Code_Aster. Viscoplastic constitutive law VISC_DRUC_PRAG

Transcription

1 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 1/17 Viscoplastic costitutive law VISC_DRUC_PRAG Summarized: This documet the mod describes viscoplastic costitutive law VISC_DRUC_PRAG based o astoplastic of Drucker-Prager ad takig ito accout viscosity accordig to a mod power of the Perzya type. Its scope of applicatio is the mudstoe which is the rock host of the cocept of storage. The mod proposed comprises oly oe viscoplastic mechaism. The criterio is hammer-hardeed with the viscoplastic strai cumulated via three thresholds: astic, of peak ad ultimate. Flow is oassociated, the flow potetial beig a potetial of Drucker-Prager beig hammer-hardeed accordig to three levs: astic, of peak ad ultimate. Betwee the thresholds, hardeigs are liear. This mod ca be used i a pure mechaical modizatio as it ca be used i a modizatio THM. It is available i 3D, plae strais ad axisymmetric. It is itegrated by the solutio of oly oe scalar equatio oliear.

2 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 2/17 Cotets 1 Notatios Itroductio Formulatio of viscoplastic mod VISC_DRUC_PRAG Equatios of the mod Itegratio i Decompositio of the tesor of déformatio Update of stresses taget Operator cohéret Data matériaux local variables Abstract of the algorithm of résolutio Results of test triaxial Features ad vérificatio Référeces Descriptio of the versios of the documet17...

3 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 3/17 1 Notatios idicate the tesor of the effective stresses i small disturbaces, oted i the shape of the followig vector: Oe otes: asticity tesor D e I 1 =tr first ivariat of the stresses s= I 1 3 I tesor of the stresses déviatoires s II = S.S secod ivariat of the tesor of the stresses déviatoires = 3 eq 2 s s I 1 s = I 1 3 I = 3 eq 2 s s equivalet stress traces astic predictio of the stresses tr = 3 I deviator of the strais v =tr volumial strai ṗ= 2 3 vp vp tesor of the stresses déviatoires of the astic predictio of the stresses equivalet stress of the astic predictio of the stresses cumulated viscoplastic deviatoric strais f viscoplastic surface of load G potetial of viscoplastic flow 0, R 0 ad hardeig parameters correspodig to the threshold of 0 asticity ( p=0 ) pic, R pic ad pic p= hardeig parameters correspodig to the peak ult, R ult ad ult threshold p= p ult hardeig parameters correspodig to the ultimate amplitude vocity of the urecoverable deformatios A parameter of creep power of the creep mod pressure of referece

4 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 4/17 2 Itroductio This documet describes the itegratio of viscoplastic costitutive law VISC_DRUC_PRAG i. This mod comprises oly oe viscoplastic mechaism. The viscoplastic criterio is hammer-hardeed with the deviatoric viscoplastic strai cumulated via three thresholds: astic for a viscoplastic strai ull, a threshold kow as of peak for a viscoplastic strai kow as of peak (parameter of the mod) ad a ultimate threshold for a viscoplastic strai kow as ultimate (parameter of the mod). Betwee the thresholds, the fuctios of hardeig are liear. I there exists aother mod based o the mod of Drucker_Prager ad used i élastoplasticié i a form associated i ame DRUCK_PRAGER or ot associated uder ame DRUCK_PRAG_N_A (see [R ]). 3 Formulatio of viscoplastic mod VISC_DRUC_PRAG 3.1 Equatios of the mod This mod is based o a viscoplastic formulatio of the Drucker-Prager type, where the surface of load is defied by: f = 3 2 s II p I 1 R p p ad R p are fuctios of the cumulated viscoplastic strai p, Oe itroduces a viscoplastic flow potetial G : G= 3 2 s II p I 1 ; For the evolutio of the criterio f ad potetial G we distiguish three thresholds distict correspodig to three values from the variable from hardeig: a astic threshold, a threshold of peak ad a ultimate threshold. Betwee these thresholds, hardeig is liear. Betwee the astic threshold ad the threshold of peak, hardeig is positive, after the peak hardeig is egative ad becomes costat after the ultimate threshold. The fuctios rated to cohesio are writte i the followig form: p= pic 0 p 0 for 0 p p= ult pic p ult p pic p p ult p= ult for p p ult the fuctios rated to dilatacy are writte i the followig form: p= pic 0 p 0 for 0 p p= ult pic p ult p pic p p ult p= ult for p p ult

5 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 5/17 the fuctios of hardeig are writte: R p= R pic R 0 pr 0 for 0 p R p= R R ult pic p ult p R pic p p ult R p=r ult for p p ult the stresses are coected to the strais by the Hooke's law: =D e vp Whe the viscoplastic threshold is reached, of the viscoplastic urecoverable deformatios are geerated ad expressed accordig to the theory of Perzya by: d vp f = A G dt ; f beig the criterio of viscoplasticity; A ad are parameters of the mod; a pressure of referece. vp with, ad from where G = 3 s II I p 1 ad 2 ṗ= 2 3 vp vp the deviator of the strai tesor, s II = s II s kl = s kl s kl s II ik jl 1 3 kl = s s II I 1 = tr = G = 3 2 s s II p

6 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 6/17 Summarized equatios: The criterio: f = 3 2 s II [ pic 0 p 0] I 1 [ R R pic 0 pr 0] for 0 p f = 3 2 s II [ ult pic p ult p pic] I 1 [ R ult R pic p ult p R pic] p p ult f = 3 2 s II ult I 1 R ult for p p ult : Flow potetial: G= 3 2 s II [ pic 0 p 0] I 1 for 0 p G= 3 2 s II [ ult pic p ult p pic ] I 1 p p ult G= 3 2 s II ult I 1 for p p ult 0, R 0 ad 0 : hardeig parameters correspodig to the threshold of asticity ( p=0 ) pic, R pic ad pic : hardeig parameters correspodig to the parameter ult, R ult ad ult : hardeig parameters correspodig to the parameter p ult the Hooke's law: =D e vp f, p 0 fid of asticity; vp =0 f, p 0 viscoplasticity ; vp =A f G ; ṗ= 2 3 vp vp 4 Itegratio i 4.1 Decompositio of the strai tesor the decompositio of the icremet of total deflectio is writte: = e vp where e ad vp are the icremets of the astic ad viscoplastic tesors. 4.2 Update of the stresses

7 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 7/17 the followig Notatios are adopted: A -, A ad A respectivy idicatig the quatities at the begiig, time step ad its icremet lastig the step. Oe expresses the stresses brought up to date at time + compared to those calculated at time -: = D e e ; s=s 2 e ; I 1 = I 1 3K v e =s I 1 3 ; = tr 3 = v 3 ; I 1 =tr ; v =tr ; Elastic predictio: = D e ; s =s 2 ; I 1 =I 1 3K v Elastic solutio Computatio of the icremet of the stresses i astic mode: = s I 1 3 ; = v 3 =2 3K v 3 =2 K =2 v tr 3 K tr =2 K 2G 3 tr { }=[ K K 2 K 2 ] K K K K 2 K K Viscoplastic solutio D e Oe expresses the stress fid at time +: = e D kl kl e = e D kl vp kl kl = e D kl vp kl.{ } s =s vp 2 ad I 1 = I 1 3K v vp =s I 1 3

8 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 8/17 which is writte by replacig the icrease i the viscous strais by their statemets i the form: = D kl G, p t with =A f, p where ad G the amplitude ad the directio vocity of the urecoverable deformatios characterize. f, p beig the criterio of viscoplasticity, A ad are parameters of the mod. The viscoplastic criterio at time + is writte: f, p = f e D kl G, p t, p The icremet of the viscoplastic strai beig, vp = G t= 3 2 the viscoplastic volumial strai beig, v vp =3 p t s p s t II the deviatoric compoet of the viscoplastic strai is writte i the form: vp = s II like eq = s t or vp s = 3 II 2 s eq t, s II =s s ad eq= 3 2 s s Oe writes also the followig equalities: s eq =s eq p= 2 3 vp vp p t = = A f, p éq 1 from where: p= t By meas of these equalities oe ca fid a statemet for s, eq ad I 1 accordig to I 1 ad p : s =s 2 vp =s 3 s t=s 3 s eq s =s 1 3 eq t =s 3 1 p eq t eq s eq, eq = eq 3 t= eq 3 p éq 2

9 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 9/17 I 1 = I 1 3K v vp =I 1 9K t=i 1 9K p éq Computatio of the ukow the viscoplastic icremet of cumulated strai p is the oly ukow of the problem. To determie it, the viscoplastic flow mod is writte (éq 1): p t = A eq p I 1 R p R p =R p p=r R cost p ; R cost = R p p = p p = cost p ; cost = p p = p p= cost p ; cost = p By preoccupatio with a simplificatio of the writig of the equatio i p, oe poses: C= A t Maybe, while replacig eq ad I 1 by their statemets (éq 2 ad éq 3), oe obtais: F p=c eq I 1 R 3 R cost cost I 1 9k p p=0 9k cost 9k cost p 2 9k cost cost p 3 Oe seeks p/ p=0 F p=0 is a oliear scalar equatio. The lower limit beig x if =0 ad the higher limit ca be built-i with x sup = A eq I 1 R t Oe uses the method of the ropes with a cotrol of the iterval of search while takig as a startig poit the documet [R ]. p [ x if, x sup ] ; x= p So F x if the p=x if So F x sup the p=x sup So F x if 0 the x 2 = x if ad y 2 =F x if

10 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 10/17 So F x sup 0 the oe makes a loop by cuttig out x sup by 10 util obtaiig a value of x sup for which F x sup 0 i this case oe multiplies the last solutio by 10 ad oe fixes x 1 =x sup ad y 1 =F x sup So F x sup 0 the oe makes a loop by multiplyig x sup by 10 util obtaiig a value of x sup for which F x sup 0 ad oe fixes x 1 =x sup ad y 1 =F x sup So F x if 0 the x 1 =x if ad y 1 =F x if So F x sup 0 the oe makes a loop by cuttig out x sup by 10 util obtaiig a value of x sup for which F x sup 0 i this case oe multiplies the last solutio by 10 ad oe fixes x 2 = x sup ad y 2 =F x sup So F x sup 0 the oe makes a loop while multiplyig x sup by 10 util obtaiig a value of x sup for which F x sup 0 ad oe fixes x 2 = x sup ad y 2 =F x sup Of the checks are made o the values which the limits ca take ad i particular if they are weaker tha a tolerace built-i with 1.E-12, they will be cosidered equal to 0. ad thus the solutio p also. If the limits are equal, oe makes a recuttig of time step. The values x 1 x 2, y 1 ad y 2 will be the values to be give as starter to the routie zeroco which is based o the method of the ropes. The solutio is calculated by the followig formula: x 1 =x 1 F x 1 x x 1 F x F x 1 With the followig values, oe represets the scalar fuctio to solve. eq 6,315 MPa 6, I 1 21,061 MPa 0,147 N 4,5 R 1,394 MPa t 10 s cost 13. A 1, cost 10. 0,1 MPa R cost 329,732 MPa The ukow x for which F x cacs itsf locates betwee ad who is wl betwee the lower limit x if ad the higher limit x sup which is worth i this precise case 1,

11 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 11/17 Figure 4-1: Pace of the scalar fuctio 4.3 coheret taget Operator Oe seeks to calculate: = s I 3 I 1 With, s p s 3 = 1. 3 eq I 1 = I 1 9K p p Computatio of s : s =2 I d eq s pq =2 ip jq 1 3 pq 2. p s eq 3 eq. s p Computatio of I 1 : I 1 =3K1

12 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 12/17 I 1 =3K pq pq Computatio of eq : = eq = eq s eq s = 3 2 s eq I d De = 3 2 s D e eq Computatio of p p t = A f, p : that is to say F p= At f, p p p = p to compute: p, oe uses F, p =0 F, p p = F p=c F, p p=0 p F, p F, p p eq I 1 R 3 R cost cost I 1 9k p p=0 9k cost 9k cost p 2 9k cost cost p 3 F, p =C.. f, p 1 f, p where f, p F, p p = eq I 1 cost I 1 p e =C.. f, p 1 f, p 1 p

13 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 13/17 f, p = 3R p cost cost I 1 9k 2 p 9k cost cost 3 p 2 9k cost cost eq Computatio of : eq = eq s s = 3 2 I 1 Computatio of : I 1 = tr =1 s eq. I d = 3 2 s eq 4.1 Data materials the 16 parameters of the mod are: uder ELAS E : Youg modulus ( Pa or MPa ) : Poisso's ratio uder VISC_DRUC_PRAG : pressure of referece ( Pa or MPa ) A : viscoplastic parameter (i s 1 ) : power of the mod creep : rate of hardeig o the lev of the threshold of peak p ult : rate of hardeig o the lev of the ultimate threshold 0, pic ad ult : parameters of the fuctio of cohesio p R 0, R pic ad R ult : parameters of the fuctio of hardeig R p 0, pic ad ult : parameters of the fuctio of dilatacy p 4.2 local variables v 1 = p ; v 2 = 0 ou 1 ; idicator of plasticity; v 3 = pos ; positio of the poit of load compared to the thresholds; ( pos=1 si 0 p ; pos=2 si p p ult ; pos=3 si p p ult ) v 4 ; ombre of iteratios local;

14 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 14/ Summarized algorithm of resolutio the algorithm of resolutio such as it is implemeted i : = D e The criterio: f, p = eq p I 1 R p Elasticity : f, p 0 p=0 ; Viscoplasticity: f, p =0 p 0 with p solutio of the equatio F p=0 where, p t = A eq p I 1 R p ad F = A t f, p p Put up to date of the stresses: = D e vp s=s 1 3 p eq eq = eq 3G p I 1 =I 1 9K p = A f, p =s I 3 I 1 Oce p is calculated, the stresses ad the up to date put local variables, oe checks the positio from p ratio with p ad sigs it f, p : If 0 p ; to test 1) if ot 2) if ot 3) If p p ult ; to test 2) if ot 3) If p p ult ; to test 3) If p p ; oe checks f, p 0 with R, ad correspodig to 0 p, so f, p 0 the oe updates the stress fids ad of local variables, if ot, oe cosiders that p is ot valid ad oe redécoupe time step

15 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 15/17 If p p p ult ; oe checks f, p 0 with R, ad correspodig to p p ult so f, p 0 the oe updates the stress fids ad of local variables, if ot, oe cosiders that p is ot valid ad oe redécoupe time step If p p p ult ; oe checks f, p 0 with R, ad correspodig to p p ult so f, p 0 the oe updates the stress fids ad of local variables, if ot, oe cosiders that p is ot valid ad oe redécoupe time step

16 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 16/17 5 the Results of a triaxial compressio test It acts to simulate a triaxial compressio test (see the case test ssv211) while imposig like a stress of cotaimet of 5 MPa. A uiaxial strai is imposed i compressio ad which evolves i time. The vocity of the loadig is fixed at 10 5 m/s. The deviator of the stresses ad the volumial strai accordig to the imposed axial strai are represeted Ci agaist. Figure 5-1: Deviator of the stresses accordig to the uiaxial strai Figure 5-2: Volumial strai accordig to the uiaxial strai

17 Titre : Loi de comportemet viscoplastique VISC_DRUC_PRAG Date : 29/05/2013 Page : 17/17 6 Fuctioalities ad checkig the costitutive law ca be defied by the key word VISC_DRUC_PRAG (commad STAT_NON_LINE, key word factor COMP_INCR). It is associated with material VISC_DRUC_PRAG (commad DEFI_MATERIAU). Mod VISC_DRUC_PRAG is checked by the cases followig tests: SSNV211 [V ] triaxial Compressio test draied with the mod VISC_DRUC_PRAG WTNV137 [V ] triaxial Compressio test draied with the mod VISC_DRUC_PRAG WTNV138 [V ] triaxial Compressio test ot draied with the mod VISC_DRUC_PRAG 7 Refereces [1] J. EL GHARIB ad C. CHAVANT, Chock o triaxial compressio tests of a viscoplastic costitutive law for the mudstoe based o the mod Drucker_Prager, H-T FR, [2] J. EL GHARIB ad C. CHAVANT, Implemeted i of a simplified viscoplastic mod, H-T FR, 8 Descriptio of the versios of the documet Aster Author (S) Orgaizatio (S) 10.0 J. EL GHARIB, C.CHAVANT EDF R &D/AMA Descriptio of the modificatios iitial Text