Role of the gelatinous layer on the origin of the physical properties of the tension wood

Transcription

1 J Wood Si (004) 50:97 08 The Jpn Wood Reseh Soiey 004 DOI 0.007/s ORIINA ARTICE Hioyuki Ymmoo Role o he gelinous lye on he oigin o he physil popeies o he ension wood Reeived: Augus 8 00 / Aeped: y 00 Abs To disuss he ole o he gelinous lye (lye) on he oigins o he physil popeies peuli o he ension wood ibe (TW ibe) he deomion poess o n isoled TW ibe used by ein biomehnil se hnge ws omuled mhemilly. The mehnil model used in he pesen omulion is ou-lyeed hollow ylinde hving he ompound middle lmell (C) he oue lye o he seondy wll (S) nd is middle lye (S) nd he -lye () s n innemos lye. In he omulion he einoed mix mehnism ws pplied o epesen he mehnil ineion beween he ellulose mioibil (CF) s mewok bundle nd he mophous subsne s mix skeleon in eh lye. The model omuled in he pesen sudy is hough o be useul o invesige he oigins o exensive longiudinl dying shinkge lge ensile gowh sess nd high xil elsi modulus whih e heologil popeies peuli o he TW. In his ile he deiled poess o he mhemil omulion is desibed. In subsequen ile some TW popeies om 70-ye-old Kohuhiwkede (Ae sieboldinum iq.) will be nlyzed using he newly developed model. Key wods elinous ibe Tension wood Cell wll Reion wood owh sess Inoduion Tension wood (TW) onsiss o bnoml issue lled gelinous ibe (-ibe) beuse i onins gelinous lye (-lye) s he innemos lye o he seondy wll. The TW oen shows heisi behvio h is dieen H. Ymmoo (*) Shool o Biogiulul Sienes Ngoy Univesiy Chikus Ngoy Jpn Tel ; Fx e-mil: hio@g.ngoy-u..jp om he noml wood (NW). A high-ensile gowh sess is geneed on he sue o he xylem in he TW egion h oen beomes en imes s lge s h in he NW egion. The longiudinl Young s modulus o he TW beomes signiinly highe hn h o he NW. Fuhemoe he xil shinkge in he TW ends o exeed moe hn % duing we desopion while h in he NW beomes less hn 0.5%. Some uhos ibue hese behvios o he ininsi popeies o he -ibe. On he ohe hnd moe hn ew esehes believe h he -lye is mehnilly oo omplin o be lge sess geneion. They bse hei gumen on he s h he -lye is oen onvolued in he lumen o he nsvese seion h is smpled om he we-swollen blok nd i n be esily peeled o he ligniied lye in he sme dieion duing miooming. This gives he impession h i is hed only loosely o he eminde o he seondy wll. 4 Fom hose obsevions hey onside h he vious heisis o he TW should be ibued no o he lexible -lye bu o he elively hike oue lye o he seondy wll (S) 5 nd/o o he elively hinne middle lye o he seondy wll (S) whose mioibil ngle (FA) is expeed o be moe o less dieen om h in he NW ibe. 7 I is impossible o djudge whih possibiliy holds unil we sueed in diely mesuing he mehnil popeies o he -lye nd he ligniied lye whih equies isoling hem om eh ohe. Howeve i my be quie nul o onside h he ligniied lye in he -ibe would be essenilly sme s h in he NW ibe beuse hee is no speii dieene in nomil nd hemil spes beween hem s poined by Okuym e l. Thus he uho expes h heisi behvios peuli o he TW my be ibued o he -lye. Simulion using wood ibe model is one o he mos eeive mehods o esime he inenl popeies nd ine suues o eh onsiuen meil in wood ell wlls. 89 In ou pevious sudies we developed wood ibe model onsising o he ompound middle lmell (C) S nd S. A bsi omul ws deived o simule he deomion poess o n isoled wood ibe when

2 98 ein biomehnil hnge ous suh s lignin deposiion 0 we dsopion o exenl lod induion. 94 The bsi omul onined sevel pmees h wee deived om he omposie suue o he ell wll lye. When omping he simuled esuls wih he expeimenl one onee vlues need o be ssigned o he pmees in ionl mnne. We onside h hose vlues ele ininsi inomion on he inenl popeies nd ine suue o ell wll onsiuens. The objeive o he pesen sudy ws o liy he ole o he -lye in he oigin o heisi behvios o he TW nd hen o impove ou pevious wood ibe model 5 ino ou-lyeed model hving he -lye s he innemos lye. In he pesen ile he deiled poess o he omulion on he deomion poess o he - ibe model whih is indued by ein biomehnil se hnges is desibed. In subsequen ile some TW popeies o 70-ye-old kohuhiwkede (Ae sieboldinum iq.) will be nlyzed by using he newly developed model inluding high-ensile gowh sess geneion lge longiudinl Young s modulus nd lge xil shinkge due o we desopion. Fomulion o he deoming wood Wood ibe model Figue shows shemi model o he ypil -ibe whih onsiss o he C he S he S nd he -lyes. Eh lye n be ppoximed s wo-phse suue speiilly he unidieionl einoing elemen o he polyshide mewok nd he enusing mix (T) o lignin hemiellulose ompound. The ome is minly omposed o highly ysllized ellulose mioibil (CF) whih is oiened in ein dieion o he ibe xis in eh lye exep he C. This mkes he seondy wll nd he -lye mehnilly nisoopi. On he ohe hnd in ses in whih he oienion o he CF is Fig.. ulilyeed suue o he gelinous ibe () is mehnil model (b) nd he ossu sue (). Eh ibe onsiss o he ompound middle lmell (C) he ouemos lye o he seondy wll (S) is middle lye (S) nd he gelinous lye () ndomly disibued in he C i is onsideed o be mehnilly isoopi. In he se o sowood seondy wll oiened polyose suh s (eyl-) gluomnnn is oen nged long nd ound he CF oming he polyshide mewok wih he highly ysllized CF. No oiened polyose hs been deeed in he hdwood ell wll. 8 Disoiened polyose h is minly xyln is blended wih lignin oming he isoopi skeleon o he T subsne. In he pesen sudy single -ibe is simpliied ino omplex hollow ylinde onsising o he C he S he S nd he -lyes s shown in Fig. b. Consiuive equions Unde low mgniiion he einoing elemen o he polyshide mewok is spilly dispesed uniomly in eh ell wll lye o om he mewok bundle. Similly he lignin hemiellulose ompound is diused in eh lye oming he isoopi T skeleon. Theeoe i is onsideed h boh he mewok bundle nd he T skeleon oupy he sme domin in he mosopi limi. Bsed on suh n ide Bbe nd eyln hypohesized he mehnil ineion beween he polyshide mewok nd he lignin hemiellulose T in eh lye s he ollowing ondiions: 90 m m σ σ σ ε ε ε () ij ij ij ij ij ij whee σ ij σ m ij nd σ ij e he sess ensos in he mewok bundle T skeleon nd ell wll lye s whole espeively. ε ij ε m ij nd ε ij on he ohe hnd e hei espeive sins. In he pesen sudy he deomion o he wood ibe model is ssumed o be symmei wih espe o he enl xis (see he ondiion C4). Thus we n expess he h osionl deomion o n individul ibe is ompleely esied by he oe o binding ibes inside he wood speimen. Then ylindil oodine sysem (O- whee nd epesen longiudinl ngenil nd dil omponens espeively) n be pplied o he pesen model s shown in Fig.. A visible deomion indued eh poin in he ell wll is expessed by he obsevble sin enso omponens (ε ij whee i j ) in he O- oodine sysem. The sess omponen indued in he T skeleon o eh lye (σ m ij ) is eled o he obsevble sin omponen used in he mix skeleon (ε m ij whee i j ) s he ollowing onsiuive equion: σ C ε α () m m m m ij ijkl kl kl C m ijkl is n elsi onsn enso nd α m kl is enso o inenl expnsion whih is used by ein biomehnil hnge. α m kl is obseved suh h in eh lye α m kl ε m δ kl whee δ kl is Koneke s symbol. ε m is sl. C m ijkl nd α m kl e boh ssumed o be isoopi. The elsi onsns o he isoopi T skeleon (nonzeo ems) e denoed s ollows:

3 99 gen. The einoing gen is onsideed o be he lignin hemiellulose isoopi mix. Then we my onside he siness omponens C bd * e ll nil exep C xxxx * (E). By nsoming he oodine sysem om O-xyz ino O- sysem Eq. 4 is ewien ino new expession s ollows: σ C ε α (5) ij ijkl kl kl whee C R R R R C * α R R α * () ijkl i jb k ld bd ij i jb b Fig. b. Coodine sysem used in he omulion. O- lol ohogonl oodine sysem. b Fl bod elemen o he CF mewok bundle. O (in ) is n biy poin in he ell wll. Dieion o x-xis (in b) is pllel o he CF moleul hins in he S o S lyes m m m C C C ( K S) m m m m m m C C C C C C ( K S) m m m C C C S () whee K λ µ S µ. λ nd µ e me s onsns. Figue b shows smll l-bod elemen o he mewok bundle in he S lye povided h he posiive dieion o noml xis (z-xis) is oiniden wih he dil dieion (-xis) o he ibe model. In his model he CF nd ohe oiened polyose in he S lye e ssumed o be oiened in n S-helix n ngle o θ nd he one in he S lye is ssumed o be ligned nomlly o he ibe xis. The elionship beween sess (σ * ij ) nd sin (ε * ij) omponens indued in he mewok bundle o eh lye o he seondy wll n be wien s he ollowing onsiuive equion in he O-xyz ohogonl oodine sysem: σ * C * ε* α* b bd d d (4) whee C bd * is he elsi onsn o he mewok bundle in eh lye in he O-xyz oodine sysem. α b * is n inenl expnsive sin used by ein biomehnil hnge h is digonl enso whose omponens e ( b ) dig α * ε ε ε whee ε nd ε e he inenl expnsive sins indued in he mewok bundle in he dieions pllel nd pependiul o he ellulose moleul hin espeively. We supposed h he mewok bundle is onsidebly omplin in is nsvese dieion. Theeoe ll she moduli Poisson s ios nd he Young s modulus in he nsvese dieion e smll enough o be negleed. This mens h he CF inluding ohe oiened polyshide nno be kep in bundle shpe wihou einoing R ij is nsomion mix beween boh oodine sysems. The nonzeo ems o he siness omponens o he mewok bundle (C ijkl) e expessed s 4 C E C C s E C C se 4 C s E C C s E C s E α ε ε s α ε s ε α ( ε ε )s povided h we onsideed C ijkl C jikl C jilk nd C ijkl C klij whee osφ s sinφ nd E is Young s modulus o he mewok bundle in he dieion long he ellulose moleul hin. E S ε m ε ( ou / in ) nd φ ke espeive vlues in eh lye s ssumed in Tble. They e no unknown vlues o be solved bu known onsns o be given in dvne. 8 S is he she modulus () o he mix skeleon in eh lye. In he sme wy s E S kes espeive vlue in eh lye. S is denoed s S 0 S S nd S in he C S S md -lyes espeively. By enging Eqs. using Eqs. 5 nd 7 nd onsideing he ompibiliy o sins o he xisymmeil deomion wihou osion ε ε we obin ε d d m Ï 4 σ Kε E ε Ì ( K S) E ε Ó Ï ε K S E s ε K S d Ì Ó d m Ï σ Kε Es ε Ì ( K S) E s ε Ó Ï 4 Ì ( K S Es ε ) ε ( K S) d Ó d m ε σ ε ( ) ε ( ) ε ( ) K K S K S K S d d σ Es ε ε s ε σ σ 0 ( ) whee σ σ σ σ σ σ ε ε ε ε ε ε. is dil disne om he enl xis. Among he she (7) (8)

4 00 Tble. is o he pmees in he bsi omule A nd B ye φ in ou S E P in P ou ε m ε 0 C 0 ( h) 0 S 0 E 0 (0) P P 0 (0) S 90deg m S E P P ε S θ m S E P P ε 0deg 4 m S 4 E P 4 (0) P ε φ he mioibil ngle; in inne dius o eh lye; ou oue dius o eh lye; ou / in ; S she modulus () o he mix skeleon; E Young s modulus o he mewok bundle in he dieion long he ellulose moleul hin; P in inne pessue; P ou oue pessue; see Eqs. 0; ε m inelsi sins in he mix skeleon; ε inelsi sins o he mewok bundle in he dieion pllel o ellulose moleul hins; C ompound middle lmell; E/S; E /S ; E /S ; E /S ; Q F/E ; F S 0 h/ ; S /S ; N S /S ε ε ε sess omponens only σ is no null whih is n inevible onsequene om he ssumpion o he xisymmeil deomion i.e. ε 0. In his sudy sess equilibium o xisymmeil deomion ws ssumed s ollows: σ σ σ d d Deiving he bsi equions Assumpions The dimensionl hnge o he wood ibe model n be expessed s se o noml sins nmely ε in he longiudinl dieion nd ε () 0 ε () ε () ε () nd ε () 4 in he ngenil dieions he espeive dius. In he sme wy s desibed in pevious wok 4 bsed on Eqs. 8 we solve ε ε () 0 ε () ε () ε () nd ε () 4 unde he ondiions C C C nd C4 nd ssumpions A A nd A: C. ε is onsn o ll. This is bsed on he ssumpion h he wood ibe model is n ininiely long ylinde in he -dieion. We denoe ε s ε heee. C. [σ ()] 4 P 4 [σ ()] 0 P 0. P 0 P 4 0. These boundy ondiions men hee e no inenl nd exenl pessues ing on he wood ibe model (see Tble ). C. The exenl oe (P ) indued pllel o he wood ibe model sisies he ollowing equion: 0 P p σ da σ ddθ p σ d Ú l Ú l Ú 4 ossu l sue ossu sue (9) (0) C4. The deomion o he wood ibe model is ssumed o be symmei wih espe o he enl xis. In ddiion o he ondiions he ssumpions A A nd A wee mde poessing: A. K S so h S/K is enough smll o be negligible in eh lye. This hypohesizes h he bulk modulus o he T skeleon (K) is lge hn S. This posules h he Poisson s io o he mix skeleon is lmos 0.5 simil o kind o elsome. A. I is onsideed h E S ε m ε nd ( ou / in ) end o hnge hei vlues duing ein biophysil hnge; howeve hose hnges n be negleed in he se h biomehnil hnges wee ininiesimlly smll. A. E 0 in Eq. 7 in he C. Nmely E 0 0. This does no men h hee is no oiened polyshide mewok in he C bu mens h mehnil onibuion o he ndomly disibued polyshide mewok in he C should be isoopi. Fo onveniene we ssume h S 0 mens he she modulus () o he C isel. oeove (h/ ) 0 ε 0 0 nd ε 0 m 0 in he C. These ondiions ssume h h (hikness o he C) is smlle hn he S nd he S lyes. Bsi omul A Solving he seond nd hid omul in Eqs. 8 o ε nd dε /d nd elimining he em o σ using he equilibium ondiion o Eq. 9 we obin 4 4 m [ S Es u( S Es )] ε S ε ( u) E s ε S E s u S E s u d σ [ ( )] ε ( ) d uσ [ S Es u( S Es )] d ε m E s ε d ( u) E s ε ( s u) E s ε ( u ) d σ 4 us σ d () whee u S/K nd E/S. oeove ombining hem by elimining he em o ε we obin dieenil equion o :

5 0 Tble. is o he oeiiens in he bsi omule A nd B ye oion Γ Λ X Ω F Σ C 0 Γ 0 () Λ 0 () X 0 () 0 (0) Ω 0 () F 0 () Σ 0 (0) Γ 0 () Λ 0 () X 0 () 0 (0) S Γ Λ X Ω F Σ Γ Λ X S Γ Λ X Ω F Σ Γ Λ X Γ () Λ () X () (0) Ω F () Σ 4 Γ () Λ () X () (0) 4 u d σ u d σ us σ d d 4 m E s ε u E s ε E s s u ε enel soluion o Eq. is given s σ α α 4 m C C E s ε u α ue sε E s s uε ] È ÎÍ α α α [ () () whee C nd C e inegl onsns nd α is desibed s ollows: α s u u s 4 4 S K S Bsed on he ssumpion A we impose he ollowing ondiions: u Æ 0 hen α Æ 0 whih yields lim αæ0 α Æ ln α Then we n simpliy he soluion o Eq. s ollows: 4 m σ C C E s ε E s ε s E s ε ln ] [ Unde he boundy ondiions [σ ()] in P in nd [σ ()] ou P ou we n deide he inegl onsns C nd C nd we obin he ollowing soluion: ou Ï 4 m σ Pin Pou Pin E s ε Ì Ó 4 m E s ε E s ( s ) ε ] ln} [ E s ε E s ε E s ( s ) ε ] ln in (4) [ Then we subsiue he bove soluion ino he is p o Eq. nd we obin bsi omul A: Γ ε Λ ε X ε ε whee Γ ou m ou Pin S s s s P S ln ou Λ s X s s nd s ( ) ln ou ln ou 4 ou Coeiiens Γ Λ X nd e dependen on φ ou nd povided h hese vibles ke hei espeive vlues in eh lye s shown in Tble. Bsi omul B We solve Eqs. 8 o ε nd inege i ove he ossu e o eh lye. When ineging i we ssume h ε m nd ε e independen o in he espeive lyes nd ε kes onsn vlue (ε ) ove he ossu sue o he ell wll beuse o ondiion C. As esul we obin bsi omul B: m Ω ε F ε Σ ε whee ( ) P S Λ S in ( ) Pin [ Γ] S ou [ Γ] ou in 4 Ω s s È 4 s ( s ) Í ln 4 ÎÍ

6 0 F s s È s ( s ) Í ln 4 ÎÍ È 4 Σ ( s ) s ( s ) Í ln ÎÍ σ θ p Ú dd eh lye 4 Coeiiens Ω F nd Σ e dependen on φ nd povided h hese oeiiens ke hei espeive vlues in eh lye s shown in Tble. Fomule o desibe he deoming -ibe In he -ibe model s he whole welve equions e obined bsed on he bsi omule A nd B. The equions e s ollows: Bsi omul A in C 0 (0) Bsi omul A in C (0) Bsi omul B in C (b0) Bsi omul A in S () Bsi omul A in S () Bsi omul B in S (b) Bsi omul A in S () Bsi omul A in S () Bsi omul B in S (b) Bsi omul A in () Bsi omul A in 4 () Bsi omul B in (b). These equions onsiue simulneous equions whose unknown vibles e ε ε ε ε ε ε ; 0 4 P P P P P ; Aoding o he ssumpion A Eqs. 0 nd 0 e degeneed by eh ohe whih yields ε ª ε 0 Theeoe he unknown vibles beome ε ε ε ε ε ; 4 P P P P P ; To solve Eqs. 0 nd Eqs. b b o he unknown vibles he ollowing ondiions bsed on ondiions C nd C e imposed: P 0 P The unknown vibles expliily equied in ou sudy e ε ε ε ε nd ε 4. Thus by elimining he unknown vibles P 0 P P P P 4 0 nd in he ollowing mnne we degeneed he simulneous equions ino simple ones in whih he unknown vibles e ε ε ε ε nd ε 4. The is equion Fom Eqs. 0 b0 b b b nd we elimine P 0 P P P P 4 0 nd. Thus we obin he is equion ε ε ε 4ε 5ε 4 m m m b ε b ε b ε ε ε ε d P (5) whee oeiiens 4 5 b b b nd d e espeive unions whose onee oms e omposed o θ Q(F/E ) (S /S ) nd N(S /S ). Deiled shpes o hose oeiiens e desibed in he Appendix. P snds o he exenl oe indued pllel o he wood ibe model whih is eled o 0 nd s ollows: P p p σ ddθ p Ú 0 4 σ d The seond equion ( 0 ) Ú ossu sue ε ε ε ε nd ε 4 e unknown vibles o be solved s soluions o n lgebi equion (Eq. 5). To solve Eq. 5 o ε ε ε ε nd ε 4 hee mus be les ou equions h e onsiued by he sme unknown vibles. These equions n be deived om bsi omul A. Fom Eqs. 0 nd we elimine P 0 P P P nd P 4. Thus we obin ε ε ε 4ε 5ε 4 m m m b ε b ε b ε ε ε ε d P The hid equion () Fom Eqs. 0 nd we elimine P 0 P P P nd P 4. Thus we obin ε ε ε 4ε 5ε 4 m m m b ε b ε b ε ε ε ε d P The ouh equion (7) Fom Eqs. 0 nd we elimine P 0 P P P nd P 4. Thus we obin 4ε 4ε 4ε 44ε 45ε 4 m m m b ε b ε b ε ε ε ε d P (8)

7 The ih equion Fom Eqs. 0 nd we elimine P 0 P P P nd P 4. Thus we obin 5ε 5ε 5ε 54ε 55ε 4 m m m b ε b ε b ε ε ε ε d P (9) whee oeiiens 55 b b 5 5 nd d d 5 e espeive unions whose onee oms e desibed in he Appendix. Equions 5 9 onsiue he simulneous lgebi equions whose unknown vibles e ε ε (ε ) ε (ε ) ε (ε ) nd ε 4 (ε 4 ). The vlues o ε m ε m ε m ε ε ε nd P should be given in dvne. Fom he simulneous equions he soluions should be expessed in he ollowing oms: 7 ( p) ε 7( p) P ( p) ε 7( p) P 4( p) ε 47( p) P 5( p) ε 57( p) P ε p ε p ε p ε p ε p ε p ε p P m m m 4 5 ε ε ε ε ε ε m m m 4 5 ε ε ε ε ε ε m m m 4 5 ε ε ε ε ε ε m m m ε ε ε ε ε ε m m m (0) whee oeiiens 57 e unions o p nd p is pmee veo whose omponens e θ Q nd N. Among he soluions ε epesens he sin o he -ibe model in he xil dieion. Aoding o he ondiion C4 he wood ibe model deoms xisymmeilly so h ε (ε 0 ) ε ε nd ε 4 e equivlen sins o he dimel deomions hei espeive dii. These sins e indued by ein biomehnil hnge in he -ibe. The hikness o he C is smll enough o be negligible i omped o hose o he S nd S lyes. Theeoe ε n be egded s he sin o he dimee in he wood ibe model (ε T ). Developing Eqs. 0 ino dieenil equions Aoding o ssumpion A Eqs. 0 e vlid unde he ondiion h he biomehnil hnge is ininiesimlly smll enough o be negleed. This mens h ε ε ε ε nd ε 4 in ddiion o ε m ε m ε m ε ε ε nd P in Eqs. 0 should be epled s dieenil quniies i.e. dε dε dε dε dε 4 dε m dε m dε m dε dε dε nd dp. Then eh expession in Eqs. 0 should be ewien s simple dieenil om e.g. 5 7 dε p dε p dε p dε p dε p dε p dε p dp m m m 4 0 (0) The Eqs. 0 e divided by d nd e onveed ino he ollowing dieenil equions: 7 ε P ( p) 7( p) ε P ( p) 7( p) ε P 4( p) 47( p) ε P 5( p) 57( p) ε p ε p ε p ε p ε p ε p ε p P m m m 4 5 ε ε ε ε ε ε m m m 4 5 ε ε ε ε ε ε m m m 4 5 ε ε ε ε ε ε m m m ε ε ε ε ε ε m m m () whee he ised do noion epesens he deivive by. Fuhemoe nd mong he omponens o p should be ewien ino he dieenil quniies d d nd d beuse hey depend on he elpsed ime duing ein biomehnil hnge. Due o he ssumpion o xisymmeil deomion he ollowing elions e inquied mong espeive lyes: d dε d d dε d dε d () d dε d dε d 4 d dε 4 4 Beuse d dε 4 d dε nd d dε e egded s highe ode ininiesiml quniies om Eqs. we n obin he ollowing dieenil oms: d dε dε d dε dε d dε dε 4 () These equions n be ewien s dieenil equions o s ollows: 4 ε ε ε ε ε ε () Equions nd onsiue sysem o simulneous dieenil equions whose unknown unions e ε ε ε ε ε 4 nd. On he ohe hnd ε m ε m ε m ε ε ε nd P e he unions whose -dependen shpes should be given in dvne.

8 04 Solving he simulneous dieenil equions (Eqs. nd ) Soluions Equions nd give he deomion o he wood ibe model whih is indued by ein biomehnil hnge ouing in he ell wll. In ou se exmples o biomehnil hnge e he gowh sins (muion sins) duing ell wll ligniiion whih is mesued s he elesed sins o he gowh sess; he swelling sins due o moisue dsopion; nd he elsi deomion used by exenl lod. We ssume h hnge in he physil se in he ell wll ss 0 nd ends Z. By ineging Eqs. nd om 0 o Z we n solve hem o ε ε ε ε ε 4 nd. As he iniil ondiions 0 we ssume 4 ε 0 ε 0 ε 0 ε 0 ε (4) Then we divide he inegl inevl ino n smll ps nd denoe he inegls o Eqs. in he i-h smll inevl [(i )Z/n iz/n i... n] s i ε i ε i ε i ε nd i ε 4. Fo exmple i ε is luled s i ε iz n Ú dε d d ( i) Z n iz n m m m Ú ( 4 ( i) Z n 5( p) ε ( p) ε 7( p) P ) d p ε p ε p ε p ε (5) In eh smll inegl inevl he unions ε m () ε m () ε m () ε () ε () ε () nd espeive omponens in p inluding nd mus be given in dvne. Howeve nd e unknown unions o be solved om simulneous dieenil Eqs. nd. The vlues o nd whih should be used in he i-h smll inevl n be esimed in he ollowing poess. A is we inege Eqs. in he (i )-h smll inevl. Then we obin i Z n Ú i Z n i Z n d d d Ï i Ô ε ( ) Ì Ú ( dd ÓÔ i) Z n dε d Ô d Ô I n is ken s lge enough numbe his equion gives i i ε i ε i ( ) () whee i is equl o () (i )Z/n. Thus he vlue o i whih is () in he i-h smll inevl is given s (7) i i i i i i ε ε By using simulneous euene equions (Eqs. 5 nd 7) we n inege simulneous dieenil Eqs. nd numeilly. Thus he nul sin o he deoming wood ibe model ein ime ( j Z/n j is posiive inege smlle hn n ) n be deived s ollows: ε lim εt j  i ε næ i 0 j i limâ ε næ Ú i 0 Ú dε d d dε d d (8) Sins ε () nd ε T () give nul sins o he dimensionl hnges indued in he wood ibe model. Howeve he elesed sin o he gowh sess nd he swelling due o we sopion e mesued s he nominl sins in he deoming wood speimen. Then we need o solve Eqs. nd o give nisoopi dimensionl hnges o he wood ibe model s nominl sins. The nominl sin o he deoming wood ibe model ein ime ( j Z/n) in espeive dieions n be deived s ollows: j È i α limí ( ε ) næ Î i j È i β limí ( ε ) næ Î i (9) I he vlues o α(z) nd β(z) e smlle hn % hnges o nd e smll enough o be negleed. In h se we n obin he α() nd β() om he is nd he seond omule o Eqs. whih simpliies Eqs. 9 s ollows: j j dε ε α ε β ε d d i i lim d d næ Ú limâ næ Ú i 0 i 0 Exmples o simulions eneion o he muion sins due o ell wll ligniiion (0) Soon e he deposiion o he CF nd he highly oiened polyshide mewok in he seondy wll o he heid o libiom wood ibe he mix subsne o hemiellulose nd lignin deposis mong he gps o he mewok bundle. In his poess he wood ibe ends o swell o shink nisoopilly whih genees nisoopi gowh sess in he newly omed xylem. This is beuse ee deomion o he individul ibe is esied o inside he ul xylem. This poess ws omuled by using wood ibe model o he C S S nd lyes. In his se ievesible deposiion o he T subsne nd he muion o he CF mewok e posuled o use hnge in he biomehnil se o he heid o libiom ibe wll. In he pesen sudy he ee dimensionl hnge o he -ibe is simuled duing he wll muion. P should be null in Eqs.. The inegl inevl is om he deposi-

9 05 Tble. Assumed vlues o he hemil omposiions in eh lye o he geen -ibe wll ye Cellulose ysl Oiened polyose Isoopi mix Cyslliniy in he (non-yslline) in he polyshide poly-shide mewok mewok C 5 (%) 0 (%) 85 (%) 00 (%) S S Cse Cse b Cse ions o he CF mewok ( 0) o he ompleion o he -ibe wll ( Z). In his inevl he -dependen pens o ε m ε m ε m ε ε ε nd evey omponen in p should be given in dvne. As he iniil ondiions 0 we dop Eq. 4. Aoding o he obsevions he elesed sins o he gowh sess e less hn 0.5% boh in he nd he dieions. Then he simpliied omule (Eqs. 0) n be used insed o Eqs. 9 when simuling he muion sins α() Z nd β() Z. In suh se () () nd () e egded s ppoximely onsn. As he peliminy simulion we luled he vlues o α() Z (ε ) nd β() Z (ε T ) unde he ondiions o he pmees ssumed in Tbles nd 4. The vlues in Tbles nd 4 exep hose eled o he -lye wee used in ou pevious simulions on he geneion o gowh sins in he lewood heid o sugi (Cypomei jponi). 8 In he pesen simulion he vious ses o he peenge o he ellulose ysl in he -lye e displyed in Tble nd he elionship beween ε ε T nd ε () Z ws luled o eh se o mewok yslliniy. The esul is shown in Fig.. Aoding o he obsevion in he noml ibe o he sowood o he hdwood xylem he elesed sin o he sue gowh sess beomes 0.0% o 0.0% in he longiudinl dieion nd 0.05% 0.% in he ngenil dieion. In he TW xylem on he ohe hnd he longiudinl elesed sin oen beomes sevel imes s lge s in he NW xylem nd in some speies oming onsidebly hik -lyes suh s Robini pseudoi i exeeds 0.5%. The pesen simulion explins hese phenomen well when he - lye is ssumed o shink onsidebly in is xil dieion duing he -ibe muion. ongiudinl elsi onsns A pesen he longiudinl Young s modulus (E ) o he wood ibe model is luled bsed on Eqs. nd unde he ssumpion o he moisue sedy se. Theeoe i is ssumed h evey omponen in p is onsn. oeove dε m dε m dε m dε dε nd dε e ll nil. Then om Eqs. we obin he longiudinl Young s modulus o he -ibe (E ) s ollows: E dp p dε p p () Fig.. An exmple o he simulion on he elions beween he gowh sins nd he vlues o ε Z ( T ) in he -ibe. Vlues o he pmees ssumed hee e displyed in Tble 4. Condiions o he yslliniy o he polyshide mewok in he - lye (see Tble ) solid lines se ; dshed lines se b; shded lines se The subsnil Young s modulus o he -ibe n be luled s dp E w () p( 0 4 ) dε p( 0 4 ) 7( p) In his se n inese in he sin enegy indued by exenl lod is egded s he biomehnil se hnge. As peliminy simulion he vlues o E T wee luled unde he ondiions o he pmees given in Tbles nd 4. The esul is shown in Fig. 4. In he se in whih he -lye onins ein moun o yslline ellulose luled vlues o E beome lge wih he hikness o he -lye; howeve i is lmos onsn when he -lye onins no yslline ellulose. Swelling nd shinkge sins due o he moisue dsopion In he sme wy s he gowh sess se ee dimensionl hnge o he -ibe due o moisue dsopion wee simuled. Thus P should be null in Eqs.. The inegl inevl is om he oven-died se (u 0) o he ibe

10 0 Tble 4. Assumed vlues o he pmees o he simulion o he gowh sins o he -ibe m m m m E 0 E E E S 0 S S S 0 ε 0 ε ε ε ε 0 ε ε ε θ (P) (P) (P) (P) (P) (degees) * b.9 Inese Inese Inese om 0 o.0% om 0 o. o 0.5% 0 T *.9. Ineses Inese 0 0 Inese 0 5 o 0.9 o 0.5% o 0.5% T T * Ineses o Ineses 5 ein o ein vlue vlue T T The polyshide mewok in he -ibe is ompleed 0. Deposiion o he mix subsne in he S lye ss 0 nd ends T. In he S lye deposiion o he mix subsne ss T nd ends T. Deposiion o he mix subsne in he -lye ss T nd ends T (Z) hen he -ibe is ompleed The vlue o E is 5. P (om Cse in Tble ) 80.4 P (Cse b) 07. P (Cse ) The vlue o S is 0.8 P (om Cse in Tble ) 0.5 P (Cse b) 0.7 P (Cse ) b Fig. 4. An exmple o he simulion on he elions beween he longiudinl Young s modulus nd he vlue o in he geen -ibe. Condiions o he yslliniy o he polyshide mewok in he -lye (see Tble ) solid line se ; dshed line se b; shded line se suion poin (u Z). Theeoe i is egded h he moisue onen hnge u is equivlen o he elpsed ime. In he inegl inevl om u 0 o u Z u-dependen pens o ε m ε m ε m ε ε ε nd evey omponen in p mus be given in dvne. In his se he moisue sopion in he ell wll uses he biomehnil se hnge in he -ibe. The shinkge sin o wood nno be egded s eipol o he swelling sin beuse wo dieen eeene bses e used in hei mesuemens. I he hyseesis ee beween shinkge nd swelling poesses is smll enough o be negleed we n onve he swellings α(u) nd β(u) ino shinkges α(u) nd β(u) ein moisue onen (u) by using he ollowing omule: α ( u) α ( Z) α( u) α( Z) ( Z) β( u) β( Z) β β ( u) () In he se h he oven-died shinkge o he wood beomes moe hn sevel peen in he nsvese dieion we should use Eqs. 9 when luling α(u) nd β(u) [o α(u) nd β(u)]. In he sme wy s in he se o gowh sess geneion we n esonbly expe h he longiudinl dying shinkge α (u) uz beomes onsidebly lge wih he hikness o he -lye nd wih he negive vlue o ε uz. Conluding emks In he subsequen epo uhe onee simulions will be demonsed nd he esuls will be omped wih he expeimenl esuls obined om he TW o 70-ye-old Kohuhiwkede (Ae sieboldinum iq.) whih e ledy epoed. Exmples inlude high ensile gowh sess geneion lge longiudinl Young s modulus nd lge xil shinkge due o we desopion in he -ibe.

11 07 Appendix Deiled expessions o oeiiens 55 ; b b 5 ; 5 ; d d 5 in Eqs A B C D E b H X b X Ωα α Λ Λ Γ α Λ Γ α α Λ F Γ αλ α Λ Γ α Σ αλ Γ b N I N d S Q N αλ Λ α Γ Γ p Q N b X b Λ Λ X b N d Q N 0 0 Γ Γ Q N b X b X b N Λ Λ d Q N Q Γ Γ Λ Λ N b X b X b N d Q N Q Γ Γ Λ Λ N b X b X b N d Povided h deiled shpes o unions A B C D E H I α nd β e s ollows: A Q B Q Ï Ì Ó Ω β α β α α C È Î Í Í Q N α

12 08 È D Í β α Î E ( β α( )) H I F β X Σ D β Q Reeenes. Okuym T Ymmoo H Iguhi Yoshid (990) eneion poess o gowh sesses in ell wlls. II. owh sess in ension wood. okuzi kkishi : Okuym T Ymmoo H Yoshid Hoi Y Ahe RR (994) owh sesses in ension wood. Role o mioibils nd ligniiion. Ann Fo Si 5:9 00. Ymmoo H Okuym T Sugiym K Yoshid (99) eneion poess o gowh sesses in ell wlls. IV. Aion o he ellulose mioibils upon he geneion o he ensile sesses. okuzi kkishi 8:07 4. Kollmnn FFP Coe J WA (984) Piniples o wood siene nd ehnology vol I: solid wood. Spinge Belin Heidelbeg New Yok 5. Boyd JD (977) Relionship beween ibe mophology nd shinkge o wood. Wood Si Tehnol :. Nobeg H eye H (9) Physil nd hemil popeies o he gelinous lye in ension wood ibes o spen (Populus emul.). Holzoshung : Suield (97) Reion wood. Is suue nd union. Siene 79: Ymmoo H Kojim Y Okuym T Absolo WP il J (00) Oigin o he biomehnil popeies o wood eled o he ine suue o he muli-lyeed ell wll. J Biomeh Eng Tns ASE 4: Ymmoo H Kojim Y (00) Popeies o ell wll onsiuens in elion o longiudinl elsiiy o wood. P. Fomulion o he longiudinl elsiiy o n isoled wood ibe. Wood Si Tehnol : Ymmoo H Okuym T Yoshid (995) eneion poess o gowh sesses in ell wlls. VI. Anlysis o he gowh sess geneion by using ell model hving hee lyes (S S nd I P). okuzi kkishi 4: 8. Ymmoo H (998) eneion mehnism o gowh sesses in wood ell wlls: oles o lignin deposiion nd ellulose mioibil duing ell wll muion. Wood Si Tehnol :7 8. Ymmoo H (999) A model o he nisoopi swelling nd shinking poess o wood. P. enelizion o Bbe s wood ibe model. Wood Si Tehnol : 5. Ymmoo H Sssus F Ninomiy il J (00) A model o he nisoopi swelling nd shinking poess o wood. P. A simulion o shinking wood. Wood Si Tehnol 5: Kojim Y Ymmoo H (00) Popeies o he ell wll onsiuens in elion o he longiudinl elsiiy o wood. P. Oigin o he moisue dependeny o he longiudinl elsiiy o wood. Wood Si Tehnol (in pess) 5. Ymmoo H Okuym T Yoshid (99) eneion poess o gowh sesses in ell wlls. V. odel o ensile sess geneion in gelinous ibes. okuzi kkishi 9:8 5. ing CY Bsse KH iness EA hessul RH (90) Ined spe o yslline polyshides. VII. Thin wood seions. TAPPI 4: Fushini (97) Sudy o moleul oienion in wood by luoesene mehod (in Jpnese). okuzi kkishi 9: Slmen (000) Suue popey elions o wood; om he ell wll polymei ngemen o he mosopi behvio. Poeedings o he hid pln biomehnis oneene Feibug- Bdenweile pp Bbe NF eyln BA (94) The nisoopi shinkge o wood. A heoeil model. Holzoshung 8: il J Sssus F Ymmoo H uid D (999) uion nd dying sin o wood in longiudinl dieion: single-ibe mehnil model. Poeedings o he hid wokshop Conneion beween silviulue nd wood quliy hough modelling ppohes nd simulion sowe. onde-es-ues 09 4