Workshop “Wood Structure/Function Relationships”

October 5-8, 2010, Hamburg, Germany

A Micromechanical Approach to the Nonlinear Poroelastic Behavior of Wood

in a Multi-scale Frawework

Ahmad Rafsanjani, Dominique Derome, Jan Carmeliet

EMPA Wood Laboratory, Multiscale Modeling Group

October 2010

Objective To investigate the relation between

macroscopic mechanical and moisture properties of wood

and its nano- and microscopic structure

using a multiscale approach,

thus to predict the

coupled moisture and mechanical behavior of wood

M1 Micro-mechanical modeling

of irreversible mechano-sorptive

mechanisms in wood

François GagnierETHZ Prof. Hans Herrmann

Dr. Falk Wittel

M2 Multiscale poro-elastic model:

bridging the gap from cellular to macroscopic scale

Ahmad RafsanjaniEMPA Derome

Prof. Jan Carmeliet

M3 Macroscopic model for wood including

nonlinear elasticity, moisture effects,

hysteresis, damage and time dependent effects

Martin DresslerEMPA Prof Jan Carmeliet

Derome

E1 4D experimental investigation of

hygro-mechanical behavior of wood on cellular scale

Alessandra PateraEMPA Dr. Michele Griffa

Carmeliet

Derome

E2 Multiscale experimental determination

and description of wood under

varying moisture and mechanical loading

Christian LanvermannETHZ – Prof Peter Niemz

5 PhD students working in sinergy

Multiscale model

hierarchy of submodels

description of coupled mechanical and moisture behavior

different spatial scales

interconnection of submodels

up-scaling - homogeneization

Thermodynamical approach

• Energy state in isothermal case

• Legendre transform

J. Carmeliet, R. Guyer, D. Derome, 6th Plant Biomechanics Conference, 2009 11

),( u

puup ),(),(

ddd BpMu

0d d dC B p

puup ),(),(

Empa, , 135/16

results

poroelastic modelling of wood - COST meeting October 2010

0.0 0.1 0.2 0.3

0

200

400

600

800

1000

1200

12000

14000

0.0 0.1 0.2 0.3

0

200

400

600

800

1000

ER

EL

ET

mo

du

lus o

f e

lasticity [M

Pa

]

moisture content [1]

data from Neuhaus

calculated values

ELR

ETL

ETR

moisture content [1]

F.-H. Neuhaus, “Elastizitätszahlen von Fichtenholz in Abhängigkeit von der Holzfeuchtigkeit” (elasticity numbers of spruce as a function of wood

moisture content), PhD thesis (in German), Institut für konstruktiven Ingenieurbau Ruhr-Universität Bochum (1981)

Empa, , 146/16

results

poroelastic modelling of wood - COST meeting October 2010

0.0 0.1 0.2 0.3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

LT

TL

Po

isso

n's

ra

tio

[1

]

moisture content [1]

RT

TR

moisture content [1]

LR

RL

moisture content [1]

F.-H. Neuhaus, “Elastizitätszahlen von Fichtenholz in Abhängigkeit von der Holzfeuchtigkeit” (elasticity numbers of spruce as a function of wood

moisture content), PhD thesis (in German), Institut für konstruktiven Ingenieurbau Ruhr-Universität Bochum (1981)

BdMdpdu

0d d dC B p

puup ),(),(

Thermodynamical approach

• Isothermal state of energy

• Material Properties

J. Carmeliet, R. Guyer, D. Derome, 6th Plant Biomechanics Conference, 2009 16

2

11103212

2

12

1)()(),,())1(2(

2

1),( IBIBpgpfIIINII

Ep

s

2

222 ),(,

),(,

),(

p

pM

p

pB

pC

ij

ij

klij

ijkl

Linear ElasticityNonlinearElasticity

Fluid Fluid-Solid Interactions

Incremental state equation

• Isothermal Poroelasticity

J. Carmeliet, R. Guyer, D. Derome, 6th Plant Biomechanics Conference, 2009 17

MdpBddu

BdpCdd

)2

)(()(

))((

1)]([

2

111000

1100

2

01

IBIBpMpMM

IBBpMB

N

EpuB

EC

ijij

klij

jlik

s

klij

s

ijkl

Wood microstructure

• Effective Medium

18

dpMdBdu

dpBdCd

rrrr

rrrr

dpMdBdU

dpBdCdE

homhom

homhom

uU

R

T

Micromechanics

• Strain and stress concentration tensor

• Eshelby’s solution for ellipsoidal inclusion

• Mori-Tanaka and Self-Consistent Scheme

A. Zaoui, Continuum Micromechanics: survey, J. Engineering Mechanics ASCE , 128 (2002) 19

rrrr E B A rr ,

111 )](:[:)](:[

IK:KSIIK:KSIA r

-1

0

Esh

rr

-1

0

Esh

rr

rr

hom AKAKK :: r

rf

hom

0

matrix

0 KKKK ,

Nonlinear micromechanics

• Elastic Behavior

• Linear Comparison Composites (LCC)

P. Ponte Castaneda, Linear Comparison Methods for Nonlinear Composites, 2004 20

:)K(

)A(:)K(K

K

),K()K(

hom

hom

rr

rr

r

E

E

:

)(

)B(:)C(C

C

),C()C(

hom

hom

rr

rr

r

E

:

)(

:)C(

Lumen: elliptic cylinder inclusion

21

Lumen dimensions

Radial

Tangential

RL

TL

22W. Zillig, Moisture transport in wood using a multiscale approach, PhD Thesis, 2009

Spruce cross section

Assumptions

• Elliptic Cylinder(a3→)

• Linear Elasticity (N(I1,I2,I3)=0)

• Equilibrium moisture content

• Isotropic Matrix

• Mori Tanaka Scheme

J. Qu, M. Cherkaoui, Fundamental of Micromechanics of Solids, 2006 23

)()()( ppCC ijklijkl

Esh

r

Esh

r SCS

Geometry and properties

• Earlywood Cell wall material properties

• Latewood

24

8.0

30

50

mL

mL

T

R

4.0

30

15

mL

mL

T

R

4.0,7000 MPaEs

Sorption

25

• Sorption Isotherm

0

5

10

15

20

25

30

0 0,2 0,4 0,6 0,8 1

mo

istu

reco

nte

nt(%

)

relative humidity

Free swelling test

26

• Evolution of elastic properties with moisture

0 5 10 15 20 25 30 35

500

1000

1500

2000

2500

3000

Moisture Content

Ela

stic

Modulu

sM

Pa

latewood

TE

latewood

RE

earlywood

REearlywood

TE

Swelling test: compression

27

• Sorption Isotherm: Compression -5MPa

0

5

10

15

20

25

30

0 0,2 0,4 0,6 0,8 1

mo

istu

reco

nte

nt(%

)

relative humidity

Earlywood

Latewood

-2

-1

0

1

2

3

0 20 40 60 80 100

RH [%]

rad

ial str

ain

[%

]

adsorption 25-85

desorption

adsorption 10-25

latewood

-2

-1

0

1

2

3

0 20 40 60 80 100

RH [%]

tan

ge

ntia

l str

ain

[%

]

adsorption 25-85

desorption

adsorption 10-25

latewood

-2

-1

0

1

2

3

0 20 40 60 80 100

RH [%]

tan

ge

ntia

l str

ain

[%

]

adsorption 25-85

desorption

adsorption 10-25

earlywood

-2

-1

0

1

2

3

0 20 40 60 80 100

RH [%]

rad

ial str

ain

[%

]

adsorption 25-85

desorption

adsorption 10-25

earlywood

Visualisation of rays by segmentation of voxel values

Modeling

the

influence of

rays on

swelling

Anisotropic swelling

• Restraining effect of the rays

30

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0 0,1 0,2 0,3 0,4

Sw

elli

ng

Str

ain

Moisture Content(%)

Longitudinal

Tangential

Radial

Anisotropic swelling

• Unit Cell Model including a row of ray cells

31

Stress Strain

Computational homogenization

• High Resolution Finite Element Model selected form -CT images

32

Computational homogenization• Displacement configurations related to six reference strain states

33

Summary, too early for conclusions

• Multiscale approach:

– coupled behavior (mechanical and moisture)

– appropriate modeling at different scales

– upscaling with rich information embedding

• components/material properties

• microstructure/macrostructure

• Towards study of influence of morphological and environmental effects

– for now, at steady state

34