Mechanics of Materials 79 (2014) 73–84
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Mechanics of Materials
journal homepage: www.elsevier.com/locate/mechmat
Numerical prediction of the stiffness and strength of medium
density fiberboards
Janis Sliseris a,c,⇑, Heiko Andrä a, Matthias Kabel a, Brigitte Dix b, Burkhard Plinke b,
Oliver Wirjadi a, Girts Frolovs c
a
Fraunhofer Institute for Industrial Mathematics (ITWM), Germany
Fraunhofer Institute for Wood Research (WKI), Germany
c
Riga Technical University (RTU), Latvia
b
a r t i c l e
i n f o
Article history:
Received 21 May 2014
Received in revised form 15 August 2014
Available online 16 September 2014
Keywords:
Wood-based panels
MDF
Numerical analysis
Experimental testing
a b s t r a c t
A numerical two scale method for the prediction of tensile and bending stiffness and
strength of medium density fiberboards (MDF) is proposed with the aim to study the fiber
orientation influence on mechanical properties of MDF. The method requires less experimental data to optimize MDF and to improve industrial manufacturing technology of
MDF. A new method for computing orientation tensors of the compressed fiber network
is proposed. First, the virtual microstructure is generated by simulations of a fiber laydown and a subsequent compression to obtain the necessary density. The density profile,
fiber length, thickness, and orientation are used for the microstructure generation, which
are obtained from lCT images and image analysis tools. Then a new damage model for
the wood fiber cell walls and joints is introduced. The microstructural problem is formulated as a Lippmann–Schwinger type equation in elasticity and solved by using Fast Fourier
Transformation (FFT). The macroscopic three point bending test is simulated with hexahedral finite elements and analytical methods based on Euler–Bernoulli theory. The difference between bending strength and stiffness numerically obtained and corresponding
experimentally measured values is less than 10%. This study lays a foundation for the optimal design of MDF fiber structures and the optimization of industrial manufacturing processes. The first results show an increase of up to 60% for bending stiffness in the case of
strongly oriented fibers.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Medium density fiberboards (MDF) are widely used for
furniture and in flooring. The mechanical and physical
properties requirements have significantly increased in
the last two decades. The manufacturing process of MDF
consists of log chipping, chip washing, thermo-mechanical
⇑ Corresponding author at: Fraunhofer Platz 1, Kaiserslautern 67663,
Germany. Tel.: +49 631316004738.
E-mail address: [email protected] (J. Sliseris).
http://dx.doi.org/10.1016/j.mechmat.2014.08.005
0167-6636/Ó 2014 Elsevier Ltd. All rights reserved.
pulping, defibrating, spraying of resin and wax, drying, mat
forming, hot pressing, and sawing.
Precise and fast simulation techniques are necessary to
increase understanding the deformation- fracture mechanisms and the influence of the wood fiber properties and
distribution on stiffness of the MDF. This knowledge can
be used to optimize the MDF in terms of maximal strength
and stiffness. MDF consists of a wood fiber network in
which the fibers are joined together at several contact
points. The fiber joins as well as the resin have a large
influence on the strength and stiffness (Eichhorn and
Young, 2003). Numerical investigations show that it is
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J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
important to include the effects of elasto-plasticity, failure
of the bond and frictional sliding of the fibers in the case of
large deformations (Liu et al., 2011; Wilbrink et al., 2013).
An approach to model cellulose fibers is proposed by
Heyden (2000). However, in this work the fibers are modeled by beam finite elements that do not consider the
deformation of the cross section (Wriggers and Zavarise,
1997). The direct simulation of 3D fiber network deformation and failure (under different humidity conditions)
using beam elements is proposed in Kulachenko and
Uesaka (2012). The crack growth in a cellulose fiber network is studied using implicit gradient nonlocal theory in
Isaksson et al. (2012). In addition, the multi-body interaction effect can be modeled by using continuum approach
(Zemerli, 2014).
Wood fiber consists of lignin, cellulose, hemicellulose
and extractives. The fiber diameter varies from 20 lm to
50 lm depending on the species and type of wood (hardwood or softwood). Multivariate statistical methods like
genetic algorithms can be used to predict the internal bond
strength of MDF (Andre et al., 2008). The cell wall of wood
fibers consists of a layered structure. However, the
strength and stiffness is mainly governed by the secondary
cell wall (S2) layer (Deng et al., 2012). The layered structure is formed from cellulose microfibrills with a diameter
of 4 nm. The thickness of the S2 layer is about 85% of the
total cell wall thickness and the microfibril angle (with
respect to the fiber axis) in the S2 layer varies from 10°
to 30° (Persson, 2000). To approximate the microfibril
angle in the S2 layer the log-normal probability distribution is suitable.
MDF is produced mainly with urea–formaldehyde resins
(partly modified with melamine) and to a small part with
PMDI (polymeric diisocyanate). These resins are duroplastics (thermosetting resins). The complexity of the wood
fiber network leads to complex nonlinear material behavior
(Sliseris and Rocens, 2010, 2013a,b). To calculate the
strength of MDF, the nonlinearity of the material has to be
considered. A non-uniform density profile leads to nonuniform strain–stress fields even for a simple tensile test.
The experimental investigation of material behavior for
complex strain–stress fields is time and resource consuming. To take the material complexity into account the two
scale simulation method can be used effectively, where,
instead of solving the constitutive laws at each macroscopic
point, microstructural homogenization is performed.
The computational homogenization of microstructures
is used to predict the strength and stiffness of highly heterogeneous materials such as masonry structures (Yuen and
Kuang, 2013), sandwich plates with Poisons locking effect
(Helfen and Diebels, 2014), laminates (Li et al., 2014) and
nano-composites (Liu et al., 2011). The swelling of natural
wood can be more precisely analyzed using a two scale
approach (Rafsanjani et al., 2013). To compute the cracks
and damage of composites, the FE2 method can be used
(Feyel and Chaboche, 2000; Nguyen et al., 2011; Visrolia
and Meo, 2013). However, this method is computationally
very intensive, therefore the speed can be increased by
using an analytical approach at the micro scale (Salviato
et al., 2013) or model reduction techniques (Somer et al.,
2014).
To solve the microstructural homogenization problem,
a so-called representative volume element (RVE) has to
be generated in advance. However, current knowledge for
generating RVEs of the MDF microstructure is not satisfactory. We propose an approach to generate RVEs using measured experimentally measured parameters, e.g. fiber
length, diameter and orientation. The initial fiber network
is generated by simulating the fiber lay-down process and
the resulting structure is then virtually compressed (see
Fig. 1).
The compression of the ligno-cellulose fiber network
leads to a nonlinear contact problem due to the fact that
new contacts appear between fibers. Further, finite strains
and material nonlinearity have to be taken into account.
The typical manufacturing process of MDF consists of compressing fiber mats at a high temperature for a certain
pressing time between 190 °C and 220 °C for between
4 s/mm and 8 s/mm of the board thickness, depending on
glue and other factors. Therefore, a high temperature gradient exists in the mat during the hot pressing. During
which the curing of the glue occurs. The friction coefficient
at the beginning of the compression is very small (perfect
sliding) and after the curing of the glue it significantly
increases. However, the change of the friction coefficient
is not clear and experimentally difficult to investigate.
A new approach for modeling microstructure of MDF is
proposed, where the microstructure is discretized as a
voxel (cuboid or 3d pixel) image instead of typically-used
beam finite elements. We propose to solve the contact
problem by transferring it into a relaxed formulation
where the voids between fibers are replaced by a material
with a relatively small stiffness in comparison to the fibers
(ratio is up to 105 ).
Fig. 1. Overall scheme of simulation.
J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
The mixed-phase voxel geometry is introduced to analyze the microstructure more precisely (Merkert, 2013).
The mixed-phase voxel means that a voxel is a mixture
of solid and void phase.
Finally, an improved two scale framework for simulating MDF is first time proposed. The advantage of our
method over previous methods is the use of a database
generated with microstructures that have been previously
analyzed. The database is reused in the next iteration or
load steps. At the microstructure level a new wood fiber
damage model with three damage variables is proposed.
The paper is organized as follows. The generation of
microstructure is presented in Section 2.2.1. The simulation method of strength and stiffness is presented in Section 2.2.2. The homogenization technique is presented in
Section 2.2.3. The wood fiber material damage model is
presented in Section 2.2.4. Finally, the results of microstructure generation, simulation of tensile and bending
test and simulation of optimized fiber structure are presented in Section 3.
2. Materials and methods
2.1. Materials
Experiments were carried out using MDF plates produced with target density 650 kg/m3, target thickness of
12 mm and size of 300 300 mm. The fiber structure is
created of defibrated pine fibers and fiber bundles. The
average fiber diameter and length is 35 and 100 lm,
respectively. The fiber network is formed by adding 9%
UF-resin and compressing at temperature 200 °C for
144 s. Moisture content of board was in range 7.5–8%.
The main parameters of the MDF are shown in Table 1.
Specimens for bending and tensile test were prepared
according to standard EN 310 with size of 290 50 12
mm and 50 50 12 mm, respectively.
The CT images of cylindrical samples with 12 mm diameter and 6 mm height were created. The lCT images with a
voxel length of 4 lm with the following X-ray parameters:
voltage 6 kV, current 360 lA and 0.1Cu filter were produced.
2.2. Computational methods
2.2.1. Construction of a virtual fiber network
The RVE is a small part of the MDF that represents the
behavior of the microstructure (see Fig. 2). Instead of computing the MDF with a resolved microstructure one can
Table 1
Technical parameters of manufactured MDF.
a
Parameter
Value
Wood species
Glue
Glue contenta
Paraffin contenta
Press temperature
Pressing time
Size of board
Pine
UF-resin KauritÒ337
9%
1.5%
200 °C
12 s/mm
500 500 mm
The content is calculated based on dry fibers.
75
Fig. 2. Macro and micro structures.
compute an RVE. The size of the RVE should be large
enough to represent the actual behavior of microstructure.
The RVE’s are constructed using experimental data of
fiber length, diameter and orientation. The overall scheme
of the main steps in generating the RVE is shown in Fig. 1.
Fibers are separated from a 2D image using appropriate
image processing algorithms (Schirp et al., 2014; Schmid
and Schmid, 2006) and a probability distribution of the
fiber length and diameter is obtained. The average orientation tensor of the fiber network is obtained from lCT
images of compressed MDF samples. Previous studies
(Sliseris, 2013) have indicated that it is necessary to create
the uncompressed voxel image of 106 voxels. To obtain the
orientation tensor of lCT image, the second order derivative matrix (Hessian) of the smoothed (with Gaussian kernel) image is computed (Redenbach et al., 2012; Wirjadi
et al.). The eigenvector corresponding to the smallest
eigenvalue of the Hessian matrix is interpreted as the local
direction. The orientation tensor of the fibers is obtained
by averaging the local directions over small sub-volumes,
in which only foreground (fiber) voxels are considered.
This method gives reasonable results in the places without
fiber contacts.
The stochastic fiber network is generated using a fiber
lay-down process that is governed by the Fokker–Plank
equation (Götz et al., 2007; Grothaus and Klar, 2008). The
commercial software GeoDict (2014) was used to perform
this. An elliptic hollow fiber cross section is used with a
thickness of the fiber cell wall of 4 lm. The ratio of the
largest and the smallest fiber diameter is assumed to be
2. The curved shape of cellulose fibers is obtained by
assembling several straight segments.
Simulations indicate that the average density of the
fiber network generated by the lay-down process is up to
450 kg/m3. To get the fiber network with the higher density, the compression of the fiber network is simulated.
Due to the compression the orientation angles (direction of normal vector n) of the fibers change (see Fig. 3)
and fiber orientation in the plane orthogonal to the direction of compression increases.
The orientation angles of the fibers are updated after
compression. Note that since the material is transversally
isotropic, we have to use only two Euler angles here.
The updated fiber orientation angles are obtained by
minimizing the sum of distances between each voxel and
the orientation vector of the fiber that goes through the
geometrical center x of a fiber segment. The distance dk
between voxel xk (with center coordinates xk ) and the
center line of the fiber segment is computed by the following equation:
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J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
Fig. 3. Deformation scheme of a fiber segment X0i .
dk ¼ kx xk þ ntk:
ð1Þ
The parameter t is obtained by minimizing the distance
dk ¼ dk ðtÞ, i.e.
t ¼ ðx xk Þ n:
ð2Þ
The components of the fiber orientation unit vector
n ¼ ðnx ; ny ; nz Þ are computed by solving a minimization
problem over all voxels N i in the domain Xi :
min
n
Ni
X
dk ðnÞ:
ð3Þ
k¼1
The necessary conditions for minimum of dk ðnÞ requires
that values of first order derivatives is zero
rn
Ni
X
dk ðnÞ ¼ 0:
ð4Þ
k¼1
This leads to a system of three nonlinear algebraic equations, which is solved numerically using the Newton–
Raphson iteration method.
The orientation angles of a multi-phase voxel that consists of the solid volume fraction of two or more differently
orientated fibers are computed using the weighted averaging (arithmetic averaging) technique.
As a result, the RVE is discretized in voxels with a
matching solid volume fraction (SVF), fiber length and
orientation.
2.2.2. Two scale simulation methodology
The stiffness for macroscopic problem can be obtained by
computing the effective stiffness of the microstructures. The
analytical approach for a wood fiber network can not be
used in case of high stress. The stiffness of MDF depends
on the strain–stress state and this leads to the two scale
(micro–macro) simulation (Souza et al., 2011; Spahn et al.,
2014a,b). The scheme of the proposed coupled simulation
is shown in Fig. 4. To perform the macroscopic simulation
Abaqus/standard finite element code (Abaqus, 2014) is
used. To reduce the number of simulations at the microscale,
a database of strain and the appropriate stiffness tensor are
created for each microstructure. The database is updated
after each simulation at the microscale. The material stiffness tensor at each integration point is obtained by searching for a similar strain tensor in the database. If it is not
found then a microscale simulation with a specified macroscopic strain tensor is performed. The simulation at the
microscale level is performed by using software FeelMath
(2014).
An Abaqus UMAT subroutine is used to interface with
the micro simulation and the database. A special Python
script is called from the Abaqus UMAT subroutine. This
script checks the database for an appropriate stiffness
tensor. If it is not available then a simulation of microstructure (solution of problem (9)) is performed to
compute the stiffness. The script calls six loadcases Ei for
a prescribed macroscopic strain load E
E ¼ E0 ¼ ðE11 ; E22 ; . . . ; E12 Þ;
E1 ¼ ðE11 þ DE; E22 ; . . . ; E12 Þ;
...
ð5Þ
E6 ¼ ðE11 ; E22 ; . . . ; E12 þ DEÞ:
The Python script computes the stiffness tensor when all
loadcases are successfully executed. The fiber constitutive
law is implemented in another UMAT file which is called
by FeelMath at each solid or multi-phase voxel.
If such a strain tensor Ei (index i indicates the number
in database) exists that is ‘‘similar’’ to given macroscopic
strain tensor E then the stiffness tensor is taken from the
database. In this context ‘‘similar’’ means that the Euclidean norm of the two strain tensors is smaller than the macroscopic strain increment
kE Ei k 6 DE:
ð6Þ
The average stress tensors are obtained at each load
(macroscopic strain) case:
hrij ðEk Þi ¼
1
V
Z
rij ðEk ; xÞdx; k ¼ 0; 1; . . . ; 6:
ð7Þ
V
Finally, we obtain the stiffness matrix (using the principle
of strain equivalence) in a simple way (due to fact that
strain difference vector contains only one non-zero
element):
J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
77
Fig. 4. Framework of two scale simulation.
2
Dr111
6
1 6 Dr122
C¼
6
DE 4 . . .
Dr112
Dr211
Dr222
...
Dr212
. . . Dr611
3
. . . Dr 7
7
7;
...
... 5
6
22
ð8Þ
. . . Dr612
where Drkij ¼ hrij ðEk Þi hrij ðE0 Þi; k ¼ 1; 2; . . . ; 6.
We use this methodology of 6 different loadcases
because it allows us to compute all loadcases in parallel,
and so the computational speed is significantly increased.
The strain increment should be chosen small enough to
obtain the stiffness at the current strain state. However, if
it is too small then the stress difference is also very small
and this may lead to inaccurate results. Current simulations show that DE ¼ 104 105 gives the best results.
To perform an uncoupled two scale simulation, microstructures with different density and typical strain tensors
are computed in advance, and the stress–strain curves are
used in the macro computations. The stress–strain curves
are computed using typically observed macroscopic strain
tensors for linear elastic analysis.
2.2.3. Nonlinear homogenization problem
The nonlinearity in homogenization of MDF results from
the nonlinear wood fiber behavior. Wood fiber is a brittle
material in short time tension loads. Typically the damage
occurs at the fiber joints or at fiber. A new continuum damage model is introduced to capture the nonlinear behavior.
The homogenization is performed using small strain theory
(strains does not exceed 3%). The computation of the
stress–strain fields of the RVE leads to a boundary value
problem in elasticity. It is solved using Fast Fourier Transformation (FFT) method.
Solution of the elasticity problem. The microstructure of
the fiber network is discretized using voxels. The initial
stiffness tensor C and strength of each voxel are specified.
The homogenization problem on a cuboid x 2 R3 with
periodic displacement and anti-periodic stress field boundary conditions on boundary @ x is solved (9). The macroscopic strain field E is applied to the structure.
div rðxÞ ¼ 0; x 2 x;
rðxÞ ¼ Cððe; d; kÞðxÞÞ : eðxÞ; x 2 x;
eðxÞ ¼ E þ
1
ru ðxÞ þ ðru ðxÞÞT ; x 2 x;
2
u ðxÞ periodic; x 2 @ x;
ð9Þ
rðxÞ nðxÞ antiperiodic; x 2 @ x:
Instead of stress field rðxÞ, the stress polarization field sðxÞ
is introduced
sðxÞ ¼ ðCðe; d; kÞðxÞ C0 Þ : eðxÞ:
ð10Þ
The problem (9) can be reformulated into an equivalent integral equation, the so called Lippmann–Schwinger equation
eðxÞ ¼ E ðC0 sÞðxÞ;
ð11Þ
78
J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
where C0 denotes the Green operator. The convolution
operator ⁄ is defined by
C0 sðxÞ ¼
Z
C0 ðx yÞ : sðyÞdy:
ð12Þ
X
The Lippmann–Schwinger equation is solved using the FFT
method Moulinec and Suquet, 1998.
The rate of convergence depends on the contrast in
stiffness of the phases. The voids are considered as a material with a significantly smaller stiffnesses comparing to
the fiber stiffness. In simulations presented here we set
the void stiffness to be 105 times smaller than fiber material. For linear elastic material the stopping criteria
tol ¼ 105 is usually reached after 200 to 500 iterations.
However, in the case of nonlinear material behavior the
maximal value of the damage variables is limited to 0.95.
This restriction avoids the occurrence of differences
between the phase stiffness being too large.
2.2.4. Constitutive law
In order to simulate the material behavior of a wood
fiber, an appropriate constitutive law has to be defined.
The wood fiber is assumed to be a transverse isotropic
material. The glue is assumed to be an isotropic material
that partly penetrates into the fiber cell wall (about 50% of
glue is penetrated in cell wall Xing et al., 2005). The thickness of the glue layer is relatively small compared to the
thickness of the wood cell wall. Different material properties are specified for the fiber joint and the fiber voxels. However, experimental results for the mechanical properties of
fiber joints are not available. Therefore, the same material
properties are used for joint and fiber voxels. Due to the penetration of the glue in a joint of two fibers, a different
strength and stiffness are used in the joint voxels.
Experimental investigations (Isaksson et al., 2012;
Matsumoto and Nairn, 2009) of cellulose fiber networks
show that damage can occur due to axial stress, shear
stress or when transversal stress of a fiber exceed critical
value. The critical stresses have been obtained experimentally by Joffe et al. (2009) and Yoshihara (2012). This motivates us to use three damage criteria.
For the fiber cell wall, the following axial failure criterion is used (the local coordinate system is shown in Fig. 5):
r11 ðxÞ < X A;i :
ð13Þ
The transversal normal stress criterion of fiber cell wall is
used:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r22 ðxÞ2 þ r33 ðxÞ2 < X T;i k1 ðxÞ:
Fig. 5. Locale coordinate system of fiber network.
Table 2
Mechanical properties of wood fiber and joints.
Name of property
Valueb
Reduction
coefficienta
Young’s modulus E1 (Persson, 2000)
Young’s modulus E2 (Persson, 2000)
Shear modulus G23 (Persson, 2000)
Poisson’s ratio v 12 (Persson, 2000)
Poisson’s ratio v 13 (Persson, 2000)
Damage Hardening modulus Hi
Initial damage threshold Y i;0
Axial strength X A;i (Joffe et al., 2009)
Transversal strength X T;i
Shear strength X S;i
Density of wood fiber (Isaksson et al.,
2012)
Density of UF glue (Osemeahon and
Barminas, 2007)
50 GPa
3 GPa
3 GPa
0.3
0.3
100 MPa
0 MPa
500 MPa
40 MPa
40 MPa
1500 kg/
m3
1500 kg/
m3
–
0.01
0.01
–
–
–
–
–
0.025
0.05
–
a
The reduction coefficient is taken into account in the axial tensile test.
All properties without reference are obtained by fitting experimental
results.
b
di ðx; eÞ ¼ 1 eHi ðnY 0;i Þ ;
2
1
where X A;i ; X T;i ; X S;i are axial, tangential and shear
strength, index i ¼ 1 denotes a non-fiber-joint region,
i ¼ 2 denotes a fiber joint region, k1 ðxÞ and k2 ðxÞ strength
reduction coefficients due to existence of residual stress
in fiber. Values of critical stress are given in Table 2.
If the strength criteria are not satisfied, then the damage variables di ðxÞ are computed using an exponential
law Spahn et al., 2014a:
i ¼ 1; 2; 3
where Hi is a damage hardening
modulus, Y 0;i is the initial
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
damage threshold and n ¼ e : C : e is the equivalent
strain measurement.
The stiffness is reduced using damage variables (compliance is increased, similar to Qing and Mishnaevsky, 2010):
ð14Þ
ð15Þ
di ðx; eÞ 2 ½0; dmax Þ;
ð16Þ
The following shear strength criterion is used:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r12 ðxÞ2 þ r13 ðxÞ2 þ r23 ðxÞ2 < X S;i k2 ðxÞ;
–
Cððe; d; kÞðxÞÞ
1
Ee1
6
6 sym
6
6
6 sym
16
¼ 6
k6
60
6
60
6
4
0
v 21
v 31
sym
Ee1
1
Ee2
Ee2
1
Ee2
0
Ee1
v 31
0
0
0
0
0
0
0
0
0
0
1
G
f
23
0
0
1
G
f
23
0
0
0
0
0
0
0
0
3
2ð1þv 12 Þ
7
7
7
7
7
7
7;
7
7
7
7
7
5
Ee2
ð17Þ
E1 ¼ E1
where k is a solid volume fraction in hybrid voxel, f
g
E2 ¼ E2 ð1 d2 ðx; eÞÞ; G
ð1 d1 ðx; eÞÞ; f
is
23 ¼ G23 ð1 d3 ðx; eÞÞ; E1
axial Young’s modulus of fiber, E2 ¼ E3 is the transversal Young’s
J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
modulus of fiber, G23 is shear modulus of fiber and v 12 ;
son’s ratio.
v 31 is Pois-
It is assumed that the damage does not affect the Poisson’s ratio. To avoid too large contrast in phase stiffness,
the maximal value of the damage variable is restricted to
dmax ¼ 0:95. The fiber buckling effect is not taken into
account.
2.3. Experimental procedure
To increase the practical usage of this work, the experimental procedure of mechanical tests were carried out
using European standards related to MDF.
According to the MDF European standard EN 622, the
bending strength, stiffness and transversal tensile strength
are necessary for to obtain. Therefore, tensile and bending
tests are performed numerically and experimentally. The
details of the tensile test is specified in EN 319:1993 and
the bending test in EN 310.
Bending tests were performed for 8 samples and tensile
test for 7 samples and results were statistically analyzed. A
test machine ‘‘Zwick 1474, 100 kN’’ was used.
The lCT images of cylindrical samples with 12 mm
diameter and 6 mm height were prepared. The lCT with
a voxel length of 4 lm and 8 lm with the following
X-ray parameters: voltage 6 kV, current 360 lA and
0.1Cu filter were produced.
To obtain the density profile, MDF with different densities are manufactured in a laboratory and the density profile is measured using stepwise X-ray scanning parallel to
the board plane.
Fiber length and diameter are obtained by creating 2d
image of fiber mat with 1200 dpi resolution and than
applying commercially available image processing algorithm, which segments individual fibers.
3. Results and discussion
The numerical simulation and physical experiments of
tensile and three point bending tests are considered here.
The two-scale simulations using a database of previous
results are performed. The mechanical properties of the
wood fibers and fiber joints are summarized in Table 2.
The overall workflow to get numerical results is
following:
(1) Analyze lCT images and obtain density profiles and
fiber orientations. Scan fiber mat and obtain probability distribution of fiber length and diameter.
Results shown in Section 3.1.
(2) Use the results from previous step as input parameters to virtual fiber network generator. Generate cellulose fibers by simulating fiber lay-down process.
Compress fiber network to obtain necessary density.
Results of this step are shown in Section 3.2.
(3) Compute unknown fiber parameters, which are not
specified in literature, by fitting computed micro
structure results with experiments.
(4) Simulation of tensile test using two scale approach
(see Section 3.3).
79
(5) Simulation of bending test using a two scale
approach. The microstructure is resolved at integration points where the maximal stress on macro
model appears, to reduce computational time. See
results in Section 3.4.
(6) Get average orientation tensor of manufactured MDF
fiberboards with oriented fiber network. Generate
virtually a database with fiber networks with the
same average orientation tensor and density. Simulate bending and tensile test. Results are shown in
Section 3.5.
3.1. Quantitative fiber microstructure image analysis
The obtained density profile is quite symmetric. The
regions with increased density are about 15% of the thickness of the surfaces. The fibers are more affected by high
temperature in the high density region. Despite the
increase in density, there is a decrease in fiber strength
and stiffness. This effect is not investigated sufficiently at
the moment. We assume that the stiffness and strength
are not significantly affected by high temperature and temperature gradients.
The total size of the gray-scale image is about 109 voxels. The image with 8 lm voxel length is created of a cylindrical sample with the full thickness of the MDF board.
Image analysis identified that 8 lm resolution is not
enough to compute the fiber orientation tensor. However,
it was used to get the variation of the density in the plane
of the board. This information is used to generate the macroscopic model with density variations. The lCT image
with 4 lm voxel length (see Fig. 6) was used to compute
the average fiber orientation tensor and to analyze the
fiber network qualitatively. We can see that most of the
fibers are completely separated. However, there still exist
a few fiber bundles (non-defibrated fibers). The fiber bundles were not explicitly taken into account in the generation of the fiber network.
The fiber length and thickness were analyzed using
image processing tools of 2D images. The cumulative probability distributions of the fiber length and thickness (both
weighted by particle area) are shown in Table 3. According
to experimental measurements, the log-normal distribution is the most suitable for approximation of fiber length
distribution.
3.2. Generation of virtual microstructures
The fiber network is generated using the virtual material
laboratory in GeoDict (module PaperGeo) (GeoDict, 2014).
The generated fiber network is compressed by applying
the macroscopic strain tensor using the software FeelMath.
We found out that for large strains it is necessary to use a
multi-phase voxel image (see Fig. 7(b)). The solid volume
fraction
of
the
uncompressed
image
is
0.22
(100 100 150 voxels, voxel length = 4 lm) and 0.4 for
the compressed image (100 100 75 voxels, voxel
length = 4 lm). The compression is done by applying a macroscopic strain of 50%. The light brown color if Fig. 7 corresponds to solid voxel and the gray color to multi-phase
80
J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
7% variation of SVF in MDF plane. A good agreement with
literature is observed.
The fiber orientation tensor (FOT (see Lin et al., 2012 for
FOT)) of the generated voxel image was reconstructed
using the method given in Eq. (3). The difference between
the orientation tensor of the generated fiber network and
the transversal isotropic fiber network is less than 10%.
3.3. Simulation of tensile test
Fig. 6. A cut in horizontal plane of
lCT image.
Table 3
Fiber length and diameter cumulative probability distribution.
Cumulative probability, %
Length, lm
Thickness, lm
0
5
10
50
90
95
100
50.6
51.0
51.8
92.0
312.7
930.0
>7 mm
17.1
21.2
23.1
36.4
72.1
84.7
>1 mm
voxels. A lighter color corresponds a smaller solid volume
fraction in this multi-phase voxel. The voxel image is
smoothed for a better visualization.
The density profile is analyzed using a binary voxel
image. The comparison with information from literature
and approximated density profiles (expressed in terms of
solid volume fraction (SVF)) is shown in Fig. 8. To analyze
the density deviation in a plane of MDF, the binary voxel
image is divided into 7 slices through the thickness of
the board. Each slice is divided into 9 segments. The inplane variation of the solid volume fraction is about 5%
to 10%. The macroscopic structure was generated using
To perform an simulation, it is necessary first to obtain
the database of microstructures. We created three randomly
generated microstructures by a fiber lay-down process with
size 100 100 150 voxels and then compressed these
structures by 50%, 60% and 70% of original thickness. The
stress–strain curves of the tensile test of the microstructures
are shown in Fig. 9.
It is observed that the ultimate stress (when stiffness is
reduced more than 50%) depends approximately linearly
on the density of the microstructure. For example, the second fiber structure has an ultimate stress of 0.4 MPa if the
density is 531 kg/m3 and 0.52 MPa if the density is 697 kg/
m3.
The structure at the macroscopic level is built by assigning the different microstructures stochastically to each
integration point in the macrostructure by taking the density profile into account. The stress load is applied to simulate the tensile test. Due to the density profile, the core
part of the MDF is more deformed than the outer part.
The normal distribution of strength was observed and
compared with data experimentally obtained (see Fig. 10).
The experimentally measured average strength is
0.368 MPa and the equivalent measure simulated
0.390 MPa. The difference between the computed average
tensile strength and that experimentally measured is less
than 5%. The standard deviation of the experimentally measured strength is 0.028 MPa and the simulated 0.0035 MPa.
This implies that the actual structure is more heterogeneous
(with defects and fiber bundles) than the simulated one.
Experimental testing shows that the transversal stiffness is about 5 times smaller when the strain is less than
1%. A significant increase of stiffness is observed when
Fig. 7. (a) Generated fiber network voxel image by simulated fiber lay-down process, (b) fiber multi-phase voxel image after compression (50% strain).
J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
Fig. 8. Comparison of computed density profiles (average density = 600 kg/m3).
Fig. 9. Tensile test of generated microstructures.
Fig. 10. Probability density functions of transversal strength.
81
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J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
Fig. 11. Stress strain plot in outer surfaces of MDF.
the strain varies from 1% to 3%. Above a strain of 3% brittle
damage occurs. Numerical simulation shows that, up to
0.1% strain, the stiffness is very high and a linear stress–
strain relationship can be observed. From 0.1% to 0.3%
strain there is a decrease of the stiffness. From 0.3% to 2%
there is increase of the stiffness and a linear stress strain
relationship up to the failure strain (2%). The main reason
of the difference between the simulation and the experiments is because of the residual stresses. The fiber network
and fiber cross sections at joint regions are compressed and
fixed by glue in the deformed state. However, the residual
stresses remain. If the fiber network is stretched, the residual stresses facilitate the stretching of the material as it is
observed during the experiments.
Fig. 12. Orientation technique of wood fibers (Lippe, 2013).
J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84
3.4. Simulation of bending test
The maximal stress and strains are computed using linear Euler–Bernoulli theory according to the force–displacement curve, which was obtained by measurements. The
experimental and simulation results are shown in Fig. 11.
We can not see decrease in stiffness at small strains as it
was in the tensile test. The maximal stress is reached at a
strain of 0.8%. Up to strain of 0.5% a linear elastic regime
is observed. The stress–strain plot obtained by microstructural simulation is in good agreement with the experimental data.
3.5. Bending stiffness of optimized MDF
The increase in stiffness of oriented MDF fiber network
is studied in this section. The fibers are generated with a
different FOT (see Fig. 12). The bending modulus of elasticity is computed for the generated fiber networks (using
two scale uncoupled simulation), where linear elastic simulations are performed.
To construct the correct virtual fiber network of oriented fibers, the investigations of experimental fiber orientation devices are done parallel to simulation. Current
research shows that the most effective method to orient
wood fibers is to lay down the fibers on a waved shell
(see Fig. 12(a)). Then the shell is compressed in perpendicular direction of the fiber orientation. Strips of oriented
fibers are obtained (see Fig. 12(b)). The obtained fiber
mat (see Fig. 12(c)) is analyzed using image processing
tools (Fur et al., 2006) and obtained the average fiber orientation. The fiber orientation plot is shown in Fig. 12(d).
The numerical simulation shows that the bending stiffness can be increased up to 60%. However, the simulation
of the fiber network with the same degree of anisotropy
as the experimentally produced shows about 15% increases
in the bending stiffness. The structure of the optimized
fiber network shows that the transversal stiffness of fiber
joints is increased. This is due to the fact that for transversal isotropic orientation there are only point joints of
fibers. An increased number of line joints for the oriented
fiber network is achieved, and this type of joints promote
a higher stiffness and strength.
4. Conclusions and outline
The present study has shown the application of a twoscale simulation technique for MDF. The macroscopic
strength and stiffness mainly depend on the micro structure of a wood fiber network. An explicit constitutive law
of MDF structure for various fiber orientations and densities contains a lot of empirical data which are time and
resource consuming to obtain. However, in multiscale
method the only input parameters are fiber and fiber–fiber
joint mechanical properties. Therefore, multiscale method
can be effectively used to optimize fiber network to obtain
better strength and stiffness. The multiscale method presented here uses previously computed microstructures as
a special database of results and is less computational time
consuming than traditional coupled multiscale simulation
83
methods. Therefore, it pretend to be a good tool for industrial applications.
Currently, the microstructure of MDF is not well understood. Therefore, the lCT images of MDF were created to
analyze the average fiber orientation tensor, density profile
and fiber bundles. This experimental work is presented
here and used to propose a framework to virtually generate
representative volume elements (RVEs) for MDF plates. In
order to effectively simulate realistic microstructure and
directly use voxel images X-ray from computer tomography, a fast FFT-based solver of the Lippmann–Schwinger
equation was used.
Typically, three different damage scenarios can occur in
wood fiber network- fiber cell wall axial damage, transversal damage or fiber–fiber joint damage. Therefore, a continuum damage model with three damage variables for the
microscale simulation is proposed in this contribution.
The multiscale simulation method allows to simulate
MDF with various fiber structure, orientation and limited
number of experiments are necessary. The tensile test
and a three point bending test were simulated and the difference comparing to experimentally obtained stress–
strain curves is less than 5%.
The compression of fibers in high temperature produce
residual stress. A method that takes into account the residual stress using strength and stiffness reduction coefficients
in the transversal direction is presented here. It turned out
that this simplified method gives reasonable results.
Finally, the simulation and experimental results of an
optimized fiber network is presented. It shows that bending stiffness can be increased up to 60%. The experimentally obtained optimized fiber structures have about 25%
increase in bending stiffness.
The more precise procedure of the computation of the
reduction coefficient due to residual stresses should be
done in future. Fiber bundles are not simulated and could
have a large effect on the results. The temperature influence on wood fibers should be taken into account in future
research.
Acknowledgments
The project is promoted through the AiF and the International Association for Technical Issues related to Wood
e.V. in the programme for promoting industrial joint
research and development (IGF) of the Federal Ministry
of Economics and Technology (BMBF) on the basis of a
decision of the German Bundestag and the grant ‘‘Simulationsgestützte Entwicklung von mitteldichten Faserplatten
für den Leichtbau’’ number IGF 17644N. The responsibility
for the content of this publication lies with the authors.
This project of the Baltic-German University Liaison
Office is supported by the German Academic Exchange Service (DAAD) with funds from the Foreign Office of the Federal Republic Germany.
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