Abstract
Composite materials used in technically advanced structures are subjected to constant aging from exposure to changing environmental conditions. Studying the effects on such materials due to exposure to varying temperature, humidity, ultraviolet radiation, etc. reveals the impact on their mechanical behavior. This study assesses alterations in static, dynamic, and viscoelastic response of polymer matrix woven carbon fiber lamina composites upon exposure to varying environmental conditions recreated in a climatic chamber. Therein, specimens suffered temperature changes from –35 to +40℃ and humidity variations from <10% to 95% RH (noncondensing) over a period of up to 30 days. Alternating cycles simulating conditions of actual 3–4 h flights were specified. Additionally, specimens of the same material were subjected to thermal shock under similar (to the aging scenario) temperature extremes. All specimens were comparatively assessed via experimental procedures involving three-point bending tests performed in both static and dynamic mechanical analysis for a range of temperatures and frequencies, frequency and thermal scans, and finally impact tests. Results indicate that aged materials exhibit increased dynamic stiffness (expressed by the storage moduli) and decreased material damping ability (expressed by the tan δ parameter). Macroscopic assessment of impact test data was performed via stochastic model-based damage detection methodologies. Results indicate that differences in the impact behavior between pristine and aged specimens are statistically detectable and quantifiable, without input from mechanical testing analysis. More importantly, this assessment of aging-induced effects on the specimens corroborates the findings on storage moduli and tan δ from mechanical testing analysis, thus validating the latter.
Keywords
Introduction
The issue of aging of composite structures due to environmental conditions is not really new. Since the appearance of composites for aerospace structures (in the 1960s), their properties have been studied against environmental aging factors such as ultraviolet radiation which causes matrix crazing and microcracking (Hodzic et al., 1994; Mouzakis et al., 2006, 2008), humidity sorption which leads to matrix–fiber interface deterioration (Hodzic et al., 1994), and thermal shock which can lead to microcracking and delaminations (Hodzic et al., 1994; Mouzakis et al., 2008). To date, mankind is benefiting from the appearance of commuter and passenger airplanes in which composites represent at least 50% of their dead weight. This can achieve up to 25% fuel reduction per flight, which, in turn, leads to reduced flight costs and a significantly increased range for such aircraft without the need for refueling. However, aging of such structures and its impact on the performance of the composite material is not easy to predict. This is due to the multiple mechanisms that act during the environmental aging of such materials (Hodzic et al., 1994; Mouzakis et al., 2006, 2008).
Modeling of damage and associated mechanisms in composites can be achieved by employing numerous methodologies. Elasto-plastic analysis coupled with continuum mechanics has proven to work quite well for long fiber reinforced thermoplastics in predicting their stress–strain response (Nguyen and Kunc, 2009).
Data-driven (analytical or computational) approaches have always been useful for damage description and assessment in composites. Such interesting approaches based on a statistical/probabilistic-based framework have been proposed in Louhghalam and Arwade (2010), where classifiers are used for the prediction of incipient damage in a 2D model for fiber or particle reinforced composites. Probabilistic micromechanical damage mechanics were proposed to predict the effects of manufacturing and residual stresses in four-phase metal matrix composites (Ju and Yanase, 2011). Again, in Rinaldi et al. (2007) a statistical-based framework is used for the description of damage tolerance in multiscale and multiphase materials, and in Chandrashekhar and Ganguli (2009) and Pawar and Ganguli (2003) damage detection in beams and helicopter rotor blades has been accomplished by introducing a genetic fuzzy system analysis of relevant data and Monte Carlo simulations. In Liu et al. (2012), damage detection in carbon fiber reinforced plastics exploits a wavelet-based time–frequency scheme, whereas a slightly different data-driven approach may be found in Barbero and Damiani (2003), which proposes the use of phenomenological models derived from curve fits to data in order to assess tensile strength of composites having suffered aging or stress corrosion. Progressive damage modeling should also be mentioned as a means to predict damage initiation in large composite structures where numerical modeling involves millions of degrees of freedom (Labeas et al., 2012).
Another interesting class of data-driven approaches is based on the use of a stochastic framework for data analysis or modeling. In Wu and Li (1995) such a framework is used for modeling multiple cracking processes in discontinuous fiber reinforced composites. Similarly, Curtin (1999) performs a stochastic-based analysis in order to stress the role of the interphasial damage in unidirectional fiber composites, whereas in Kaminski (2008) the use of stochastic models tuned via Monte Carlo simulations is proposed in order to achieve a homogenous description of the damaged composite structure during aging procedures. Shear lag modeling in combination with Monte Carlo was also employed to predict the damage plasticity, cyclic behavior, and fatigue life in similar metal matrix fiber reinforced composites (Zhang and Wang, 2010). Damage detection has also been performed by vibration response analysis of composite structures subjected to stochastic excitation (Yang et al., 2009) and even by stochastic model-based analysis of polymer epoxy structural members subjected to periodic excitation (Dimogianopoulos and Mouzakis, 2010; Dimogianopoulos et al., 2011). The authors, in the latter contributions (Dimogianopoulos and Mouzakis, 2010; Dimogianopoulos et al., 2011), have illustrated the use of stochastic data analysis on structures via the example of polymer specimens with preexisting damage in the form of circular holes.
Impact damage mechanics and related mechanism analysis are also of paramount significance for advanced composites especially in the transport section where minor or major impact incidents are of a high probability. Hilbert–Huang transform was used to detect and assign failure mechanism frequency regimes in co-woven-knitted epoxy matrix/glass-polyester composites (Ma et al., 2012). In 3D cellular woven composites it was also shown that finite element analysis can be performed in order to predict the impact behavior of such materials (Tang et al., 2011).
The aim of this work was to test carbon fiber reinforced plaques in both pristine and two different aging schemes and, despite using different methodologies for test data analysis, to achieve a universal characterization of the influence of aging effects on the material’s properties. An initial study of test data allows for assessing crucial mechanical properties of the materials, thus drawing a clear picture of the influence of aging on their mechanical, viscoelastic, and impact response. Subsequently, an additional macroscopic assessment of impact test data is performed via stochastic model-based damage detection methodologies. These utilize a completely different theoretical framework from that used in mechanical testing analysis, as performed to get the usual mechanical impact parameters for composites (Tan et al., 2012), in order to statistically detect and quantify differences in the impact behavior of pristine and aged specimens. They are thus validated as an additional but independent means of verifying conclusions from the mechanical testing analysis and corroborate its experimental findings.
Materials and testing
Composite materials manufacturing
Composite laminates were prepared by vacuum assisted resin infusion molding in stacked woven carbon fiber layers, cured for 12 h at room temperature. Resin type was R2940/Infusion RTM (E = 3.6 GPa, σult≈70–85 MPa, Tg≈67–80oC). Carbon fabric used was a C400P (TR50S) plain weave (E = 240 GPa, σult = 4.9 GPa). Supplier of both room temperature curing resin and carbon fabric was Fibermaxcomposites Ltd. The fabric was hand laid in six layers of the following stacking sequence: [(0/90)2/±45]s so as to result in quasi-isotropic plates of 400 × 400 × 2.50 mm3. These plates were subsequently postcured to meet the manufacturer’s requirements. The layers were vacuum bagged and readily mixed resin/accelerator mixture infiltrated them by a flexible tube placed in one side of the vacuum bag. Specimens for testing prior and after aging were cut away from these plaques by a diamond-coated saw blade.
Composite plaques—aging procedures
Two different aging procedures were performed for the composite plaques. A mild one, which did not include the effect of humidity, and a more realistic flight-simulating scenario including humidity. These aging scenarios were performed as follows:
(Scenario A—thermal shock) composite plaques were aged for 12 days by indwelling for 24 h at 50oC, RH = 65% inside a climatic chamber of Instron mod. 3119-409 (High Wycombe, UK) and were immediately afterwards removed and placed in a dry air freezer for 24 h at 40oC. (Scenario B—environmental aging): In order to investigate the combined effects of temperature and humidity on composites subjected to changes of temperature from –35 to +40℃ and humidity variations from <10% to 95% RH (noncondensing), specimens were stored in a climatic chamber (Vötsch 4060, Balingen-Frommern, DE) for 30 days. Alternating cycles were chosen as if actual flight cycles of 3–4 h of an airplane flight were to be simulated.
Experimental procedures
Three-point bending (3pB) testing was performed on an Instron 3382 as per EN ISO 178:1996, equipped with a specially manufactured high-stiffness bending rig as shown in Figure 1. Specimens’ dimensions were length = 50 mm, width = 15 mm, t = 2.50 mm, whereas the crosshead speed v was set at 2.1 mm/min. Support span was 40 mm. Five specimens per sample were used for property averaging. A similar specimen geometry (50×12×2.50 mm3) was used for the dynamic mechanical analysis (DMA) in 3pB mode on a DMA Q800 system of TA Instruments. Temperature was kept isothermally at T = 25oC, and frequencies scanned were f = 1–200 Hz. Maximum dynamic displacement was set at 25 µm. Impact testing analysis was performed on an Instron CEAST 9340 Drop tower, with a semispherical impactor head of Ø20 mm at an incident energy of 18 J. The impactor energy value was set just above the bibliographic threshold of critical impact energy for such laminates (Klaus and Reimerdes, 2009; Stavropoulos et al., 1998) of ca. 18/2.5 = 7.2 J/mm thickness. Force versus time impact signals were digitally recorded for further statistical analysis. Slabs from 3pB tests were further examined by scanning electron microscopy (SEM) in order to examine their failure characteristics. A Jeol 5200 SEM was used to examine the fractured specimens of aged and pristine samples.
Stress–strain graphs from three-point bending specimens: (a) pristine, (b) 12 days T-shock, and (c) 30 days of accelerated aging.
Results and discussion
Three-point bending tests
Flexural properties for all composite specimens.
Dynamic mechanical analysis
It becomes obvious by observing the DMA curves in Figure 2 that aging has a strong effect on the storage modulus E′. With increasing aging time, the dynamic storage of the woven carbon composite deteriorates whereas the glass transition area (Tg) increases when only the effect of temperature is imposed as aging and tends to return back when the combined scenario of temperature and humidity is applied. This probably owes to some postcuring effect in the first scenario, whereas probable plasticization of the matrix due to humidity is responsible for the Tg return in the case of the combined effects of temperature and humidity. It should be reported here that the trend for dynamic E′ modulus reduction (Figure 2(a)) confirms the results from static 3pB testing.
DMA parameters: Storage modulus (a) and tan (δ), (b) as functions frequency.
Damping coefficient (tan δ parameter) is plotted in Figure 2(b). It can be seen that tan δ verifies the initial migration of the Tg of the composites in the case of temperature-only aging scenario, whereas it also shows the return of the Tg when both temperature and humidity are cycled during the aging procedures. Another interesting finding here is that with increasing aging time the values of tan δ are reduced. This is a very important effect, since it implies that the capability of the composite material to damping is also reduced, and moreover it signals for eigenfrequency shifting to higher values verified by the authors in earlier work (Mouzakis et al., 2008). Hence, the important finding is that aged composites lose part of their ability to dampen vibrations and those structures involving such composites shall suffer from eigenfrequency shifting (Mouzakis et al., 2008).
The conclusions on tan δ and storage modulus are also validated later on (see “Validation of experimental findings on aging effects via the multimodel approach” section) via a stochastic model-based analysis, and, thus, constitute guidelines to be taken seriously into account by designers of composite structures.
DMA demonstrated the effects of aging in a very clear manner. Frequency scans in 3pB mode have shown that dynamic stiffness as expressed by the storage moduli (E′) is being reduced as in the static 3pB experiments, for 12 days of thermal shock. The 30-day exposure to alternating-combined temperature/humidity cycles results in the lower dynamic stiffness. A resonant frequency effect can be observed in all E′(f) curves right after 50 Hz but this is a specimen geometry dependent effect. Also, tan δ(f) curves show distinctively that the 30-day aged specimens exhibit the highest damping coefficients in comparison to all other specimens possibly due to their increased internal damage. This parameter did not deliver significant differences with respect to pristine and thermal-shock treated laminates.
Impact testing
In Figure 3, the typical force–displacement, F–t (a) and force–deformation, F–x (b) graphs are shown for all specimen types. Note that no specimen was fully penetrated. The F–t curves were stored for all specimens for further processing through statistical algorithms. The differences among the tested specimens are interesting. For instance, the pristine specimens exhibit the highest force maxima at different time instants with respect to the main fracture event.
Typical (a) force–time impact curves and (b) force–displacement curves for the three types of impacted specimens.
Impact properties of the composites studied.
Scanning electron microscopy
Delamination appears to be the governing failure mechanism in pristine specimens as seen in SEM pictures in Figure 4(a) and (b). This seems to be facilitated by the presence of fiber bundles in the woven layers as they too appear to delaminate across their borders. Matrix–fiber interphase appears relatively good as seen in Figure 4(c) and (d) in the pristine specimens since the fibers appear to be relatively well embedded in the matrix.
SEM scans from the fractured three-point bending pristine specimens.
On the other hand, delamination corroborated by intralayer cracking appears now as the main failure mechanism in the aged composites (Figure 5(a) and (b)). Matrix–fiber interphase appears to be relatively fair as fibers exhibit a relatively smooth surface, which accounts for interphase corruption (Figure 5(c) and (d)). Similar SEM pictures with the same damage mechanisms are obtained for the 30-day cycled specimens.
SEM scans from the fractured three-point bending specimens after 12 days of thermal shock aging.
Validation of experimental findings via stochastic model-based fault diagnosis methods
The experimental findings reported in “three-point bending tests, dynamic mechanical analysis, impact testing, and scanning electron microscopy” sections suggest that the mechanical properties of specimens from carbon fiber reinforced composites are significantly affected by aging. In this section, stochastic model-based fault diagnosis methods are used to evaluate the impact test data of nine such specimens (three from each aging procedure, see Table 2). The aim is to detect impact behavior changes, which are consistent with the experimental findings and, thus, indirectly validate the latter. Similar stochastic-model based diagnosis methods have been tailored over the last 20 years for use with health monitoring applications, with (a nonexhaustive list of) examples involving aerospace systems (Dimogianopoulos et al., 2012), medical orthopedic implants (Mouzakis et al., 2009), or even smart structures with integrated contact-free sensors (Dimogianopoulos, 2012).
The pristine specimens are considered as systems with intact internal structure (health). Each system undergoes a standard impact testing procedure, and its force data during the (short) impact time are recorded. Any aging-induced effects in a specimen’s health will lead to obtaining force data quite different than those from a pristine specimen. Hence, comparing the aging-related information (dynamics) in the impact data from an “unknown health” specimen with that of a pristine one allows for qualifying the (hitherto) unknown health. According to the framework of stochastic model-based diagnosis, two approaches may be considered (Mouzakis and Dimogianopoulos, 2008):
The single-model approach: This postulates that a discrete-time stochastic model is identified on force data from one (or a group of) pristine specimen(s). The model accounts for the dynamics of the typical force data of pristine specimens, meaning that any aging-induced effects in the dynamics are not captured. The model is, then, used for processing force data from unknown health specimens. The more these specimens suffer from aging, the less their force data will be accurately represented by the stochastic model, since this is optimized for the behavior of pristine specimens. Hence, the model’s statistically evaluated “lack of fit” to the data set examined is an indirect indicator of the aging-induced effects in the specimen, whose testing provided the data set. The multimodel approach: This postulates that one discrete-time stochastic model is identified on data of each specimen or a group of specimens of similar health, so that the number of identified models is equal to that of (groups of) specimens. Then, statistically comparing the characteristics (parameters, damping factors, natural frequencies, etc.) of models identified on pristine specimen data with those of models corresponding to unknown health specimens allows for qualifying the unknown health state.
Note that one stochastic model describing a (group of) signal(s) may consist of several submodels operating in tandem, as described in “Identification of stochastic time-series model schemes” section. The term “single-model” simply refers to using one such model for assessing all specimen groups, thus limiting the required time and computational effort for processing data from all specimens. Naturally, if more (computational) effort can be invested in the diagnostic task, the multimodel approach allows for better insight in characteristics of the force data signals, and, thus, for more detailed conclusions. The current analysis, as presented in “Identification of stochastic time-series model schemes and Evaluation of impact force data of tested specimens via the single-model approach” sections, undertakes the single-model approach for assessing the changed impact behavior due to aging of the tested specimens. Finally, in “Validation of experimental findings on aging effects via the multimodel approach” section, the differences in impact behavior among specimens are evaluated via the multimodel approach and checked for consistency with the experimental findings on aging-affected mechanical properties.
Identification of stochastic time-series model schemes
The force versus time data signals from nine similarly tested specimens (three from each aging procedure, see Table 2) are considered. Following the single-model approach of “Validation of experimental findings via stochastic model-based fault diagnosis methods” section, a discrete-time stochastic pooled nonlinear representation is used for building (identifying) a model describing the force signals from the group of three pristine specimens. As previously stated, throughout the analysis the stochastic model will be shown to include two submodels. The starting point is the basic representation used for the model, which has the following form
The set of equation (1) states that one model (admitting this representation) with constant coefficients θi may describe output (force) data from any of the three impact tests, using the principle that any current output yj[t] is a linear regression of nonlinear products of lagged yj values, augmented by a noise term representing the underlying uncertainties. Such constant coefficient pooled nonlinear autoregressive (CCP-NAR) models have been successfully used in demanding applications, such as the representation of nonlinear relationships among jet engine variables (Dimogianopoulos et al., 2012). Finally, note that the special case of a representation with linear regressors pi,j[t] (i.e. values of yj[t–1] up to yj[t–ny]) is referred to as constant coefficient pooled autoregressive (CCP-AR) representation. Note that if the coefficients θi are known then, at each time instant t, the best prediction yj[t]
pr
of the real value yj[t] is given by the sum involving terms θi·pi,j[t] in equation (1). Then, yj[t]–yj[t]
pr
= ej[t], which is the unknown noise term. Hence, the accuracy of fit of a given model to a data set may be assessed by using these data for computing the difference yj[t]–yj[t]pr at each instant t. If the residual’s ej[t] sum of squares (RSS) over the signal’s yj[t] sum of squares (SSS) is “large,” then the model’s lack of fit to the current data is significant, and the model is not representative of the signal. A good model is characterized by small values of RSS/SSS and residuals as uncorrelated as possible. The degree of correlation of ej[t] is checked by means of the autocorrelation coefficients ρi (see Box et al., 1994, p. 26) at lag i of the residuals, which reflect the dependency (or correlation) between the ej[t] and the ej[t–i] values. For this purpose, an N-sample long data sequence [ej[t–(N–1)]... ej[t]] is formed, and the following binary composite hypothesis testing problem is constructed
The portmanteau (as is often referred to) hypothesis test at the risk level α (defining the probability of rejecting ℋ0 given that ℋ0 is true) is then formulated as
Statistical hypothesis test for checking the correlation of the sequence ej[t] using ρi … ρr and the test statistic Q in equation (5) (the shaded area corresponds to accepting ℋ0 at risk level α).
The identification of the stochastic model is performed along the guidelines given in Dimogianopoulos et al. (2009) and is presented in detail in Appendix 1. This two-stage model includes a CCP-NAR submodel with nl = 4, ny = 15, and 132 terms (listed in Appendix 1) and a CCP-AR submodel with 115 terms. Figure 7 shows the structure of the two-stage model, which accurately represents the differenced force data (i.e. yj[t]≡f[t]–f[t–1], for j = 1, 2, 3 and ∀t) from impact with any of the three pristine specimens. Main dynamics are represented by the CCP-NAR submodel, which produces the first-stage residual signal ej[t]. The portmanteau hypothesis test in equations (3) and (4) asserts that the signal ej[t] is not completely uncorrelated, meaning that some of the yj[t] dynamics are not yet modeled. Then, the CCP-AR submodel is charged with modeling all such unmodeled dynamics in ej[t], as attested by the resulting second-stage residuals ɛj[t]. Using the portmanteau test, these are found as significantly less correlated than their first-stage counterparts ej[t] (see Appendix 1), meaning that the two-stage model is representative of signal yj[t]. Typical plots of the force versus time, the differenced force, and the first- and second-stage residuals (derived from the CCP-NAR and the CCP-AR models, respectively) are shown in Figure 8(a) to (d).
The modeling scheme comprising of two stages and equivalent number of residuals (the subscript “pr” indicates predicted quantity). Experiment with the second (out of three) pristine specimen: (a) Force versus time, (b) differenced force y2[t] and CCP-NAR prediction y2[t]pr, (c) first-stage residual e2[t], and (d) second-stage residual ɛ2[t].
Evaluation of impact force data of tested specimens via the single-model approach
Given the force data from impact tests, the diagnosis should involve the detection of specimens affected by aging and the estimation of the severity of inflicted damage. These are performed as follows:
Values of test statistic Q (and upper limit at α = 0.0001 for ensuring uncorrelated sequence), hypothesis accepted, and RSS/SSS values computed on second-stage residuals obtained from tests with all specimens.
Severity estimation: The statistical hypothesis tests state that, when aged specimens are tested, the resulting second-stage residuals ɛj[t] are more correlated than their counterparts from tests with pristine specimens. It is important to note that the magnitudes of test statistic Q values have no specific trend related to the degree of aging-induced damage in the specimens (see Table 3). On the other hand, the RSS/SSS values are clearly a quantitative indicator of “lack of fit” of a data sequence when used with the two-stage model. Figure 9 shows the first- and second-stage residuals from “aged 2” and “aged 5” specimens, which have suffered a 12- and a 30-day aging procedure, respectively. The different aging procedures have resulted in specimens behaving quite differently in similar impact tests, meaning that the relationship (or dependency) among samples in the force signals is not the same as in the case of pristine specimens. Consequently, when these force signals are forwarded through the differencing and, subsequently, the CCP-NAR and CCP-AR blocks (Figure 7), the resulting residuals can never have the same characteristics as those resulting from pristine specimens, for a simple reason: The CCP-NAR or CCP-AR models perform predictions as if pristine specimens were used, whereas the actual signals come from aged specimens. This is practically shown by the first- and second-stage residuals indicating large spikes of different magnitude at around 1.2 s for each (aged) specimen in Figure 9. For comparison, spikes of these (aged) specimens are notably larger than those featured by pristine specimens in Figure 8(c) and (d). Furthermore, the spike magnitudes in first- or second-stage residuals of the “aged 2” specimen are notably smaller than those of the “aged 5” specimen, meaning that the severity of aging-induced damage is related to the spike magnitude from each specimen. These spikes are, in turn, indicative of the degree of “lack of fit” of each force data set on the (common) two-stage model, as described by the RSS/SSS values in Table 3. If the two-stage model could perfectly exploit (fit) the force signals considered, then no large variations between actual and predicted values (hence, spikes in residuals of Figure 9) would have been obvious. Consequently, the RSS/SSS values provide an indirect estimation of the severity of aging-induced damage on the specimen, with this being verified in all but the “aged 4” specimen, as given in Table 3. Thus, using the single-model approach one may relate the degree of the two-stage model’s inability of explaining the considered data to the (existence and) severity of aging-induced effects in the corresponding specimens.
Experiment with “aged 2” (12-day aging) and “aged 5” (30-day aging): (a) first-stage residuals from “aged 2” data, (b) second-stage residuals from “aged 2” data, (c) first-stage residuals from “aged 5” data, and (d) second-stage residuals from “aged 5” data. Regressors of optimal CCP-NAR model with nl=4 and ny=15.
Note that a larger set of tests per aging procedure would enable a clearer representation of variations due to noise, test condition repeatability, and various uncertainties in experimental data. Nonetheless, the use of stochastic modeling for each data set allows for estimating the true system dynamics (those corresponding to the specimen response to the impact) as opposed to dynamics related to parasitic phenomena (those represented by the RSS/SSS, see Table 3). If a stochastic model is identified on pristine data, then its RSS/SSS quantifies (in a way) the parasitic phenomena in the testing procedure. These phenomena appear, thus, quite similar for the pristine samples given the limited RSS/SSS differences between them in Table 3. At the same time, the respective phenomena for the aged samples are also quite similar in RSS/SSS terms in all but one case (aged 4) while, at the same time, radically different (RSS/SSS values from 4× to 50× higher) from those of the pristine ones. Such consistent differences between pristine and aged specimens may only be attributed to changed internal structure of the latter (see also “Validation of experimental findings on aging effects via the multimodel approach” section), even though more tested specimens per aging procedure should help to quantify better the underlying parasitic phenomena.
Validation of experimental findings on aging effects via the multimodel approach
In “Evaluation of impact force data of tested specimens via the single-model approach” section, the severity of aging-induced damage is estimated by linking the “abnormal” specimen’s behavior with the statistical “lack of fit” of the specimen’s data to the given two-stage model. There is no direct link to the aging-induced degradation of mechanical properties reported in “Three-point bending tests, dynamic mechanical analysis, impact testing, and scanning electron microscopy” sections. Such link will now be established using the multimodel approach to show that the aged specimen’s test behavior is consistent with the aging-induced changes in mechanical properties such as the storage modulus. An initial problem is that using the multimodel approach would involve identifying three two-stage models, each one for a group of specimens (pristine, 12- and 30-day aging). Given the complexity of CCP-NAR identification the task could soon prove very complicated. However, if the identified CCP-NAR model is kept common for all the three groups, then the identification of a CCP-AR model for each of the three groups of tested specimens is a trivial task. This compromise means that only the part of the yj[t] dynamics remaining in ej[t] will be studied, but this is sufficient for concluding on the test behavior of specimens affected by aging, as will be shown later on.
At this point, it would be useful to recall some basic facts of linear system theory (see Figure 10). Any “large” linear model may be considered as a superposition of first- and second-order submodels, with each one representing a specific part of the dynamics of the modeled signal (Dorf and Bishop, 2008, pp. 294–296). In turn, each of these subsystems is completely defined by the location of its poles on the complex plane. Figure 10 presents the most interesting case of a second-order subsystem Gi(s) with its pair of poles characterized by a damping factor ζ and a natural frequency ωn. The subsystem with pole pair located closer to the imaginary axis than all other subsystems is responsible for the major part of the dynamics of the modeled signal. It is, therefore, referred to as dominant subsystem (and pole pair, see Figure 10). Then, studying only the dominant pair provides a comprehensive explanation of the signal dynamics modeled via the “large” linear model.
Basics of linear system analysis: A system shown as a superposition of first- or second-order subsystems, with poles of a second-order subsystem and their characteristics shown.
Consequently, in the current case, given the first-stage residuals ej[t] (produced via the standard CCP-NAR submodel) for each group of specimens, one may identify one second-stage CCP-AR submodel on ej[t] residuals of each group. As already stated this is a trivial procedure using the “IDENT” Graphical User Interface in MATLAB(R). Each of these “large” linear models may be converted to its continuous-time counterpart (via the MATLAB command d2c.m) and studied with respect to the damping factors and natural frequency characteristics of the dominant pair of poles. Differences in these dominant pair of poles are directly related to differences in the modeled first-stage residuals ej[t] of each group of specimens. The multimodel-based analysis may, thus, give rise to the following algorithmic procedure:
For each group of specimens, the force data are used for producing first-stage residuals ej[t] by means of the common CCP-NAR model identified on data from pristine specimens (see “Identification of stochastic time-series model schemes” section). For each group of specimens, the first-stage residuals ej[t] are used for identifying a CCP-AR model using the IDENT Graphical User Interface in MATLAB(R). The three identified CCP-AR models are then converted to their continuous-time counterparts via the command d2c.m in MATLAB(R). The dominant pair of continuous-time poles is extracted for each model and their damping factor ζ and natural frequency ωn are computed. Major differences in ζ and ωn among the three pairs of dominant poles (one per group of specimens) indicate different characteristics of first-stage residuals ej[t] among the three groups and, consequently, different test behavior. The product <ζ*ωn> from dominant CCP-AR poles for each group of specimens for growing CCP-AR model orders, showing a clear trend for CCP-AR orders larger than 32.
A critical point for the validity of results of this algorithmic procedure is the accuracy of representation of the ej[t] dynamics achieved by the CCP-AR model. A large model order ny does not always lead to the most accurate model, as attested by the CCP-NAR model identification (see Appendix 1). For this reason, CCP-AR model orders ny ranging from a small value of 20 to the (optimal for the pristine specimen case of) 115 have been tested, with the five-step procedure applied for each value of ny. The results shown in Figure 11 are highly significant, because they indicate that the product <ζ*ωn> (with ζ and ωn associated to the dominant pair of poles for each CCP-AR model examined) has a uniform trend for the majority of CCP-AR orders considered. Specifically, all CCP-AR dominant poles from pristine specimens exhibit the lowest <ζ*ωn> values, followed by the slightly larger <ζ*ωn> values from specimens having suffered a 12-day aging procedure and, finally, with the largest <ζ*ωn> values corresponding to specimens having suffered a 30-day aging procedure. This trend is uniform in the sense that it is valid for CCP-AR model orders ranging from 33 to 115, whereas all dominant poles (whose products <ζ*ωn> are shown in Figure 11) admit ωn values from 75 to 150 r/s (11–23 Hz). This result essentially states that even if one selects a less-than-perfect representation of the ej[t] dynamics, the <ζ*ωn> values for the dominant poles will still respect the previously defined trend.
Now, the product <ζ*ωn> is critically related (among others) to the time duration of the specimen’s vibration under impulse testing (Palani, 2010). The impulse response of the specimen is characterized by lightly damped oscillations, with a decay factor proportional to the term [exp(–ζ*ωn*t)] (see Palani, 2010, p. 5.35), and t indicating time. This means that larger <ζ*ωn> values indicate vibratory responses of shorter duration. Hence, the vibration of the most aged (i.e. the 30-day aged) specimen will have the shortest duration of all, followed by those of the 12-day aged and the pristine specimens. These results validate findings of the experimentally determined storage modulus for the three specimen cases shown in Figure 2(a). This figure presents the storage modulus values as function of the frequency of the sinusoidal loading imposed to the specimen under test. Focusing on the region of interest of 11–25 Hz, it is obvious that the part of the energy provided to the specimen by the impact and transformed into vibratory motion is large for the pristine, notably smaller for the 12-day aged and even smaller for the 30-day aged specimen. Then, the duration of vibration should be maximal for the pristine, shorter for the 12-day aged, and the shortest for the 30-day specimens.
The same conclusion is drawn from Figure 2(b), which presents the tan δ coefficients as function of the frequency. In the region of interest (11–25 Hz), the pristine specimen is characterized by the lowest values of tan δ, followed by slightly larger tan δ values for the 12-day aged, with the largest tan δ values from the 30-day aged specimens. According to Osinski (1998, pp. 129–130) the damping decrement characterizing the lightly damped oscillations of a beam is a linear function of the material loss factor η (or tan δ). On the other hand, the damping decrement is directly proportional to the term ln(x[t]/x[t+k]), with x[t] and x[t+k] indicating the oscillation peaks of the beam’s response at time instants [t] and [t+k], respectively, and ln(.) indicating natural logarithm. Since the decay of x[t] is proportional to the term [exp(–ζ*ωn*t)], the term ln(x[t]/x[t+k]) is proportional to <ζ*ωn> and, thus, so is the tan δ parameter. Then, based on this last remark, Figure 11 confirms the experimentally derived results of Figure 2(b), since the largest tan δ values should be exhibited for 30-day aged specimens, followed by smaller tan δ values for 12-day aged specimens and finally the smallest tan δ values for 30-day aged specimens. Thus, in terms of both the storage modulus and the tan δ, the results of the stochastic model-based analysis validate the dynamic mechanical testing analysis.
Conclusions
Extensive experimental testing has shown that in the early stages of environmental aging in woven carbon fabric/epoxy reinforced composites, the following effects may be noted:
Flexural modulus and strength show a deteriorating behavior as aging time increases and humidity is combined with temperature cycling. Viscoelastic property analysis has shown that dynamic moduli are monotonously decreasing with aging time and scheme, whereas the 30-day aged specimens showed increased damping coefficients hinting for increased internal damage. It should be noted here that the findings on dynamic storage modulus reduction due to aging are in concert with the findings of the static flexural testing. Impact analysis has shown that the behavior of the aged specimens is inferior to the pristine ones due to probable internal damage. Embrittlement and strength deterioration under impact loading are typical changes in the composite material behavior. SEM revealed interphase damage in all aged specimens and increased interlaminar cracking during loading.
Stochastic model-based analysis of the impact data confirmed the inflicted damage due to aging effects. This damage resulted in significant differences in the behavior of aged specimens under impact testing with respect to their pristine counterparts. The analysis showed that, using impact test data, the aged specimens may be statistically detected and the severity of damage effects on the specimen may be evaluated. More importantly, the analysis also showed that the impact test behavior of aged specimens is consistent with the aging-induced (and experimentally verified) alterations of their mechanical properties and especially the dynamic ones.
Footnotes
Funding
The author(s) received no financial support for the research and/or authorship of this article.