H. SRBOVÁ: COMPARISON OF HOMOGENIZATION APPROACHES USED FOR THE IDENTIFICATION ...
373–378
COMPARISON OF HOMOGENIZATION APPROACHES USED FOR
THE IDENTIFICATION OF THE MATERIAL PARAMETERS OF
UNIDIRECTIONAL COMPOSITES
PRIMERJAVA HOMOGENIZACIJSKIH PRIBLI@KOV ZA
UGOTAVLJANJE PARAMETROV MATERIALA ENOSMERNIH
KOMPOZITOV
Hana Srbová, Tomá{ Kroupa, Vladimír Luke{
University of West Bohemia in Pilsen, NTIS – New Technologies for the Information Society, Univerzitní 22, 306 14, Plzeò, Czech Republic
hsrbova@kme.zcu.cz
Prejem rokopisa – received: 2015-07-01; sprejem za objavo – accepted for publication: 2016-06-06
doi:10.17222/mit.2015.177
The paper is aimed at the identification of the material parameters of the constituents of unidirectional long-fiber carbon/epoxy
composites. The ratios between the real fiber and the matrix volume of unidirectional fiber composites were identified from the
images obtained with scanning electron microscopy (SEM). Fiber and matrix areas were analyzed using Matlab and its
image-processing toolbox. Determined volume ratios were used to propose the geometry of a micromechanical representative
volume element. An algorithm ensuring an irregular fiber distribution in the representative volume element cross-section and
periodicity in the cross-section plane were used. A single-layer representative volume element with a mesh was built in
Abaqus/CAE. Experimentally obtained force-elongation dependencies for the tensile tests of the specimens with different fiber
orientations were processed and effective moduli were determined. In the first approach, the material parameters of phases were
identified using the OptiSlang optimization software and finite-element code Abaqus. A periodically constrained representative
volume element was loaded according to the experiments where specimens with different fiber orientations were loaded by
tensile force. In the second approach, the SfePy software and asymptotic homogenization were used for the numerical compu-
tation on the microscopic scale, resulting in the homogenized material parameters that were afterwards used at the macroscopic
level. The optimization function characterized the sum of the differences of the effective moduli and the difference of Poisson’s
ratios of the whole composite.
Keywords: composite, unidirectional, micro-model, constituents, material parameters, cross-section, homogenization
Namen ~lanka je ugotavljanje materialnih parametrov sestavin epoksi kompozitov z enosmernim dolgimi ogljikovimi vlakni.
Razmerje realnih vlaken in volumna osnove kompozitov z usmerjenimi vlakni je bilo dolo~eno iz slik, dobljenih na vrsti~nem
elektronskem mikroskopu (SEM). Vlakna in podro~ja osnove so bili analizirani s pomo~jo Matlaba in njegovega orodja za
obdelavo slik. Dolo~ena volumska razmerja so bila uporabljena za predlog geometrije reprezentativnih mikromehanskih volum-
skih elementov. Uporabljen je bil algoritem, ki zagotavlja neenakomerno razporeditev vlaken na preseku reprezentativnega
volumskega elementa in periodi~nost na ravnini preseka. Plast reprezentativnega volumskega elementa z mre`o, je bila
postavljena z Abaqus/CAE. Izvedeni so bili eksperimenti za dolo~anje odvisnosti sila-raztezek pri nateznih preizku{ancih z
razli~no orientacijo vlaken in dolo~eni so bili efektivni moduli. V prvem pribli`ku so bili ugotavljani materialni parametri faz s
pomo~jo OptiSlang programske opreme za optimizacijo in kodo kon~nih elementov Abaqus. Periodi~no omejen reprezentan~ni
volumski element je bil obremenjen, skladno s preizkusi. Pri ~emer so bili vzorci z razli~no orientacijo vlaken, obremenjeni z
natezno silo. Pri drugem pribli`ku je uporabljena programska oprema SfePy in asimptoti~na homogenizacija, za numeri~ne
izra~une na mikroskopskem nivoju, kar se je odrazilo v homogenih parametrih materiala, ki so bili kasneje uporabljeni na
makroskopskem nivoju. Optimizacijska funkcija je karakterizirana z vsoto razlik efektivnega modula in razliko Poissonovega
razmerja za celoten kompozit.
Klju~ne besede: kompozit, enosmernost, mikromodel, sestavine, parametri materiala, presek, homogenizacija
1 INTRODUCTION
The experiments are the main tools and play a domi-
nant role in the characterization of mechanical behavior
of materials.1–4 Nevertheless, in some cases, it is
advantageous to support experimental measurements
with the knowledge of the behavior of the considered
material at the micro- or meso-scale level (low scale).5–7
A precisely calibrated low-scale model can be helpful in
the cases where all the necessary material parameters for
a macro-scale model are impossible to be characterized
or in the case when the initiation or propagation of the
processes observed on the micro-scale are not clear.
Anisotropy and heterogeneity of the inner structure
and homogeneity of virtual macro-scale models of com-
posite materials call for low-scale-model processing. The
presented paper is aimed at a comparison of two metho-
dologies for the identification of the material parameters
of the substituents of unidirectional carbon-epoxy
composites. There are several different approaches for
homogenization and low-scale modelling, the presented
paper deals with two of these. The first approach uses the
advantages of commercial software Abaqus 6.14-2 and
its wide range of pre- and post-processing tools includ-
ing the possibility of an automatic model construction
using the Python software.5 The second approach uses
Materiali in tehnologije / Materials and technology 51 (2017) 3, 373–378 373
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
UDK 67.017:620.1:621.385.833 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 51(3)373(2017)
the well established theory of asymptotic homogeniza-
tion,6,7 which is implemented in the SfePy finite-element
software.8–10
2 SPECIMENS AND THE EXPERIMENTAL
PART
Carbon-epoxy unidirectional continuous-fiber com-
posite coupons were subjected to tensile loading.
Rectangular specimens (coupons) were cut by a waterjet
from a composite plate made of eight equally oriented
layers, the so-called prepregs with 40 % of epoxy
HexPly 913C with fiber Tenax HTS 5631 cured for 120
minutes at a temperature of 125 °C and a pressure of 6
bar. There were 5 specimens tested for each of the
following fiber orientations (10, 20, 30, 40, 50, 60, 70,
80 and 90)° with respect to the loading direction. Only
two specimens were successfully tested up to the allow-
able type of rupture with fiber orientation 0°.
Force-elongation dependencies were obtained using a
Zwick Roell/Z050 test machine where force values were
estimated by a force cell and the gage area elongation
was measured using a uniaxial macro-extensometer. The
tested coupons with glass textile (0°) or aluminum tabs
(10, 20, 30, 40, 50, 60)° were bonded with high-strength
and tough epoxy adhesive Spabond 345. In the case of
the specimens with fiber orientations (70, 80 and 90)° a
non-bonded emery cloth was used as the interface bet-
ween the grip and the coupon. The tab configuration
summarized in Table 1 led to the required gage-section
tensile failure (Figure 2).
Table 1: Experiment configuration
Tabela 1: Konfiguracija preizkusov
Fiber
orientation (°) Tab material
Tab length
(mm)
Extensometer
length (mm)
0 glass textile 60.0 10.0
10, 20, 30, 40,
50, 60 aluminum 25.0 60.0
70, 80, 90 emery cloth 25.0 60.0
3 IDENTIFICATION
Two linear material models were supposed to be cali-
brated using a combination of finite-element analyses,
optimization methods and experiments.11,12 The targets
for the identification process were calculated from the
experimental results before the identification process in
Python began. The nominal stress was calculated from
the applied tensile force F as:
 =
F
A
(1)
where A is the cross-section of a non-loaded specimen.
The strain was calculated as

=
F
A
(2)
Where l is the elongation of the gage area (between
the arms of the extensometer) and l is the original gage
length.
H. SRBOVÁ: COMPARISON OF HOMOGENIZATION APPROACHES USED FOR THE IDENTIFICATION ...
374 Materiali in tehnologije / Materials and technology 51 (2017) 3, 373–378
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 1: Geometry of coupons (mm), the cross-section (A-B)
Slika 1: Geometrija kuponov (mm), presek A-B)
Figure 3: Tensile dependencies (dashed lines) and calculated targets
for further identification (solid lines)
Slika 3: Natezne odvisnosti (~rtkana linija) in izra~unani cilji za
nadaljnjo identifikacijo (polna ~rta)
Figure 2: Cracked specimens, the final crack direction is parallel to
the fibers for directions 10°–90° and perpendicular for direction 0°
Slika 2: Poru{eni vzorci, usmeritev kon~ne razpoke je vzporedna z
vlakni za smeri 10°–90° in pravokotna za smer 0°
The target for each tensile test was calculated as
least-squares fit of the straight line through the interval
of the nominal strain 0, 0.25 %
 in the case of the
concave dependency of force on elongation (Figure 3).
The only exceptions are the experimental results with 0°
direction of the fibers where the target was calculated as
least-squares fit of the straight line through the whole
force-elongation dependency (Figure 3). The results for
0° fiber orientation are influenced by the uncertainties
caused by the penetration of the glass textile by the jaws
of the machine at the beginning of the measurement (the
loading process).
The tangents kE of the fitted straight lines in the
force-elongation diagram for each fiber orientation were
taken as the values to be achieved in the numerical
model. The only additional condition, which should be
met in the model, is that Poisson’s ratio v C12 of the whole
composite should be as close as possible to the value of
0.28 (the measured value and the value obtained from the
manufacturer). The residual function r to be minimized
in the identification processes, regardless of the used
homogenization approach (homogenization using the
commercial software Abaqus or asymptotic homogeni-
zation performed in SfePy), has the following form:
r
k
k
v C
= −
⎛
⎝
⎜
⎞
⎠
⎟ + −
⎛
⎝
⎜
⎞
⎠
⎟∑ 1 1
0 28
2
12
2
N
E
( )
( ) .


(3)
Where  is the fiber angle (the type of specimen) and
kN is the tangent of the numerically obtained results
(Figure 3). Gradient algorithms implemented in the
OptiSLang and Python software were used for the identi-
fication.12
4 CREATING A REPRESENTATIVE UNIT CELL
The representative volume element with a random
distribution of fibers was built with an event-driven
algorithm for simulating chaotically colliding billiard
balls13 in Matlab. Each component (a fiber in this case)
moves along a straight line until it collides with another
component. Events (collisions) are processed in a non-
decreasing order of time and interactions are scheduled
using pre-integrated equations of the evolutions of the
components. In the case of a uniform-component distri-
bution, the evolution space is subdivided into small
sectors so that only the components in the neighboring
sectors have to be checked to determine the immediate
collision and the time for processing the expected event
is reduced. The defined fiber volume fraction is ensured
with the termination criterion controlled by an increase
in the volume fraction of the components.14
Periodicity of the volume element is ensured by a
component disappearing at a boundary and reappearing
at the opposite side (Figure 4).
The calculation of the fiber volume fraction was
performed in the Matlab software. Digital images
obtained with scanning electron microscopy (SEM) were
analyzed (Figure 5) and the identified fiber volume
fraction of the composite material is Vf
image
= 0.59 The
average fiber radius was identified as Rf = 6.488 μm. The
rectangular area with 15 fibers and dimensions of
28.7334 × 28.7334 μm was discretized in Abaqus/CAE
by hexahedral 8-node linear isoparametric elements and
wedge 6-node linear isoparametric elements (Figures 5
and 7).
The finite-element mesh was built with only one
layer of elements in the fiber direction. The fiber direc-
tion and loading direction are shown in Figure 6.
5 MATERIAL MODELS
The infinitesimal strain theory was considered when
proposing the material models. The material model of
fibers was linear, transversely isotropic and the behavior
in the plane perpendicular to the fiber axis was con-
sidered to be isotropic. Five material parameters define
this type of material. A material model of the matrix was
proposed to be linear isotropic with two parameters
necessary for the description of its behavior. The
material model can be written in the matrix form as
H. SRBOVÁ: COMPARISON OF HOMOGENIZATION APPROACHES USED FOR THE IDENTIFICATION ...
Materiali in tehnologije / Materials and technology 51 (2017) 3, 373–378 375
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 5: Image of cross-section (A-B) of unidirectional long fiber
composite obtained with SEM (left) and unit cell with generated finite
elements (right) and depicted fiber edges (black circles)
Slika 5: SEM-posnetek preseka (A-B) enosmernih dolgih vlaken v
kompozitu (levo) in enotna celica z ustvarjenimi kon~nimi elementi
(desno) in prikaz robov vlakna (~rni krogi)
Figure 4: Parts of the fiber at the area boundaries in accordance with
periodic boundary conditions
Slika 4: Deli vlakna na podro~ju mej v skladu s pogoji periodi~nosti
meje









1
2
3
12
13
23
11 12 13
12 22
0 0 0⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
=
S S S
S S S
S S S
S
S
S
23
13 23 33
44
55
66
0 0 0
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⋅
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥






1
2
3
12
13
23
(3)
Where 
i is the normal engineering strain in direction
i, ij is the engineering shear strain in the ij plane, all
being the elements of the strain tensor 
, i is the normal
stress in direction i and ij is the shear stress in the ij
plane, all being the elements of the stress tensor . The
elements of the compliance matrix S, the elasticity tensor
in the inverse shape, are, in the case of isotropic material,
as follows in Equation (4):
S S S
E
S S S
v
E
S S S
E
v
11 22 33
12 13 23
44 55 66
1
2 1
= = =
= = = −
= = =
+( )
(4)
and in the case of transversely isotropic material in
Equation (5):
S
E
S S
E
S S
v
E
S
v
E
S S
11
1
22 33
2
12 13
12
1
23
23
2
44 55
1 1
= = =
= = − = −
= =
1 2 1
12
66
23
2G
S
v
E
=
+( )
(5)
The meanings of the material constants (parameters)
of the models mentioned above are given in Table 2.
Table 2: Description of material constants
Tabela 2: Opis konstant materiala
Constant Unit Meaning
Fibers
E1 (Pa) Young’s modulus in the axis direction
E2 (Pa) Young’s modulus in the radial direction
v12 (-) Poisson’s ratio in the planes parallel tothe fiber axis
v23 (-)
Poisson’s ratio in the planes perpendi-
cular to the fiber axis. This parameter
was defined by the manufacturer.
G12 (Pa) Shear modulus in the planes parallel tothe axis of the fiber
Matrix
E (Pa) Young’s modulus
v (-) Poisson’s ratio
5.1 Micro-model in the commercial software
The geometry of the finite-element model (a micro-
model) of the periodically repeated volume (a unit cell)
of the unidirectional fiber composite material was
created in Abaqus/CAE 6.14-2.
The global coordinate system (xyz) is given by the
force direction (x) and the direction perpendicular to the
composite surface (z). The local coordinate system (123)
is defined by the unit-cell edges where the axis directions
correspond to the fiber direction (1) and the directions
perpendicular to it (Figure 7).
Assuming a uniaxial stress across the whole speci-
men, the behavior of the material can be simulated by
loading the unit cell with the normal stress correspond-
ing to the external tensile force F.
The loading force is represented by stress x obtained
with Equation (1) and transformed to the local coordi-
nate system using the following form:



   
 
1
2
12
2 2
2 2
2
2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= −
cos sin sin cos
sin cos sin cos
sin cos sin cos cos sin
 
     

−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⋅
2 2
0
0
x⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(6)
where  is the angle of rotation between the local and
global coordinate systems, e.g., between the fiber direc-
tion and force direction in the experiment.1,2
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376 Materiali in tehnologije / Materials and technology 51 (2017) 3, 373–378
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 6: Material principal and loading-force directions
Slika 6: Glavni material in smeri sile obremenitve
Figure 7: Equivalently deformed opposite boundaries of a heteroge-
neous unit cell
Slika 7: Ekvivalentno deformirani nasprotni meji heterogene enotne
celice
The results of the finite-element analysis (strains) are
transformed back to the global coordinate system using
the following form:5





   
 
x
y
xy
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
−cos sin sin cos
sin cos sin
2 2
2 2  
     





cos
sin cos sin cos cos sin2 2 2 2
1
2
1−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⋅
2
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
(7)
The unit cell must also follow the periodic boundary
conditions:5–7
Δ
Δ
Δ
u u u
v v v
w w w
B A
B A
B A
= −
= −
= −
(8)
where u, v and w are the translation differences of
the opposing pair of nodes in directions 1, 2 and 3,
respectively. These differences must remain constant for
all the pairs of the corresponding nodes on opposite
sides.5
The periodical boundary conditions were implemen-
ted in Abaqus/CAE 6.14-2 using equation constraints.
6 ASSYMPTOTIC HOMOGENIZATION
The two-scale asymptotic homogenization method is
used to identify the material parameters of the periodic
composite structure. The linear elastic medium (occupy-
ing domain ) whose microstructure can be represented
by a periodic (representative) volume element associated
with region Y and consists of the matrix part Ym and
fibers Yf (Figure 8). Using the standard method of
asymptotic expansion8 applied to the linear elastic
problem, we get a microscopic equation formulated in
terms of the characteristic response function and a ma-
croscopic equation that involves the homogenized elasti-
city parameters.6,8
The problem at the microscopic level can be for-
mulated to find the characteristic response function
 rs H Y∈ # ( ) so that:
D Y
D
ijkl kl
y
Y
rs
ij
y
ijkl kl
y
Y
rs
ij
y

  
 

  
 
∫
∫
=( ) ( )
( ) ( )
d
dY H Yin #∀ ( )
(9)
where 
y (·) is the linear strain tensor with respect to the
microscopic coordinate y,  i
rs
s riy= , H Y# ( ) is the set of
Y-periodic functions with the zero average value in Y.
The elasticity tensor D = S–1 is defined as:
D D y
D Y
D
f
f
m
= =
=
( )
in (transvers. isotropic fibers)
in (isotropic matrix)Y m
⎧
⎨
⎩
(10)
The solution of the microscopic equation, which has
to be solved with the periodic boundary conditions, is
used to evaluate the homogenized elasticity tensor:
D
Y
D Yijkl
H
pqrs rs
y kl kl
pq
y ij ij
Y
= − −∫
1

   
  ( ) ( )d (11)
The homogenized (macroscopic) elasticity problem
can be expressed with Equation (12):
D Y v tijkl
H
kl
x
ij
y
Y i it

 
 

( ) ( )u v0 0 0d d∫ ∫= (12)
which is solved for the unknown macroscopic displace-
ment field u0  H0(). The macroscopic linear strain
tensor is denoted by 
x (·)and H0() is the set of
admissible displacements with a zero value at a boun-
dary where the Dirichlet conditions are imposed.
The homogenization procedure, i.e., the computation
of the microscopic and macroscopic equations, is
embedded in an optimization loop in order to find the
material parameters of the components constituting the
composite structure. The optimization module of SciPy
(Python library for scientific computing)10 and the
Python-based finite-element solver SfePy9 for the nume-
rical solution of the optimization problem are used.
7 COMPARISON OF THE RESULTS
The identified material parameters obtained using
both homogenization approaches are in Table 3.
Table 3: Identified material parameters
Tabela 3: Opredeljeni parametri materiala
Parameter Unit Commercial Asymptotic
Fibers
E1 (GPa) 165.99 162.55
E2 (GPa) 16.13 18.52
v12 (-) 0.18 0.16
G12 (GPa) 44.86 45.65
Matrix
Em (GPa) 3.28 3.21
vm (-) 0.43 0.39
H. SRBOVÁ: COMPARISON OF HOMOGENIZATION APPROACHES USED FOR THE IDENTIFICATION ...
Materiali in tehnologije / Materials and technology 51 (2017) 3, 373–378 377
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 8: Macroscopic domain  and microscopic domain Y (repre-
sentative volume element) consisting of fibers Yf and matrix Ym
Slika 8: Makroskopska domena  in mikroskopska domena Y (repre-
zentativni volumski element), ki sestoji iz vlaken Yf in osnove Ym
Poisson’s ratio in plane 23 was considered to be
constant, v23 = 0.4. Both methodologies converged to
similar values of the identified parameters.
8 CONCLUSION
Both methodologies provide comparable results. The
asymptotic homogenization implemented in SfePy
exhibits a less time-consuming performance taking into
account the whole process of the calculation and pre-
and post-processing of the results in each optimization
design. Both approaches provide a possibility for an
inverse process, calculating the stress distribution at the
microscopic level using the macroscopic stress distribu-
tion. Nevertheless, commercial software Abaqus 6.14-2
with its ability to apply user subroutines provides (at the
present state) a wider range of possibilities for more
complicated constitutive relations (e.g., nonlinear elasti-
city, plasticity, damage).
Acknowledgement
This publication was supported by the project
LO1506 of the Czach Ministry of Education, Youth and
Sports.
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MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS