Poincaré s Equations for Cosserat Media: Application to Shells

Transcription

1 Pincaré s Equains fr Cssera Media: Applicain Shells Frédéric Byer, Federic Renda T cie his versin: Frédéric Byer, Federic Renda. Pincaré s Equains fr Cssera Media: Applicain Shells. Jurnal f Nnlinear Science, Springer Verlag, 2016, pp < /s >. <hal v3> HAL Id: hal hps://hal.archives-uveres.fr/hal v3 Submied n 3 Oc 2016 HAL is a muli-disciplinary pen access archive fr he depsi and disseminain f scienific research dcumens, wheher hey are published r n. The dcumens may cme frm eaching and research insiuins in France r abrad, r frm public r privae research ceners. L archive uvere pluridisciplinaire HAL, es desinée au dépô e à la diffusin de dcumens scienifiques de niveau recherche, publiés u nn, émanan des éablissemens d enseignemen e de recherche français u érangers, des labraires publics u privés.

2 Pincaré's equains fr Cssera media : applicain shells Frederic Byer 1 Federic Renda 2 Absrac In 1901 Henri Pincaré discvered a new se f equains fr mechanics. These equains are a generalizain f Lagrange's equains fr a sysem whse cngurain space is a Lie grup which is n necessarily cmmuaive. Since hen, his resul has been exensively rened hrugh he Lagrangian reducin hery. In he presen cnribuin, we apply an exended versin f hese equains cninuus Cssera media, i.e. media in which he usual pin paricles are replaced by small rigid bdies, called micr-srucures. In paricular, we will see hw he Shell balance equains used in nnlinear srucural dynamics, can be easily deduced frm his exensin f he Pincaré's resul. In fuure, hese resuls will be used as fundains fr he sudy f squid lcmin, which is an emerging pic relevan sf rbics. Acknwledgmen The nal publicain is available a: Byer, F. & Renda, F. J Nnlinear Sci di: /s Frédéric Byer EMN, IRCCyN, La Chanrerie 4, rue Alfred Kasler B.P Nanes Cedex 3 France. Tel.: , Fax: , frederic.byer@emn.fr 2 Federic Renda KUSTAR, KURI, Abu Dhabi campus, Abu Dhabi, UAE. Tel: , Fax: , federic.renda@kusar.ac.ae

3 Nmenclaure ime E 3-dimensinal gemeric space f classical mechanics B 3-dimensinal maerial space f a classical cninuus medium D Maerial p dimensinal, p < 3 reference subspace M Rigid micrsrucure B = D M Maerial space f a Cssera medium O, E 1, E 2, E 3 Maerial frame aached B, e 1, e 2, e 3 Spaial frame aached E x = x i e i Pins f gemeric space X = X i E i Maerial pins f B X = X α E α Maerial pins f D Φ Transfrmain a ime frm maerial gemeric space Φ B Defrmed cngurain f B Φ B Reference cngurain f B Φ ed Defrmed cngurain f D Φ ed Reference cngurain f D rx Psiin f Φ ex RX SO3 Rain ensr mapping E 1, E 2, E 3 n 1, 2, 3 X g 1, g 2, g 3 X Cnveced basis n Φ B a Φ X h 1,...h p X Cnveced basis n Φ ed a rx 1, 2, 3 X Orhnrmal spaial basis aached he X-micrsrucure g ij g i g j X Euclidean meric ensr in he cnveced basis f Φ B h αβ h α h β X Euclidean meric induced n Φ ed in is cnveced basis ν, ν, ν Oriened uni nrmal vecr he maerial, reference and defrmed surface elemen f D ds, ds, ds Area f he maerial, reference and defrmed surface elemen f D C Cngurain space f a Cssera medium D M G and g Grup f ransfrmain and ransfrmain f micrsrucure g, g Lie algebra f G and is dual Ad and Ad Adjin and cadjin acin map f G n g and g ad and ad Adjin and cadjin acin map f g n g and g η and ξ α Lef invarian elds alng ime and space-variables L, L and L Densiy f lef-reduced Lagrangian f a Cssera medium per uni f is maerial, reference and defrmed vlume L η, L L η and η Densiies f maerial -cnjugae kineic mmenum, per uni f maerial, [ reference and defrmed vlume and Densiies f maerial X α -cnjugae sress mmenum, L, L L ] per uni f maerial, reference and defrmed vlume F ex, F ex, and F ex, Densiies f maerial exernal frces per uni f maerial, reference and defrmed vlume F ex, F ex, and F ex, Densiies f exernal frces per uni f maerial, reference, defrmed bundary vlume

4 D R + Space-ime f a p dimensinal Cssera medium X 0 + Xα X Pin in space-ime wih = X 0 α Υ Space-ime 1-frm eld wih value in g Λ, Λ and Λ Densiy f a space-ime vecr eld wih value in g, per uni f maerial, reference, defrmed vlume <.,. > and.,. Dualiy prduc in g and space-ime h Ad L g 1 η Densiies f spaial in he xed frame kineic wrench, per uni h f defrmed vlume Ad L g 1 Densiies f spaial in he xed frame sress wrench, per uni f defrmed vlume SE3 Special Euclidean Grup in R 3 wih Lie algebra se3 R, r Transfrmain f SE3 Ω T, V T T se3 Maerial ime-rae f ransfrmain velciy f he micrsrucure frames ω T, v T T se3 Spaial ime-rae f ransfrmain velciy f he micrsrucure frames Σ T, P T T se3 Densiy f maerial kineic wrench per uni f defrmed vlume σ T, p T T se3 Densiy f spaial in he micrsrucure frame kineic wrench per uni f defrmed vlume Kα T, Γα T T se3 Maerial X α -rae f ransfrmain f he micrsrucure frames kα T, γα T T se3 Spaial X α -rae f ransfrmain f he micrsrucure frames Mα,, T Nα, T T se3 Densiy f maerial sress wrench per uni f defrmed vlume m T α,, n T α, T se3 Densiy f spaial sress wrench per uni f defrmed vlume ρ, ρ, ρ and J, J, J Densiies f mass and f maerial angular ineria ensr per uni f maerial, reference, defrmed vlume I, I, I Densiies f spaial ineria ensr per uni f maerial, reference, defrmed vlume ɛ αβ, ρ αβ, τ α Eecive srain measures sreching, bending, ransverse shearing f a classical shell N αβ, M αβ, Q α Densiies f eecive sress f a classical shell per uni f defrmed vlume 1 Inrducin In cnras classical cninuus media where he basic cnsiuive elemen f maer is he pin paricle, Cssera media are dened by small rigid bdies, called micr-srucures, cninuusly sacked alng maerial dimensins [1]. This fundamenal dierence has srng cnsequences n he w heries classical vs Cssera. In he classical hery, he gemeric mdel f nie rains disappears frm he mdel, nly re-appearing as a kinemaic cnsequence f he ranslains e.g. hrugh he curl f he linear velciy eld, while in he

5 Cssera mdel he rains have a saus similar ha f ranslains frm he beginning he end f he dynamic frmulain. As a resul, he Lie grup srucure naurally appears in he inrinsic deniin f he cngurain space f a Cssera medium hrugh he rigid ransfrmains in SO3, and mre generally SE3 undergne by is cnsiuive micr-srucures. Thus, he mdel f Cssera media shuld be recverable frm he absrac variainal calculus develped by Henri Pincaré [2], knwn day as he Pincaré r Euler-Pincaré equains [3], [4]. These equains can be cnsidered as a generalizain f Lagrange's equains sysems whse cngurain space is dened as a nn cmmuaive Lie grup. As Pincaré remarked himself, hey are paricularly relevan when he Lagrangian f he sysem is lef r righ invarian by he grup ransfrmains, a prpery which is relaed he symmery f space lef invariance and maer righ invariance as Arnld and Marsden discvered laer hrugh he Lagrangian reducin hery [3], [4]. In his cnex, he righ invariance has been shwn be a key cncep include he Eulerian pin f view f uid mechanics wihin he Euler-Pincaré's apprach [5]. Arnld demnsraed ha he ideal uid is he innie righ invarian cunerpar f he nie lef invarian Euler and Pincaré's rigid bdy [6]. Anher case ha mivaes us shif frm he nie he innie dimensinal case is ha f Cssera media [7]. While in he case f he ideal uid, he ransfrmains live in an innie dimensinal grup, in he case f Cssera media he ransiin ward innie dimensin is dramaically dieren. In his her case, in each pin f a cninuus maerial medium D, a nie dimensinal grup G acs n a micrsrucure M, i.e., a rigid bdy f inniesimal size. Applying Pincaré's variainal calculus his cnex requires a shif frm he basic picure f rdinary dierenial Pincaré equains f classical mechanical sysems, a se f parial dierenial equains in a eld hereical apprach as i has been develped in he seminal wrks [8] and [9]. Ging back he riginal aim f he Cssera brhers [7], in [8], he elds equains f an unbunded Cssera medium are derived in he cnex f he absrac frmal hery f sysems f parial dierenial equains [10], wih n reference he Pincaré picure. Frm a mre gemerical viewpin, in [9], he riginal variainal calculus f Pincaré is exended frm a cngurain Lie grup a principal ber bundle derive a se f cvarian Euler-Pincaré equains bu wih n relain Cssera media. In [11], bh elds equains and bundary cndiins f a bunded mulidimensinal Cssera medium are derived in he cnex f Euler-Pincaré reducin. In his apprach, he dynamics f he Cssera medium are deduced frm a unique Lagrangian densiy lef invarian by he ransfrmains f G, he ransfrmains being parameerized by he ime and he maerial crdinaes Lagrangian labels f he medium D. Thugh i may seem absrac a rs, his variainal calculus, which generalises Pincaré's calculus frm ne parameric dimensin he ime axis several space-ime, is in fac a pwerful alernaive l Newn's laws and Euler's herems fr deriving in a blind manner he balance equains f Cssera media. Furhermre, revealing he inrinsic gemeric naure f hese media, he apprach can assis in he develpmen f numerical mehds able cpe wih

6 nie rains. In his laer cnex, Cssera media have been prmed in he eld f he Finie Elemen Mehd, under he name f he "gemerically exac apprach" by J.C. Sim and c-auhrs [12], [13] bu wih n reference he Euler-Pincaré reducin hery. In paricular, hugh he Hamilnian srucure f he gemerically exac balance equains f rds and plaes is revealed in [14] hrugh he derivain f an apprpriae bracke, hese equains are cnsidered as a saring pin in [14], and derived frm Newn's laws and Euler's herems. Mre recenly, he relains beween Lagrangian reducin and gemerically exac beam hery, have been esablished and explied in [11], [15] and [16], wih furher exensin he case f mlecular srands [17], [18]. In [18], several ses f reduced min equains, ranging frm Euler-Pincaré Lagrange-Pincaré equains, are develped fr mdelling mlecular srands subjeced nnlcal elecrsaic frces, while in he same reference, he case f mulidimensinal media in his case, mlecular membranes, is evked as a furher perspecive by he auhrs. In he rbics cmmuniy, he Pincaré equains fr Cssera beams have als raised a grwing ineres by prving ha hey are an ecien l mdel he lcmin f nvel cninuus hyperredundan and sf rbs inspired frm sh [19], snakes and wrms [20], as well as he manipulain by sf rbs inspired frm cpus arm [21]. Mre recenly, cephalpds have drawn he aenin f birbiciss wih he lng erm aim f designing squid-like swimming rbs able prpel by cyclically cnracing a sf shell caviy inspired frm he manle f hese animals [22]. Originally driven by he need mdel sf rbs inspired frm squids, he presen aricle aims a applying he Pincaré picure he case f mulidimensinal Cssera media and especially, he Cssera shells. T ha end, we will resar frm he general cnsrucin f [11], and will remind hw ne can derive frm a Lagrangian densiy relaed he space f maerial labels f D, a rs se f Pincaré equains ha will be cnsequenly named "Pincaré equains f Cssera media in he space f maerial labels", r mre cncisely, "in he maerial space D". In his general cnex, we will cnsider bh he eld equains and he bundary cndiins f a Cssera medium D, M, wih D f arbirary dimensin p, and M a full nn-degeneraed hree-dimensinal 3D micrsrucure. This rs se f equains being relaed D, i.e., a space discnneced frm he physical gemeric space, i is physically incnsisen and pracically unusable fr shells. As a resul, we will need derive w furher ses f Pincaré equains, ne relaed he reference cngurain, and he secnd, relaed he defrmed curren cngurain f he medium, bh being embedded in gemeric space. T derive hese new equains, we will sar frm he equains in he maerial space D f [11]. Then, lying n he cncep f dualiy, we will idenify he inrinsic gemeric naure f all he bjecs hey handle, and esablish hw hey ransfrm frm maerial space D he reference and defrmed cngurain f he medium. In parallel his rs apprach f derivain, we will shw ha hese w ses f equains can be derived sraighfrwardly by exending he variainal calculus f [11] Lagrangian densiies

7 relaed reference and defrmed cngurains f he Cssera medium. Alng derivain f hese equains, we will prgressively relae he bjecs naurally prduced by he Pincaré calculus he physics f he Cssera media, and will recver in a pure deducive manner, several f he key cnceps f he micrplar hery [1], as hse relaed he maerial bjeciviy f heir cnsiuive laws, heir kinemaic and kineic mdels, several mdels f srain and sress, and heir balance equains in he gemerically exac frm. A he end, he apprach will give a srucured picure f he mdel f Cssera media, while pening prmising perspecives fr fuure. Fr he purpse f illusrain, we will hen shw hw hese furher general frmulains in gemeric space, allw recvering he s called gemerically exac balance equains f he exising shell hery as hey have been develped ver he years by her means in wrks by Reissner [23], Green and Naghdi [24], Anman [25], Libai [26] amng hers. Applying ur general equains in he case p = 2, will give he gemeric exac balance equains f a micrplar shell [26], wih several mdaliies f expressin depending n wheher hey are relaed he reference r curren cngurain f D, and, in he maerial frame r he spaial frame f M. Based n hese nminal ses f equains, we will shif frm micrplar classical shell, while sressing he rle f he cnsiuive laws in a generic reducin prcess allwing recver he gemerically exac frmulain f classical shells in which he micrsrucure degenerae in a direcr [23],[24],[25],[12]. Fllwing [12], in he resuling reduced dynamic mdel, he angular velciy eld arund he direcrs, referred as he "drilling rain" in he shell lieraure [27], is arbirarily frced zer. In a nal sep, we will reincrprae he drilling degrees f freedm and derive a furher clsed frmulain hlding fr classical Cssera shells wih drilling rains. While his apprach is inspired f wrks in nie-elemens [28], i is here saed in a new frm which explis he sae-space frm f he Pincaré picure, as a funcinal space f surfaces parameerized by he labels f D in SE3 se3. The aricle is srucured as fllws. Secin 2 inrduces all he basic deniins and saemens required by he exensin f he Pincaré picure a Cssera medium f arbirary dimensin. In secin 3, we derive he Pincaré equains f a Cssera medium in he space f is maerial labels. Secin 4 recnsider hese rs equains frm he viewpin f dualiy and analyses he inrinsic gemeric naure f he bjecs hey handled, and hw hey ransfrm frm he maerial he gemeric space. Based n his analysis, in secin 5, he Pincaré equains f p-dimensinal Cssera medium in he reference and defrmed cn- gurain are derived, and heir underlaying mdel f sress is deailed in relain cninuus media mechanics. In secin 6, his general picure is applied 2-dimensinal Cssera media derive he gemerically exac balance equains f micrplar shells. Frm hese rs se f equains, we deduce her ses f equains, hse f classical shells wih n mdel f he drilling rains secin 7, and wih a mdel f he drilling rains secin 8. Fr he pur-

8 pse f illusrain, he full Pincaré picure is applied frm is beginning axisymeric shells in secin 9. Secin 10 summarizes and pens perspecives fr applicains squid lcmin in he cnex f sf Rbics. 2 Basic saemens and deniins In his secin we sae he basic deniins required fr he res f he aricle. We invie he reader wh is familiar wih he gemeric pin f view f nie elasiciy [29] g direcly subsecin 2.5. The key infrmain f secins are essenially: he deniin f he cngurain space and he basic kinemaics f a Cssera medium eq. 3 and 8, he expressins f he area elemen in he defrmed cngurain eq. 12, alng wih he gure 1 which illusraes he gemeric cnex used in he aricle. 2.1 Deniin and space f cngurain f a classical medium Accrding he mechanics f cninuus media, a classical hree-dimensinal medium B is a cmpac se f maerial pins f Euclidean space R 3 labelled by 3 parameers {X i } i=1,2,3 in a Caresian frame O, E 1, E 2, E 3 named maerial frame. A cngurain f B is he deniin f he psiin x = x i X j e i f all he maerial pins X = X j E j f B in an inerial frame, e 1, e 2, e 3 f he ambien Euclidean space E = R 3. Frmally, we dene such a cngurain as ΦB his is he se f he x = ΦX fr X running ver B, where Φ is a smh inverible map frm R 3 R 3 which preserves rienain, i.e., an elemen f DiR 3. The space f cngurains f B in E is hus dened as: C = {ΦB E, Φ DiR 3 }, 1 and a min f B in E is dened as a curve f cngurain, i.e., a mapping: R + Φ B E, 2 where Φ denes a ime-parameerized curve f DiR 3. Amng all he pssible cngurains accessible B, we disinguish ne f hem as a reference cngurain, dened Φ B in which B is inernally energeically a res 3. In pracise we will assume O, E 1, E 2, E 3 =, e 1, e 2, e 3 and will inerchangeably speak abu he "inerial" r "maerial frame", depending n he cnex. Fr he purpse f cmpuain, ne may cnsider in all subsequen develpmens ha E 1 = 1, 0, 0 T, E 2 = 0, 1, 0 T, and E 3 = 0, 0, 1 T. Finally, ne ha hese deniins can be exended frm hree, w, and ne-dimensinal classical media, wih a maerial index i running frm 1 n wih n = 2 and n = 1 respecively. Mrever, i is wrh ning ha {X i } i=1,...n denes a maerial char 3 Here ne ha Φ is n a mechanical ransfrmain f he bdy in he ambien space bu raher a gemeric ransfrmain r mre exacly a parameerizain f B. The mechanical ransfrmain beween he reference and he defrmed cngurains is in fac dened as Φ Φ 1.

9 n he pen se B B, and ha fr plgical reasns, i may be cnvenien prvide B wih an alas f several such maerial chars, even if, in he fllwing, we will cnsider nly ne f hem. 2.2 Deniin and space f cngurain f a Cssera medium A Cssera medium D, M is a classical p-dimensinal p 3 medium D, in each pin, ned X, f which, a Lie grup G f rigid bdy mechanics SO3, SE2, SE3... acs n a rigid slid f small dimensins a "micr-slid" r "micrsrucure", dened M, generae all he pssible cngurains f D, M. The cngurain space f a Cssera medium D, M can hus be dened as he fllwing se f parameerized maps in G: C = {g : X D gx G}. 3 Mins f D, M in C are dened as he ime-parameerized curves f cngurain: R + g D = gd, G. 4 where he nain gd, indicaes ha he space and ime variables play similar rles. This basic cnex can be used describe dieren physical siuains depending n he meaning we aribue D and M. Fr insance, a nn-classical hree dimensinal 3D medium cnsiued f hree-dimensinal micr-srucures, als called micr-plar medium, beys his deniin if we ake M D R 3. In all he aricle, we will preferenially use he abve deniin as a reduced mdel describe a classical 3D medium B fr which B = D M where D is a sub-manifld f B ver each pin X f which, M is ransfrmed by an elemen f G = SE3, represened by an hmgeneus ransfrmain f he general frm: RX rx gx =, wih RX SO3 and rx R 3 being he rain and ranslain cmpnens f g respecively. Fllwing secin 2.1, he parameerizain f B = D M is chsen in such a manner ha D and M are crdinaized by {X α } α=1,2..p and {X γ } γ=p+1,..3 respecively. Hence, any pin in D is inrinsically dened as X = X 1, X 2,...X p and he map e : X ex = X, 0 3 p, denes an embedding frm D B R 3. This embedding allws any cngurain f D in E be dened as he submanifld Φ ed f ΦB. In E, a cngurain f he micrsrucure M abve X will be dened as ΦX, M see gure 1. Wih his paramerizain he reducin f B in D M is mivaed by cnsidering maerial media as beams and shells, having dimensins alng D far larger han he hers alng M, i.e., media fr which he 3D cngurains f 1 can be expanded in he fllwing Taylr series in which γ = p + 1,..3: ΦX = ΦX, 0 3 p + Φ X, 0 3 p X γ + X 2, 6 X γ

10 wih X = X ex he vecr cmpnen f X alng M. Based n his expansin, he Cssera based apprach cnsiss in reducing he kinemaics 6 is rs rder apprximain wih respec X while neglecing he defrmains f he maerial abve each X, 0 3 p, a cndiin which denes he rigid micrsrucure M. These apprximains allw 6 be rewrien as: ΦX = rx + γ X X γ 7 wih X rx = Φ ex he eld f psiin f he maerial pins f D in E, and γ X = Φ/ X γ ex a se f vecrs lying in ΦX, M. Furhermre, M being rigid, i is always pssible chse is parameerizain {X γ } γ=p+1,..3 such ha γ = RX.E γ wih RX SO3, and rewrie he reduced kinemaics 7 as: ΦX = rx + γ X X γ = rx + RX.Xγ E γ = rx + RX.X, 8 which explicily makes he grup ransfrmains gx SE3 f 3-5 appear, wih RX dened as he w-pin ensr [30]: RX = i X E i, and: rx = r i Xe i. As in rigid-bdy mechanics, hese ransfrmains ac n he maerial frame O, E 1, E 2, E 3 cnsidered as rigidly aached M. Hwever hey d n ransfrm i in a single frame, bu in a eld f rhnrmal mbile frames 1, 2, 3 X = RX.E 1, RX.E 2, R.XE 3 based in each pin Φ ex as illusraed in gure 1. In [11], he reduced kinemaics 8 are applied beams, while in he secnd par f he aricle, hey are applied shells wih D R 2 dening he maerial reference shell's mid surface. Fr a shell, he micrsrucure M mdels a generic rigid ber acrss each pin f is midsurface D, i.e. a degeneraed ne-dimensinal rigid bdy named "direcr" in he shell's lieraure [24]. This is in cnras wih beams, where M sands fr a full 3D rigid bdy mdelling he beam crss-secins. As a resul, fr shells, shifing frm 7 8, i.e. replacing 3 by R in he basic kinemaics inrduces an indeerminacy in he mdel f a classical shell which shuld be remved by reducing is cngurain space frm 3 {r, 3 : X D rx, 3 X R 3 S 2 }. Hwever, we will inially ignre his fac and cnsider shells as full- Cssera 2D media wih cngurain space 3, i.e. media fr which M is a full nn-degeneraed rigid bdy which a full 3D-rhnrmal frame can be aached. This will allw us see ha X RX is univcally dened by a dynamic mdel bained by applying Pincaré's picure n he cngurain space 3. Frm a physical pin f view, such a mdel hlds fr micr-plar shells, i.e. 2D-media wih inrinsic kineic spin and cuple sress alng 3 [1]. In a secnd sep, we will see hw he mdel f 2D-full Cssera media can be adaped, and he indeerminacy n R can be remved, when cnsidering he classical mdel f shell in which he frame aached M is degeneraed in a single direcr. 2.3 Cnveced frame and c-frame In all he aricle, we will use cnveced frames express he ensr elds relaed he mechanical sae f B and D. A cnveced frame eld is de-

11 E 3 g 1 g dg 1 dg gx G g dg D X dx M E 2 Grup : G X, M 3 dg 2 h 2 H 1 E 1 d X Maerial space : B E e 3 3 e D h 1 1 O E e 1 1 E e 2 2 h 1 gx g X dx Gemeric space : E Fig. 1. Lef and righ bm: Parameerizain f he defrmed cngurain Φ B hrugh a ransfrmain f B. Righ: Kinemaics in E bm and n G p f frames rigidly aached he micr-srucures. ned in each pin f he curren cngurain f B as he naural basis angen a se f maerial crdinae lines drawn n B and adveced by is curren defrmain. Le us cnsider a min as dened by 2 wih Φ = Φ =0. A any ime, we may dene he eld f he cnveced frames cvering he manifld Φ B as a map Φ B T Φ B which assigns any pin Φ X E, he frame g 1, g 2, g 3 X = Φ / X i X i=1,2,3. In his eld f frame, he Euclidean meric f E is dened as he fundamenal ensr g i.g j g i g j = g ij g i g j f deerminan g, where Φ X g 1, g 2, g 3 X denes a eld f c-frame such ha g i.g j = δj i. Applying he same cnsrucin he sub-manifld Φ ed Φ B allws inrduce w her elds f cnveced frame and c-frame whse base pins lie in Φ ed respecively dened as h α α=1,2..p X = r/ X α X α=1,2..p, and h α α=1,2..p X such ha h α.h β = δβ α, wih h = h αβh α h β he fundamenal meric ensr n Φ ed whse deerminan is dened by h. Using he expressin f he reduced kinemaics 7 in hese deniins, i is sraighfrward shw ha g i X, 0 i=1,2,3 = h 1,..h p, p+1,.. 3 X. and ha, fr a nedimensinal Cssera medium beam, h 1 X 1 is a angen vecr he defrmed line f he beam cenrids in is maerial abscissa X 1, while 2, 3 X 1 span he curren cngurain f X 1 -crss-secins, i.e., Φ X 1, M. Fr a 2-

12 dimensinal Cssera medium a shell, h 1, h 2 X 1, X 2 denes a basis f he angen planes he shell's mid-surface in he base pin f maerial crdinaes X 1, X 2 while 3 X 1, X 2 span he curren cngurain f X 1, X 2 -bers, i.e., Φ X 1, X 2, M. 2.4 Pull-back and push-frward Our Cssera medium is dened as B = D M where M and D are ransfrmed hrugh dieren kinemaics. This means ha we can dene w kinemaically independen push-fward/pull-back prcesses, ne relaed he rigid ransfrmains f M, he secnd relaed he defrmains f D. We nw presen hese w prcesses and refer he reader gures 1 and 2 which prvide a parial illusrain f he cnex. Pull-back and push-frward by he rigid ransfrmains f M: Due he presence f he micrsrucure in heir basic cnsiuive deniin, Cssera media inheri frm he gemeric picure f he rigid bdy [31], in which rhnrmal frames play a crucial rle. In paricular, any vecr eld Φ ex vx f T Φ ed can be inerpreed as a vecr eld f E, i.e. a "spaial vecr eld" expressed in he eld f mbile rhrnrmed frame accrding v = V i i r pulled-back in he unique maerial frame f B hrugh V = R T.v = V i E i. Remarkably, he cmpnens f a spaial vecr in is mbile frame are hse f is pull-back, named he "maerial vecr", in he maerial frame. Due he rhgnaliy f R, he same relains apply c-vecr elds f T Φ ed and nally any Euclidean ensr eld angen Φ ed. Fr he purpse f illusrain see als gure 2, le us cnsider he case f shells fr which p = 2, and cnsider he eld f frames and c-frames dened by h 1, h 2 X and h 1, h 2 X. They can be ransfrmed in X Γ 1, Γ 2 X = R T.h 1, R T.h 2 X and X Γ 1, Γ 2 X = R T.h 1, R T.h 2 X, which dene midsurface cnveced frame and c-frame elds respecively, pulled back in he maerial frame. Ne ha he change f space is idenical fr he w elds because R 1 = R T, while dualiy impses Γ α.γ β = δ β α. Pull-back and push-frward by defrmains f D: Due he presence f he classical medium D in he deniin f he Cssera medium D M, a secnd pull-back/push-frward prcess hlds beween T ed and T Φ ed. Using he ransfrmain Φ, any ensr eld n ed can be pushed frward n Φ ed, and reciprcally pulled back frm Φ ed ed by using he resriced linear angen maps Φ = h α E α, Φ 1 = E α h α, Φ T = E α h α and Φ T = h α E α. In paricular, he frames and c-frames cnveced by he ransfrmain are relaed {E α } α=1,..p and {E α } α=1,..p hrugh: h α = Φ.E α, E α = Φ T.h α. 9

13 M E D RE. RE. h. E X E T Rh X, M e D Maerial space : B Gemeric space : E Fig. 2. Gemeric picure f frames and heir push-frward/pull-back relains. Hwever, sme gemeric angen bjecs as he "exerir frms" and "mulivecrs" als invlve h in his pull-back/push-frward prcess. This is paricularly he case when cnsidering he riened maerial vlume elemen dx 1 dx 2...dX p f D, which is changed in he meric defrmed vlume f same rienain h dx 1 dx 2...dX p by he ransfrmain Φ e. Similarly, he riened maerial area elemen in any pin X f D is dened as he p 1 frm dy 1 dy 2... dy p 1, where {Y γ } γ=1,2...p 1 is a Caresian char, f naural rhnrmal basis {H γ = / Y γ } γ=1,2..p 1, cvering a maerial hyperplane crssing X. This frm is ransfrmed in h 1/2 dy 1 dy 2...dY p 1 by he defrmain, wih h he deerminan f he fundamenal ensr f E in he cnveced basis {h γ = Φ.H γ = r/ Y γ } γ=1,2..p 1. Expliing dualiy f p 1-frms and p 1-vecrs, we may dene he maerial riened area elemen as he p 1-vecr H 1 H 2... H p 1, which can be represened by he cnjugae rue vecr f T ex [32]: ν = ν α E α 1 = p 1! ɛ αα 1...α p 2α p 1 X α1, X α2..., X αp 1 Y 1, Y 2, Y 3..., Y p 1 Eα, 10,,.., which denes he uni nrmal he elemen, where,,.., denes a funcinal deerminan, while fr any ineger n, ɛ i1i 2...i n is equal zer if w indices

14 are idenical, equal +1 respecively 1, if i 1, i 2,...i n is an even respec. dd permuain f 1, 2,...n. Similarly, nrmalizing he cnjugae vecr f h 1 h 2... h p 1, denes he uni nrmal he riened defrmed elemen as he cvecr: ν = ν,α h α = h h 1/2 1 p 1! ɛ αα 1...α p 2α p 1 X α1, X α2..., X αp 1 Y 1, Y 2, Y 3..., Y p 1 hα. 11 Ne ha he w vecrs ν and ν whse cmpnens are relaed by ν,α = h / h 1/2 ν α, cnain all he infrmain abu he rienain f he riginal p 1-frms. Mrever, dening he measure area f he maerial and defrmed elemens as ds = dy 1...dY p 1 and ds = h 1/2 dy 1...dY p 1 respecively, where each dy γ represens he cmpnens f an inniesimal vecr dy γ H γ wih n summain n γ = 1, 2...p 1, we als have he relain beween he c-vecrs ν ds and νds which will be used in he subsequen develpmens insead f he riginal p 1 frms: ν ds = ν,α ds h α = h ν α ds Φ T.E α = h Φ T.νdS. 12 Taking ds = H 1/2 dy 1...dY p 1, 12 hlds fr a relain beween maerial and defrmed riened area elemens f any hyper-surface f D parameerically dened by X α Y γ, wih H he deerminan f he meric f D in he naural basis {H γ } γ=1..p 1 cvering he hyper-surface. 2.5 Lagrangian f a Cssera medium On he deniin 3 f he cngurain space, he Lagrangian f a Cssera medium B = D M a curren ime is dened as he fllwing funcinal: L = L g, g, g h dx 1 X α dx 2... dx p, 13 D where L h dx 1... dx p is he Lagrangian vlume-frm f he Cssera medium, and L is he densiy f Lagrangian per uni f meric vlume h dx 1 dx 2... dx p f he curren defrmed cngurain Φ ed. Alernaively, L can be relaed he vlume f reference cngurain Φ ed as: L = D L g, g, g h X α dx 1 dx 2... dx p, 14 r direcly he vlume f parameric maerial space D: L = L g, g, g X α dx 1 dx 2... dx p. 15 D In 14 respecively 15 L, respec. L, is he densiy f he Lagrangian f B per uni f meric vlume h dx 1 dx 2... dx p respec. nn-meric vlume dx 1 dx 2... dx p f Φ ed respec., f D.

15 2.6 Reducin f he Lagrangian Physically, he Lagrangian L depends n g/ hrugh he kineic energy f B, and n g/ X α hrugh he inernal srain energy f is maerial ha is assumed be hyperelasic. The hree Lagrangian densiies L, L and L are lef invarian in he sense ha subsiuing g by kg, in any f hem wih k a cnsan ransfrmain in G ver each pin f space-ime D R +, des n change is value. Physically, his reecs he fac ha bh he densiies f kineic and inernal srain energy are he same when bserved frm any frame f gemeric space. The rs prpery is a key resul f rigid bdy mechanics [31], while he secnd is a cnsequence f he maerial frame indierence f mechanics f cninuus media [33]. As a resul, aking k = g 1, allws he ransfrmain f he abve hree Lagrangian in a unique reduced Lagrangian, which can be wrien, if, fr insance, we sar frm 13, as: L r = D L η, ξ α h dx 1 dx 2... dx p, 16 where we inrduced he fllwing lef invarian vecr elds leaving in he Lie algebra g f G cnsidered as a grup f marices: 1 g η = g, ξ α = g 1 g, α = 1, 2...p 17 Xα In 16, L denes he reduced Lagrangian vlume densiy per uni f meric vlume f defrmed cngurain Φ ed, while h is by deniin, he deerminan f he marix h αβ = h α.h β = R T. r/ X α.r T. r/ X β = Γ α.γ β. Then, using 5 in 17, shws ha Γ α is he linear ppsed angular cmpnen f ξ α. Thus h nly depends n he lef invarian elds ξ α, and 16 des dene, as a whle, a reduced Lagrangian L r in he Lie algebra g. Alernaively, applying he same reducin prcess 14 r 15 insead f 13, permis he reduced Lagrangian densiies be dened per uni f meric vlume f reference cngurain Φ ed and per uni f nn-meric maerial vlume D, dened L and L respecively. As L, hese w furher densiies nly depend n η, ξ α. Mrever, in each f hese w her cases, he vlume elemen des n depend n he curren cngurain and he crrespnding Lagrangian is lef invarian. Finally equaing he reduced versins f 13, 14 and 15, allws saing he fllwing relains beween he hree reduced Lagrangian densiies: L = L h = L h, 18 In he fllwing, we sar frm he densiy L relaed he maerial space and will inrduce laer he case f densiies L and L.

16 3 Pincaré's equains f Cssera media in he maerial space In his secin we quickly remind he cnsrucin f [11] leading he balance equains f muli-dimensinal Cssera in he maerial space, i.e., in he space f he labels f he micr-srucures. These equains will be a rs sep ward heir exensin he gemeric space, namely in he reference and defrmed cngurain, in which hey will ake he cnsisen frm required by heir pracical use. Fllwing Pincaré's apprach [2], a Cssera medium B = D M subjeced a se f exernal frces is gverned by he exended Hamiln principle, which can be saed direcly n he deniin 3 f C α running frm 1 p, as: 2 2 δ L η, ξ α dx 1 dx 2... dx p d = δw ex d, 19 1 D 1 fr any δg = gδζ where δζ g is a eld f maerial variain f g achieved while and all he X α are kep xed. In 19, δw ex mdels he virual wrk f exernal frces and can be deailed as: δw ex = < F ex, δζ > dx 1 dx 2... dx p + D < F ex, δζ > H 1/2 dy 1 dy 2... dy p 1, 20 D wih <.,. > he dualiy prduc in he Lie algebra g, Y γ γ=1,2..p 1, a se f maerial crdinaes cvering he bundary D. Finally, F ex and F ex are densiies f exernal frces in g per uni f vlume f D and D respecively. Ne ha we cnsider bundaries subjeced exernal frces nly, he case f impsed mins being easily mdelled by dening w ypes f bundaries. Als ne ha fr beams, i.e. ne-dimensinal Cssera media, ne has ake: H 1/2 = δ irac D, 21 where δ irac is he Dirac disribuin, and 20 has be inegraed wihin he meaning f disribuins. Nw, le us invke he cnsrains f variain a xed ime and maerial labels: δ g = δg, δ g X α = δg X α, fr α = 1, 2..p. 22 Then insering "δg = gδζ" in 22 gives he fllwing relains, as rs derived by Pincaré [2], which play a key rle in he variainal calculus n Lie grups [34]: δη = δζ + ad ηδζ, δξ α = δζ X α + ad ξ α δζ, 23 wih ad he adjin map f g n iself. As deailed in [11], applying he sandard variainal calculus 19 wih 23 running befre he usual by par inegrain

17 in ime, and he divergence herem, gives he Pincaré equains f a Cssera medium in he maerial frame we use summain cnvenin n repeaed indices α: L ad η η L ν α = F ex, L + L L η X α ad ξ = F ex, 24a α 24b where ad is he c-adjin map f g n g, while ν = ν α E α is he uni uward nrmal ed, and L/ η, and L/ dene sme cnjugae generalized mmens ha will be deailed in he fllwing. These equains represen he dynamics f a Cssera medium reduced in he dual f is Lie algebra here idenied as he space f inniesimal maerial righ ransfrmains f G. They gvern he ime-evluin f he maerial velciies η and when he exernal frces are lef invarian, hey can be ime-inegraed separaely cmpue he velciy eld in a rs sep. In a secnd sep, he min f he medium can be recnsruced by using he s-called recnsrucin equain, which can be simply saed as: X D : g X, = gηx,. 25 In all cases symmeric r n, 25 supplemens 24, give a se f imeevluin equains in he fllwing deniin f a Cssera medium's sae space: S = {g, η : X D g, ηx G g}. 26 Remark 3.1: Unfrunaely, hese equains are n direcly expliable in pracise since hey are saed in he maerial space f micrsrucure labels D, i.e., in a nn meric space a priri discnneced frm he gemeric physical space. The purpse f wha fllws is give hem he physical cnsisency required by heir applicain. This will be dne by reexpressing he medium dynamics in erms f he meric densiies L and L. A he end, we will bain w new ses f equains hlding fr he Pincaré equains f Cssera media in he reference and defrmed cngurain Φ ed and Φ ed respecively. These furher equains saed in secin 5 will allw recvering several gemerically exac frmulains fr shells in secins 6, 7, 8. T derive hem, we need ener furher in he gemeric mdel underlaid by Gemeric mdel f Cssera media The variainal calculus leading 24 handles velciy-ype vecrs η, ξ α and frce-ype vecrs L/ η, L/, F ex, F ex f g and g respecively, which are dual f each her hrugh he dualiy prduc <.,. >. Hwever, his

18 calculus hides a furher dimensinaliy invlving he space-ime base-manifld D R + [8]. Thugh n readily apparen, his furher aspec f he hery srngly srucures he gemeric mdel f Cssera-media and especially ha f inernal sress. The purpse f his secin is inrduce his aspec and use i prepare he grund fr he mdel f Cssera shells as i will be discussed in he furher secins. T inrduce his impran pin, we will rs build n he cncep f dualiy. 4.1 Dualiy in he maerial space We cnsider in his secin a maerial Lagrangian densiy L in he frm Lη, ξ α = Tη Uξ α, wih T and U, he densiy f kineic and inernal elasic penial energy per uni f maerial vlume respecively. Le us rs remark ha in he Pincaré-Cssera picure, η and ξ α=1,2...p are n nly vecr elds in he Lie algebra bu als he cmpnens f a unique eld f 1-frm n space-ime, wih value in he Lie algebra g f G [11]. Endwing g wih a basis {l 1, l 2...l n }, such a eld, here generically ned as Υ, is dened as Υ : X, D R + Υ X, g T D R +, and may be deailed as: Υ X, = Υ j 0 l j d + Υ j β l j dx β, 27 where 0 denes he crdinae-index alng ime axis, i.e., X 0 =, while in he case f he lef-invarian elds f 17, Υ j 0 l j = η and Υ j β l j = ξ β. Similarly, he generalized mmena L/ η and L/=1,2..p. gemerically dene he cmpnens f a eld in he dual f he space f Υ, i.e. a unique vecr eld n spaceime wih cmpnens in he dual f he Lie algebra f G f basis {ω 1, ω 2,...ω n }. Such a eld, generically dened as Λ : X, D R + g T D R + is deailed as: ΛX, = Λ 0 i ω i / + Λ α i ω i / X α, 28 where in he case f ur generalized mmena, we have Λ 0 i ωi = L/ η = T/ η and Λ α i ωi = L/ = U/. Wih hese deniins, in each pin f spaceime, any Λ linearly acs n any Υ accrding he fllwing duble dualiy prduc dened <.,. >: < Λ, Υ > = Λ 0 i Υ j 0 < ωi, l j > /, d + Λ α i Υ j β < ωi, l j > / X α, dx β, wih < ω i, l j >= δj i, while.,. is anher dualiy prduc requiring cnsidering vecrs v f T D R + as linear funcinal acing n 1-frms ω f T D R + accrding vω = ωv = ω, v = v, ω. Wih hese cnsiderains, and since./ X i, dx j = δ j i wih X0 =, we simply have: < Λ, Υ > = Λ 0 i Υ i 0 + Λ α i Υ i α =< Λ 0, Υ 0 > + < Λ α, Υ α >= Λ i, Υ i, 29 which appears a he very beginning f he abve variainal calculus in he virual wrk f he inerial and inernal frces and in all is cnsequences. In he subsequen develpmens, he vecrs Υ 0 = Υ i 0l i and Υ α = Υ i αl i are velciy-ype

19 vecrs, r in rigid bdy mechanics' erminlgy, are "wiss" f g. On he dual side, Λ 0 = Λ 0 i ωi and Λ α = Λ α i ωi are frce-ype vecrs r "wrenches" f g. Mrever, frm he abve cnex, he cmpnens f he w elds 27 and 28 in heir respecive basis f g and g, can be wrien as: Υ i = Υ i 0d + Υ i αdx α and Λ i = Λ 0 i / + Λα i / Xα which dene 1-frms acing n T D R + and T D R + respecively. Kinemaically, he Υ i wis cmpnens mdel he space-ime variains f he rigid micrsrucure M-cngurain in G in any pin f D R + when shifing alng any direcin f D R +, while he Λ i wrench cmpnens mdel he kineic mmenum f each cpy f M abve D and he sress exered n i. We will deail furher hese relainships in secin 4.5 afer shifing his cnex he reference and defrmed cngurain in subsecin 4.2 and inrducing he rle f densiies and vlume frms in subsecin Dualiy in he reference and defrmed cngurain The abve cnex can be shifed he reference and defrmed cngurain by using he push-frward and pull-back perains in he cnveced frame and c-frame elds angen Φ ed and Φ ed inrduced in secin 2.3 and 2.4. In his cnex, he eld Υ = Υ 0 d + Υ β dx β ransfrms in: Υ = Υ 0 d + Υ β h β, Υ = Υ 0 d + Υ β h β, 30a 30b n he reference cngurain 30a, and he defrmed 30b. On he dual side, he eld Λ = Λ 0 / + Λ α / X α ransfrms in: Λ = Λ 0 / + Λ α h,α, Λ = Λ 0 / + Λ α h α, 31a 31b n he reference and defrmed cngurain. Ne ha in 31, he ransfrmain des n nly aec he basis vecrs f Λ by pull-back bu als is cmpnens, since as we will see in he nex secin, Λ is n a rue ensr bu a vlume densiy and is ransfrmain invlves h as menined in secin 2.4. Wih hese furher cnveced elds, he dualiy prduc 29 sill hlds, bu beween he cnveced frame and c-frame elds h α α=1,2..p and h α α=1,2...p, and we have: < Λ, Υ > = < Λ, Υ > = < Λ, Υ >. 32 Fllwing he remark a he end f he previus sub-secin, Υ and Υ sill mdel he space-ime variains f he micrsrucure cngurain in G, bu when shifing alng he meric crdinaes lines drawn n he reference and defrmed cngurains by he cnvecin f he maerial char. Similarly, Λ and Λ mdel he kineic mmenum f he micr-srucures and he sress exered n hem, bu relaed he reference and defrmed cngurain in a way we will deail in he nex secin afer inrducing he rle f densiies and vlume frms in he nex subsecin.

20 4.3 Vlume frms and densiies All he erms f he equilibrium described in 24 dene densiies f wrench in g relaed he meric vlume f D 24a, and D 24b. In paricular, he exernal vlume frces invlved in 20, are inrinsically dened as he vlumefrm eld n D wih values in g : F ex dx 1 dx 2... dx p which ransfrms in: F ex, h dx 1 dx 2... dx p, F ex, h dx 1 dx 2... dx p, 33 in he reference and curren defrmed cngurain. In he same way, he exernal bundary frces exered n D are dened by vlume-frm elds n he bundaries f D wih value in g f he ype F ex dy 1 dy 2... dy p 1, which ransfrm in: F ex, h 1/2 dy 1 dy 2... dy p 1, F ex, h 1/2 dy 1 dy 2... dy p n he reference and defrmed cngurain. Similarly, any space-ime maerial cnjugae mmenum f he ype Λ dened in 28 is in fac he unique cmpnen f a vlume-frm eld Λ dx 1 dx 2... dx p = Λ 0 / + Λ α E α dx 1 dx 2... dx p, wih values in g T D R + which ransfrms in: Λ h dx 1 dx 2... dx p = Λ 0 h / + Λ α h h α dx 1 dx 2... dx p, 35 n he reference cngurain, and in: Λ h dx 1 dx 2... dx p = Λ 0 h / + Λ α h hα dx 1 dx 2... dx p, 36 n he defrmed cngurain. The hree vlume densiies Λ, Λ and Λ are relaed he maerial vlume elemen, he meric reference and he meric defrmed vlume elemen respecively, and are apparen in he fllwing funcinal dualiy prduc beween elds: < Λ, Υ > dd = < Λ, Υ > h dd = < Λ, Υ > h dd, 37 D D which cmplees he inrinsic mdel f cnjugae mmena f 24. D 5 Pincaré's equains f Cssera media in he reference and defrmed cngurain Based n he abve gemeric mdel, his secin aims a deducing frm he Pincaré equains in he maerial space 24, w her ses f equains enjying mre physical insighs: ne relaed he reference cngurain, he her

21 relaed he defrmed cngurain. T derive hese w furher ses f equains, le us rs remark ha frm he previus secin, he exernal frces 33 and he space-ime generalized mmena 35, and 36 are vlume-densiies which behave like he Lagrangian densiies L, L and L, i.e. bey he fllwing relains, similar 18, when shifing frm he maerial he reference and curren cngurain: F ex = F ex, h = F ex, h, Λ = h Λ = h Λ. 38a 38b In he same way, he exernal surface frces 34 dene vlume-densiies n he bundaries f D which behave as: F ex H 1/2 = F ex, h 1/2 = F ex, h 1/2. 39 In 38 and 39, F ex,, F ex, and F ex,, F ex, dene he densiies f he exernal wrenches exered n B, per uni f meric-vlume f Φ ed, Φ e D, and per uni f meric vlume f Φ ed, Φ e D respecively. Similarly, as his will be deailed in he nex secin, he Λ α and Λ α respecively Λ 0 and Λ 0 dene sme densiies f inernal sress wrench respecively he densiies f kineic mmenum wrench per uni f meric vlume f Φ ed and f meric vlume f Φ ed. Fr he ime being, le us remark ha inrducing L = L h = L h in he cnjugae mmenums L/ η and L/ f he maerial Pincaré equains 24, and using he ideniies 38b, allws he fllwing relains beween he densiies Λ, Λ and Λ and he Lagrangian densiies L, L and L be saed: Λ 0 = L η, Λ0 = L η, Λ0 = L η, 40 Λ α = L, Λ α = L, Λ α = L 1 h h L. 41 where we used he fac ha h is independen f η and he ξ α 's, while h depends n he ξ α 's nly. Fr he purpse f cncisin, we will wrie he las relain f 41, as: Λ α = L 1 h h [ ] L L. 42 Nw, inrducing all hese relains frm in he riginal maerial Pincaré equains 24 allws sae:

22 The Pincaré-equains f Cssera media in he reference cngurain: 1 h L h ad L η + h η η 1 h L h h X α ad L ξ = F ex,, α 1/2 h L ν α = F ex,, 43 h The Pincaré-equains f Cssera media in he defrmed cngurain: 1 h L h ad L η + h η η [ ] [ ] 1 h L h L h X α ad ξ = F ex,, α 1/2 [ ] h L ν α = F ex,, 44 h Le us d several remarks afer hese w ses f equains. Remark 5.1: The new equains 43 and 44 prlng he physical space, he equains 24 f [11], which were relaed he maerial nn-meric space D. As expeced, hey have he physical insighs ha were missing in he maerial nes 24. This will becme mre apparen in he subsequen secins where hey will be relaed cninuus media mechanics, and especially, shell hery. Remark 5.2: Equains 43 and 44 can be direcly derived by re-saring he variainal calculus leading 24, bu wih Lagrangian and exernal frces densiies per uni f meric-vlume f reference and defrmed cngurain respecively. This calculain is dne in Appendix 1 in he defrmed cngurain. Remark 5.3: While in he reference cngurain he densiy f space-cnjugae mmenum Λ α direcly derives frm he Lagrangian densiy L, his is n he case in he defrmed cngurain fr which Λ α is relaed L hrugh: [ ] L Λ α L 0 = = Γ α L, 45 where we used he fac ha h / Γ α = h Γ α. Ne ha 45 makes appear a cnvecive erm induced by he curren defrmain f D while he Lagrangian densiy L is kep cnsan. Referring hree-dimensinal hyperelasiciy, his crrespnds he well knwn fac ha an Eulerian sress ensr, as he Cauchy sress ensr, has n cnjugae srain eld, and ha cnsequenly here is n pure Eulerian hyperelasic cnsiuive law. Ne ha his cnex can be easily circumvened, and he cnvecive erm f 45 remved, if insead f aking a

23 densiy f srain energy U, we use a U relaed he reference cngurain, and he mixed Lagrangian-Eulerian cnsiuive law: Λ α [ ] L = = which has be insered in equains 44. h h 1 2 U, 46 Remark 5.4: In 43, since h is ime-independen, we can remve i frm he p line relaed he ime dimensin. This eliminain cann be applied he rs line f 44. Hwever, invking he cnservain f mass, allws his expressin be simplied in a way we will deail in secin 7, when we will apply hese equains classical Cssera shells. Remark 5.5: We recgnize in 24, 43 and 44 he divergence f he elds Λ α E α, Λ α h α and Λ α h α in he maerial char f D and in is curvilinear defrmain by Φ e and Φ e respecively, i.e.: L DIV E α = L X α, 47 L DIV h α = 1 h L h X α, 48 [ ] L DIV h α = 1 [ ] h L h X α. 49 Similarly, using he deniins f he uni nrmals a maerial, reference, and defrmed area elemen, as hey are inrduced in secin 2.4, he bundary cndiins f 24, 43 and 44 can be rewrien in he alernaive frms, which handle uxes f sress: L E α. ν = F ex, 50 L h α. ν = F ex,, 51 [ ] L h α. ν = F ex,. 52 Ne ha frm p bm, he hree divergence perars f and he crrespnding bundary cndiins naurally appear hrugh he applicain f he divergence herem, when deriving 24, 43 and 44 frm he Hamiln principle see remark 5.2 and Appendix 1. Ne als ha hese hree perars can be exended space-ime by aking he meric g ij g i g j, i, j = 0, 1, 2..p, wih g 00 = 1, g 0α = 0, g αβ = h αβ, α, β = 1, 2...p. Finally, hese perars and he assciaed bundary cndiins underly a mdel f sress fr

24 Cssera media which is deailed in he nex secin. Remark 5.6: Equains 44, as 43 and 24, represen he reduced dynamics f a Cssera medium in he maerial frame. When F ex, is g-independen i.e., when he exernal frces are lef invarian, hese equains are rs-rder parial dierenial equains gverning he maerial velciy eld f he medium ha can be inegraed separaely frm he recnsrucin f is cngurain. This decupling beween dynamics and kinemaics is he cnsequence f he symmery prperies lef invariance f he Lagrangian densiy in he maerial seing. Alernaively, we knws frm Neher herem ha symmeries are als assciaed wih cnservain laws. This secnd pin f view n he abve reducin appears if ne chses express 44 r 24, 43 in he xed frame, e 1, e 2, e 3 f space. T perfrm his change frm he maerial he spaial seing, i suces expli he prperies f he c-adjin acin f G n g, and rewrie 44 in he equivalen frm: Ad g 1 h L η + X α [ ] h L Ad g 1 = Ad g 1 h F ex,, 53 where we inrduced he furher densiies f spaial kineic, sress, and exernal wrenches per uni f defrmed vlume, all he wrenches being relaed he xed frame, e 1, e 2, e 3 : Ad g 1 L η [ ] L, Ad g 1, Ad g 1F ex,. 54 Physically, equain 53 which has is cunerpar n D and Φ ed sands fr he spaial Pincaré equains f a Cssera medium in he "cnservain frm". In paricular, if he vlume exernal frces are zer i.e. F ex, = 0, 53 can be inerpreed as a zer-divergence cndiin in D R + wih he exended meric menined by remark 5.5, as i is expeced frm Neher herem in he cnex f eld hery. In he language f gauge hery, his is he lcal cnservain law r cninuiy equain f he generalized mmenums in space-ime. Using divergence herem, his zer-divergence cndiin can be changed in a zer-ux cndiin, which when saed n he bundaries f D [0, ] wih F ex, = 0, leads he glbal cnservain law: Ad L h dd g = 1 D η Ad L h dd g = 0, 55 1 D η which mechanically sands fr he cnservain f he al kineic wrench f he Cssera medium alng ime. Ne ha being based n a zer-ux cndiin, his cnservain law hlds fr any cmpac manifld Φ ed wihu bundary. In paricular, fr mn-dimensinal media i.e. p = 1, 55 can be inerpreed as he cnservain f he circulain f he spaial angular mmenum alng any Cssera beam clsed in a lp [18]. In his alernaive frmulain, 55

25 becmes a cnsequence f he mre general Kelvin-Neher herem which als hlds fr he ideal uid [34]. 5.1 Mdel f Cssera sress T relae he mdel f sress underlaid by he Pincaré equains 24, 43, 44 he sress' mdel f mechanics f cninuus media, we cnsider G = SE3 represened by marices 5, and idenify g = se3 wih R 6 endwed wih is six dimensinal crss prduc [35], wih basis {E i, 0, 0, E i } i=1,2,3 where 0 and E i are he zer and basis vecrs f B R 3. Mrever, using he meric f R 6 allws us idenify se3 se3 and rewrie 30b and 31b in he equivalen frm: Ωi E Υ = i Kiα E V i E i d + i h α, 56a Σ i Λ = E i P i E i Γ iα E i E i N iα E i M + iα h α. 56b The w ensr elds Υ and Λ n nly ac n each her hrugh he dualiy prduc <.,. >, bu als perae n T Φ ed R + and is dual respecively. In paricular, he ime 0-cmpnens f 56 perae as fllws n he vecrs and c-vecrs f he ime axis: Ω V Γ α Ω d / = V / d = P Σ, 57a. 57b P Σ In he same way, he space α-cmpnens f 56 perae n he vecrs dx β h β and he c-vecrs ds ν,β h β = ds ν f 12 respecively, as fllws: Kα h α.dx β Kβ h β = dx β, 58a M α N α h α.ds ν,β h β = Γ β M β N β ν,β ds. 58b The perains 57a and 58a, give he rae f rain R T.dR f axis and displacemen R T.dr f he base pin f he rhnrmal mbile spaial frames RX,.E i = i i=1,2,3 X, due a small ime variain 57, and a small displacemen dx β h β alng Φ ed 58 see gure 1. In paricular, ne ha he vecrs Γ α are hse dened in secin 2.4 as he pulled back f he cnveced basis {h α } α=1,2..p in he maerial frame. The perain 57b gives he densiy f kineic wrench per uni f defrmed vlume f Φ ed a ime. The perain 58b mdels he maerial resulan dn in, and he maerial mmen dm in f inernal cnac frces exered acrss an riened defrmed

26 surface elemen ds ν = ds ν,β h β f Φ ed by he piece f defrmed maerial ward which ν pins, n is cmplemen par see gure 3 in he case f a shell. Repeaing his cnex n D and Φ ed, i.e. by using 27,28 and 30a, 31a respecively insead f 30b, 31b, allws us inrduce hree ensr densiy elds f Cssera sress whse cmpnens are relaed he space α- cnjugae mmena f 24, 43 and 44 as fllws: M α L N α E α = E α, 59a M α L N α h α = h α, 59b [ ] M α L h α = h α. 59c N α I is wrh ning here ha hese hree densiies f ensr are cnsisen wih he hree divergence perars and bundary cndiins 47-50, 48-51, f remark 5.5. The rs 59a respecively, secnd 59b and hird 59c f hese hree ensr-densiies, allws dening he densiy f inernal sress wrench F in = Min T, N in T T exered acrss an riened maerial surface elemen respecively, a reference and defrmed surface elemen f nrmal ν respecively, f nrmal ν and ν, by he piece f maerial respecively, he piece f reference and defrmed cngurain ward which ν, respecively ν and ν pins, per uni f is maerial respecively per uni f reference and defrmed area, as fllws: df in ds = M β N β ν β, df in M β = ds N β ν,β, df in = ds M β N β ν,β. 60 Remark 5.7: The abve sress ensrs have heir lef leg in he maerial frame. Referring secin 2.4, we can push frward hem frm he maerial he micrsrucure frames dene he hree spaial ensr elds: m m α = n n α E α, 61a m n m n = m α n α = m α n α h α, h α, 61b 61c where m α.,, = R.M.,, α = M.,, iα i and n α.,, = R.N.,, α = N.,, iα i are densiies f spaial frces and cuples per uni f maerial, reference and defrmed vlume. These hree ensr-densiies map a maerial 61a, a reference 61b, and a defrmed 61c riened area elemen subscriped wih.,, respecively, n he spaial inernal wrench df in = dm T in, dnt in T g exered n i. Referring elasiciy f classical 3D media, he ensr 61b generalizes he rs Pila-Kirchh since i acs n he reference area elemen give he resulan