n n ee (for index notation practice, see if you can verify the derivation given in class)

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EN24: Computatioal mthods i Structural ad Solid Mchaics Homwork 6: Noliar matrials Du Wd Oct 2, 205 School of Egirig Brow Uivrsity I this homwork you will xtd EN24FEA to solv problms for oliar

1 EN24: Computatioal mthods i Structural ad Solid Mchaics Homwork 6: Noliar matrials Du Wd Oct 2, 205 School of Egirig Brow Uivrsity I this homwork you will xtd EN24FEA to solv problms for oliar matrials. You ca choos whthr to implmt a hypolastic matrial (simpl or a cyclic plasticity modl (much hardr. Solv ithr problm or problm 2 (you ar wlcom to solv both, if you wish.. Extd EN24FEA to solv problms with th simpl hypolastic matrial modl discussd i class. Th strss-strai rlatio is: 2 σ = σ + Kkkδ whr 2 = kkδ = σ ( 0 = σ0 / 0 0 ad K is th bulk modulus (usually tak to b larg, to modl a ar-icomprssibl matrial. Th tagt modulus is dσ 4 kl 2 δikδ jl + δilδ jk δδkl = ( Et Es + E ( 9 2 s + Kdkld dkl 2 whr Es = s/, ad Et = dσ/ d (for idx otatio practic, s if you ca vrify th drivatio giv i class To implmt this matrial modl you will d to do complt th followig stps (show for a D lmt; th vctors/matrics will chag for 2D:. You ca ithr ( crat a w lmt subrouti (g copy your B-bar lmt from last wk ito a w fil or ls add a coditioal statmt i your xistig B-bar lmt from last wk that will allow th usr to choos btw a stadard liar lastic matrial or a hypolastic matrial (if you lik, you ca us th valu of th first Elmt Proprty to idtify th matrial modl, or you ca simply cout th umbr of matrial proprtis 2. Writ a subrouti to calculat th strss vctor [ σ, σ22, σ, σ2, σ, σ 2] ad th matrial tagt stiffss matrix [ D ] giv a valu for th total strai vctor (i.. strai+dstrai i th cod [ ε, ε22, ε,2 ε2,2 ε,2 ε2] + [ ε, ε22, ε,2 ε2,2 ε,2 ε2]. Th D matrix ca b calculatd as follows

2 E 2 2 [ ] ( s E D E s = 2 t Es + + K whr [, 22,, 2,, 2] = is th total dviatoric strai vctor (calculatd usig th total strai at th d of th stp ot thr is o factor of 2 i th shar strais. You ca comput th dyadic product of two vctors i FORTRAN usig _dyadic_ = sprad(,dim=2,copis=6*sprad(,dim=,copis=6 You will d to trap for th spcial cas of ε = 0 - i this limitσ = 0, ad Et = Es so th first trm i D vaishs, ad you ca st Es = ds / d i th scod ad third trms.. Us th calculatd [D] ad [ σ, σ22, σ, σ2, σ, σ 2] to calculat th lmt stiffss ad rsidual (you wo t d to chag th cod i ay way othr tha to calculat [D] ad th strss corrctly. Us your B-bar matrix codd last wk, sic you ar itrstd i solvig problms for ar-icomprssibl matrials. 4. You will d to modify your fild variabls subrouti to comput th strsss for th w matrial modl. 5. Crat a iput fil to tst your lmt/matrial modl. Ru a sigl lmt tst (you ca copy th simpl lmt D liar lasticity iput fil. Th matrial proprtis for your w lmt ar th valus of { σ0, ε 0, K, }. You will d to us th oliar solvr. For this you will d to chags th lis i th iput fil that rads SOLVER, DIRECT, LINEAR to SOLVER, DIRECT, NONLINEAR,.d-06, 5 (th first umbr is th tolrac that will b usd to dcid that th itratios hav covrgd, th scod is th maximum Nwto itratios that should b attmptd bfor rducig th tim stp siz ad startig a scod attmpt to fid a solutio. 6. Ru th chck stiffss procdur (stop th calculatio just aftr th CHECK STIFFNESS to chck that your matrial stiffss is corrct. Us σ0 =, ε0 = 0.0, = 0, K = 000. You will d to mak a small chag i th way th chck stiffss procdur computs th umrical drivativ for this problm: i th fil amd chckstiffss.f90, fid th lis that rad lmt_dof_icrmt(icout = lmt_dof_icrmt(icout +.D-07 umrical_stiffss(:iu,icout = -(rsid(:iu-rsid0(:iu/.d-07 lmt_dof_icrmt(icout = lmt_dof_icrmt(icout -.D-07 ad chag th.2-07 to.d-2. Th aalytical ad umrical stiffss should th agr to all th dcimal placs pritd. 7. Ru a tst i which a sigl lmt is subjctd to a prscribd displacmt, ad plot th prdictd strss-v-strai bhavior for th lmt. You ca us th subrouti providd i th usr_prit.f90 fil that computs th volum avrag of th strss ad strai i a lmt for this purpos. Try th followig matrial paramtrs: σ0 =, ε0 = 0.0, = 0, K = 000. Apply a cycl of displacmt to th lmt that cycls th strai btw limits of ± 0.. You ca chck your calculatios agaist th aalytical solutio. You could us th usr prit subrouti to prit a strss-v-strai curv (but you will hav to modify th comput_lmt_volum_avrag_d subrouti to calculat th strss stat, sic for a hypolastic matrial strss is ot stord as a stat variabl 8. You ca ru othr mor complicatd tsts at your discrtio; you could also cod a 2D vrsio (for pla strai or axisymmtry this is fairly simpl; pla strss is mor challgig. Prhaps complld is a bttr word tha itrstd

3 2. I this problm, your missio (should you choos to accpt it is to implmt a oliar kimatic hardig plasticity modl that is oft usd to idaliz th cyclic plastic rspos of polycrystalli mtals. Its mai fatur is that it is abl to prdict cyclic crp or ratchttig bhavior, i.. th tdcy of a tsil spcim to progrssivly icras i lgth wh subjctd to a tsio/comprssio cycl with a tsil ma strss, as show i th figur. For simplicity, w focus oly o small strais. Th Armstrog-Frdrick oliar kimatic hardig modl has strss-strai rlatios giv by p = + ( S α + ν ν p = σ σkkδ = E + ν 2 σ 2 p p S = σ σkkδ / σ = ( S α ( S α = 2 Hr, α is a matrial stat variabl that dscribs how th matrial hards. Th yild critrio is ad α obys a hardig law σ Y p α = c γ α You ca fid mor backgroud radig o plasticity ad hardig laws hr. Your solutio should iclud th followig stps: Strss Updat Algorithm: ( Dvis a mthod for calculatig th strss σ + at th d of a strai icrmt ε. Us a fully implicit computatio, i which th yild critrio ad hardig law ar xactly satisfid at th d of th load icrmt. Your drivatio should follow closly th procdur discussd i class, xcpt that. Aftr computig th lastic prdictor * S = S + E = kkδ +

4 for th strss, you should chck ad s if th strsss ar blow yild (us th yild critrio. If so, th lastic prdictor is th corrct strss. 2. If th lastic prdictor xcds yild, th plastic strai magitud must b calculatd by solvig (usig Nwto-Raphso itratio isid your strss calculatio subrouti * E c * α * α F(, S = Y0 + + S S = 0 2 ( + + γ 2 + γ + γ Plas driv this rsult. You ca us th approach discussd i class start by showig + + * E ( S α + S = S σ ( + + * E ( S α S α = S α σ You ca th solv th hardig rlatio forα +. Substitutig this rsult for + + ( S α + + = + c + σ α α γ α α S α o th right had sid of th scod quatio i ( ad rarragig th rsult (collct o th lft had sid, squar both sids ad multiply by /2, ad simplify usig th yild critrio givs th aswr you d.. Nxt, show that + * S = λ + ηα λ = + η η = c α α S + = + ( + ( γ + c/ Y0 Y0 so that E c E ( S 2 Y ( + ν Y 2 ( + ν ( Y + ( γy + c + * ( E = S + + kk kk + ( 2 σ λ ληα σ δ ε δ 4. Fially, calculat th lastic-plastic tagt matrix. This is a tdious, but straightforward calculatio. Th followig rsults ar hlpful: * S E δikδ jl + δ jkδil = δδ kl kl ( + ν 2 It is also hlpful to ot that 0 kk F F * * S * F(, S = 0 = S α =. If I did th calculatio corrctly, you should fid that

5 whr ( + le ik jl jk il σ δ δ + δ δ E = δδ kl + δδ kl kl ( + ν 2 ( 2 ν l * * l η αklα SklS E F + β η + l 2 ( + + γ * l η α * l kls + η + l Sklα + γ E c β = Y ( ν γ + + This ca probably b simplifid a lot mor, but I am too lazy. Implmtig th w matrial modl i your cod Th xt stp is to add this matrial modl to your EN24FEA fork. Som suggstios: If you lik, you ca add th cod to your D small-strai B-bar lmt. To do this, you could add a itgr valud flag to th list of lmt proprtis that will allow th usr to slct ithr a liar lastic matrial modl, or your w plasticity modl; altrativly, you could simply chck th umbr of proprtis to cotrol th slctio. Your matrial modl is ow history dpdt, so you will d to stor valus for stat variabls that volv with th dformatio. You will d to stor th dviatoric strss; th hydrostatic strss; ad th compots of α. You will d to stor valus for ths variabls for ach itgratio poit i th lmt. Not that α = α ji so it ca b stord as a -D vctor usig th sam covtio that is usd to stor strsss [ α, α22, α, α2, α, α 2]. If you stor dviatoric strss, prssur, ad α, you will hav stat variabls pr itgratio poit for a 4 odd hxahdro with 8 itgratio poits this would com to 04 total stat variabls pr lmt. I EN24FEA, all stat variabls for a lmt ar stord i two -D vctors: th usr subrouti will provid th valus of stat variabls at th start of th stp i a vctor calld iitial_stat_variabls; your usr-subrouti must rtur thir valus at th d of th stp i a variabl calld updatd_stat_variabls. You ca us ay storag schm you lik for your variabls as log as thy ar cosistt. For xampl, i my cod I xtract th stat variabls for th first itgratio poit as sdv0 = iitial_stat_variabls(:6 prss0 = iitial_stat_variabls(7 alpha0 = iitial_stat_variabls(8: Th scod itgratio poit starts at iitial_stat_variabls(4, ad so o. You must also spcify th umbr of stat variabls pr lmt i th list of lmt paramtrs i th iput fil. You will d to writ a subrouti that calculats th strss vctor ad th matrial tagt matrix. To calculat th strss ad tagt, first comput th lastic prdictor; th chck for yild; th updat th strss ad stat variabls ad fially comput th matrial tagt matrix D. Although th xprssio for D is quit log you ca assmbl it vry quickly by usig outr

6 * * products of th strss ad α vctors (g Sαkl S α. Not that i Fortra90 you ca calculat th outr product of two 6 dimsioal vctors S α as follows sprad(s,dim=2,copis=6*sprad(alpha,dim=,copis=6 Codig this lmt is quit tricky a small sig rror will caus vrythig to blow up (if it is ay cosolatio, it took m a tir day to fid a sig rror i th formula giv abov for λ. Chck your cod with th CHECK STIFFNESS i th iput fil. I foud it hlpful to add a fw lis i th cod that pritd th valu of σ = ( S + α + ( S + α + 2 If yild is xcdd, th rsult should b xactly qual to Y 0 if you typd vrythig i corrctly. B vry carful with calculatig trms lik Sα - it is tmptig to itrprt this as th dot product of th strss ad α vctors, but this is ot th cas, bcaus th off-diagoal trms oly appar oc i th dot product but twic i th tsor ir product. Tst your cod by calculatig th strss-strai rspos of o or two lmts to a cycl of uiaxial tsio. A sampl iput fil for this purpos (for D lmts has b providd i cyclic_plastic_d.i. Also, a usr-subrouti (i usr_prit.f90 has b providd that calculats th volum avragd strai ad volum avragd stat variabls i a lmt (you ca choos th lmt i th iput fil, ad prits th tim history of ths volum avragd quatitis to a fil that ca b rad with TECPLOT (us th XY-li optio to plot th rsults. Ru this tst i stags: first choos Y 0 to b much largr tha th applid strss, ad chck that you gt th corrct lastic bhavior. Th ru a simulatio with E = 500, ν = 0., Y0 = 0, c= 50, γ = 0, ad strsss varyig from 0 < σ < 2.5. You should s that th strss-strai curv is a closd cycl of plastic strai, ad th hardig rat is liar, as show i th figur. Fially, ru a simulatio with E = 500, ν = 0., Y0 = 0, c= 50, γ = : this should produc th ratchttig bhavior show at th top of th homwork. If this works, you could st up som mor itrstig boudary valu problms to xplor as wll (.g. cyclic bhavior ahad of a fatigu crack tip; cyclic crp i a prssur vssl, tc.

MONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx

MONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of