Lecture #4: Integration Algorithms for Rate-independent Plasticity (1D)

Transcription

1 5-0735: Damic behavior of materials a structures Lecture #4: Itegratio Algorithms for Rate-ieeet Plasticit (D) b Dirk Mohr TH Zurich, Deartmet of Mechaical a Process gieerig, Chair of Comutatioal Moelig of Materials i Maufacturig 205

2 Recall: Imortat ifferece : Damic behavior of materials a structures lastic loaig LASTO-PLASTIC (e.g. metals, cocrete, thermolastics ) 2 lasto-lastic loaig 3 lastic uloaig NON-LINAR LASTIC (e.g. rubbers, foams) lastic loaig 2 lastic uloaig e e 2 2 2

3 5-0735: Damic behavior of materials a structures Rate-ieeet erfect lasticit Simlifie rheological moel: INITIAL CONFIGURATION frictioal evice liear srig DFORMD (CURRNT) CONFIGURATION The strai is slit ito a elastic a a lastic art e i.e. the elastic strai is e 3 3 3

4 5-0735: Damic behavior of materials a structures Rate-ieeet erfect lasticit - Summar i. Costitutive equatio for stress ii. Yiel fuctio ( ) f [, k k iii. Flow rule sig [ iv. Loaig/uloaig coitios 0 if 0 if 0 if f f f a a f f 0 0 Material moel arameters: () Youg s moulus, a (2) flow stress k

5 5-0735: Damic behavior of materials a structures Rate-ieeet erfect lasticit - Alicatio Total strai Plastic strai k / time k k / time Stress 5 k time

6 5-0735: Damic behavior of materials a structures Rate-ieeet isotroic hareig lasticit 2 lasto-lastic loaig 5 lasto-lastic loaig lastic loaig 3 lastic uloaig 4 lastic re-loaig The magitue of the stress icreases ue to strai hareig whe the material is eforme i the elasto-lastic rage. For isotroic hareig materials, it is escribe through a evolutio equatio for the flow stress k

7 5-0735: Damic behavior of materials a structures Rate-ieeet isotroic hareig lasticit Firstl, we itrouce a scalar value o-egative fuctio to measure the amout of lastic flow (sli). This measure is ofte calle equivalet lastic strai. Ulike the lastic strai, the magitue of the equivalet lastic strai ca ol icrease! k t It is the assume that the flow stress is a mootoicall icreasig smooth ifferetiable fuctio of the equivalet lastic strai k[ This equatio escribes the isotroic hareig law

8 5-0735: Damic behavior of materials a structures Rate-ieeet isotroic hareig lasticit Frequetl use arametric forms of the fuctio Swift a Voce laws: k k[ are the Hareig saturatio k k k k 0, k k0 Q k A ) S ( k V k Q ex[ Swift Voce Swift-Voce k ( ) k V k S 8 8 8

9 5-0735: Damic behavior of materials a structures Rate-ieeet isotroic hareig lasticit I egieerig ractice, the isotroic hareig fuctio is ofte reresete b a iece-wise liear fuctio k [MPa PQ k [ 9 9 9

10 i. Costitutive equatio for stress ( ) ii. Yiel fuctio f, k[ [ : Damic behavior of materials a structures Isotroic hareig lasticit - Summar iii. Flow rule sig[ iv. Loaig/uloaig coitios 0 if f 0 0 if f 0 a f 0 0 if f 0 a f 0 v. Isotroic hareig law k k[ with t 0 0 0

11 First-orer oriar ifferetial equatio ( ) sig[ : Damic behavior of materials a structures Differetial equatio to be solve sig [ t [ t Iitial coitio [ t 0 0 Prescribe loaig [t Multilier to satisf the costraits 0 if 0 if 0 if f f f a a f f 0 0 with f [, k[, ( ) a t

12 5-0735: Damic behavior of materials a structures Numerical solutio of Differetial quatios Without the loaig a uloaig coitios, the lasticit roblem reuces to solvig a oriar first-orer ifferetial equatio for the lastic strai, cosierig time as the ol ieeet variable: D.. sig [ t [ t g[ t I.C. [ t 0 0 [ t 0 0 Such equatios are solve umericall usig itegratio algorithms. Istea of the calculatig the exact aaltical solutio, we limit our attetio to calculatig the aroximate solutio [ t at equall-sace istats t, =,,N with the time ste Dt, Dt t t 2 2 2

13 5-0735: Damic behavior of materials a structures Numerical solutio of Differetial quatios A first oular metho is the so-calle forwar (exlicit) uler algorithm: Dtg Dtg[ 0 [ Dtg[ Startig with the iitial coitio, the aroximatios ca be rogressivel calculate

14 5-0735: Damic behavior of materials a structures Numerical solutio of Differetial quatios Recall that ' g[ a thus the forwar (exlicit) uler algorithm ma also be writte as 0 0 Dt 0 2 Dt Dt exact erivative aroximatio t t + I other wors, the time erivative at time t is give b the aroximatio [ t Dt 4 4 4

15 5-0735: Damic behavior of materials a structures Numerical solutio of Differetial quatios A seco oular metho is the so-calle backwar (imlicit) uler algorithm: 0 2 which is obtaie from solvig the imlicit equatio which is obtaie from solvig the imlicit equatio 0 Dtg[ 2 Dtg[ 2 which is obtaie from solvig the imlicit equatio Dtg[ Startig with the iitial coitio, the aroximatios ca be rogressivel calculate. However, at each time ste t i, a ofte imlicit equatio ees to be solve

16 5-0735: Damic behavior of materials a structures Numerical solutio of Differetial quatios Accorig to the backwar (imlicit) uler algorithm, the time erivative at time t is give b the aroximatio g[ Dtg[ [ t Dt aroximatio exact erivative t - t 6 6 6

17 xamle: t [ t : Damic behavior of materials a structures Illustratio (ifferetial equatio) (iitial coitio) ex[t (exact solutio) The aroximate solutio with forwar (exlicit) uler algorithm for a time ste of Dt= reas (we have g[= a 0 =): 0 0 Dtg[ 0 Dtg[ [ Dtg Dtg[ Dtg[ exact Observe from the grah that the metho coverges for Dt Dt 0.0 Dt 0. Dt

18 5-0735: Damic behavior of materials a structures Comariso imlicit vs. exlicit The aroximate solutios with forwar (exlicit) uler a backwar (imlicit) uler algorithms for a time ste of Dt= imlicit exact exlicit 8 8 8

19 5-0735: Damic behavior of materials a structures back to the lasticit roblem Differetial equatio: sig [ t [ t Dtg[ t Dt sig D D sig Iitial coitio: 0 0 lus iscrete evolutio costraits: f 0 D 0 ( D ) 0 f State variable: 0 0 if if D Deeet variables: ( ) k D 0 the f 0 f 0 the f D 0 [ k k

20 5-0735: Damic behavior of materials a structures back to the lasticit roblem Differetial equatio: sig [ t [ t Dtg[ t Dt sig D D sig Iitial coitio: 0 0 lus iscrete evolutio costraits: f 0 D 0 ( D ) 0 f State variable: 0 0 if if D Deeet variables: ( ) k D 0 the f 0 f 0 the f D 0 [ k k Mai ukow: Plastic multilier D which makes our roblem more comlex tha solvig a oriar first orer ifferetial equatio!

21 5-0735: Damic behavior of materials a structures Retur Maig Algorithm We solve the lasticit roblem assumig a strai-rive rocess, i.e. for a give icremet i the alie total strai, D we etermie umerical aroximatios of the corresoig stress a state variables at time t + base o their values at time t. Alie total strai icremet D OUTPUT: State variables at time t, RTURN MAPPING ALGORITHM State variables at time t +, Stress at time t + ( ) 2 2 2

22 5-0735: Damic behavior of materials a structures Retur maig roceure Whe comutig the solutio at time t +, we first comute the elastic state b assumig that the material resose is urel elastic (o lastic evolutio) whe alig D : ( ) ( D ) ( D ),, f k[ if f 0 f if 0 the elastic loaig ste the lastic loaig ste

23 5-0735: Damic behavior of materials a structures Retur maig roceure ( D ) f 0 f 0 f 0 f 0 f 0 elastic ste elastic ste lastic ste lastic ste elastic ste

24 5-0735: Damic behavior of materials a structures Retur maig roceure lastic loaig ste: D 0 f f 0 State variables at time t, Alie total strai icremet D Calculate Trial State, f f 0 OUTPUT: State variables at time t + Stress at time t + D 0 ( )

25 5-0735: Damic behavior of materials a structures Retur maig roceure Plastic loaig ste: f 0 D 0 I a lastic loaig ste, the lastic multilier D>0 must be etermie such that the iel coitio at time t + is full fille. f k () Firstl, we exress the absolute value of the stress + as a fuctio of the ukow lastic multilier: while D ( D ( ) ) D A hece D sig ( ) ( D )sig[

26 sig[ sig[ sig[ : Damic behavior of materials a structures Retur maig roceure ( D )sig[ ( D ) observe that sig[ sig[ D (2) Secol, we exress the flow stress k + as a fuctio of the ukow lastic multilier: D k k[ D (3) The, usig the results (2) a (3) i (), we obtai the so-calle iscrete cosistec coitio: f k D k[ D

27 5-0735: Damic behavior of materials a structures Retur maig roceure f D k[ D 0 ( D ) k k D D e k[ D D

28 : Damic behavior of materials a structures Solvig the iscrete cosistec coitio 0 ) ( ) ( ) ( 0 0 D D D D D H f H H k H k f xamle #: Liear hareig law H k k 0 [ with costat hareig moulus H The iscrete cosistec coitio the reas H f D from which we etermie the lastic multilier f D x H f ) (

29 5-0735: Damic behavior of materials a structures Solvig the iscrete cosistec coitio xamle #2: Geeral o-liear cocave hareig law k[ k The iscrete cosistec coitio the reas f f D k[ k D D D k[ D k k[ D k 0 k D Which corresos to seekig the root of the covex fuctio f [x f x k[ x k D x

30 5-0735: Damic behavior of materials a structures Solvig the iscrete cosistec coitio x0 x x2 x Seekig the root of a C -cotiuous fuctio is a staar roblem i alie mathematics. For examle, it ca be fou usig a Newto-Rahso scheme: x 0 [ x0 [ x [ x x 0 x0 x2 x x x '[ x '[ x '[ x iterate util x TOL the D x [

31 : Damic behavior of materials a structures lasto-lastic Taget Moulus The erivative / is calle elasto-lastic taget moulus. Durig lastic tesile loaig, we have 0 ) ( k k f a thus 0) 0, 0, ( k k k ) ( k k

32 5-0735: Damic behavior of materials a structures lasto-lastic Taget Moulus Formall, we ote the icremetal stress-strai resose as e ( ) with if 0 e e k k if

33 State variables at time t, : Damic behavior of materials a structures Summar: Retur Maig Algorithm Solve: State variables at time t + ( D )sig[ D Alie total strai icremet D Calculate Trial State, f f 0 f 0 D k[ D D 0 OUTPUT: OUTPUT: State variables at time t + Stress at time t + ( ) Stress at time t + ( ) D 0

34 5-0735: Damic behavior of materials a structures Reaig Materials for Lecture #4 J.C. Simo a T.J.R. Hughes, Comutatioal Ielasticit (first chater): htt://lik.sriger.com/book/0.007%2fb98904 M.. Gurti,. Frie, L. Aa, The Mechaics a Thermoamics of Cotiua, Cambrige Uiversit Press,