epoxy composite subjected to accelerated environmental ageing

Composite Structures 111 (2014) 179–192

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Composite Structures journal homepage: www.elsevier.com/locate/compstruct

Multi-factorial models of a carbon fibre/epoxy composite subjected to accelerated environmental ageing Enrique Guzmán, Joël Cugnoni, Thomas Gmür ⇑ Faculté des Sciences et Techniques de l’Ingénieur (STI), Ecole Polytechnique Fédérale de Lausanne, Station 9, CH-1015 Lausanne, Switzerland

a r t i c l e

i n f o

Article history: Available online 6 January 2014 Keywords: Carbon fibre epoxy composite Accelerated ageing protocol Aero-structures Moisture absorption Design of experiments Structural health monitoring

a b s t r a c t Among materials being introduced in the aerospace industry, the carbon fibre reinforced plastics (CFRP) have a place of privilege because of their exceptional stiffness-to-mass ratio. However, the polymerbased matrix is vulnerable to damages by environmental conditions. This work exposes the experimental results of several accelerated environmental ageing protocols on CFRP panels. The main concern is to justify or reject by statistical means that a significant degradation of mechanical properties does occur over the time, and to establish a basic model to quantify the effects of different environmental factors of the composite ageing. The results considered here are the elastic properties evaluated over several weeks of accelerated artificial ageing. The stiffness degradation of the samples subjected to the aforementioned ageing protocols is statistically described by a non-linear multi-factorial model inspired by the Design of Experiments (DoE) theory. The evolution of constitutive properties (namely mass and elastic properties) over the time exhibits an asymptotic exponential increasing (or decreasing) pattern over the time. The usefulness of these mathematical models is their predictability, based only on theoretical considerations on moisture absorption. This path is further investigated in this paper, clearing up the way to a methodical prediction of ageing models. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Composite materials are ultra-light structural materials, massively introduced in the recent years in aeronautical applications. In spite of their exceptional mechanical performances, they are vulnerable to aggressive natural ageing factors such as rough temperature changes, chemical corrosion, moisture and solar radiation. It is widely accepted that special caution needs to be observed when using these materials to manufacture mechanically critical airframe components inside a full-scale structure. Although an online health monitoring is strongly suggested by authors [1–3] to verify the state of a composite structure, a model could be useful to estimate the degradation extent when subjected to some frequently confronted weathering agents. Carbon fibre reinforced plastics (CFRP) are currently the most used composites in the aeronautical industry (for example in the Boeing 787 and the currently in development Airbus 350 XWB). There is currently in scientific literature some lack of understanding of polymer-based composite materials behaviour to weathering. Research has been mainly centred about the effects of mechanical fatigue [4] and chemical corrosion [5,6]. A quantitative model of the material ageing under other usual natural agents ⇑ Corresponding author. Tel.: +41 21 693 2924; fax: +41 21 693 7340. E-mail address: thomas.gmuer@epfl.ch (T. Gmür). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.12.028

such as heat, moisture or solar radiation, when applied cyclically on a sample, has not yet been established. This is frequently due to the elevated number of factors that can potentially affect the life cycle of polymers. Considering the case of CFRP [7–10], the polymer-based matrix and the fibre/polymer interface are the most vulnerable components of the composite. Specialised researchers in the field of composite ageing have described qualitatively the failure mechanisms [6,9,10] or presented empirical quantitative evidence of changes in constitutive properties of polymers [7,8,11,12] when subjected to natural and/or artificial weathering. In more extensive treatises by Carraher [13] and Brinson [14], some chemical mechanisms that explain the degradation are reviewed. These changes lead to a progressive macroscopic degradation of the elastic properties of the composite, which could turn out to be critical if the proper safety precautions are not considered. A method to evaluate systematically the ageing of polymerbased composites and establish a mathematical model is proposed, which can be interpreted to understand the contributions of isolated or combined factors on the ageing process of aerospace composite. It will be verified if it is possible to estimate quantitatively the extent of the ageing by estimating the parameters of the mathematical model, based only on theoretical assumptions and basic information about the subject material. According to the specialised literature in the design of experiments theory [15,16], a multi-factorial model is useful to compare quantitatively the influence

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of the existent ageing factors on the composite panels. The elastic properties, which are the object of this study, are measured by indirect methods, using modal testing to obtain the natural frequencies of the working specimens [17], which in turn are processed by a mixed numerical–experimental identification algorithm to obtain the experimental elastic properties [18]. Physically, the modal testing is done using mainly integrated piezoelectric sensors and accelerometers, which can be inscribed in a global structural health monitoring (SHM) method, which in summary allows surveying the state of a structure using networks of dynamic sensors. The choice of a dynamic measurement method is motivated by its flexibility, robustness, readiness and non-destructiveness when compared to more classic static methods (tensile tests, bending tests, ultrasound), qualities that are appreciated in aeronautical applications. The mathematical pattern of the aforementioned model can be hinted by visual inspection of the experimental pre-modelling plots after the identification of the constitutive properties. The notion of Prony series and asymptotically exponential increase or decrease are inspired from previous works on composite testing [11,13,14]. A deeper statistical analysis shows a correlation between the absorbed moisture mass and the loss of stiffness. This statement leads to the replacement of the number of cycles by the water concentration as the state variable in the model. The usefulness of this reasoning is evident in the final lines of this paper: it leads to a generalised model for mechanical parts with more complex shapes, and contributes to the future research on CFRP with an early estimation of the extent of their ageing using only some basic information about the material’s initial properties. 2. Accelerated ageing protocol due to environmental conditions In order to reveal any changes in stiffness due to exposition to aggressive ageing factors, progressively demanding experimental campaigns took place. The ageing factors included temperature, relative humidity (RH) and ultraviolet (UV) radiation, since they are frequently met by full-scale CFRP aeronautical structures during a life cycle. The ageing protocols were inspired from previous works [7–10] as well as in ASTM standard guidelines for cyclic ageing protocols [19], for the combined hygro-thermal testing [20–22] and for UV radiation testing [23,24]. These factors, alone or combined, have several effects on polymer-based pieces inside longtime operating aircraft: water diffusion, polymer molecules cross-linking/de-linking, alternate dilatation/contraction cycles, photo-oxidation, post-curing, residual stress, temperature gradients, and many others. In summary, a suitable ageing protocol would have the following characteristics: a. Cyclic conditions are more likely to induce damage than steady conditions. Indeed, at a microscopic level, cyclic thermal gradients and water concentration gradients continuously contribute to the rise of mechanical stress around the fibres and between the composite layers. Cyclic changes induce thus additional damage due to the mechanical fatigue of the components. Moreover, this is far closer to reality since aerospace components are constantly exposed to cyclic environments. b. Temperature peaks are usually fixed above the start point of the glass transition zone. In this paper, this range of temperatures was estimated from the supplier’s data sheet and after the corresponding curing process (curing rate is higher than 99%). Physically, dilatation, vitreous transition and thermal oxidation are likely to occur at elevated temperature. c. Highly humid environments can contribute to composite ageing as well, facilitating fibre de-bonding, de-lamination, embrittlement, polymer chemical weakening, inner stressing,

etc. The mass absorption is to be verified by weighing periodically the total mass of the samples, in order to evaluate the diffusion coefficient. d. High UV radiation is generally more specific to structures continuously dwelling at high altitudes, leading to photooxidation and polymer chain dissociation, among others. Consequently, the specimens were subjected to cyclic environments, with the following factors controlled: the surrounding temperature (T), the relative humidity (RH) and the intensity of an A-class ultraviolet radiation (UV) lamp on one face of the plate samples (Fig. 1). The cycles are longer for the humid protocols due to technical reasons. The scheduled campaigns and the series, with the respective codes and samples included, the protocol parameters and the durations are summarised in Table 1. Hardware included a Weiss TechnikÒ WK180/40 climatic chamber, with the required control and data acquisition software. Normalised variables are worked with: for a real factor u (that can be either T; RH or UV), the corresponding normalised factor x is given by

x¼

u  umin umax  umin

ð1Þ

where umin and umax are respectively the minimum and maximum value adopted by u. Thus, the value of x is always between 0 and 1, inclusively. The normalised temperature, relative humidity and UV radiation are denoted x1 ; x2 and x3 respectively. These are shown along with the complementary time parameters for each ageing protocol in Table 1. After an estimation based on data provided by the supplier and in the literature, the thermal and the water diffusion coefficients, the thermal equilibrium due to convective transfer inside the climate chamber is reached after between 1 and 2 min in the range of temperature 25–125 °C, so the saturation is easily reached. On the other hand, the moisture saturation is not reached in one cycle, in the case of 4 mm-thick plate samples. In fact, it is not reached before 200 h in steady-state conditions. This is expected, because of the tests carried out prior to the cyclic ageing tests (as detailed in Section 6.1). However, the water concentration is accumulative, since most of the water absorbed in each cycle remains inside the structure during the dry phase of the cycle. At the end of the protocol, there is a significant amount of water inside the bulk body of the sample (as shown in Section 6). Concerning the penetration of UV radiation, the same problem can be observed. After the information available in the literature, the equilibrium of radiant heat on plates by UV radiation (340 nm wavelength, 0.35 W/m2/nm intensity, as recommended by the G155 ASTM standard, corresponding to a direct sunlight beam at sea level and 0° latitude) is estimated to take about 20 min. This time period is less than any of the periodic exposure times faced in the protocols (the shortest is 1800 s = 30 min). 3. Sample manufacturing 3.1. CFRP specimens The material for experimentation is the Carbon-PrePreg PR-UD CST 125/300 FT109, supplied by Suter-KunstoffeÓ AG (Switzerland). It is originally a scroll of unidirectional (UD) carbon fibre tissue (ToraycaÓ T700S carbon fibre), pre-impregnated in unhardened epoxy polymer (PREDOÓ FT109) with an areal weight of 125 g/m2 (60% of fibre volume fraction). The nominal after-curing elastic properties of this material are given in Table 2. A total of six different 30  30 cm2 surface plates were manufactured in autoclave (curing at 85 °C and under 5 bar for 10 h, followed by curing at 90 °C and under 5 bar for 4 h). Each one of these square

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Table 1 Summary of ageing protocol parameters for all the series: nominal values (T; RH and UV), normalised values (x1 ; x2 and x3 ), the number of cycles N cyc , the time period Dt ¼ Dtþ þ Dt  and the number of measurements q. Series

Samples

Temperature T (°C)

D EF1 EF2 G1 G2 H1 H2 J a

DI-DVI EI, FI EII, FII GI,GIII GII HI,HIII HII JI-JV

Relative humidity x1

Min

Max

5 45 45 5 5 45 45 5

95 135 135 95 95 135 135 95

RH (%)

0 1 1 0 0 1 1 0

UV radiation 2

x2

Min

Max

0 0 0 0 0 0 0 0

95 70 70 0 0 0 0 95

UV (W/m )

1 0.73 0.73 0 0 0 0 1

N cyc

Dt (min)

q

Dt  a Dt þ

800 900 900 1100 1000 1100 1100 800

90 90 90 60 60 60 60 90

8 9 9 12 11 12 10 8

2:1 1:1 1:1 1:1 1:1 1:1 1:1 1:1

x3

Min

Max

0 0 0 0 0 0 0 0

300 300 0 0 300 300 0 0

1 1 0 0 1 1 0 0

Dt þ is the part of the cycle at high temperature and Dt is the part at low temperature.

Table 2 Nominal elastic properties after layup and curing (1: longitudinal direction, 2: transverse direction, 3: normal direction). Over 63 measurements, l is average value and r is the corresponding standard deviation. Young’s modulus (GPa)

l r a

Poisson’s ratio (–)

Shear modulus (GPa)

E1

E2

E3

m12

m13

m23

G12

G13

G23

96.00 0:067

7.67 0:013

8.70 –a

0.38 –a

0.3 –a

0.03 –a

3.60 0:021

3.59 –a

2.24 –a

Nominal values as supplied in the data sheet.

Table 3 Specimen dimensions and initial mass.

3.2. Sensors and integration

Sample

Length l (mm)

Width w (mm)

Thickness h (mm)

Mass m (g)

DI-II-III-IV-V-VI EI-II FI-II GI-II-III HI-II-III JI-II-III

150 300 300 300 300 300

99.0 98.5 81.6 92.0 95.4 98.2

4.0 4.4 4.0 3.7 4.2 4.3

83:49  0:43 188:5  4:5 134:0  2:0 153:0  3:0 179:0  2:0 181:0  1:5

plates was divided, giving six different groups of specimens, named D–J and scheduled to be tested following the ageing protocols described in Section 2. The layup configuration was purely UD, with the carbon fibres in the longitudinal direction. The dimensions and corresponding nominal masses of the plates were measured using an electronic Vernier and Mettler ToledoÓ weigh balances respectively (a 0.01 g resolution for the D-series, a 0.1 g resolution for all the others, see Table 3).

Fig. 1. Example of ageing protocol profile: the temperature T (in blue), the relative humidity RH (in green) and UV radiation intensity UV (in red). The mean values are in dashed line. Temperature and relative humidity profiles exhibit finite slopes in the transient phases, while UV radiation can be turned on/off instantaneously. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

In SHM, dynamic tests are preferred since they are non-destructive and easily applicable. A combination of accelerometers and polyvinylidene fluoride (PVDF) integrated film sensors has been used for obtaining the dynamic signals. The minimum number of modes to be observed is determined using a FE model of the sample, as described in Section 4. While accelerometers are fixed and removed between two measurements, PVDF sensors are glued to or embedded inside the composite plate, implying that they are part of the ageing process as well (as in Fig. 2). It has been demonstrated that PVDF sensors are able to survive the ageing process without seeing their dynamic measurement capabilities affected [25]. 4. Monitoring tools and methods 4.1. Non-destructive test: modal analysis There is a wide range of solutions to measure directly or indirectly the elastic properties of a structure. In the interest of SHM, it is imperative to restrict the choice to non-destructive, fast, robust techniques. Modal analysis is a simple but powerful tool to obtain information about the state of a structure, based on the measurement of the natural frequencies and the corresponding modes. Among the available technologies, it was found that piezoelectric transducers provide remarkable quality signals, suitable for dynamic analyses [26–28]. For this reason, an adequate modal testing bench was chosen for experimentation. Hardware includes a Hewlett Packard HP 35670 Dynamic Signal Analyser, which is capable to directly compute and display the FRFs from time waveforms supplied by sensors. The complete measurement chain hardware is shown in Fig. 3. It is important to remember that an additional parameter to deduce the sample stiffness from its natural frequencies is the mass, which would potentially change in case of moisture absorption/evaporation between the environment and the structure. As the expected change in the mass is not negligible, it is necessary to monitor it carefully in order to avoid systematic measurement errors. The modal extraction software ME-ScopeÓ,

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at least 2 elements along the thickness (5 nodes). The stiffness matrix is computed using the reduced integration technique. The UD plates were modelled assuming a homogeneous orthotropic elasticity model in which the parameters E1 (longitudinal Young’s modulus), E2 (transversal Young’s modulus) and G12 (transversal shear modulus) are to be identified, while the remaining parameters have been assumed as constant. Thus, it is not necessary to include in the variable vector all of the nine independent engineering constants that determine the elastic behaviour of an orthotropic material. A simple sensitivity study based on a FE model links the lowest frequency modes to the elastic parameters having a significant influence on them. A sensitivity parameter is defined as:

S¼

Fig. 2. Schemes of sensor mounting: (a) glued PVDF sensor, (b) embedded PVDF sensor and (c) accelerometer with wax.

developed by Vibrant TechnologiesÒ, is used to find the modal parameters from the supplied FRFs. 4.2. Finite element model and structural identification algorithm A finite element (FE) model provides accurate results when the dynamic behaviour of real plates is simulated. This approach was used to build a mixed numerical–experimental identification algorithm, based upon a non-linear least squares method combined to ABAQUSÓ FE simulations in order to minimise the error between estimated and experimental frequencies, and to find the most suitable values for the elastic properties. A structured mesh with C3D20R (in ABAQUSÓ nomenclature) quadratic hexahedral elements of 2 mm was used, in order to have

Fig. 3. Equipment used for the modal analysis: (1) composite plate, (2) accelerometer, (3) PVDF integrated sensor, (4) impact hammer, (5) charge amplifier B&K 2635 and (6) HP 35670 Dynamic Signal Analyser.

Df =f DE=E

ð2Þ

S is a normalised derivative of the frequency f with respect to the elastic parameter E. For example, in Fig. 4, it can be seen that the basic vibration modes (flexural and torsional modes) for a D-series specimen depend essentially on two tensile moduli (E1 and E2 ) and one shear modulus (G12 ). An algorithm for the minimisation of an objective function based on the discrepancy between the estimated and experimental natural frequencies has been developed. This algorithm is based on the one proposed by [18], which uses a Levenberg–Marquardt optimisation strategy. The initial guesses for E1 ; E2 and G12 are estimated by the values provided in CFRP prepreg supplier’s data sheet, and shown in Table 2. The descending direction is obtained by the finite difference method on the FE model. Only positivity of the elastic parameters is a constraint for this optimisation problem. 5. Results: setup of a multi-factorial statistical model 5.1. Introduction to experimental results For the sake of uniformity and clarity, the changes in the constitutive properties of the samples are tracked with respect to their initial value:

l¼

m ; m0

1 ¼

E1 ; E10

2 ¼

E2 ; E20

c12 ¼

G12 G120

ð3Þ

Fig. 4. Example of sensitivity analysis between engineering constants and natural frequencies for a D-series sample. The latter are mainly influenced by the longitudinal and transverse tensile moduli E1 and E2 , and the transverse shear modulus G12 .

E. Guzmán et al. / Composite Structures 111 (2014) 179–192

The 0 index denotes the nominal initial values of the elastic properties, summarised in Table 2. At first glance, the graphs suggest an exponential asymptotic behaviour (Fig. 5). An increase in the mass can be observed in the ‘‘humid’’ protocols (D-, EF1-, EF2- and J-series). As stated before, in a UD configuration, the influence of the polymer matrix and the fibres on the elastic properties are partially decoupled. Among the identified properties, the longitudinal tensile modulus E1 is mostly associated to the carbon fibres, while E2 and G12 are related to the epoxy matrix. On one hand, the value of 1 exhibits a rather more irregular variation around a steady value. Globally, it was observed a slight increase in the longitudinal tensile modulus, at least during the first phase of the ageing process, although not statistically representative (error bars represent the standard deviation). On the other hand, an opposite evolution of 2 and c12 can be observed in Fig. 5, and shows a decrease of stiffness. However, the extent of the loss depends on the ageing protocol parameters. In all cases, the loss of transverse bending and shear stiffness is much more significant that the scattering. 5.2. Mathematical model inspired from the Prony series From the preliminary observations in the previous section, a multi-factorial mathematical model following the pattern of a Prony series is proposed. The mathematical expression is given by the following equation

Fig. 5. Example of evolution of relative mass

y¼

X b Y ¼1þ ai /i ðx1 ; x2 ; x3 Þð1  en=10 i Þ Y0 i

183

ð4Þ

/0 ¼ ð1  x1 Þð1  x2 Þð1  x3 Þ /1 ¼ x1 ð1  x2 Þð1  x3 Þ /2 ¼ ð1  x1 Þx2 ð1  x3 Þ /3 ¼ ð1  x1 Þð1  x2 Þx3 /12 ¼ x1 x2 ð1  x3 Þ /13 ¼ x1 ð1  x2 Þx3 /23 ¼ ð1  x1 Þx2 x3 /123 ¼ x1 x2 x3 In Eq. (4), each function /i ðx1 ; x2 ; x3 Þ represents a series of samples under a given set of environmental conditions x1 ; x2 and x3 (see Table 1), Y represents either m; E1 ; E2 or G12 , and y represents l; 1 ; 2 or c12 . Physically, the identified constant coefficients ai are the final asymptotic values of the loss/gain, and 10bi are the corresponding time constants (in number of cycles). In other words, this expression is basically a model expressing an exponential decay (or rise) of the constitutive properties of the samples subjected to an accelerated ageing. Given the strong non-linearity of the model, the result of the identification process can be sensitive to the initial estimations. The Levenberg–Marquardt leastsquare optimisation algorithm was used to find the best fitting

l, Young ’s moduli 1 ; 2 and shear modulus c12 for (a) D-, (b) EF1-, (c) EF2- and (d) H1-series.

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parameters. The initial conditions were chosen on a basis of multiple Monte Carlo simulations, in order to find a global minimum for the least-square fitting error. The fitting results are shown in Fig. 6, with the corresponding residuals and coefficients of determination R2 . As it can be seen, the fitting quality is very high for l; 2 and c12 , while it is considerably lower in the case of 1 . As it is shown in the following sections, 1 (which represents the stiffness along the carbon fibres direction) shows a behaviour that cannot be statistically determined. 5.3. Validation of the mathematical model The validation of a model has been carried out by graphical and analytical means. For the former, a simple way to do so is by establishing a scatter plot displaying a graphic comparison between the estimated curve and the experimental results of aged samples as shown in Figs. 7 and 8. Visually, it can be observed that the experimental results are globally well predicted by the model, especially concerning the ageing of the polymer matrix (tensile modulus 2 and shear transverse modulus c12 ). For the analytical validation, a lack-of-fit test is also performed on the global data, in order to show the goodness of fit of the mathematical model, as in the supplement document about the Lack-of-fit (LoF) test, of which the test hypothesis can be formulated as follows:



H0 : The model does not induce any significant error: H1 : The model induces a significant error:

In other words, if H0 is accepted, the model pattern (in this case given by Eq. (4)) is appropriate for the measured dataset. If H0 is rejected, the systematically induced error is inherent to the chosen model pattern. The parameters of the dataset for the hypothesis test are: 1. A total of N ¼ 193 measurements (one measurement per series over 100 cycles). 2. The averaged measurements (one averaged measurement per series over 100 cycles) were Q ¼ 79. 3. The model in Eq. (4) has an order p ¼ 16. 4. m1 ¼ Q  p ¼ 63 degrees of freedom (DoF) for the ‘‘lack-of-fit’’ error. 5. m2 ¼ N  Q ¼ 114 degrees of freedom (DoF) for the experimental measurement error. The quotient k between the squared lack-of-fit normalised error and the squared experimental normalised error serves as observed value for a Fischer’s distribution law Fðm1 ; m2 Þ. For such a distribution and a significance level of a ¼ 1%, the critical value of lambda is kcrit ¼ F 0:01 ð63; 114Þ ¼ 0:583. If k is above this value, the null hypothesis H0 is rejected with a fail probability of 1%. It can be seen in Table 4 that H0 is accepted in all cases except for the observed k corresponding to 1 . This confirms the statement made about the R2 values for the dependent variables: for l; 2 and c12 , which exhibit high values for the coefficient of determination, it can thus be concluded that the models are accurate enough to represent the ageing of T700S/FT109 specimens, in the frame of the given conditions, while the model is not appropriate to predict 1 . 5.4. Interpretation of the validation A statement after inspecting Fig. 5 is that there is indeed a temporal evolution of the elastic properties and the mass of the samples. It is clear that over a long-term weathering, there is a limit

Fig. 6. Bar diagrams for identified model coefficients ai for relative (a) mass Young’s modulus 1 , (c) Young’s modulus 2 and (d) shear modulus c12 .

l, (b)

E. Guzmán et al. / Composite Structures 111 (2014) 179–192

Fig. 7. Scatter plot for measurements of D-series samples: (a)

for the degradation of properties. The exhibited trend of 1 is not very clear. On the contrary, l shows a clear increase, and 2 and c12 show a clear decrease. Physically, 1 is associated to the stiffness of the carbon fibres in a UD composite. The carbon fibres are rather chemically and thermally inertial, thus the observed increase of 1 would not be due to a change in the inner structure of the carbon. The changes in 2 and c12 are less surprising since they were expected, as stated in the review of the state-of-the-art in accelerated ageing. The loss of stiffness by the polymer matrix is a combination of several phenomena detailed in the literature, in particular due to polymer chain delinking, photo-oxidation, water corrosion, etc. From the observations made on the scatter plots, the determination coefficient R2 and the corresponding Fischer quotients F in the previous section, it is safe to state that the evolution of l; 2 and c12 can be explained by such a mathematical model. On the other hand, the model is not an accurate predictor for 1 .

185

l, (b) 1 , (c) 2 and (d) c12 .

moisture absorption, for determination of ‘‘equivalent’’ cyclic constants of diffusivity and saturation mass. 6.1. A model for water diffusion in isothermal conditions Water absorption is possible only in the presence of some humidity in the surrounding atmosphere (RH – 0%). Thus, to simplify the calculation, the following hypothesis is going to be adopted: any change of mass in a ‘‘dry’’ protocol (RH ¼ 0%) is neglected. Concerning the water absorption, a simple mathematical reasoning can be adopted in order to estimate the water concentration absorbed by a sample when subjected to accelerated ageing conditions. This amount is generally not equal between periodic or isothermal conditions. The mathematical approach to model the water absorption comes from Fick’s law of diffusion in combination with the mass conservation principle, as expressed in the following equation:

6. Equivalent moisture absorption in cyclic ageing protocols

  @c @ @c D ¼ @t @x @x

Unfortunately, mathematical models based on statistical treatment remain uninteresting from the experimental point of view. Indeed, they cannot be directly applied to more general cases since the geometry plays a determining role in several of the ageing mechanisms (moisture absorption, heat diffusion, penetration depth of UV radiation, etc.). To remove this dependency, it is proposed to replace time by a state variable, such as moisture concentration. To do so, it is necessary to establish a link between theory and experiments in the

where c is the volume concentration, D the diffusion coefficient, t the time coordinate and x the space coordinate along the thickness. Under the hypothesis of absence of swelling, the volume concentration c is proportional to the mass concentration M, defined by M ¼ l  1. This case can be assimilated to an infinite plate with thickness h with two exposed boundary conditions (since l  h and w  h). The solution of Eq. (5) in such conditions is obtained by application of the Fourier’s series theory, giving the following result

ð5Þ

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Fig. 8. Scatter plot for measurements of J-series samples: (a)

Table 4 Summary of the lack-of-fit test. To accept the null hypothesis, k 6 kcrit ¼ 0:583. Model

2

R k Decision

ES

Ms ¼ M s0 eRT

Mass

Elastic moduli

l

1

2

c12

0.997

0.853

0.966

0.981

0.047 H0

9.938 H1

0.515 H0

0.215 H0

1 MðtÞ 8 X 1 ð2k þ 1Þ ¼1 2 exp  2 Ms p k¼0 ð2k þ 1Þ2 h

2

p2 Dt

! ð6Þ

where Ms is the saturation mass. However, this expression is complicated, and a simpler function is used to approximate it

 0:75 MðtÞ t  1  exp 1:31 Ms s

! ¼H

  t

s

ð7Þ

2

where s ¼ h =ðp2 DÞ is a time constant for this specific exponential growth and H symbolises a function. Diffusivity depends on temperature following an Arrhenius-type law: ED

D ¼ D0 e RT

lnðDÞ ¼ lnðD0 Þ 

ð8Þ ED RT

l, (b) 1 , (c)2 and (d)c12 .

ð9Þ

where D0 is the diffusion coefficient when T ! 1; ED the diffusion activation energy and R the universal gas constant. Similarly, the saturation mass Ms depends exponentially on the inverse of the temperature by Eq. (10)

lnðM s Þ ¼ lnðM s0 Þ 

ð10Þ ES RT

ð11Þ

where M s0 is the saturation mass when T ! 1 and ES the corresponding diffusion activation energy. Concerning the influence of the relative humidity RH on the saturation mass, it has been proposed by some authors [12,29,30] that a power law model can be used to determine the relation between both

Ms ¼ M smax ðRHÞf

ð12Þ

where f is an exponent depending on the temperature and the absorbing composite and Msmax the maximum saturation mass possible. After [29], in the case of an epoxy matrix, this value is 1 in the current temperature range, so this hypothesis is adopted in this paper. The thermodynamic constants D0 ; ED ; Ms ; M s0 defined in Eqs. (8)–(11) can be estimated experimentally by linear regression, if datasets of measured (T; M s ) and (T; D) are available (see Fig. 9). In a previous experimental campaign [31], T700S/FT109 samples were subjected to ageing protocols in steady-state isothermal conditions with constant RH  95%, at 60 °C, 80 °C and 100 °C, with the results summarised in Table 5. From this, and supported by information in literature [12,30], the associated diffusion activation energy ED ¼ 9:067 kJ/mol, and the saturation enthalpy ES ¼ 1:68 kJ/mol can be deduced in a range of temperature between 5 °C and 135 °C. With these two linear models, it is possible to estimate quite accurately D and the M s in steady conditions.

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6.2. Diffusion for periodic ageing protocols

Table 5 Computed diffusion coefficients in preliminary tests with test series A and B [31].

The behaviour under periodic protocols is different from that under constant conditions. Despite this, the base hypothesis is that, after a ‘‘transient’’ phase, the amount of absorbed water oscillates around a local moving mean value, which is a fraction of M s at steady conditions. For the first charge and discharge, we have

Temperature

Diffusion 2

Saturation mass

T (°C)

T (K)

D (mm /s)

Ms (%)

60 80 100

333 353 373

2.21e7 2.67e7 3.14e7

8.84 9.02 9.44

  M 01 Dt  ¼H Ms s

ð13Þ

  M1 Dt þ ¼ 1  H sþ M 01

where dt n is the time it would take to have a concentration of Mn1 in isothermal conditions, and it is defined by Eq. (18):

ð14Þ

dt n

2

2

where s ¼ h =ðp2 D Þ and sþ ¼ h =ðp2 Dþ Þ; Dt  and Dt þ are the corresponding durations of the low temperature and the high temperature part of the cycle (including the transient state), and D and Dþ are the corresponding diffusion coefficients. Combining Eqs. (13) and (14) gives

     M1 Dt þ Dt  ¼ 1H H Ms sþ s

ð15Þ

Similarly, for the nth charge and discharge, we have

M 0n Ms

 ¼H

Dt þ dtn

s

  Mn Dt þ ¼ 1  H sþ M 0n

 ð16Þ

ð17Þ

Fig. 9. Graphs of linear regressions (1=T; ln M s ) and (1=T; ln D) as in Eqs. (9) and (11) for measured values of A- and B-series. The slopes give respectively the values of (a) ED and (b) ES .

s

¼ H1

  M n1 Ms

ð18Þ

where H1 is the inverse function of H and Mn1 is the mass left after the ðn  1Þth discharge. The physical meaning of dt is shown in Fig. 10. A mathematical sequence can then be defined to determine the absorbed mass after the nth cycle:

       Mn M n1 Dt þ Dt  þ dt n ¼ 1H ¼f H Ms Ms sþ s       Dt þ Dt  M n1 ¼ 1H H þ H1 sþ s Ms

ð19Þ

where f denotes a recurrence relation. The mass concentration M can be determined recursively after the nth cycle as a function of the concentration after the ðn  1Þth cycle. In Fig. 10, the amplitude of the oscillation is exaggerated in order to show the concentrations and time variables. From a simulation after Eq. (19), it can actually be deduced that the absorbed moisture mass depends essentially on the ratio Dt þ =sþ and Dt  =s . The final value limn!1 Mn ¼ Ms;eq (where eq stands for equivalent) of the mass absorption is given by the solution of x ¼ f ðxÞ. After using MATLAB to solve this recursive equation, the results for the ageing protocols are summarised in Table 6. The ratio Ms;eq =Ms is asymptotic, representing the fraction of water absorbed by the specimen under a cyclic protocol Ms;eq with respect to the absorbed mass under a steady protocol Ms . Concerning the results, it is interesting to see in Table 6 that independently from the thickness, the equivalent saturation concentrations M s;eq are similar between the EF1-, EF2- and J-series. The equivalent saturation mass for D is on the other hand much higher. The difference comes mostly from the difference in the cyclic timing of the protocols, since the cycles in the D-series have a 2:1 ratio between Dt =Dt þ (instead of 1:1 for EF1-, EF2- and Jseries).

Fig. 10. Cyclic mass absorption. The variables are those described in Eqs. (13)–(19) in Section 6.2: the saturation mass in constant conditions is M s , while M1 ¼ M s;eq is the equivalent saturation mass in cyclic conditions.

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6.3. Equivalent diffusivity and equivalent saturation mass

Table 7 Comparison between theoretical and experimental estimations of the equivalent diffusivity.

The idea is now to compare the results from the two models for moisture absorption inside a structure: an experimental empiric model as defined in Section 5.2, and a theoretical analytic recurrent model as in Section 6.2. To do so, the ‘‘equivalent’’ diffusivity Deq , for cyclic conditions, needs to be defined. Let us consider the Prony series approximation of the diffusion 0:75 function HðxÞ ¼ 1  e1:311x  1  e1:3181x , which has the same exponential form as the terms of the ageing mathematical model, as it is described in Eq. (4). A link between the theoretical diffusion and the statistical experimental model can then be established. For the ith protocol, we have b

1  e1:3181t=si ¼ 1  en=10 i

where t is the time (in [s]), n is the corresponding number of cycles,

si is the diffusion time constant (defined in Eq. (7)) and 10bi is the time constant in number of cycles of the ith term in the mathematical model (4). For a single cycle, n ¼ 1 and t ¼ Dt. It can easily be deduced the following equation:

ð21Þ

After the definition of s, the equivalent diffusivity Deq for the ith protocol can be defined as:

Deqi ¼

h

2

h

2

ð23Þ

13:01Dt  10bi

Rel. error %

D EF1 EF2 J

1.27e12 4.45e13 4.61e13 4.44e13

1.54e12 5.08e13 4.93e13 5.39e13

21.5 14.2 7.1 21.5

The corresponding Pearson’s correlation coefficients q for each series is given in Table 8. If the absolute value jqj is close to one, the linear regression is appropriate (for further detail, see the supplementary document Statistical correlation between water concentration and loss of stiffness). A hypothesis test based on the value of a Student’s t-distribution parameter h is carried out to find out if the correlation, for each series and each elastic variable, is statistically significant, and can be formulated as follows

There are now at our disposal two ways to determine the value of Deq : a theoretical value Deq;th obtained from simulated results and experimental Deq;exp obtained from the cyclic experimental results:

Deq;exp ¼

Theoretical Deq;th (from isothermal data)

7.2. Correlation test

ð22Þ

p2 si

Fitted model Deq;exp (cyclic test)

relative humidity, the evolution of the mass depends essentially on the existence/absence of moisture in the ageing environment. In the graphs, the ageing track of all of the 8 series can be distinguished. More particularly, it shows that the pattern of the stiffness evolution over the time is similar between ageing protocols that exhibit the same relative humidity profile (closeness between the J- and D-, EF1- and EF2-, G1- and G2-, and H1- and H2-series), although the intensity of the loss grows deeper when UV radiation and high temperatures are applied. However, the main result may be the visible correlation between two a priori independent variables: the water concentration and the elastic moduli. In Fig. 12 and Table 8, linear regression models for the series are graphically represented (by straight lines) and numerically computed (by the corresponding slopes).

ð20Þ

si ¼ 1:3181Dt  10bi

Series

All taken into account, the theoretical prediction of the saturation mass and diffusivity, and their experimentally determined counter-parts are quite close. A fair prediction of absorption can be done only with theoretical considerations and isothermal diffusion data as shown in Tables 6 and 7 and in Fig. 11. The advantage of these modelling procedure is that the ageing problem can finally be seen freed from the influence of geometry, and then applicable to a much wider range of composite parts.



H0 : There is no correlation between variables: H1 : There is a correlation between variables:

qffiffiffiffiffiffiffiffi q2 The observed parameter is defined as h ¼ jqj 1 q2 , following a Student’s t-distribution, with q  2 DoF, where q is the number of observations per series. The null hypothesis is accepted if h > hcrit ¼ t a=2 ðq  2Þ. In Table 8, it can be seen that the null hypothesis is constantly rejected for 2 and c12 , while it is almost constantly accepted for 1 . This reinforces our past statements: the modelling of 2 and c12 is possible, while 1 ’s is statistically incorrect. To sum up, this shows that the ageing by water absorption and the decrease of stiffness inside the polymer matrix are correlated. The details of the computations are in the supplementary document. Since, by definition, the fitted straight lines (see Fig. 12) start always at (1, 1), the regression formula takes invariably the form described in Eq. (24):

7. Relationship between moisture concentration and loss of stiffness 7.1. Linear regression between mass and elastic properties In Fig. 12, a pattern of correlation between the mass variation and the elastic properties can be observed. Depending on the

Table 6 Estimated mass increase for the four series subjected to humid protocols. M s;eq is the equivalent saturation mass for the cyclic protocol and M s the corresponding saturation mass under isothermal conditions. Series

D EF1 EF2 J a b c d

Cycles N

800 900 900 800

Temperature (K)

Time period (s)

Thickness (mm)

Mass ratio

Equivalent sat. mass (cyclic) M s;eq a

T

Tþ

Dt 

Dt þ

h

Ms;eq =M s

–

278 318 318 278

368 408 408 368

3600 2700 2700 2700

1800 2700 2700 2700

4.04 4.23 4.02 4.29

0.135 0.110 0.112 0.124

0.0106 0.0059 0.0064 0.0063

Predicted from the fitted model. Predicted from the theory. Mean residual mass, obtained by drying of the samples. Predicted from the experimental result.

b

c

Saturation mass (isothermal) M s

–

–

–a

–b

–d

0.0100 0.0066 0.0067 0.0066

0.0101 0.0060 0.0064 0.0061

0.079 0.054 0.058 0.051

0.074 0.060 0.060 0.054

0.075 0.055 0.057 0.049

E. Guzmán et al. / Composite Structures 111 (2014) 179–192

189

Fig. 11. Graphic comparison between the theoretical and fitted models, for (a) D-, (b) EF1-, (c) EF2- and (d) J-samples.

M ¼ l  1 ¼ Ai ðyi  1Þ where i ¼ 1; 2; 3, namely yi stands for either denotes the slope.

ð24Þ

1 ; 2

or c12 and Ai

7.3. A linear model for Ai There is then a linear relationship between the mass absorption and the ageing of the polymer matrix, represented by a decay of 2 and c12 . Since the time variable is included in l, the proportionality coefficient Ai should depend only on the weathering conditions x1 (temperature), x2 (relative humidity) and x3 (UV radiation). A linear model is suggested in the following equation:

Aðx1 ; x2 ; x3 Þ ¼ a0 þ a1 x1 þ a2 x2 þ a3 x3 þ a12 x1 x2 þ a13 x1 x3 þ a23 x2 x3

ð25Þ

where Aðx1 ; x2 ; x3 Þ represents any of the coefficients Ai in Eq. (24). The coefficients Ai identified by linear regression are summarised in Table 9, along with the corresponding Pearson’s q. Under the hypothesis that the coefficients of the linear regression follow a normal law, an analysis of variance (ANOVA) can be then carried out, in order to check the significance of the different factors in the ageing of the composite. This multi-factorial analysis of variance can be performed as described by [15,16], and is detailed in the supplementary document about Statistical correlation between water concentration and loss of stiffness. As it can be seen in Table 9, the null hypothesis (the factor is statistically significant) is

accepted for a0 ; a1 ; a2 and a12 (in the case of c12 ). Physically, this can be interpreted as follows: 1. Since a0 – 0, there is a natural ageing in ‘‘normal conditions’’ (no humidity nor UV radiation, temperature is cyclic but always under the glass transition threshold). 2. The polymer matrix is sensitive to temperature (a1 ) and humidity (a2 ), and eventually a combination of both (a12 ). 3. The part of UV radiation is close to negligible, at least with the intensity of the lamps in the ageing processes analysed in this paper. Conclusively, since ageing and water absorption are correlated, Eq. (24) shows that the time (or the number of cycles) can be replaced by the water concentration as an ageing variable. The time variable is then implicitly included in the water diffusion model, which can be predicted theoretically from the models in Section 6.3. On the other hand, the Ai coefficients depend only on the material and environmental conditions of the ageing protocol (T; RH and UV) and can be estimated by the linear model in Eq. (25) with the parameters ai in Table 9. 8. A predicting method for composite ageing From a scientific point of view, the multi-factorial model presented in this paper can prove to be useful in two ways: for

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Fig. 12. Correlation plots between the relative mass

l and (a) 1 , (b) 2 and (c) c12 .

E. Guzmán et al. / Composite Structures 111 (2014) 179–192 Table 8 Results of hypothesis testing on the correlation between l and the elastic properties. If H0 is accepted, there is no correlation, while if H1 is accepted, there is a correlation. Series

q

J D EFI EFII GI GII HI HII

8 8 9 9 12 11 12 10

Ai

Decision

1

2

c12

1

2

c12

0.48 0.62 0.20 0.31 0.13 0.12 0.03 0.03

0.14 0.18 0.06 0.08 0.02 0.03 0.01 0.01

0.13 0.16 0.06 0.07 0.02 0.03 0.01 0.01

H0 H1 H1 H0 H0 H0 H0 H0

H1 H1 H1 H1 H1 H1 H1 H1

H1 H1 H1 H1 H1 H1 H1 H1

Table 9 Summary table of data yi ¼ Ai ðx1 ; x2 ; x3 Þ used to set up a linear model with interactions, along with the corresponding correlation coefficients q. ANOVA

a0 a1 a2 a3 a12 a13 a23

Effects

p-Value

2

c12

2

0.0305 0.0294 0.1754 0.0127 0.0822 0.0280 0.0200

0.0287 0.0252 0.1663 0.0049 0.0804 0.0132 0.0168

0.51 8.58 0.14 69.67 12.16 42.88 64.20

(%)

p 6 10%

c12 (%)

2

c12

0.11 3.10 0.03 62.72 3.33 50.35 52.40

Yes Yes Yes No No No No

Yes Yes Yes No Yes No No

characterisation and design of experiments, for the experimenter to follow a similar method to characterise other composite materials using different combinations of fibre/resin, and for prediction, using this model to estimate the ageing of a composite structure with variable geometries, since the effect is known. For characterisation/design of experiments: Based on all that has been exposed in this paper, a simple method for establishing a statistical ageing model for a given material is proposed: 1. By isothermal tests or from providers datasheet, establish the D and M s Arrhenius’ laws, as in Eqs. (8) and (10). 2. Determine the parameters of the ageing protocol: time duration of the cycle (Dt  ; Dtþ ; Dt) and the corresponding temperature and relative humidity levels, in order to determine Ms ; D and Dþ by using the model described in Section 6.2. 3. Build a water absorption model using the recursive Eq. (19), determining by least squares and Eq. (7) the equivalent diffusivity Deq and equivalent saturation mass concentration M s;eq . 4. Design and carry out an experimental campaign, using 2k sample series (where k is the number of factors to be considered; in this paper, k ¼ 3). 5. The parameters a of the simplified ageing model (25) for the elastic moduli of the composite can be then determined by a linear regression, while the pertinence of the model can be statistically assessed by an ANOVA. Prediction: If the ageing model for a material is available, the experimenter can use it to predict the ageing of the analysed structure: 1. Simulate the evolution of water concentration l using an analytical or numerical model that takes into account the geometry of the part, since the equivalent diffusion parameters (Deq and Ms;eq ) are known. 2. Using the linear model (24), determine the level of ageing for a given water concentration state.

191

9. Conclusions The hereby presented macroscopic model, based on the Prony series, proposed a characterisation method for a set of simultaneous physical/chemical phenomena that intervene on the CFRP ageing. The parameters of that model, fitted on the basis of the NLLS criterium, were determined from experimental results. However, the application of these results was restricted to samples with the same geometrical dimensions and proportions. Thus, the main contribution of this document is not only to suggest the form of a mathematical model, but also to set this model free from the specific dimensions of the physical samples that were experimented with. In order to do so, the first step was to study the link between the theoretical and the experimental results of water diffusion and mass absorption. Such analysis gave as a result ‘‘equivalent’’ diffusivities and saturation masses. The second step proved by statistical means that there is a correlation between the absorbed moisture (a state variable) and the ageing in cyclic conditions. In this case, the model has gotten free from the dimensions influence. In other words, based only on the constitutive properties of a composite material and the features of a cyclic ageing protocol, the ageing can be estimated. This method can be used in the future to predict the material ageing in more complex CFRP demonstrators. Acknowledgements The authors would like to acknowledge the partial financial support from the Swiss National Science Foundation, Grant No. 200020-143968/1. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.compstruct.2013. 12.028. References [1] Giurgiutiu V. Structural health monitoring with piezoelectric wafer active sensors. 1st ed. Elsevier; 2008. [2] Sohn H, Farrar C, Hemez F, Shunk D, Stinemates D, Nadler B, et al. A review of structural health monitoring literature: 1996–2001. Tech. Rep. LA-13976-MS, Los Alamos National Laboratories; 2004. [3] Lynch J, Loh K. Summary review of wireless sensors and sensor networks for structural health monitoring. Shock Vib Digest 2006;38(2):91–128. [4] Pupurs A, Varna J. Modeling mechanical fatigue of UD composite: multiple fiber breaks and debond growth. IOP C Ser Mater Sci Eng 2009;5(1):12–7. [5] Liao K, Schultheisz C, Hunston D, Brinson L. Long-term durability of fiberreinforced polymer matrix composite materials for infrastructure applications: a review. J Advan Mater 1998;30(4):3–40. [6] White J. Polymer ageing: physics, chemistry or engineering? Time to reflect. CR Chim 2006;9(11–12):1396–408. [7] Mouzakis D, Zoga H, Galiotis C. Accelerated environmental ageing study of polyester/glass fiber reinforced composites (GFRPCs). Compos Part B – Eng 2008;39(3):467–75. [8] Dao B, Hodgkin J, Krstina J, Mardel J, Tian W. Accelerated ageing versus realistic ageing in aerospace composite materials – IV. Hot/wet ageing effects in a low temperature cure epoxy composite. J Appl Polymer Sci 2007;106(6):4264–76. [9] Fox B, Lowe A, Hodgkin J. Investigation of failure mechanisms in aged aerospace composites. Eng Fail Anal 2004;11(2):235–41. [10] Bondzic S, Hodgkin J, Krstina J, Mardel J. Chemistry of thermal ageing in aerospace epoxy composites. J Appl Polym Sci 2006;100(3):2210–9. [11] Tong M, Singhal S, Chamis C, Murthy P. Simulation of fatigue behavior of high temperature metal matrix composites. Tech. Rep. NASA-CR-204605, NASA Glenn Research Center; 1996. [12] Lin Y, Chen X. Investigation of moisture diffusion in epoxy system: experiments and molecular dynamics simulations. Chem Phys Lett 2005; 412:322–6. [13] Carraher C. Polymer chemistry. New York, NY: CRC Press Inc.; 2010. [14] Brinson H, Brinson L. Polymer engineering science and viscoelasticity: an introduction. 1st ed. New York, NY: Springer; 2008.

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