R. ZEM^ÍK et al.: ANALYSIS OF THE FIBER DISTRIBUTION, SIZE, AND VOLUME RATIO OF UNIDIRECTIONAL ...
59–64
ANALYSIS OF THE FIBER DISTRIBUTION, SIZE, AND VOLUME
RATIO OF UNIDIRECTIONAL COMPOSITE PLATES WITH
DIFFERENT THICKNESSES
ANALIZA RAZPOREDITVE VLAKEN, VELIKOSTI IN
VOLUMSKEGA DELE@A V ENOSMERNIH, RAZLI^NO DEBELIH
KOMPOZITNIH PLO[^AH
Robert Zem~ík1, Hana Srbová2, Kamil Ek{tein3, Ivan Pirner4, Rostislav Medlín5
1University of West Bohemia in Pilsen, NTIS – New Technologies for the Information Society, Univerzitní 22, 301 00 Pilsen, Czech Republic
2University of West Bohemia in Pilsen, Department of Mechanics, Univerzitní 22, 301 00 Pilsen, Czech Republic
3University of West Bohemia in Pilsen, Department of Computer Science and Engineering, Univerzitní 22, 301 00 Pilsen, Czech Republic
4University of West Bohemia in Pilsen, Department of Cybernetics, Univerzitní 22, 301 00 Pilsen, Czech Republic
5University of West Bohemia in Pilsen, New Technologies – Research Centre in the West Bohemian Region, Univerzitní 22, 301 00 Pilsen,
Czech Republic
zemcik@kme.zcu.cz
Prejem rokopisa – received: 2015-07-02; sprejem za objavo – accepted for publication: 2015-12-18
doi:10.17222/mit.2015.195
This work focuses on an assessment of the real fiber and matrix volume ratios of unidirectional fiber composites that can be
used for the design of corresponding numerical models on the micro-scale. Samples with polished cross-sections were prepared
from three composite plates and they were previously analyzed using scanning electron microscopy (SEM). All the plates were
manufactured from the same prepreg material using autoclave technology. Each plate consisted of a different number of plies.
Images of the various parts of the composite cross-sections obtained with SEM are analyzed using several image-processing
techniques programmed in Matlab, its Image Processing Toolbox, and C code. The results of this analysis are the center
positions and radii of all fibers within the image. The fiber and matrix volume ratios are determined subsequently and mutually
compared for different locations across the plates’ thicknesses.
Keywords: composite, unidirectional, micromechanics, constituents, image, detection, microscopy, cross-section
Delo je osredoto~eno na oceno realnih vlaken in volumenskega dele`a v osnovi, v enosmernih kompozitih iz vlaken, ki jih je
mogo~e uporabiti za postavitev ustreznih numeri~nih modelov na mikropodro~ju. Pripravljeni vzorci, s poliranim presekom, iz
treh kompozitnih plo{~, so bili pregledani s pomo~jo vrsti~nega elektronskega mikroskopa (SEM). Vse plo{~e so bile izdelane iz
enakega, v avtoklavih izdelanega, predimpregniranega materiala. Vsaka plo{~a je bila sestavljena iz razli~nega {tevila plasti.
Posnetki razli~nih delov preseka kompozita, dobljeni s SEM, so bili analizirani z ve~ tehnikami za obdelavo slik, programi-
ranimi v Matlab, Image Processing Toolbox in C-kodo. Rezultati teh analiz so pozicije sredine in radiji vseh vlaken na posnetku.
Pozneje so bili dolo~eni, in medsebojno primerjani, volumenski dele`i vlaken na razli~nih lokacijah preko debeline plo{~e.
Klju~ne besede: kompozit, enosmeren, mikromehanika, gradniki, posnetek, odkrivanje, mikroskopija, pre~ni presek
1 INTRODUCTION
All methods of fabricating unidirectional fiber-rein-
forced composites result in a non-uniform distribution of
fibers in the matrix. The fiber distribution affects the
mechanical properties of the composite. In 1 the local
stress fields and damage initiation in dependence on the
fiber distribution are investigated. Although the fiber
volume ratio (usually designated as vf) determines the
geometry of composite micromodels and subsequently
the identified homogenized material properties2, it is a
problem to determine it precisely. For example, a fiber
composite structure manufactured by transfer molding
method typically ranges between approximately vf = 0.5
and vf = 0.6.3 The actual value can be different in various
parts of the structure and it can significantly influence
the local properties, such as the stiffness, strength, or
durability.
2 EXPERIMENTAL PART
Three specimens were cut out from three different
carbon/epoxy plates (designated as A, B, and C) using a
water jet. Each plate was made of a different number (1,
2, and 4) of similar autoclaved prepreg layers. The spe-
cimens were fixed in epoxy resin and their cross-sections
were polished on a low-speed polishing machine.
Gray-scale images of various parts of the cross-sec-
tions were obtained using scanning electron microscopy
(SEM). The images were obtained using a magnification
factor of 4000 with 15.9 px/μm (pixels per micrometer),
as shown in Figure 1. They were taken at the top, middle
and bottom regions of the plate’s thickness. An example
of the image in Figure 2 with a full thickness view
(obtained using magnification factor of 150) shows the
layers and approximate position of the three investigated
areas.
Materiali in tehnologije / Materials and technology 51 (2017) 1, 59–64 59
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967-2017) – 50 LET/50 YEARS
UDK 620.17:62-49:621.385.833 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 51(1)59(2017)
In general, the quality of the obtained SEM images is
quite low and the contrast varies between the different
areas and samples. Therefore, the following automated
image analysis includes several quality-improving steps.
3 IMAGE PROCESSING
The design stage of the image-processing algorithm
was preceded by a thorough analysis of the obtained
gray-scale microscopic images of the material cuts. Each
image is a matrix f(x,y) with pixel intensity values rang-
ing from 0 to 255 and x and y being the pixel coordinates
(x = 1..M, y = 1..N, where M×N defines the image size).
All these images demonstrated a substantial, strong
common trait, which is a rather small contrast gradient
over the whole image. Thus, the standard computer
vision methods for both edge detection and binarization
provide poor results or fail completely, which makes the
object detection impossible.
The analysis of the image collection proved that the
tonal distribution of the gray-scale images (represented
by a histogram of pixel values) is of a bimodal nature. It
clearly indicates that the image can be successfully seg-
mented so that two global regions are identified: a back-
ground and the desired set of foreground objects, in this
case the fibers of the composite material.
However, due to the small contrast gradient the
bimodal histogram – or rather its region of the highest
interest where the binarization threshold is located – is
very narrow and even a minor inaccuracy in the position
of the threshold can spoil the binarization. Unfortunately,
the traditional reliable techniques for threshold detection,
like, e.g., the Otsu method, set the binarization threshold
in this case completely incorrectly. The Otsu method is a
robust efficient technique for binarization-threshold esti-
mation used in most of the computer-vision toolboxes. It
is used, for instance, by MATLAB’s im2bw function for
image binarization.4
Thus, a new robust method for threshold estimation
on images with bimodal histograms was needed.
4 THRESHOLD ESTIMATION
Before computing the threshold estimate the original
image f(x,y) is filtered by a median filter over a 3×3
neighborhood. Then, a histogram h(i) of pixel intensities
i = 0…255 is computed. The obtained histogram must be
weighted before further processing as it has usually
rather extreme values at both ends, i.e., both very low
and very high intensities are contained heavily in the
image (mostly due to the black-and-white state informa-
tion added by the microscope software). The weighting
is performed by a Hann window. The weighted histo-
gram is then smoothed by Gaussian smoothing, a convo-
lution filtering with a Gaussian kernel. The neighbor-
hood used for the discrete convolution is an odd ceiling
of 2.5 % of the whole histogram length, i.e., for the
256-bin histogram seven neighboring values are used.
The value of the variance is set to 1.
The weighted smoothed histogram is numerically
differentiated in order to get an estimate of the first deri-
vative dh/di with respect to the intensity. The derivative
is smoothed by the same Gaussian filter as mentioned
above. In the next step, the histogram derivative is gated.
A list of threshold position candidates is assembled
by searching for zero values of the histogram. Since the
histogram is discrete a zero-crossing detection algorithm
is applied.
Depending on the character of the input image, the
candidate list should have 2 or 3 items (although under
real conditions, it might have more than three and the
R. ZEM^ÍK et al.: ANALYSIS OF THE FIBER DISTRIBUTION, SIZE, AND VOLUME RATIO OF UNIDIRECTIONAL ...
60 Materiali in tehnologije / Materials and technology 51 (2017) 1, 59–64
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Figure 1: Original image of bottom area of plate C, intensity values
range from 82 to 201
Slika1: Originalni posnetek spodnjega dela plo{~e C, vrednosti
intenzitete so med 82 in 201
Figure 2: Cross-section of unidirectional composite plate (C) made of
4 prepregs obtained using SEM, approximate locations of investigated
areas (top, middle, and bottom) are marked
Slika 2: SEM-posnetek preseka enosmerne kompozitne plo{~e (C),
izdelane iz 4 predimpregniranih plasti, ozna~eni so pribli`ni polo`aji
preiskovanega podro~ja (vrh, sredina, dno)
algorithm is robust to this case). One of the candidate
items is always equal to the position of the maximum
value in the histogram. Then, the threshold position t is
set to the next candidate in the direction towards the side
where there is more remaining candidates – in the case
with three or more candidates in the list. If there are only
two candidates, the threshold position is set to the can-
didate not equal to the maximum position (such situation
is depicted in Figure 3).
5 FIBER CENTER DETECTION
The original gray-scale image f is binarized to fbin
(Figure 4) using the threshold value t obtained above:
f x y
f x y t
f x y tbin
( , )
, ( , )
, ( , )
=
<
≥
⎧
⎨
⎩
0
1
(1)
and then processed by dilation.5 The dilated image
(Figure 5) is obtained as:
f x y f g x y
f x x y y g x y x y
( , ) ( )( , )
max ( ' , ' ) ( ' ' )( ' ' )
= ⊕ =
− − +
bin
{ }∈DB
(2)
where g is binary structuring matrix (5×5) of ones, with
rectangular domain DB.4
The preprocessed image, a circle binary mask of the
minimum fiber cross-section size and a discrete spatial
matrix6 are then transformed with two-dimensional
discrete Fourier transform:
F u v f x y e
m
M
n
N i
nx
N
my
M( , ) ( , )= ⋅
=
−
=
− − ⋅ +
⎛
⎝
⎜ ⎞
⎠
⎟
⎛
⎝
⎜
∑∑
0
1
0
1 2π⎜
⎞
⎠
⎟⎟
(3)
where F is the image matrix in spatial frequencies u and
v. Matrix G in spatial frequencies is obtained by discrete
convolution:
( )G u v F u v H u v Q u v( , ) ( , ) ( , ) ( , )= ∗ ⋅ (4)
where H is a binary mask and Q is the matrix trans-
formed by the Fourier transform (3). Its local maxima
are considered as the fiber center positions (xj, yj).4,6
R. ZEM^ÍK et al.: ANALYSIS OF THE FIBER DISTRIBUTION, SIZE, AND VOLUME RATIO OF UNIDIRECTIONAL ...
Materiali in tehnologije / Materials and technology 51 (2017) 1, 59–64 61
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967-2017) – 50 LET/50 YEARS
Figure 5: Dilated image of bottom area of plate C
Slika 5: Raz{irjena slika spodnjega dela plo{~e C
Figure 3: Threshold estimation from histogram
Slika 3: Prag, dolo~en iz histograma
Figure 6: Clustered image of bottom area of plate C, in total, 56
clusters were detected
Slika 6: Slika gru~ v spodnjem delu plo{~e C, skupaj je bilo odkritih
56 gru~
Figure 4: Binary image of bottom area of plate C, threshold intensity
was determined to be i = 118
Slika 4: Binarna slika spodnjega dela plo{~e C, intenziteta praga je
bila dolo~ena kot i = 118
6 RADIUS AND AREA DETECTION
Further, a cluster analysis was applied to the binary
image fbin. The cluster analysis is an iterative process
where a set of similar objects (pixels with similar
intensity) is assigned to a group (cluster). The center
positions. In this case a hierarchical clustering method
was used. Each cluster belongs to one of the center
positions (xj,yj). The fiber area Sj is calculated as the
number of pixels assigned to each cluster j (Figure 6)
and the corresponding fiber radius rj is then approxi-
mated by Equation (5):
r
S
j
j
=
π
(5)
The fiber volume ratio vf is then calculated as the
ratio of pixels corresponding to all the fibers and the
total number of pixels (in the selected rectangular area).
R. ZEM^ÍK et al.: ANALYSIS OF THE FIBER DISTRIBUTION, SIZE, AND VOLUME RATIO OF UNIDIRECTIONAL ...
62 Materiali in tehnologije / Materials and technology 51 (2017) 1, 59–64
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967-2017) – 50 LET/50 YEARS
Figure 9: Analyzed bottom area of plate A, gray-scale from SEM with
maximized contrast, circles denote detected fibers (position and size)
Slika 9: Analiziran spodnji del plo{~e A, SEM-posnetek s pove~anim
kontrastom, krogi ozna~ujejo odkrita vlakna (pozicijo in velikost)
Figure 11: Analyzed middle area of plate B, gray-scale from SEM
with maximized contrast, circles denote detected fibers (position and
size)
Slika 11: Analiziran srednji del plo{~e B, SEM-posnetek s pove~anim
kontrastom, krogi ozna~ujejo odkrita vlakna (pozicijo in velikost)
Figure 8: Analyzed middle area of plate A, gray-scale from SEM with
maximized contrast, circles denote detected fibers (position and size)
Slika 8: Analizirano srednje podro~je plo{~e A, SEM-posnetek s
pove~anim kontrastom, krogi ozna~ujejo odkrita vlakna (pozicijo in
velikost)
Figure 10: Analyzed top area of plate B, gray-scale from SEM with
maximized contrast, circles denote detected fibers (position and size)
Slika 10: Analiziran zgornji del plo{~e B, SEM posnetek s pove~anim
kontrastom, krogi ozna~ujejo odkrita vlakna (pozicijo in velikost)
Figure 7: Analyzed top area of plate A, gray-scale from SEM with
maximized contrast; circles denote detected fibers (position and size)
Slika 7: Analiziran vrhnji del plo{~e A, SEM-posnetek s pove~anim
kontrastom; krogi ozna~ujejo odkrita vlakna (pozicijo in velikost)
7 RESULTS
All the images were analyzed using the above-des-
cribed process. The positions and sizes of the detected
fibers (cyan circles with cluster IDs) are shown in
Figures 7 to 15. The gray-scale images are shown with
maximized contrast. The yellow rectangle denotes an
area for which the fiber volume ratio and the average
fiber radius were calculated. The calculated values (to-
gether with standard deviation "SD") are summarized in
Table 1. It was found that the volume ratio is slightly
larger in the top and bottom areas in all the plates. The
average fiber radius in all the plates was found to be
54.5276 px, which corresponds to 3.4294 μm.
Table 1: Properties obtained from images
Tabela 1: Lastnosti, dobljene iz posnetkov
Image /
area
Fiber volume
ratio
vf
Averaged fiber
radius
(r)
Fiber radius
SD
sd(r)
(-) (px) (px)
A-top 0.62378 55.4685 3.3918
A-middle 0.61156 55.2918 3.5781
A-bottom 0.65256 53.8739 2.6742
B-top 0.63991 56.2806 3.6586
B-middle 0.53645 54.2127 3.6388
B-bottom 0.60706 58.3071 4.7778
C-top 0.58752 53.1380 4.1013
C-middle 0.43809 52.6896 4.1948
C-bottom 0.54454 51.4864 3.6767
8 CONCLUSIONS
A new approach was proposed for an accurate
estimation of the threshold value that is necessary for the
creation of a proper binary map and subsequent cluster-
ing of the objects in images. The method was used for
the detection of fibers in images of cross-sections of
unidirectional composite plates obtained by SEM. The
R. ZEM^ÍK et al.: ANALYSIS OF THE FIBER DISTRIBUTION, SIZE, AND VOLUME RATIO OF UNIDIRECTIONAL ...
Materiali in tehnologije / Materials and technology 51 (2017) 1, 59–64 63
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Figure 15: Analyzed bottom area of plate C, gray-scale from SEM
with maximized contrast, circles denote detected fibers (position and
size)
Slika 15: Analiziran spodnji del plo{~e C, SEM-posnetek s pove~anim
kontrastom, krogi ozna~ujejo odkrita vlakna (pozicijo in velikost)
Figure 12: Analyzed bottom area of plate B, gray-scale from SEM
with maximized contrast, circles denote detected fibers (position and
size)
Slika 12: Analiziran spodnji del plo{~e , SEM-posnetek s pove~anim
kontrastom, krogi ozna~ujejo odkrita vlakna (pozicijo in velikost)
Figure 14: Analyzed middle area of plate C, gray-scale from SEM
with maximized contrast, circles denote detected fibers (position and
size)
Slika 14: Analiziran srednji del plo{~e C, SEM-posnetek s pove~anim
kontrastom, krogi ozna~ujejo odkrita vlakna (pozicijo in velikost)
Figure 13: Analyzed top area of plate C, gray-scale from SEM with
maximized contrast, circles denote detected fibers (position and size)
Slika 13: Analiziran zgornji del plo{~e C, SEM-posnetek s pove~anim
kontrastom, krogi ozna~ujejo odkrita vlakna (pozicijo in velikost)
method provided acceptable results, even in the case of
relatively low-quality (low-contrast) images. The ana-
lysis also showed that the fiber volume ratio tends to
have larger values at the surfaces of the composite, rather
than in the center of the plates. Such data are important
for reliable micromechanical models.7
Acknowledgement
The work was supported by the European Regional
Development Fund (ERDF), through project "NTIS –
New Technologies for Information Society", European
Centre of Excellence, CZ.1.05/1.1.00/02.0090, by pro-
ject LO1506 PUNTIS, and by the student research pro-
ject of Ministry of Education of Czech Republic
SGS-2013-036.
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