Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 2, pp. 503-516, Warsaw 2016
DOI: 10.15632/jtam-pl.54.2.503

ANALYSIS OF THE PROCESS OF WOOD PLASTICIZATION

BY HOT ROLLING

Marian Dudziak, Ireneusz Malujda, Krzysztof Talaśka,

Tomasz Łodygowski, Wojciech Sumelka

Politechnika Poznańska

e-mail: marian.dudziak@put.poznan.pl; ireneusz.malujda@put.poznan.pl; krzysztof.talaska@put.poznan.pl;

tomasz.lodygowski@put.poznan.pl; wojciech.sumelka@put.poznan.pl

In this research, a mathematical model is derived to enable analytical determination of
effective ultimate forces in the process of plasticization of the surface layer of wood. The
experimentally determined thermo-mechanical properties of the material subjected to the
process of plasticization are used in defining the structure of the model. The analysis of
plastic strain in the layer in consideration is based on a generalised model of an ideally
rigid-plasticmedium, including certainmodifications.Considering the anisotropicproperties
of wood, the Azzi-Tsai-Hill (ATH) strength criterion is applied which takes into account
variation in the response of the loaded material depending on the direction. The article
presents also results of FEM analysis of the same process of hot rolling of wood.

Keywords: yield point, temperature, moisture content, porosity, orthotropy

1. Properties of porous anisotropic natural polymers

The focus is on designingmachines and devices of newly developed techniques for modification
of the internal structure and surface characteristics of products made of various engineered
materials. These materials include wood, which is considered the oldest structural material,
through generally usedmetals to the cutting edge composites. The inspiration to undertake this
research are the machines designed to improve properties of natural wood veneered furniture
components by application of hot rolling technique.Wood has been used to analyse the complex
process of plasticisation of superficial layers of suchmaterials with the objective to improve their
quality as well as their strength and functional performance.

The efficiency of plasticisation and densification of such materials depends on the desired
ultimate load values (Ashby and Jones, 1996). Hence, an extensive testing program has been
carried out to determine mechanical properties of wood and their dependence on temperature
andmoisture content. The results were used to set up constitutive equations of plasticity and to
build models describing these processes. This was the basis to determine the effective ultimate
load values used to define themachine design criteria (Mackenzie-Helnwein et al., 2005;Malujda,
2006).

Properties of natural polymers are defined by specific and unique characteristics of the plant
tissue of which they are made (Forest Products Laboratory, 1999; Kokociński, 2004). Some of
themmaybe determinedwith organolepticmethods, others require highly specialised test appa-
ratus to measure oftenmutually dependent physical quantities (Malujda andMarlewski, 2011).
The values dependon several factors andphenomenawhich occur inside thesematerials exposed
simultaneously to mechanical and thermal loading. Considering the number and complexity of
characteristics of natural polymers such as wood, for the sake of clarity, a few important groups
of properties and physical quantities have been identified, as schematically illustrated in Fig. 1.



504 M. Dudziak et al.

Fig. 1. The main thermo-mechanical properties of wood dependent on temperature andmoisture
content.Rm is the ultimate strength andRe is the yield point, both in the three orthotropic directions:

longitudinalL, tangential T and radialR

There is a number of important complex biochemical phenomena involved in plasticization
processes highly relevant to the desiredmodification of geometrical parameters of their surfaces
and internal structures.These phenomenahave not been analysed in detail, as thiswould exceed
the scope of this study. However, it can be assumed that the effect of these phenomena on the
thermo-mechanical properties is reflected in the material functions determined experimentally
through tests carried out on a macroscopic scale (Sumelka et al., 2013).

The structural component of wood is cellulose, and it is the specific anatomical structure
of cellulose that is responsible for the completely different structure and appearance of wood
depending on the cutting direction. Skeleton-forming substances are based on cellulose and owe
it its strength.

2. Experimental motivation

2.1. Compression test

The test specimens were subjected to compression load in the direction parallel to the grain
and two perpendicular directions: tangential and radial. The specimens were made of beech
wood (Fagus silvatica). The dimensions of test specimens are presented in Fig. 2. Fig. 3 shows
the annual growth rings.The tests specimenswere produced in sufficient number for the planned
tests.

Fig. 2. Dimensions of the compression test specimen

The specimens were made of selected wood logs to obtain representative and reproducible
resultsof testingasperPolishStandardPN-81/D-04107 (Kokociński, 2004). Specimens including
structural features such as snags were rejected.

The specimenswere compressed in the respective directions (L,T ,R) and the output results
enabled relating the determined strength to temperature andmoisture gradients. The tests were
carried out at three temperatures: 20◦C, 50◦C, 80◦C and three moisture contents: 9%, 18%
and 27%. The highest test temperature was 80◦C because above that point chemical reactions



Analysis of the process of wood plasticization by hot rolling 505

Fig. 3. Specimens: (a) radial direction, (b) tangential direction, (c) longitudinal direction

take place in beech wood, which brings unrecoverable changes in the composition and structure
of wood. The moisture content was limited to 27%, which in the case of wood is the practical
limit for absorption ofmoisture fromair.The testing program for one fibredirection is presented
in Table 1.

Table 1.Testing program for one fibre direction

Temperature
20◦C 50◦C 80◦C

Percent
moisture

9% 3 3 3
18% 3 3 3
27% 3 3 3

Before the test, the specimens were conditioned in a climatic chamber (Fig. 4b) to attain
the equilibrium moisture content. After subsequent storage in ambient conditions at the labo-
ratory, the moisture content of specimens was 9%. The threshold values of 18% and 27% were
achieved by placing the specimens in the desiccator above the saturated solution of NaCl and
water respectively (Fig. 4a). The moisture content was checked by weighing of samples during
conditioning until the desired value was obtained. The percent moisture at the point when the
sample weight has stabilised is called the saturation moisture content. The weight was checked
on a moisture balance with 0.001g accuracy. Before each test, specimens of a specific percent
moisture were heated up to the specified test temperature. The compression tests were carried
out using a strength tester with 50kN load cell andMTSmechanical extensometer resistant to
high temperature and moisture. The strength tester was integrated with the climatic chamber
placedwithin theMTSworking space tomaintain the specifiedambient conditions.The strength
tester incorporating the climatic chamber is presented in Figs. 5 and 6.

Fig. 4. (a) Desiccator with specimens placed inside, (b) stationary climatic chamber

The stress-strain curveswere obtained for each compression test on the basis of extensometer
displacement.The cross-sectional area of the specimenwasdeterminedbefore the test.Theother
input parameters, namely temperature andpercentmoisture, weremeasured right after the test.



506 M. Dudziak et al.

Fig. 5. MTS strength tester incorporating climate chamber 1 – strength tester, 2 – climate chamber,
3 – load cell (measuring the applied force)

Fig. 6. Climatic chamber with holding jaws assembly designed for compression test: 1 – specimen placed
between holding jaws, 2 – extensometer, 3 – force/ specimen alignment fixture to ensure that the force

is applied perpendicular to the specimen cross-section

An example of a typical curve obtained from a compression test carried out on a specimen of
beech wood in the tangential direction is presented in Fig. 7.

Fig. 7. The stress-strain curve for beech compressed in the tangential direction

The yield point values Re in the respective orthotropic directions L, T , R were taken as
the wood strength criterion used to set up constitutive equations describing the plasticization
model of wood. For the compression test, those parameters were designated RLec, R

Rec, R
T
ec
,

respectively. Their values were determined by the point where the tangent “departs” from the
curve by more than 1% of the ultimate strength value. This point, in turn, was determined by



Analysis of the process of wood plasticization by hot rolling 507

solvinga systemof equations comprisinganequationof a straight line tangent to the stress-strain
curve in tension and an equation of the actual stress-strain curve in compression (Fig. 7).
The results of each test type were recorded in single test record sheets and test logs which

included all the relevant strength test information. Table 2 presents an example of a single test
record sheet from a compression test carried out on a beech wood specimen with 9% moisture
at 20◦C. The test logs include the results obtained for the whole test series and accompany the
respective test descriptions. An example of such a test log is presented in Table 3.

Table 2. Single test record sheet (compression of beech in the tangential direction, 9% mois-
ture, 20◦C

Wood variety beech

Direction tangential

Moisture content [%] 9

Temperature [circC] 20

Cross-section area [mm2] 402

Young’s modulusE [MPa] 1282.857

Breaking force [M] 4894.27

Ultimate strengthRTc [MPa] 12.175 Test No. 1

Shortening atRTc [mm] 0.407

Strain at RTc 0.016

Yield pointRTec [MPa] 4.495

Shortening atRTec [mm] 0.09225

Strain at RTec 0.00369

RTec/R
T
c 0.369

Gauge length l0 [mm] 25

Table 3.One series of tests: beech – tangential direction

Test No.
1 2 3 avg.

Wood variety beech

Direction tangential

Moisture content [%] 9

Temperature [◦C] 20

Cross-section area [mm2] 402 406 404 404

Young’s modulusE [MPa] 1282.857 1244.392 1267.786 1265.01

Breaking force [N] 4894.27 5309.5 5239.05 5147.60

Ultimate strengthRTc [MPa] 12.175 13.078 12.969 12.7406

Shortening atRTc [mm] 0.407 0.565 0.475 0.48233

Strain at RTc 0.016 0.023 0.019 0.01929

Yield pointRTec [MPa] 4.495 3.541 4.21 4.082

Shortening atRTec [mm] 0.09225 0.073 0.08675 0.084

Strain at RTec 0.00369 0.00292 0.00347 0.00336

RTec/R
T
c 0.369 0.271 0.325 0.32152

Havingall the results, itwas possible todefine the relationbetween strengthand temperature
for the respective moisture content levels.
An example of the test results is presented in form of curves in Fig. 8. They relate the yield

pointRe to the effect of temperature for three differentmoisture content levels in the tangential
direction.



508 M. Dudziak et al.

Fig. 8. Yield pointRT
ec
vs. temperature T for different moisture content levels (beech wood,

tangential direction)

2.2. Experimental determination of thermal conductivity

The thermal tester used in the determination of the thermal conductivity of wood was
designed specifically for testingmaterials of low thermal conductivity (Fig. 9a). As the first step,
specimens of the same density and thickness were prepared. The test set-up components were
aligned coaxially and then load was applied to the top of the heat sink. Then the whole set-up
was thermally isolated and the remaining parts, namely temperature sensors, power supply unit
and themeasuring device were connected. The systemwas energised and the heat sink switched
on. Measurements were taken upon reaching the test temperature and thermal equilibrium.

Themeasurementswere carriedoutonbeechwood (Fig. 9b) in the threedirections in relation
to the grain: L, T , R (longitudinal, tangential and radial) at test temperatures of 40, 60, 80,
100, 120, 140 and 150◦C to obtain arithmetic average of five measurement results for each test
point.

Fig. 9. (a) Experimental set-up for measuring thermal conductivity: 1 – thermal conductivity tester,
2 – amplifier, 3 – recording computer; (b) specimens of beech wood used for thermal conductivity

measurement tests

The measurement results for beech wood were used to derive the curves (Fig. 10) relating
the thermal conductivity coefficient λ to temperature for the three directions in relation to the
grain:L, T ,R.

The relationship between the thermal conductivity and temperature is described with ap-
plication of an arctan approximation model λarctan, specially developed for this purpose by the
authors. This description is consistent with the approximation methods developed by several
researchers working in different research centres (Harada et al., 1998; Gu and Zink-Sharp, 2005;
Yang, 2001).Malujda andMarlewski (2011) demonstrated that the arctan approximationmodel
provided more accurate approximation that the methods proposed by other researchers.



Analysis of the process of wood plasticization by hot rolling 509

Fig. 10. Thermal conductivity vs. temperature for different fibre directions, approximation with the
arctanmodel

3. Analysis of plasticization of an anisotropic material

In anisotropic materials, specific directions can be found in which plastic properties reach the
extremevalues. If these directions are oriented orthogonally one to another,we call it orthotropy,
and the material with such characteristics is referred to as an orthotropic material (German,
2001). Wood is an example of such an orthotropic material.
The problem of defining the yield criterion of the surface layer of wood is approached as a

two-dimensional problem (Fig. 11).

Fig. 11. Treatment to improve properties of the surface layer of the natural material by hot rolling:
1 – adjustable upper roller, 2 – layer of wood, 3 – chipboard,Rw – roller radius, s – roller/material
contact length, h – depth of the layer of wood, l – board width,∆h – compression degree, directions:

L(1) – longitudinal, T(2) – tangential,R(3) – radial

Let us use the anisotropic yield criterion of Azzi-Tsai-Hill (ATH) to analyse the ultimate
stress-strain state of the analysed layer of wood, which considers its multi-directional reactions
resulting fromthe components of the complex stress state.Hill (1956) generalized theHMHyield
criterion to make it hold true for orthotropic materials. Wood is a natural polymer for which
the reference system of co-ordinates can be set up so that the directions of axes coincide with
the orthotropic directions L, T , R (Fig. 11). Having taken this assumption, the yield criterion
may be represented by

(M+N)σ211+(L+N)σ
2
22+(L+M)σ

2
33−2Nσ11σ22−2Mσ11σ33

−2Lσ22σ33+2Pσ
2
23+2Qσ

2
31+2Rσ

2
12 =1

(3.1)



510 M. Dudziak et al.

whereL,M,N are theproportional limitswith thevaluesdependingon theorthotropic direction
and P,Q,R are the respective shear strength proportional limits.

The yield criterion formulated in thisway is the strength criterion of an anisotropicmaterial,
andthecritical strength is reachedat theproportional limit. It is a characteristic featureofbrittle
materials, such as wood, for which these limits coincide with the yield and elasticity limits, and
this has been confirmed by the results of experimental research.

The process of plasticization of natural materials having such complex properties as, for
example, wood has not yet been described by non-linear models. Thus, the engineers cannot
use amathematical model whose structure is built of complex engineering constants, typical for
natural polymers.

Now, let us take the process of hot rolling of wood (Fig. 11) and try to derive constitutive
relations for natural materials by analytical method, assuming applicability of the theory of
plasticity taking into account the plastic compressibility mechanism and allowing for the effect
of heat andmoisture (Bordia et al., 2006; Fujii et al., 2003; Malujda, 2006; Müllner et al., 2004;
Nairn, 2005).

The key parameter for the modelling of the process in consideration is the ultimate stress
corresponding to the lower limit of statically determined strength of the layer subjected to
loading (Ashby and Jones, 1996). At this point, having in hand the mathematical model, we
shall try to determine the ultimate load resulting in plasticization of the material. The value of
this load is of primary importance to the efficiency of the analysed process.

The establishing of a model can be defined as the identification of the process occurring
in the material. Therefore, it is necessary to carry out in-depth analysis of these elements of
the model which are critical to obtaining a satisfactory solution thereof. In the case of wood,
these are: complex thermo-mechanical properties, porosity and process parameters including in
particular the ultimate force F which depends on temperature and process duration (Malujda,
2006).Wood is ahighlyporousmaterial and, for this reason, theultimate stress dependsnot only
on the second but also on the first invariant of the stress tensor. This effect must be considered
in the final form of the yield criterion, similarly to the effect of direction in relation to the
grain (Tsai and Wu, 1971). For this reason, the Huber von Mises-Hencky yield criterion for
isotropicmaterials was used (German, 2001; Hill, 1956;Malujda andTalaśka, 2010), generalized
by Azzi-Tsai-Hill (ATH) to hold true for anisotropic materials.

Scalar functions of porosity, temperature andmoisture are thenfitted into this yield criterion
formula, generalised to hold true for anisotropic materials. Subsequent analyses are carried out
for the assumed physical model of the unsupported rigid-plastic medium.

Now, let us formulate the general yield criterion expressing the yield criterion for an aniso-
tropic andporousmaterial allowing for the effect of temperature andmoisture content expressed
generally as a function describing the yield surface

F(σ,fv,T,w)= 0 (3.2)

where σ is the stress tensor and the other terms are scalar functions of porosity fv, temperatu-
re T and moisture content w (Malujda and Talaśka, 2010). The structure of the yield criterion
formulated in this way is created by interrelated physical quantities which have direct effect on
the value of the ultimate yield force.Wood features porosity (Malujda andTalaśka, 2010), which
affects its yield point. This effect has been allowed for in the formulated mathematical model
by introducing the following scalar functions of porosity

A=
fv
1−fv

B=1−fν (3.3)



Analysis of the process of wood plasticization by hot rolling 511

which, while meeting the following limit conditions:

if fv → 0, A→ 0, then B→ 1

if fv → 1, A→∞, then B→ 0

reduce criterion (3.2) to HMH criterion.
Introducing porosity functions (3.3) accorindg to equation (3.2), we obtain

A(σ1+σ2)
2+(σ1−σ2)

2 =BY 21 (3.4)

where σ1 and σ2 are the principal stresses in the orthotropic directions (L-T).
In yield criterion (3.4), the complex ultimate stress-strain state is equated to the stress-

-strain state expressed by the non-zero main stress determined during a uniaxial compression
test as a function of temperature and moisture content Y1 = R

L
ec
. At the critical state, this

stress obtains a constant value at the strength limit of wood. The compressive yield strength
has been taken as the parameter defining the critical stress of a natural polymer, i.e. the stress
created by application of the ultimate force. This parameter is used as the basis for evaluating
the characteristics of composite materials, such as wood, and its value depends on temperature
andmoisture content as well as on the direction of load application in relation to the grain.
In the first phase of pressure application to the thin layer of wood, permanent strains are

producedwithout a significant effect of heat. Nevertheless, they are necessary in order to obtain
the roller/ surface contact time sufficient to increase the temperature to approximately 120◦C
depending on the wood variety. As it has been demonstrated in the experimental research, this
process takes very little time, namely ca 0.04-0.1s. During that time, heat penetrates into the
thin layer of wood by ca 0.1-0.12mm which is the key element of the process of improving the
material properties.
Hence, the process in considerationmay be divided into two stages. The first stage concerns

permanent strains created upon exceeding theproportional limit and it is related to densification
of the structure of wood. In the second stage, smoothing and consolidation of the structure of
the heated surface layer takes place. The yield load is reduced as a result of a decreased strength
of wood resulting from an increase in temperature in the processing zone.
Introducing the orthotropy coefficients α= (Y1/Y2)

2 and β = (Y1/Y3)
2, established on the

basis of experimentally determined strength of wood Y1, Y2, Y3 into equation (3.4), gives the
following yield criterion

σ21 +ασ
2
2 − (1+α−β)σ1σ2+Aσ

2
1 +Aσ

2
2α+2A(1+α−β)σ1σ2 =BY1 (3.5)

which is then rearranged to

σ21(1+A)+ασ
2
2(1+A)− (1−2A)(1+α−β)σ1σ2 =BY1 (3.6)

Some additional assumptions have been taken and simplifications have been made due to
complexity of the process of hot rolling of wood. This process concerns plasticization of a thin
layer of the processedmaterial in the case when two dimensions aremuch higher than the third
one. The depth of this layer h is small (Fig. 11) as compared to the lateral dimensions and hence
the process is analysed in the in-plane strain state. The high value of thewidth do depth ratio of
l/h> 10 and porosity of the material structure allow us to consider the strain in the tangential
direction T(2) ignorable (dε22 = 0) (Bednarski, 1995). This is supported by the approach used
in real-life industrial applications. The second of the important simplifications is leaving out the
stress σ1 justified by the low value of the rolling resistance (ca 2% of the normal force) in the
rolling directionL(1) (Bednarski, 1995). This is due to a very small contact angle (large diameter
of the roller is required for process-related reasons) and hence it is justified to assume that the



512 M. Dudziak et al.

roller pressure p=F/(sl) (Fig. 11) is almost entirely directed on the component perpendicular
to the planeL(1)-T(2) (Fig. 11).
In the next step of building up themodel, we shall use the associated plastic flow rule which

relates the strain increase deviator dεij to the plastic potential function F(σij) represented by
the following relationship

dεij = dς
∂F(σij)

∂σij
(3.7)

where ς is a coefficient determined from the increase of work of plastic strain.
Hence, with the assumed value of dεij = 0, the associated plastic flow rule is expressed by

the following formula

dε22 = dς
∂F

∂σ22
=0 (3.8)

Differentiating equation (3.5)

∂F

∂σ2
=2ασ2(1+A)− (1−2A)(1+α−β)σ1 =0 (3.9)

yields

σ2 =
(1−2A)(1+α−β)

2α(1+A)
σ1 (3.10)

Substituting expression (3.10) described by equation (3.6) to get rid of σ2, we get

σ21(1+A)+α(1+A)
[(1−2A)(1+α=β)

2α(1+A)

]2
σ21

− (1−2A)(1+α−β)
(1−2A)(1+α−β)

2α(1+A)
=BY 21

(3.11)

Rearranging equation (3.11), we get the following relationship for calculating the stress σ1

σ21 =
B

(1+A)−
(1−2A)2(1+α−β)
4α(1+A)

Y 21 (3.12)

Now let us use yield criterion (3.6) in order to determine the isotropic state stress

Ap2+Aαp2+2A(1+α−β)p2+βp2 =BY 21 (3.13)

and thus, upon rearranging equation (3.13), we obtain

p2 =
B

A(3+3α−2β)+β
Y 21 (3.14)

IfA→ 0 then p→∞, where fv → 0means incompressibility of the analysed material.
Finally, we get the following equation describing the yield criterion, expressing the ultimate

stress state derived through transition from the in-plane strain state to the equivalent in-plane
stress state

p=

√

√

√

√

√

[

B

(1+A)−
(1−2A)2(1+α−β)
4α(1+A)

−

B

A(3+3α−2β)+β

]

Y 21 (3.15)

Theabove expression, describing theultimate roller pressureapplied to theprocessed layer of
wood allows for the porosity and anisotropic properties of the analysed material. Experimental
determination of the compressive yield strength in the longitudinal directionY1 =R

L
ec
takes into

account the effect of heat andmoisture content.



Analysis of the process of wood plasticization by hot rolling 513

4. Analytical and numerical solutions of the model of the thin layer plasticization

process

In the obtained analytical model, functions of material characteristics are applied including the
arctangent thermal conductivity function. The relevant data are given in Table 1. Since all the
functions ofmaterial characteristics required for thedetermination of the critical stress according
to (3.15) depend on the temperature, the values calculated with equation (3.15) are compared
to the results of FEM analysis by determining the stress value for a given temperature at a
given point during the rolling processmodelledwith the FEMsoftware. The process parameters
assumed as the input for calculations correspond to the values applied in wood improvement
processes in furniture manufacturing applications. Figure 12 illustrates geometry of the model
derived for the purpose of FEM analysis.

Fig. 12. Model derived for the purpose of FEM analysis: 1 – adjustable upper roller, 2 – lower roller,
3 – processedmaterial, processed layer of wood showing nodes in the depth-wise direction

A 0.8mm deep surface layer of the processed material is sub-divided into twenty finite ele-
ments. The entire model comprising of the roller and processed material is “encases” with air
at temperature of 20◦C. The purpose is to imitate the real-life ambient temperature of the sur-
rounding air which is about 20◦C.The roller/material contact parameters are defined assuming
that the exchange of heat between these surfaces is close to ideal. The abutting surfaces of the
roller and the processed material are assumed to have the same temperature, namely 220◦C.
The duration of contact between each point located in the thin surface whose location is

defined by the co-ordinates of the node of a rectangular grid component, and the roller is
assumed at ca 0.06s. This time corresponds to ca 6m/min linear speed of the roller (process
parameter). The thermo-mechanical properties of beech wood and geometric parameters of the
analysed wood improvement process are presented in Tables 4 and 5.
The curves in Figs. 13a,b,c represent the relation between the stress value and the depth

below the surface of the rolled material determined with the use of analytically determined
relationship (3.15) andFEManalysis. Figure 14 presents the relative percent difference between
the values calculated with equation (3.15) and the results of FEM analysis.

5. Conclusions

The processes of plasticization of natural polymers in real-life conditions are governed by seve-
ral parameters. The relevant factors include external mechanical loads, temperature, moisture
content and process duration. Consequently, the established constitutive relations have a more
general form.
The analysis of plastic strain in the analysed layer of wood is based on a generalisedmodel of

an ideally rigid-plastic medium with certain simplifications and modifications. Taking account



514 M. Dudziak et al.

Table 4.Material characteristics – thermo-mechanical properties of beech

Physical quantity Designation Unit Value/function

Arctangent thermal conducti- λLarctan λ
L
arctan =133.806arctanT

2
−209.712

vity in theL,T,R directions λRarctan W/(m
◦C) λRarctan =57.723arctanT

2
−90.462

in relation to the grain λTarctan λ
T
arctan =12.67arctanT

2
−19.739

EL
c−9%

=−78.48T +17809

EL
c−18%

=−21.514T +11574

EL
c−27%

=−34.454T +8530.9

E
L,R,T
c−9%

ER
c−9%

=−8.363T +2047.2

Elastic constant E
L,R,T
c−18%

MPa ER
c−18%

=−7.757T +2022

E
L,R,T
c−27%

ER
c−27%

=−1.3884T +1114

ET
c−9%

=−4.0966T +1256.5

ET
c−18%

=−5.775T +1278.2

ET
c−27%

=−3.598T +832.78

RLec−9% =−0.06T +17.451

RLec−18% =−0.047T +13.213

RLec−27% =−0.0312T +9.5012

Y1 =R
L
ec

RRec−9% =−0.0201T +6.0151

Yield strength Y2 =R
T
ec

MPa RRec−18% =−0.014T +4.6398

Y3 =R
R
ec

RRec−27% =−0.0102T +3.4249

RTec−9% =−0.015T +4.2452

RTec−18% =−0.0128T +3.9331

RTec−27% =−0.0101T +2.5629

Specific heat capacity
cpo kJ/(kgK) cpo =0.1031+0.003867T

of dry wood

cp =
cpo+0.01wcpw
1+0.01w+Ak

w – moisture content (9%,18%,27%)

Specific heat capacity cpw – specific heat of water

allowing for the effect cp kJ/(kgK) Ak – coefficient

of moisture content Ak =−0.07592 (9%)

Ak =−0.17338 (18%)

Ak =−0.29238 (27%)

Density ρ kg/m3 810

Moisture content m % 9, 18, 27

the anisotropic properties of wood, the Azzi-Tsai-Hill (ATH) strength criterion is used as it
allows for the variation of reaction to the loading depending on the direction. The strength
criterion is based on the experimentally determined compressive yield strength allowing for the
effect of temperature andmoisture content.Theultimate loaddetermined in thiswaydependson
the established strength criterion as well as factors defining porosity and orthotropic properties
of the material which relate the yield strength values in the respective orthotropic directions.



Analysis of the process of wood plasticization by hot rolling 515

Table 5.Geometric parameters of the analysed wood improvement process

Physical quantity Designation Unit Value

Roller radius Rw m 0.1565

Thickness of layer before rolling h1 m 0.0008

Linear speed v m/s 0.1

Roller temperature T ◦C 220

Fig. 13. Stress depending on depth inwards the processedmaterial with (a) 9%, (b) 18%, (c) 27%
moisture content

Fig. 14. Percent difference between the results of analytical and numerical calculations as a function of
depth of the layer of the plasticized material

The ultimate stresses determined with the above-mentioned model are used in simulations
and numerical calculations. The equations presented herein may be used to ensure the desired
properties and dimensions of products and to define input assumptions and design machines
used to realise the process in consideration and other similar processes.

According to the results of the process lasting ca 0.06s, the calculated stresses generated by
the roller in the plasticized layer of wood decrease nearly two times. This is due to an increase
in temperature up to the range of 110-130◦Catwhich plasticization of the surface layer of wood
to ca 0.1mm in depth occurs.

It can be seen that an increase in the moisture content results in a decrease in the value of
stress. With the moisture content of 27%, the stress values are almost two times lower at the
same depth.



516 M. Dudziak et al.

The percent differences between the results obtained by FEM analysis and calculated using
the ultimate roller pressure formula are small. This confirms that the assumptions and simplifi-
cations made in the process of deriving constitutive relationships relating to the rolling process
are appropriate.

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Manuscript received June 23, 2015; accepted for print September 15, 2015