Mechanics of Materials 79 (2014) 73–84

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Mechanics of Materials

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Numerical prediction of the stiffness and strength of mediumdensity fiberboards

http://dx.doi.org/10.1016/j.mechmat.2014.08.0050167-6636/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Fraunhofer Platz 1, Kaiserslautern 67663,Germany. Tel.: +49 631316004738.

E-mail address: janis.sliseris@itwm.fraunhofer.de (J. Sliseris).

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), Germanyb Fraunhofer Institute for Wood Research (WKI), Germanyc Riga Technical University (RTU), Latvia

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 May 2014Received in revised form 15 August 2014Available online 16 September 2014

Keywords:Wood-based panelsMDFNumerical analysisExperimental testing

A numerical two scale method for the prediction of tensile and bending stiffness andstrength of medium density fiberboards (MDF) is proposed with the aim to study the fiberorientation influence on mechanical properties of MDF. The method requires less experi-mental data to optimize MDF and to improve industrial manufacturing technology ofMDF. A new method for computing orientation tensors of the compressed fiber networkis proposed. First, the virtual microstructure is generated by simulations of a fiber lay-down and a subsequent compression to obtain the necessary density. The density profile,fiber length, thickness, and orientation are used for the microstructure generation, whichare obtained from lCT images and image analysis tools. Then a new damage model forthe wood fiber cell walls and joints is introduced. The microstructural problem is formu-lated as a Lippmann–Schwinger type equation in elasticity and solved by using Fast FourierTransformation (FFT). The macroscopic three point bending test is simulated with hexahe-dral finite elements and analytical methods based on Euler–Bernoulli theory. The differ-ence between bending strength and stiffness numerically obtained and correspondingexperimentally measured values is less than 10%. This study lays a foundation for the opti-mal design of MDF fiber structures and the optimization of industrial manufacturing pro-cesses. The first results show an increase of up to 60% for bending stiffness in the case ofstrongly oriented fibers.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Medium density fiberboards (MDF) are widely used forfurniture and in flooring. The mechanical and physicalproperties requirements have significantly increased inthe last two decades. The manufacturing process of MDFconsists of log chipping, chip washing, thermo-mechanical

pulping, defibrating, spraying of resin and wax, drying, matforming, hot pressing, and sawing.

Precise and fast simulation techniques are necessary toincrease understanding the deformation- fracture mecha-nisms and the influence of the wood fiber properties anddistribution on stiffness of the MDF. This knowledge canbe used to optimize the MDF in terms of maximal strengthand stiffness. MDF consists of a wood fiber network inwhich the fibers are joined together at several contactpoints. The fiber joins as well as the resin have a largeinfluence on the strength and stiffness (Eichhorn andYoung, 2003). Numerical investigations show that it is

Fig. 1. Overall scheme of simulation.

74 J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84

important to include the effects of elasto-plasticity, failureof the bond and frictional sliding of the fibers in the case oflarge deformations (Liu et al., 2011; Wilbrink et al., 2013).An approach to model cellulose fibers is proposed byHeyden (2000). However, in this work the fibers are mod-eled by beam finite elements that do not consider thedeformation of the cross section (Wriggers and Zavarise,1997). The direct simulation of 3D fiber network deforma-tion and failure (under different humidity conditions)using beam elements is proposed in Kulachenko andUesaka (2012). The crack growth in a cellulose fiber net-work is studied using implicit gradient nonlocal theory inIsaksson et al. (2012). In addition, the multi-body interac-tion effect can be modeled by using continuum approach(Zemerli, 2014).

Wood fiber consists of lignin, cellulose, hemicelluloseand extractives. The fiber diameter varies from 20 lm to50 lm depending on the species and type of wood (hard-wood or softwood). Multivariate statistical methods likegenetic algorithms can be used to predict the internal bondstrength of MDF (Andre et al., 2008). The cell wall of woodfibers consists of a layered structure. However, thestrength and stiffness is mainly governed by the secondarycell wall (S2) layer (Deng et al., 2012). The layered struc-ture is formed from cellulose microfibrills with a diameterof 4 nm. The thickness of the S2 layer is about 85% of thetotal cell wall thickness and the microfibril angle (withrespect to the fiber axis) in the S2 layer varies from 10�to 30� (Persson, 2000). To approximate the microfibrilangle in the S2 layer the log-normal probability distribu-tion is suitable.

MDF is produced mainly with urea–formaldehyde resins(partly modified with melamine) and to a small part withPMDI (polymeric diisocyanate). These resins are duroplas-tics (thermosetting resins). The complexity of the woodfiber network leads to complex nonlinear material behavior(Sliseris and Rocens, 2010, 2013a,b). To calculate thestrength of MDF, the nonlinearity of the material has to beconsidered. A non-uniform density profile leads to non-uniform strain–stress fields even for a simple tensile test.The experimental investigation of material behavior forcomplex strain–stress fields is time and resource consum-ing. To take the material complexity into account the twoscale simulation method can be used effectively, where,instead of solving the constitutive laws at each macroscopicpoint, microstructural homogenization is performed.

The computational homogenization of microstructuresis used to predict the strength and stiffness of highly heter-ogeneous materials such as masonry structures (Yuen andKuang, 2013), sandwich plates with Poisons locking effect(Helfen and Diebels, 2014), laminates (Li et al., 2014) andnano-composites (Liu et al., 2011). The swelling of naturalwood can be more precisely analyzed using a two scaleapproach (Rafsanjani et al., 2013). To compute the cracksand damage of composites, the FE2 method can be used(Feyel and Chaboche, 2000; Nguyen et al., 2011; Visroliaand Meo, 2013). However, this method is computationallyvery intensive, therefore the speed can be increased byusing an analytical approach at the micro scale (Salviatoet al., 2013) or model reduction techniques (Somer et al.,2014).

To solve the microstructural homogenization problem,a so-called representative volume element (RVE) has tobe generated in advance. However, current knowledge forgenerating RVEs of the MDF microstructure is not satisfac-tory. We propose an approach to generate RVEs using mea-sured experimentally measured parameters, e.g. fiberlength, diameter and orientation. The initial fiber networkis generated by simulating the fiber lay-down process andthe resulting structure is then virtually compressed (seeFig. 1).

The compression of the ligno-cellulose fiber networkleads to a nonlinear contact problem due to the fact thatnew contacts appear between fibers. Further, finite strainsand material nonlinearity have to be taken into account.The typical manufacturing process of MDF consists of com-pressing fiber mats at a high temperature for a certainpressing time between 190 �C and 220 �C for between4 s/mm and 8 s/mm of the board thickness, depending onglue and other factors. Therefore, a high temperature gra-dient exists in the mat during the hot pressing. Duringwhich the curing of the glue occurs. The friction coefficientat the beginning of the compression is very small (perfectsliding) and after the curing of the glue it significantlyincreases. However, the change of the friction coefficientis not clear and experimentally difficult to investigate.

A new approach for modeling microstructure of MDF isproposed, where the microstructure is discretized as avoxel (cuboid or 3d pixel) image instead of typically-usedbeam finite elements. We propose to solve the contactproblem by transferring it into a relaxed formulationwhere the voids between fibers are replaced by a materialwith a relatively small stiffness in comparison to the fibers(ratio is up to 105).

Fig. 2. Macro and micro structures.

J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84 75

The mixed-phase voxel geometry is introduced to ana-lyze the microstructure more precisely (Merkert, 2013).The mixed-phase voxel means that a voxel is a mixtureof solid and void phase.

Finally, an improved two scale framework for simulat-ing MDF is first time proposed. The advantage of ourmethod over previous methods is the use of a databasegenerated with microstructures that have been previouslyanalyzed. The database is reused in the next iteration orload steps. At the microstructure level a new wood fiberdamage model with three damage variables is proposed.

The paper is organized as follows. The generation ofmicrostructure is presented in Section 2.2.1. The simula-tion method of strength and stiffness is presented in Sec-tion 2.2.2. The homogenization technique is presented inSection 2.2.3. The wood fiber material damage model ispresented in Section 2.2.4. Finally, the results of micro-structure generation, simulation of tensile and bendingtest and simulation of optimized fiber structure are pre-sented in Section 3.

2. Materials and methods

2.1. Materials

Experiments were carried out using MDF plates pro-duced with target density 650 kg/m3, target thickness of12 mm and size of 300 � 300 mm. The fiber structure iscreated of defibrated pine fibers and fiber bundles. Theaverage 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 for144 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 � 12mm and 50 � 50 � 12 mm, respectively.

The CT images of cylindrical samples with 12 mm diam-eter and 6 mm height were created. The lCT images with avoxel 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 networkThe RVE is a small part of the MDF that represents the

behavior of the microstructure (see Fig. 2). Instead of com-puting the MDF with a resolved microstructure one can

Table 1Technical parameters of manufactured MDF.

Parameter Value

Wood species PineGlue UF-resin Kaurit�337Glue contenta 9%Paraffin contenta 1.5%Press temperature 200 �CPressing time 12 s/mmSize of board 500 � 500 mm

a The content is calculated based on dry fibers.

compute an RVE. The size of the RVE should be largeenough to represent the actual behavior of microstructure.

The RVE’s are constructed using experimental data offiber length, diameter and orientation. The overall schemeof the main steps in generating the RVE is shown in Fig. 1.Fibers are separated from a 2D image using appropriateimage processing algorithms (Schirp et al., 2014; Schmidand Schmid, 2006) and a probability distribution of thefiber length and diameter is obtained. The average orienta-tion tensor of the fiber network is obtained from lCTimages of compressed MDF samples. Previous studies(Sliseris, 2013) have indicated that it is necessary to createthe uncompressed voxel image of 106 voxels. To obtain theorientation tensor of lCT image, the second order deriva-tive matrix (Hessian) of the smoothed (with Gaussian ker-nel) image is computed (Redenbach et al., 2012; Wirjadiet al.). The eigenvector corresponding to the smallesteigenvalue of the Hessian matrix is interpreted as the localdirection. The orientation tensor of the fibers is obtainedby averaging the local directions over small sub-volumes,in which only foreground (fiber) voxels are considered.This method gives reasonable results in the places withoutfiber contacts.

The stochastic fiber network is generated using a fiberlay-down process that is governed by the Fokker–Plankequation (Götz et al., 2007; Grothaus and Klar, 2008). Thecommercial software GeoDict (2014) was used to performthis. An elliptic hollow fiber cross section is used with athickness of the fiber cell wall of 4 lm. The ratio of thelargest and the smallest fiber diameter is assumed to be2. The curved shape of cellulose fibers is obtained byassembling several straight segments.

Simulations indicate that the average density of thefiber network generated by the lay-down process is up to450 kg/m3. To get the fiber network with the higher den-sity, the compression of the fiber network is simulated.

Due to the compression the orientation angles (direc-tion of normal vector n) of the fibers change (see Fig. 3)and fiber orientation in the plane orthogonal to the direc-tion of compression increases.

The orientation angles of the fibers are updated aftercompression. Note that since the material is transversallyisotropic, we have to use only two Euler angles here.

The updated fiber orientation angles are obtained byminimizing the sum of distances between each voxel andthe orientation vector of the fiber that goes through thegeometrical center x of a fiber segment. The distance dk

between voxel xk (with center coordinates xk) and thecenter line of the fiber segment is computed by the follow-ing equation:

Fig. 3. Deformation scheme of a fiber segment X0i .

76 J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84

dk ¼ kx� xk þ ntk: ð1Þ

The parameter t is obtained by minimizing the distancedk ¼ dkðtÞ, i.e.

t ¼ ðx� xkÞ � n: ð2Þ

The components of the fiber orientation unit vectorn ¼ ðnx;ny; nzÞ are computed by solving a minimizationproblem over all voxels Ni in the domain Xi:

minn

XNi

k¼1

dkðnÞ: ð3Þ

The necessary conditions for minimum of dkðnÞ requiresthat values of first order derivatives is zero

rn

XNi

k¼1

dkðnÞ ¼ 0: ð4Þ

This leads to a system of three nonlinear algebraic equa-tions, which is solved numerically using the Newton–Raphson iteration method.

The orientation angles of a multi-phase voxel that con-sists of the solid volume fraction of two or more differentlyorientated fibers are computed using the weighted averag-ing (arithmetic averaging) technique.

As a result, the RVE is discretized in voxels with amatching solid volume fraction (SVF), fiber length andorientation.

2.2.2. Two scale simulation methodologyThe stiffness for macroscopic problem can be obtained by

computing the effective stiffness of the microstructures. Theanalytical approach for a wood fiber network can not beused in case of high stress. The stiffness of MDF dependson 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 simulationis shown in Fig. 4. To perform the macroscopic simulationAbaqus/standard finite element code (Abaqus, 2014) isused. To reduce the number of simulations at the microscale,a database of strain and the appropriate stiffness tensor arecreated for each microstructure. The database is updated

after each simulation at the microscale. The material stiff-ness tensor at each integration point is obtained by search-ing for a similar strain tensor in the database. If it is notfound then a microscale simulation with a specified macro-scopic strain tensor is performed. The simulation at themicroscale level is performed by using software FeelMath(2014).

An Abaqus UMAT subroutine is used to interface withthe micro simulation and the database. A special Pythonscript is called from the Abaqus UMAT subroutine. Thisscript checks the database for an appropriate stiffnesstensor. If it is not available then a simulation of micro-structure (solution of problem (9)) is performed tocompute the stiffness. The script calls six loadcases Ei fora prescribed macroscopic strain load E

E ¼ E0 ¼ ðE11; E22; . . . ; E12Þ;E1 ¼ ðE11 þ DE; E22; . . . ; E12Þ;. . .

E6 ¼ ðE11; E22; . . . ; E12 þ DEÞ:

ð5Þ

The Python script computes the stiffness tensor when allloadcases are successfully executed. The fiber constitutivelaw is implemented in another UMAT file which is calledby FeelMath at each solid or multi-phase voxel.

If such a strain tensor Ei (index i indicates the numberin database) exists that is ‘‘similar’’ to given macroscopicstrain tensor E then the stiffness tensor is taken from thedatabase. In this context ‘‘similar’’ means that the Euclid-ean norm of the two strain tensors is smaller than the mac-roscopic strain increment

kE� Eik 6 DE: ð6Þ

The average stress tensors are obtained at each load(macroscopic strain) case:

hrijðEkÞi ¼ 1V

ZVrijðEk; xÞdx; k ¼ 0;1; . . . ;6: ð7Þ

Finally, we obtain the stiffness matrix (using the principleof strain equivalence) in a simple way (due to fact thatstrain difference vector contains only one non-zeroelement):

Fig. 4. Framework of two scale simulation.

J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84 77

C ¼ 1DE

Dr111 Dr2

11 . . . Dr611

Dr122 Dr2

22 . . . Dr622

. . . . . . . . . . . .

Dr112 Dr2

12 . . . Dr612

26664

37775; ð8Þ

where Drkij ¼ hrijðEkÞi � hrijðE0Þi; k ¼ 1;2; . . . ;6.

We use this methodology of 6 different loadcasesbecause 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 toobtain the stiffness at the current strain state. However, ifit is too small then the stress difference is also very smalland this may lead to inaccurate results. Current simula-tions show that DE ¼ 10�4 � 10�5 gives the best results.

To perform an uncoupled two scale simulation, micro-structures with different density and typical strain tensorsare computed in advance, and the stress–strain curves areused in the macro computations. The stress–strain curvesare computed using typically observed macroscopic straintensors for linear elastic analysis.

2.2.3. Nonlinear homogenization problemThe nonlinearity in homogenization of MDF results from

the nonlinear wood fiber behavior. Wood fiber is a brittlematerial in short time tension loads. Typically the damageoccurs at the fiber joints or at fiber. A new continuum dam-age model is introduced to capture the nonlinear behavior.

The homogenization is performed using small strain theory(strains does not exceed �3%). The computation of thestress–strain fields of the RVE leads to a boundary valueproblem in elasticity. It is solved using Fast Fourier Trans-formation (FFT) method.

Solution of the elasticity problem. The microstructure ofthe fiber network is discretized using voxels. The initialstiffness tensor C and strength of each voxel are specified.The homogenization problem on a cuboid x 2 R3 withperiodic displacement and anti-periodic stress field bound-ary conditions on boundary @x is solved (9). The macro-scopic 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 Eþ 12ru�ðxÞ þ ðru�ðxÞÞT� �

; x 2 x;

u�ðxÞ � periodic; x 2 @x;rðxÞ � nðxÞ � antiperiodic; x 2 @x:

ð9Þ

Instead of stress field rðxÞ, the stress polarization field sðxÞis introduced

sðxÞ ¼ ðCðe;d;kÞðxÞ � C0Þ : eðxÞ: ð10Þ

The problem (9) can be reformulated into an equivalent inte-gral equation, the so called Lippmann–Schwinger equation

eðxÞ ¼ E� ðC0 � sÞðxÞ; ð11Þ

Fig. 5. Locale coordinate system of fiber network.

Table 2Mechanical properties of wood fiber and joints.

Name of property Valueb Reductioncoefficienta

Young’s modulus E1 (Persson, 2000) 50 GPa –Young’s modulus E2 (Persson, 2000) 3 GPa 0.01Shear modulus G23 (Persson, 2000) 3 GPa 0.01Poisson’s ratio v12 (Persson, 2000) 0.3 –Poisson’s ratio v13 (Persson, 2000) 0.3 –Damage Hardening modulus Hi 100 MPa –Initial damage threshold Yi;0 0 MPa –Axial strength XA;i (Joffe et al., 2009) 500 MPa –Transversal strength XT;i 40 MPa 0.025Shear strength XS;i 40 MPa 0.05Density of wood fiber (Isaksson et al.,

2012)1500 kg/m3

–

Density of UF glue (Osemeahon andBarminas, 2007)

1500 kg/m3

–

a The reduction coefficient is taken into account in the axial tensile test.b All properties without reference are obtained by fitting experimental

results.

78 J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84

where C0 denotes the Green operator. The convolutionoperator ⁄ is defined by

C0 � sðxÞ ¼Z

XC0ðx� yÞ : sðyÞdy: ð12Þ

The Lippmann–Schwinger equation is solved using the FFTmethod Moulinec and Suquet, 1998.

The rate of convergence depends on the contrast instiffness of the phases. The voids are considered as a mate-rial with a significantly smaller stiffnesses comparing tothe fiber stiffness. In simulations presented here we setthe void stiffness to be 105 times smaller than fiber mate-rial. For linear elastic material the stopping criteriatol ¼ 10�5 is usually reached after 200 to 500 iterations.However, in the case of nonlinear material behavior themaximal value of the damage variables is limited to 0.95.This restriction avoids the occurrence of differencesbetween the phase stiffness being too large.

2.2.4. Constitutive lawIn 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 materialthat partly penetrates into the fiber cell wall (about 50% ofglue is penetrated in cell wall Xing et al., 2005). The thick-ness of the glue layer is relatively small compared to thethickness of the wood cell wall. Different material proper-ties are specified for the fiber joint and the fiber voxels. How-ever, experimental results for the mechanical properties offiber joints are not available. Therefore, the same materialproperties are used for joint and fiber voxels. Due to the pen-etration of the glue in a joint of two fibers, a differentstrength and stiffness are used in the joint voxels.

Experimental investigations (Isaksson et al., 2012;Matsumoto and Nairn, 2009) of cellulose fiber networksshow that damage can occur due to axial stress, shearstress or when transversal stress of a fiber exceed criticalvalue. The critical stresses have been obtained experimen-tally by Joffe et al. (2009) and Yoshihara (2012). This moti-vates us to use three damage criteria.

For the fiber cell wall, the following axial failure crite-rion is used (the local coordinate system is shown in Fig. 5):

r11ðxÞ < XA;i: ð13Þ

The transversal normal stress criterion of fiber cell wall isused:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r22ðxÞ2 þ r33ðxÞ2q

< XT;ik1ðxÞ: ð14Þ

The following shear strength criterion is used:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir12ðxÞ2 þ r13ðxÞ2 þ r23ðxÞ2

q< XS;ik2ðxÞ; ð15Þ

where XA;i; XT;i; XS;i are axial, tangential and shearstrength, index i ¼ 1 denotes a non-fiber-joint region,i ¼ 2 denotes a fiber joint region, k1ðxÞ and k2ðxÞ strengthreduction coefficients due to existence of residual stressin fiber. Values of critical stress are given in Table 2.

If the strength criteria are not satisfied, then the dam-age variables diðxÞ are computed using an exponentiallaw Spahn et al., 2014a:

diðx; eÞ ¼ 1� e�Hiðn�Y0;iÞ; diðx; eÞ 2 ½0; dmaxÞ; i ¼ 1;2;3

ð16Þ

where Hi is a damage hardening modulus, Y0;i is the initialdamage threshold and n ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie : C : ep

is the equivalentstrain measurement.

The stiffness is reduced using damage variables (compli-ance is increased, similar to Qing and Mishnaevsky, 2010):

Cððe;d;kÞðxÞÞ�1 ¼ 1k

1eE1

�v21eE2

�v31eE10 0 0

sym 1eE2

�v31eE10 0 0

sym sym 1eE20 0 0

0 0 0 1fG230 0

0 0 0 0 1fG230

0 0 0 0 0 2ð1þv12ÞeE2

26666666666664

37777777777775;

ð17Þ

where k is a solid volume fraction in hybrid voxel, fE1 ¼ E1

ð1� d1ðx; eÞÞ; fE2 ¼ E2ð1� d2ðx; eÞÞ; gG23 ¼ G23ð1� d3ðx; eÞÞ; E1 isaxial Young’s modulus of fiber, E2 ¼ E3 is the transversal Young’s

J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84 79

modulus of fiber, G23 is shear modulus of fiber and v12; v31 is Pois-son’s ratio.

It is assumed that the damage does not affect the Pois-son’s ratio. To avoid too large contrast in phase stiffness,the maximal value of the damage variable is restricted todmax ¼ 0:95. The fiber buckling effect is not taken intoaccount.

2.3. Experimental procedure

To increase the practical usage of this work, the exper-imental procedure of mechanical tests were carried outusing European standards related to MDF.

According to the MDF European standard EN 622, thebending strength, stiffness and transversal tensile strengthare necessary for to obtain. Therefore, tensile and bendingtests are performed numerically and experimentally. Thedetails of the tensile test is specified in EN 319:1993 andthe bending test in EN 310.

Bending tests were performed for 8 samples and tensiletest for 7 samples and results were statistically analyzed. Atest machine ‘‘Zwick 1474, 100 kN’’ was used.

The lCT images of cylindrical samples with 12 mmdiameter and 6 mm height were prepared. The lCT witha voxel length of 4 lm and 8 lm with the followingX-ray parameters: voltage 6 kV, current 360 lA and0.1Cu filter were produced.

To obtain the density profile, MDF with different densi-ties are manufactured in a laboratory and the density pro-file is measured using stepwise X-ray scanning parallel tothe board plane.

Fiber length and diameter are obtained by creating 2dimage of fiber mat with 1200 dpi resolution and thanapplying commercially available image processing algo-rithm, which segments individual fibers.

3. Results and discussion

The numerical simulation and physical experiments oftensile and three point bending tests are considered here.The two-scale simulations using a database of previousresults are performed. The mechanical properties of thewood fibers and fiber joints are summarized in Table 2.

The overall workflow to get numerical results isfollowing:

(1) Analyze lCT images and obtain density profiles andfiber orientations. Scan fiber mat and obtain proba-bility distribution of fiber length and diameter.Results shown in Section 3.1.

(2) Use the results from previous step as input parame-ters to virtual fiber network generator. Generate cel-lulose 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 notspecified in literature, by fitting computed microstructure results with experiments.

(4) Simulation of tensile test using two scale approach(see Section 3.3).

(5) Simulation of bending test using a two scaleapproach. The microstructure is resolved at integra-tion points where the maximal stress on macromodel appears, to reduce computational time. Seeresults in Section 3.4.

(6) Get average orientation tensor of manufactured MDFfiberboards with oriented fiber network. Generatevirtually a database with fiber networks with thesame average orientation tensor and density. Simu-late bending and tensile test. Results are shown inSection 3.5.

3.1. Quantitative fiber microstructure image analysis

The obtained density profile is quite symmetric. Theregions with increased density are about 15% of the thick-ness of the surfaces. The fibers are more affected by hightemperature in the high density region. Despite theincrease in density, there is a decrease in fiber strengthand stiffness. This effect is not investigated sufficiently atthe moment. We assume that the stiffness and strengthare not significantly affected by high temperature and tem-perature gradients.

The total size of the gray-scale image is about 109 vox-els. The image with 8 lm voxel length is created of a cylin-drical sample with the full thickness of the MDF board.Image analysis identified that 8 lm resolution is notenough to compute the fiber orientation tensor. However,it was used to get the variation of the density in the planeof the board. This information is used to generate the mac-roscopic model with density variations. The lCT imagewith 4 lm voxel length (see Fig. 6) was used to computethe average fiber orientation tensor and to analyze thefiber network qualitatively. We can see that most of thefibers are completely separated. However, there still exista few fiber bundles (non-defibrated fibers). The fiber bun-dles were not explicitly taken into account in the genera-tion of the fiber network.

The fiber length and thickness were analyzed usingimage processing tools of 2D images. The cumulative prob-ability distributions of the fiber length and thickness (bothweighted by particle area) are shown in Table 3. Accordingto experimental measurements, the log-normal distribu-tion is the most suitable for approximation of fiber lengthdistribution.

3.2. Generation of virtual microstructures

The fiber network is generated using the virtual materiallaboratory in GeoDict (module PaperGeo) (GeoDict, 2014).The generated fiber network is compressed by applyingthe macroscopic strain tensor using the software FeelMath.We found out that for large strains it is necessary to use amulti-phase voxel image (see Fig. 7(b)). The solid volumefraction of the uncompressed image is 0.22(100 � 100 � 150 voxels, voxel length = 4 lm) and 0.4 forthe compressed image (100 � 100 � 75 voxels, voxellength = 4 lm). The compression is done by applying a mac-roscopic strain of 50%. The light brown color if Fig. 7 corre-sponds to solid voxel and the gray color to multi-phase

Fig. 6. A cut in horizontal plane of lCT image.

Table 3Fiber length and diameter cumulative probability distribution.

Cumulative probability, % Length, lm Thickness, lm

0 50.6 17.15 51.0 21.210 51.8 23.150 92.0 36.490 312.7 72.195 930.0 84.7100 >7 mm >1 mm

80 J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84

voxels. A lighter color corresponds a smaller solid volumefraction in this multi-phase voxel. The voxel image issmoothed for a better visualization.

The density profile is analyzed using a binary voxelimage. The comparison with information from literatureand approximated density profiles (expressed in terms ofsolid volume fraction (SVF)) is shown in Fig. 8. To analyzethe density deviation in a plane of MDF, the binary voxelimage is divided into 7 slices through the thickness ofthe board. Each slice is divided into 9 segments. The in-plane variation of the solid volume fraction is about 5%to 10%. The macroscopic structure was generated using

Fig. 7. (a) Generated fiber network voxel image by simulated fiber lay-down p

7% variation of SVF in MDF plane. A good agreement withliterature is observed.

The fiber orientation tensor (FOT (see Lin et al., 2012 forFOT)) of the generated voxel image was reconstructedusing the method given in Eq. (3). The difference betweenthe orientation tensor of the generated fiber network andthe transversal isotropic fiber network is less than 10%.

3.3. Simulation of tensile test

To perform an simulation, it is necessary first to obtainthe database of microstructures. We created three randomlygenerated microstructures by a fiber lay-down process withsize 100 � 100 � 150 voxels and then compressed thesestructures by 50%, 60% and 70% of original thickness. Thestress–strain curves of the tensile test of the microstructuresare shown in Fig. 9.

It is observed that the ultimate stress (when stiffness isreduced more than 50%) depends approximately linearlyon the density of the microstructure. For example, the sec-ond fiber structure has an ultimate stress of 0.4 MPa if thedensity is 531 kg/m3 and 0.52 MPa if the density is 697 kg/m3.

The structure at the macroscopic level is built by assign-ing the different microstructures stochastically to eachintegration point in the macrostructure by taking the den-sity profile into account. The stress load is applied to sim-ulate the tensile test. Due to the density profile, the corepart of the MDF is more deformed than the outer part.

The normal distribution of strength was observed andcompared with data experimentally obtained (see Fig. 10).The experimentally measured average strength is0.368 MPa and the equivalent measure simulated0.390 MPa. The difference between the computed averagetensile strength and that experimentally measured is lessthan 5%. The standard deviation of the experimentally mea-sured 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 stiff-ness is about 5 times smaller when the strain is less than1%. A significant increase of stiffness is observed when

rocess, (b) fiber multi-phase voxel image after compression (50% strain).

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.

J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84 81

Fig. 11. Stress strain plot in outer surfaces of MDF.

82 J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84

the strain varies from 1% to 3%. Above a strain of 3% brittledamage occurs. Numerical simulation shows that, up to0.1% 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 strainrelationship up to the failure strain (2%). The main reason

Fig. 12. Orientation technique of

of the difference between the simulation and the experi-ments is because of the residual stresses. The fiber networkand fiber cross sections at joint regions are compressed andfixed by glue in the deformed state. However, the residualstresses remain. If the fiber network is stretched, the resid-ual stresses facilitate the stretching of the material as it isobserved during the experiments.

wood fibers (Lippe, 2013).

J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84 83

3.4. Simulation of bending test

The maximal stress and strains are computed using lin-ear Euler–Bernoulli theory according to the force–displace-ment curve, which was obtained by measurements. Theexperimental and simulation results are shown in Fig. 11.We can not see decrease in stiffness at small strains as itwas in the tensile test. The maximal stress is reached at astrain of 0.8%. Up to strain of 0.5% a linear elastic regimeis observed. The stress–strain plot obtained by microstruc-tural simulation is in good agreement with the experimen-tal data.

3.5. Bending stiffness of optimized MDF

The increase in stiffness of oriented MDF fiber networkis studied in this section. The fibers are generated with adifferent FOT (see Fig. 12). The bending modulus of elastic-ity is computed for the generated fiber networks (usingtwo scale uncoupled simulation), where linear elastic sim-ulations are performed.

To construct the correct virtual fiber network of ori-ented fibers, the investigations of experimental fiber orien-tation devices are done parallel to simulation. Currentresearch shows that the most effective method to orientwood fibers is to lay down the fibers on a waved shell(see Fig. 12(a)). Then the shell is compressed in perpendic-ular direction of the fiber orientation. Strips of orientedfibers are obtained (see Fig. 12(b)). The obtained fibermat (see Fig. 12(c)) is analyzed using image processingtools (Fur et al., 2006) and obtained the average fiber orien-tation. The fiber orientation plot is shown in Fig. 12(d).

The numerical simulation shows that the bending stiff-ness can be increased up to 60%. However, the simulationof the fiber network with the same degree of anisotropyas the experimentally produced shows about 15% increasesin the bending stiffness. The structure of the optimizedfiber network shows that the transversal stiffness of fiberjoints is increased. This is due to the fact that for transver-sal isotropic orientation there are only point joints offibers. An increased number of line joints for the orientedfiber network is achieved, and this type of joints promotea higher stiffness and strength.

4. Conclusions and outline

The present study has shown the application of a two-scale simulation technique for MDF. The macroscopicstrength and stiffness mainly depend on the micro struc-ture of a wood fiber network. An explicit constitutive lawof MDF structure for various fiber orientations and densi-ties contains a lot of empirical data which are time andresource consuming to obtain. However, in multiscalemethod the only input parameters are fiber and fiber–fiberjoint mechanical properties. Therefore, multiscale methodcan be effectively used to optimize fiber network to obtainbetter strength and stiffness. The multiscale method pre-sented here uses previously computed microstructures asa special database of results and is less computational timeconsuming than traditional coupled multiscale simulation

methods. Therefore, it pretend to be a good tool for indus-trial applications.

Currently, the microstructure of MDF is not well under-stood. Therefore, the lCT images of MDF were created toanalyze the average fiber orientation tensor, density profileand fiber bundles. This experimental work is presentedhere and used to propose a framework to virtually generaterepresentative volume elements (RVEs) for MDF plates. Inorder to effectively simulate realistic microstructure anddirectly use voxel images X-ray from computer tomogra-phy, a fast FFT-based solver of the Lippmann–Schwingerequation was used.

Typically, three different damage scenarios can occur inwood fiber network- fiber cell wall axial damage, transver-sal damage or fiber–fiber joint damage. Therefore, a contin-uum damage model with three damage variables for themicroscale simulation is proposed in this contribution.

The multiscale simulation method allows to simulateMDF with various fiber structure, orientation and limitednumber of experiments are necessary. The tensile testand a three point bending test were simulated and the dif-ference comparing to experimentally obtained stress–strain curves is less than 5%.

The compression of fibers in high temperature produceresidual stress. A method that takes into account the resid-ual stress using strength and stiffness reduction coefficientsin the transversal direction is presented here. It turned outthat this simplified method gives reasonable results.

Finally, the simulation and experimental results of anoptimized fiber network is presented. It shows that bend-ing stiffness can be increased up to 60%. The experimen-tally obtained optimized fiber structures have about 25%increase in bending stiffness.

The more precise procedure of the computation of thereduction coefficient due to residual stresses should bedone in future. Fiber bundles are not simulated and couldhave a large effect on the results. The temperature influ-ence on wood fibers should be taken into account in futureresearch.

Acknowledgments

The project is promoted through the AiF and the Inter-national Association for Technical Issues related to Woode.V. in the programme for promoting industrial jointresearch and development (IGF) of the Federal Ministryof Economics and Technology (BMBF) on the basis of adecision of the German Bundestag and the grant ‘‘Simula-tionsgestützte Entwicklung von mitteldichten Faserplattenfür den Leichtbau’’ number IGF 17644N. The responsibilityfor the content of this publication lies with the authors.

This project of the Baltic-German University LiaisonOffice is supported by the German Academic Exchange Ser-vice (DAAD) with funds from the Foreign Office of the Fed-eral Republic Germany.

References

Abaqus 6.13. <http://www.3ds.com/products-services/simulia/portfolio/abaqus/latest-release/> (accessed: 2014-04-02).

Andre, N., Cho, H., Baek, S., Jeong, M., Young, T., 2008. Prediction ofinternal bond strength in a medium density fiberboard process using

84 J. Sliseris et al. / Mechanics of Materials 79 (2014) 73–84

multivariate statistical methods and variable selection. Wood Sci.Technol. 42 (7), 521–534.

Deng, Q., Li, S., Chen, Y., 2012. Mechanical properties and failuremechanism of wood cell wall layers. Comput. Mater. Sci. 62, 221–226.

Eichhorn, S., Young, R., 2003. Deformation micromechanics of naturalcellulose fibre networks and composites. Compos. Sci. Technol. 63,1225–1230.

Feelmath. <http://www.itwm.fraunhofer.de/abteilungen/stroemungs-und-materialsimulation/festkoerpermechanik/feelmath.html>(accessed: 2014-04-02).

Feyel, F., Chaboche, J.-L., 2000. Fe2 multiscale approach for modelling theelastoviscoplastic behaviour of long fibre SiC/Ti composite materials.Comput. Methods Appl. Mech. Eng. 183 (3–4), 309–330.

Fur, X.L., Thole, V., Plinke, B., 2006. Assessing and optimizing strandgeometry and orientation to produce strengthened OSB. In: FifthEuropean Wood-Based Panel Symposium: Proceedings. Hannover, 4.–6.10.2006.

Geodict, 2014. <http://geodict.de/> (accessed: 2014-04-02).Götz, T., Klar, A., Marheineke, N., Wegener, R., 2007. A stochastic model

and associated Fokker–Planck equation for the fiber lay-down processin nonwoven production processes. SIAM J. Appl. Math. 67 (6), 1704–1717.

Grothaus, M., Klar, A., 2008. Ergodicity and rate of convergence for a non-sectorial fiber lay-down process. SIAM J. Math. Anal. 40 (3), 968–983.

Helfen, C.E., Diebels, S., 2014. Computational homogenisation ofcomposite plates: consideration of the thickness change with amodified projection strategy. Comput. Math. Appl. 67 (5), 1116–1129.

Heyden, S., 2000. Network Modelling for the Evaluation of MechanicalProperties of Cellulose Fiber Fluff (Ph.D. thesis). Division of StructuralMechanics, LTH, Lund University.

Isaksson, P., Dumont, P., du Roscoat, S.R., 2012. Crack growth in planarelastic fiber materials. Int. J. Solids Struct. 49 (13), 1900–1907.

Joffe, R., Andersons, J., Sparnins, E., 2009. Applicability of Weibull strengthdistribution for cellulose fibers with highly non-linear behaviour. In:ICCM 17, Edinburgh: 17th International Conference on CompositeMaterials, 27 Jul 2009–31 Jul 2009, Edinburgh InternationalConvention Centre, Edinburgh, UK.

Kulachenko, A., Uesaka, T., 2012. Direct simulations of fiber networkdeformation and failure. Mech. Mater. 51, 1–14.

Li, H., Wang, W., Matsubara, T., 2014. Multiscale analysis of damageprogression in newly designed UACS laminates. Compos. Part A: Appl.Sci. Manuf. 57, 108–117.

Lin, J., Liang, X., Zhang, S., 2012. Numerical simulation of fiber orientationdistribution in round turbulent jet of fiber suspension. Chem. Eng.Res. Des. 90 (6), 766–775 (Special issue on the Third European Processintensification Conference).

Lippe, D., 2013. Process development for the experimental proof ofproperty changes of mdf with oriented fibers (in German). Master’sthesis, Fraunhofer-Institut für Holzforschung, Wilhelm-Klauditz-Institut and University of Applied Sciences in Eberswalde (HNEE),Germany.

Liu, J., Chen, Z., Wang, H., Li, K., 2011. Elasto-plastic analysis of influencesof bond deformability on the mechanical behavior of fiber networks.Theor. Appl. Fract. Mech. 55, 131–139.

Liu, A., Wang, K., Bakis, C.E., 2011. Effect of functionalization of single-wallcarbon nanotubes (SWNTs) on the damping characteristics of SWNT-based epoxy composites via multiscale analysis. Compos. Part A:Appl. Sci. Manuf. 42 (11), 1748–1755.

Matsumoto, N., Nairn, J., 2009. The fracture toughness of medium densityfiberboard (MDF) including the effects of fiber bridging and crack–plane interference. Eng. Fract. Mech. 76, 2748–2757.

Merkert, D., 2013. Voxel-based fast solution of the Lippmann–Schwingerequation with smooth material interfaces. Master’s thesis, TUKaiserslautern, Germany.

Moulinec, H., Suquet, P., 1998. A numerical method for computing theoverall response of nonlinear composites with complexmicrostructure. Comput. Methods Appl. Mech. Eng. 157 (12), 69–94.

Nguyen, V.P., Lloberas-Valls, O., Stroeven, M., Sluys, L.J., 2011.Homogenization-based multiscale crack modelling: From micro-diffusive damage to macro-cracks. Comput. Methods Appl. Mech.Eng. 200 (9–12), 1220–1236.

Osemeahon, S., Barminas, J., 2007. Study of some physical properties ofurea formaldehyde and urea proparaldehyde copolymer compositefor emulsion paint formulation. Int. J. Phys. Sci. 2 (7), 169–177.

Persson, K., 2000. Micromechanical Modeling of Wood and FiberProperties (Ph.D. thesis). Department of Mechanics and Materials,Lund University.

Qing, H., Mishnaevsky, L., 2010. 3D constitutive model of anisotropicdamage for unidirectional ply based on physical failure mechanisms.Comput. Mater. Sci. 50, 479–486.

Rafsanjani, A., Lanvermann, C., Niemz, P., Carmeliet, J., Derome, D., 2013.Multiscale analysis of free swelling of norway spruce. Compos. Part A:Appl. Sci. Manuf. 54, 70–78.

Redenbach, C., Rack, A., Schladitz, K., Wirjadi, O., Godehardt, M., 2012.Beyond imaging: on the quantitative analysis of tomographic volumedata. Int. J. Mater. Res. 103 (2), 217–227.

Salviato, M., Zappalorto, M., Quaresimin, M., 2013. Plastic shear bands andfracture toughness improvements of nanoparticle filled polymers: amultiscale analytical model. Compos. Part A: Appl. Sci. Manuf. 48,144–152.

Schirp, A., Mannheim, M., Plinke, B., 2014. Influence of refiner fibre qualityand fibre modification treatments on properties of injection-mouldedbeech wood–plastic composites. Compos. Part A: Appl. Sci. Manuf. 61,245–257.

Schmid, H.G., Schmid, G.P., 2006. Characterisation of high aspect ratioobjects using the powdershape quantitative image analysis system.In: European Congress and Exhibition on Powder Metallurgy (EUROPM), Diamond tooling, vol. 3, Ghent, Belgium, 23–25.10.2006.

Sliseris, J., 2013. Numerical prediction for the modulus of elasticity of l-mdf plates. In: Young Researcher Symposium (YRS) 2013, November8, Fraunhofer-Zentrum Kaiserslautern, pp. 42–47.

Sliseris, J., Rocens, K., 2010. Curvature analysis for composite withorthogonal, asymmetrical multilayer structure. J. Civ. Eng. Manage.16 (2), 242–248.

Sliseris, J., Rocens, K., 2013a. Optimal design of composite plates withdiscrete variable stiffness. Compos. Struct. 98, 15–23.

Sliseris, J., Rocens, K., 2013b. Optimization of multispan ribbed plywoodplate macrostructure for multiple load cases. J. Civ. Eng. Manage. 19(5), 696–704.

Somer, D., Peric, D., deSouzaNeto, E., Dettmer, W., 2014. On thecharacterisation of elastic properties of long fibre composites usingcomputational homogenisation. Comput. Mater. Sci. 83, 149–157.

Souza, F., Castro, L., Camara, S., Allen, D., 2011. Finite-element modeling ofdamage evolution in heterogeneous viscoelastic composites withevolving cracks by using a two-way coupled multiscale model. Mech.Compos. Mater. 47 (1), 133–150.

Spahn, J., Andrä, H., Kabel, M., Müller, R., 2014a. A multiscale approach formodeling progressive damage of composite materials using fastFourier transforms. Comput. Methods Appl. Mech. Eng. 268, 871–883.

Spahn, J., Andrae, H., Kabel, M., Mueller, R., Linder, C., 2014b. Multiscalemodeling of progressive damage in elasto-plastic compositematerials. In: Proceedings of the 11th World Congress onComputational Mechanics.

Visrolia, A., Meo, M., 2013. Multiscale damage modelling of 3d weavecomposite by asymptotic homogenisation. Compos. Struct. 95, 105–113.

Wilbrink, D., Beex, L., Peerlings, R., 2013. A discrete network model forbond failure and frictional sliding in fibrous materials. Int. J. SolidsStruct. 50, 1354–1363.

Wirjadi, O., Godehardt, M., Schladitz, K., Wagner, B., Rack, A., Gurka, M.,Nissle, S., Noll, A., 2014. Characterization of multilayer structures infiber reinforced polymer employing synchroton and laboratory X-rayCT. Int. J. Mater. Res. 105 (7), 645–654.

Wriggers, P., Zavarise, G., 1997. On contact between three-dimensionalbeams undergoing large deflections. Commun. Numer. Methods Eng.13, 429–438.

Xing, C., Riedl, B., Cloutier, A., Shaler, S., 2005. Characterization of urea–formaldehyde resin penetration into medium density fiberboardfibers. Wood Sci. Technol. 39, 374–384.

Yoshihara, H., 2012. Interlaminar shear strength of medium-densityfiberboard obtained from asymmetrical four-point bending tests.Constr. Build. Mater. 34, 11–15.

Yuen, Y., Kuang, J., 2013. Fourier-based incremental homogenisation ofcoupled unilateral damage-plasticity model for masonry structures.Int. J. Solids Struct. 50 (20–21), 3361–3374.

Zemerli, C., 2014. Continuum Mechanical Modeling of Dry GranularSystems: From Dilute Flow to Solid-like Behavior (Ph.D. thesis). TUKaiserslatuern.