A new formulation of internal forces for non-linear hypoelastic constitutive models verifying conservation laws

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1 A ew formulatio of iteral forces for o-liear hypoelastic costitutive models verifyig coservatio laws Ludovic NOELS 1, Lauret STAINIER 1, Jea-Philippe PONTHOT 1 ad Jérôme BONINI. 1 LTAS-Milieux Cotius et Thermomécaique, Uiversity of Liège, Chemi des Chevreuils 1, 4 Liège, Belgium SNECMA Moteurs Egieerig Divisio, cetre de Villaroche, 7755 Moissy-Cramayel, race. Secod MIT Coferece o luid ad Solid Mechaics K.J. Bathe (ed) 17- Jue 3 Bosto, USA, Vol. 1 pages

2 57 A ew formulatio of iteral forces for o-liear hypoelastic costitutive models verifyig coservatio laws L. Noels a,, L. Staiier a, J.-P. Pothot a, J. Boii b a Uiversity of Liège, LTAS-Thermomechaics, Chemi des Chevreuils 1, Bât. B5/3, 4 Liège 1, Belgium b SNECMA-Moteurs, Egieerig Divisio, cetre de Villaroche, 7755 Moissy-Cramayel, race Abstract This paper proposes a ew formulatio of the iteral forces for hypoelastic costitutive models esurig that the elastic work of deformatio ca be restored by the scheme. Moreover, we demostrate that the work of the plastic deformatio is positive ad cosistet with the material model. Keywords: Eergy-mometum; Coservig algorithm; Dyamics; Hypoelastic material; Large strai plasticity; Costitutive models 1. Itroductio A ew kid of itegratio algorithms for dyamics has recetly appeared, which verifies the mechaical laws of coservatio (i.e. coservatio of the liear ad agular mometum ad of the total eergy) ad which remais stable i the o-liear rage. The first algorithm verifyig these properties was described by Simo ad Tarow 1. This algorithm was called Eergy Mometum Coservig Algorithm or EMCA. It cosists i a mid-poit scheme with a adequate evaluatio of the iteral forces, ad was derived for a Sait Veat Kirchhoff hyperelastic material. Laurse geeralized this approach to other hyperelastic models. His method ivolves resolvig iteratively a ew equatio for each material poit i order to compute the adequate Piola Kirchhoff stress tesor. Aother solutio, that avoids this iterative procedure ad leads to a geeral formulatio of the Piola Kirchhoff stress tesor, was proposed by Gozalez 3. The EMCA was recetly exteded to dyamic fiite deformatio plasticity by Meg ad Laurse 4. To our kowledge, the EMCA coservig algorithm was ever exteded to hypoelastic materials. This paper proposes such a extesio, through a ew expressio of the iteral forces, which verifies the coservatio laws. Moreover, we prove that this expressio remais cosistet whe plastic deformatios occurs. A umerical example illustrates the good accuracy of the proposed formulatio. Correspodig author. Tel.: ; ax: ; L.Noels@ulg.ac.be. The hypoelastic model Let the cofiguratio be the cofiguratio computed after time steps (i.e. at time t ). Let x be the deformatio mappig (coordiates) i the cofiguratio, adletx ξ be the odal coordiates for the ode ξ. With ϕ ξ the shape fuctio liked to ode ξ, it comes (Eistei s otatios are used) x = ϕ ξ x ξ (1) The gradiet of deformatio (two poit tesor) betwee cofiguratios m ad is idicated by m. The tesor f represets 1. This gradiet tesor ca be decomposed ito a rotatio tesor R ad a symmetric deformatio tesor U. The determiat of m is deoted by the scalar J m.the atural strai tesor E m, the Gree Lagrage strai tesor GL m, ad the Almasi strai tesor A m are respectively defied as E m = 1 l T m m GL m = 1 T m m I A m = 1 I f T m f m () The Cauchy stress tesor is evaluated i the cofiguratio ad is referred to as. or the hypoelastic costitutive laws, the Cauchy stress tesor is computed from a stress icremet deduced from the icremetal atural strai tesor. With H the Hooke fourth order tesor ad with the operatio H : E defied by H ijkl E kl, the fial rotatio scheme 3 Elsevier Sciece Ltd. All rights reserved. Computatioal luid ad Solid Mechaics 3 K.J. Bathe (Editor)

3 58 L. Noels et al. / Secod MIT Coferece o Computatioal luid ad Solid Mechaics 5,6 is defied by the followig relatio +1 = R +1 + H : E +1 s c (ε p ) R +1 T (3) I this relatio s c, resultig from the plastic deformatio, has a trace equal to zero ad is give by the radial retur mappig (for isotropic materials with J -plasticity) 6. The scalar ε p is the equivalet plastic strai. 3. The eergy-mometum coservig scheme 3.1. The mid-poit scheme 1 or a itegratio from time t to time t + t = t +1, if the variable z represets the positio or the velocity, it leads ż +1/ = z+1 z t = ż+1 + ż The balace law for ode ξ is M ξµ ẍ +1/ µ = +1/ ext +1/ it (4) ξ (5) where M is the mass matrix, +1/ ext ad +1/ it are respectively the exteral ad iteral forces evaluated i cofiguratio + 1. This expressio idirectly depeds o the positios i cofiguratio + 1(i.e.x +1 ). The goal of this work is to evaluate the expressio of the iteral forces i cofiguratio + 1 for hypoelastic models. The system of Eqs. (4) ad (5) is resolved iteratively. 3.. The coservatio coditios Eq. (5) has to verify the coservatio of the liear mometum: ξ =, (6) ξ +1/ it the coservatio of the agular mometum: x +1/ξ +1/ ξ it =, (7) ad the coservatio of the eergy 1: ξ +1/ it x +1 x ξ = W +1 it W it + it (8) where W it ad it are respectively the iteral eergy ad the plastic dissipatio. The operator deotes the vector product ad the operator deotes the scalar product. 4. Iteral forces expressio for the hypoelastic model Let D be the derivative of the shape fuctio i the referece cofiguratio D ξ = ϕ ξ / x. The, the followig expressio is proposed ξ 1 = +1/ it it + 1 ξ 1 = V V ξ 1 it = + 1 V { I + +1 V it + ξ it { I + +1 f T D ξ J dv c : GL +1 GL +1 : GL +1 f T D ξ J dv { I + f f +1 { I + f +1 c : +1 A +1 : A +1 T D ξ J +1 dv A +1 f +1 T D ξ J +1 dv (9) where c ad c are tesors yet to be determied. This expressio ca be show to coserve the properties of the fial rotatio scheme ad of the radial retur mappig scheme Verificatio of coservatio laws The coservatio of the liear mometum (6) is directly ( reached usig the properties of the shape fuctios ξ Dξ K = ). The coservatio of the agular mometum (7) is a cosequece of the symmetry of the Cauchy stress tesor 7. Sice o potetial ca be defied for a hypoelastic material model we defie a loadig uloadig cycle, from cofiguratio 1 to 3, where the iitial Cauchy stress tesor correspods to the fial Cauchy stress tesor for ay rotatio Q 3 = Q 1 Q T (1) Durig the loadig phase from cofiguratio 1 to, plastic deformatios occur. Durig the uloadig phase from cofiguratio to 3, there is o subsequet plastic deformatio. The Eq. (9) is cosistet with Drucker s Postulate if the reversible work of the first step is recovered durig the secod step. Therefore, the eergy balace betwee cofiguratios 1 ad 3 ca be expressed as 3/ it ξ x x 1 ξ + 5/ it ξ x 3 x ξ = it (11) Let E el 1 be the elastic atural strai tesor defied such that H : E el 1 H : E 1 sc 1 (1) The we defie U el 1 a symmetric tesor such that E el 1 1 U l el 1 Uel 1 (13) The elastic ad plastic Gree Lagrage strai tesor (GL el 1 ad GL pl 1 ), ad the elastic ad plastic Almasi strai tesor

4 L. Noels et al. / Secod MIT Coferece o Computatioal luid ad Solid Mechaics 59 (A el 1 ad Apl 1 ) are defied from Uel 1 as GL el 1 1 U el1 Uel1 I GL pl 1 GL 1 GLel 1 A el 1 1 R 1 I U el 1 1 U el 1 1 R 1T A pl 1 A 1 Ael 1 (14) Therefore, sice the tesors c ad c ca be take equal to zero whe o plastic deformatio occurs 7, we ca rewrite Eq. (11) as 7: it = 1 {GL pl1 + c : 1 J,1 V + A pl1 + c : J, dv (15) If D it is the aalytical expressio of the volumic dissipatio durig trasformatio from cofiguratio 1 to 3, the followig expressio of the tesors D it c = 1 J 1 : J J GL 1 + GLel 1 D it c = 1 J 1 : 1 J 1 J A 1 + Ael 1 (16) leads to it = {D it dv (17) V which is cosistet with the eergy coservatio laws 7. Moreover, the magitude of the tesors c ad c is of secod order i the icremet of the plastic deformatio 7 ad thus ca be omitted if the time step size is small eough. 5. Numerical example Results obtaied with the coservative algorithm (EMCA) are compared with the results obtaied with the Newmark algorithm (NMK) (first Newmark parameter β equal to.5 ad secod Newmark parameter γ equal to.5) Taylor s bar problem This example was first simulated with a coservative algorithm for a hyperelastic Sait Veat Kirchhoff material by Meg ad Laurse 4. It cosists i a cylidrical bar (Table 1), discretized by 576 elemets. It has a iitial velocity ẋ. The time step size is equal to.1 µs. ig. 1 represets the evolutio of the total eergy (sice o potetial exists the total eergy is the work of the iteral ad of the iertial forces, iteral dissipatio icluded). ig. represets the evolutio of the iteral dissipatio. The fial plastic strais Table 1 Properties of the Taylor bar problem Property Value Exteral diameter d e = 6.4 mm Legth l = 3.4 mm Desity ρ = 893 kg/m 3 Youg modulus E = N/m Poisso ratio ν =.35 Yield stress σ = 4 N/mm Hardeig parameter h = 1 N/mm Iitial velocity ẋ = 7 m/s 6 Total eergy (N m) NMK EMCA,E+,E-5 4,E-5 6,E-5 8,E-5 Time (s) ig. 1. Evolutio of the total eergy for the Taylor bar problem.

5 53 L. Noels et al. / Secod MIT Coferece o Computatioal luid ad Solid Mechaics 6 Iteral dissipatio (N m) NMK EMCA,E+,E-5 4,E-5 6,E-5 8,E-5 Time (s) ig.. Evolutio of the total iteral dissipatio for the Taylor bar problem. NMK EMCA ig. 3. Equivalet plastic strai for the Taylor bar problem after 8 µs. are illustrated i ig. 3. Results obtaied with both schemes are similar. The Newmark scheme requires 1951 iteratios ad is.5% more expesive tha the coservative scheme (194 iteratios). 6. Coclusios A ew expressio of the iteral forces for a hypoelastic material was preseted. Whe used with the mid-poit scheme, this expressio leads to a eergy-mometum coservative scheme. Moreover, the iteral dissipatio, resultig from the plastic deformatio is cosistet with the laws of thermodyamic. The solutio obtaied with this coservative scheme was compared with the results obtaied with the Newmark scheme for the Taylor bar impact, leadig to similar results. Note that the EMCA is expected to show a improved performace for larger time steps, whe the secod order terms gai importace, but this is part of a future work.

6 L. Noels et al. / Secod MIT Coferece o Computatioal luid ad Solid Mechaics 531 Ackowledgemets L. Noels ad L. Staiier wish to ackowledge support from the Belgia Natioal ud for Scietific Research (.N.R.S.). Refereces 1 Simo JC, Tarow N. The discrete eergy-mometum method. Coservig algorithms for oliear elastodyamics. ZAMP 199;43: Laurse T, Meg X. A ew solutio procedure for applicatio of eergy-coservig algorithms to geeral costitutive models i oliear elastodyamics. Comput Methods Appl Mechaics Egrg 1;19: Gozalez O. Exact eergy ad mometum coservig algorithms for geeral models i oliear elasticity. Comput Methods Appl Mechaics Egrg ;19: Meg X, Laurse T. Eergy cosistet algorithms for dyamic fiite deformatio plasticity. Comput Methods Appl Mechaics Egrg 1;191: Nagtegaal J, Veldpaus. O the implemetatio of fiite strai plasticity equatios i a umerical model. I: Pittma et al. (Eds), Numerical Aalysis of ormig Processes, Joh Wiley, Pothot JP. Uified stress update algorithms for the umerical simulatio of large deformatio elasto-plastic ad elastoviscoplastic processes. It J Plasticity ;18: Noels L, Staiier L, Pothot JP. Eergy-mometum coservig algorithm for o-liear hypoelastic costitutive models. It J Numer Methods Egrg 3; submitted.