Contitutive model art latoplatic
latoplatic material model latoplatic material are aumed to behave elatically up to a certain tre limit after which combined elatic and platic behaviour occur. laticity i path dependent the change in the material tructure are irreverible
Stre-train curve of a hypothetical material Idealized reult of one-dimenional tenion tet force / initial ngineering tre area Johnon limit 50% of oung modulu value ield point ield tre ε Δl 0 l ngineering train
Real life 1D tenile tet, cyclic loading Where i the yield point? Conventional yield point Lin. elat. limit Hyterei loop move to the right - racheting
Mild carbon teel before and after heat treatment Conventional yield point 0.%
he platicity theory cover the following fundamental point ield criteria to define pecific tre combination that will initiate the non-elatic repone to define initial yield urface Flow rule to relate the platic train increment to the current tre level and tre increment Hardening rule to define the evolution of the yield urface. hi depend on tre, train and other parameter
ield urface, function F(, ε, K...) 0 ield urface, defined in tre pace eparate tre tate that give rie to elatic and platic (irrecoverable) tate For initially iotropic material yield function depend on the yield tre limit and on invariant combination of tre component A a imple example Von Mie F effective yield 0 ield function, ay F, i deigned in uch a way that F < 0 tre tate within the urface F 0 on the urface F > 0 outide, inadmiible for analytical platicity
hree kinematic condition are to be ditinguihed Small diplacement, mall train material nonlinearity only (MNO) Large diplacement and rotation, mall train L formulation, MNO analyi K tre and GL train ubtituted for engineering tre and train Large diplacement and rotation, large train L or UL formulation Complicated contitutive model
Rheology model for platicity Ideal or perfect platicity, no hardening
Loading, unloading, reloading and cyclic loading in 1D train - new yield tre 1 new yield tre initial yield tre tre new yield tre 1 Iotropic hardening reloading loading unloading
Iotropic hardening in principal tre pace 3 1 3 1, 0 ) ( tree and 1D yield tre in tenion reca expreed by principal > > F 0 ] ) ( ) ( ) [( tree and 1D yield tre in tenion von Mie expreed by principal 1 3 3 1 F π - plane arcco (/qrt(3))
Loading, unloading, reloading and cyclic loading in 1D initial yield tre tre Kinematic hardening reloading loading new yield tre 1 unloading train
Kinematic hardening in principal tre pace intead of F( ) 0 (a in cae of iotropic hardening) we take F( α ) 0, whereα cε, c... contant
Von Mie yield condition, four hardening model 1. erfect platicity no hardening. Iotropic hardening 3. Kinematic hardening 4. Iotropic-kinematic
F around the platicity region. Definition of where K Different type of yield function perfect platicity mean no hardening, material tart to flow and i inclided to do o forever. It practice it i tabilized by the ' healthy' latic material flow i caued by motion of dilocation... Generally, the hardening depend on blocking the motion of F F F which depend on the permanent platic train ε. kinematic hardening and c i a contant. non - iotropic hardening hardening depend on every component of ε iotropic hardening Generally, which i not general at all, we could have F F( ) whereα c ε L F( α ) F(, ε ) F(, K) F(, ε, K) L L L material tructure which exit dilocation. in a different way K( ε ) i a calar function of ε, uually an invariant. dilocation (free flow)
laticity model phyical relevance Von Mie - no need to analyze the tate of tre - a mooth yield ufrace - good agreement with experiment reca - imple relation for deciion (advantage for hand calculation) - yield urface i not mooth (diadvantage for programming, the normal to yield urface at corner i not uniquely defined) Drucker rager a more general model
1D example, bilinear characteritic tre dε dε dε d β tan β d d d tangent modulu elatic platic Strain hardening parameter α total H / 1 tanα elatic modulu dε dε dε train mean total or elatoplatic
Strain hardening parameter again Upon unloading and reloading the effective tre mut exceed Initial yield latic train removed Geometrical meaning of the train hardening parameter i the lope of the tre v. platic train plot
How to remove elatic part
1D example, bar (rod) element elatic and tangent tiffne L δ A F F > L A F k δ ( )L A L A F k d d d d d d d ε ε ε ε δ 1 / d / d / d L A L A k latic tiffne angent tiffne
Reult of 1D experiment mut be correlated to theorie capable to decribe full 3D behaviour of material Incremental theorie relate tre increment to train increment Deformation theorie relate total tre to total train
Relation for incremental theorie iotropic hardening example 1/9 Relation between increment and rate : Let the yield urface i F(, ε ) 0 lim t 0 d & dt arameter only increment of deformation depend on F if F < 0 elatic F 0 and F 0 and F 0 and & & & eff eff eff < 0 elatic > 0 elatoplatic F > 0 go back to yield urface and & eff 0 neutral - it mean that & ε 0
Relation for incremental theorie iotropic hardening example /9 Flow rule i aumed in the form (Drucker,1947) F & ε λ λq where λ F q { 11 q. (i) increment of platic deformation ha a direction normal to F while it magnitude (length of vector) i not yet known i o far unknown calar and F define outer normal to F L } in ix dimenional tre pace 31 F can be expreed a a total differential df F d F ε dε F which mut be zero during platic deformation, o df F & & ε ε 0
Relation for incremental theorie iotropic hardening example 3/9 Denoting p F { ε the condition df 0 can be expreed in the form q d dε tre increment are & ε& p ( ε& ε& 11 q & L p F ε ε& 31 } 0 eq. (ii) ) eq. (iii) elatic total platic deformation matrix of elatic moduli
Relation for incremental theorie iotropic hardening example 4/9 Combining the relation for flow rule (i), df 0 (ii) and for tre increment (iii) Row vector we get λ p q ε& q q q Column vector Still to be determined Dot product and quadratic form calar Lambda i the calar quantity determining the magnitude of platic train increment in the flow rule
Relation for incremental theorie iotropic hardening example 5/9 Now, for the tre increment we can & with ε& ( ε& a a function of where p till ha ε& ) q ( q) p q q q with ε& to be determined λ q write Subtituting for λ we get the tre increment & & ε total train increment in the form diadic product equal to zero for perfect platicity
t Relation for incremental theorie F ε iotropic hardening example 6/9 Determination of where J D f ( W F t 1 F to evaluate ε Chain rule ), W t W F p { ε we need xperiment ugget that W ε 11 dε 3 L Aume von Mie yield condition f t F ( ε ) F uing t J t W 31 D t 3 1 3 i the econd deviatoric invariant L F } ε t work done by platic increment A and 0 A new contant defined W ε At time t
Relation for incremental theorie iotropic hardening example 7/9 ε ε 0 t 0 ε ε t W 31 11 0 0 t 0 1 } { o finaly 3 3 3 ) ( 1 ) ( 1D bilinear characteritic ) ( work done in1d the elatic ε ε L A A W W W t t t t t t t t p W
Relation for incremental theorie iotropic hardening example 8/9 ε bb b q q q q b q p p q ε & & & & 31 3 1 33 11 31 3 1 33 11 31 3 1 m 33 m m 11 31 3 1 33 11 33 11 3 1 m,, } { 3 } { } { } { ) ( a follow we can compute and and For given Summary. c a c a A A
{ } { } { ε} { } or J theory, perfect platicity 1/6 alternative notation example of numerical treatment ( tre deviator econd invariant of J m D [ ]{ ε}...hooke' law { { ε 1 3 { J J xx xx xx ε xx 1 yy 1 yy ε ( { } m xx zz yy zz γ [ M τ xy yy xy γ ) } } mean tre tre deviator yy τ yz zz yz γ m τ zx zz zx zz xy m τ ]{ }, with [ M ] xy τ yz yz τ zx } zx ) diag(1,1,1,,,)
J theory, numerical treatment /6 one can prove that { } [ M ]{ } { } [ M ]{ }, ince xx yy zz 0 and alo [ ][ M ]{ } G{ }, with G /(1 μ) von Mie effective tre yield criterion for perfectly platic behaviour eff 3J 3{ } [ M eff ]{ }/
J theory, numerical treatment 3/6 Flow rule according to randtl - Reu hypothei { & ε} { } { & } [ ]{ & ε} [ ]{ & ε } [ ]{ & ε} G{ }... it time derivative, increment no platic deformation in elatic region can be expreed by if eff endif F λ { } [ ]{ ε } < λ[ M ]{ }... λ i o far unknown parameter then λ ele λ [ ]{ ε ε }... 0, 0 Hooke' law Six nonlinear differential equation one algebraic contraint (inequality) here i exact analytical olution to thi. In practice we proceed numerically
J theory, numerical treatment 4/6 Differentiating platicity condition & eff Gλ we get finally eff Subtituting for Mε& 3 Mε& 4G with and realizing that λ & M M ε& & 3 M & M 3 ε& eff 0 and alo Gλ 4GλJ 0 M& 4Gλ eff 0 / 3 3G eff 4Gλ / 3 Sytem of ix nonlinear differential equation to be integrated
J theory, numerical treatment 5/6 predictor-corrector method, firt part: predictor 1. known tre t 3b. platic part of increment ( 1 r) Δ. tet tre (elatic hot) t Δ t Δε 4. r Δ c t 5. Δλ 3(1 r) Δε c /( ) 6. Δλ ' t Δt G c 3a. elatic part of increment r Δ
J theory, numerical treatment 6/6 predictor-corrector method, econd part: corrector Correction For eff eff β ' tδt ( ( β eff ( ) ' tδt ' tδt eff ' tδt ) ) ' tδt ' tδt ) (1 β ) (1 β ) find β in uch a way that ' tδt and ince the pherical part of the tre tenor doe not enter into platicity conideration we have tδt tδt tδt β ( tδt β ' tδt
Secant tiffne method and the method of radial return