A micromechanical Approach to the Nonlinear Poroelastic...
Transcript of A micromechanical Approach to the Nonlinear Poroelastic...
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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
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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
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M1 Micro-mechanical modeling
of irreversible mechano-sorptive
mechanisms in wood
François GagnierETHZ Prof. Hans Herrmann
Dr. Falk Wittel
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M2 Multiscale poro-elastic model:
bridging the gap from cellular to macroscopic scale
Ahmad RafsanjaniEMPA Derome
Prof. Jan Carmeliet
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M3 Macroscopic model for wood including
nonlinear elasticity, moisture effects,
hysteresis, damage and time dependent effects
Martin DresslerEMPA Prof Jan Carmeliet
Derome
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E1 4D experimental investigation of
hygro-mechanical behavior of wood on cellular scale
Alessandra PateraEMPA Dr. Michele Griffa
Carmeliet
Derome
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E2 Multiscale experimental determination
and description of wood under
varying moisture and mechanical loading
Christian LanvermannETHZ – Prof Peter Niemz
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5 PhD students working in sinergy
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Multiscale model
hierarchy of submodels
description of coupled mechanical and moisture behavior
different spatial scales
interconnection of submodels
up-scaling - homogeneization
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Thermodynamical approach
• Energy state in isothermal case
• Legendre transform
J. Carmeliet, R. Guyer, D. Derome, 6th Plant Biomechanics Conference, 2009 11
),( u
puup ),(),(
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ddd BpMu
0d d dC B p
puup ),(),(
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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)
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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)
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BdMdpdu
0d d dC B p
puup ),(),(
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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
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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
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Wood microstructure
• Effective Medium
18
dpMdBdu
dpBdCd
rrrr
rrrr
dpMdBdU
dpBdCdE
homhom
homhom
uU
R
T
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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 ,
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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(
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Lumen: elliptic cylinder inclusion
21
Lumen dimensions
Radial
Tangential
RL
TL
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22W. Zillig, Moisture transport in wood using a multiscale approach, PhD Thesis, 2009
Spruce cross section
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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
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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
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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
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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
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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
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-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
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Visualisation of rays by segmentation of voxel values
Modeling
the
influence of
rays on
swelling
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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
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Anisotropic swelling
• Unit Cell Model including a row of ray cells
31
Stress Strain
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Computational homogenization
• High Resolution Finite Element Model selected form -CT images
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Computational homogenization• Displacement configurations related to six reference strain states
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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