ORIGINAL ARTICLE

J Wood Sci (2004) 50:197–208 © The Japan Wood Research Society 2004DOI 10.1007/s10086-003-0556-4

H. Yamamoto (*)School of Bioagricultural Sciences, Nagoya University, Chikusa,Nagoya 464-8601, JapanTel. 81-52-789-4152; Fax 81-52-789-4150e-mail: hiro@agr.nagoya-u.ac.jp

Hiroyuki Yamamoto

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

Received: August 28, 2002 / Accepted: May 6, 2003

Abstract To discuss the role of the gelatinous layer (G-layer) on the origins of the physical properties peculiar tothe tension wood fiber (TW fiber), the deformation processof an isolated TW fiber caused by a certain biomechanicalstate change was formulated mathematically. The mechani-cal model used in the present formulation is a four-layeredhollow cylinder having the compound middle lamella(CML), the outer layer of the secondary wall (S1) and itsmiddle layer (S2), and the G-layer (G) as an innermostlayer. In the formulation, the reinforced matrix mechanismwas applied to represent the mechanical interaction be-tween the cellulose microfibril (CMF) as a frameworkbundle and the amorphous substance as a matrix skeleton ineach layer. The model formulated in the present study isthought to be useful to investigate the origins of extensivelongitudinal drying shrinkage, large tensile growth stress,and a high axial elastic modulus, which are rheologicalproperties peculiar to the TW. In this article, the detailedprocess of the mathematical formulation is described. In asubsequent article, some TW properties from a 70-year-oldKohauchiwakaede (Acer sieboldianum Miq.) will be ana-lyzed using the newly developed model.

Key words Gelatinous fiber · Tension wood · Cell wall ·Reaction wood · Growth stress

Introduction

Tension wood (TW) consists of abnormal tissue calledgelatinous fiber (G-fiber) because it contains a gelatinouslayer (G-layer) as the innermost layer of the secondary wall.The TW often shows characteristic behavior that is different

from the normal wood (NW). A high-tensile growth stress isgenerated on the surface of the xylem in the TW region thatoften becomes ten times as large as that in the NW region.1,2

The longitudinal Young’s modulus of the TW becomes sig-nificantly higher than that of the NW.1,3 Furthermore, theaxial shrinkage in the TW tends to exceed more than 1%during water desorption,3 while that in the NW becomesless than 0.5%. Some authors attribute these behaviors tothe intrinsic properties of the G-fiber.1,3

On the other hand, more than a few researchers believethat the G-layer is mechanically too compliant to bear alarge stress generation. They base their argument on thefacts that the G-layer is often convoluted in the lumen of thetransverse section that is sampled from the water-swollenblock, and it can be easily peeled off the lignified layer in thesame direction during microtoming. This gives the impres-sion that it is attached only loosely to the remainder of thesecondary wall.4 From those observations, they considerthat the various characteristics of the TW should be attrib-uted not to the flexible G-layer but to the relatively thickerouter layer of the secondary wall (S1),5 and/or to the rela-tively thinner middle layer of the secondary wall (S2) whosemicrofibril angle (MFA) is expected to be more or lessdifferent from that in the NW fiber.6,7

It is impossible to adjudge which possibility holds untilwe succeed in directly measuring the mechanical propertiesof the G-layer and the lignified layer which requires isolat-ing them from each other. However, it may be quite naturalto consider that the lignified layer in the G-fiber would beessentially same as that in the NW fiber, because there is nospecific difference in anatomical and chemical aspects be-tween them as pointed by Okuyama et al.1 Thus, the authorexpects that characteristic behaviors peculiar to the TWmay be attributed to the G-layer.

Simulation using a wood fiber model is one of the mosteffective methods to estimate the internal properties andfine structures of each constituent material in wood cellwalls.8,9 In our previous studies, we developed a wood fibermodel consisting of the compound middle lamella (CML),S1 and S2. A basic formula was derived to simulate thedeformation process of an isolated wood fiber when a

198

certain biomechanical change occurs, such as lignin deposi-tion,10,11 water adsorption,12,13 or external load induction.9,14

The basic formula contained several parameters that werederived from the composite structure of the cell wall layer.When comparing the simulated results with the experimen-tal one, concrete values need to be assigned to the param-eters in a rational manner. We consider that those valuesreflect intrinsic information on the internal properties andfine structure of cell wall constituents.

The objective of the present study was to clarify the roleof the G-layer in the origin of characteristic behaviors of theTW, and then, to improve our previous wood fiber model12,15

into a four-layered model having the G-layer as the inner-most layer. In the present article, the detailed process ofthe formulation on the deformation process of the G-fiber model which is induced by a certain biomechanicalstate changes is described. In a subsequent article, someTW properties of 70-year-old kohauchiwakaede (Acersieboldianum Miq.) will be analyzed by using the newlydeveloped model, including a high-tensile growth stressgeneration, a large longitudinal Young’s modulus, and alarge axial shrinkage due to water desorption.

Formulation of the deforming wood

Wood fiber model

Figure 1 shows a schematic model of the typical G-fiberwhich consists of the CML, the S1, the S2, and the G-layers.Each layer can be approximated as a “two-phase structure”,specifically the unidirectional reinforcing element of thepolysaccharide framework and the encrusting matrix (MT)of lignin–hemicellulose compound. The former is mainlycomposed of highly crystallized cellulose microfibril(CMF), which is oriented in a certain direction to the fiberaxis in each layer except the CML. This makes the second-ary wall and the G-layer mechanically anisotropic. On theother hand, in cases in which the orientation of the CMF is

randomly distributed in the CML, it is considered to bemechanically isotropic.

In the case of softwood secondary wall, oriented polyose,such as (acetyl-) glucomannan, is often arranged along andaround the CMF, forming the polysaccharide frameworkwith the highly crystallized CMF. No oriented polyose hasbeen detected in the hardwood cell wall.16–18 Disorientedpolyose, that is mainly xylan, is blended with lignin, formingthe isotropic skeleton of the MT substance.

In the present study, a single G-fiber is simplified into acomplex hollow cylinder consisting of the CML, the S1, theS2, and the G-layers, as shown in Fig. 1b.

Constitutive equations

Under low magnification, the reinforcing element of thepolysaccharide framework is spatially dispersed uniformlyin each cell wall layer to form the framework bundle. Simi-larly, the lignin–hemicellulose compound is diffused in eachlayer forming the isotropic MT skeleton. Therefore, it isconsidered that both the framework bundle and the MTskeleton occupy the same domain in the macroscopic limit.Based on such an idea, Barber and Meylan hypothesizedthe mechanical interaction between the polysaccharideframework and the lignin–hemicellulose MT in each layeras the following conditions: 12,19,20

σ σ σ ε ε εij ijf

ij ij ij ij m f m, (1)

where σfij, σ

mij , and σij are the stress tensors in the framework

bundle, MT skeleton, and cell wall layer as a whole, respec-tively. εf

ij, εmij , and εij, on the other hand, are their respective

strains.In the present study, the deformation of the wood fiber

model is assumed to be symmetric with respect to the cen-tral axis (see the condition C4). Thus, we can express thefact that torsional deformation of an individual fiber is com-pletely restricted by the force of binding fibers inside thewood specimen. Then a cylindrical coordinate system (O-tr, where , t, and r represent longitudinal, tangential, andradial components, respectively) can be applied to thepresent model as shown in Fig. 2a.

A visible deformation induced at each point in the cellwall is expressed by the observable strain tensor compo-nents (εij, where i, j , t, r) in the O-tr coordinate system.The stress component induced in the MT skeleton of eachlayer (σm

ij ) is related to the observable strain componentcaused in the matrix skeleton (εm

ij , where i, j , t, r) as thefollowing constitutive equation:

σ ε αij ijkl kl klCm m m m ( ) (2)

Cmijkl is an elastic constant tensor, and αm

kl is a tensor ofinternal expansion which is caused by a certain biomechani-cal change. αm

kl is observed such that in each layer, αmkl

εmδkl, where δkl is Kronecker’s symbol. εm is a scalar. Cmijkl and

αmkl are both assumed to be isotropic. The elastic constants of

the isotropic MT skeleton (nonzero terms) are denoted asfollows:

Fig. 1. Multilayered structure of the gelatinous fiber (a), its mechanicalmodel (b), and the crosscut surface (c). Each fiber consists of thecompound middle lamella (CML), the outermost layer of the second-ary wall (S1), its middle layer (S2), and the gelatinous layer (G)

199

C C C K S

C C C C C C K S

C C C S

mttttm

rrrrm

ttm

ttrrm

ttm

rm

rrm

rrttm

trtrm

r rm

t tm

13

2

13

12

( )

( )

,

,

(3)

where K 3λ 2µ, S 2µ. λ and µ are Lame’s constants.Figure 2b shows a small flat-board element of the frame-

work bundle in the S2 layer, provided that the positivedirection of normal axis (z-axis) is coincident with the radialdirection (r-axis) of the fiber model. In this model, the CMFand other oriented polyose in the S2 layer are assumed to beoriented in an S-helix at an angle of θ, and the one in the S1layer is assumed to be aligned normally to the fiber axis.The relationship between stress (σf*

ij ) and strain (εf*ij) compo-

nents induced in the framework bundle of each layer of thesecondary wall can be written as the following constitutiveequation in the O-xyz orthogonal coordinate system:

σ ε αab abcd cd cdCf f f f* * * * ( ) (4)

where Cf*abcd is the elastic constant of the framework bundlein each layer in the O-xyz coordinate system. αab

f* is aninternal expansive strain, caused by a certain biomechanicalchange, that is a diagonal tensor whose components are

diag *f f f fα ε ε εab( ) ( )¢ ¢ , ,

where εf and εf are the internal expansive strains induced inthe framework bundle in the directions parallel and perpen-dicular to the cellulose molecular chain, respectively.

We supposed that the framework bundle is considerablycompliant in its transverse direction. Therefore, all shearmoduli, Poisson’s ratios, and the Young’s modulus in thetransverse direction are small enough to be neglected. Thismeans that the CMF including other oriented polysaccha-ride cannot be kept in a bundle shape without a reinforcing

agent. The reinforcing agent is considered to be the lignin–hemicellulose isotropic matrix. Then, we may consider thestiffness components Cf*abcd are all nil except Cf*xxxx (E).

By transforming the coordinate system from O-xyz intoO-tr system, Eq. 4 is rewritten into a new expression asfollows:

σ ε αij ijkl kl klCf f f f ( ) (5)

where

C R R R R C R Rijkl ia jb kc ld abcd ij ia jb abf f f f* * , α α (6)

Rij is a transformation matrix between both coordinate sys-tems. The nonzero terms of the stiffness components of theframework bundle (Cf

ijkl) are expressed as

C c E C C c s E C C c sE

C s E C C cs E C c s E

c s s c

ftt

fttf

tf

tf

ttttf

tt tf

tttf

t tf

f f fttf f f

4 2 2 3

4 3 2 2

2 2 2 2

, , ,

, , ,

, ,α ε ε α ε ε α¢ ¢tf f f ε ε ¢( )cs

(7)

provided that we considered Cfijkl Cf

jikl Cfjilk and Cf

ijkl Cf

klij, where c cos, s sin, and E is Young’s modulus ofthe framework bundle in the direction along the cellulosemolecular chain. E, S, εm, εf, r (rout/rin), and take respec-tive values in each layer as assumed in Table 1. They are notunknown values to be solved but known constants to begiven in advance.8,13 S is the shear modulus (2) of thematrix skeleton in each layer. In the same way as E, S takesa respective value in each layer. S is denoted as S0, S1, S2, andS3 in the CML, S1, S2, amd G-layers, respectively.

By rearranging Eqs. 1 using Eqs. 2, 3, 5, and 7, and con-sidering the compatibility of strains for the axisymmetricaldeformation without torsion,

ε εε

r tt r

ddr

we obtain

σ ε ε ε

εε

σ ε ε ε

K Ec K S Ec

K S Ec s K S rddr

K Es K S Ec s

K S Es

m f

tt

tm f

2 4

2 2

2 2 2

4

13

2

23

13

13

13

2

( )ÏÌÓ

¸˝˛

( )ÏÌÓ

¸˝˛

( )

( )ÏÌÓ

¸˝˛

( )ÏÌÓ

¸˝

,

˛( )

( ) ( ) ( )

( )

εε

σ ε ε εε

σ ε ε ε σ σ

tt

rm

tt

t tf

tr r

13

13

13

213

2

02 2

K S rddr

K K S K S K S rddr

Ecs c s

,

,

,

(8)

where σ σ, σt σtt, σr σrr, ε ε, εt εtt, εr εrr. r isa radial distance from the central axis. Among the shear

Fig. 2a,b. Coordinate system used in the formulation. a O-tr localorthogonal coordinate system. b Flat board element of the CMF frame-work bundle. O (in a) is an arbitrary point in the cell wall. Direction ofx-axis (in b) is parallel to the CMF molecular chains in the S1 or S2layers

200

stress components, only σt is not null, which is an inevitableconsequence from the assumption of the axisymmetricaldeformation, i.e., εt 0. In this study, stress equilibrium foraxisymmetrical deformation was assumed as follows:

σ σσ

t rr r

ddr

(9)

Deriving the basic equations

Assumptions

The dimensional change of the wood fiber model can beexpressed as a set of normal strains, namely, ε in the longi-tudinal direction, and εt(r)|rr0

, εt(r)|rr1, εt(r)|rr2

, εt(r)|rr3,

and εt(r)|rr4 in the tangential directions at the respective

radius. In the same way as described in previous work,11,12,14

based on Eqs. 8, we solve ε, εt(r)|rr0, εt(r)|rr1

, εt(r)|rr2,

εt(r)|rr3, and εt(r)|rr4

under the conditions C1, C2, C3, andC4, and assumptions A1, A2, and A3:

C1. ε is constant for all r. This is based on the assumptionthat the wood fiber model is an infinitely long cylinder inthe -direction. We denote ε as εL hereafter.

C2. [σr(r)]rr4 P4, [σr(r)]rr0

P0. P0 P4 0. Theseboundary conditions mean there are no internal and ex-ternal pressures acting on the wood fiber model (seeTable 1).

C3. The external force (PL) induced parallel to the woodfiber model satisfies the following equation:

P L dA rdrd r r drl l lr

r

L

crosscutsurface

crosscutsurface

2 24

0p pσ σ θ σÚ Ú Ú ( )

(10)

C4. The deformation of the wood fiber model is assumed tobe symmetric with respect to the central axis.

In addition to the conditions, the assumptions A1, A2,and A3 were made at processing:

A1. K S, so that S/K is enough small to be negligible ineach layer. This hypothesizes that the bulk modulus ofthe MT skeleton (K) is larger than S. This postulates

that the Poisson’s ratio of the matrix skeleton is almost0.5, similar to a kind of elastomer.

A2. It is considered that E, S, εm, εf, and r (rout/rin) tend tochange their values during a certain biophysical change;however, those changes can be neglected in the casethat biomechanical changes were infinitesimally small.

A3. E 0 in Eq. 7 in the CML. Namely, E0 0. This doesnot mean that there is no oriented polysaccharideframework in the CML, but means that mechanicalcontribution of the randomly distributed polysaccha-ride framework in the CML should be isotropic. Forconvenience, we assume that S0 means the shear modu-lus (2) of the CML itself. Moreover, (h/r1)

2 0, εf0

0 and ε0m 0 in the CML. These conditions assume that

h (thickness of the CML) is smaller than the S1 and theS2 layers.

Basic formula A

Solving the second and third formula in Eqs. 8 for εt andrdεt/dr, and eliminating the term of σt using the equilibriumcondition of Eq. 9, we obtain

2 2 3 1 2

2 1 2

3

2 2 3

2

4 4 2

2 2 2 2

4 4 4

S Es u S Es S u E s

S Ec s u S Ec s u rddr

u

S Es u S Es rddr

E s

u

( )[ ] ◊ ( ) ◊ ◊

( )[ ] ( )

( )[ ] ◊ ◊

ε ε ε

εσ

σ

εε

tm f

Lr

r

t m

,

(( ) ◊ ◊ ( ) ◊ ◊

( )

E s s u E s

u rddr

Gus

2 2 2

4

2 3

2 3

ε ε

σσ

fL

rr

(11)

where u S/K and G E/S. Moreover, combining them byeliminating the term of εt, we obtain a differential equationof r:

Table 1. List of the parameters in the basic formulae A and B

Layer rin rout r S E Pin Pout L εm εf

CML – r1 r0 (r1 h) r00

1

rr

ÊËÁ

ˆ¯ S0 E0 (0) P1 P0 (0) L0 0 0

S1 90deg r2 r1r1

1

2

rr

ÊËÁ

ˆ¯ S1 E1 P2 P1 L1 ε1

m ε1f

S2 θ r3 r2r2

2

3

rr

ÊËÁ

ˆ¯ S2 E2 P3 P2 L2 ε2

m ε2f

G 0deg r4 r3r3

3

4

rr

ÊËÁ

ˆ¯ S3 E3 P4 (0) P3 L3 ε3

m ε3f

, the microfibril angle; rin, inner radius of each layer; rout, outer radius of each layer; r rout/rin; S, shear modulus (2) of the matrix skeleton;E, Young’s modulus of the framework bundle in the direction along the cellulose molecular chain; Pin, inner pressure; Pout, outer pressure; L, seeEqs. 10; εm, inelastic strains in the matrix skeleton; εf, inelastic strains of the framework bundle in the direction parallel to cellulose molecularchains; CML, compound middle lamella; G E/S; G1 E1/S1; G2 E2/S2; G3 E3/S3; Q F/E1; F S0·h/r1; M S2/S1; N S3/S2

201

1 2 3 1 2 3

3 2 2 3

22

24

4 2 2 2

u rddr

u rddr

Gus

E s u E s E s s u

( ) ( )◊ ( ) ◊ ◊ ( )

σ σσ

ε ε ε

r rr

m fL

(12)

General solution of Eq. 12 is given as

σα

ε

ε ε

α α

α α

α

rm

fL

C r C ru

E s

u E s E s s u

r

11

21 4

2 2 2

1

11 2

12

3

2 2 3

11

11

◊ ◊[( ) ◊ ◊ ( ) ]

ÊËÁ

ˆ¯

È

ÎÍÍ

˘

˚˙˙

(13)

where C1 and C2 are integral constants, and α is described asfollows:

α

1 31 2

1 32

4 4G su

uG s

SK S

◊ ◊

Based on the assumption A1, we impose the followingconditions:

u then 1Æ Æ 0 0α

which yields

limα

α

αÆÆ

1 0

1 11

rr ln

Then, we can simplify the solution of Eq. 13 as follows:

σ ε ε

ε

rm f

L

3

ln

¢ ◊ ◊[( ) ◊ ]

C C r E s E s

s E s r

1 22 4 2

2 2

12

2

2 3

Under the boundary conditions [σr(r)]rrin Pin and

[σr(r)]rrout Pout, we can decide the integral constants C1

and C2, and we obtain the following solution:

σ ε

ε ε ε

ε ε

r inout2 2

out inm

fL

m

fL

in

ln

ln

Pr r

P P E s

E s E s s E s

E s E s srr

rr

r

2

24

2 2 2 4

2 2 2

112

3

2 2 312

3

2 2 3

◊[ÏÌÓ

◊ ◊ ( ) ] ◊[◊ ◊ ( ) ]

(14)

Then, we substitute the above solution into the first part ofEq. 11, and we obtain basic formula A:

Γ ε Λ ε X ε ∆ ε◊ ◊ ◊ ◊ ( )ÊËÁ

ˆ¯

( )ÊËÁ

ˆ¯

L tm f out

2out

out2

in

21

1

21

1

2 2

2 2

r

r

rr

PS

rr

PS

where

Γ

Λ X

∆

112

2 31

2 332

31

21

4 2 22 2

4 4 42 2

22 2

G s G s srr

G s G s G srr

G srr

◊ ◊ ( ) ( )ÊËÁ

ˆ¯

◊ ◊ ◊ ( )ÊËÁ

ˆ¯

◊ ( )ÊËÁ

ˆ¯

ln

ln

andln

out2

out2

out2

rr

rr

rr

,

, ,

Coefficients Γ, Λ, X, and ∆ are dependent on G, , r, rout,and r, provided that these variables take their respectivevalues in each layer as shown in Table 2.

Basic formula B

We solve Eqs. 8 for εL and integrate it over the crosscut areaof each layer. When integrating it, we assume that εm and εf

are independent of r in the respective layers, and ε takes aconstant value (εL) over the crosscut surface of the cell wallbecause of condition C1. As a result, we obtain basic for-mula B:

Ω ε ε Σ εΛ

Γ Γ

◊ ◊ ◊◊ ( )

( ) [ ] ( ) [ ]

Lm f

in2

out in

S Sout in

Fr

rr r

23 1

23 1

23

1

1

2

2

2 2

LS r

P Pr r r r

where

Ω

123

1 3 3

13

2 31

14

2 4

2 4 2 2

2

2

G s s

G s s

( )

◊ ( ) ÊËÁ

ˆ¯

È

ÎÍÍ

˘

˚˙˙

( )

rr

rln ,

Table 2. List of the coefficients in the basic formulae A and B

Layer Location Γ Λ X ∆ Ω F Σ

CML r r0 Γ0 (1) Λ0 (2) X0 (3) ∆0 (0) Ω0 (1) F0 (1) Σ0(0)r r1 Γ0 (1) Λ0 (2) X0 (3) ∆0 (0)

S1 r r1 Γ1 Λ1 X1 ∆1 Ω1 F1 Σ1

r r2 Γ1 Λ1 X1 ∆1

S2 r r2 Γ2 Λ2 X2 ∆2 Ω2 F2 Σ2

r r3 Γ2 Λ2 X2 ∆2

G r r3 Γ3 (1) Λ3 (2) X3 (3) ∆3 (0) Ω3 F3 (1) Σ3

r r4 Γ3 (1) Λ3 (2) X3 (3) ∆3 (0)

202

1

2 31

14

2 2 2

2 6 22

2

G s c s

G s s

◊ ( )◊ ( ) Ê

ËÁˆ¯

È

ÎÍÍ

˘

˚˙˙

F

rr

rln ,,

,Σ

σ θ

13

2 323

2 31

14

12

2 2 4 22

2

G s G s s

L r r

( ) ◊ ( ) ÊËÁ

ˆ¯

È

ÎÍÍ

˘

˚˙˙

Ú

rr

r

p

ln

d deach layer

Coefficients Ω, F, and Σ are dependent on G, , and r,provided that these coefficients take their respective valuesin each layer as shown in Table 2.

Formulae to describe the deforming G-fiber

In the G-fiber model as the whole, twelve equations areobtained based on the basic formulae A and B. The equa-tions are as follows:

Basic formula A in CML at r r0 (a0),Basic formula A in CML at r r1 (a0),Basic formula B in CML (b0),Basic formula A in S1 at r r1 (a1),Basic formula A in S1 at r r2 (a1),Basic formula B in S1 (b1),Basic formula A in S2 at r r2 (a2),Basic formula A in S2 at r r3 (a2),Basic formula B in S2 (b2),Basic formula A in G at r r3 (a3),Basic formula A in G at r r4 (a3),Basic formula B in G (b3).

These equations constitute simultaneous equations whoseunknown variables are

ε ε ε ε ε εL t t t t t

1 2 3 4 0 1 2 3

, , , , , ;

, , , , ; , , ,r r r r r r r r r r

P P P P P L L L L 0 1 2 3 4

0

According to the assumption A3, Eqs. a0 and a0 are degen-erated by each other, which yields

ε εt tr r

r r0 1ª

Therefore, the unknown variables become

ε ε ε ε εL t t t t

1 2 3 4 0 1 2 3

, , , , ;

, , , , ; , , ,r r r r r r r r

P P P P P L L L L 1 2 3 4

0

To solve Eqs. a0–a3, and Eqs. b1–b3 for the unknownvariables, the following conditions based on conditions C2and C3 are imposed:

P P L L L L L0 4 0 1 2 30 1 0 2 c c c3( ) ( ) ( ), ,

The unknown variables explicitly required in our study areεL, εt|rr1

, εt|rr2, εt|rr3

, and εt|rr4. Thus, by eliminating the

unknown variables P0, P1, P2, P3, P4, L0, L1, L2, L3, and L inthe following manner, we degenerated the simultaneous

equations into simpler ones in which the unknown variablesare εL, εt|rr1

, εt|rr2, εt|rr3

, and εt|rr4.

The first equation

From Eqs. a0, b0, a1, b1, a2, b2, b3, c1, c2, and c3, weeliminate P0, P1, P2, P3, P4, L0, L1, L2, L3, and L. Thus, weobtain the first equation

a a a a a

b b b c c c d P

r r r r r r r r11 12 13 14 15

11 1 12 2 13 3 11 1 12 2 13 3 11

1 2 3 4ε ε ε ε ε

ε ε ε ε ε ε

L t t t t

m m m f f fL

(15)

where coefficients a11, a12, a13, a14, a15, b11, b12, b13, c11, c12, c13,and d11 are respective functions whose concrete forms arecomposed of θ, r1, r2, r3, Q(F/E1), G1, G2, G3, M(S2/S1),and N(S3/S2). Detailed shapes of those coefficients aredescribed in the Appendix. PL stands for the external forceinduced parallel to the wood fiber model, which is related toL0, L1, L2, and L3 as follows:

P L L L L L rdrd

rdrr

r

L

crosscutsurface

2 2

2

0 1 2 3

4

0

p p

p

( ) Ú

Ú

σ θ

σ

The second equation

εL, εt|rr1, εt|rr2

, εt|rr3, and εt|rr4

are unknown variables to besolved as solutions of an algebraic equation (Eq. 15). Tosolve Eq. 15 for εL, εt|rr1

, εt|rr2, εt|rr3

, and εt|rr4, there must

be at least four equations that are constituted by the sameunknown variables. These equations can be derived frombasic formula A.

From Eqs. a0, a1, a2, a3, c1, and c2, we eliminate P0, P1,P2, P3, and P4. Thus, we obtain

a a a a a

b b b c c c d P

r r r r r r r r21 22 23 24 25

21 1 22 2 33 3 21 1 22 2 23 3 21

1 2 3 4ε ε ε ε ε

ε ε ε ε ε ε

L t t t t

m m m f f fL

(16)

The third equation

From Eqs. a0, a1, a2, a3, c1, and c2, we eliminate P0, P1, P2,P3, and P4. Thus, we obtain

a a a a a

b b b c c c d P

r r r r r r r r31 32 33 34 35

31 1 32 2 33 3 31 1 32 2 33 3 31

1 2 3 4ε ε ε ε ε

ε ε ε ε ε ε

L t t t t

m m m f f fL

(17)

The fourth equation

From Eqs. a0, a1, a2, a3, c1, and c2, we eliminate P0, P1, P2,P3, and P4. Thus, we obtain

a a a a a

b b b c c c d P

r r r r r r r r41 42 43 44 45

41 1 42 2 43 3 41 1 42 2 43 3 41

1 2 3 4ε ε ε ε ε

ε ε ε ε ε ε

L t t t t

m m m f f fL

(18)

203

d f d f d f d f d

f d f d f dP

ε ε ε ε ε

ε ε

Lm m m f

f fL

11 1 12 2 13 3 14 1

15 2 16 3 17

p p p p

p p p

( ) ( ) ( ) ( )( ) ( ) ( ) (20)

The Eqs. 20 are divided by dt, and are converted into thefollowing differential equations:

˙ ˙ ˙ ˙ ˙ ˙

˙ ˙ε ε ε ε ε ε

ε

Lm m m f f

fL

f f f f f

f f P

11 1 12 2 13 3 14 1 15 2

16 3 17

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

˙ ˙ ˙ ˙ ˙ ˙

˙ ˙ε ε ε ε ε ε

ε

tm m m f f

fL

r f f f f f

f f P

121 1 22 2 23 3 24 1 25 2

26 3 27

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

˙ ˙ ˙ ˙ ˙ ˙

˙ ˙ε ε ε ε ε ε

ε

tm m m f f

fL

r f f f f f

f f P

231 1 32 2 33 3 34 1 35 2

36 3 37

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

˙ ˙ ˙ ˙ ˙ ˙

˙ ˙ε ε ε ε ε ε

ε

tm m m f f

fL

r f f f f f

f f P

341 1 42 2 43 3 44 1 45 2

46 3 47

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

˙ ˙ ˙ ˙ ˙ ˙

˙ ˙ε ε ε ε ε ε

ε

tm m m f f

fL

r f f f f f

f f P

451 1 52 2 53 3 54 1 55 2

56 3 57

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

(21)

where the raised dot notation represents the derivative by t.Furthermore, r1, r2, and r3 among the components of pshould be rewritten into the differential quantities dr1, dr2,and dr3, because they depend on the elapsed time during acertain biomechanical change. Due to the assumption ofaxisymmetrical deformation, the following relations are in-quired among respective layers:

r r r

r r r

r r r

1 11 1

2 21

2 22 2

3 32

3 33 3

4 43

11

11

11

1

2

2

3

3

4

dr drr dr

dd

dr drr dr

dd

dr drr dr

dd

r

r

r

r

r

r

ε

ε

ε

ε

ε

ε

t

t

t

t

t

t

,

, (22)

Because dr3dεtr4, dr2dεt

r3, and dr1dεtr2 are regarded as higher

order infinitesimal quantities, from Eqs. 22, we can obtainthe following differential forms:

d d d d d d

d d d

r r r r

r r

r r r r

r r

1 1 2 2

3 3

1 2 2 3

3 4

ε ε ε ε

ε ε

t t t t

t t

( ) ( )( )

, ,(22)

These equations can be rewritten as differential equationsof t as follows:

˙ ˙ ˙ , ˙ ˙ ˙ , ˙ ˙ ˙r r r r r r1 1 2 2 3 31 2 2 3 3 4 ε ε ε ε ε εt t t t t t

r r r r r r( ) ( ) ( )(23)

Equations 21 and 23 constitute a system of simultaneousdifferential equations whose unknown functions are εL, εt

r1,εt

r2, εtr3, εt

r4, r1, r2, and r3. On the other hand, ε1m, ε2

m, ε3m, ε1

f, ε2f,

ε3f, and PL are the functions whose t-dependent shapes

should be given in advance.

The fifth equation

From Eqs. a0, a1, a2, a3, c1, and c2, we eliminate P0, P1,P2, P3, and P4. Thus, we obtain

a a a a a

b b b c c c d P

r r r r r r r r51 52 53 54 55

51 1 52 2 53 3 51 1 52 2 53 3 51

1 2 3 4ε ε ε ε ε

ε ε ε ε ε ε

L t t t t

m m m f f fL

(19)

where coefficients a21–a55, b21–b53, c11–c53, and d11–d51 are re-spective functions whose concrete forms are described inthe Appendix.

Equations 15–19 constitute the simultaneous algebraicequations whose unknown variables are εL, εt

r1(εt|rr1),

εtr2(εt|rr2

), εtr3(εt|rr3

), and εtr4(εt|rr4

). The values of ε1m, ε2

m,ε3

m, ε1f, εf

2, εf3, and PL should be given in advance. From the

simultaneous equations, the solutions should be expressedin the following forms:

ε ε ε ε ε ε

ε

Lm m m f f

fL

f f f f f

f f P

11 1 12 2 13 3 14 1 15 2

16 3 17

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

ε ε ε ε ε ε

ε

tm m m f f

fL

r f f f f f

f f P

121 1 22 2 23 3 24 1 25 2

26 3 27

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

ε ε ε ε ε ε

ε

tm m m f f

fL

r f f f f f

f f P

231 1 32 2 33 3 34 1 35 2

36 3 37

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

ε ε ε ε ε ε

ε

tm m m f f

fL

r f f f f f

f f P

341 1 42 2 43 3 44 1 45 2

46 3 47

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

ε ε ε ε ε ε

ε

tm m m f f

fL

r f f f f f

f f P

451 1 52 2 53 3 54 1 55 2

56 3 57

p p p p p

p p

( ) ( ) ( ) ( ) ( )( ) ( )

(20)

where coefficients f11–f57 are functions of p, and p is a param-eter vector whose components are θ, r1, r2, r3, Q, G1, G2, G3,M, and N. Among the solutions, εL represents the strain ofthe G-fiber model in the axial direction. According tothe condition C4, the wood fiber model deformsaxisymmetrically so that εt

r1(εtr0), εt

r2, εtr3, and εt

r4 are equiva-lent strains to the diametral deformations at their respectiveradii. These strains are induced by a certain biomechanicalchange in the G-fiber. The thickness of the CML is smallenough to be negligible if compared to those of the S1 andS2 layers. Therefore, εt

r1 can be regarded as the strain of thediameter in the wood fiber model (εT).

Developing Eqs. 20 into differential equations

According to assumption A2, Eqs. 20 are valid under thecondition that the biomechanical change is infinitesimallysmall enough to be neglected. This means that εL, εt

r1, εtr2, εt

r3,and εt

r4 in addition to ε1m, ε2

m, ε3m, ε1

f, ε2f , ε3

f, and PL in Eqs. 20should be replaced as differential quantities, i.e., dεL, dεt

r1,dεt

r2, dεtr3, dεt

r4, dε1m, dε2

m, dε3m, dε1

f, dε2f, dε3

f, and dPL. Then, eachexpression in Eqs. 20 should be rewritten as a simple differ-ential form, e.g.,

204

Solving the simultaneous differential equations (Eqs. 21and 23)

Solutions

Equations 21 and 23 give the deformation of the wood fibermodel, which is induced by a certain biomechanical changeoccurring in the cell wall. In our case, examples of biome-chanical change are the growth strains (maturation strains)during cell wall lignification, which is measured as the re-leased strains of the growth stress; the swelling strains dueto moisture adsorption; and the elastic deformation causedby external load.

We assume that change in the physical state in the cellwall starts at t 0 and ends at t Z. By integrating Eqs. 21and 23 from t 0 to t Z, we can solve them for εL, εt

r1, εtr2,

εtr3, εt

r4, r1, r2, and r3. As the initial conditions at t 0, weassume

ε ε ε ε εL t t t t0 0 0 0 0 0

0 0 0

1 2 3 4

1 1 2 2 3 3

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

r r r r ,

ˆ , ˆ , ˆr r r r r r (24)

Then, we divide the integral interval into n small parts, anddenote the integrals of Eqs. 21 in the i-th small interval[(i 1)Z/n t iZ/n, i 1, 2, . . . , n] as ∆iεL, ∆iεt

r1,∆iεt

r2, ∆iεtr3, and ∆iεt

r4. For example, ∆iεL is calculated as

∆ εε

ε ε ε ε

ε ε

i

i Z n

i Z n

i Z n

i Z n

ddt

dt

f f f f

f f f P dt

LL

m m m f

f fL

∫

( ) ( ) ( ) ( )(

( ) ( ) ( ) )

( )

◊

( )

◊

Ú

Ú

1

11 1 12 2 13 3 14 1

1

15 2 16 3 17

p p p p

p p p

˙ ˙ ˙ ˙

˙ ˙ ˙

(25)

In each small integral interval, the functions ε1m(t), ε2

m(t),ε3

m(t), ε1f(t), ε2

f(t), ε3f(t), and respective components in p, in-

cluding r1, r2, and r3, must be given in advance. However,r1, r2, and r3 are unknown functions to be solved fromsimultaneous differential Eqs. 21 and 23. The values of r1,r2, and r3, which should be used in the i-th small interval,can be estimated in the following process. At first, we inte-grate Eqs. 23 in the (i 1)-th small interval. Then, weobtain

ddt

dtddt

ddt

t dti Z n

i Z ni

r r

i Z n

i Z nr r r11

t t1

2

11

2

11 2

( )

( )

( )

( )

Ú Ú∫( ) ÊËÁ

ˆ¯

◊ ( )ÏÌÔ

ÓÔ

¸˝ÔÔ

∆ε ε

If n is taken as a large enough number, this equation gives

∆ ∆ ε ∆ εi i r i r i 1 1 11

21 2r r1 t t( ) ◊ (26)

where r1i2 is equal to r1(t) at t (i 2)Z/n. Thus, the

value of r1i1, which is r1(t) in the i-th small interval, is given

as

r r r r11

12 1

11 1

121 1 2i i i i r i r i ∆ ∆ ε ∆ εt t( ) ◊ (27)

By using simultaneous recurrence equations (Eqs. 25 and27), we can integrate simultaneous differential Eqs. 21 and23 numerically.

Thus, the natural strain of the deforming wood fibermodel at a certain time t ( j · Z/n, j is positive integersmaller than n ) can be derived as follows:

ε ∆ εε

ε ∆ εε

L LL

T tt

lim

lim

tddt

dt

tddt

dt

n

i

i

j t

n

i r

i

j rt

( ) ∫

( ) ∫

Æ•

Æ•

 Ú

 Ú

1 0

1 0

1

1

,

(28)

Strains εL(t) and εT(t) give natural strains of the dimensionalchanges induced in the wood fiber model. However, thereleased strain of the growth stress and the swelling due towater sorption are measured as the nominal strains in thedeforming wood specimen. Then, we need to solve Eqs. 21and 23 to give anisotropic dimensional changes of the woodfiber model as nominal strains.

The nominal strain of the deforming wood fiber model ata certain time t ( j · Z/n) in respective directions can bederived as follows:

α ∆ ε

∆ ε

t lim

t lim

L

t

( ) ( )È

ÎÍ

˘

˚˙

( ) ( )È

ÎÍ

˘

˚˙

Æ•

Æ•

’

’

n

i

i

j

n

i r

i

j

1 1

1 1

1

1

1

,

(29)

If the values of α(Z) and (Z) are smaller than 1%, changesof r1, r2, and r3 are small enough to be neglected. In thatcase, we can obtain the α(t) and (t) from the first and thesecond formulae of Eqs. 21, which simplifies Eqs. 29 asfollows:

α ∆ εε

∆ εε

tddt

dt tddt

dtn

i

i

j t

n

i r

i

j rt

( ) @ ( ) @Æ• Æ•Â Ú Â Ú lim limL

Lt

t

1 0 1 0

1

1

,

(30)

Examples of simulations

Generation of the maturation strains due to cellwall lignification

Soon after the deposition of the CMF and the highly ori-ented polysaccharide framework in the secondary wall ofthe tracheid or libriform wood fiber, the matrix substance ofhemicellulose and lignin deposits among the gaps of theframework bundle. In this process, the wood fiber tends toswell or shrink anisotropically, which generates anisotropicgrowth stress in the newly formed xylem. This is becausefree deformation of the individual fiber is restricted to in-side the actual xylem. This process was formulated by usinga wood fiber model of the CML, S1, S2, and G layers. In thiscase, irreversible deposition of the MT substance and thematuration of the CMF framework are postulated to causea change in the biomechanical state of the tracheid or libri-form fiber wall.

In the present study, the free dimensional change of theG-fiber is simulated during the wall maturation. PL shouldbe null in Eqs. 21. The integral interval is from the deposi-

205

tions of the CMF framework (t 0) to the completion ofthe G-fiber wall (t Z). In this interval, the t-dependentpatterns of ε1

m, ε2m, ε3

m, ε1f, ε2

f, ε3f, and every component in p

should be given in advance. As the initial conditions at t 0, we adopt Eq. 24. According to the observations, thereleased strains of the growth stress are less than 0.5% bothin the and the t directions. Then, the simplified formulae(Eqs. 30) can be used instead of Eqs. 29 when simulating thematuration strains α(t)|tZ and (t)|tZ. In such a case, r1(t),r2(t), and r3(t) are regarded as approximately constant.

As the preliminary simulation, we calculated the valuesof α(t)|tZ (εL) and (t)|tZ (εT) under the conditions ofthe parameters assumed in Tables 3 and 4. The values inTables 3 and 4, except those related to the G-layer, wereused in our previous simulations on the generation ofgrowth strains in the latewood tracheid of sugi (Cryptom-eria japonica).8 In the present simulation, the various casesfor the percentage of the cellulose crystal in the G-layer aredisplayed in Table 3, and the relationship between εL, εT

and ε3f(t)|tZ was calculated for each case of framework

crystallinity.The result is shown in Fig. 3. According to the observa-

tion in the normal fiber of the softwood or the hardwoodxylem, the released strain of the surface growth stress be-comes 0.02% to 0.06% in the longitudinal direction, and0.05%–0.1% in the tangential direction. In the TW xylem,on the other hand, the longitudinal released strain oftenbecomes several times as large as in the NW xylem, and insome species forming considerably thick G-layers, such asRobinia pseudoacacia, it exceeds 0.5%.2 The presentsimulation explains these phenomena well when the G-layer is assumed to shrink considerably in its axial directionduring the G-fiber maturation.

Longitudinal elastic constants

At present, the longitudinal Young’s modulus (EL) of thewood fiber model is calculated based on Eqs. 21 and 23under the assumption of the moisture steady state. There-fore, it is assumed that every component in p is constant.Moreover, dε1

m, dε2m, dε3

m, dε1f, dε2

f, and dε3f are all nil. Then,

from Eqs. 21, we obtain the longitudinal Young’s modulusof the G-fiber (EL) as follows:

Er

dPd r fL

L

L

1 1 1

02

02

17p pε p( ) (31)

The substantial Young’s modulus of the G-fiber can becalculated as

Er r

dPd r r fL

w

1 1 1

02

42

02

42

17p p( ) ( ) ( )L

Lε p(31)

In this case, an increase in the strain energy induced byexternal load is regarded as the biomechanical state change.

As a preliminary simulation, the values of EL at t T3

were calculated under the conditions of the parametersgiven in Tables 3 and 4. The result is shown in Fig. 4. In thecase in which the G-layer contains a certain amount ofcrystalline cellulose, calculated values of EL become largerwith the thickness of the G-layer; however, it is almostconstant when the G-layer contains no crystalline cellulose.

Swelling and shrinkage strains due to the moistureadsorption

In the same way as the growth stress case, free dimensionalchange of the G-fiber due to moisture adsorption weresimulated. Thus, PL should be null in Eqs. 21. The integralinterval is from the oven-dried state (u 0) to the fiber

Table 3. Assumed values for the chemical compositions in each layer of the green G-fiber wall

Layer Cellulose crystal Oriented polyose Isotropic matrix Crystallinity in the(non-crystalline) in the polysaccharidepoly-saccharide framework framework

CML 15 (%) 0 (%) 85 (%) 100 (%)S1 15 5 80 75S2 30 10 60 75G

Case a 40 40 20 50Case b 60 20 20 75Case c 80 0 20 100

Fig. 3. An example of the simulation on the relations betweenthe growth strains and the values of ε3

f at t Z ( T3) in the G-fiber.Values of the parameters assumed here are displayed in Table 4. Con-ditions for the crystallinity of the polysaccharide framework in the G-layer (see Table 3), solid lines, case a; dashed lines, case b; shaded lines,case c

206

Tab

le 4

.A

ssum

ed v

alue

s of

the

par

amet

ers

for

the

sim

ulat

ion

of t

he g

row

th s

trai

ns o

f th

e G

-fibe

r

E0

E1

E2

E3

S 0S 1

S 2S 3

r 0r 1

r 2r 3

ε 0mε 1m

ε 2mε 3m

ε 0fε 1f

ε 2fε 3f

θ(G

Pa)

(GP

a)(G

Pa)

(GP

a)(G

Pa)

(deg

rees

)

t 0

T

1a0

20.1

40.2

*b3.

92In

crea

se0

01.

025

1.1

1.5

1.5

0In

crea

se0

00

Incr

ease

00

15fr

om 0

to 1

.0%

from

0to

1.1

3to

0.

15%

t T

1

T2a

020

.140

.2*

3.92

1.13

Incr

ease

s0

1.02

51.

11.

51.

50

–In

crea

se0

0–

Incr

ease

015

to 0

.933

to 0

.5%

to

0.15

%t

T2

T

3a0

20.1

40.2

*3.

921.

130.

933

Incr

ease

s to

1.02

51.

11.

51.

50

––

00

––

Incr

ease

s15

a ce

rtai

nto

a c

erta

inva

luec

valu

e

aT

he p

olys

acch

arid

e fr

amew

ork

in t

he G

-fibe

r is

com

plet

ed a

t t

0. D

epos

itio

n of

the

mat

rix

subs

tanc

e in

the

S1

laye

r st

arts

at

t 0

, and

end

s at

t

T1.

In t

he S

2 la

yer,

dep

osit

ion

of t

he m

atri

x su

bsta

nce

star

ts a

t t

T1,

and

ends

at

t T

2. D

epos

itio

n of

the

mat

rix

subs

tanc

e in

the

G-l

ayer

sta

rts

at t

T

2, an

d en

ds a

t t

T3

(Z

), t

hen

the

G-fi

ber

is c

ompl

eted

bT

he v

alue

of

E3

is 5

3.6

GP

a (f

rom

Cas

e a

in T

able

3),

80.

4G

Pa

(Cas

e b)

, 107

.2G

Pa

(Cas

e c)

cT

he v

alue

of

S 3 is

0.8

GP

a (f

rom

Cas

e a

in T

able

3),

0.5

33G

Pa

(Cas

e b)

, 0.2

67G

Pa

(Cas

e c)

saturation point (u Z). Therefore, it is regarded that themoisture content change u is equivalent to the elapsed timet. In the integral interval from u 0 to u Z, u-dependentpatterns of ε1

m, ε2m, ε3

m, ε1f, ε2

f, ε3f, and every component in p

must be given in advance. In this case, the moisture sorptionin the cell wall causes the biomechanical state change in theG-fiber.

The shrinkage strain of wood cannot be regarded as areciprocal of the swelling strain because two different refer-ence bases are used in their measurements. If the hysteresiseffect between shrinkage and swelling processes is smallenough to be neglected, we can convert the swellings α(u)and (u) into shrinkages α(u) and (u) at a certain mois-ture content (u) by using the following formulae:

¢( ) ( ) ( )( ) ¢( ) ( ) ( )

( )αα α

α

u

Z u

Zu

Z u

Z

1 1, (32)

In the case that the oven-dried shrinkage of the wood be-comes more than several percent in the transverse direc-tion, we should use Eqs. 29 when calculating α(u) and (u)[or α(u) and (u)].

In the same way as in the case of growth stress genera-tion, we can reasonably expect that the longitudinal dryingshrinkage αL(u)|uZ becomes considerably larger with thethickness of the G-layer and with the negative value of ε3

f|uZ.

Concluding remarks

In the subsequent report, further concrete simulations willbe demonstrated, and the results will be compared with theexperimental results obtained from the TW of a 70-year-oldKohauchiwakaede (Acer sieboldianum Miq.), which arealready reported.3 Examples include high tensile growthstress generation, large longitudinal Young’s modulus, andlarge axial shrinkage due to water desorption in the G-fiber.

Fig. 4. An example of the simulation on the relations between thelongitudinal Young’s modulus and the value of r3 in the green G-fiber.Conditions for the crystallinity of the polysaccharide framework in theG-layer (see Table 3), solid line, case a; dashed line, case b; shaded line,case c

207

Appendix

Detailed expressions of coefficients a11–a55; b11–b53; c11–c53; d11–d51 in Eqs. 15–19.

a A B C a D a E a a

b HXM

bX

11 2 12 13 1422 2 2 2 15

111

2 222

12

12 12 2

2

22 2

16

2 0

62

11

1 16

2

Ω αα

Λ Λ Γ

αΛ Γ α α Λ

, , , , ,

,

r

rr

rF

r

( )

( )ÊËÁ

ˆ¯ÊËÁ

ˆ¯

◊ ΓΓ αΛ

α∆Λ Γ α Σ

∆α Λ Γ

2 13 222

32

32

111

2 222

12

12 12 2

2

22 2 2 13

12

11

1

62

11

1 16

2

( ) ÊËÁ

ˆ¯ÊËÁ

ˆ¯

( )ÊËÁ

ˆ¯ÊËÁ

ˆ¯

◊ ( )

, ,

, ,,

b N

c IM

c c

rr

r

rr

r r

113

11

1

23 1

1 12

12

12

1 12

222

32

32 3

112 2

2

22

1 22 21 1

12

12 1

22

22 2

N G

dM S r

a G QM

MN

αΛ

Λα Γ Γ

rr

r

rr p

rr

rr

ÊËÁ

ˆ¯ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

,

, rr

rr

rr

r rr

rr

32

22 1

2312

12 1 24

22

22 2 3

225 21

12

12 1 22

22

22

1 2

12

12

11 0

12

12

1

( )ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

( ) ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

, ,

, , , ,

a G Q

a aM

MN a b X bM

Λ Λ ˜

( ) ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

( ) ¢

X

b MN c cM

c d

a G QM

MN

2

23 32

2112

12 1 22

22

22 2 23 21

31 112

12 1 2

22

32

112

12

10 0

12

12

112

,

, , , , ,

r rr

rr

rr

r

∆ ∆

Γ Γ rr

rr

r r rr

r

r

32

32 1

3312

12 1 2

22 34 3

235 31

12

12 1 32 2

22

33 32

1 2

12

12

1 1 012

12

1

32

( )ÊËÁ

ˆ¯

( ) ( ) ÊËÁ

ˆ¯

( ) ¢

, ,

, , , , ,

a G Q

aM

a MN a b X bM

X

b MN

Λ Λ

112

12

1 0 0

12

12

112

1 212

1

3112

12 1 32 2

22 33 31

41 1 12

1 22

2 32

42 1 12

( ) ÊËÁ

ˆ¯

( ) ¢

( ) ¢ ( ) ¢ ( ) ( )

, , , , ,

,

c cM

c d

a G QM

MN a G Q

rr

r

r r r r

∆ ∆

Γ Γ ΛΛ Λ

∆

∆

1 43 22

2

44 32

45 41 12

1 42 22

2 43 32

41 12

1

42 22

2

21

1 012

12

132

112

1

21

, ,

, , , , , ,

,

aM

a MN a b X bM

X b MN c

cM

r

r r r r r

r

( )( ) ( ) ¢ ( ) ¢ ( ) ( ) ¢

( ) ¢ cc d a G QM

MN

a G Q a aM

43 41 51 112

12 1

22

22 2

32

32

52 1 5312

12 1 54

22

0 012

12

1 12

1

212

12

1

, , ,

, ,

rr

rr

rr

rr

rr

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

Γ Γ

Λ222 2 55

32

32 51

12

12 1

5222

22 2 53

32

32 51

12

12

1 12

1

21 3

21 1

21

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

Λ

∆

, , ,

, ,

a MN b X

bM

X b MN c

rr

rr

rr

rr

rr 11 52

22

22 2 53 512

10 0, , ,c

Mc d

rr

ÊËÁ

ˆ¯

∆

Provided that detailed shapes of functions A, B, C, D, E, H, I, α, and are as follows:

AM

G Q

BM

G Q

11

123

2

13 1

21

2

1

22

22 1

21 1

21 1 1

2 22

22 1

12

12 1

¢ÊËÁ

ˆ¯ÊËÁ

ˆ¯

( ) ¢( )ÏÌÓ

¸˝˛

¢ÊËÁ

ˆ¯ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

GG

rr

r r G L

G rr

rr

G

Ω

α,¢¢Ê

ËÁˆ¯ÊËÁ

ˆ¯

¢ÊËÁ

ˆ¯ÊËÁ

ˆ¯

( ) ÊËÁ

ˆ¯ÊËÁ

ˆ¯ÊËÁ

ˆ¯

ÊË

GG

LL

GG

LL

L G Gr

Gr

rr

2

1

1

2

2

1

1

2

22

22

1

22

12

12

1 1

216

16

11

1 13

1

, , α

α

C

M M2 ÁÁ ˆ¯ÊËÁ

ˆ¯

È

ÎÍÍ

˘

˚˙˙

◊ ◊ÊËÁ

ˆ¯ÊËÁ

ˆ¯ÊËÁ

ˆ¯

11

12

11

11

23

22 1

222

32

32 3

r

Lr

rr

G Q

N Gα ,

208 Ë ¯ Ë ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯ ¢ ¢2

31

11

22

22 2

22

1

rr G L G

DM

11

112

2 2 1

1

22

12

12 2 2

22 2

2 22

22

2

16

11

12 2

1

Gr G L

Lr

rr

G r L G

G rr

ÊËÁ

ˆ¯

( )È

ÎÍ

˘

˚˙

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

¢ ( )( )¢Ê

ËÁˆ¯

ÊËÁ

α

α

G Q

EM

HM

,

,

ˆ¯

ÊËÁ

ˆ¯

◊ÊËÁ

ˆ¯

¢ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

◊ÊËÁ

ˆ¯

rr

FG

r

G rr

rr G

r D

12

12

1

112

1

2 22

22

12

12

1

112

1

1 13

11 1

3

Σ

X

IM

,

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