A New I -Based Hypeelastic Model fo Rubbe Elastic Mateials Osca Lopez-Paies SES Octobe -4, Evanston, IL
Neo-Hookean odel W ìï ï ( I - ) = ( if l + l + l - ) lll = = í ï + othewise ïî (*). Matheatical siplicity (aenable to analytical solutions fo fundaental BVP and ipleentation in FEM codes). Physical intepetation (Gaussian statistics ). Good ageeent with expeients (fo soe elastoes and up to easonably lage defoations) Eve since its deivation in the 94 s, thee have been nueous (oe than 4!!!) efineents of the Neo-Hookean odel (*). Many of the do not confo with featues,, and/o Vahapoglu & Kaadeniz ()
Poposed constitutive odel A oe appopiate basic easue of stain -a a a j( I ; a) = ( I - ), a Î a Note: i) it genealizes the linea NH easue ii) it lineaizes popely (unlike ( I - ) a ) iii) it neglects I fo atheatical siplicity Poposed odel WI M -a a a ( ) = å ( I - ) = a, a eal-valued ateial paaetes j ( I ; ) = ( I - ) M tes to be included in the suation Lopez-Paies ()
. Matheatical siplicity The esulting Cauchy stess is siply given by M W æ ö -a a- T T = - pi = I å FF - pi F ç è = ø with pincipal stesses æ M ö -a a- t = I i ç å l - p ( i =,, ) i çè = ø And the inceental tangent odulus by M M W æ ö æ ö -a a- -a a- L = = ( a - ) I FÄF+ I F çå è çå = ø è= ø These closed-fo quantities ae needed fo the solution of BVPs, and fo the ipleentation in nueical codes such as ABAQUS.
. Physical significance of ateial paaetes Non-Gaussian statistical echanics odel of Beatty () T T =Y( I )FF -pi Hee, Y( I ) contains infoation about the statistical distibution of the undelying polyeic chains Exaple: Auda-Boyce odel with enegy Y = AB nkt W I C I ( ) = ( ) AB å - = - ( I / h ) nkt I / h - Note: By choosing = nktc and a = WI educes to the AB odel M -a a a ( ) = å ( I - ) = a
. Pedictive capabilities The poposed enegy adits the polynoial epesentation: -i é M æi- ö ù WI ( ) = å ( a j) - ( I- ) i= i êåç ú! = è ë j= ø û Thus, in the liit of sall defoations it educes to a - M M WI ( ) = å ( I- ) + å ( I- ) + O( I-) = = å = i ( ) In the sall defoation egie: it lineaizes popely and has no dependence on a with M = In the odeate defoation egie: polynoial dependence on ( I - ) In the lage defoation egie: stong dependence on a povides the eans to captue the typical lock-up
Copaisons with elastoes Fo deonstation puposes, conside the two-te odel -a -a a a a a WI ( ) = ( I - ) + ( I - ) a a Note: the ateial paaetes, a,, a will be deteined fo uniaxial data only by least-squae fitting and by enfocing the pope lineaization condition + = Fo copaison puposes, we also conside the esponse of the Gent odel J I W ( I ) ln é - ù =- - G ê J ú ë û whee and J ae ateial paaetes
Copaisons with a vulcanized ubbe S un (MPa) 6 5 4 Uniaxial Tension Two - Te Model Gent Model Data S (MPa) 4.5.5.5.5 Biaxial Tension Pue Shea Tension l = Two - Te Model Gent Model Shea Data Biaxial Data 4 5 6 7 8 l 4 5 6 l Teloa (944)
Copaisons with a silicone ubbe Uniaxial Tension & Copession Biaxial Tension.5.6.4. Two - Te Model Gent Model Data S un (MPa) -.5 S bi (MPa).8.6 - Two - Te Model Gent Model Data.5.5 l.4...4.6.8. l Meunie et al. (8)
Copaisons with a silicone ubbe Pue Shea Copession l = Pue Shea Tension l = -.4.8 Two - Te Model Gent Model Data S ps (MPa) -.8 -. S ps (MPa).6.4 -.6 Two - Te Model Gent Model Data.5.6.7.8.9 l...4.6.8. l Meunie et al. (8)
Copaisons with a Michelin elastoe Uniaxial Tension Siple Shea 4.5.5 S un (MPa).5.5 S ss (MPa).5 Two - Te Model Gent Model Data....4.5.6.7.8 l.5 Two - Te Model Gent Model Data....4.5.6.7.8 g Lahellec et al. (4)
Constitutive estiction on and Necessay and sufficient conditions fo stict polyconvexity a W '( I ) > and W '( I ) + IW ''( I ) > Necessay and sufficient conditions fo stong ellipticity W '( I ) > and ( l l - ) W '( I ) + I - - W ''( I ) > ( i =,, ) i i Sufficient conditions fo stict polyconvexity (and theefoe fo stong ellipticity) in tes of the ateial paaetes > and a > ( =,,..., M)
Cavitation instabilities Undefoed FE odel l Mesh nea cavity l l x 5 l l l = = Soe specifics: 64,8 8-node bick eleents f 5x 4 Initial volue faction of cavity: f = p/6».5-9 -9 f Cavitation ensues wheneve f Nakaua & Lopez-Paies () 5 = f.5.5.5 s/
FEM Results t > t t= > t t= t f / f 5 unde fixed f / f = 5 Nea Cavity
Onset-of-cavitation suface: Michelin Rubbe Axisyetic Loading.5..5 s.5..5. s t/ - - - - - - t /.5 t = t = t - - t / Hydostatic stess: Shea stesses: t s t + t + t = t -t t -t =, t = Nakaua & Lopez-Paies ()
Genealizations The poposed incopessible, isotopic odel constitutes a pactical platfo fo which to account fo oe levels of coplexity to odel elastoes Although weakly, W is expected to depend on the second invaiant I. At the expense of sacificing atheatical siplicity it ay be of inteest to conside M -a N -bs a a bs bs = å - + u å - s = a s= bs WI (, I) ( I ) ( I ) Copessibility effects ay be eadily added in a nube of diffeent ways, e.g., M -a M a a ' WI J I J J å å = a = (, ) = ( - )- ln + ( -) Othe effects that can be added include Mullins effect, hysteesis, as well as ate and theal effects
Final eaks We have poposed a new constitutive odel fo ubbe like ateials that: i) is atheatically siple, ii) contains paaetes that ay be given a physical significance, and iii) chaacteizes and pedicts accuately the esponse of a vaiety of elastoes. In view of its functional siplicity, it is staightfowad to ipleent it in coecial finite eleent packages (e.g., ABAQUS) fo the study of stuctual pobles. In addition, the poposed I -based odel peits to eadily check fo conditions of polyconvexity and stong ellipticity, needed to undestand the developent of instabilities