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\begin{document}

\title{Evaluation of damage in composites by using Thermoelastic Stress
Analysis: a promising technique to assess the stiffness degradation}



\author[1,2]{Rosa De Finis}%
\author[2]{Davide Palumbo}%
\author[2]{Umberto Galietti}%
\affil[1]{Polytechnic Institute of Bari}%
\affil[2]{Politecnico di Bari}%


\vspace{-1em}



  
  \date{June 14, 2024}


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\selectlanguage{english}
\begin{abstract}
The stiffness degradation represents one of the most interesting
phenomena used for describing the fatigue behaviour of composites. In
this regard, in literature, several works have been presented for
modelling the fatigue life by studying the stiffness degradation. A
critical aspect of modelling damage fatigue is represented by the
difficulties in simulating the whole behaviour of material and then in
describing the damage progression in all its stages. In addition, the
validation of models requires the measurement of stiffness variations by
means of experimental techniques. Above all for real components, the
difficulties in defying proper models are accompanied by the
difficulties in measuring stiffness degradation due to inapplicability
of classic experimental techniques. In this work, the stiffness
degradation of quasi-isotropic carbon-fibre-reinforced-polymer obtained
by automated fiber placement, has been assessed by means of
Thermoelastic Stress Analysis. The amplitude of temperature signal at
the mechanical frequency (thermoelastic signal) was considered as an
indicator of material degradation and compared to the data provided by
an extensometer. The correlation between thermoelastic and mechanical
data allowed to build a new experimental model for evaluating and
predicting material stiffness degradation by just using thermoelastic
data. The proposed approach seems to be very promising for stiffness
degradation assessment of real and complex mechanical components
subjected to actual loading conditions.%
\end{abstract}%



\sloppy


\textbf{Nomenclature}

\emph{E\textsubscript{0,} initial Young's modulus (MPa)}

\emph{E, actual Young's modulus (MPa)}

\(\frac{E}{E_{0}},\) \emph{Stiffness degradation (MPa/MPa)}

\emph{(E/E\textsubscript{0})\textsubscript{mod} stiffness degradation
modeled by using thermoelastic data}

\emph{f,mechanical loading frequency (Hz)}

\emph{N, actual loading cycles}

\(N_{f}\)\emph{, total loading cycles}

\emph{N0, initial loading cycles of a specific thermal sequence}

\emph{N/N\textsubscript{f} ,cycles-to-total cycles ratio}

\(S\)\emph{, thermal signal (Unit Signal)}

\emph{S\textsubscript{mean ,} mean thermal signal (Unit Signal)}

\emph{S1, the first amplitude harmonic of thermal signal (thermoelastic
signal) (Unit Signal)}

\emph{S2, the second amplitude harmonic of thermal signal (Unit Signal)}

\emph{(S1\_filt)\textsubscript{N}/(S1\_filt)\textsubscript{N0 ,}filtered
thermoelastic signal evaluated at cycles N normalised by initial cycles
N0}

\emph{(S1\_98prc)\textsubscript{N}/(S1\_98prc)\textsubscript{N0 ,} 98th
percentile of thermoelastic signal evaluated at cycles N normalised by
initial cycles N0}

\emph{(S1\_2prc)\textsubscript{N}/(S1\_2prc)\textsubscript{N0} , 2nd
percentile of thermoelastic signal evaluated at cycles N normalised by
initial cycles N0}

\(\varphi\) , \emph{Phase shift of the first harmonics (rad)}

\(\varphi_{d}\)\emph{, phase shift of the second harmonics (rad)}

\selectlanguage{greek}\emph{σ\textsubscript{max ,} maximum stress (MPa)}\selectlanguage{english}

\(\omega\)\emph{, system pulsation (rad/s)}

\textbf{Introduction}

Fibre reinforced polymer (FRP) materials are widely exploited in the
weight-critical structural applications due to the high
strength-to-weight ratio, which allows the advantage of a great fuel
saving \textsuperscript{{[}1{]}}. Despite this advantage, their
intrinsic anisotropy and heterogeneity play a remarkable role in the
assessment of mechanical properties by making complex the damage
mechanisms \textsuperscript{{[}2{]}}. In this regard, residual life and
fatigue damage assessment are the prime concerns when the materials or
components are subjected to fatigue loading. It follows that, as
composites represent primary structural members in various fields of
industry (aerospace, automotive\ldots{}), extensive research campaigns
and suitable investigations on in-service components need to be
performed \textsuperscript{{[}1-3{]}}. In addition, the data analysis
requires the knowledge of several typical phenomena (i.e. damage
initiation and propagation in layer and at interface
regions\textsuperscript{{[}4{]}}. In this regard, different approaches
have been developed \textsuperscript{{[}5-15{]}} to describe the fatigue
damage mechanisms based on macroscopic failure, strength degradation,
actual damage mechanisms, and stiffness reduction in terms of
degradation of elastic properties during fatigue loading. By considering
these damage mechanisms, various studies were carried
out\textsuperscript{{[}18-23{]}} to understand their influence on
stiffness degradation that can be described as the ratio of actual
Young's modulus (\emph{E} ) and the undamaged modulus
(\emph{E\textsubscript{0}} ) and depends on the imposed stress (the
dependence is a power function)\textsuperscript{{[}1{]}, {[}3{]},
{[}8-15{]}}. Stiffness degradation of a laminate is caused by transverse
cracks and delamination. The matrix cracking is the first mechanism that
appears in the plies with transverse fibres when load is applied. Even
if, it does not determine a sudden failure, it can be detrimental to the
strength as it produces a mechanical properties reduction. Matrix
cracking enhances resin-dominated damage modes that involve a local
delamination\textsuperscript{{[}16{]}}. Matrix cracking and delamination
affect the load carrying capability of the material
\textsuperscript{{[}17{]}}, they can also occur sometimes independently
and sometimes interactively\textsuperscript{{[}9{]}} making difficult
any prediction.

Focused on determining the material deterioration, Kobayashi et
al.\textsuperscript{{[}24{]}}, proposed an analytical model for
predicting the formation of cracks by considering an average stress
distribution for each ply. However, the crack formation is a local
phenomenon hence a more local analysis is required to understand its
effect on mechanical properties.

Another approach adopted the shear-lag
theory\textsuperscript{{[}25-28{]}} for describing the effect of
micro-cracking and micro-crack induced delamination on material
behaviour. The study \textsuperscript{{[}28{]}} was focused on isolated
cracks while another analytical model \textsuperscript{{[}29{]}},
overpassed the problem of isolated cracks by considering interacting
cracks in any ply of a symmetric laminate.

The computational cost, the assumptions on damage mechanisms and their
appearance (isolated, multiple, interacting) make these approaches
difficult to be performed. In all the cases, the experimental validation
is essential for understanding the real behaviour of the specific
material.

Besides analytical approaches, several empirical/semi-empirical methods
to study the stiffness degradation of the material have been
proposed\textsuperscript{{[}30-34{]}}. Crammond et
al.\textsuperscript{{[}31{]}}, proposed an experimental analysis of the
stresses and strains in double butt strap joint in GFRP composite by
using digital image correlation that required an accurate speckle
pattern painted. Packdel \textsuperscript{{[}34{]}}, performed optical
microscopy to study mechanical properties. In the same way, Hosoi et
al.\textsuperscript{{[}2{]}}, performed the evaluation of inner and
outer crack density and delamination by using microscope and soft X-ray
tomography while in \textsuperscript{{[}35{]}} used ultrasonic C-SCAN
for assessing delaminated areas. Chen \textsuperscript{{[}36{]}} ,
measured the mechanical properties variations of a composite wind
turbine blade by installing strain gauges. O'Brien et
al.\textsuperscript{{[}23{]}}, proposed a method to predict stiffness
loss at failure from a secant modulus criterion by measuring stress by
means of strain gages. The technique requires a careful installation and
the related measurement is punctual.

All these techniques and methods to evaluate damage parameters require
an accurate setup and/or post-mortem inspections to determine the
typical damage mechanism present and the number of crack sites.

A full-field technique, capable of providing a map of signal
proportional/correlated to material degradation would be suitable for
studying material behaviour in laboratory and in-situ on real
components. In this way, the thermography has already demonstrated its
capability in the assessment of mechanical behaviour of
metals\textsuperscript{{[}37-39{]}}.

In the field of composites, Montesano et al. \textsuperscript{{[}33{]}},
and Gagel et al. \textsuperscript{{[}30{]}}, adopted thermography to
estimate the strength at specific number of cycles and to determine
qualitatively the sites of final failure in fatigue loaded GF-NCF-EP in
an early stage of the fatigue life. Even if this technique is useful, it
has already been demonstrated that temperature is a parameter influenced
by several factors \textsuperscript{{[}40{]}}.

The Thermoelastic Stress Analysis (TSA) technique can be used to assess
the amplitude of the thermal, under adiabatic conditions, that linearly
depends on the sum of the principal
stresses/strains\textsuperscript{{[}37{]}, {[}40-43{]}}. In this regard,
Emery et al.\textsuperscript{{[}37{]}}, showed qualitatively the
possible relationship between the component of thermoelastic signal and
the stiffness degradation. The advantage of this approach is such that
thermal signal provides full field information related to damage with a
simple set-up.

By following this approach, the aim of the research is to present a
novel experimental model, based on thermoelastic data, capable of
describing the stiffness degradation of a quasi-isotropic composite
undergoing fatigue constant amplitude tests.

No similar models based on thermoelastic data have been presented in
literature yet. In particular, by correlating mechanical and thermal
data at a specific cycles number, material damage state was assessed
during the test by means of a contactless technique requiring a simple
setup.

The advantage of the proposed approach lies in the possibility to
implement the procedure and analysis on in-service
structures/components.

\textbf{Theoretical Framework}

\emph{2.1} \emph{Mathematical} \emph{Models for stiffness degradation}

Residual strength and stiffness are commonly indicated as damage
metrics\textsuperscript{{[}43{]}}. Depending on loading conditions they
decrease through the cycles until achieving a certain critical value
which determines the failure of material \textsuperscript{{[}43, 44{]}}.

Under cyclic loading, the stiffness of the whole fatigue life is
characterised by three typical behaviours
Fig.1\textsuperscript{{[}10,17-18,41{]}}. The first trend lasting
roughly 10-20\% of the whole life is characterised by inner and outer
matrix cracking. This latter produces edge delamination and/or local
delamination in the second stage \textsuperscript{{[}2{]}}. The
appearance of delaminations is the consequence of the achievement of a
specific damage state where crack density
saturates\textsuperscript{{[}27-30{]}}. This phase is characterised by a
succession of micromechanics equilibrium stages. It is slow due to the
multiplication of cracks in the matrix and the coalescence of
delaminations which reduces the rate of
damage\textsuperscript{{[}34{]}}. In the third phase, a widespread
fibres breakage governs the failure of the material.

As the major of stiffness reduction of an off-axis dominated laminate
appears from first to second stage, it becomes interesting to evaluate
the amount of mechanical properties loss. Ogin et
al.\textsuperscript{{[}10{]}}, proposed a power dependence between the
stiffness reduction rate \emph{dE/dN} , maximum
stress\selectlanguage{greek}\emph{σ\textsubscript{max}}\selectlanguage{english} and\emph{N} the cycles to failure at
specific\selectlanguage{greek}\emph{σ\textsubscript{max}}\selectlanguage{english} :

\(-\frac{1}{E_{0}}\frac{\text{dE}}{\text{dN}}=A^{*}{\left(\frac{{\sigma_{\max}}^{2}}{E^{2}(1-\frac{E}{E_{0}})}\right)\mathrm{\ }}^{n}\ \)(1)

where \emph{A*} and \emph{n} are material constants
and\emph{E\textsubscript{0}} is the initial Young modulus in undamaged
conditions.

By integrating Eq. (1), it is possible to obtain the stiffness reduction
expression:

\(\frac{E}{E_{0}}=1-\left[{K^{{}^{\prime}}}^{\frac{1}{n+1}}\left(\frac{{\sigma_{\max}}^{2}}{{E_{0}}^{2}}\right)^{n/(n+1)}\ {(N)}^{1/(n+1)}\right]\)(2)

where \emph{K'} and is a material constant. In a compacted form, as
demonstrated by Ogin \textsuperscript{{[}10{]}}, it becomes:

\(\frac{E}{E_{0}}=1-A\left(\frac{\sigma_{\max}}{E_{0}}\right)^{b}\left(N\right)^{d}\)(3)

The Young's modulus variation, Eq. (3), is a function at the same time
of material coefficients \emph{A} , \emph{b} , \emph{d} , the reached
cycles and the specific stress level, making complex the prediction of
stiffness reduction especially in those applications where imposed
stress is unknown.

Another form of stiffness degradation was recently proposed
by\textsuperscript{{[}3{]}, {[}17{]}} as a function of cycles-to-total
cycles ratio\emph{N/N\textsubscript{f}} :

\(\frac{E}{E_{0}}=K\left(\sigma_{\max}\right)\ {(\frac{N}{N_{f}})}^{k}\)(4)

where \emph{K} and \emph{k} are material constants, and
specifically,\emph{K} depends on imposed stress.

The material coefficients are obtained by fitting the mathematical model
to the experimental data and depend on several variables: stacking
sequence, ply thickness, material properties, load, and stress
ratio\textsuperscript{{[}34{]}, {[}43-44{]}}.

\emph{2.2} \emph{Thermoelastic Stress Analysis technique for composites}

In order to study the stiffness degradation, the temperature is
particularly promising as it is related to the energy involved in
fatigue damaging \textsuperscript{{[}39{]}}. In particular, the
thermoelastic temperature component is strictly correlated to elastic
properties of material \textsuperscript{{[}37-39{]}} as it represents
the reversible response of the material to the external mechanical
excitation under adiabatic conditions. The amplitude of thermoelastic
component can be described by the well-known
form\textsuperscript{{[}37{]}} :

\(T=\frac{-T_{0}}{\rho C_{p}}(\alpha_{1}{\sigma}_{1}+\alpha_{2}{\sigma}_{2})\)(5)

where \emph{T\textsubscript{0}} is the environment temperature,
\selectlanguage{greek}\emph{ρ}\selectlanguage{english}the density, \emph{C\textsubscript{p}} is the specific heat at
constant pressure, \selectlanguage{greek}\emph{α\textsubscript{i}}\selectlanguage{english} and
\selectlanguage{greek}\emph{Δσ\textsubscript{i}}\selectlanguage{english}respectively the linear thermal diffusivity
and peak-to-peak stress variations in the principal material directions.

Pitarresi et al. \textsuperscript{{[}45{]}}, modelled the thermoelastic
behaviour of a composite where the resin rich layer acted like a strain
witness. For outer lamina detected by infrared detector, it is likely
that the role of the resin is influent in the stress analysis especially
in the first part of loading cycles where, as found by
Nijessen\textsuperscript{{[}43{]}}, stiffness degradation is
matrix-driven.

By assuming the laminate is in plane strain conditions, the surface
strain field is identical through the thickness. The relation between
the peak-to-peak temperature variations and stress amplitude variations
under the hypothesis of isotropic resin and adiabatic conditions are
{[}\textsuperscript{45{]}}:

\(T^{r}=-T_{0}\left(\frac{\alpha^{r}}{\rho^{r}\text{Cp}^{r}}\right)\left(\frac{E^{r}}{E_{l}^{c}}\right)\left(\frac{1-v_{\text{lt}}^{c}}{1-v^{r}}\right)\left[{\sigma}_{l}^{c}+\left(\frac{E_{l}^{c}}{E_{t}^{c}}\frac{1-v_{\text{tl}}^{c}}{1-v_{\text{lt}}^{c}}\right){\sigma}_{t}^{c}\right]\)(6)

where upper the script \emph{c} indicates the composite while
\emph{r}the resin contribution to Young's moduli, Poisson's
moduli\selectlanguage{greek}\emph{υ\textsubscript{lt}}\selectlanguage{english} , \selectlanguage{greek}\emph{υ\textsubscript{tl}}\selectlanguage{english} , the
subscript\emph{l} stands for longitudinal and \emph{t} for transverse.
Eq. 6 allows the assessment of the thermoelastic temperature signal of
resin\selectlanguage{greek}\emph{ΔT\textsuperscript{r}}\selectlanguage{english} related to the sum of longitudinal and
transverse stress variations\({\sigma}_{l}^{c}+\left(\frac{E_{l}^{c}}{E_{t}^{c}}\frac{1-v_{\text{tl}}^{c}}{1-v_{\text{lt}}^{c}}\right){\sigma}_{t}^{c}\), through thermo-physical
properties of resin\(\frac{\alpha^{r}}{\rho^{r}\text{Cp}^{r}}\), \(T_{0}\), and a
combination of Young's and Poisson's moduli ratios\(\left(\frac{E^{r}}{E_{l}^{c}}\right)\left(\frac{1-v_{\text{lt}}^{c}}{1-v^{r}}\right)\)of
resin and composite.

Eq. (6) provides a tool for studying the relationship between
temperature and stresses and describes a local phenomenon strongly
related to mechanical properties variation throughout laminae.

In the case of local damaged areas, the stress values change with
respect to the initial conditions. As the damage growths several
phenomena appear as described by \textsuperscript{{[}41{]}}, producing
an opposing behaviour in the signal \textsuperscript{{[}37{]}}:
stiffness/strength variations.

Damage mechanisms can be basically imputable to matrix cracking of
off-axis laminae \textsuperscript{{[}44{]}} due to Poisson's ratio
mismatch between plies and a shear mismatch at interfaces. The
appearance of a transverse crack involves the change of the
cross-section area with the consequent changes of stress distributions
and the reduction of the load carrying
capability\textsuperscript{{[}8{]},{[}37{]}}. Moreover, in the lamina
several zones are interested by higher stress values and some others by
lower stresses.

Due to variety of fatigue mechanisms occurring in the material and their
random appearance that affects locally certain regions of material, a
great advantage of using thermoelastic stress analysis would be to
assess a parameter leading a local analysis.

\textbf{Material and Methods}

The samples tested in this paper were obtained by Automated Fibre
Placement technology \textsuperscript{{[}46{]}} where robotic system can
depose each layer of the laminate with different orientations. Each tape
is pressed to the mould by a roller which provides the proper compacting
pressure \textsuperscript{{[}47{]}.}

The specimens were obtained from a panel made of sixteen plies of
epoxy-type resin reinforced by carbon fibres with a stacking sequence of
{[}0/-45/45/90/90/45/-45/0{]}\textsubscript{2}. The panel dimensions
were 560 mm (weight) and 695 mm (length) while sample were 25 mm width,
250 mm length and 3.5 mm thick. All the specimens were tested on an
INSTRON 8850 (250 kN capacity) a servo-hydraulic loading frame.

Tensile tests were preliminarily performed in order to evaluate the
ultimate tensile strength of material (824 MPa, standard deviation 84.57
MPa). The tests were carried out at 1 mm/min of displacement rate
according to the Standard \textsuperscript{{[}48{]}}.

Constant stress amplitude tests were performed (S/N curve, run-out at
2*10\textsuperscript{6} cycles, load control) at stress ratio of 0.1 and
at loading frequency of 7 Hz. All the stress levels are reported in
Table I in terms of maximum and mean stresses applied. In Table I, the
values marked with an asterisk indicate the test with the acquisition of
the thermal signal. For each stress level applied reported in Table I,
one sample was tested. An extensometer with clamping length of 25 mm was
used for strain measurements. The acquisition of stress/strain values
from loading system were sampled at 100 Hz. An optical microscope Nikon
SLZ1000 was used for post-mortem damage investigations.

Fig. 2a shows the results in terms of \emph{S/N} curve and the 90\%
prediction interval bounds. The endurance limit at
2*10\textsuperscript{6} cycles was obtained in correspondence of a
maximum stress of 482 MPa.

Infrared sequences were acquired by a cooled In-Sb detector FLIR X6540
SC (640X512 pixel matrix array, thermal sensitivity NETD \textless{} 30
mK) with a frame rate of 177 Hz. The spatial resolution in terms of
millimetre-to-pixel ratio was roughly 0.35. Each thermal sequence
contained 1770 frames that corresponds to 170 loading cycles acquired.
Temperature and stress/strain data were sampled at the same cycles.

Fig. 2b reports the equipment layout in terms of loading frame and IR
detector and Fig. 2c reports the sample and extensometer setup.

\textbf{Signal Processing}

In this section, the algorithms for processing both thermal and
mechanical data series are presented to assess the metrics to represent
the stiffness reduction of the material. Extensometer provided averaged
stress/strain data in the gage length while the analysis of thermal
signal provided full field maps with local information as demonstrated
in \textsuperscript{{[}39{]}}.

\emph{4.1 Processing of the data provided by extensometer}

Mechanical data were processed by using Matlab\selectlanguage{ngerman}® software according to
the procedure represented in Fig. 3. For each test, the values of the
number of cycles \emph{N} , the force \emph{F} and displacement
\emph{s}allowed the assessment of stress \selectlanguage{greek}\emph{σ} \selectlanguage{english}and strain \selectlanguage{greek}\emph{ε} .
\selectlanguage{english}In particular, for each stress/strain curve, the slope of the elastic
trends (conventionally indicated as rising and descending) was
calculated automatically by taking into account a minimum of 12 data.
Such the analysis provided Young's moduli:
\emph{E\textsubscript{load}}and \emph{E\textsubscript{unload}} . For
each cycle \emph{N} , the mean value \emph{E} of these latter was
evaluated and divided to Young's modulus in undamaged condition
\emph{E\textsubscript{0}} , (\emph{E/E\textsubscript{0}}
)\emph{\textsubscript{N}} . To reduce the noise of the data series a
five-points moving average filter was applied to
(\emph{E/E\textsubscript{0}} )\emph{\textsubscript{N}} values of the
different stress levels.

\emph{4.2 Thermal data processing}

Thermal sequences provided by the IR camera, were processed according to
the procedure presented in \textsuperscript{{[}40{]}}\textbf{.}

A signal reconstruction algorithm based on the least squares method was
performed in order to extract pixel by pixel from the generic thermal
signal \emph{S} , Eq.7, its components:

\(S\left(\frac{N}{f}\right)=S_{\text{mean}}(\frac{N}{f})+S1\sin\left(\omega\frac{N}{f}+\varphi\right)+S2\cos\left(2\omega\frac{N}{f}+\varphi_{d}\right)\)(7)

where the \emph{S\textsubscript{mean}} is the term describing the mean
temperature while \emph{S1} and \emph{S2} \(\ \)are
respectively the first and second amplitude harmonics,
\(\varphi\) and \(\varphi_{d}\)are phase shift of first and
second harmonics respectively and the system pulsation is
\(\omega=2\pi f\).

In this research, the first amplitude harmonic signal representative of
the thermoelastic signal and the related map were considered for the
analysis according to the procedure of Fig.4. The first consideration on
matrixes in Fig. 4, obtained by frequency domain analysis, is
that\emph{S\textsubscript{mean}} does not provide local information
\textsuperscript{{[}40{]}} while \emph{S1}allows for assessing more
local information\textsuperscript{{[}39-40{]}}. The algorithm involves
the following steps according to Fig. 4:

- the size reduction to the gage length of
maps\emph{S1\textsubscript{N}}, in order to refer thermoelastic signal
to the area controlled by extensometer, the output map
is\emph{(S1\_red)\textsubscript{N}}. The subscript \emph{N} represents
the number of cycles at which thermal sequence was recorded. - a 2D
median filtering of \emph{(S1\_red)\textsubscript{N}} maps. Each output
pixel represents the median value in a 3-by-3 neighbourhood around the
corresponding pixel of the input
image,\emph{(S1\_filt)\textsubscript{N}}. - the extraction of the mean
value, 98\textsuperscript{th} and 2\textsuperscript{nd}
percentiles,\emph{(S1\_mean)\textsubscript{N}},
(\emph{S1\_98prc)\textsubscript{N}} and
(\emph{S1\_2prc)\textsubscript{N}} respectively, to make a preliminary
analysis of the behaviour of thermoelastic signal. - the normalization
of each \emph{(S1\_filt)\textsubscript{N}} map by initial value (number
of
cycles\emph{N0}),\emph{(S1\_filt)\textsubscript{N}}/\emph{(S1\_filt)\textsubscript{N0}}.
- the extraction of the mean value, 98\textsuperscript{th} and
2\textsuperscript{nd}
percentiles,\emph{(S1\_mean)\textsubscript{N}}/\emph{(S1\_mean)\textsubscript{N0}},
(\emph{S1\_98prc)\textsubscript{N}/}(\emph{S1\_98prc)\textsubscript{N0}}and
(\emph{S1\_2prc)\textsubscript{N}/}(\emph{S1\_2prc)\textsubscript{N0}}respectively.
These latter represent the thermal metrics used for evaluating the
damage. The use of percentile instead of maximum/minimum values allows
to avoid outliers in the analysis.

In Fig. 5, the evolution of percentiles and mean values of thermoelastic
signal \emph{(S1\_mean)\textsubscript{N}} ,
(\emph{S1\_98prc)\textsubscript{N}} and
(\emph{S1\_2prc)\textsubscript{N}} are reported. Each test was named as
sample/50-60-65-70/\% UTS. The index (\emph{S1\_98prc)\textsubscript{N}}
, Fig. 5a, presents a variable trend (increasing/decreasing)
demonstrating the complexity of damage mechanisms and their effects on
composites. Fig. 5b reports the mean values
\emph{(S1\_mean)\textsubscript{N}} which seem more affected by the
influence of maximum signal. The signal
(\emph{S1\_2prc)\textsubscript{N}} in Fig. 5c exhibits a decreasing
behaviour through the cycles that is reproducible for each test. This
latter parameter seems to be more robust to follow the stiffness
degradation than others as confirmed by the analysis
of\textsuperscript{{[}37{]}}.

\textbf{Results}

\emph{5.1 Stiffness degradation obtained by extensometer}

In Fig. 6, the data from extensometer processed according to the
algorithm of section 4.1, are presented.

Fig. 6a shows \emph{E/E\textsubscript{0}} versus \emph{N} . At a first
sight, it is possible to see that the entity of stiffness reduction is
the same at each stress level, and the metrics achieves a steady-state
value of 0.80 for each test. At a specific ratio value
of\emph{N/N\textsubscript{f}} , Fig. 6b, all the data exhibits the same
value of \emph{E/E\textsubscript{0}} and present a steady state
condition. The stiffness reduction occurs up
to\emph{N/N\textsubscript{f}} of 0.3 according to
literature\textsuperscript{{[}10{]}, {[}17{]}}.

The stress dependence is appreciable in Fig. 6a, but at
fixed\emph{N/N\textsubscript{f}} there is a slight effect on stiffness
reduction.

In this research, the models of literature (Eq. (3) and (4)) have been
considered as a reference for analysing the stiffness loss trend.

Table IIa reports the coefficients \emph{A} , \emph{b} , \emph{d} of Eq.
(3) obtained by the fitting of the data provided by extensometer and the
squared correlation coefficient \emph{R\textsuperscript{2}} .

Eq. (4) can be rewritten by indicating the stress dependence of
\emph{K}coefficient, Eq. (8):

\(\frac{E}{E_{0}}=\left(C\sigma_{\max}\right)^{a}\left(\frac{N}{N_{f}}\right)^{k}\)(8)

with the coefficients \emph{C} , \emph{a} and \emph{k} experimentally
determined.

Table IIb reports the coefficients of Eq. (8) together with
the\emph{R\textsuperscript{2}} coefficient. This latter is higher in the
case of the model described by Eq. (8) indicating a better capability of
this model in describing the stiffness degradation of the investigated
material.

Fig. 7 reports graphically for each stress level the comparison between
the two models and experimental data in terms
of\emph{E/E\textsubscript{0}} . The model of Eq. (8) (black dotted line)
fits better than the one of Eq. (3) (black solid line) the experimental
data. However, it is noteworthy to highlight that all the literature
models are not capable of describing all the damage stages of the
materials. In effect, the steady state conditions are not taken into
account in the model even if the stabilisation of the stiffness is an
important stage of the damage \textsuperscript{{[}5-15{]}}.

The model of Eq. (3) fits very well the behaviour of the material at
initial cycles but in some cases (Fig. 7a and c) it does not model the
major stiffness degradation that occurs between 0-0.4
of\emph{N/N\textsubscript{f}} . On the other hand, the model of Eq. (8)
seems to be a good compromise for describing all the damage stages. This
latter model will be used as reference to represent stiffness
degradation of the material in the next section.

\emph{5.2 Thermoelastic Data}

In this section, the data processed according to the procedure indicated
in section 4, are presented.

Fig. 8 reports for three imposed stress values,
the\emph{(S1\_filt)\textsubscript{N}}
/\emph{(S1\_filt)\textsubscript{N0}} normalised thermoelastic signal
values.

The sample at 50\%UTS exhibits a widespread lower signal except in some
localised areas where the signal is high in presence of the transverse
cracks. The samples at 60-65\%UTS, at the
same\emph{N/N\textsubscript{f}} value present a similar behaviour of the
thermoelastic signal. In this case, the thermoelastic maps present a
more evident effect of the transverse cracks due to the higher values of
the imposed stress.

By resuming, the normalised thermoelastic maps at the initial stage of
the test present a characteristic behaviour related to the stress/strain
of the laminate, as previously described. During the cycles, the damage
produced by the transverse cracks determines a stress/strain
redistribution within laminae that, in turn, leads to a stress/strain
increase or decrease with respect to initial conditions. It follows
that, the thermoelastic signal increases in correspondence of the cracks
and decreases in the rest of the gage area due to the reduction of the
load carrying capability of the laminate. In this regard, the minimum
thermoelastic signal (2\textsuperscript{nd} percentile) was adopted as
the proper metrics for describing the stiffness degradation.

\emph{5.3 Modelling the stiffness degradation by using Thermoelastic
Data: A calibration by using mechanical data}

Fig. 9a reports the\emph{(S1\_2prc)\textsubscript{N}}
/\emph{(S1\_2prc)\textsubscript{N0}} data plotted versus \emph{N} while
in Fig. 9b, the data are compared in terms of
\emph{N/N\textsubscript{f}} . In Fig. 9a, the thermoelastic data show a
stress dependence from lower to higher imposed stress. Such stress
effect is slight when data are compared in terms
of\emph{N/N\textsubscript{f}} , Fig. 9b.

By observing both Fig. 9a-b, it appears that mechanical and thermal data
follow the same trend. In particular, the thermoelastic signal metrics
exhibits a steady state from \emph{N/N\textsubscript{f}} =0.4 at a value
of 0.4. (Unit Signal/ Unit Signal).

In Fig. 10, the data of \emph{E/E\textsubscript{0}}
and\emph{(S1\_2prc)\textsubscript{N}}
/\emph{(S1\_2prc)\textsubscript{N0}} are reported in the same plot for
each test as function of\emph{N/N\textsubscript{f}} . The extensometer
data in terms of\emph{E/E\textsubscript{0}} have been opportunely
sampled, in order to compare mechanical and thermal data at the
same\emph{N/N\textsubscript{f}} values. In Fig.10a-b-c-d, a slightly
higher sensitivity of thermoelastic data than mechanical data is
observed: for each stress level the initial stiffness loss described
by\emph{(S1\_2prc)\textsubscript{N}}
/\emph{(S1\_2prc)\textsubscript{N0}}is characterised by a more severe
decrease. This can be attributed to the fact that the thermoelastic
signal provides a local information of the behaviour if compared to the
mean value of the stiffness provided by extensometer.

The following step was the correlation of thermal and mechanical data at
fixed \emph{N/N\textsubscript{f}} values. By using
for\emph{(S1\_2prc)\textsubscript{N}}
/\emph{(S1\_2prc)\textsubscript{N0}} the same mathematical model used
for \emph{E/E\textsubscript{0}} , we obtain Eq. (9):

\(\frac{({S1\_2\text{perc})}_{N}}{{(S1\_2\text{perc})}_{N0}}=\left(C^{\prime}\sigma_{\max}\right)^{a^{\prime}}\left(\frac{N}{N_{f}}\right)^{b^{\prime}}\)(9)

where the symbols \emph{a'} , \emph{b'} and \emph{C'} are empirical
constants.

By including Eq. (9) in Eq.(8), a direct relation
between\emph{E/E\textsubscript{0}} and\emph{(S1\_2prc)\textsubscript{N}}
/\emph{(S1\_2prc)\textsubscript{N0}} can be obtained, Eq.(10).

\(\left(\frac{E}{E_{0}}\right)_{\text{mod}}=\left(C\sigma_{\max}\right)^{a}{\left(\frac{1}{C^{\prime}\sigma_{\max}}\right)^{a^{{}^{\prime}}k/b^{\prime}}\left(\frac{({S1\_2\text{perc})}_{N}}{{(S1\_2\text{perc})}_{N0}}\right)}^{k/b^{\prime}}\)=\(\overset{\overline{}}{A\ }{\ \left(\frac{({S1\_2\text{perc})}_{N}}{{(S1\_2\text{perc})}_{N0}}\right)}^{\overset{\overline{}}{b}}\)(10)

Eq. (10) represents a mathematical relation between stiffness
degradation and thermoelastic signal. It shows also that to
obtain\emph{E/E\textsubscript{0}} values from thermoelastic data one
can:

\begin{itemize}
\tightlist
\item
  To find the coefficients \(\overset{\overline{}}{A}\) and\(\overset{\overline{}}{b}\)
  indirectly estimating \emph{a} , \emph{k} ,\emph{a'} , \emph{b'} ,
  \emph{C} , \emph{C'} .
\item
  To find \(\overset{\overline{}}{A}^{\prime}\) and `\(\overset{\overline{}}{b}\)' directly by
  plotting\emph{E/E\textsubscript{0}}
  versus\emph{(S1\_2prc)\textsubscript{N}}
  /\emph{(S1\_2prc)\textsubscript{N0}} .
\end{itemize}

The coefficients are reported in Table III for each stress level. They
are slightly similar at each stress level, as confirmed by the
experimental data (both mechanical and thermal), Fig.11.

In Fig. 11, for each stress level the \emph{E/E\textsubscript{0}} data
are reported compared to\emph{(S1\_2prc)\textsubscript{N}}
/\emph{(S1\_2prc)\textsubscript{N0}} data. Specifically, in Fig.11a-b,
the high number of acquired data close to the value 0.80 of
\emph{E/E\textsubscript{0}} , is due to the presence of a plateau of
stiffness degradation. The test run at the stress level of 70\%UTS, Fig.
11d, is the one with a smaller number of thermal acquisitions, but they
are well distributed for each value of\emph{E/E\textsubscript{0}} .

An advantage of the present material is such that it does not present a
marked stress dependence when the metrics \emph{E/E\textsubscript{0}}and
\emph{(S1\_2prc)\textsubscript{N}} /\emph{(S1\_2prc)\textsubscript{N0}}
are compared to \emph{N/N\textsubscript{f}} . So, in order to describe
the material behaviour one can choose the coefficients\({}^{\prime}\overset{\overline{}}{A}^{\prime}\)
and `\(\overset{\overline{}}{b}^{\prime}\) of a specific stress level from Table III. If
the test at 70\%UTS is adopted as representative of the relation between
thermal and mechanical data, the model is:

\(\left(\frac{E}{E_{0}}\right)_{\text{mod}}\)=\(1.01{\ \left(\frac{({S1\_2\text{perc})}_{N}}{{(S1\_2\text{perc})}_{N0}}\right)}^{0.22}\)(11)

In the next section, the validation of the model for the data at
different stress levels will be performed and an estimation of errors
between measured data and modelled data is provided.

\textbf{Discussion}

In this section \emph{(E/E\textsubscript{0}}
)\emph{\textsubscript{mod}}values obtained by Eq. (11) have been
expressed as a function of\emph{N/N\textsubscript{f}} by using a power
law in of Eq. (4) where the coefficient depending from stress
\(K\left(\sigma_{\max}\right)\ \) is 0.81 and the exponent \emph{k} is -0.03.

In Fig.12 are reported for each stress level, the experimental data
compared with the models according to Eq. (3)-(8) and the model obtained
by calibrating thermoelastic data\emph{(E/E\textsubscript{0}}
)\textsubscript{\emph{mod}} .

The results in Fig. 12 show a promising correlation between the
stiffness degradation obtained from thermoelastic data-based model and
extensometer data. In particular, at stress levels higher than the
endurance limit (Fig. 12b-c-d) the\emph{(E/E\textsubscript{0}}
)\textsubscript{\emph{mod}} data match the experimental data at the
steady state better than the other models.

The capability of the proposed approach can be also assessed by
evaluating absolute errors between the values forecast by each model and
experimental data \emph{(E/E\textsubscript{0}} ), at each stress level
in terms of maximum stress:

\(\text{Absolute\ Error}_{1}=\left|{\left(\frac{E}{E_{0}}\right)_{Eq.3}-\left(\frac{E}{E_{0}}\right)}_{\frac{N}{N_{f}}}\ \right|_{\sigma_{\max}}\)(12)

\(\text{Absolute\ Error}_{2}=\left|{\left(\frac{E}{E_{0}}\right)_{Eq.8}-\left(\frac{E}{E_{0}}\right)}_{\frac{N}{N_{f}}}\ \right|_{\sigma_{\max}}\)(13)

\(\text{Absolute\ Error}_{3}=\left|{\left(\frac{E}{E_{0}}\right)_{\text{mod}}-\left(\frac{E}{E_{0}}\right)}_{\frac{N}{N_{f}}}\ \right|_{\sigma_{\max}}\)(14)

where \emph{(E/E\textsubscript{0}} )\emph{\textsubscript{Eq.3}}
,\emph{(E/E\textsubscript{0}} )\emph{\textsubscript{Eq.8},
(E/E\textsubscript{0}} )\emph{\textsubscript{mod}} are respectively the
stiffness degradation modelled using Eq.(3), Eq.(8) for just mechanical
data and Eq. (11) for thermoelastic data.

The results reported in Fig.13b demonstrate as the model described by
Eq.(8) reproduces in a better way the experimental behaviour at specific
stress level if compared to the model of Eq. (3), Fig.13a.

In the case of Fig.13b, the absolute error reduces
as\emph{N/N\textsubscript{f}} increases while is it is high at initial
cycles.

Fig.13c represents the absolute error evaluated between the values
of\emph{(E/E\textsubscript{0}} )\emph{\textsubscript{mod}} and
experimental data. The \emph{Absolute Error\textsubscript{3}} is
comparable with the other two estimated errors specifically, it is
generally lower than\emph{Absolute Error\textsubscript{1}} . This
demonstrates the capability of thermoelastic data in describing the
stiffness degradation of the material.

\textbf{Conclusions}

In the present work, the stiffness reduction of a quasi-isotropic CFRP
laminate obtained by automated fibre placement innovative process was
studied by means of a thermographic approach.

A total of ten specimens were tested at four different stress levels,
50-60-65-70\% of UTS, and monitored by using infrared thermal camera.

The analysis of thermal signal provided the thermoelastic signal used as
metrics for evaluating stiffness degradation. At the same
cycles-to-total cycles ratio, thermoelastic data were compared to the
data of the extensometer that provided averaged stress/strain data in
the gage length of the sample.

Specific processing algorithms were used to extract the mechanical and
thermal metrics.

The major outcome of the present research is represented by the
modelling of the stiffness degradation by using thermoelastic data.

The other results achieved by performing present research are:

\begin{itemize}
\tightlist
\item
  A demonstration of the capability of the thermoelastic signal to
  provide local information and making possible the quantitative
  estimation of the damage in the material.
\item
  A demonstration of narrow correlation between mechanical and thermal
  behaviour
\item
  A qualitative major capability of the thermoelastic metrics for
  describing mechanical properties variations
\end{itemize}

This approach is useful also because it can be adopted to estimate
stiffness degradation on in-situ applications on operating components
where stress state and damage behaviour are unknown and where the
extensometer cannot be used.

\textbf{Acknowledgements}

The authors would like to thank Novotech Aerospace Advanced Technology
S.R.L. for the manufacturing of the samples and Professor Riccardo
Nobile and Mr. Andrea Saponaro for the support during the experimental
activity performed in this work.

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\textbf{Figures and Tables}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

Fig. 1. Typical behavior of a laminate containing both longitudinal and
transverse plies, during fatigue

\textbf{Hosted file}

\verb`image2.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

(a) (b)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

(c)

Fig. 2. (a) S/N curve, (b) equipment and layout (c) details of setup.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

Fig. 3. Procedure for analyzing data from extensometer\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

Fig. 4. Procedure for analyzing thermoelastic data\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

(b)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}

\end{center}
\end{figure}

(c)

Fig. 5. (a) Maximum (\({S1\_98prc)}_{N}\), (b) Mean (\({S1\_mean)}_{N}\),
(c) Minimum (\({S1\_2prc)}_{N}\), values of thermoelastic signal\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}

\textbf{Hosted file}

\verb`image11.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

(a) (b)

Fig. 6. \({E/E}_{0}\) Experimental data (a) versus cycles, (b)
versus N/N\textsubscript{f}

\textbf{Hosted file}

\verb`image12.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

\textbf{Hosted file}

\verb`image13.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

(a) (b)

\textbf{Hosted file}

\verb`image14.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

\textbf{Hosted file}

\verb`image15.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

(c) (d)

Fig. 7. \({{}^{\prime}E/E}_{0}^{\prime}\ \)experimental data versus empirical models (a)
50\%UTS (b) 60\%UTS (c) 65\%UTS (d) 70\%UTS\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image16/image16}

\end{center}
\end{figure}

Fig. 8 Qualitative analysis of thermoelastic signal
maps\({{}^{\prime}(S1\_filt)}_{N}\mathrm{/}{(S1\_filt)}_{N0}^{\prime}\) for three stress levels.

\textbf{Hosted file}

\verb`image17.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

\textbf{Hosted file}

\verb`image18.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

(a) (b)

Fig. 9. \({{}^{\prime}(S1\_2prc)}_{N}\mathrm{/}{(S1\_2prc)}_{N0}^{\prime}\)thermoelastic data versus (a) cycles, (b)
`N/N\textsubscript{f}'.

\textbf{Hosted file}

\verb`image19.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

\textbf{Hosted file}

\verb`image20.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

(a) (b)

\textbf{Hosted file}

\verb`image21.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

\textbf{Hosted file}

\verb`image22.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

(c) (d)

Fig. 10. \({{}^{\prime}(S1\_2prc)}_{N}\mathrm{/}{(S1\_2prc)}_{N0}^{\prime}\) and\({{}^{\prime}E/E}_{0}^{\prime}\ \)at fixed
`N/N\textsubscript{f} ` for samples: (a) 50\%UTS (b) 60\%UTS (c) 65\%UTS
(d) 70\%UTS\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image23/image23}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image24/image24}

\end{center}
\end{figure}

(a) (b)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image25/image25}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image26/image26}

\end{center}
\end{figure}

(c) (d)

Fig. 11.\(\ {}^{\prime}\frac{E}{E}_{0}^{\prime}\) versus\({{}^{\prime}(S1\_2prc)}_{N}\mathrm{/}{(S1\_2prc)}_{N0}^{\prime}\) data and data
fitting (solid line) for each stress level.

\textbf{Hosted file}

\verb`image27.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

\textbf{Hosted file}

\verb`image28.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

(a) (b)

\textbf{Hosted file}

\verb`image29.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

\textbf{Hosted file}

\verb`image30.emf`  available at \url{https://authorea.com/users/176512/articles/436282-evaluation-of-damage-in-composites-by-using-thermoelastic-stress-analysis-a-promising-technique-to-assess-the-stiffness-degradation}

(c) (d)

Fig. 12. Comparison between experimental data and modeled data at (a)
50\%UTS (b) 60\%UTS (c) 65\%UTS (d) 70\%UTS .\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image31/image31}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image32/image32}

\end{center}
\end{figure}

(b)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image33/image33}

\end{center}
\end{figure}

(c)

Fig. 13. Errors between experimental data and models: (a) Eq. (12), (b)
Eq. (13), (c) Eq.(14).

Table I. Loading table (stress values rounded up to the nearest unit)\selectlanguage{english}
\begin{longtable}[]{@{}llllllllll@{}}
\toprule
\begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\textbf{Sample}\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\textbf{\%UTS}\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\selectlanguage{greek}\textbf{σ\_mean {[}MPa{]}}\selectlanguage{english}\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\selectlanguage{greek}\textbf{σ\_max {[}MPa{]}}\selectlanguage{english}\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\textbf{N} \textbf{Cycles to Failure}\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\textbf{Sample}\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\textbf{\%UTS}\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\selectlanguage{greek}\textbf{σ\_mean {[}MPa{]}}\selectlanguage{english}\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\selectlanguage{greek}\textbf{σ\_max {[}MPa{]}}\selectlanguage{english}\strut
\end{minipage} & \begin{minipage}[b]{0.10\columnwidth}\raggedright\strut
\textbf{N} \textbf{Cycles to Failure}\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
1\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
80\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
363\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
661\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
1765\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
6\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
*65\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
295\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
537\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
291610\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
2\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
80\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
363\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
661\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
2875\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
7\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
75\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
341\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
620\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
6589\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
3\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
*70\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
318\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
578\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
60041\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
8\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
75\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
341\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
620\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
12323\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
4\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
70\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
318\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
578\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
75461\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
9\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
*50\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
227\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
413\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
2*10\textsuperscript{6}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
5\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
*60\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
273\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
496\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
300334\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
10\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
50\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
227\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
413\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
2*10\textsuperscript{6}\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

* infrared thermal sequences acquired

Table II. Parameters for mechanical data modeling (a) Eq. (3), (b) Eq.
(8).

(a)\selectlanguage{english}
\begin{longtable}[]{@{}lllllll@{}}
\toprule
\begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
\textbf{UTS \%}\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
\textbf{A} \textbf{{[}1/cycles{]}}\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
\textbf{b}\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
\textbf{d}\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
\[\mathrm{E}_{\mathbf{0}}\] \textbf{{[}MPa{]}}\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
\[\mathbf{\sigma}_{\mathbf{\max}}\] \textbf{{[}MPa{]}}\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
\[\mathbf{R}^{\mathbf{2}}\] \textbf{{[}/{]}}\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
50\%\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
19.4\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
1.53\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
0.23\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
69616\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
413\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
0.95\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
60\%\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
14.2\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
1.59\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
0.33\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
87754\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
496\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
0.86\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
65\%\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
13.1\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
1.62\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
0.34\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
92717\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
536\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
0.88\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
70\%\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
19.4\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
1.70\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
0.31\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
59515\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
577\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
0.87\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

(b)\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
\begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
\textbf{UTS \%}\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
\textbf{C} \textbf{{[}1/MPa{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
\textbf{a} \textbf{{[}/{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
\textbf{k} \textbf{{[}/{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
\[\mathbf{\sigma}_{\mathbf{\max}}\] \textbf{{[}MPa{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.16\columnwidth}\raggedright\strut
\[\mathbf{R}^{\mathbf{2}}\] \textbf{{[}/{]}}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
50\%\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
0.002\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
1.66\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-0.02\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
413\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
0.95\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
60\%\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
0.002\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
2.07\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-0.04\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
496\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
0.97\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
65\%\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
0.002\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
1.91\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-0.04\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
536\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
0.98\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
70\%\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
0.001\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
0.39\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
-0.05\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
577\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
0.98\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

Table III. Coefficients for thermoelastic data calibration by using
experimental \({}^{\prime}{E/E}_{0}^{\prime}\ \)data\selectlanguage{english}
\begin{longtable}[]{@{}lllllllllll@{}}
\toprule
\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\textbf{\%UTS}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\textbf{a} \textbf{{[}/{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\textbf{k} \textbf{{[}/{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\textbf{C} \textbf{{[}1/MPa{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\textbf{a'} \textbf{{[}/{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\textbf{k'} \textbf{{[}/{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\textbf{C'} \textbf{{[}1/MPa{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\[\overset{\overline{}}{\mathbf{A}\mathrm{\ }}\] \textbf{{[}/{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\[\overset{\overline{}}{\mathbf{b}}\] \textbf{{[}/{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\[\mathbf{\sigma}_{\mathbf{\max}}\] \textbf{{[}MPa{]}}\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\textbf{UTS} \textbf{{[}MPa{]}}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
50\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.36\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
-0.04\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.001\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.23\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
-0.21\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
7.7E-06\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
1.04\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.22\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
413\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
825\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
60\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.45\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
-0.04\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.001\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.46\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
-0.21\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
1.4E-04\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
1.02\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.22\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
495\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
825\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
65\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.27\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
-0.03\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.001\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.80\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
-0.17\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
4.0E-04\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
1.01\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.22\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
536\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
825\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
70\%\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.68\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
-0.04\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.001\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.90\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
-0.17\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
5.1E-04\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
1.01\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
0.22\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
577\strut
\end{minipage} & \begin{minipage}[t]{0.07\columnwidth}\raggedright\strut
825\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig-1-Typical-behavior-of-laminate/Fig-1-Typical-behavior-of-laminate}

\end{center}
\end{figure}\selectlanguage{english}
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\includegraphics[width=0.70\columnwidth]{figures/Fig-2a-SN-curve/Fig-2a-SN-curve}

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\end{figure}\selectlanguage{english}
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\includegraphics[width=0.70\columnwidth]{figures/Fig-2b-equipment-and-layout/Fig-2b-equipment-and-layout}

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\end{figure}\selectlanguage{english}
\begin{figure}[H]
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\includegraphics[width=0.70\columnwidth]{figures/Fig-2c-detail-of-gripping-system/Fig-2c-detail-of-gripping-system}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
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\includegraphics[width=0.70\columnwidth]{figures/Fig-3-Procedure-for-analyzing-experimental-data/Fig-3-Procedure-for-analyzing-experimental-data}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig-4-Procedure-for-analyzing-thermoelastic-data/Fig-4-Procedure-for-analyzing-thermoelastic-data}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig/Fig}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fih/Fih}

\end{center}
\end{figure}\selectlanguage{english}
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\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fii/Fii}

\end{center}
\end{figure}\selectlanguage{english}
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\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fij/Fij}

\end{center}
\end{figure}\selectlanguage{english}
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\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fik/Fik}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fil/Fil}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fim/Fim}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fin/Fin}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fio/Fio}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fip/Fip}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fiq/Fiq}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fir/Fir}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fis/Fis}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fit/Fit}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fiu/Fiu}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fiv/Fiv}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fiw/Fiw}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fix/Fix}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fiy/Fiy}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fiz/Fiz}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fja/Fja}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fjb/Fjb}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fjc/Fjc}

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\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fjd/Fjd}

\end{center}
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\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fje/Fje}

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\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fjf/Fjf}

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\begin{figure}[H]
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\includegraphics[width=0.70\columnwidth]{figures/Fjg/Fjg}

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\selectlanguage{english}
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