F1 = eqns_cylind_tube_elast([eps_rr, eps_qq],param); % (Alternative method using cgrelax, require gradient evaluation)% [U dU_depsrrqq] = grad_cylind_tube_elast([eps_rr, eps_qq],param);% acc = 1e-6; maxfn = 40000; dfpred = 1e-5; n = length([eps_rr, eps_qq]);% [U,eps_rrqq_rlx] = cgrelax('grad_cylind_tube_elast',n,acc,maxfn,dfpred,...% [eps_rr,eps_qq],param);% eps_rr = eps_rrqq_rlx(1:end/2); eps_qq = eps_rrqq_rlx(end/2+1:end); % stress fieldsig_rr = (lambda + 2*mu) * eps_rr + lambda * eps_qq;sig_qq = (lambda + 2*mu) * eps_qq + lambda * eps_rr;sig_zz = lambda * (eps_rr + eps_qq); pp=p0*a^2/(b^2-a^2); E = 2*mu*(1+nu);sig_rr_anl = pp*(1-b^2./r.^2);sig_qq_anl = pp*(1+b^2./r.^2);eps_rr_anl = (1-nu^2)/E*sig_rr_anl - nu*(1+nu)/E*sig_qq_anl;eps_qq_anl = (1-nu^2)/E*sig_qq_anl - nu*(1+nu)/E*sig_rr_anl;

4/29/13 10:04 AM D:\My Documents\Courses\ME342-Inelastic...\cylind_tube_elast.m 2 of 2

% plot solutionfs = 17;figure(1);plot(r, eps_rr /p0*mu, '.', r, eps_qq/p0*mu, 'd', ... r, eps_rr_anl/p0*mu, r, eps_qq_anl/p0*mu);xlim([0 max(r)*1.5]); set(gca,'FontSize',fs);xlabel('r'); ylabel('\epsilon \mu / p_0');legend('\epsilon_{rr}^{num}','\epsilon_{\theta\theta}^{num}',... '\epsilon_{rr}^{anl}','\epsilon_{\theta\theta}^{anl}');figure(2);plot(r, sig_rr/p0, '.', r, sig_qq/p0, 'o', r, sig_rr_anl/p0, r, sig_qq_anl/p0);xlim([0 max(r)*1.5]); set(gca,'FontSize',fs);xlabel('r'); ylabel('\sigma / p_0');legend('\sigma_{rr}^{num}','\sigma_{\theta\theta}^{num}',... '\sigma_{rr}^{anl}','\sigma_{\theta\theta}^{anl}');

4/29/13 9:54 AM D:\My Documents\Courses\ME342-Inela...\eqns_cylind_tube_elast.m 1 of 1

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2D Example: solve strain field in pressurized cylindrical tube % elastic deformation only% Construct the equations to be solved by cylind_tube_elast.m%% Wei Cai caiwei@stanford.edu% Created 03/24/2013% Last Modified 04/29/2013%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function F = eqns_cylind_tube_elast(vars, param)% F(1:N-1): equilibrium equation% F(N:2N-2): compatibility equation% F(2N-1:2N): boundary condition a = param(1);b = param(2);dr = param(3);mu = param(4);nu = param(5);p0 = param(6); N = length(vars)/2; F = zeros(1,2*N);eps_rr = vars(1:N); eps_qq = vars(N+1:end); lambda = 2*mu*nu/(1-2*nu);r = [a:dr:b]; r_avg = (r(2:end) + r(1:end-1)) / 2; % stress fieldsig_rr = (lambda + 2*mu) * eps_rr + lambda * eps_qq;sig_qq = (lambda + 2*mu) * eps_qq + lambda * eps_rr;sig_zz = lambda * (eps_rr + eps_qq); dsigrrdr = (sig_rr(2:end) - sig_rr(1:end-1)) / dr;sig_rr_avg = (sig_rr(2:end) + sig_rr(1:end-1)) / 2;sig_qq_avg = (sig_qq(2:end) + sig_qq(1:end-1)) / 2; % equilibrium equationF(1:N-1) = dsigrrdr + (sig_rr_avg-sig_qq_avg)./r_avg; % strain fielddepsqqdr = (eps_qq(2:end) - eps_qq(1:end-1)) / dr;eps_qq_avg = (eps_qq(2:end) + eps_qq(1:end-1)) / 2;eps_rr_avg = (eps_rr(2:end) + eps_rr(1:end-1)) / 2; % compatibility conditionF(N:2*N-2) = depsqqdr - (eps_rr_avg-eps_qq_avg)./r_avg; % boundary conditionF(2*N-1:2*N) = [sig_rr(1) + p0, sig_rr(end)];

5/10/13 8:39 AM D:\My Documents\Courses\ME342-Inelastici...\cylind_tube_plast.m 1 of 3

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2D Example: solve strain field in pressurized cylindrical tube % pressure large enough to cause plastic deformation% % Wei Cai caiwei@stanford.edu% Created 03/24/2013% Last Modified 04/29/2013%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mu = 100;nu = 0.3;k = 4; % yield stress / sqrt(3)E = 2*mu*(1+nu); % Young's modulusK = 2*mu*(1+nu)/3/(1-2*nu); % bulk modulus a = 1; b = 2;dr = 0.02; drho = dr;r = [a:dr:b]; rho = r;p0 = 1; S_rr_sol = zeros(length(r), length(rho));S_qq_sol = zeros(length(r), length(rho));sig_rr_sol = zeros(length(r), length(rho));sig_qq_sol = zeros(length(r), length(rho));sig_zz_sol = zeros(length(r), length(rho));eps_rr_sol = zeros(length(r), length(rho));eps_qq_sol = zeros(length(r), length(rho)); % analytic solution in elastic regionfor i=1:length(r), for j=1:length(rho), % limit to elastic region if r(i) >= rho(j), ppp=k*((1-2*nu)^2/3 + b^4/rho(j)^4)^(-1/2); sig_rr_1 = ppp*(1-b^2./r(i).^2); sig_qq_1 = ppp*(1+b^2./r(i).^2); sig_zz_1 = nu*(sig_rr_1 + sig_qq_1); S_rr_1 = sig_rr_1 - (sig_rr_1+sig_qq_1+sig_zz_1)/3; S_qq_1 = sig_qq_1 - (sig_rr_1+sig_qq_1+sig_zz_1)/3; eps_rr_1 = (1-nu^2)/E*sig_rr_1 - nu*(1+nu)/E*sig_qq_1; eps_qq_1 = (1-nu^2)/E*sig_qq_1 - nu*(1+nu)/E*sig_rr_1; S_rr_sol(i,j) = S_rr_1; S_qq_sol(i,j) = S_qq_1; sig_rr_sol(i,j) = sig_rr_1; sig_qq_sol(i,j) = sig_qq_1; sig_zz_sol(i,j) = sig_zz_1; eps_rr_sol(i,j) = eps_rr_1; eps_qq_sol(i,j) = eps_qq_1; end endend

5/10/13 8:39 AM D:\My Documents\Courses\ME342-Inelastici...\cylind_tube_plast.m 2 of 3

% find solution in plastic regiondisp('find solution in plastic region...');for j=2:length(rho), disp(sprintf('j = %d / %d',j, length(rho))); for i=j-1:-1:1, % find solution at (r, rho+drho) eps_rr_1 = eps_rr_sol(i,j-1); eps_qq_1 = eps_qq_sol(i,j-1); S_rr_1 = S_rr_sol(i,j-1); eps_rr_2 = eps_rr_sol(i+1,j); eps_qq_2 = eps_qq_sol(i+1,j); S_rr_2 = S_rr_sol(i+1,j); param = [r(i), dr, mu, nu, k, eps_rr_1, eps_qq_1, ... S_rr_1, eps_rr_2, eps_qq_2, S_rr_2]; %options = optimset('Display','iter','TolFun',1e-8); options = optimset('Display','off','TolFun',1e-8); %F0 = eqns_cylind_tube_plast([eps_rr_1, eps_qq_1, S_rr_1], param); %sol = fsolve('eqns_cylind_tube_plast', [eps_rr_1, eps_qq_1, S_rr_1], ... % options, param); [sol,Fval,exitflag] = fsolve('eqns_cylind_tube_plast', ... [eps_rr_1, eps_qq_1, S_rr_1], options, param); switch exitflag case 0, fprintf(['Solution incorrect: ' ... 'Maximum number of function evaluations or iterations reached.\n']); case -1, fprintf(['Solution incorrect: ' ... 'Algorithm terminated by the output function.\n']); case -2, fprintf(['Solution incorrect: ', ... 'Algorithm seems to be converging to a point that is not a root.\n']); case -3, fprintf(['Solution incorrect: ', ... 'Trust region radius became too small.\n']); case -4, fprintf(['Solution incorrect: ' ... 'Line search cannot sufficiently decrease the residual '... 'along the current search direction.\n']); end %F1 = eqns_cylind_tube_plast(sol, param); eps_rr_sol(i,j) = sol(1); eps_qq_sol(i,j) = sol(2); S_rr_sol(i,j) = sol(3); S_qq_sol(i,j) = (-S_rr_sol(i,j) + sqrt(4*k^2-3*S_rr_sol(i,j)^2))/2; sig_rr_sol(i,j) = S_rr_sol(i,j) + K*(eps_rr_sol(i,j)+eps_qq_sol(i,j)); sig_qq_sol(i,j) = S_qq_sol(i,j) + K*(eps_rr_sol(i,j)+eps_qq_sol(i,j)); sig_zz_sol(i,j) = -S_rr_sol(i,j)-S_qq_sol(i,j) ... + K*(eps_rr_sol(i,j)+eps_qq_sol(i,j)); endend % verify solutiondisp('verify solution in plastic region...');for j=2:length(rho), for i=j-1:-1:1, % find solution at (r, rho+drho) eps_rr_1 = eps_rr_sol(i,j-1); eps_qq_1 = eps_qq_sol(i,j-1); S_rr_1 = S_rr_sol(i,j-1); eps_rr_2 = eps_rr_sol(i+1,j); eps_qq_2 = eps_qq_sol(i+1,j);

5/10/13 8:39 AM D:\My Documents\Courses\ME342-Inelastici...\cylind_tube_plast.m 3 of 3

S_rr_2 = S_rr_sol(i+1,j); param = [r(i), dr, mu, nu, k, eps_rr_1, eps_qq_1, ... S_rr_1, eps_rr_2, eps_qq_2, S_rr_2]; F = eqns_cylind_tube_plast([eps_rr_sol(i,j), eps_qq_sol(i,j), ... S_rr_sol(i,j)], param); disp(sprintf('i = %d j = %d F = %e %e %e',i,j,F(1),F(2),F(3))); endend % plot solutionfs = 17;figure(1);skip = find(r/r(1)==1.2,1,'First')-1;plot(r/a, sig_rr_sol(:,1:skip:end)/(2*k), '.-');set(gca,'FontSize',fs);xlabel('r / a');ylabel('\sigma_{rr} / (2k)');legend('\rho/a = 1','1.2','1.4','1.6','1.8','2.0','Location','SouthEast'); figure(2);plot(r, sig_qq_sol(:,1:skip:end)/(2*k), '.-');set(gca,'FontSize',fs);xlabel('r / a');ylabel('\sigma_{\theta\theta} / (2k)');legend('\rho/a = 1','1.2','1.4','1.6','1.8','2.0','Location','NorthWest'); figure(3);plot(r, sig_zz_sol(:,1:skip:end)/(2*k), '.-');set(gca,'FontSize',fs);xlabel('r / a');ylabel('\sigma_{zz} / (2k)');legend('\rho/a = 1','1.2','1.4','1.6','1.8','2.0','Location','SouthEast');

4/29/13 10:07 AM D:\My Documents\Courses\ME342-Inel...\eqns_cylind_tube_plast.m 1 of 2

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2D Example: solve strain field in pressurized cylindrical tube % pressure large enough to cause plastic deformation% Construct the equations to be solved by cylind_tube_plast.m% % Wei Cai caiwei@stanford.edu% Created 03/24/2013% Last Modified 04/29/2013%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function F = eqns_cylind_tube_plast(vars, param)% F(1): equilibrium equation% F(2): compatibility equation% F(3): plastic flow rule eps_rr = vars(1);eps_qq = vars(2);S_rr = vars(3); r = param(1);dr = param(2); drho = dr;mu = param(3);nu = param(4);k = param(5);eps_rr_1 = param(6);eps_qq_1 = param(7);S_rr_1 = param(8);eps_rr_2 = param(9);eps_qq_2 = param(10);S_rr_2 = param(11); K = 2*mu*(1+nu)/3/(1-2*nu); % bulk modulus F = [0 0 0]'; S_qq = (-S_rr + sqrt(4*k^2-3*S_rr^2 ))/2;S_qq_1 = (-S_rr_1 + sqrt(4*k^2-3*S_rr_1^2))/2;S_qq_2 = (-S_rr_2 + sqrt(4*k^2-3*S_rr_2^2))/2; dSrr_dr = (S_rr_2-S_rr)/dr;dSqq_dr = (S_qq_2-S_qq)/dr;depsrr_dr = (eps_rr_2-eps_rr)/dr;depsqq_dr = (eps_qq_2-eps_qq)/dr; depsrr_drho = (eps_rr - eps_rr_1)/drho;depsqq_drho = (eps_qq - eps_qq_1)/drho;dSrr_drho = (S_rr - S_rr_1)/drho;dSqq_drho = (S_qq - S_qq_1)/drho;

4/29/13 10:07 AM D:\My Documents\Courses\ME342-Inel...\eqns_cylind_tube_plast.m 2 of 2

% equilibrium conditionF(1) = dSrr_dr + K*( depsrr_dr + depsqq_dr ) ... + ( (S_rr_2+S_rr)/2 - (S_qq_2+S_qq)/2 ) / (r+dr/2); % compatibility conditionF(2) = depsqq_dr - ( (eps_rr_2+eps_rr)/2 - (eps_qq_2+eps_qq)/2 ) / (r+dr/2); % plastic flow ruleF(3) = 2*mu*( (2/3*depsrr_drho - 1/3*depsqq_drho)*(S_qq+S_qq_1)/2 ... -(2/3*depsqq_drho - 1/3*depsrr_drho)*(S_rr+S_rr_1)/2 ) ... - dSrr_drho*(S_qq+S_qq_1)/2 + dSqq_drho*(S_rr+S_rr_1)/2 ;

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o(tf"*>qr/

',-'f ra-' rr L i /| / 7 r

.- . . . .- \ '11'

A lut t roLulion;: ':'.afljlturfuA U3 canL;r,;ng " {e-"

SUp {,n-n-, Jtvt&:Q l. e{e^Wor*t S"hr*ouZ

t g,f we

%pt<;oX), *.1,! twl,/y {l^tt*L S^il",!tI d \ r I

- - - l - - '

Arfrlru ths' ,2ntn4 nltn-Rfl-p SM/n u " i

- l

nn U"!- S,frtz:" ponq,,U.& bo Sux$'ra .

2 , Artrn',:,rJ!',y r"t;tlv tnz fa*"';(y ^4 c/t4ce'd6L

hl 0z:5.

)AU-rr:-. p-t;rrzs a'uL slo^;,Vl"t , t"

F-

* r - (= i - *= t " (u rwd- )

j - , , - 2Lrp+8")

IA 1,,;

-W,',44,,f-l- 7/a^.r a/?^d/i GL

frv,'r SunJaczt

3- /-ogaif 'c. : l i r^t

It

t\AE 7+zL n r ' t ' . /

32. ? I,at ,tn,4*al"o{ ( t ;qn'd ,t- - * " / _ _ -

lro^J L l's so(n.,h-w( karJ.^nt", P ztt 1

P ! Arut- ,,": v-a) n X Co.rl

a

lflwJr"tw gflPjy

6-: -k-Urt t )

6 a y = e - k t = - A ( l T + z )

Q x : . + l r - - k l t

l,^rr;+ {oad is P : - ZG qJ

p .=,. Zq R_ ( fi-+ -l )

s-lota,. BDt, 6FA,A c t 3

2

Reg;* B pE'\7'\/

Regir" CDFA6v---zA

\ /\ /

\ /V.

t--

ALog B i l , f :+ .6==-k-eklry a-l;oL, G-'., L+ : cn-^A't-

flvt ,-t- 'il-or1 B c 1i : - k- t zk.(_T I= - k ( 7 t + t 1

SInq{" s*ut s-fiJ|z , A7c, BC P

t<" y y - u

&-x:2K6::trt

fiy: oWx:2k

6 : -A

Q=-&4'!=T9\'. -- ,r < f = K

( a*sWr',-e L.^br; c"tt dt

d.rr^.ta+t , tw cl'rzo^/Y;kvt rn AB )

lv+x-

Rt 3ior AB C

5o thl.

lVt73*Z ?

D

D€- i",t u\athc-pla*hz bown dtu-*

itrgcanqrr.- ur.q cA4r1,.?"r[. <)o"t +''>^r.(p{]r4 r1 r iqr6.i .J I '

\,zp =-r:l 6rL p C

[3ecs,ur.y';, dt': o - o.i-[ F" h*s (srra$v1 t

Vp=.,;> rrt e/4"1t &Pg nqit-.

So t{t-q *ektftb r'J l,\tt-q. DF cli}'t-rfs"vr

!q,( rzr-eqc{ t f"1 iy1+qtY*1T:lCD b ztiar\AL*plan+Z fuwnJ-e',,6

Vp =,O fY1 C D

Ysor^^*,- p );nM an-c- Y[3sgi's1,

Vp--.:) o"n *ha €zntiv€ BCD tejitr^

ic +h{ ur.{o"h d,rrt^*ir'n f otb'rr} a'fc-'l.w ^ffcs

R e 3 * ' + B C

Al( +4^rr ,ZTh^n- mearl.f

Be ca,,n >tr".- bcfA d- 1; n-pts o,,"a p-'!"'utt *o tW\I

Vd re.rnai"n Otfrwllavr''k o1't a - L;y'lz i

W f g,na,-t c:.^:uvdv*'f t*n 1l' t'<l-01

A rr"rnff rhd,ic^Ie "d'oU;Fy J.lrt.e'4"n G'os-L

1/'.0-{"oa't*t M\-e laurvlrr.,q"aura "+ A C onrt CB

lfcrw<,vaq fu-p- ruo'p+4|v'!-u "t'+an ""J'o $lymu,,M le carut*'wYd,^lAlvr( Udh t'lYtl'ry (lv'e.

bu:t n^tf ,uc*n"^46- *b fou^+ *Yw{*'t '

Yef Lflfd A Vy .==', u?l/r,f (nga'tM) ., A I

Ne twrt d..p*e*w,),.* tl^-{- ve.f,"a\-f (;eU *:1

1$s6ott-f:r*".6'ar={.;7O c^ndnlnt,, f

nu^- Upd,/: g "J"rJ a-k"r-t

' ol uF t va d f :o "tu"g f3 * iui"*

o'nd bw-^O*V w,d;flrn

rRe6.-urr 13f fI-YE

vP =:: V, an ABr - J

Mfj{Z P(o.nu" Sfmw: $f ] ,([*,|tt irnf,rt4ir "*= nl,= , ,p= #

=-i. -7-. f

-T:. z l^ Tl+,,:a m{o,nl 6jt'

),/ o-'''v7' "' ./' ' .*--\ Jx

rrLove 64 r lpic{

B"'t +{";z Sol,,.fic:rt

J-{i{t P*Foio-d ^^ a1-fiemnhveg,jw-tAyan&l 6 lo#nf

';offi'or1-,,^l*irh U.a.e*r*ixl'-L'.y

or1 oB

dyr A C>

'r--04

&;tr-r,"y; t"4riit{ ,J

t"i'w"pvJnt b[o rk sJ

{llrj'r'%,

;\ nof ytntqv-L .L.

*,,,,,('t )o [.rrln'orn {-zqda ta tlnilN{

Sl"rv* /-'uUo ,B,ft,, 6 F'A. ?il_, u

o = = _ K

t:"tc" 9' - o-= -lq 6\ BY/6 = -R(/t-+r ) rn Bc"

ra')c" b', AC'o,- o:-ktn-t\ ){YY ='-k (n+z I6 x x : - k n

$ t1;ttf io[nr.A;-o"n kcnda 't^ th;- sannL

l"tr*t Lo^a p := z^ h- (7t+z I

B*/[- +i^"r- plan'fit- ?crLa- is srno.IArrrz, a,,.d

+h* r,3)d blntl:a c'"tt mov,S 6'f,

{^i./iil +M spmd "+ Pnr^"Jt {'s &t"^ffivl

.1Y

1,,k = -,E 4

',t/p ff vJ

l-tiil NU^td dirf nhmag ",,,{ou.(d sfyzl- t.c C-u:,vg{W r-.tatr- Ttxta

A , i ( s;y^t*r' s:+nos? ,',+v t /, tsm'nqt in <i/s-'-'t-7'r'ltg ) rt so f1e

WI_'"M,@ a-l ftu*lttS n1^r1hln **^_".-b;m thott ,"fu /rzhrL wtt,{t' ll^'llk sotufiw^wfil*!

tv\E3W n / / - r r a

frcvf,JL 5t1,,u/t S Ca,t'

9g Dou,bLe Crack;

4

Tu.zrury tlLs f/^* inds,",lt*

i4 /e,r-*ltrn- ( wrrL 3{"" 6ffr;e* t

arnd tl,'-pt" rcflwcL +h'1, i'nr/t^*-t't

by tluz m')rray' rerLk,taw +tl,a soa,y(t i-t'dt, Lr)e N+A!- aLo"ilL ctrark 6"/,

f*.tivYt

Md s^o1nirmft$J,, lhz nl*rf ioni"^ol< ",, dot' {o"F-q it'ifts soin''fi?rt :{ p{wa mirva:', refe'-*itn }

Rrpinn .oc p

/e\-o' * ''*- -tr,

6 y y : 7 k

6 y y : o

6 : K

6- (fitp g

qf, (r+u kffi:* 7f k

-rh*" &:*"f Cca* dY f: ,?(tttz) L k-

Cnn\fu:tLfid Pri{lr n*iftwa,fft"X i.,,vuii* loetct iS

P n - r + I -_ - - 8 | I a

f,ft- AY1 C

i>lltlS> {,t;€fu wrc}+h zf.d

p n * e L " J eU

ftt* solutian B -I va,h/. jtrv sutr1+*e#h1 deeT notz,te4,)

ME3+L Plarlu Sf-to'^;" lt C*; 6

Other Slip Line Field lSolutions for Notches

L. M. Ka.chanov, Fundamentals of il-'e Theory of plasticity, Dover 2004

' - wr'llllli llllllli|lll]lll *

Ir

JIII

I

Fis. 107. Fis. 108.

fV\EWzi=* l,t )edqe { n d<^-{-art irn-

( kacha^orz

Q<"oa**v, 7 :a6 )

l' il, ea'l; -r,'alrtgYL

toB&- f -+ zoF $.: 7d- tr/_A FC:7o'- |

, : l - ' t

?(Mtg- S'tv,t*n lf Cc)

FIr )

lF

pyoof ,f LlcA= Y;t

L BPA = tPT' )_ r)r7f=fc i lo/ D A E _ a fLEAI ='(-f /

= lSoo- l dF - /AFe=t8oo- '(qoo +0) - (?f- Y )= h g

AB: A C- I'oB: l t

fl 7J , I

Y

re.!vnyla.,>1",'c

h,ylx s+rr,o

((qcAaav Frj t+8 ) ^fu^

tII

IL

l,/rt;3+z

\

plw.L:ia;l*t"* 5 -_rfi&

6tf ' -ZH

4"so = o

dyr"5a ;- 0

6 = - k

ReVrrt fuAEA f {"nra *''v W&'h' 5o

\

\ * - # = 5 : f " ( c > * f f )

o\ \_ * r ,on6.AE, o: -k-1)rY 4 l , a n 6 A D , A , * - k t e k G S )

= : _ k ( i t e - f " t )

Reg;ott B A D'S l "^ [ " e i - -kb-s+( )

3

6trr 7i= 6-k-

o/u/2" 6 loy , :6+k : *Lkc>

(' Let . Len6+h,+ ll7 : 4 , lN4Jwrt- ozr 4g : f

\ * = f

- 2 k t o ' r t )

Toto.l, irt&eza J-o=r {"^"

F

tWd, t, ote*-au.*-rza- & i7"Vfu, /

Awlwuolnv, aNafoFh) = 0"4e^ (A FC 1a/re,^(c1T;: tt@,t---Nyea G* t :Z@e S,Gc)

tle Sht(

vah*ryL

trr AF| \ _ =

, ' r - , . _ =

:,r.(Qno+gt f;n{l/-85

t rT At r- ! ->-ca'e sal0l.gl

f'y.rqo"-v)

,{*ut

fft?!

{,/tt3+Z Flan t- S*raun I"

Lr _ .! a>$l L'

t"t} {

Aq : 1s ,> r (Y -9 )

crY€a.( AFc ) -- * -{ J,rr(y- 8)

AF = (- s',uu- o2c6s y

F B : A B _ A F - l i I , -

C;*,

!cn9 !1tnt(t/-o)a>:o(ey 2 co: y

(]

,85nG0)dny

- oB: FB-c , ty - : - / (q l5 , "U-S) )a

Wiu-.r,h(/-(t))< A D f / - - u L

c F = f E ' [ * t / - . { ( p s \ ( r ; , q ) ) s n /a r l y ) " " ' "

- .{.' Go).f Jn(tr'o)lct',ft -6y

!.' snrl-ol a>c9z c.t>/

5 th ( /-0)

z hatr

co)B _ (*,,y- Jtvl u-U)' s,h/f<>R

J . >

A

(a,11

t . t

2 . O

(zy : 0 r a6i 'b.yLtf -*l )

\ f q(Aa r',N, p.2y7 /

lrl'l^lrl""h,

f : in L;ns ( S ratg-t1.x co:& ) - (cor {3.r - 5in $-t l) i Z. x g f t i ( l ) l 7

' t , , , g , ) ,

3 : I o : o o 2 : l . s 1 ;

1 = [ * l^ rc(@cxt {u , l t ) , 5) i

I

r t

L f (r^d'tIt?/h5uva. ai-,Ydaadw -f"*,

P

hAba;\g frl^r. ntr

er: {(9+l)s,n/

IfL -2&: - 0 + I

Jo' / o00

+-1^,L

Mu3+L V{o,w Si r,r-n'* C*,, {o

a(v,-,o"Lz* pa*avn,

mifrnut d'rna;-'

OBDO, EC

ODE d

_>

-)

u"jrw dtlnt',w$r{l

Wsh,eptfr^J

A E C

A'Dg A

0+ A ,

otA

dnrta,rrtt

t{rt,fe sh-p*,r

It

/ r \J.gJ

/ a l 6 \t l \

J \

, ' 9 \t I r I | , '

A , n , t . ! J J r

Ve3*z Ptarut S*?'*o: f

$'5 iy

' t

t f '

i La4 ttC Pro3w8-i-tod5t, f l6r )

al.t fU6"-r"* 0, 0, 9. @

arrc in u,vnfm sJrgry5

lr-nw 7ft,08, 0O, oP,-gw

Qen+,,*ac^1'. rehkuYrt,

Y=T-* ft* p**r)Y'= T r ;{,

\}*l''^'' /

Y+ Y'= T: . LAop = lBoC= $

Ac,rutl hnZ ,f JXtUr*'*r&4-, e'9- oB

6ttf 6.yrrt' s/''/{ 'lrwnt 1'bL cnrvlirtuowv l r r . u / , I

D / " { i

^l^k (fyyt ca,n k- /4c.o,v4irww4, so F -o7,-b"i'e ol':r'tti+*^rva ' ' / J

I- 1 O n t t r t - t ' , - - t - lgO 4**z k be on MoA,a atoLut,

I t | ^ : , ' r i ' ' t Io,f rardtu ,k . io t> f i

J ' - ' r

4t the Jo+,< hnt ,

617; "9_* : \ - ) (

__,6rtry,r: _ u?r,,-- k

oo(

l

tVd,

I

I

R-ry"x'{o*7*,

- ka

-\

[ , r

oy" Y,-r$ |=v \/t,l,t

iI

T/It a^qls. d' rotnRIYL h mx.Nl-trl,Lm sh%'/ rt^wffiwtA a { \ I , , f i \ ^ r l J - } !

kerrroe- f.h-e

- ^ ' d d l , A \ . \ r t t i

0u , 0'a 'T,trrrt+

re gfact rh ̂ Iff*qu *a ;o,,rPo$h- it' l',fu- t'

iI t ' u

, , :

I

r- -LCl:-ilAt3(L

lr*f 4 6ar""d'iL6!, re(a,fint ".

Wr,)n(yrr)+$=xn

'iI

z y + d = f

Y=+-+.r . T rl t d = t ,4r

t = u

Et

(g^,)r [{ -v'->r" t 6+y,,= [+

rI

J i l f

iva

AB ^A C?U6=f f i , .UL=rgr ,7

1',I

,1frnt

!-t'

+\

7 l

\

c;!ty,r =

Altgr.:-

6i,y" --k-ksa":.6

6xo?', : ,Q c",zr6

F = - k - zbs , , nE

eV+vn@ soc

f l ,rt, = -Zk

6Oc^, =O

6{ y , =o

o --*4

-,'\K22-{a-n tU

"JPnJe'

' t \I

?(*,Y S*a'r': 'I"

I Q, o0h4/ 4

\ f f i ' ' f#

yof6liurl ang.tt -f*r" !' t" x" ot

Mrtl "onctu- z6ty) = T+ S

' k- k s,>6k asE

Q i - : 6 + k - - 2 P s r t " f

6 y v :

: /&. i r+J,n I "t6x:c : 6-+ lr,. : 2t:s;56jy = 7- H = -zk: ( l -9,h5 s - -7-

I L

3\o tt,-t

Y'/

?=tu(T-v ' t=W,=#{*=t##.=r#,:]#

?-_.3!-_!4r 'l;lfJ

tvlE3p L'..rtif Anoriysr j Grr'.J

Tlu- sot^r.no-rr" h tks rrjid -plaa:n*c rnod-uL Cal^ be trard t^

C,oM+n^,rt Yf/ nrnd t^ww boi^m{4 f* t\L had rru q{,{04+ir- PWftc,ttj l:{a,>hc_ rq?.rt?ri0.(, feee-{^;19 a C/ri},.col 5+aftt.

5 l i

( f u a 3 v v - F ( o d g e . $ 3 3 , P . 1 t 3 )

ASlr-r..c.- "f*jr-* -ko*rN.'tq^r,i TJ ({) inprra,tp4 mnnofor,j C^\ trrrh {r}rre bJ

arvt atiptr o,JJ nr,r.l,/-.''pl,^ c^.i.W ornlta,,',.d: , A Sfu&. { ,fugrld^tf p(orf,tif "^

hlitf €iiQr^*\{al4 be rtq.u,wc\ , cn l.-q^a- p(orh\ rtrat- rhc,,rep,ag a/,thor,ntJ

inc.{eaM$ 5,1, I f r.e. }zirreafyivted pl,r,mt ft"*).

@tMnq 4 stn,aa Ya,u" af ,i^yetd^,,'g p(^n+'t f t{,", . Sj :g alw1h,h^,-e

thrb ca^ be Ptb.Jed uz!;nq Drcukrzr ineq(ol,ifY Pos+, {h&' J J -

6t' &,f 2" ('b, be d.sc^weJ (rra.'v >J t

i;l**ve'l^'d f* botfu up",r- Aw"d'r.'n3 a,,rJ ?t4",h\ .la+-c chtttt

Tke- Sjo.trl.wUnt 6i,':o iS rund-rd b ps-le fi{ +/.r0 tkln-ur\a b&Lvrll.

Looe^r Bo, ''d Tt',4!.-arrr

A*j s+*L'c^tly adr..s:rblZ (s4rw f.rptJ ) -toL^fibr4 nr5,,,r-o,k a c.

Le'. )t/y baw,nd "f filw (,oqd od- .inc,r9 i%"t pt"th'"tf,2

A srur"t^Q ^d,r."i sx bl-s s.|nA fl r tJ r, i s ̂ t *4, fl.tL {"thrdnq c,snd^-+,-u.rra ..-

0 eqwi,hbriw" c,cnd^*+.-rn eJvvlw{-RlLc . gr,j^,,. t Fj : o

@ bom'd^n1 cxn&i{.-on , A5 hr. : T).

/osru,*-n- -"f tu tgu of B,C.: tf 4 T1 o1,--.{.ed )

\ aN\ J S,^ w#,t/LL t,t; :9 /.t th'e(d fneq*o.l^\ tAe"jt"'4'rnr .

(o.^ _ 6yyl+ +oi, _+k € o

S*qc*- T;Cr.l o'n 5r is tpea$'ed uq to a.n o.ra*U u>nztv.rvf , th-u larrqezt :J n - , I t

. | ^ , , J - ' . ' , IcoY1a."k f-n ",e^e ^ :+-o*'c*\ ad-rnrp>-;'bLz Sf-,*ry> ft ld Cj c*n" s].;ll

be{--"4 i5 ;+r'(( a L&^,Q/r t'"-"J -fo #ul conS+or,* + ,fd pbuLc ft,u"r

vfi+zi

i " - * l { | " * i f \ulrv\r h.h,l"v,twu) : (na " . - - - W

t l - n i * l

WW boutytd tf.!fi-Lwt f

lcircno"h*tb ^C'a,isi,;bl& fujnqfrtd I So{p.hort- /rrkl,ra AA

b**"td ,f +lu- htad o,t thr"'ltlvnt p(affi

A !4Ln4At;ceUA/ a**W;uk "ehrt'l;A fyT *fr S"+nfnt

A, f-U^try os,\U;|fibtuS ' ,4fr--sr

O i/to{r"fTvb;Mh ,,,1,, n ui,l = o 4ry"t{"t""e,

rl t/'t"V*i"f' l^^* fr"b- s^,bre.lrvna

ry^fuJ r"V /u'n-,,t,"J"*^-,,^yt (^tr^r- 4 tuyfill,u!""% Ctvf" b-a cla)annnv,r"tl.

@ O^,^-ru ar?d{+}lt? , Vl: o Nt g^ tuL'.-e- tr * e

O pdgtl+ve N&k\-^&, [, !.(zr2

urrcxt JS > o

T/LL ov€z-aU "",h/^'c.GJ11,Q

t.o"tzft-vf 11 TA)

a o61b)nel /^-

t/,a C{yd^-.+ib.' ( t, A'{vt ^ ^,flM bts^d ;

ry^fw

. fN = Jsr

r f _

Tu,&,4s : lr l (^t; ; l ;L ' N

J

V

dv + [,'*'ol

^tJ : i

i \ , * ,nL*,1 'h"ff*.'dU^.:c.',+t^^'7 ,N}"rrb

if :1 ("{ .c,)

Y;fr p*r'J"aA o' P{rt;lrt-L scona+io ( (-*!^"rr-(t\ra7ftaf ptwr*'"ft,r-)

74,e- ̂*rrr- equa{it* g|'ez .51,.t* (LQ- uouk d/v'!- bt @ Ofler-*{-

h^d m,uvf bz s^.fuv^* 6 ,b^!nnue- /<9 rhfut-rr^r' d-[7t;f^th6>L

+ +AL {^,r.t-,t-y ga.a,ztt)o

Mtb fh^* 4 +r^4..e- So()^hv,t to tla e{a,ttty -/arnz- noleyf?^l dl-

iaaVft",'t pldnhztF, 4d.l N sff?t)t ft'/d rLa* a :n*ut.f/,yyod-r,n.e*rb@ ;r( a' ,)erlno*,y r*td +l'^* A khum;h'a|L9

nC rn&S;bLi- , S;rr-+l*"+o,n17

ME3+L .l,.'*;i- Ano..!ur,,>

9 z Prr'"c^ Il{-e- of virl*..d ,^rm/t ( hqgw - tio*e", .[sz p.b? )

3C*,

Ib drrrurad-r'or" ,f t{,r!, /^IuW aunl ,ryyw bow.,d f^ts,oarr4 ,V**n -tL&

f)yltclni- ,l vV{r,^'n[ v'Nr]K, ̂+"'uh. h jtaf,ed tLa're ( in zD plwru jh.?*^)/

<^.ey^@ ute- a+e nnt t,,ry ft fu*",e-+lre hwV,y and atrty+r bunl

{Lwt-p.m'1.

6Yr,q&-,- 6, Stri/A $^eH 6xX, 6yl, Ary^/t4 ^, nLhuh, 6.eU \/x. rlt/t fe?"v" R {ho* o,tts- ccn*iruwota a,nd l,orrt-

d,,A; n;u-t^ /-W asa*vova+nJ

Tlp sfrvys fiu ld a..nd thy "tJa'",W, fUtl ca,t bLcJ'ne-r :!9e,"fu-*? "f ea* art^n

k4hn4- " tyac,t*m f"-t " N, bn*rd.*^n-- l- = 6r- o^,

Sb"i4 rdfiL - c,, : tCv;; + u1,; )

PrinqyLl- "f vir+n^l' '^tu".,k..

f , *6*2 '^ . - r dv l tyv* 2 t r ry&y) d{ - SrGV.r t Ty" t ) dS

q4 tonj o-) a

su,h}f'ea elutAby}r'* crn$a1.rxa r)n R

MtW C-.cn*'d*zt, K <",rr1^i-rv.- 4^>\-t *J ,1,$c^-,-tn*1*-

tAa+ oL;-r,i,a ,4' ,r^rl-',, ̂ {-t';C' i^'"1* "J staYyl-w) , O

R,,, Rr,, -" , Rn. fi^'.)^tAct Ok2, fVy, O-xJAJL+ C'i>\$'^a.o't^.t a.-vrd Awv<- c..o-trlfa,ryLJvr2

t* clfuv+",Att+r+zl

itr aae'L Stt'href'a'vn.

Wo ^nr. ^-I[r"u +o,Vr-.av-L fuub b b+ /4ecnn'axary

"tf !^j\t- + {i\,w"4,pu^9";.

PnrrqFs oft - r , , - r l l :! f

*^ Lxt 6yyr^r1tzt r*y) lA.. I (eqtTtvy) lS +Zl7,ro,(r l_vf>d(t tR ^ D , L h k

*> I,cng a-\ -) S^**f,Vl eqW'0;bWwu c,{r\dj+tuh rn R

lchRr v\ Shs"r ;h+2.' 'fro^urnyttd o.crrvra Ltp- fr* na to Rt

Cr+- V/) i.s thz dDcon*irw^? * +"ye^*,'a..l uu{*crl1-o,<.r-!pi LL,q

Ivr.b

Rp

tt"rF

\ , - o

\K1 \

h / 1 t r ? / n r . \ ' At:iL F-l- L/frht Attwi,y+r* L ^ /

f a . l !b J Nofched Dar in B"l:!:t

Fnd M a*' rnf er<i;ag plw>*'c+-6 i

lqux"t bo\" d g5fi'm'f,*

^. s+-* c"-Q-&X. alr,r,adl*Sta"*> ffutri

o*^= -zA

6ir. = zRo

fx' = o

* r r< ao < x < +

- Q ( x c o

lil, 6il't J- s't'rUs osryon;,'d: *ur,\

( t a ' i - o l i ) ' + + r " / - 4 R ' < o )

l v 1 = : z ' z k + . * . z v

U?pw bou.rnd e.a+r-rnof-a.

C-o-tr,l+wtcf, a. k rqlmrJr rr.lJy

a+y flerd /

lsgu/yttL a " m p e[4aq'5rn / )A #r\r{.

*: ^ v lg id se3nd/\rh rotote o,u-r-w,n{

a r\^^9e

fabuf extu .n^ l -u t te j< , W:2M t "

vn*;o-t e?'fu irl'y^ a:l--'^1e^frr.cl a q l : r w 2 # n

' ( r s i n q = e

"O^-Xb[

7 )4\ of ,lncx'n+r**ry Ll?rQ- . Z LX,.f - #+t"+"-0 r'A-Ftrr.n0.I* d&tipai-rTn :

u == Rffi) ," #* zb =zka'b6#

Mre j47 DrYit Ans.1a|,.:r,t L q,4

M - #;. lz. a'b

ob+o;,* th-t qh+*f rf flr- brcynrt.ch$s}& dx thoi rrvin\hsqa

ft

lhi,t $ = l38ol ^.t c**116o <d<lr )'h o{

;+*J)h 6 \ '

: / . J t karb

{3flto.

t. l8o I

-b

l,a/e

of =i.1555

f = int * ( 'x .7surxt.^r ' , , ; lXo=fmlrpov.A(t , r ) if rxot I

_li l1/- LMrxl: ::Lktb

fkt1fu, t/,A- ",,"rwut U a* fuytc^d.hf pl^+z fM,l,o ka'b < M

t\AEH-2 Lrruf ftnat,ysis lat _5_

,l,e

r**"t F rnovQ. art ^.

t;X^J b to et<

\lelpdfiA 1^rr.1.,s,'-.;f';l^\^-e-

f = t.,L'nc('z*srn(zxx) -stu>b,]

{ z e r o ( € , o . t ) )

q *L -

r -T -

F

atyswmy a. ryid bLoe.K shd"'y "ff. r . f i

f-oJe- of {;r4tr'ng.{-v$sul< . W = f . V.F

tnry*t ot '^rh*$ j*T , lovrl-- vtPrY{-h of Jlir"*="n"^'fu A-"t I i a' E

, f. {9{ k'tevurrr-c-Q-' rid}9pnt'rrn,, {D = " k V . 3 a E

r l

\ / : f , r J . " * = k . \ r . . 3 o . J T = D

| = 6 A q

frnd F ^f irrtfr2nJinq ilaM/ J ' J

/ t lbwe.r boun d, ez {;l.nofz

{rtfiA{rr^}"t A

Sh%. Nd

feca.{l &-r- gr-rneAv co>ta/..Ji-vv

AB t - s i n ITD =

i+rnT

AB =2(za- * f l

c D : + q

Lh ( t -e;n6)

+ a . L - 3 ( t - s , n 3 ) k q- 5 -06 kq

\Lrtnetr boutnd uinbfu.I t

r-'on tfruct a, /<ila'rn^'h''r. 0{y. "d/'{t5"11+v"hcf futJ

a iI

-rA-rfu. tLr- f **a-t* rnpaa&;y frnr*z *N . f.o6 kq < F < 6 A^

IIII

i

i Lq^

|.rt 'I

v FI t(J"f.lg on Cen nscvrutk

|n-* p"?e)

MFgr lrnu t Aneii,t1i,;

a- t,TWrrr ryPw pc,rord cqn'' bg ofutp"i-nrayAn r,q C'f ;hp {;ne Srl^d lrn

v -

L0 3s.'(i-{h^,, C sp i r *15

5+t{/6fvf,J, 6rr:o ".4 f-G.

6yr =- z k- l,* * G( fC zo.

coo = zk + LL!^ +

wJ*"+V f*td.

Th-c- ar"tr+z"4 ve/rc"t%v "J+Lrl- r;X"d veg'lstu5

sl?eaflu borr"da'y o6nd"7.6n) f*lz'",. fuq Sp i ra,L A B

a'rtol fAY

ir,'a *V spiua-t A'B

( \ry^'f,)^l uo{nuYU csv bL d464h\^^^.o1^4oLf, d'q- "Td

-p/asnv b""rnW)

Luh-r1: du6a-' l /pdf :o ^1-"? 4-L"n+dup : *VndP :o ^ r y

f -C " ro

fh* e^*rv*- /<laot7 PEA W, vy can bz aryulrwvfed-f-, ,Tn ,4 13 A/ ( I * e.(+ya>L )

Ag a,td fllt9 a/t Awa ,t' dna*',w"y

TIA tLf l|,l2 rrt,rrrat fVd ; k"t't''no,hujb ad*W;bl*be f f i - o , * r y l r ; l l r yw

fl thaLllvte,- vle bo*<nA^ft "ilw,t 4=o

0 ( r , v i d J > oJB . / r

l 1 l v

I

7

ME 3+L hrrut Anat,yua CuTo *"f-Lu t{u d,wpadt>n ra.ts-i f"trt,rz ve}na$ {-,utl.,

Ne M!, tl,-o prnot/* vf ,rW'fu,.r,L u$t{, w;iln t/te- sLp kl,t!*

3

Sf,t<,ts /orc t^hfury efu;\Lrot* an.l B of,

{ ( f f f / L B = 4 B A / + C B C ,

o<f,p-np 5=1-n".

fj a tfu skp k, f;et,t ),LA\n

lrec*'t* Ai ud;tfnr ?fht&hn"/,^

= 3 !^Z 'kq :, S-S;.f ka, (^1fry batnJ )

5.o6 ka < trr

( nal,Jfer' LY/W brnd eh'*, l.j-/e (6 ka )

4i ̂ rt "f A^ravu^ry 51 :AE t A,g+ cE+ r /g

ota + lo lo'l,l ds J

c t a , , ^ < d ) \ t r r r t t y v l ) c r . )

ltry h f,!^tuL ------>B'n^J6't'

o',bs-t/+ r- Z V I r u " l S) 1 "ks

t I C .I= 2V ) Tv d5

G Z

D L

= { v I 6ay '15PE

= + v l 6 o o d JvB

: f k ' v ' I r I ^ I 1 ' ^A l c "

= ? k w . ( z ^ ! " 2 1

/\4r}rtt

r t t . t

a

HL.xdyvtr:4 Louut$ ,, CAiJ

t*t; n-c-Ld +" bu s^J{"^e,-"!r

h {at" tl.;q- },-a"ixl.. ^-^*',ir*roo{. ela',,+;t--penptt ptufir. tnndufa ,h uu{*;,,,'r.

'rnu yhl& ;*'fi:i" Tt/rnnrr* u^cA*ny'J, a'g . n q shx;^ h a,n do,,rg .

\de *t-l tX"",ro,j,^rU uon^ 6p1, *p t" tn d-Al Straj-. hw-* d*r,Vy ,f,,}o

Siq- C^'1"/ytitrr*:utl" eq,.t*tnrn,

bt Tlornc Inst* i i {*a h kzn,tnr ' (R,Hit{ ,p-rr)

. >k:lt- tiorulo"t6'rn odr4 t

- 6- Kh- Pt^r"t'c

We, wi[\ srhm.r't th^f, ^ pryq{$ p(ar"flc rno*en,iv.-l* ( no hand'-vuny;

i S u,n Sta.ibL in *arnansn .t | . e. A€ oflg wi tl.. rwrrru-d"-u.i-v]

o#* fetdtg.

, s*far}^ honfu"rUnrl

*PJ-", -fu..tSVmS {..v st",bt{ft / ,'. t . {c prerrxrnx- vre uk-\ .

MoSt dAc{*L.g- vqu|-e.nnv'{zl to crdgd m {+znz4rr'1 €'v9'^'b''{^L-li^/-trAl

t ne vk-ig , d/^-a- -{rc vio la*irrn of Stb-hh|ft c,-'l-t ,A&

C r'. ( - rnS*ff-'u e""f hondrr*lf ) .

Larl ft. , )r h +lno- cYfl4 r+-.'A.-art^!- o'/rtA a'-,. d A^t* * nt"

ba^r pri'w 1;l &fuurna.tr-o-rt (i.*- w'pt *'nc* sf"/*';.

LVf A, L bS thr- v'nfruq ovt C^ ivLt'Z^?t- ylo"*e t L+fnl- "'l't'\t^w1 )

MelV**- +rlarrtt 9fra,^rvr , becarvr'Q- plw*-c strc''vt't cs nk'uvel ny'u*''r'

hlg fr-awq- A , I == A, .1.

d G' l) == A J, i- + ),dA : o

#-.=-+

ME3+L Hara.e,,r*1t ^| t u r r \ i

n >e U

I a l -(\,> " -* \ /

L o*" L&fw, t . €ztlA'inp"pz,fnry S+ra./-/L

r t f I _

* . +.rt-s- St-at'-:

JNz )dtT )

s - !-L-- --T ,:

l- !.u+ t)

(^ * , , . . f l=o , € :\ , ' )c t=)1" , f=

S'{"tfbfrd Ye\lwYw lrur &w (a't^l F *

)nOr.eoant-t ,atfth- Pr\fi/e. d/,,

et= !*.+ :,A"(t+g)

- t f t : - r c l) - l

I / € t : ' hZ /

F ()n &* c,rurs Y-,Axw; ? d F, ' - ' f r > ' o

lf d/'A A n,,of 'tr1a2 cat>e .,

//J7e,( d^r- {-r.- WVr)/

€ - ---*q----A- @ - r,' <-!-- ' - t .- ' r

fi e-{l!-+ F-

F:--f:--l3ra F

I ninz ^ u irfua!- ,LJn*^"tm "(futti- PNf A

?\,e. d-L'7 <o. ,/4@ = -{pQ > o

^ l CS<^pfose- f f .o-, i l -sm F,>f, Fr<F

l/A*,L poft^t 4 ^ttL^l

MJlebk-..

il\evt W+v* k tA, k-J+.

ohe. 9,041l.-* caww tf{wl@

b6' S$"ar,h-laro,.l"z+Nvn

/nt^XzXa tarc SP-p4;r'^d "t'{rht ro"{.

mdw blL

F : A - o -

otF : d (A'c-) : Adtr'-1' o-d A

: 11 (o{r+ * nl _ A (d6-- * -;

dF = A (do- r# fu ; : Ad t (+ f

o l F - A ( d , r - o - d i l t ) : A d t t ( S

o t l L d € d t iT=W)=q. l

-(I-l i^ luns ol II+t:- ) nX"n ur;(5 ,

t r ) , ,n **1+n^{ st}?,v-

*8, fa

MFI+L

St'*t{,%

vtTl"'su

Ht-tdr,no;, ,-uMlS

g6lirv^ nrrU S ,",

'U'$f , o,

dtra"^, " {-*;r\ "{ alnt'ui Cd^afud t,1q;

6- o-* cx' |+ t )tt €,5fuaTAr Av.u p"tr ry ( t'=-r, 6-=. o)

fTw"lkta + c(.

"Jl fh4tt'u u'/w{A

SrffifU thL c&dtrhl-ii-

# - o ( : f zrtt tfu- irtv,lr-oh]r)4" pfiXl- lon^,te+n+ht tl&i SJrai^ aarwe" rCt'.) aord

*:l d"-+'-\d -!'+Y,s. tf $F trt*9fu/Y) g,h6"; uNv\/*- ia Lwy*n' tfu'.t

gr"f+ 'ftL d",AAd AYq, ahta

tle- glv,t a gh"tl-e

O= 4I o,s coJJed srat hon/i--*,., q-6r,o{L

(rc"te the p,nt 0:-l tZ qs'Pvwkt*\ t

t t I

A( 80t'viWJt'r6uyL 0

Tl,D c^tnd,i&un 4^4 ^ grTh ,'cJ rp.Pr-e'aarta-*ivnt'r/rr4fdt-c"

J -,

CrA'

r-e dl ,od l f

al

EAWA''W "+,"ftg

\

d*r, ," t^^dL, o{ c"r*ertQ4J/^{"d UY-{-)

tl:-

[ = d e € *

Crecolr( L+-- lncr+t , l )

{egarrdl,W ut ^, ilv\ t\t*U'tnl"eA SMtl fhr- cd'd^"1itrn

#; = ue*: o-

At tt^s in{ezk,fitrt Po''lrl|,

t tF + +h" -!'|'ed4-stra.h ,UN,he- i t l6nA//v tht/* lJyr-o4 +* ,rlo,rtl.^zd c'wv&,,

I{* ,pot\rft,Z J+abq,

./'.// - /- 4 - g

Ite

t t : -rc )

rAESq.L flandau,,"E ra"p$ (-^'

tywr,{^r nko,"rdl-r.;na', yetyvttitt\ {.:r. >iaeia,'t";, r^n *aznr,i{n ?

4^1#.9Wt+r\*

i:-* =-:/!6! - e*Gvgla,r+''*r-- €f = /ott 1,4 fa

q

l'fuurf

(MPa)

/oo

t ( a / 9

E = Z o " 6 f a

S't'min ,4 6 '6V. 6'rE t = Y : 0 . 9 x t i } . o o f /

Ey =o ^r tl,w" |c*,Lc

Tfu requ;e ho'ad'un,u,1rc'fu, a Os * 6, = /<:a plfc\ .

Th izmp,a - , v j ̂ t f - k1 , 6> / . /Cy

1'.(. *

tztr iAup-M-L *\ fi"u^tt* {* (a% plar+1 54la;

Tha 'urylar;nt ,U nal+uJ nr^/U. tr*ft/ d'4- l4'^th s+o'nli$

ei7<n;aw " ,|',Xtn'c*n sh(( be bNX/ Y?tr4tllryt+d bV +h*

t/r'Wr.- y,x4.**0, p(o,r4+u r,',n c/,tJ,

IAr:rz P{ondz*t,* Lau;t0

I / A t 'v v w J 5

f Z Jso {rrttst'c Had**;rr.1y ttl,a &a"tr.-__-----tr

r/z-u 1e'*e""{* ;;{reaa.54r^*." otlrLA,e i3 n + "s*f/'A'et,'t h fllf J^e-un;k-

th-+ plua:c h>dwin'. Ng ,n*J- & pa** hnil th* z''-t*p- TvtJ'@

"A"'ry". ,r/rttv y{a't'l+t ok t'-;rl ana^',|€;m

TA-.e- is.h"P;c. Aandn ';nft- -'^/-t-r!' a-,o( +fu-I

A/YV t^ft' caznn'sJh u"ttd apTnXrw*hu"e

4 rea-L rrrJt u^LLl

74 DeJrTr,)^w han-tU"-,Y m'ed'e''|, u'/e r"tr'fu' fh!- a7't^eft c<rnd^'{'rrn M

f f r ,1 ' ; * f ( t l &"oha"nsv P'&o)

Nf^-r/L!- ,y ia, +ha d-e,utorzfnx-c p*-t 4 ihL st\<,v! T' f Si: or--af1.,')

, (s,nrt-- ,x..,v(d s),w^td rwt- ,t1u,rrd.cr-y. @s*L s*eot 6 )

t > o A -9 urzue- mt o'+.wLz- 4

iJdfnl>tY h o"t-oLe'^.:a3 ,

A c-rr'ntn,zc;-t "r"s^^"*,J, of ,;L h fui- w<rre .1 plartt;c ^ryrr^^,+;:cvL"

L: I oy d €, jL

A*+t^ln , LUny fi"1"-,rf , rYr4t-A)-tttt4-, ,\ C)tq^*<*ftna *

aCc,wrrvv.al-r;tad

plo*4eu lhfte.a.^n

l+'nznt-t-L h a ta+rt'1'12,r"o/4

h, VhL c--r,wlta b<.lvwrva,/

L: IttqTfWa r*r. fAft raa.u th.^- rfrn r\4^-:.t,l czvrNfg,rxrsru t"' fv"a.-r/+sa f

fu^5> :

(rL* S y c.,, dl c*,-a- ,t y*-l*- y:tw*z+u6. ( no ^ ̂ o6r,t,,! )\ tt? 11 = k-- ( cnr'wt-a'""t )

^)

Fq h"l,"tUttrry yrro Ja-L, 4te- c.vrt cA."*X- f (L ) to feprcJ"ucZ

4ro 4+tSl* S6e'ts-S1va>n otaav€.

* s,',- t,.,.r _ J Jt,

cxwAyond.+

VE7+uk o'-t ora.mfa t

w,al f CtL t

a'".& (t*'> finl

Ho,nrl-w*r,-ry Laa

l * . r { cs , . i t : tS j t , j , ? .=

: k-+ h 1,.

ou-f h",A 1i;hA tOvt*+b S'h-eA'-5+rv,rv,h c-ururL Ut"Y4 kk!

Lo.1

f r i

J o y l l j -

r-)

h rtnnoX;o-! det"t;r* |Ircx ) ,7, 6yy - 6yy:a , f: * O^,

S^x == qr-G : i o^r, Syy = -

tt*", )'tt : -t6yy

l (^t t = t ty rJ,= + ( to** l t rn}+ to*7) = *d*

flna+rc fkN 'rr^l-(,

' , i l . P t - A* ' ) z) t

h-anc/'ttu,rg { cu,r/ .J ( , ,

f( I'J') = i t;b : <le) = k"+ p L

*u i - k '+ PLL ^

i 6 u J 6 1 1 = F . l Z :

J o^^ = i; * 6xx 4.*

olfux 3 n +fI = ^|H *dg} ._ t_ 3 Hana*'-aa1t0iz-,

@ = ?P, J rTxt - -d,6u t r f f i '= En g, , -= &^

*r = ( e r f t f

No@ th^* *Ajtn hand.p'nir7;-i4 1tesa,vf, X cr-*, be dbtttn"-tutJ {rt4A*/at wa'zi n*' fvWULc,n pA'f<^,4{,rfpl**- mo+4+,u-"*l .

&, h"d tha A-,/ua6> n,&. A ,,vuroq;>-l Lr*^ ,L Z - I,F${f) t 't 'vtJ ta*eal'

ctrJ

c. fLc 4 i

*j

^ (1"- \ 6", .d

l )

d,l == * tl

r 3: ) sL fi,tikdr = i; ! a,i* a.r

d't d tl = h: 6*x d,*

B ̂ - . +, z/\ )o"j, a*

f\Aev(ft Lland-t-rrrtv,1. L

93 h'lu, ^Jq H4"d43\^.,L_ftto"lsL

CTro,^l"nm"!-

f c ti - *i)

oii =

l t\ F I

tfxx

d & x=,'T'*r =o l5d

z .J -

catuLzlTt-*vt-gL t f Ao fiv,p1- c h a,c aar^'ry &,f,'7' rP/u?/1tr1 l^o o,d^1"4K - | J d -

W(A ftflw, in ra/?/$t lrvtd;turotao iacne.a&.J

a

rw Baws..t*y 4fu

hn,Ldo.-"ttn rrn Ka tAon al )

I L= R .

. t . d' ' j

L gt^ {_ln t ott- .. ul arn,1{'.e,,nce in,,-ru oxt"r^d]'tmn''\.

Sxr: ? no, \f =-\ t*" , h+ - - ! {^n'

d f{t=- } sr lt

aq ' i=od l =-Ldt* '

r#= [J* '= - t#xipa= c t* , dyy = - ! c fn l : , d*r : - I r_ *

tt t '-cri ) = i ( io-" -, *1)"*(- i o* +;u!)i) '+ (- s6rn+i,at"; ]\ / - 3 ^ o P L \ a r n a .=

5 \ , + o C - f . ' x x ) - r K

= 'F k + !, t{*

Hndanirj ratr-. P = ffi,

= 3 r-fE )

ME34z llo"','rt"^,y L*IIJ C*'NY*nu- "f P;as,rnnth'c fLo.?-obli^f nolul- ra fUJervtt- k,ada'al ,

8

tBa^s"a."l^?zv .lW

,t e.suY!t, ,6. c,^t""btn^*r'frn ,f i>o}ruyiu a^nd h"r,vnlotln! ha,"rrf*nrg- rlroU.rL i

cllr\ b€- ok/A\td , u3

f , t ; - x i ) : H l + p ?x,J = . r;", l: I o; ,t i t

Q-,- WWK o,r.li- t+S oa^/te-f)$^r4- r'^ un"'crrto.l ft^ru!\n^ h€,b''t

Me%\ l1rur{o*'lto1, L^s/ ,9t ASiouvfrz,{ fU"^t 1g{_-fhe,

ft- rt*[.c- tw-l''a^'f- fu-unttt>"'ift'' p L +t;. : i rii

r,4 d/n Ai6OO,-,nneJ f-t""f L^''r (a)t+\". +Vrz- u'ur, Ma'rza ,6-t'e\d gf,^thh\r,-

\r*,, == lti t) : 9( t ):

, 0n t4rL *lfgi, h.and, it t, nnt a9roc.r"urftilr€- dith th-r l

frtacrr getd annd,, {i-r''/\ lq - Ot l = q rZ ) , 6, >.6t >. 6t i; r

l

| :-To 'd t^tP\"rt rh m''enn,l ., We 1h"X.l intvod.r't"c-a ^ f-^er>gfti)ls1^<.h +hak ttrt- fu* ywLz- 6.a'n b'€- ovnlk'- ^4 (plw>n ftk yh>l )

6t : h, i#,

^derJ- lL t4 a srala'v .L'|\ A/h-',v' :

', I+ g --

+_ , +hzn tlw f h','v rui-a A a44 a c,;a'hv{.^

\", ^r ,^ r r ( w;+h thl- Te lJ s^n da-{'-}^ )/ ty th* 1! t,uttr*"fuu -

4d;cfhw2 A fu^V nownJ Io yll s"-l=^* i

f-w ay uvwyLt-, f* tlrt vv'4 MrW'l a/'eld cad+t-;urr ' ( - t t'j S)l ' r - a -

tl,,,L x+t o azuted f ln-J Yw(z tz

: p L , . E f t ? / ' t . It , f t ' T q : \ ; t , 1 t r 5 ; s l ; 1 ) : h ' 5 ; ;J J

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18

6/4/13 12:38 PM D:\My Documents\Courses\ME342-Inelasti...\fcc_slip_system_map.m 1 of 3

% FCC slip systems for single crystal under uniaxial tension% plane slip vectorslip_system = [ 1 1 1 1 -1 0 1 1 1 1 0 -1 1 1 1 0 1 -1 1 1 -1 1 -1 0 1 1 -1 1 0 1 1 1 -1 0 1 1 1 -1 1 1 1 0 1 -1 1 1 0 -1 1 -1 1 0 -1 -1 -1 1 1 -1 -1 0 -1 1 1 -1 0 -1 -1 1 1 0 1 -1 ];N_slip_system = length(slip_system(:,1)); %domain_type = 1; % a simple parameterization covering several trianglesdomain_type = 2; % parameterization of the standard triangle % G. Y. Chin and W. L. Mammel, % Computer Solutions of the Taylor Analysis for Axisymmetric Flow % Trans. Metall. Soc. AIME 239, 1400-1405 (1967). if domain_type == 1 theta = linspace(1e-4,1-1e-4,50)*acos(1/sqrt(3)); phi = linspace(-0.2+1e-4,1-1e-4,50)*(pi/2);elseif domain_type == 2 theta = linspace(1e-4,1-1e-4,50)*(pi/4); phi = linspace(1e-4,1-1e-4,50)*(pi/4);else disp('unknown domain_type');end X = zeros(length(theta), length(phi)); Y = X; x = X; y = X; z = X;maxS = X; maxSind = X;secondS = X; secondSind = X; T_select = [ 0 0 1 1 0 1 1 1 1 0 1 1 1 1 2 2 1 3 ]; for ai = 1:length(theta), for bi = 1:length(phi), if domain_type == 1 x(ai,bi) = sin(theta(ai))*cos(phi(bi)); y(ai,bi) = sin(theta(ai))*sin(phi(bi)); z(ai,bi) = cos(theta(ai));

6/4/13 12:38 PM D:\My Documents\Courses\ME342-Inelasti...\fcc_slip_system_map.m 2 of 3

elseif domain_type == 2 factor = atan(sin(theta(ai)))/(pi/4); x(ai,bi) = sin(theta(ai))*cos(phi(bi)*factor); y(ai,bi) = sin(phi(bi)*factor); z(ai,bi) = cos(theta(ai))*cos(phi(bi)*factor); end X(ai,bi) = 2*x(ai,bi)/(1+z(ai,bi)); Y(ai,bi) = 2*y(ai,bi)/(1+z(ai,bi)); T = [x(ai,bi) y(ai,bi) z(ai,bi)]; S = zeros(N_slip_system,1); for i = 1:N_slip_system n = slip_system(i,1:3); b = slip_system(i,4:6); S(i) = abs(dot(T,n)/norm(T)/norm(n) * dot(T,b)/norm(T)/norm(b)); end [sortedS ind] = sort(S,1,'descend'); maxS(ai,bi) = sortedS(1); maxSind(ai,bi) = ind(1); secondS(ai,bi) = sortedS(2); secondSind(ai,bi) = ind(2); endend X_select = zeros(length(T_select(:,1)),1);Y_select = X_select;maxS_select = X_select;maxSind_select = X_select;secondS_select = X_select;secondSind_select = X_select; for si=1:length(X_select), T = T_select(si,:); T = T / norm(T); X_select(si) = 2*T(1)/(1+T(3)); Y_select(si) = 2*T(2)/(1+T(3)); S = zeros(N_slip_system,1); for i = 1:N_slip_system, n = slip_system(i,1:3); b = slip_system(i,4:6); S(i) = abs(dot(T,n)/norm(T)/norm(n) * dot(T,b)/norm(T)/norm(b)); end [sortedS ind] = sort(S,1,'descend'); maxS_select(si) = sortedS(1); maxSind_select(si) = ind(1); secondS_select(si) = sortedS(2); secondSind_select(si) = ind(2);end %plot resultsfigure(1);contourf(X,Y,maxS,20);hold onplot3(X_select,Y_select,ones(size(X_select)),'ko','LineWidth',2);hold offxlabel('x');

6/4/13 12:38 PM D:\My Documents\Courses\ME342-Inelasti...\fcc_slip_system_map.m 3 of 3

ylabel('y');axis equaltitle('highest Schmid factor'); figure(2);mesh(X,Y,maxSind);hold onplot3(X_select,Y_select,13*ones(size(X_select)),'ko','LineWidth',2);hold offview([0 90]);xlabel('x');ylabel('y');axis equaltitle('slip system ID with highest S'); figure(3);mesh(X,Y,secondSind);hold onplot3(X_select,Y_select,13*ones(size(X_select)),'ko','LineWidth',2);hold offview([0 90]);xlabel('x');ylabel('y');axis equaltitle('slip system ID with second highest S'); figure(4);contourf(X,Y,maxS./secondS-1,20);colorbarhold onplot3(X_select,Y_select,ones(size(X_select)),'ko','LineWidth',2);hold offxlabel('x');ylabel('y');axis equaltitle('S_2/S_1 - 1');

6/4/13 1:55 PM D:\My Documents\Courses\ME342-I...\fcc_slip_system_polycrystal.m 1 of 3

% FCC slip systems for poly-crystal under uniaxial tension% Find the 5 slip systems activated for each tensile axis% following Taylor 1938% G. Y. Chin and W. L. Mammel, % Computer Solutions of the Taylor Analysis for Axisymmetric Flow% Trans. Metall. Soc. AIME 239, 1400-1405 (1967). % plane slip vectorslip_system = [ 1 1 1 1 -1 0 1 1 1 1 0 -1 1 1 1 0 1 -1 1 1 -1 1 -1 0 1 1 -1 1 0 1 1 1 -1 0 1 1 1 -1 1 1 1 0 1 -1 1 1 0 -1 1 -1 1 0 -1 -1 -1 1 1 -1 -1 0 -1 1 1 -1 0 -1 -1 1 1 0 1 -1 ];N_slip_system = length(slip_system(:,1)); theta = [0:0.01:1]*(pi/4);phi = [0:0.01:1]*(pi/4);dtheta = theta(2)-theta(1);dphi = phi(2)-phi(1);Ntrial = 3;dTfactor = 1e-4;X = zeros(length(theta), length(phi)); Y = X; x = X; y = X; z = X;activeS = zeros(length(theta), length(phi), N_slip_system);dissip = zeros(length(theta), length(phi)); weight = dissip;Taxis = zeros(length(theta), length(phi), 3); netrot = Taxis; dTaxis = Taxis;Taxisnew = Taxis; Xnew = X; Ynew = Y; dX = X; dY = Y; strain = zeros(N_slip_system,6);rotation = zeros(N_slip_system,3);for i = 1:length(slip_system(:,1)), n = slip_system(i,1:3); n = n/norm(n); b = slip_system(i,4:6); b = b/norm(b); tmp = (n'*b + b'*n)/2; strain(i,:) = [tmp(1,1) tmp(2,2) tmp(3,3) tmp(2,3) tmp(3,1) tmp(1,2)]; rotation(i,:) = cross(n,b)/2;end % compute participation ratio of each slip systemfor ai = 1:length(theta), if mod(ai,10)==0 disp(sprintf('ai = %d / %d ', ai, length(theta))); end for bi = 1:length(phi),

6/4/13 1:55 PM D:\My Documents\Courses\ME342-I...\fcc_slip_system_polycrystal.m 2 of 3

factor = atan(sin(theta(ai)))/(pi/4); x(ai,bi) = sin(theta(ai))*cos(phi(bi)*factor); y(ai,bi) = sin(phi(bi)*factor); z(ai,bi) = cos(theta(ai))*cos(phi(bi)*factor); X(ai,bi) = 2*x(ai,bi)/(1+z(ai,bi)); Y(ai,bi) = 2*y(ai,bi)/(1+z(ai,bi)); T = [x(ai,bi) y(ai,bi) z(ai,bi)]; Taxis(ai,bi,:) = T; tmp = (T'*T - (T*T'/3)*eye(3))*(3/2); tensile_strain = [tmp(1,1) tmp(2,2) tmp(3,3) tmp(2,3) tmp(3,1) tmp(1,2)]; %options = optimset('Display','iter','TolFun',1e-8); options = optimset('Display','off','TolFun',1e-8); % double # of slip systems so that coeff is non-negative Aeq = [strain', -strain']; beq = [tensile_strain']; lb = zeros(length(slip_system(:,1))*2,1); f= ones(length(slip_system(:,1))*2,1); coeff = linprog(f, [], [], Aeq, beq, lb, [], [], options); % half # of coeff to store as positive and negative values compact_coeff = coeff(1:end/2)-coeff(end/2+1:end); activeS(ai,bi,:) = compact_coeff; dissip(ai,bi) = sum(abs(compact_coeff)); netrot(ai,bi,:) = rotation'*compact_coeff; endend % compute change of Taxisfor ai = 1:length(theta), for bi = 1:length(phi), dTaxis(ai,bi,:) = cross(netrot(ai,bi,:),Taxis(ai,bi,:)); Taxisnew(ai,bi,:) = Taxis(ai,bi,:) + dTaxis(ai,bi,:)*dTfactor; Xnew(ai,bi) = 2*Taxisnew(ai,bi,1)/(1+Taxisnew(ai,bi,3)); Ynew(ai,bi) = 2*Taxisnew(ai,bi,2)/(1+Taxisnew(ai,bi,3)); endenddX = Xnew - X; dY = Ynew - Y; % compute area of mesharea = zeros(size(weight));for ai = 1:length(theta)-1, for bi = 1:length(phi)-1, dr1 = [x(ai+1,bi) y(ai+1,bi) z(ai+1,bi)] - [x(ai,bi) y(ai,bi) z(ai,bi)]; dr2 = [x(ai,bi+1) y(ai,bi+1) z(ai,bi+1)] - [x(ai,bi) y(ai,bi) z(ai,bi)]; dr3 = [x(ai+1,bi) y(ai+1,bi) z(ai+1,bi)] ... - [x(ai+1,bi+1) y(ai+1,bi+1) z(ai+1,bi+1)]; dr4 = [x(ai,bi+1) y(ai,bi+1) z(ai,bi+1)] ... - [x(ai+1,bi+1) y(ai+1,bi+1) z(ai+1,bi+1)]; area(ai,bi) = (norm(cross(dr1,dr2))+norm(cross(dr3,dr4)))/2; endend

6/4/13 1:55 PM D:\My Documents\Courses\ME342-I...\fcc_slip_system_polycrystal.m 3 of 3

% compute integration (quadrature) weightfor ai = 1:length(theta), for bi = 1:length(phi), weight(ai,bi) = area(ai,bi)/4; if ai > 1 weight(ai,bi) = weight(ai,bi) + area(ai-1,bi)/4; end if bi > 1 weight(ai,bi) = weight(ai,bi) + area(ai,bi-1)/4; end if ai > 1 && bi > 1 weight(ai,bi) = weight(ai,bi) + area(ai-1,bi-1)/4; end endendmean_dissip = sum(sum(dissip.*weight))/(4*pi/48); % plot resultsfs = 17;figure(1);contourf(X,Y,dissip,50);colorbarset(gca,'FontSize',fs);xlabel('x'); ylabel('y');axis equaltitle(sprintf('mean dissip = %.4f (Taylor Factor)',mean_dissip)); figure(2);mesh(X,Y,dissip);set(gca,'FontSize',fs);xlabel('x'); ylabel('y');axis equalview([30 80]);title(sprintf('mean dissip = %.4f (Taylor Factor)',mean_dissip)); figure(3);skip = 10;contour(X,Y,dissip,50); hold onquiver(X(1:skip:end,1:skip:end), Y(1:skip:end,1:skip:end), ... dX(1:skip:end,1:skip:end),dY(1:skip:end,1:skip:end));hold offset(gca,'FontSize',fs);xlabel('x'); ylabel('y');axis equaltitle('rotation of tensile axis');t001 = text(-0.01,0.02,'001');t101 = text( 0.80,0.02,'101');t111 = text( 0.70,0.70,'111');