Determination of Stresses in Drying Wood by Means of a Viscoelastic Relaxation Model

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Deeminaion of Sesses in Dying Wood by Means of a Viscoelasic Relaxaion Model Oma Saifouni, Rosand Mouou Pii, Jean-Fançois Desebecq To cie his vesion: Oma Saifouni, Rosand Mouou Pii,

Challenges in Mechanics of Time-Dependen Maeials and Pocesses in Convenional and Mulifuncional Maeials, Poceedings of he 2012 Annual Confeence on Expeimenal and Applied Mechanics,

f/hal-01616935 Submied on 17 Oc 2017 HAL is a muli-disciplinay open access achive fo he deposi and disseminaion of scienific eseach documens, whehe hey ae published o no.

1 Deeminaion of Sesses in Dying Wood by Means of a Viscoelasic Relaxaion Model Oma Saifouni, Rosand Mouou Pii, Jean-Fançois Desebecq To cie his vesion: Oma Saifouni, Rosand Mouou Pii, Jean-Fançois Desebecq. Deeminaion of Sesses in Dying Wood by Means of a Viscoelasic Relaxaion Model. Challenges in Mechanics of Time-Dependen Maeials and Pocesses in Convenional and Mulifuncional Maeials, Poceedings of he 2012 Annual Confeence on Expeimenal and Applied Mechanics, 2, pp.29-36, 2013, < / _5>. <hal > HAL Id: hal hps://hal.achives-ouvees.f/hal Submied on 17 Oc 2017 HAL is a muli-disciplinay open access achive fo he deposi and disseminaion of scienific eseach documens, whehe hey ae published o no. The documens may come fom eaching and eseach insiuions in Fance o aboad, o fom public o pivae eseach cenes. L achive ouvee pluidisciplinaie HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau echeche, publiés ou non, émanan des éablissemens d enseignemen e de echeche fançais ou éanges, des laboaoies publics ou pivés.

2 DETERMINATION OF STRESSES IN DRYING WOOD BY MEANS OF A VISCOELASTIC RELAXATION MODEL Oma SAIFOUNI, Rosand MOUTOU PITTI, Jean-Fançois DESTREBECQ Clemon Univesié, Univesié Blaise Pascal, Insiu Pascal, BP 10448, F CLERMONT-FERRAND, Fance CNRS, UMR 6602, Insiu Pascal, F AUBIERE, Fance osand.mouou.pii@polyech.univ-bpclemon.f ABSTRACT. Sess saes caused in wood by dying peiods ae ofen he souce of consideable sucual disodes, when his maeial is used as sucual maeial. The oigin of hese sesses is geneally due o he viscoelasic behaviou of he wood combined wih he dimensional vaiaions (shinkage) elaed o he moisue conen. In ode o bee undesand his phenomenon, a slice of geen wood is submied o naual dying in sable envionmenal condiions. The wood slice is placed on an eleconic balance so as so o measue he moisue conen vaiaion duing he dying peiod. Simulaneously, he displacemens caused in he slice by he dying ae capued by a video camea. An incemenal elaxaion model based on he genealized Maxwell s chain is used o analyze he evoluion of he sesses induced by he dying pocess wihin he wood slice. Numeical esuls show he developmen of ensile sesses in he maeial. Analyzing hese esuls leads o he conclusion ha he sesses ae due o he ohoopic behaviou of he wood maeial combined wih anisoopic dying shinkage. A ensile sess concenaion is evidenced in a zone whee a cack was finally obseved duing he es. 1. Inoducion The naual o he aificial dying of wood is commonly esponsible fo lage sain and can lead o he cacking and he final collapse of he imbes sucues. These defecs ae much moe maked on he wood pieces cu and soed oudoos he long peiods ago [1]. The imbe is composed of polymes sensiive o he empeaue and he humidiy, heefoe, is dying involves dimensional changes due o he shinkage. Hence, he behaviou of wood duing dying depends songly in he one pa, of he geomey, he moisue conen gadien, he empeaue, he sopion of wood hisoy, bu also, of he heeogeneous chaace, he ohoopic behaviou, and he viscoelasic effecs of his maeial. I appeas ha, he phenomenon of shinkage and he mechanical behaviou of wood duing dying play an impoan ole in he developmen of wood defecs. In his conex, i is essenial o undesand he conibuion of diffeen pocesses in he sain of he geen wood duing he dying phase. In he lieaue, vaious sudies concening he sain and he sess evoluions caused by dying of wood have been conduced. These woks ae geneally based on complex analyical appoaches solved by finie elemen models aking ino accoun he changes of he mechanical popeies of wood duing he dying phase [2,3]. In addiion, seveal auhos have pesened dying models ha can assess he disibuion of moisue inside he wood duing he shinkage wih moisue conen nea he fibe sauaion poin (FSP) [4]. Also, he mechanical behaviou of wood and ee has been poposed in ode o know he impac of he biomechanical and he micosucues paamees in he old hisoy [5]. Recenly, he sudies on he heological behaviou of wood duing gowh phases have shown he effec of he viscoelasic effecs in he ecovey of he ee duing is gowh [6] wihou insis on he conibuion of moisue conen in he manifesaion of his complex phenomenon. In his wok, an incemenal viscoelasic elaxaion model adaped o ohoopic maeial aking ino accoun he hygoscopic vaiaions of wood duing he dying pocess is developed. The analyical model is based on he Bolzmann inegal equaions ha can be solved by an incemenal fomulaion expessed in em of ceep [7] o elaxaion [8,9] appoach. The fis pa descibes he mechanism of dying sesses in wood and he expeimenal deails consideed in he numeical sample. Then, he incemenal law applied o viscoelasic ohoopic maeial and he algoihm esoluion implemened in he finie elemen code Cas3m, ae deailed ogehe. The numeical esuls, in ems of sess evoluions, wih and wihou viscoelasic behaviou ae compaed ae discussed.

exising in he wood maeial [2]. Fo he wood species, i appeas ha above he fibe sauaion poin (FSP), he wood is no submied o any dimensional changes.

3 2. Expeimenal echnique 2.1 Wood dying mechanical behaviou and expeimenal seup The objecive of he dying pocess is o educe he moisue conen of wood avoiding in he same ime he poenial losses of qualiy caused by he diffeen defecs exising in he wood maeial [2]. Fo he wood species, i appeas ha above he fibe sauaion poin (FSP), he wood is no submied o any dimensional changes. Howeve, below he FSP unil he anhydous sae, hee is an almos linea elaionship beween moisue conen and he dimensional changes [10]. The FSP ae is slighly vaiable and depends on he consideed species. Accoding o he cylindical geomey of he ee, he cicula pah of he annual ings and he axial oienaion of cells, an axisymmeic mechanical chaaceisics o cylindical ohoopic maeial (longiudinal L, angenial T, and adial R diecions) can be applied. In fac, he shinkage is due o he dimensional changes duing he vaiaion of moisue conen. Thus, each species is chaaceized by hee coefficiens of shinkage (axial, adial and angenial shinkages) ha expess he dimensional vaiaions of a wood piece fo a humidiy change of 1% in he hee diecions menioned above. The dying sainis highly anisoopic and he axial shinkage can be negleced compaed wih he angenial and he adial shinkage [10]. Howeve, fo some species, he adial and angenial shinkage ae almos idenical, consequenly, hese species ae less sensiive o dying cacks. The expeimenal seup is based on he mak acking mehod consising of an acquisiion sysem dae ecoding he displacemen of age posed on he wood slice duing he dying phase, Figue 1 (a). The diamee and hickness of he slice ae abou 10 cm and 2.5 cm especively, and he envionmenal condiions of he oom es ae consan. Simulaneously, he sample is placed on an eleconically balance poviding he weigh measue vesus ime, Figue 1 (b).!%#$!"#$!"#$ Figue 1 (a) Geen wood slice. (b) Expeimenal device. 2.2 Expeimenal esuls Moisue zone vaiaion Figue 2 shows he evoluion of he moisue conen vesus ime obained fo a slice of geen wood subjeced o naual dying. Figue 2 Evoluion of moisue vesus ime Compued [11].

The slice was placed on a high-sensiiviy weigh whose values wee ecoded evey wo hous unil he sabilizaion of he inenal moisue aound 11% fo a FSP of abou 22%.

4 The slice was placed on a high-sensiiviy weigh whose values wee ecoded evey wo hous unil he sabilizaion of he inenal moisue aound 11% fo a FSP of abou 22%. The ange ime of obseved hygoscopic vaiaion is 14h. Accoding o he maked posed on he slice along he adius R i, he displacemens of he age 27 (see Figue 3 (a)) vesus he moisue conen ae shown by Figue 3 (b). I obseved also ha, he PSF is abou 22% due o he low vaiaion of displacemens. (a) R 5 R 4 R 3 R 6 R 2 R 1 R 7 R 1 2 R 8 R 11 1,5 R 9 Hoizonal displacemens R 10 Veical displacemens 1,0 Displacemen (mm) 0,5 0,0-0,5 (b) -1, Moisue conen (%) Figue 3 (a) Tage on wood slice. (b) Displacemen evoluions vesus moisue conen. 3. Viscoelasic fomulaion and algoihm 3.1 Viscoelasic incemenal appoach The behaviou of a viscoelasic solid maeial can be wien in he couse of ime in he fom of a Volea s inegal as follows: In he geneal ohoopic case, he viscoelasic elaxaion maix R, 0 [ ( )] depends on nine independen ime funcions 0 (ime loading applicaion) and (cuen ime). This numbe is educed o hee in he one-dimensional case (plane sess o plane sain) if he ceep o he elaxaion akes place a consan Poisson s coefficien. Accoding o he addiional assumpion ha hese quaniies change popoionally o he same dimensionless elaxaion funcion, he viscoelasic maix elaxaion is given by [ R(, 0 )] = ρ(, 0 )[ K 0 ] (1)

[ ] is he ohoopic elasic siffness maix; ρ(, 0 ) is a non-dimensional funcion ha is expessed In his expession, K 0 wih a Diichle seies (specal epesenaion equivalen o a genealized Maxwell model, Figue

5 [ ] is he ohoopic elasic siffness maix; ρ(, 0 ) is a non-dimensional funcion ha is expessed In his expession, K 0 wih a Diichle seies (specal epesenaion equivalen o a genealized Maxwell model, Figue 4): ( ) = γ 0 + γ µ e α µ 0 ρ, 0 ( ) wih γ µ µ=1 µ=0 =1 and ρ(, 0 ) 1,γ 0 [ [ (2) Figue 4 Genealized Maxwell model. The behaviou law of a viscoelasic maeial is given by he following Bolzmann consiuive equaion wien in he elaxaion appoach { σ( ) } = [ K 0 ] ρ(,τ ) Consideing ha he oal sain is given by { ε( τ )} = ε m ( τ ) 0 { }+ ε * τ { ε m ( τ )}dτ (3) { ( )} (4) whee{ ε m ( τ )} is he oigin mechanical sain, { ε *( τ )} is he non dependen fee sain of he maeial sess sae (dying moisue). Accoding o he equaion (3), he sess incemen { Δσ} fo a finie ime ineval [, + Δ] akes he following fom: +Δ { ε m ( τ )}dτ + ρ( + Δ,τ ) ρ,τ { Δσ} = [ K 0 ] ρ( + Δ,τ ) 0 ( ( )) ε { m ( τ )}dτ (5) We suppose ha he sain { ε m (τ )} can be appoached by a linea fom duing he ime finie ineval [, + Δ] as follow: { ε m ( τ )} ε m ( τ ) { }+ τ { Δε m } wih ε m τ Δ { { ( )} ε m } Δ (6) Finally, inoducing he equaions (6) and (4) in he elaion (5), he ohoopic linea viscoelasic law akes he fom of an incemenal maix equaion: wih [ ] Δε { Δσ} = K { }+{ σ his ( ) } Δσ * { },Δ (7)

6 [ K ] = γ 0 + µ=1 γ µ K 0 { σ his ( ) } = σ his µ µ=1 1 e [ ] α µ Δ = γ 0 + γ µ µ=1 ( ) { ( )} = 1 e α µ Δ µ=1 α µ Δ σ µ K 0 [ ] and [ ] Δε * { ( )} e { Δσ *} = K { } (8) { ( )} is he em hisoy wich summay he pas loading effecs unil he fis whee [ K ] is he ficiious igidiy maix. σ his loading on he acual esponse. These values depends of he paameesα µ,γ µ and of he ime incemen Δ ; i depends also of he inenal vaiables wich mus be acualized a he end of each incemen calculaion: { σ µ ( + Δ) } = σ µ { ( )}+ [ K µ ] Δε his ({ } { Δε *})+ σ µ The enso{ Δε} epesens he sain incemen fo he sep pas ime. 3.2 Finie elemen implemenaion { ( )} µ [ 1,] : (9) The poposed viscoelasic incemenal model is implemened in he finie elemen sofwae Cas3m following he ouine pesened in Figue 5. Beginning Daa: Geomey, mechanical paamees (Maxwell), moisue loading Viscoelasic calculaion: Moisue loading Ficiious igidiy maix Finie elemen compuing Vaiable acualizaion Resuls: End Figue 5 Viscoelasic incemenal ouine.

Numeical simulaion and discussion 4.1 Wood slice viscoelasic paamees The wood geen slice (Douglas fi, see Figue 4) is modelled wih he finie elemen sofwae Cas3M.

7 Each sep viscoelasic calculaion begins by he evaluaion of he ficiious igidiy maix and he heediay sess { σ his ( ) }. Afe, he peceden daa ae consideed in he solving of he equivalen hemoelasic poblem accoding o equaion (7). Finally, he inenal sesses egading equaion (8) ae acualized befoe he following sep. [ K ] 4. Numeical simulaion and discussion 4.1 Wood slice viscoelasic paamees The wood geen slice (Douglas fi, see Figue 4) is modelled wih he finie elemen sofwae Cas3M. Consideing he Small hickness, he plane sess hypohesis is consideed. The hygoscopic bounday condiions ae fee and he veical displacemens ae zeoed a he boom of he specimen. The modelling is conduced on he basis of he incemenal fomulaion pesened in he secion 3.1 by consideing a Zene s model composed of wo Maxwell s cells posed in Figue 4. The ee elasic moduli ( E, E and G ), he Poisson s coefficien υ he adial and angenial shinkage coefficiens (α and α ) and he paamees of he non-dimensional elaxaion funcion ρ 0, ( ) ae pesened in he following Table 1: Table 1 Paamees of he elaxaion funcion. (GPa) (GPa) (GPa) ( h -1 ) Resuls and discussion The numeical simulaion shows he sess onse duing he developmen shinkage by he heeogeneiy of wood due o he ohoopic chaace o he shinkage pocess Figue 6 Sess a he ime =14h. (a) Radial sesses. (b) Cicumfeenial sesses. The sess analysis shows ha he slice is compessed almos eveywhee in he adial diecion Figue 6 (a). In he cicumfeenial diecion (Figue 6 (b)), he sign of he sesses changes: hee ae ensile sesses a he peiphey and compession sesses when appoaching he cene. We noe also he exisence of wo zones of cicumfeenial acion concenaion sesses due o he asymmey of he slice. Fo he fuhe, we sudy he evoluion of cicumfeenial sesses vesus he adius noed R, which connecs he cene of he slice aound P1 a he poin P4, whee he cicumfeenial sess is maximal.

Figue 7 shows ha he cicumfeenial sesses occu in a self-balanced manne along he adius R, he compessed zone a he cene balancing he ension zone a he peiphey.

8 Figue 7 shows ha he cicumfeenial sesses occu in a self-balanced manne along he adius R, he compessed zone a he cene balancing he ension zone a he peiphey. Radius R (cm) Figue 7 Disibuion of cicumfeenial sesses along he adius R Figue 8 shows he evoluion of he cicumfeenial sesses vesus ime fo each poin P1 o P4 in he case of elasic and viscoelasic behaviou. The boh figues show ha he sesses incease ove ime wih he shinkage incease unil he ime = 14h. In addiion, he sesses sabilize and educe due o elaxing effec (Figue 8). The compaison beween elasic and viscoelasic calculaions shows ha he elasic elasic calculaion oveesimaes he sesses. In he viscoelasic case, he calculaions yield sesses lowe han in he elasic case. Figue 8 Evoluion of cicumfeenial sesses along he adius (P1 o P4).

9 7. Conclusion The hydo-mechanical behaviou of wood duing dying and he heeogeneous and ohoopic chaace of he maeial have been ecalled. The expeimenal seup consised of composed of a Douglas fi geen wood slice, posed on a sensiively weigh, and a video camea ecoded he displacemens of ages have been used. The PSF aound 22/% has been obained by his echnique. Then he decomposiion of he elaxaion enso on he basis of a Diichle seies has been inoduced. This povided, accoding o he Bolzmann equaion, o wie he consiuive viscoelasic maix unde a fee incemenal sain. An algoihm incopoaing he poposed incemenal fomulaion, he humidiy and he ohoopic behaviou of wood has been poposed and implemened using he finie elemen sofwae Cas3m. This model was used o simulae a geen wood slice subjeced o naual dying. Thee is he appeaance of sesses due o shinkage and he wood ohoopic chaace. The combined effec of wae loss and he viscoelasiciy of wood gives sesses below hose obained by elasic calculaion. The ensile sesses ae obseved in he cicumfeenial diecion wih disance fom he hea of he slice, he eaching peak values ae obseved a he slice bounday. These sesses can lead o adial cacking fequenly obseved in he dying phase. The chaaceizaion and he sudy of hydo-viscoelasic effecs of wood ae needed o pedic is behaviou and avoid he isk of cacking duing he dying pocess. 8. Refeences [1] Omasson S., Dahlblom O., Peesson H. Numeical sudy of he shape of Sawn imbe subjeced o moisue vaiaion, Pa 1 : Theoy. Wood Sci Technol, 32, , [2] Mouee M. Modélisaion du compoemen mécanique du bois au cous du séchage, PhD Theses, Univesié Laval, Québec, [3] Diawanich P., Maan N., Kyokong B. Evoluion of inenal sess duing dying, cooling and condiioning of ubbewood lumbe. Eu J Wood Pod, , [4] Husson J.M., Dubois F., Sauva N. Elasic esponse in wood unde moisue conen vaiaions: analyic developmen. Mech Time-Depend Mae, 14, , [5] Thibaua B., Gil J., Founie M. Mechanics of wood and ees: some new highlighs fo an old soy, Mécanique du bois e biomécanique des abes : nouveaux egads su une vieille quesion. CR Mecanique, 329, , [6] Couand C., Mahias J.D., Jeonimidis G., Desebecq J.F. TWIG: A model o simulae he gaviopic esponse of a ee axis in he fame of elasiciy and viscoelasiciy, a ina-annual ime scale. J Theo Biology, 273, , [7] Chazal C., Mouou Pii R. Incemenal consiuive fomulaion fo ime dependen maeials: ceep inegal appoach. Mech Time-depend Mae, 2011, DOI /s z. [8] Jukiewiez B., Desebecq J.F., Vegne A. Incemenal analysis of ime-dependen effecs in composie sucues, Comp Suc, 73, , [9] Saifouni O, Mouou Pii R., Desebecq J.F. An incemenal consiuive law fo damaging viscoelasic maeials. SEM Annual Confeence & Exposiion on Expeimenal and Applied Mechanics, Mohegan Sun, Uncasville, USA, Spinge New Yok DOI / [10] Navi P., Hege F. Compoemen hemo-hydomécanique du bois, Pesses polyechniques e univesiaies omandes, CH-1015 Lausanne, 2005.

Orthotropic Materials

Orthotropic Materials Kapiel 2 Ohoopic Maeials 2. Elasic Sain maix Elasic sains ae elaed o sesses by Hooke's law, as saed below. The sesssain elaionship is in each maeial poin fomulaed in he local caesian coodinae sysem. ε