Finit Strain Elastic-Viscoplastic Modl Pinksh Malhotra Mchanics of Solids,Brown Univrsity Introduction Th main goal of th projct is to modl finit strain rat-dpndnt plasticity using a modl compatibl for high strain rats. In such scnarios, Johnson-ook modl is vry usful. Th modl is usd in adiabatic dynamic simulations, for xampl, prssur-shar plat impact xprimnts and machining. Th modl incorporats tmpratur ffcts as wll, using a powr-law dpndnc. Tmpratur dpndnc is ignord in th prsnt study. In th prsnt study, th modl is dvlopd kping in mind th futur us for prssur-shar impact simulations. Kinmatics of prssur-shar xprimnt is introducd to giv an ida about th typ of dformation involvd. This is followd by introduction to th modl and simulations on two lmnts. Th FEA formulation is don in EN234FEA. 2 Govrning Equations 2. Kinmatics Dformation in a prssur-shar xprimnt can b writtn as: x = λt)x ) x 2 = X 2 κt)x 2) x 3 = X 3 3) Th dformation gradint and vlocity gradints ar, thrfor: λt) 0 0 F = κt) 0 4) 0 0 λλ 0 0 L = ḞF = κ/λ 0 0 5) 0 0 0 2.2 Finit Strain Viscoplastic Matrial Modl F = F F p 6) L = Ḟ F ) + F Ḟ p F p ) F ) 7) = L + L p = D + W ) + D p + W p ) 8) Hr, W p = 0 is considrd. onsidr th Kirchoff strss τ = J σ) as th strss-masur for this study. Rat of chang of Kirchhoff strss is dfind as: τ = ˆτ + W τ τ W ) 9)
2 GOVERNING EQUATIONS 2 whr ˆτ is Jaumann strss rat and is givn as: ˆτ = : D + D τ + τ D ) 0) Th scond trm of th Kirchoff strss rat is usually takn car of by ABAQUS, so w nd to find th Jaumann strss rat only. Plasticity Equations D p = 3 ɛ τ D ) 2 τ 3 τ = 2 τ ij Dτ ij D 2) 2 ɛ = 3 Dp ij Dp ij 3) A constitutiv law govrning ɛ is rquird. On of th laws particularly usful for impact problms is th Johnson/ook modl, whr th yild strss is givn as: ) ) m σ y = A + Bɛ n ɛ T T0 ) + ln 4) T m T 0 3 whr σ = 2 s ijs ij = 3 J 2 τ ij Dτ ij D. A is th static shar strngth, B is th strain-hardning modulus, is th rat-snsitivity cofficint, m is th thrmal-softning xponnt, n is th strain-hardning xponnt, T is th currnt tmpratur, T 0 is th room tmpratur and T m is th mlting tmpratur. Ignoring th ffcts of tmpratur, i.. assuming T = T 0, ɛ = σ A+Bɛ n ) 5) Sinc th strain rat grows xponntially with ffctiv shar strss, it is ncssary to limit th strain rat to dal with high strsss during initial lastic rspons. A limiting strain rat ɛ lim is usd as follows to dfin th actual plastic strain rat: ɛ ɛ + ɛ lim ɛ ff = ɛlim Johnson-ook Dynamic Failur ritrion A damag paramtr is calculatd at th intgration points and failur is assumd to occur whn this paramtr is qual to. Th damag paramtr,ω is givn as: 6) ω = ɛ ɛ,f 7) whr ɛ,f is th failur plastic strain. Th summation is prformd ovr all th tim incrmnts in th analysis. Th failur plastic strain is assumd to b dpndnt on th plastic strain rat in similar fashion as th yild strss and is formulatd as blow: ) ) ɛ,f = d + d 2 d3 p σ ) ɛ T T0 ) + d 4 ln + d 5 8) T m T 0 Th paramtrs d to d 5 ar failur paramtrs dtrmind using xprimnts.
3 NEWTON-RAPHSON FOR ɛ E 3 Simplification of th Jaumann strss rat: τˆ E ij = + ν D Eν + ν) 2ν) D 3 E kkδ ij 2 + ν) M ij = D ik τ kj + τ im D mj 3 2 Hnc, th Kirchoff strss rat, calculatd xplicitly is givn as: τ n+) ij = τ n) E tn+ Eν D ij dt + + ν t n + ν) 2ν) 3 E t ɛ ffn) tn+ τ Dn) 2 + ν) τ ɛ ff τij D + M ij 9) τ ɛ ff τ τ D ik τ kj + τ im τ D mj) 20) t n tn+ t n D n) kk δ ijdt 2) tn+ M n) ij dt + Q n) ij dt t n whr Q ij = W ik τ kj τ im W mj 22) Drivation ˆ τ = : D + M 23) : D = : D : D p 24) : D) ij = E ijkld kl = 2 + ν) δ Eν ikδ jl + δ il δ jk ) + + ν) 2ν) δ ijδ kl D kl 25) E = + ν) D Eν + ν) 2ν) D kkδ ij 26) : D p E 3 ɛ ) ij = τij D + τji D ) = E 3 ɛ τij D ) 27) 2 + ν) 2τ + ν) 2τ M ij = D ikτ kj + τ ik D kj 28) 3 Nwton-Raphson for ɛ Hr, ɛ ff is addrssd simply as ɛ and ɛ p ij and Dp ij ar th sam thing. This lads to ɛ ij = ɛ ɛ p ij 29) Elastic Prdictor ṡ ij = s n+) ij E + ν ė ij 30) = s n) E + ν ij ɛ p ij ) 3) s n+) ij σ n+) = = s n) E + ν n) ij 32) 3 2 s n+) ij s n+) ij 33)
4 FE FORMULATION 4 orrction: Lt s n+) ij Now w try to solv for ɛ. = βs n+) ij. On solving, β can b found to b β = 3E 2 + ν) ɛ σ n+) 34) ɛ = t ɛlim ɛ lim + F = ɛlim t ɛ df d ɛ = ɛlim t ɛ 2 ) σ n+) A+Bɛ n+)n ) = σ n+) + ɛlim A+Bɛ n+)n + ɛlim F can also b formulatd as: 4 FE Formulation + ɛlim σn+) A+Bɛ n+)n ) ) βσ n+) A+Bɛ n) + ɛ) n F = σ n+) ɛ lim t σn+) A+Bɛ n+)n 3E 2 + ν) ) 35) A + Bɛ n) 36) + ɛ ) + βσ n+) Bn ɛ n) + ɛ ) n n A + Bɛ n) + ɛ ) n ) 2 37) ) A + B ɛ n+)n ɛ ) + ln 38) Finit Elmnt formulation usd is similar to Gurson modl implmntd in Assignmnt-0. Th UEL is writtn for finit strain using L-bar mthod. Maximum tim stp that can b usd is givn by: t max = L c whr L is lngth of th lmnt and c is longitudinal wav spd in th matrial. Th matrial paramtrs usd in th simulations corrspond to Al-606-T6: E =70 GPa, A =324. MPa, B =3.8 MPa, =0.002 MPa, n =0.42, =, ɛ lim =, d =-0.77, d 2 =.45, d 3 =0.47, d 4 =0. Similarly, for Stl 4340, th following paramtrs ar availabl: E =200 GPa, A =792 MPa, B =50 MPa, =0.04 MPa, n =0.26, =, ɛ lim =, d =0.05, d 2 =3.44, d 3 =2.2, d 4 =0.002. A nw subroutin, strss updat prssurshar is writtn and th plastic strain incrmnt is solvd implicitly using Nwton-Raphson. It can b solvd by ithr of th two formulations of F prsntd abov. Damag variabl, ω is stord as th ninth of th 0 stat variabls th first six bing Kirchhoff strsss followd by th old accumulatd plastic strain, nw accumulatd plastic strain and ). NOTE:Elmnt dltion is includd in th subroutin l prssurshar. Th valus at intgration points ar projctd onto th nods using f ildvars prssurshar. Th corrsponding input fils ar P rssurshar 3D.in and notch f ractur dynamic.in 5 Rsults Th modl is tstd on two-lmnts. A unit displacmnt is applid in x-dirction at t = 0. Strss and strain in x-dirction,damag and accumulatd plastic strain ar plottd in Figur. Th tim stp chosn is.d 5. 39) )
6 REFERENES 5 Nwton-Raphson chcks ar prformd using a D MATLAB cod prior to ths simulations. Th modl is also applid to Mod-I notch fractur problm for Gurson Modl as shown in Figur 2. 6 Rfrncs. Stphn.E.Grunschl, Prssur-Shar Plat Impact Exprimnts on High-Purity Aluminum at Tmpraturs Approaching Mlt, PhD Thsis, 2009. 2. ABAQUS Documntation, 8.2.7, Johnson ook plasticity. 3. ABAQUS Tchnology Brif, Dassault Systms. Simulation of th ballistic prforation of aluminum plats with Abaqus/Explicit.
6 REFERENES 6 a) s b) c) Damag paramtr, ω d) Accumulatd Plastic Strain Figur : 80,000 tim stps. Each tim stp qual to.d 5
6 REFERENES 7 a) s b) c) Damag paramtr, ω d) Accumulatd Plastic Strain Figur 2: 3000 tim stps. Each tim stp qual to.d 5