Parameter identification of hyperelastic and hyper-viscoelastic models

Transcription

1 Parameer defcao of hyperelasc ad hyper-vscoelasc models *Y Feg Wu ), A Qu L ) ad Hao Wag 3) ), ), 3) School of Cvl Egeerg, Souheas Uversy, Najg, 0096 ) Bejg Uversy of Cvl Egeerg ad Archecure, Bejg,00044 ABSTRACT Based o he Ogde model ad he Leveberg-Marquard olear opmzao algorhm, a professoal mehod ha ca realze he comprehesve fg of he uaxal eso, baxal eso, plaar eso ad smple shear expermeal daa of rubbers was developed. The experme daa from Treloar(944) was fed very well, ad he deermed parameers by usg hs mehod were proved o be correc ad praccal he umercal verfcao ANSYS. The, he cosuve model of he hyper-vscoelasc maerals whch combes he Ogde model wh he geeralzed Maxwell model was explaed deal, ad he parameer defcao algorhm was proposed based o he egrao of he relaxao modulus. Moreover, he resrcos of he al values se for he udeermed parameers he hyper-vscoelasc model were vesgaed, ad he valdy ad he praccably of he esmaed parameers was also verfed ANSYS.. INTRODUCTION Wh he rapd developme of moder dusralzao, rubbers are oe of he mos remarkable maerals havg a wde rage of applcaos cvl egeerg, aerospace egeerg, mechacal egeerg, auomove egeerg, ec. I order o mee varous requremes of he dusry, specal fllers, lke carbo black or slca, wh dffere proporo are usually added durg vulcazao for mprovg he sregh ad oughess properes, whch ur makes dffcul o accuraely characerze he mechac properes of rubbers(am e al. 006). Rubbers usually prese a umber of eresg feaures lke hyperelascy ad hyper-vscoelascy. The sress-sra relaoshp of he hyperelascy ca be llumaed by a sra eergy fuco W, he sra-vara-based models of W maly clude he Mooey-Rvl model, he Yeoh model ad he Ge model, ad he prcpal-srech-based model maly clude he Ogde model(yag e al. 004). I s worh meog ha he Ogde model breaks hrough he lmao of he sra eergy fuco beg he eve power of he sreches, ad s capable of more accuraely fg he expermeal daa whe rubbers udergo large deformao(ogde 973; Treloar 975). The cosuve ) PH.D Caddae ) Professor 3) Assocae Research Fellow

2 relaos for he vscoelascy ca be characerzed by combg he elasc compoes wh he vscous compoes. The relaxao-modulus-based geeralzed Maxwell model s oe of he mos wdely used models he commercal fe eleme (FE) sofware, lke ANSYS, ABAQUS, LS-DYNA, ec. I order o ucover he vscoelascy properes uder fe deformao, Las(963) proposed a relaxao-modulus cosuve fuco for he compressble soropc maerals. I he framework of a mulplcave decomposo of he deformao grade esor, Huber e al.(000) geeralzed he applcao rage of he so-called hree-parameer solds o he fe deformao. Yoshda e al.(004) preseed a cosuve model combed wh a elasoplasc body wh a sra-depede soropc hardeg law ad a hyperelasc body wh a damage model, whch cocded well wh he expermeal resuls. Am e al.(00; 006a; 006b) proposed a mproved hyperelasc cosuve equao for rubbers compresso ad shear regme, he parameers he equao were defed by he daa from he rae-depede saaeous expermes ad equlbrum expermes, ad he olear vscous coeffce was roduced o represe he rae-depede properes of rubbers. Wh regard o he olear maeral parameer esmao of he hyperelasc model, Gedy e al.(000) developed a professoal scheme amed COMPARE o ge he parameers by usg uaxal eso(st), baxal eso(et) ad plaar eso (PT) daa, whch was based o he Ogde model ad cossed of a sesvy aalyss as well as a opmzao procedure. Wag e al.(007) provded a hyper-vscoelascy relao o smulae he hgh dampg maerals, ad he parameers of he model were defed from he experme daa. The vesgaos herebefore show ha here have bee few researches coduced o he parameer defcao of he hyper-vscoelasc model. I hs paper, a professoal mehod based o he Ogde model ad he Leveberg-Marquard (L-M ) opmzao algorhm was developed o f he ST, PT, ET, ad smple shear(ss) expermeal daa of hyperelasc maerals. The, Takg he combao of he geeralzed Maxwell model wh he Ogde model as he cosuve model of hyper-vscoelasc maerals, ad he egrao of relaxao modulus whch s based o he Bolzma superposo prcple was employed as he fudameal algorhm, he mehod ha ca defy he parameers he hyper-vscoelasc models by usg SS daa wh dffere sra loadg veloces was developed. All he esmaed parameers he hyperelasc ad he hyper-vscoelasc models were verfed by ANSYS.. PARAMETER IDENTIFICATION OF THE HYPERELASTIC MODEL The cosuve models represeg he hyperelasc properes of rubbers maly clude he sascal models, he sra-vara-based models ad he prcpal-srech-based models, Amog whch he Ogde model s able o f he

3 experme daa whe rubbers udergo large deformaos( <7). I he followg secos, he aalycal heores of he rubbers subjeced o ST, ET, PT ad SS deformaos ad he umercal verfcao wh he experme daa are frsly preseed deal.. Theores of he Ogde model Accordg o he heores of he couum mechacs, here exss a sra eergy fuco W for he hyperelascy properes of rubbers. The sresses ca be obaed by he paral dervave of he varable W wh respec o he sra. S W C () Where C s he rgh Cauchy-Gree deformao grade esor, S s he secod Pola-Krchhoff sress. Afer rasformg he secod Pola-Krchhoff sress o he Cauchy sress (La e al. 00), we wll ge W W T pi B B I I B () T FF (3) Where F s he deformao grade esor, B s he lef Cauchy-Gree deformao grade esor, I, I, I3 are he hree sra varas of B, p s he udeermed hydrosac pressure whch ca be decded by he uderlyg equlbrum ad boudary codos of he parcular problem. I vew of he compressble properes of he volume of rubbers, he sra eergy of he Ogde model ca be defed wh he Eq.(4) whou regard for he corbuo of he volume sra. W ( 3 3) (4) I Eq. (4), s he umber of erms cosdered for he Ogde model, udeermed parameers of he model, s he prcpal srech., are he I he ST experme, he deformao grade esor F ad he lef Cauchy-Gree deformao grade esor B are descrbed as F 0 / 0 B 0 / / 0 0 / ad he hree sra varas of he esor B are (5)

4 I I I3 (6) Takg he boudary codo T T33 0o cosderao, he expresso for Cauchy sress T ca be expressed by usg he Eq. () as T W W ( )( ) I I (7) The, subsug he Eq. (4) ad he Eq. (6) he Eq. (7), we wll oba T / ( ) (8a) ad he correspodg omal sress s f T / ( ) (8b) / I he ET experme, he boudary codos T T33, T 0, 3. Afer he smlar dervave processes of he uxal eso, he Cauchy sresst, as well as T 33, ca be expressed as T ( ) 33 ad he correspodg omal sress s T (9a) f T / ( ) (9b) Smlarly, Cauchy sress T of he PT experme s T ( ) (0a) where he boudary codo s T33 0,, ad he omal sress s f T / ( ) 3 (0b) As for he SS experme, he dreco of he prcpal srech does o cosse wh he dreco of he appled deformao, raher volves a roao of axes (Am e al., 006). The deformao grade esor F ad he lef Cauchy-Gree deformao esor B are expressed as

5 0 F ad he hree varas of he esor B are I I B he Cauchy sress T s () I () T ( W W ) I I (3) where s he shear sra. The, afer subsug he Eq. (4) ad he Eq. () he Eq. (3), we wll ge he shear sress ad he omal sress as show he Eq.(4). f T 4 ( ) / where he prcpal srech ca be obaed by calculag he egevalues of he esor B. (4) (5) 4. The objec fuco for parameer defcao Prevous researches have show ha he cosuve parameers of a hyperelasc model defed form a parcular deformao mode may be vald for oher modes. For example, Charlo e al.(994) clamed ha he parameers defed from he uaxal es daa fal predcg baxal or plaar eso resposes. So, s of much sgfcace o ake he four ypes of experme daa lsed above o cosderao. Ths goal ca be acheved by he use of he leas square procedure o make he devaos of he experme daa ad he fed daa he leas. The objec fuco ca be defed as Where m : 4 q q ( (( fq ) ( q ) ) ) q (6) q s he wegh of dffere experme ype, f here exss o experme daa for a specfc ype, we jus eed o se s o be zero, f we wa o emphass he experme daa of a specfc ype, we ca se s q q larger ha ohers'. he umber of he experme daa for each ype of experme, ( f ) q q represes s he heorecal

6 value whch s calculaed by he Eq.8(b), Eq.9 (b), Eq.0(b) ad Eq.4, ( ) experme daa. q s he.3 The defed parameers ad he umercal verfcao The objec fuco Eq. (6) ca be solved by he L-M olear opmzao algorhm(leveberg, 944). I hs paper, he ST, ET ad PT daa from Treloar(944) are ake as he experme daa, he precedg algorhm s realzed by MATLAB sofware(00), ad he Ogde model s chose as he hyperelasc model owg o s favorable performaces uder large deformaos. Whe he umber of erms cosdered for he Ogde model are respecvely se as 3 ad 4, he resuls of he udeermed parameers ad he devao S are lsed Table, he esmaed parameers preseed by Treloar(944) are lsed as well, he u of Table s Mpa. Table. Resuls of he defed parameers Ogde model Treloar =3 = E E E E S=0.056 S= S=0.08 As he Table shows, hese hree groups of parameers are all capable of fg he experme daa very well, ad he las group s he bes. Sce here s o eed o work ou he globally opmal soluo for he Eq.(6), he esmaed parameers are o uque as log as hey make he devao lower ha a cera lm. However, order o make he age sffess marx o be posve defe, s ecessary o esure ha he produc of ad s posve. For he purpose of verfyg he esmaed parameers, he las group of he defed parameers s ake as he example o predc he respose of rubbers dffere ypes of expermes ANSYS. The resuls of he umercal smulao, he fed daa obaed by he use of he proceedg algorhm, alog wh he experme daa from Treloar(944) are all ploed Fg..

7 Treloar ST Malab ST f (Mpa) 3 ET PT ST ANSYS ST Treloar PT Malab PT ANSYS PT Treloar ET Malab ET ANSYS ET Fg. Verfcao of he deermed parameers Ogde model As llusraed he Fg., The resuls of he algorhm developed MATLAB f he experme daa very well. Besdes, he umercal smulao ess, he resuls of ST ad PT cocde well wh he experme daa (whe he prcpal srech s larger ha 5, he FE aalyss of ST ca' coverge due o some ucera reasos). However, he resuls of ET dffers a b from he experme daa, he reaso for he errors les ha he boudary codos of he ET experme ca' be accuraely sasfed whe he prcpal srech rages from.5 o.9 he smulao es. The FE al models of ST, PT, ET experme ad her deformed shapes wh he srech beg 5, 6.4, 5.04 are preseed respecvely Fg. (a)-fg. (f). (a) al model of ST experme (b) deformed model of ST experme( =5 ) (c) al model of PT experme (d) deformed model of PT experme( =6.64 )

8 (e) al model of ET experme (f) deformed model of ET experme( =5.04 ) Fg. The FE models for umercal verfcao 3. PARAMETER IDENTIFICATION OF THE HYPER-VISCOELASTIC MODEL I he prevous seco, he parameer defcao of he Ogde model for hyperelasc maerals was explaed deal, hs model s suable for he umercal smulao of aural rubbers wh low dampg. As for he hgh dampg rubbers, hey ca be smulaed by he hyper-vscoelasc model whch combes he Ogde model wh he geeralzed Maxwell model. I he followg, he egrao of he relaxao modulus was employed as he fudameal algorhm, ad he parameer defcao of he hyper-vscoelasc model for hgh dampg rubbers was expouded. 3. Theores of he hyper-vscoelasc model The egrao of he relaxao modulus for vscoelascy s expressed Eq. (7) ( ) Y( ) d( ) (7) where Y ( ) s he relaxao modulus fuco for characerzg he vscoelasc properes, d( ) s he mcro-sra of maerals. Geerally, whe 0, he ouer aco s assumed o be 0, so he Eq. (7) wll be chaged o If he sra loadg speed s cosa, amely ( ) ( ) Y( ) d( ) (8) 0 0, he ( ) Y( ) d (9) where s he sra loadg velocy. The Eq. (9)ca be calculaed by dvdg he me o equal poros, he we wll ge

9 ( ) ( ) Y d (0) Whe (, ), f he Youg modulus of maerals s assumed o be a cosa E, ad he Proy seres s roduced as he relaxao coeffce, he he Eq. (0) wll become where m ( )/ k 0 k k ( ) E ( e ) d () k s he wegh of modulus correspodg o he relaxao me k, m s he umber of erms cosdered for he Proy seres, 0 s he wegh of he equlbrum modulus. I s obvous for he equao Eq. () s preseed Fg. 3. m k 0 o be rue, ad he physcal model of he k u0e ue η uke ηk ume ηm Fg. 3 Physcal model of he hyper-vscoelasc model I Fg. (3), E, 0, k shares he same meag wh ha Eq. (), he vscosy coeffce ad k k E k k represes, k s he relaxao me cluded Eq. (). Ths physcal model cludes wo ma feaures, he frs oe s ha subsues he srech-depede modulus sprg for he cosa modulus sprg of he geeralzed Maxwell model, whch s based o a assumpo ha he modulus correspoded o he vscosy coeffce modulus E ad he wegh k s obaed by mulplyg he saaeous u k. The secod oe s ha he saaeous modulus ca be decded by he hyperelascy properes of maerals, ad he wegh udeermed parameers whch s me depede. If we spulae ha k 0 k LS-DYNA(007). The, () ca be expressed as E uk s he, whch s adoped by he FE sofware m ( )/(0 k ) 0 k k ( ) E ( e ) d ()

10 As meoed before, E represes he saaeous modulus of maerals wh he me rage from o. I ca be solved by he followg procedures. Frsly, he fas srech-sress experme daa s obaed by coducg experme a suffcely hgh speed o exclude he vscous effec. The hey wll be fed by he Ogde model o decde he udeermed parameers, whch s explaed prevous seco. Afer acqurg he srech-sress curve based o Ogde model, E wll be solved by he dervao of he sress wh respec o he prcpal srech, amely,, or akg he seca modulus as a subsuo for he age modulus, amely, ( ) / ( ). I hs seco, he SS es s ake as a example, ad he secod way o ge he saaeous modulus s adoped, he he Eq. () wll be covered o T ( e ) d (3) m m ( )/(0 k ) k k k k where, s he correspodg shear sress ad shear sra wh he me, ad he shear sress ca be solved by Eq. (4) ad (5). The Eq. (3) s he compuaoal formula for he SS experme of hyper-vscoelasc maerals, ad s he foudao for he ex seco o realze he parameer defcao of hyper-vscoelasc maerals. I should be meoed ha he calculao error may be up o 0% whe he shear sra s larger ha 300%, whch resuls from eglecg he fluece of large deformao. 3. The parameer defcao algorhm for he hyper-vscoelasc model Accordg o he Eq. (3), he umber of he udeermed parameers s m+, cludg uk ad. Smlarly o he Eq. (6), he objec fuco for he parameer defcao of hyper-vscoelasc model ca be defed as follows, p j ( ) (4) j m : f T where p represes he umber of he ess wh dffere loadg speed, j s he umber of he experme daa for he jh group of experme, ad hs paper, j s se as a cosa r, for smplcy. T s he experme daa, f s he heorecal value whch s calculaed by he Eq. (3). The process of he parameer defcao for he hyper-vscoelasc model s summarzed o wo seps. Frsly, he saaeous shear modulus G a each me

11 ode s calculaed by he mehod proposed prevous seco. The, o he bass of he L-M olear algorhm, he p groups of experme daa are fed o realze he parameer defcao. Owg o he complexy of he Eq. (4), I s ecessary here o expla how o ge he sesvy marx he L-M algorhm. As we ca see from he Eq. (4), he objec fuco s a summao of p r fucos, ad he umber of he udeermed parameers s m+, so, he sesvy marx A ca be defed as he Eq. (5). f f f f u uk u m f f f f A u uk um f pr f pr f pr f pr u u u k m pr ( m ) (5) f where u k ca be solved by Eq. (3). For he lack of he approprae experme daa for SS experme, we assume a ype of hgh dampg rubbers whose hyperealsc model s characerzed by he las group of parameer Table, ad he vscoelasc model s characerzed by he followg Proy seres, P( ) 0.4e 0.e ( )/0 ( )/00 (6) The, we wll geerae four groups of experme daa by ANSYS sofware, he loadg speed for each experme s 0.5/ s,0./ s,0.0/ s ad 0.00/ s, he larges shear sra s se o be 00%, ad he umber r s se o be 9. All he fgures geeraed are lsed Table, Table. The geeraed fgures o be fed 0.5 / s 0./ s 0.0/ s 0.00/ s T T T T

12 I s ecessary here o expla how o se he al values for all of he udeermed parameers Eq.(3). As for he wegh from 0 o, ad he summao of u k, should be wh he rage u k, excludg u 0, should also be wh he same rage. Whle for he relaxao me, seems o be a b more complcae. The suggesed al value for s he ma relaxao me of maerals, ad he ma relaxao me s he po correspodg o he relaxed sress he creep ess, s defed as 0.368( ) (7) where 0 s he saaeous sress ad s he equlbrum sress. The creep experme of he maerals, whch s characerzed by he Eq. (6) ad he las group of parameers Table, s coduced ANSYS, ad he ma relaxao me s decded as.54s. Wh regard o he umber of he erms cosdered for he Proy seres, he fgures Table wll be fed more accuraely f he umber becomes larger, however, hs may leads o ureasoable esmaed parameers. So he umber o be decded s suggesed o make he larges relaxao me Eq. (3) he same order of magude wh he larges experme me. For example, he larges experme me Table s 000s, so, he umber of he erms cosdered for Proy seres s 3 sce he larges relaxao me s 54s. Wh all he resrcos preseed above, he al values for he udeermed parameers are se as [,,, 3] [.54,0.3,0.3,0.3]. 3.3 The defed parameers ad he umercal verfcao O he bass of he proceedg L-M olear algorhm for hyper-vscoelasc maerals, a group of parameers are obaed by usg he daa Table, ha s [,,, 3] [9.853, , , ]. The devao of he experme daa ad he fed daa s 3.43E-4, whch meas he error s suffcely small. Smlarly, order o verfy he accuracy of he esmaed parameers, hese parameers are employed ANSYS o predc he respose of he hgh dampg maerals. The resuls calculaed ANSYS ad he daa Table s ploed Fg. 4.

13 s s 0.4 s T (Mpa) s Fg. 4 Verfcao of he deermed parameers hyper-vscoelasc model I Fg. 4, he sold les represe he fgures Tab. (), ad he doed le represe he daa calculaed ANSYS. Obvously, he errors are very small sce he larges oe s oly abou 3.%, whch meas he defcao algorhm proposed hs paper s capable of esmag he parameers hyper-vscoelasc models whe he shear sra s lower ha 00%, ad he reaso for he errors has bee explaed he prevous seco. The FE al model for SS experme ad s deformed shape wh he shear sra beg 00% are preseed Fg. 5. (a) al model (b) deformed model ( 00% ) Fg. 5 The FE models for umercal verfcao There exs some eresg resuls whe usg dffere al values for he udeermed parameers. I some crcumsaces, some defed resuls of he weghs uk are egave, s ureasoable he sese of he physcal model preseed Fg. 3. However, f we gore he physcal meags of hese weghs, seems accepable. Moreover, mos cases he defed egave weghs are proved o be praccal ANSYS ad he fg effec s comparavely beer. A las, deserves o be emphaszed ha he parameer defcao algorhm for he SS experme s o he assumpo ha he Ogde model s able o characerze he smple shear properes of maerals. I he same way, hs algorhm ca be also

14 used o defy parameers he hyper-vscoelasc model wh he ST experme daa, ET experme daa as well as PT daa, he oly hg eeds o be doe s subsug E deermed by correspodg saaeous expermes for he erm Eq. (3). 4. CONCLUSIONS I hs sudy, we focus o he parameer defcao of he hyperelasc ad hyper-vscoelasc maerals, ad he followg coclusos have bee made: () O he bass of he Ogde model ad he L-M olear opmzao algorhm, he comprehesve fg of he mul-ype experme daa, cludg ST, PT, ET, SS, of rubbers has bee realzed. () Accordg o he fg resuls of he experme daa form Treloar ad he verfcao resuls of he umercal smulao ANSYS, he mehod for defyg parameers Ogde model proposed hs paper s praccal ad effce. (3) The parameer defcao mehod for he hyper-vscoelasc maerals whch s characerzed by combg he Ogde model ad he geeralzed Maxwell model was developed. (4) I hs paper, he SS es was ake as he example, ad he proposed algorhm for defyg parameers he hyper-vscoelasc model was proved praccal ad effce whe he shear sra s lower ha 00%. REFERENCES Am, A.F.M.S., Alam, M.S., ad Oku, Y.(00), "A mproved hyperelascy relao modelg vscoelascy respose of aural ad hgh dampg rubbers compresso: expermes, parameer defcao ad umercal verfcao", Mech. Maer., 34, Am, A.F.M.S., Wragua, S.I., Bhuya, A.R., ad Oku, Y. (006), "Hyperelascy model for fe eleme aalyss of aural ad hgh dampg rubbers compresso ad shear", J. Eg. Mech., 3, Am, A.F.M.S., Lo, A., Seka, S., ad Oku, Y. (006), "Nolear depedece of vscosy modelg he rae-depede respose of aural ad hgh dampg rubbers compresso ad shear: expermeal defcao ad umercal verfcao", I. J. Plascy.,, Charlo, D.J., Yag, J., ad Teh, K.K. (994), "A revew of mehods o characerze rubber elasc behavor for use fe aalyss", Rubber. Chem. Techol., 67, Gedy, A.S., Saleeb, A.F. (000), "Nolear maeral parameer esmao for characerzg hyperelasc large sra models", Compu. Mech., 5,

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