A VISCOELASTIC-VISCOPLASTIC-DAMAGE CONSTITUTIVE MODEL BASED ON A LARGE STRAIN HYPERELASTIC FORMULATION FOR AMORPHOUS GLASSY POLYMERS

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1 0 h Inernaonal Conferene on Comose Maerals Coenhagen, 9-4 h July 05 A VISCOELASTIC-VISCOPLASTIC-DAMAGE COSTITUTIVE MODEL BASED O A LARGE STRAI HYPERELASTIC ORMULATIO OR AMORPHOUS GLASSY POLYMERS V.-D. guyen, X. Morelle,. Lan, T. Pardoen, C. Bally 3 and L. oels Comuaonal & Mulsale Mehanal of Maerals (CM3), Dearmen of Aerosae and Mehanal Engneerng (LTAS), Unrsy of Lège Quarer POLYTECH, allée de la Déoure 9, 4000 Lège, Belgum Emal: {vandung.nguyen, l.noels}@ulg.a.be, web age: h:// Maeral and Proess Engneerng (IMAP), Insue of Mehans, Maerals and Cvl Engneerng (MMC), Unrsy of Louvan Plae Sane Barbe be L5.0.0 à 348 Louvan-la-eu, Belgum Emal: {xaver.morelle, freder.lan, homas.ardoen}@ulouvan.be, web age: h:// 3 Bo and Sof Maer (BSMA), Insue of Condensed Maer and anosenes (IMC), Unrsy of Louvan Crox du Sud be L à 348 Louvan-la-eu, Belgum Emal: hrsan.bally@ulouvan.be, web age: h:// Keywords: Hyerelas, Vsoelasy, Vsolasy, Sauraon damage, Glassy olymers ABSTRACT The am of hs work s o delo an effen large-sran hyerelas onsu model for amorhous olymers n he glassy sae. These maerals exhb a omlex rae- and ressure-sensble behavor n boh elas and las regmes. Afer an nal lnear elas regon, a nonlnear sage onnues unl reahng a eak sress, whh s followed by a sofenng sage. A large srans, when he sofenng s sauraed, a re-hardenng sage s reahed. The vsoelas effe s aured usng he generalzed Maxwell model. The vsolas effe s onsdered usng a Perzyna-ye flow rule nororang a ressure sens yeld surfae and a non-assoaed flow oenal. Ths yeld surfae s exended from he Druker-Prager one. The sauraed sofenng henomenon s modeled usng an soro numeral damage varable rogressed by a sauraed sofenng law. Wh he nroduon of he damage arameer, a non-loal ml graden damage model s used o avod he loss of he soluon unqueness. Through exermenal omarsons, s shown ha he roosed model has he ably o model he omlex mehanal resonses of amorhous glassy olymers. ITRODUCTIO The amorhous olymers n he glassy sae are ofen used as marx maerals n olymer renfored omoses, whh aear n a large range of alaons n he aerosae and auomo ndusres. In her glassy sae, hese maerals exhb rae-, me-, ressure-, and emerauredeenden resonse, see [] for more deals n ase of he RTM6 eoxy resn. Moreor, dfferen sages an be denfed n he sress-sran urs from exermens []. Afer an nal lnear elas regon, a nonlnear sage onnues unl reahng a eak sress, whh s followed by a sofenng sage. A large srans, when he sofenng s sauraed, a re-hardenng sage s reahed. The rae effes are sgnfan en a low sran lel. As a onsequene, a onsu model nororang he vsoelas, vsolas, and damage ngredens s neessary o aure he whole range of he olymer resonse. Usng a hyerelas formulaon, he onsu model s based on an elas oenal, from whh he frs Pola-Krhhoff an be obaned by dervaon wh rese o s work onjugaed measure, whh s he deformaon graden. The vsoelas effe s modeled usng he generalzed

2 V.-D. guyen, X. Morelle,. Lan, T. Pardoen, C. Bally and L. oels Maxwell model onsrued from he oroaonal Krhhoff sress and he elas logarhm sran. The vsolas effe s based on a Perzyna-based flow rule [3] nororang a ressure-sens yeld funon and a non-assoaed flow oenal. Ths yeld surfae s exended from he Druker- Prager one. A quadra flow oenal s used o orrely aure he volumer deformaon durng he las roess wh a onsan las Posson rao [4]. The sauraed sofenng henomenon s nrodued o he onsu law hrough an soro damage varable, whh s sauraed by a lm value. The rehardenng sage s naurally obaned afer he damage sauraon beause of he knema and soro hardenng henomena, whh are sll delong. Wh he nroduon of he damage, he non-loal ml graden damage model [5] s used o avod he loss of he soluon unqueness. Wh hese ngredens, he roosed model s shown o aure he full range of he mehanal behavor of he amorhous glassy olymers a varous sran raes, as omared o he exermens. COSTITUTIVE MODEL ollowng he sandard mulla deomoson n elasolas maerals a fne srans, he deformaon graden ensor s deomosed no a reorable (vsoelas) omonen and an rreorable (vsolas) omonen, followng v v, () where and are he vsoelas and vsolas ensors, resely. Based on he onnuum damage mehans, he damage reresens he relaon beween he sresses n he aaren body and n s undamaged reresenaon [6]. Ths relaon an be wren as ˆ P P, () D where P s he frs Pola-Krhhoff sress n he aaren body, Pˆ s he effe (undamaged) ounerar n he undamaged reresenaon, and 0 D s he numeral damage varable used o reresen he sofenng resonse. In hs work, he sauraon damage henomenon s onsdered wh a lm value D. The hyerelas formulaon s based on he exsene of an elas oenal. The sress measures are derd from hs oenal funon by he relaon ˆ P : κˆ :( ) : E, (3) where Pˆ s he effe frs Pola-Krhhoff sress, where κˆ s he effe Krhhoff sress and where s he effe oroaonal Krhhoff sress, whh s he sress measure onjugaed o he vsoelas logarhm sran E ln C wh he vsoelas rgh Cauhy sran ensor T C. The effe Cauhy sress an be omued from he relaon σ Pˆ T ˆ J J κˆ, wh J de.. Vsoelas ar gure : Srngs/dashos nework of he generalzed vsoelas Maxwell model.

3 0 h Inernaonal Conferene on Comose Maerals Coenhagen, 9-4 h July 05 The vsoelas behavor s modeled usng he generalzed Maxwell model. Ths model onans + elas srngs and dashos as shown n gure. The oal elas oenal s gn by D, (4) where wh,,..., are he elas oenals ang on he srngs and where D wh are he dssang funons ang on he dashos. In eah srng, he elas oenal s defned as,..., K and G K ln J G deve : deve wh,,...,, (5) where are resely he bulk and shear modul and where J de 0 s he vsoelas Jaoban. rom he elas oenal gn by Eq. (5), he relaed effe oroaonal Krhhoff sress s derd as q E E D E D E K r( E ) I G deve q wh K r( E ) I G deve,...,, (6) where s he nernal varables gornng he vsoelas behavor. The vsoelas effe s added hrough he nernal varables by a reardaon aon. or he srng/dasho branh, he vsoelas equaons are gn as follows where g and k dev ˆ r τ devq, rq,,...,, (7) g k are he haraers reardaon mes for he devaor and volumer ars, resely. Equaons (6) and (7) lead o he soluon of he effe oroaonal Krhhoff sress on eah srng/dasho branh. The oal effe oroaonal Krhhoff sress he onvoluon form n me as ( s omued from ˆ τ C ( s) : E ( s) ds, (8) s where he fourh order Hook ensor s gn by C jkl K( K G( G ( K( G( j kl K ex G ex k g k,. jl l jk j kl, 3 nally, he effe frs Pola-Krhhoff sress s esmaed usng Equaons (3) and (8) as lnc C. (9) vt P ˆ :. (0)

4 V.-D. guyen, X. Morelle,. Lan, T. Pardoen, C. Bally and L. oels. Vsolas ar In order o model he vsolas behavor, a Perzyna-ye vsolas flow rule [3] s used. The vsolas deformaon evoluon an be esmaed wh he followng relaon D D P, 0 f 0, () f 0 where s he vsolas sran rae, where s he yeld funon, where P s he flow oenal, where s he vsosy, and where s he rae sensvy exonen. In he vsolas range, a new yeld ondon an be defned as whh s omleed by he Kuhn-Tuker ondon, - 0, wh, () (3) To model he ressure sens effe n amorhous glassy olymers, a generalzed rson of he lassal Druker-Prager s onsdered wh a ower-enhaned oahedral erm of exonen, suh as eq m r( χ) m m, r( χ) χ m m, eq 3 devχ : devχ, m. (4) In Eq. (4), χ b s he ombned sress ensor, s he effe oroaonal Krhhoff sress ensor exressed n Eq. (8), s he bak-sress ensor, whh s used o aure he knema hardenng effe, and are he soro omresson and ensle yeld sresses, resely, and m s he raon/omresson yeldng ansoroy. The arbrary value of he exonen an be used leadng o a new lass of yeld surfae, so-alled ower yeld surfaes. If, he lassal Druker-Prager one s reored; f, he arabolodal one s obaned. Boh of hem an be used for amorhous glassy olymers [4]. A non-assoaed flow rule s assumed n hs work. The flow oenal n Eq. () s sefed wh a quadra funon of he oroaonal Krhhoff sress as b r( χ) P eq 3, (5) where s he flow arameer. Usng hs non-assoaed flow, he onsu model has he ably o aure he volumer las deformaon wh a onsan las Posson rao, whh s drely relaed o he flow arameer hrough he relaon [4] 9. (6) 8 The equvalen las deformaon rae s defned from he vsolas sran rae ensor n Eq. () by he followng relaon [4] v v D : D. (7)

5 0 h Inernaonal Conferene on Comose Maerals Coenhagen, 9-4 h July 05.3 Sauraed damage evoluon Usng he ouled damage one of Eq. (), he remanng ar s o rovde he evoluon of he soro numeral damage arameer. Ths work uses he non-loal aumulaed las sran o omue he damage evoluon, as D D,, (8) wh he assumon ha he damage deends only on he sran sae and on he non-loal varable. In hs non-loal enhaned model, he non-loal aumulaed las sran s omued from he ml formulaon [5], (9) where s he square of he haraers non-loal lengh. The relaon (0) s omleed by he naural boundary ondon n 0 The evoluon of damage follows a sauraon law D. (0) 0 f D, s () h ( D D) f where s he sauraon damage value, h s he sofenng sloe, s s he sofenng exonen and s he las hreshold for he damage evoluon. 3 COMPARISO WITH EXPERIMETS The numeral omresson and ree redons of eoxy resns a he room emeraure (3 o C) usng he urren vsoelas-vsolas-damage onsu model are llusraed n gure, where hey are omared o he exermenal urs reored n []. The obaned resuls are n good agreemen wh he exermens showng he ably of he roosed model for redng he behavor of amorhous glassy olymers. gure : Model redons omared o he exermenal daa [] a room emeraure: unaxal omresson resuls a varous sran raes n he full sran range (lef mage), zoom a small sran lels (mddle mage); ree es resuls (rgh mage). 4 COCLUSIOS The deloed model an orrely aure he orall resonse of amorhous glassy olymers wh he eoxy resns as a sef ase hrough a oule vsoelas-vsolas-damage onsu model. The ml non-loal las model s emloyed o avod he loss of unqueness. In he near fuure, an arorae ressure-deenden falure reron s nororaed. The onsu model wll be aled o he olymer resns of omoses, so ha he mul-sale fraure of omoses wll be

6 V.-D. guyen, X. Morelle,. Lan, T. Pardoen, C. Bally and L. oels suded usng he Dsonnuous Galerkn/Cohes Zone Mehod (DG/CZM) as roosed n [7]. ACKOWLEDGEMETS The auhor graefully aknowledge he fnanal suor from. R. S... R. S. under he roje number PDR T REERECES []. Lan, X. Morelle, C. Bally, and T. Pardoen, Charaerzaon and modelng of he sranrae, emeraure and ressure deendene of he deformaon of a hghly rosslnked aerosae grade eoxy resn, n 0h Inernaonal Conferene on Comose Maerals ICCM0, 05. [] X. Morelle,. Lan, M. Melhor, S. Andre, C. Bally, and T. Pardoen, The elaso-vsolasy and fraure behavor of he RTM6 sruural eoxy and ma on he resonse of won omoses, n 5h Euroean Conferene on Comose Maerals, 0. [3] P. Perzyna, Thermodynam heory of vsolasy, vol.. Elsever, 97. [4] A. R. Melro, P. P. Camanho,. M. Andrade Pres, and S. T. Pnho, Mromehanal analyss of olymer omoses renfored by undreonal fbres: Par I Consu modellng, Inernaonal Journal of Solds and Sruures, vol. 50, no., , Jun. 03. [5] R. H. J. Peerlngs, R. De Bors, W. a M. Brekelmans, and J. H. P. De Vree, Graden enhaned damage for quas-brle maerals, Inernaonal Journal for umeral Mehods n Engneerng, vol. 39, no. 9, , 996. [6] J. Lemare, Couled elaso-lasy and damage onsu equaons, Comuer Mehods n Aled Mehans and Engneerng, vol. 5, no. 3,. 3 49, Se [7] L. Wu, D. Tjahjano, G. Beker, A. Makrad, A. Jérusalem, and L. oels, A mro mesomodel of nra-lamnar fraure n fber-renfored omoses based on a dsonnuous Galerkn/ohes zone mehod, Engneerng raure Mehans, vol. 04,. 6 83, May 03.