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].