A Three-Dimensional Anisotropic Viscoelastic Generalized Maxwell Model for Ageing Wood Composites

Transcription

1 A Three-Dimeioal Aioropic Vicoelaic Geeralized Maxwell Model for Ageig Wood Compoie Jea Deeix, George Djouma, Adré Fori Groupe Ierdicipliaire de Recherche e Éléme Fii (GIREF) Dépareme de mahémaique e aiique, Uiverié Laval, Québec Caada Alai Clouier Cere de recherche ur le boi (CRB) Dépareme de ciece du boi e de la forê, Uiverié Laval, Québec Caada Pierre Blache FPIovaio-Foriek Diviio ad Dépareme de ciece du boi e de la forê, Uiverié Laval, Québec Caada Abrac I hi paper, he developme of a umerical model for he ageig liear vicoelaic behaviour of maerial i he geeral hree-dimeio coex i addreed. More preciely we propoe a approach allowig he iegraio i pre-exiig liear vicoelaic umerical ool (pecifically he fiie eleme mehod) ad avoidig ime ep rericio. The coiuive law baed o he geeralized Maxwell model i repreeed uig a Dirichle erie where he relaxaio ime fucio deped o he rae ad he hiory of variaio of a arbirary ageig fucio. The hardeig ad ofeig phae of he maerial are characerized by hi fucio ad he vicoelaic law ake io accou o moooou variaio of he phae. The aumpio of lieariy of he rai i avoided by he ue of a ordiary differeial equaio givig more freedom i he choice of he ime ep. A parameerized ime eppig cheme i ued o approximae he oluio of he ordiary differeial equaio. The umerical procedure combie he fiie-eleme mehod wih a icremeal formulaio. The reul for he umerical experime o illurae he performace of he propoed approach how good agreeme wih aalyical reul. Key Word: aioropic, o moooou agig, rheological geeralized Maxwell model, fiie-eleme mehod, vicoelaiciy. Paper AP-4 of 9

2 Iroducio The ieracio bewee moiure variaio ad he mechaical behaviour of wood i a impora iue ha impac he durabiliy ad erviceabiliy of wood produc. The hygrohermal ageig iduce a depedece of he rheological parameer upo moiure coe ad emperaure. A beer uderadig of he mechaical behaviour of wood produc ubjeced o chagig eviromeal codiio could help for he deig of wood compoie ad imber rucure. A review of he approache for modelig he creep pheomeo reveal (Hahijarvi 995) ha rai could be decribed uig a rheological model (oe or more dahpo combied wih prig) acivaed by moiure variaio. The fiie eleme modellig of o ageig vicoelaic ioropic maerial ha bee widely udied i he lieraure (Ghazla 988). For he aioropic cae, he lieraure i maily cocer wih o ageig maerial (Zocher e al 997, Poo e al 998 ad 999). I he ageig cae, Duboi e al (25), have developed a oe dimeioal vicoelaic model accordig o he hermodyamic priciple baed o he geeralized Kelvi-Voig model. I a rece paper, (Chaage e al. 26), a hree-dimeioal model i preeed baed o a geeralized Maxwell model wih dahpo depedig o re level. Our work ca be coidered a coiuaio of he work i Fafard (2) ice we ry o cover he ageig of he prig ad he dahpo. Our purpoe i hreefold, fir o develop, baed o a geeralized Maxwell model (GMM), a hree-dimeioal aioropic model which hould be hermodyamically admiible (repec he hermodyamic priciple). Secodly, baed o he liear vicoelaiciy, he formulaio hould ake io accou ageig, maifeed by chage of he vicoelaic properie a fucio of ime. Laly, i hould be eaily iegraed io pre-exiig fiie eleme (FE) code ad hould avoid impoig limiaio o he legh of he ime ep of he umerical mehod. Saeme of he Problem 3 Le Ω be a regular ope bouded domai of, repreeig a vicoelaic body ubjeced o mechaical loadig (ad poibly hermal ad hygromeric variaio). The ae variable are: he diplaceme vecor u=u i (x,), he re eor σ = σ ij (x, ) ad he rai eor ε= ε ij (x, ), where x i he poiio vecor, i he ime variable ad i,j=,2,3. Throughou hi work we will omi he pace ad ime variable x ad whe o cofuio i poible. The goverig equaio for he decripio of he liear vicoelaic repoe o he applied loadig i he hree-dimeioal (icremeal quai-aic ae) equaio of equilibrium: σ = f i Ω ], [ () b F Paper AP-4 2 of 9

3 where F i a o zero poiive real umber, fb i he body force ad he rai eor ε i relaed o he diplaceme, u meaured alog he x i direcio: u u i j εij ( u) = + i, j 3 2 xj x i (2) Equaio () compleely defie he diplaceme if we have a coiuive law (relaio bewee σ ad ε(u)) ad iiial ad boudary codiio. oe ha we omi o pecify he iiial ad boudary codiio ice i i of o impac o he followig. We iroduce he relaxaio ime fucio R (Chriee 97) ad he coiuive equaio i give by dεkl σ = + R (, ) d, (3) d ij σij where σ i he iaaeou repoe correpodig o he elaic ae of he maerial. We defie ageig a he ime depedecy of he properie of he maerial, le M be a arbirary calar fucio repoible for he agig of he maerial. The coefficie R deped M. Obviouly we ca coider more elaborae problem, for example a hermo-hygromechaical problem (M i o a calar bu a pair of calar fucio, emperaure ad moiure coe) ad (Eq. ()) i coupled wih a hermal rafer equaio (uually baed o Fourier law) ad a ma rafer equaio (uually baed o Fick law) for he moiure coe. Of coure, i uch cae (Eq. (3)) i modified by addig erm for he hermal ad/or moiure iduced rai. Sice hi preeaio will be ceered o he reame of he relaxaio fucio R we will o rea couplig erm a i hermohygro-machaical problem. However he approach propoed here eaily cover hoe ype of addiio o he coiuive law (Eq. (3)). We are iereed i a hermodyamically admiible model which implie ha he coiuive equaio (Eq. (3)) mu aify cerai codiio (Duboi e al 25, Baza 979). Our formulaio i baed o he Dirichle erie correpodig o he GMM. Uig he codiio impoed for a hermodyamically admiible Maxwell model we will eablih he codiio ha mu be impoed o he coefficie of he erie for a hermodyamically admiible model. The Dirichle Serie for he Relaxaio Fucio ad Ageig Pheomeo We iroduce a Dirichle erie (Baza 988) o defie R i (Eq. (3)). Each compoe of he relaxaio fourh-order eor R i repreeed by: Paper AP-4 3 of 9

4 = λ ( r) dζ ( r) R (, ) = C ( ) e (4) where i he umber of cell i he Maxwell model. For wood maerial, ageig i due o variable climae codiio. Thi implie ha he maerial properie deped o moiure coe ad emperaure; hece he maerial properie vary i ime. Sice we wa a erie which i equivale o he GMM here i a relaio bewee he eor C µ ad λ µ i (Eq. (4)) ad he more claical iffe ad vicoiy of each prig ad dahpo. Rheological Model For hi model o be hermodyamically admiible i i eceary ha every rheological eleme (a prig ad a dahpo i erie) i hermodyamically admiible. Thu all prig ad dahpo compoig a rheological model mu aify he poiive diipaio codiio. Fulfilig uch codiio ca be guaraeed by impoig cerai codiio o prig moduli ad vicoiie. Whe he prig modulu E i age-depede, Baza (979) ha prove ha wo diic coiuive law are required o aify he hermodyamic poiive diipaio codiio. The claical Hooke law for ofeig prig behaviour ad he age law for he hardeig prig. The elaic repoe of a agig prig i defied by uig hoe wo law: σ prig E ε = E ε+ E ε hardeig (Baza) ofeig (Hooke) (5) There i o eed o modify he coiuive law for he dahpo (ewo law) (Duboi e al 25). For =,, coider, a mo 8 Maxwell eleme, where he prig coefficie i E ad he dahpo vicoiy η ad iroduce λ ( θ) dθ dεkl σ () = C () e () d (6) dθ if hardeig C C C = E Θ ( M) = λ = Θ ( M) if ofeig η C (7) Uig he defiiio give i (Eq. (7)) ad formally derivig i (Eq. (6)) we ca how ha σ + λ σ = C εkl (8) Paper AP-4 4 of 9

5 Equaio (8) i he differeial form of he coiuive equaio for a ageig Maxwell eleme ad i decribe a re i direcio ij produced by a rai i direcio kl. Sice (Eq. (8)) i baed o (Eq. (5)), hi eleme i hermodyamically admiible. Clearly σ aifyig (Eq. (3)) i he uperpoiio of he elemeary re σ σ = σ + σ (9) ij ij kl The σ i he re of a aioropic ageig hree-dimeioal hermodyamically admiible GMM. Thu, he re defied by (Eq. (3)) wih he ue of he Dirichle erie (Eq. (4)) i equivale o he ue of a geeralized Maxwell model ad i accordace wih he hermodyamic priciple provided ha he coefficie of he erie aify (Eq. (7)). Muliple Parameer Ageig Formulaio I he piri of (Duboi e al 25, Baza 979), we propoe o formulae he ageig,ref properie a follow: for each cell =, 2,..., we deoe E,ref η he iffe ad vicoiy a referece ageig M ref, ad ue wo calar fucio o ake io accou he ageig b ( x, M( x, )) = b ( x, ) : Ω, b ( x, Mref ) = x Ω () l ( x, M( x, )) = l ( x, ) : Ω, l ( x, M ) = x Ω () ref, ref, ref E (,) x b (,) x E η (,) x l (,) x η = = (2) I he ioropic/orhoropic cae, if he Poio coefficie are coa i ime, a oeparameer formulaio ca be juified for ioropic maerial. For orhoropic maerial (uch a wood), we eed a lea 4 parameer (oe for he ormal re ad hree for he hearig re). Each pair ( E, η ) i a uiaxial Maxwell eleme. I order o aify he hermodyamic we ue (Eq. (5)), which ca be decribed uig he derivaive of b. Deoig, ref b b ( x, ), ref E Θ ( M ) = d ( x, ), ref,..., b = λ = = l ( x, ) η (3) d correpod o he reduced ime, become,ref λ o he referece relaxaio ime ad (Eq. (7)) Paper AP-4 5 of 9

6 Iroducig b C = b E λ = d λ Θ ( M) b, ref, ref (4) b Δ φ (, ) = d ( xd, ) Δ ψ (, ) = d b (5), ref λ Δφ (,) Θ ( M) Δψ (,) dεkl D ( x, ) = b ( x, ) e e ( x, ) d d (6) he eor D ha oly mior ymmery ( D = Djikl = Dijlk ) o ha here are a mo 36 calar o rack i he geeral aioropic cae. Thi lead o a rewriig of (Eq. (3)-(4)) uig (Eq. (6)) σ = σ + E D (7),ref ij ij kl A Differeial Formulaio Thi formulaio i ipired by he paper of Poo e al (999) o produce a umerical model baed o a ordiary differeial equaio comig from he defiiio of D. We defie, ref λ Δφ (,) Θ ( ) Δψ (,) ( ) M dεkl = d () b () b () e e () d (8) ˆ ε ( x, ) = ε ( x, ) ε ( x,) (9) kl kl kl The derivaive of give ( ) ˆ ( x, ) = b ( x, ) + b ( x, ) λ ( x, ) ε ( x, ) λ ( x, ) ( x, ) (2) kl I ummary, goig back o D i (Eq. (6)), he re will aify, ref ij σij E D = kl σ = + (2) D = b ˆ ε (22) kl wih oluio of he differeial equaio Paper AP-4 6 of 9

7 ( ) ˆ = b + b λ ε λ (23) kl ( x,) = (24) I he o-ageig cae hi correpod o he model preeed i he paper of Poo e al (998). The ceral poi of hi preeaio, from a algorihmic poi of view, i (Eq. (23)- (24)). Coider a pariio of he ime ierval [, F ] io Q ubierval Q [, ] = [, ], Δ = =,..., Q (25) F = I Baza (979), Ghazla (988) ad Zocher (997) he coiuive equaio i dicreized, uig a icremeal approach, aumig ha he rai i liear over he ime ierval [, + ], which eceiae ufficiely mall for accuracy ad umerical abiliy. The ue (Eq. (23)-(24)) allow more flexibiliy for he ime ep a we ca choe a uiable ime iegraio mehod baed o accuracy. More igificaly, i remove he eed for aumpio uch a liear ime variaio of rai hroughou a ime ep. Applyig he imple θ-cheme family: θ =,, /2 (correpodig o Euler-explici, Euler-implici ad Crak-icholo cheme repecively) o (Eq. (23)), uig he oaio f for f ( x, ), give Deoig,,,,, (( b ) ˆ b kl ) (( ) ),, ( ), ˆ,, + θ b + b λ εkl λ, +, Δ + = θ + λ ε λ (26) E E b, +, +, +, +, ref, + = Δ + θ, + = +Δ+ θλ b b + b + b,,, 2, +, ref = Δ + ( θ), + = +Δ+ θλ E E b + b λ E, +, +, + +, ref, ij = Δ + θ, + kl = +Δ+ θλ λ λ (27) (28) (29) We have, from (Eq. (2)), he ime-dicree coiuive law Paper AP-4 7 of 9

8 σ = E ˆ ε + σ E ˆ ε (3), , + ij kl ij ij kl Cocluio We propoe a umerical algorihm for hree-dimeioal aioropic ageig Maxwell model. The model ca be coupled wih hermal ad hygromeric rafer. The model i hermodyamically admiible ad ca be eaily iegraed i pre-exiig FE code a how by (Eq. (3)). The model allow he ue of muliple reduced ime, ad ca be viewed a a exeio of he work of Poo e al (998) o ageig maerial. We are workig o he relaio of hi model wih he work of Zocher (997), ryig o eablih if he liear aumpio o he rai ca be relaed o a pecific ime eppig cheme for (Eq. (23)-(24)). We performed ome prelimiary e o a ioropic cailever beam ubjeced o ip loadig coupled wih moiure rafer. The umerical reul were i accordace wih he lieraure. Mo imporaly, we had a oiceable gai i performace ice we ued ime ep almo e ime larger for comparable reul obaied by Zocher (997). Thee ecouragig reul ugge uderakig furher experime wih realiic daa. The mai difficuly i he deermiaio of parameer for he hree-dimeioal cae. We are preely doig experimeal work o maple. Referece Baza Z.P., 979, Thermodyamic of olidifyig or melig vicoelaic maerial, J. Eg. Mech. Div. 5(EM6): Baza Z.P., 988, Mahemaical modelig of creep ad hrikage of cocree, Joh Wiley ad So Ld. Chaper 2, 988. Chaage P., Bou-Said E., Jullie J. F. ad Galimard P., 26, Three-dimeioal creep model for wood uder d variable humidiy-umerical aalye a differe maerial cale, Mechaic of ime-depede maerial, 9: Chriee R.M., 97, Theory of vicoelaiciy. A iroducio, ecod ediio. Academic Pre, ew York. Duboi F., Radriambololoa H., Pei C., 25, Creep i wood uder variable climae codiio: umerical modellig ad experime validaio, Mechaic of imedepede maerial, 9: Fafard M.; Boudjelal M. T.; Bioee B.; Clouier A., 2, Three-dimeioal vicoelaic model wih ocoa coefficie, J. of Eg. Mech., Vol. 27, o. 8, pp Paper AP-4 8 of 9

9 Ghazla G., Caperaa S., Pei C., 988, A icremeal formulaio for he liear aalyi of hi vicoelaic, rucure uig geeralized variable, I. J. umer. Mehod Eg. 38: Hahijarvi A., 995, Modellig of creep deformaio mechaim i wood, Techical Reearch Ceer of Filad Epoo, Fidlad, VTT Publicaio, 23. Poo H., ad Ahmad M. F., 999, A fiie eleme coiuive updae cheme for aioropic, vicoelaic olid exhibiig o-lieariy of he Schapery ype, I. J. umer. Meh. Egg., 46: Poo H. ad Ahmad M.F., 998, A maerial poi ime iegraio procedure for aioropic, hermo-rheological imple, vicoelaic olid, Comp. Mech., 2: Zocher M.A., Grove S.E., Alle D. H., 997, A hree-dimeioal fiie eleme formulaio for hermovicoelaic orhoropic media, I. J. umer. Meh. Egg., 4: Ackowledgeme The auhor would like o hak he aural Sciece ad Egieerig Reearch Coucil of Caada (SERC), FPIovaio-Foriek Diviio, Boa-Frac ad Uiboard Caada for fudig hrough he SERC Sraegic Gra Parerhip program. Paper AP-4 9 of 9