International audienceIn a recent work, Böl and Reese have introduced a discrete model for polymer networks by means of a finite element modeling. They have also provided a comparison with real experiments. A key parameter of their model is the size h of the finite element mesh, that is meant to be small in practice. The aim of the present work is to study the asymptotic behaviour (and the convergence of the finite element method) when the meshsize goes to zero. In particular, we address the properties satisfied by the model at the limit, depending on the properties of the mesh

Alicandro, Roberto

Cicalese, Marco

Gloria, Antoine

HAL Descartes

Convergence analysis of the Böl-Reese discrete model for rubber

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HAL Id: inria-00166846https://hal.inria.fr/inria-00166846v4Submitted on 23 Oct 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Convergence analysis of the Böl-Reese discrete modelfor rubberRoberto Alicandro, Marco Cicalese, Antoine GloriaTo cite this version:Roberto Alicandro, Marco Cicalese, Antoine Gloria. Convergence analysis of the Böl-Reese discretemodel for rubber. Continuum Models and Discrete Systems CMDS 11, Jul 2007, Paris, France. ￿inria-00166846v4￿inria-00166846, version 3 - 13 Aug 2007ISSN 0249-6399appor t  de recherche Thème NUMINSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUEConvergence analysis of the Böl-Reese discretemodel for rubberR. Alicandro1, M. Cicalese2, A. Gloria3N° 6270August, 2007Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le ChesnayCedex (France)Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30Convergene analysis of the Böl-Reese disrete model forrubberR. Aliandro1, M. Cialese2, A. Gloria3 ∗Thème NUM  Systèmes numériquesProjet MICMACRapport de reherhe n° 6270  August, 2007  9 pagesAbstrat: In [1℄, Böl and Reese have introdued a disrete model for polymer networksby means of a nite element modeling. In [2, 3℄, they present details on the method anda omparison with real experiments. A key parameter of their model is the size h of thenite element mesh, that is meant to be small in pratie. The aim of the present work is tostudy the asymptoti behaviour (and the onvergene of the nite element method) whenthe meshsize goes to zero. In partiular, we address the properties satised by the model atthe limit, depending on the properties of the mesh.Key-words: nonlinear elastiity, rubber, disrete system, nite element method∗ 1 DAEIMI, Università di Cassino, via Di Biasio 43, 03043 Cassino (FR), Italy ; 2 Dipartimento diMatematia e Appliazioni `R. Caioppoli', Università di Napoli, via Cintia, 80126 Napoli, Italy ; 3 CER-MICS - ENPC & INRIA Paris-Roquenourt, 6 et 8 avenue Blaise Pasal - Champs sur Marne, Frane ;Contat: aliandrunias.it, ialeseunina.it, antoine.gloriainria.frPropriétés de onvergene du modèle disret deBöl-Reese pour le aouthouRésumé : Dans [1℄, Böl et Reese ont introduit un modèle disret de réseau de polymèrespar éléments nis. Dans [2, 3℄, les auteurs détaillent leur méthode et omparent les résultatsnumériques obtenus à des expérienes réelles. Un des paramètres lés de leur modèle est lataille du pas h du maillage, qui est petit en pratique. L'objetif de e travail est d'étudierle omportement asymptotique (ainsi que la onvergene de la méthode des éléments nis)lorsque h tend vers zéro. En partiulier, nous étudions les propriétés satisfaites par le modèleà la limite, en fontion, des propriétés du maillage.Mots-lés : élastiité non linéaire, aouthou, système disret, méthode des élémentsnisConvergene of a disrete model for rubber 3Phenomenologial onstitutive laws for rubber-like materials often involve parameterswhih lak of physial motivation. In addition, they are usually diult to t in pratie.Muh attention has been paid in the reent years to mirosopially-based models. Thiskind of models aims at pointing out the mirosopi features that govern the marosopibehavior of the material, suh as the geometry of the underlying polymer network. Suh amodel has been introdued by Böl and Reese in [1℄. In addition to the physial parametersprovided by physis at the sale of the polymer hains (suh as the free energy of a hain),this model involves two geometri parameters: a typial distane h > 0 (whih is meant tobe small) and a tetraedral mesh.The aim of the present work is to study the asymptoti properties of the model when hgoes to zero for dierent assumptions on the mesh.This note is organized as follows:Contents1 The BöL-Reese model 41.1 The geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Convergene analysis 52.1 Assumptions on the energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The periodi ase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 The stohasti ase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Conlusion 8RR n° 62704 R. Aliandro, M. Cialese, A. Gloria1 The BöL-Reese model1.1 The geometryFor the sake of simpliity, let Ω denote the unit ube of R3 and Th be a tetraedral mesh ofΩ. The typial distane we assoiate with the mesh is dened as follows:h = 3√1Nel ,where Nel denotes the number of elements (tetraedras) of Th.From a modeling point of view, eah edge of the mesh represents an elasti spring withan energy related to the underlying polymer network.Let d be a displaement eld of the mesh dened at eah vertex xi of Th by di. One anassoiate with d a ontinuous displaement eld on Ω as follows: vh is the unique pieewiseane funtion on Th suh that for eah vertex xi of Th, vh(xi) = xi + di.The energy of the system deformed by vh is then given byEh(vh) =∑T∈Thh3∑(xi,xj)∈TfTijWTij( |vh(xi) − vh(xj)||xi − xj |)+∫ΩWvol(∇vh), (1)where T denotes elements of Th, (xi, xj) are distint pairs of verties in the tetraedra T ,WTij is a pair potential, fTij is a saling fator and Wvol is an energy density aounting forvolume hanges of the deformed system.Let us now desribe the pair-potential energy.1.2 The energyThe energy of the elasti springs is related to the energy of a polymer hain as follows:WTij (r) =kβnTijr√nTijL−1r√nTij + lnL−1(r√nTij)sinhL−1(r√nTij)− cβ, (2)where β is the inverse of the absolute temperature, L−1 is the inverse of the Langevinfuntion, k and c are onstants and nTij is a typial number of segments of a polymer hain.WTij an be interpreted as the free energy of a polymer hain made of nTij segments at thelength √nTij l, where l denotes the length of a segment.The fator fTij is a measure of the number of polymer hains per unit of volume. We willtypially onsider it as onstant on Ω.INRIAConvergene of a disrete model for rubber 5Finally, the energy aounting for volume hanges is given byWvol(ξ) =K4(J2 − 1 − lnJ), (3)where K > 0, ξ is a deformation gradient and J = det(ξ). The physial origin of thisontribution is the Van der Waals fores, whih prevent atoms from being too lose to oneanother. The sale of this fores is far larger than the sale of the polymer network, whihexplains why this ontribution is already oarse-grained and somewhat unorrelated withthe desription of the network.In the following setion, we study the onvergene when h → 0 of the sequene ofminimization problemsinf {Eh(vh), vh ∈ Vh + BC} ,where Vh denotes the spae of pieewise ane funtions on Th and BC stands for theboundary onditions (let say mixed imposed displaement and free tration for instane).In partiular, we address both the onvergene of the inmum of the energy and the on-vergene of the minimizers (the deformation eld) when h → 0. This onvergene analysisis performed using Γ-onvergene (see [4, 5℄ e.g.).2 Convergene analysis2.1 Assumptions on the energiesFor tehnial reasons, in what follows, we will assume that: WTij satises the following standard growth ondition of order p > 1 for all T and ij:there exist C ≥ c > 0 suh that for all r,c|r|p − 1 ≤ WTij (r) ≤ C(|r|p + 1), (4) Wvol satises (4) from above.In partiular, one annot diretly deal with the inverse of the Langevin funtion. Tosatisfy the growth ondition (4), one may use a trunated series expansion of L−1, as it isdone in [1, 2, 3℄:L−1(ρ) = 3ρ + 95ρ3 +297175ρ5 +1539875ρ7 + O(ρ9).Replaing L−1 by its development up to order 7 in (2), we obtain a free energy density fora polymer hain whih satises (4) for p = 8.The energy density Wvol given by (3) does not satisfy (4) sine limdet(ξ)→0 Wvol(ξ) = +∞while det(ξ) → 0 does not imply |ξ| → ∞. One possible solution onsists in taking a ut-o,RR n° 62706 R. Aliandro, M. Cialese, A. Gloriawhih amounts to relaxing the onstraint of inompressibility. A typial ut-o reads asfollows:Wηvol(ξ) ={Wvol(ξ) if det(ξ) > ηK4 (η2 − 1 − ln η) otherwise. (5)For η > 0 xed, W ηvol satises (4) for Cη large enough. To reover the inompressibilitybehavior, one an rst perform the onvergene analysis on the disrete to ontinuum proessfor η > 0 and then let η go to zero at the ontinuum level. We refer the reader to [6, 7℄ fortehnial details.We are now in position to state our onvergene results.2.2 The periodi aseIn this paragraph, we address the onvergene of the Böl-Reese model in the periodi ase.To this end, let us assume that the edges of Th are obtained by the periodi repliation ofa unit ell of edges (up to border eets on the boundary), as illustrated in two dimensionson Figure 1.Figure 1: Two-dimensional periodi mesh, the unit ell is in boldLet us further assume that the energy of the elasti springs does not depend of the unitell and on the mesh lengthsale h. As a partiular ase, one an take nTij to be onstanton Th, as it is done in [1, 2, 3℄. Then, [8, Theorem 4.1℄ shows there exists an homogeneousquasionvex energy density Whom, whih is frame-invariant and satises a growth onditionof order p, suh that: limh→0inf {Eh(vh), vh ∈ Vh + BC} = inf {Ehom(v), v ∈ V + BC}, where V is the Sobolevspae W 1,p(Ω), andEhom(v) =∫ΩWhom(∇v); Whom is given by the asymptoti disrete homogenization formulaWhom(ξ) = limh→0inf{Eh(vh), vh ∈ Vh, v(xi) = ξ · xi if d(xi, ∂Ω) ≤ 2h}.INRIAConvergene of a disrete model for rubber 7In addition, minimizers uh of Eh on Vh + BC weakly onverge in W 1,p(Ω) to minimizersuhom of Ehom on V + BC. This proves in partiular the onvergene of the solutions of thenite element modelling of rubber introdued by Böl and Reese.In the periodi ase, the energy density at the limit is frame-invariant. However it is notisotropi in general. To ensure the isotropy, one may use a stohasti framework. The endof this paragraph is dediated to the study of a simple example for whih the energy densityat the limit is not isotropi.Let us onsider simple linear springs by settingWTij (vh) = K( |vh(xi) − vh(xj)||xi − xj |)2 and Wvol = 0on the mesh skethed on Figure 1. Then, realling [8, Remark 5.2℄, the homogenized energydensity is quadrati and we end up with a onvex minimization problem on one singleperiodi ell. The inmum is trivial and shows that the homogenized energy is not isotropi.It is atually enough to see that the material is stier in the diretion e2 − e1 than in thediretion e1 + e2 (where e1 and e2 denote the anonial basis of R2).2.3 The stohasti aseLet us quikly reall the onept of admissible stohasti network on a probability spae(Ξ,F , P ) used in [9, 7℄, whih is a partiular ase of the stohasti networks introdued byBlan, Le Bris and Lions in [10, 11℄.Let Λ = {yi}i∈Zd ∈ (Rd)Zd be a set of points. We say that Λ is an admissible set ofpoints if it satises the two following onditions:i. there exists R > 0 suh that #Λ ∩ B(y, R) > 0 for all y ∈ Rd ;ii. there exists r > 0 suh that d(yi, Λ \ {yi}) ≥ r for all i ∈ Zd.In partiular, to eah admissible set of points Λ one an assoiate a Delaunay triangulationD(Λ). A stohasti lattie L : Ξ → (Rd)Zd is said to be admissible, if for P -almost everyω ∈ Ξ, L(ω) is an admissible set of points, and if the Delaunay triangulation is regular inthe sense of the interpolation theory.Given a realization Λ(ω) of a stohasti lattie, one may resale the lattie by a fator h,setting yhi (ω) = hyi(ω). Let us then dene Λ(ω)h(Ω) = {yhi ∈ R3, yi ∈ Λ(ω) and yhi ∈ Ω},whih is the intersetion of the resaled lattie with Ω. Using the notations of Böl and Reese,one may think of a mesh Th as the Delaunay triangulation of Λ(ω)h(Ω).In what follows, we assume that Th is related to a stohasti network in the way desribedabove and we add an index ω to make the stohasti dependene more expliit. Futhermorewe will make some hypotheses related to the stohasti network and the probability spae:RR n° 62708 R. Aliandro, M. Cialese, A. Gloria(a) there exists a stationary translation group that ats on the stohasti lattie and whihis ergodi for P ;(b) there exists a stationary group of rotations that ats on the stohasti lattie (thestohasti lattie is then said to be rotation invariant on average).Roughly speaking, the translation invariane allows us to obtain a deterministi limit whereasthe invariane by rotation implies that the energy density at the limit is isotropi. We referthe interested reader to [7℄ for the preise formulation of the assumptions and the proofs ofthe following result.Within hypothesis (a), there exists an homogeneous quasionvex energy density Whom,whih is frame-invariant and satises a growth ondition of order p, suh that: For almost all ω ∈ Ξ,limh→0inf {Eωh (vωh ), vωh ∈ V ωh + BC} = inf {Ehom(v), v ∈ V + BC},where V is the Sobolev spae W 1,p(Ω), andEhom(v) =∫ΩWhom(∇v); Whom is given by the asymptoti disrete homogenization formulaWhom(ξ) = limh→0∫Ξinf{Eωh (vωh ), vωh ∈ V ωh , v(xi) = ξ · xiif d(xi, ∂Ω) ≤ 2hR}dP (ω).In addition, minimizers uωh of Eωh on V ωh + BC weakly onverge in W 1,p(Ω) to minimizersuhom of Ehom on V + BC. This proves in partiular the onvergene of the solutions of thenite element modelling of rubber introdued by Böl and Reese.In addition, if (b) holds, the stohasti network is rotation invariant on average, andWhom is isotropi.3 ConlusionIn the rst setion, we have realled the nite element modelling of rubber developed by Böland Reese and we have pointed out some onvergene issues related to their formulation.In the following setions, we have analyzed the asymptoti behaviour of the model. Inpartiular, we have identied two ases for whih their model provides a nite elementapproximation of an underlying ontinuous model. The ontinuous models obtained bothsatisfy the frame invariane property and the usual inompressibility behaviour used forrubber modelling (up to some tehnialities detailed in [7℄). However, only the stohastimodel is ensured to yield an isotropi energy density at the limit. We have also provided aperiodi example whih is not isotropi at the limit. INRIAConvergene of a disrete model for rubber 9Referenes[1℄ M. Böl and S. Reese. Finite element modelling of polymer networks based on hainstatistis. In J. Buseld and A. Muhr, editors, Constitutive models for rubber III (Pro-eedings of the third European onferene on onstitutive models for rubber), pages203211, Lisse, 2003. A.A. Balkema.[2℄ M. Böl and S. Reese. Finite element modelling of rubber-like polymers based on hainstatistis. Int. J. Sol. Stru., 43:226, 2006.[3℄ M. Böl and S. Reese. Finite element modelling of rubber-like materials - a omparisonbetween simulation and experiment. Journal of Materials S., 40:59335939, 2005.[4℄ A. Braides. Γ-onvergene for beginners, volume 22 of Oxford Leture Series in Mathe-matis and Its Appliations. Oxford University Press, 2002.[5℄ A. Braides. A handbook of Γ-onvergene. volume 3 of Handbook of Dierential Equa-tions: Stationary Partial Dierential Equations, pages 101213. Elsevier, Amsterdam,2006.[6℄ A. Gloria. Méthodes numériques et modélisation multiéhelle en élastiité non linéaire.PhD thesis, ENPC, 2007.[7℄ R. Aliandro, M. Cialese, and A. Gloria. Integral representation results for energiesdened on stohasti latties and appliation to nonlinear elastiity. In preparation.[8℄ R. Aliandro and M. Cialese. A general integral representation result for the ontinuumlimits of disrete energies with superlinear growth. SIAM J. Math. Anal., 36(1):137,2004.[9℄ R. Aliandro, M. Cialese, and A. Gloria. Mathematial derivation of a rubber-likestored energy funtional. Submitted.[10℄ X. Blan, C. Le Bris, and P.L. Lions. Du disret au ontinu pour des réseaux aléatoiresd'atomes. C. R. Aad. Si. Paris, Série I, 342:627633, 2006.[11℄ X. Blan, C. Le Bris, and P.L. Lions. The energy of some mirosopi stohasti latties.Arh. Rational Meh. Anal., to appear.RR n° 6270Unité de recherche INRIA RocquencourtDomaine de Voluceau - Rocquencourt - BP 105 - 78153 Le ChesnayCedex (France)Unité de recherche INRIA Futurs : Parc Club Orsay Université- ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)Unité de recherche INRIA Rennes : IRISA, Campus universitaie de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Ch snay Cedex (France)http://www.inria.frISSN 0249-6399

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Convergence analysis of the Bo¨l-Reese discrete modelfor rubberRoberto Alicandro, Marco Cicalese, Antoine GloriaTo cite this version:Roberto Alicandro, Marco Cicalese, Antoine Gloria. Convergence analysis of the Bo¨l-Reesediscrete model for rubber. Continuum Models and Discrete Systems CMDS 11, Jul 2007,Paris, France. 2008, Mines Paris ParisTech. <inria-00166846v4>HAL Id: inria-00166846https://hal.inria.fr/inria-00166846v4Submitted on 23 Oct 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.inria-00166846, version 3 - 13 Aug 2007ISSN 0249-6399ap por t  de  r ech er ch e Thème NUMINSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUEConvergence analysis of the Böl-Reese discretemodel for rubberR. Alicandro1, M. Cicalese2, A. Gloria3N° 6270August, 2007Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30Convergene analysis of the Böl-Reese disrete model forrubberR. Aliandro1, M. Cialese2, A. Gloria3 ∗Thème NUM  Systèmes numériquesProjet MICMACRapport de reherhe n° 6270  August, 2007  9 pagesAbstrat: In [1℄, Böl and Reese have introdued a disrete model for polymer networksby means of a nite element modeling. In [2, 3℄, they present details on the method anda omparison with real experiments. A key parameter of their model is the size h of thenite element mesh, that is meant to be small in pratie. The aim of the present work is tostudy the asymptoti behaviour (and the onvergene of the nite element method) whenthe meshsize goes to zero. In partiular, we address the properties satised by the model atthe limit, depending on the properties of the mesh.Key-words: nonlinear elastiity, rubber, disrete system, nite element method∗ 1DAEIMI, Università di Cassino, via Di Biasio 43, 03043 Cassino (FR), Italy ;2Dipartimento diMatematia e Appliazioni `R. Caioppoli', Università di Napoli, via Cintia, 80126 Napoli, Italy ;3CER-MICS - ENPC & INRIA Paris-Roquenourt, 6 et 8 avenue Blaise Pasal - Champs sur Marne, Frane ;Contat: aliandrunias.it, ialeseunina.it, antoine.gloriainria.frPropriétés de onvergene du modèle disret deBöl-Reese pour le aouthouRésumé : Dans [1℄, Böl et Reese ont introduit un modèle disret de réseau de polymèrespar éléments nis. Dans [2, 3℄, les auteurs détaillent leur méthode et omparent les résultatsnumériques obtenus à des expérienes réelles. Un des paramètres lés de leur modèle est lataille du pas h du maillage, qui est petit en pratique. L'objetif de e travail est d'étudierle omportement asymptotique (ainsi que la onvergene de la méthode des éléments nis)lorsque h tend vers zéro. En partiulier, nous étudions les propriétés satisfaites par le modèleà la limite, en fontion, des propriétés du maillage.Mots-lés : élastiité non linéaire, aouthou, système disret, méthode des élémentsnisConvergene of a disrete model for rubber 3Phenomenologial onstitutive laws for rubber-like materials often involve parameterswhih lak of physial motivation. In addition, they are usually diult to t in pratie.Muh attention has been paid in the reent years to mirosopially-based models. Thiskind of models aims at pointing out the mirosopi features that govern the marosopibehavior of the material, suh as the geometry of the underlying polymer network. Suh amodel has been introdued by Böl and Reese in [1℄. In addition to the physial parametersprovided by physis at the sale of the polymer hains (suh as the free energy of a hain),this model involves two geometri parameters: a typial distane h > 0 (whih is meant tobe small) and a tetraedral mesh.The aim of the present work is to study the asymptoti properties of the model when hgoes to zero for dierent assumptions on the mesh.This note is organized as follows:Contents1 The BöL-Reese model 41.1 The geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Convergene analysis 52.1 Assumptions on the energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The periodi ase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 The stohasti ase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Conlusion 8RR n° 62704 R. Aliandro, M. Cialese, A. Gloria1 The BöL-Reese model1.1 The geometryFor the sake of simpliity, let Ω denote the unit ube of R3 and Th be a tetraedral mesh ofΩ. The typial distane we assoiate with the mesh is dened as follows:h = 3√1Nel,where Neldenotes the number of elements (tetraedras) of Th.From a modeling point of view, eah edge of the mesh represents an elasti spring withan energy related to the underlying polymer network.Let d be a displaement eld of the mesh dened at eah vertex xi of Th by di. One anassoiate with d a ontinuous displaement eld on Ω as follows: vh is the unique pieewiseane funtion on Th suh that for eah vertex xi of Th, vh(xi) = xi + di.The energy of the system deformed by vh is then given byEh(vh) =∑T∈Thh3∑(xi,xj)∈TfTijWTij( |vh(xi)− vh(xj)||xi − xj |)+∫ΩWvol(∇vh), (1)where T denotes elements of Th, (xi, xj) are distint pairs of verties in the tetraedra T ,WTij is a pair potential, fTij is a saling fator and Wvol is an energy density aounting forvolume hanges of the deformed system.Let us now desribe the pair-potential energy.1.2 The energyThe energy of the elasti springs is related to the energy of a polymer hain as follows:WTij (r) =kβnTij r√nTijL−1 r√nTij+ ln L−1(r√nTij)sinhL−1(r√nTij)− cβ , (2)where β is the inverse of the absolute temperature, L−1 is the inverse of the Langevinfuntion, k and c are onstants and nTij is a typial number of segments of a polymer hain.WTij an be interpreted as the free energy of a polymer hain made of nTij segments at thelength√nTij l, where l denotes the length of a segment.The fator fTij is a measure of the number of polymer hains per unit of volume. We willtypially onsider it as onstant on Ω.INRIAConvergene of a disrete model for rubber 5Finally, the energy aounting for volume hanges is given byWvol(ξ) =K4(J2 − 1− lnJ), (3)where K > 0, ξ is a deformation gradient and J = det(ξ). The physial origin of thisontribution is the Van der Waals fores, whih prevent atoms from being too lose to oneanother. The sale of this fores is far larger than the sale of the polymer network, whihexplains why this ontribution is already oarse-grained and somewhat unorrelated withthe desription of the network.In the following setion, we study the onvergene when h → 0 of the sequene ofminimization problemsinf {Eh(vh), vh ∈ Vh +BC} ,where Vh denotes the spae of pieewise ane funtions on Th and BC stands for theboundary onditions (let say mixed imposed displaement and free tration for instane).In partiular, we address both the onvergene of the inmum of the energy and the on-vergene of the minimizers (the deformation eld) when h → 0. This onvergene analysisis performed using Γ-onvergene (see [4, 5℄ e.g.).2 Convergene analysis2.1 Assumptions on the energiesFor tehnial reasons, in what follows, we will assume that: WTij satises the following standard growth ondition of order p > 1 for all T and ij:there exist C ≥ c > 0 suh that for all r,c|r|p − 1 ≤WTij (r) ≤ C(|r|p + 1), (4) Wvol satises (4) from above.In partiular, one annot diretly deal with the inverse of the Langevin funtion. Tosatisfy the growth ondition (4), one may use a trunated series expansion of L−1, as it isdone in [1, 2, 3℄:L−1(ρ) = 3ρ+ 95ρ3 +297175ρ5 +1539875ρ7 +O(ρ9).Replaing L−1 by its development up to order 7 in (2), we obtain a free energy density fora polymer hain whih satises (4) for p = 8.The energy densityWvol given by (3) does not satisfy (4) sine limdet(ξ)→0Wvol(ξ) = +∞while det(ξ)→ 0 does not imply |ξ| → ∞. One possible solution onsists in taking a ut-o,RR n° 62706 R. Aliandro, M. Cialese, A. Gloriawhih amounts to relaxing the onstraint of inompressibility. A typial ut-o reads asfollows:Wηvol(ξ) ={Wvol(ξ) if det(ξ) > ηK4 (η2 − 1− ln η) otherwise. (5)For η > 0 xed, W ηvol satises (4) for Cη large enough. To reover the inompressibilitybehavior, one an rst perform the onvergene analysis on the disrete to ontinuum proessfor η > 0 and then let η go to zero at the ontinuum level. We refer the reader to [6, 7℄ fortehnial details.We are now in position to state our onvergene results.2.2 The periodi aseIn this paragraph, we address the onvergene of the Böl-Reese model in the periodi ase.To this end, let us assume that the edges of Th are obtained by the periodi repliation ofa unit ell of edges (up to border eets on the boundary), as illustrated in two dimensionson Figure 1.Figure 1: Two-dimensional periodi mesh, the unit ell is in boldLet us further assume that the energy of the elasti springs does not depend of the unitell and on the mesh lengthsale h. As a partiular ase, one an take nTij to be onstanton Th, as it is done in [1, 2, 3℄. Then, [8, Theorem 4.1℄ shows there exists an homogeneousquasionvex energy density Whom, whih is frame-invariant and satises a growth onditionof order p, suh that: limh→0inf {Eh(vh), vh ∈ Vh +BC} = inf {Ehom(v), v ∈ V +BC}, where V is the Sobolevspae W 1,p(Ω), andEhom(v) =∫ΩWhom(∇v); Whom is given by the asymptoti disrete homogenization formulaWhom(ξ) = limh→0inf{Eh(vh), vh ∈ Vh, v(xi) = ξ · xi if d(xi, ∂Ω) ≤ 2h}.INRIAConvergene of a disrete model for rubber 7In addition, minimizers uh of Eh on Vh + BC weakly onverge in W1,p(Ω) to minimizersuhom of Ehom on V +BC. This proves in partiular the onvergene of the solutions of thenite element modelling of rubber introdued by Böl and Reese.In the periodi ase, the energy density at the limit is frame-invariant. However it is notisotropi in general. To ensure the isotropy, one may use a stohasti framework. The endof this paragraph is dediated to the study of a simple example for whih the energy densityat the limit is not isotropi.Let us onsider simple linear springs by settingWTij (vh) = K( |vh(xi)− vh(xj)||xi − xj |)2and Wvol = 0on the mesh skethed on Figure 1. Then, realling [8, Remark 5.2℄, the homogenized energydensity is quadrati and we end up with a onvex minimization problem on one singleperiodi ell. The inmum is trivial and shows that the homogenized energy is not isotropi.It is atually enough to see that the material is stier in the diretion e2 − e1 than in thediretion e1 + e2 (where e1 and e2 denote the anonial basis of R2).2.3 The stohasti aseLet us quikly reall the onept of admissible stohasti network on a probability spae(Ξ,F , P ) used in [9, 7℄, whih is a partiular ase of the stohasti networks introdued byBlan, Le Bris and Lions in [10, 11℄.Let Λ = {yi}i∈Zd ∈(Rd)Zdbe a set of points. We say that Λ is an admissible set ofpoints if it satises the two following onditions:i. there exists R > 0 suh that #Λ ∩B(y,R) > 0 for all y ∈ Rd ;ii. there exists r > 0 suh that d(yi,Λ \ {yi}) ≥ r for all i ∈ Zd.In partiular, to eah admissible set of points Λ one an assoiate a Delaunay triangulationD(Λ). A stohasti lattie L : Ξ → (Rd)Zd is said to be admissible, if for P -almost everyω ∈ Ξ, L(ω) is an admissible set of points, and if the Delaunay triangulation is regular inthe sense of the interpolation theory.Given a realization Λ(ω) of a stohasti lattie, one may resale the lattie by a fator h,setting yhi (ω) = hyi(ω). Let us then dene Λ(ω)h(Ω) = {yhi ∈ R3, yi ∈ Λ(ω) and yhi ∈ Ω},whih is the intersetion of the resaled lattie with Ω. Using the notations of Böl and Reese,one may think of a mesh Th as the Delaunay triangulation of Λ(ω)h(Ω).In what follows, we assume that Th is related to a stohasti network in the way desribedabove and we add an index ω to make the stohasti dependene more expliit. Futhermorewe will make some hypotheses related to the stohasti network and the probability spae:RR n° 62708 R. Aliandro, M. Cialese, A. Gloria(a) there exists a stationary translation group that ats on the stohasti lattie and whihis ergodi for P ;(b) there exists a stationary group of rotations that ats on the stohasti lattie (thestohasti lattie is then said to be rotation invariant on average).Roughly speaking, the translation invariane allows us to obtain a deterministi limit whereasthe invariane by rotation implies that the energy density at the limit is isotropi. We referthe interested reader to [7℄ for the preise formulation of the assumptions and the proofs ofthe following result.Within hypothesis (a), there exists an homogeneous quasionvex energy density Whom,whih is frame-invariant and satises a growth ondition of order p, suh that: For almost all ω ∈ Ξ,limh→0inf {Eωh (vωh ), vωh ∈ V ωh +BC} = inf {Ehom(v), v ∈ V +BC},where V is the Sobolev spae W 1,p(Ω), andEhom(v) =∫ΩWhom(∇v); Whom is given by the asymptoti disrete homogenization formulaWhom(ξ) = limh→0∫Ξinf{Eωh (vωh ), vωh ∈ V ωh , v(xi) = ξ · xiif d(xi, ∂Ω) ≤ 2hR}dP (ω).In addition, minimizers uωh of Eωh on Vωh + BC weakly onverge in W1,p(Ω) to minimizersuhom of Ehom on V +BC. This proves in partiular the onvergene of the solutions of thenite element modelling of rubber introdued by Böl and Reese.In addition, if (b) holds, the stohasti network is rotation invariant on average, andWhom is isotropi.3 ConlusionIn the rst setion, we have realled the nite element modelling of rubber developed by Böland Reese and we have pointed out some onvergene issues related to their formulation.In the following setions, we have analyzed the asymptoti behaviour of the model. Inpartiular, we have identied two ases for whih their model provides a nite elementapproximation of an underlying ontinuous model. The ontinuous models obtained bothsatisfy the frame invariane property and the usual inompressibility behaviour used forrubber modelling (up to some tehnialities detailed in [7℄). However, only the stohastimodel is ensured to yield an isotropi energy density at the limit. We have also provided aperiodi example whih is not isotropi at the limit.INRIAConvergene of a disrete model for rubber 9Referenes[1℄ M. Böl and S. Reese. Finite element modelling of polymer networks based on hainstatistis. In J. Buseld and A. Muhr, editors, Constitutive models for rubber III (Pro-eedings of the third European onferene on onstitutive models for rubber), pages203211, Lisse, 2003. A.A. Balkema.[2℄ M. Böl and S. Reese. Finite element modelling of rubber-like polymers based on hainstatistis. Int. J. Sol. Stru., 43:226, 2006.[3℄ M. Böl and S. Reese. Finite element modelling of rubber-like materials - a omparisonbetween simulation and experiment. Journal of Materials S., 40:59335939, 2005.[4℄ A. Braides. Γ-onvergene for beginners, volume 22 of Oxford Leture Series in Mathe-matis and Its Appliations. Oxford University Press, 2002.[5℄ A. Braides. A handbook of Γ-onvergene. volume 3 of Handbook of Dierential Equa-tions: Stationary Partial Dierential Equations, pages 101213. Elsevier, Amsterdam,2006.[6℄ A. Gloria. Méthodes numériques et modélisation multiéhelle en élastiité non linéaire.PhD thesis, ENPC, 2007.[7℄ R. Aliandro, M. Cialese, and A. Gloria. Integral representation results for energiesdened on stohasti latties and appliation to nonlinear elastiity. In preparation.[8℄ R. Aliandro and M. Cialese. A general integral representation result for the ontinuumlimits of disrete energies with superlinear growth. SIAM J. Math. Anal., 36(1):137,2004.[9℄ R. Aliandro, M. Cialese, and A. Gloria. Mathematial derivation of a rubber-likestored energy funtional. Submitted.[10℄ X. Blan, C. Le Bris, and P.L. Lions. Du disret au ontinu pour des réseaux aléatoiresd'atomes. C. R. Aad. Si. Paris, Série I, 342:627633, 2006.[11℄ X. Blan, C. Le Bris, and P.L. Lions. The energy of some mirosopi stohasti latties.Arh. Rational Meh. Anal., to appear.RR n° 6270Unité de recherche INRIA RocquencourtDomaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)http://www.inria.frISSN 0249-6399

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Convergence analysis of the Böl-Reese discrete model for rubber

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