σmax , maximum stress (MPa)
\(\omega\), system pulsation (rad/s)
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 [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 [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…), extensive research campaigns
and suitable investigations on in-service components need to be
performed [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[4]. In this regard, different approaches have
been developed [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[18-23] to understand their influence on stiffness
degradation that can be described as the ratio of actual Young’s modulus
(E ) and the undamaged modulus (E0 ) and
depends on the imposed stress (the dependence is a power function)[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[16]. Matrix cracking and delamination affect the
load carrying capability of the material [17],
they can also occur sometimes independently and sometimes interactively[9] making difficult any prediction.
Focused on determining the material deterioration, Kobayashi et al.[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[25-28] for describing the effect of
micro-cracking and micro-crack induced delamination on material
behaviour. The study [28] was focused on isolated
cracks while another analytical model [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[30-34]. Crammond et al.[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 [34], performed optical
microscopy to study mechanical properties. In the same way, Hosoi et al.[2], performed the evaluation of inner and outer
crack density and delamination by using microscope and soft X-ray
tomography while in [35] used ultrasonic C-SCAN
for assessing delaminated areas. Chen [36] ,
measured the mechanical properties variations of a composite wind
turbine blade by installing strain gauges. O’Brien et al.[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[37-39].
In the field of composites, Montesano et al. [33],
and Gagel et al. [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 [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[37], [40-43]. In this regard, Emery et al.[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.
Theoretical Framework
2.1 Mathematical Models for stiffness degradation
Residual strength and stiffness are commonly indicated as damage metrics[43]. Depending on loading conditions they
decrease through the cycles until achieving a certain critical value
which determines the failure of material [43, 44].
Under cyclic loading, the stiffness of the whole fatigue life is
characterised by three typical behaviours Fig.1[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 [2]. The
appearance of delaminations is the consequence of the achievement of a
specific damage state where crack density saturates[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[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.[10], proposed a power dependence between the
stiffness reduction rate dE/dN , maximum stressσmax andN the cycles to failure at
specificσmax :
\(-\frac{1}{E_{0}}\frac{\text{dE}}{\text{dN}}=A^{*}{\left(\frac{{\sigma_{\max}}^{2}}{E^{2}(1-\frac{E}{E_{0}})}\right)\mathrm{\ }}^{n}\ \)(1)
where A* and n are material constants andE0 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 K’ and is a material constant. In a compacted form, as
demonstrated by Ogin [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 A , b , 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[3], [17] as a function of
cycles-to-total cycles ratioN/Nf :
\(\frac{E}{E_{0}}=K\left(\sigma_{\max}\right)\ {(\frac{N}{N_{f}})}^{k}\)(4)
where K and k are material constants, and specifically,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[34], [43-44].
2.2 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 [39]. In particular, the
thermoelastic temperature component is strictly correlated to elastic
properties of material [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[37] :
\(T=\frac{-T_{0}}{\rho C_{p}}(\alpha_{1}{\sigma}_{1}+\alpha_{2}{\sigma}_{2})\)(5)
where T0 is the environment temperature, ρthe density, Cp is the specific heat at constant
pressure, αi and Δσirespectively the linear thermal diffusivity and peak-to-peak stress
variations in the principal material directions.
Pitarresi et al. [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[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
[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 c indicates the composite while rthe resin contribution to Young’s moduli, Poisson’s moduliυlt , υtl , the subscriptl stands for longitudinal and t for transverse. Eq. 6
allows the assessment of the thermoelastic temperature signal of resinΔTr 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 [41], producing
an opposing behaviour in the signal [37]:
stiffness/strength variations.
Damage mechanisms can be basically imputable to matrix cracking of
off-axis laminae [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[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.
Material and Methods
The samples tested in this paper were obtained by Automated Fibre
Placement technology [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 [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]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 [48].
Constant stress amplitude tests were performed (S/N curve, run-out at
2*106 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 S/N curve and the 90%
prediction interval bounds. The endurance limit at
2*106 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 < 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.
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 [39].