Dr Antoine Kuczkowiak
email: antoine.kuczkowiak@edf.fr

Scott Cogan
email: scott.cogan@univ-fcomte.fr

Morvan Ouisse
email: morvan.ouisse@femto-st.fr

Emmanuel Folt
email: emmanuel.foltete@femto-st.fr

Dr Mathieu Corus
email: mathieu.corus@edf.fr

Experimental validation of an info-gap uncertainty model for a robustness analysis of structural responses

The aim of this work is to propose an indicator, based on the info-gap approach, which 1 assesses the robustness of the dynamic response of the model of a structure to lack of knowledge in the associated eigensolutions. The info-gap uncertainty model for the eigensolutions is constructed mode by mode based on a set of experimentally identified eigensolutions obtained from tests on a set of nominally identical structures. A robustness analysis is then performed which provides a useful bound to the maximum response levels that are consistent with the defined uncertainty, thus allowing to extrapolate to a population of untested structures. The proposed methodology is validated experimentally on a simple structure composed of two plates clamped together on one side. Uncertainty is introduced by adding lumped masses at random locations. A subset of these test results is used to construct the info-gap model of the eigenproperties and the remaining data is used to confirm that the robustness curve usefully bounds the observed maximum responses.

INTRODUCTION

Structural dynamic behaviors must be well understood in order to ensure the reliable and safe operation of complex systems. When available, validated numerical models are useful for predicting vibratory levels under both tested and untested conditions. However, the complexity and the degree of epistemic uncertainty, due to the difficulty in modeling poorly known physics, can render the model validation process excessively time consuming and costly. Moreover, experimental modal analyses are often performed in a configuration which differ significantly from real operating conditions. For example, the configuration of GVT (Ground Vibration Testing) used to identify normal modes on a aircraft differs from the conditions encountered during flight. Another example is an alternator in a power plant where experimental modal analyses are performed on a non-operating machine while it is known that thermal and nonlinear mechanical effects of the functioning alternator modify the dynamic behavior. While in the case of the aircraft testing, operational modal analysis provides a way to obtain the eigensolutions during flight, in the case of the alternator, this is not possible.

In this paper, the identified modal model is considered to be uncertain and is assumed to be a direct result of the lack of knowledge between two configurations. Broadly speaking, the present study is directly related to the field of robustness analysis which aims is to provide decision support tools in order to support decision under deep uncertainty. The notion of robustness can take on many meanings as discussed in the papers by G öhler et al [START_REF] Eifler | Robustness metrics: Consolidating the multiple approaches to quantify robustness[END_REF] and Zhang et al [START_REF] Zhang | Nuanced robustness analysis with limited information[END_REF]. In the context of this work, a non-probabilistic notion of robustness will be employed as defined in the framework of the info-gap theory [START_REF] Ben-Haim | Information-Gap Theory: Decisions Under Severe Uncertainty[END_REF]. Knight [START_REF] Knight | Risk, Uncertainty and Profit[END_REF] distinguished between "risk" based on known probability distributions and "true uncertainty" for which probability distributions are not known. In the present work, robustness is defined as an aptitude to resist to approximations or zones of ignorance in order to prevent undesirable impacts [START_REF] Roy | Robustness in operational research and decision aiding: A multi-faceted issue[END_REF]. Robustness analyses are performed within the framework of model predictions corresponding to a posteriori actions. Reliable robustness analyses can be performed only when the uncertainty in the system can be represented as a mathematical idealization. A well-established classification of uncertainty permits to make a clear distinction between aleatory and epistemic uncertainty [START_REF] Oberkampf | Error and uncertainty in modeling and simulation[END_REF][START_REF] Klir | Generalized information theory: aims, results and open problems[END_REF][START_REF] Kiureghian | Aleatory or epistemic? does it matter?[END_REF]. Aleatory uncertainty is also referred to as irreducible uncertainty since it always exists by nature and can not be suppressed even when more accurate experiments are performed. The mathematical idealization of such uncertainties is often a probability or frequency distribution [START_REF] Ghanem | Stochastic Finite Element: A Spectral Approach[END_REF][START_REF] Soize | A nonparametric model of random uncertainties for reduced matrix models in structural dynamics[END_REF][START_REF] Schu Ëller | On the treatment of uncertainties in structural mechanics and analysis[END_REF][START_REF] Adhikari | Wishart random matrices in probabilistic structural mechanics[END_REF]. Epistemic uncertainty deals with lack of knowledge about the system and/or the environment and is also referred to as reducible uncertainty since more accurate experiments could reduce them. A wide variety of approaches have been developed to treat epistemic uncertainty, including interval [START_REF] Dessombz | Analysis of mechanical systems using interval computations applied to finite element methods[END_REF], fuzzy [START_REF] Moens | Non-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: Recent advances[END_REF], Dempster-Shafer [START_REF] Dempster | Upper and lower probabilities generated by a random closed interval[END_REF][START_REF] Shafer | A mathematical theory of evidence[END_REF] or info-gap theory [START_REF] Ben-Haim | Information-Gap Theory: Decisions Under Severe Uncertainty[END_REF]. The industrial issue of interest in this study concerns a modal model of a turboalternator. This model is assumed to be uncertain as it is obtained from an experimental modal analysis under non-operating conditions. Moreover, no model is currently available to predict the evolution of the modal basis when the alternator is in operation. In this context, the uncertainty in the identified eigensolutions are considered to be epistemic. In this paper, an info-gap robustness framework will be adopted to investigate the impact of this epistemic uncertainty on the response levels of interest. As with any uncertainty propagation technique, the results depends strongly on the info-gap model of uncertainty used to represent the lack of knowledge affecting the eigensolutions. The objective of this paper is to validate experimentally, on the basis of an academic test case, the model of uncertainty selected for the robustness analysis. Very few studies are currently available in the literature that investigate this question for the info-gap approach. [START_REF] Adhikari | Uncertainty in structural dynamics: Experimental validation of a wishart random matrix model[END_REF] or [START_REF] Adhikari | Experimental case studies for uncertainty quantification in structural dynamics[END_REF][START_REF] Friswell | Experimental data for uncertainty quantification[END_REF] address an analogous problem of validation of a non-parametric model of uncertainty based on real measurements.

The paper is organized as follows. Section 2 introduces the methodology to assess the robustness of the dynamic response levels to uncertain eigensolutions. Section 3 proposes a numerical illustration. Finally, section 4 proposes a way to construct and calibrate a model of uncertainty based on a subset of test-data. Finally, the selected uncertainty model is validated using on real experiments.

ROBUSTNESS OF DYNAMIC RESPONSES TO UNCERTAIN EIGENSOLUTIONS

The approach adopted here consists in synthesizing the dynamic response of a linear elastodynamic structure based on uncertain eigensolutions. The impact of lack on knowledge in the eigenparameters is then investigated using an info-gap robustness analysis [START_REF] Ben-Haim | Information-Gap Theory: Decisions Under Severe Uncertainty[END_REF]. The info-gap methodology is a decision theory and aims to analyze the robustness of a decision under severe uncertainties. It has been applied in very diverse domains including economics, medicine and business management as its formulation is very general. In the field of structural dynamics, a wide variety of applications can be found in the literature, for example [START_REF] Atamturktur | Calibration of imprecise and inac-REFERENCES curate numerical models considering fidelity and robustness: a multi-objective optimizationbased approach[END_REF][START_REF] Stevens | Evaluating the fidelity and robustness of calibrated numerical model predictions[END_REF][START_REF] Hot | An info-gap application to robust design of a prestressed space structure under epistemic uncertainties[END_REF][START_REF] Hot | An info-gap application to robust design of a prestressed space structure under epistemic uncertainties[END_REF], and readers can refer to one of these references for a more thorough introduction to the info-gap methodology if required.

The info-gap robustness analysis requires the definition of three essential components, namely, the system model, the info-gap model of uncertainty and the performance requirements. The robustness function is based on the concept of "satisficing"1 performance requirements, that is to say, insuring a given level of performance even under a given level of uncertainty.

System model

System model

The system model establishes the functional relationship between the uncertain model parameters, x, the known model parameters, p, and the response features of interest, y. It is written as

M(p, x) = y. (1)
In many structural applications, M represents a physics-based model of a structure, x the model input parameters (eg geometric dimensions, plate thicknesses or material properties) and y the model outputs or structural responses of interest (eg frequency responses, eigenfrequencies or stresses). But the concept of a system model is very general and can be seen simply as a blackbox representation relating inputs and outputs via a more or less complex functional relationship M. In the present case, M will represent the relationship between the eigenparameters x of a mechanical structure and its frequency responses y.

In the low frequency domain, the modal behavior controls the mechanical response of a given system. Experimental modal analyses are used to obtain the spectrum of the structure. Let

ω 0 ν , ξ 0 ν , φ 0 ν ∈ R × R × R c
be an experimentally identified modal basis with ω 0 ν the eigenfrequency, ξ 0 ν the modal damping factor, φ 0 ν the eigenvector corresponding to the mode ν

(ν ∈ {1, • • • , m})
and c the number of sensors. The incomplete identified modal basis can be used to evaluate the dynamic response levels y at the frequency ω subjected to a dynamic load F ∈ R c at the sensors degrees of freedom (dofs):

y = m ν=1 φ 0 ν φ 0 ν T F -ω 2 + 2ξ 0 ν ωω 0 ν + (ω 0 ν ) 2 .
(

) 2
In what follows, equation (2) is the system model.

Info-gap model of uncertainty

Info-gap model of uncertainty

The info-gap model of uncertainty defines an unbounded family of nested sets of realizable designs. The size of these sets is controlled by the horizon of uncertainty and denoted α (α ∈ R+).

The horizon of uncertainty indicates the level of uncertainty present in the model of uncertainty.

A wide variety of info-gap uncertainty models are available in the literature [START_REF] Ben-Haim | Information-Gap Theory: Decisions Under Severe Uncertainty[END_REF]. The choice of the model of uncertainty must be performed with care since it is known that the info-gap robustness analysis is sensitive to the model of uncertainty. In this paper, the identified modal basis is assumed to be uncertain. Let (ω ν , ξ ν , φ ν ) ∈ R × R × R c be the uncertain modal basis. The uncertainty model is derived from the nominal modal basis by assuming that the uncertainty is driven by the vector x ∈ R p with p = 2m + m 2 . This column vector is defined by

ω ν = x ν ω 0 ν , ∀ν ∈ {1, • • • , m}; ξ ν = x ν+m ξ 0 ν , ∀ν ∈ {1, • • • , m}; φ ν = m k=1 x (1+ν)m+k φ 0 k , ∀ν ∈ {1, • • • , m}, (3)
where x ν is the ν-th component of x. The uncertain eigenvector is assumed to be described by the initial eigenvectors (assumption valid for small uncertainties and decoupled pairs of eigenvectors [START_REF] Pichler | A mode-based meta-model for the frequency response functions of uncertain structural systems[END_REF]). In addition, the particular column vector which corresponds to a zero-perturbation applied on the modal basis is denoted x 0 ∈ R p and is defined in such way to recover the initial modal basis

ω 0 ν , φ 0 ν , ξ 0 ν with ν ∈ {1, • • • , m}.
This vector is not assumed to be uncertain contrary to the vector

x. For example, in the case where m = 3 modes are taking into account in the analysis, the initial vector x 0 ∈ R p , with p = 15 and is defined by

x 0 =     1, 1, 1 Perturbation on ω 0 ν , ν∈{1,2,3} , Perturbation on ξ 0 ν ,ν∈{1,2,3}
1, 1, 1 , 1, 0, 0, 0, 1, 0, 0, 0, 1

Perturbation on Φ 0 =[φ 0 1 φ 0 2 φ 0 3 ]     T . (4)

Performance requirement

The lack of knowledge in the eigenparameters is represented by an absolute error model:

U α, x 0 = {x ∈ R p : x -x 0 ≤ αw}, (5)
where α ≥ 0, the horizon of uncertainty;

w ≥ 0, a vector of weighting coefficients (w ∈ R p ).

In other words, the greater the horizon of uncertainty α, the greater the Euclidean distance between the vectors x and x 0 can be. The vector of weighting coefficients, which defines the uncertainty model (equation ( 5)), expresses the relative confidence between the different eigenparameters. It can be assessed in different ways: on the basis of engineering judgment when information is lacking or by using the empirical robustness. This point will be discussed in section 4.3.

Performance requirement

The performance requirements represent the quantity of interest upon which a decision or set of decisions will be based. The performance requirement is sometimes called the decision-making model since the decision is based on the specification of a critical level. In the present case, the maximum response level of the system y at one frequency of interest ω 0 (or possibly a frequency band of interest) must be less than or equal to some critical value y c y ≤ y c .

In practice, y is built from the components of y defined by the equation [START_REF] Knight | Risk, Uncertainty and Profit[END_REF]. In what follows, the equation ( 6) is the decision-making model.

Info-gap robustness

Info-gap robustness

"Uncertainty may be either pernicious or propitious" (p. 37 in [START_REF] Ben-Haim | Information-Gap Theory: Decisions Under Severe Uncertainty[END_REF]). While this may seem like a trivial observation, info-gap decision theory addresses the problem of quantifying these two aspects of uncertainty by two "immunity functions": the robustness function, which expresses the immunity to failure, and the opportuneness function, which expresses the immunity to windfall gain. The present study only deals with robustness function. The robustness function expresses the greatest horizon (level) of uncertainty α that can be tolerated without exceeding the critical performance requirement y c . Thus, α depends intimately on y c and is formally defined by [START_REF] Ben-Haim | Information-Gap Theory: Decisions Under Severe Uncertainty[END_REF]

α(y c ) = arg max α≥0 { max x∈U (α,x 0 ) y ≤ y c }. (7)
However, for a given performance requirement, the costly non linear optimization problem ( 7) must be solved in order to obtain a value of robustness. While an analytical expression of the robustness function may be available in simple cases where the structural dynamic behavior is controlled by few well-separate modes, a more general approach is employed here. Practically speaking, rather than fix y c , it is easier to fix α. The robustness is thus discretized by a series of increasing values:

α = {0, • • • , αi , • • • }, (8)
and, for each αi an optimization problem under constraints is solved:

y c (α i ) = max x∈U ( αi ,x 0 ) y. ( 9
)
The solution is obtained by using the interior-point method which is a local optimization algorithm from the fmincon function available in the Matlab R Optimization ToolBox.

The info-gap decision theory is not directly related to uncertainty quantification/propagation. The main interest of such an approach is to study the different trade-offs involved in making decision under lack of knowledge. One of the most important trade-off is that the robustness and the performance requirement are antagonistic: if small values of y c are required then only low robustness (immunity) to uncertainty is possible. The simplest way to explore this trade-off is obtained by the robustness curve, which is the graphical representation of α = f (y c ). Nonetheless, it should be emphasized that solving equation ( 9) is equivalent to solving equation [START_REF] Ghanem | Stochastic Finite Element: A Spectral Approach[END_REF].

NUMERICAL ILLUSTRATION

In this section, an illustration, based on finite element (FE) model predictions, is proposed.

The system is composed of two 700x440x5 mm 3 aluminum plates clamped together on one edge and at three stacking points (cf. Figure 6). The system is assumed to be in free-free configuration.

The FE model (cf. Figure 1) is composed of approximately 3000 nodes. A numerical modal basis is thus obtained and a robustness analysis is performed in order to assess the maximum level of uncertainty on the modal basis that the system can tolerate without exceeding a critical response level. For the sake of clarity, only the first eigenfrequency is assumed to be uncertain. As shown

Common Reference Frame GS_2D_Updated vs GS_2D_Updated in Figure 1, a transverse unit point force is located on node 168 (direction z) while the transverse response displacement amplitude is observed at the node 1 (direction z). The modal basis used to assess the response of the system is composed of five modes and is depicted in Figure 3. The Figure 2 plots the FRF between the input/output of the system. In the present study, the output of interest is the response displacement amplitude at the frequency of interest f = 30 Hz. The robustness curve is the black curve in Figure 4 and is obtained for an horizon of uncertainty from 0 to 0.2 with a step α i = 0.001 . This curve is evaluated point by point using optimization methods. At a given horizon of uncertainty α i :

1. The input parameter in the optimization problem is the eigenfrequency of the first elastic mode that can vary between (1

-α i )f 0 1 ≤ f 1 ≤ (1 + α i )f 0 1
, where f 0 1 is the value of the frequency for the nominal model. 5. α i is incremented a step and the next optimization problem is solved over the new domain.

In the present case, the robustness curve is composed of 2 zones:

α ∈ [0; α1 ]: the displacement amplitude increases significantly with increasing horizons of uncertainty. This is evidently due to the fact that the peak response at the resonance is moving closer and closer to the frequency of interest situated at 30 Hz. The performance in this zone is clearly vulnerable to increasing uncertainty.

α ∈ [α 1 ; 0.2]: the maximal displacement amplitude does not change since the horizon of uncertainty is large enough for the first eigenfrequency to shift to f = 30 Hz. Larger horizons of uncertainty in this simple test case will not lead to a worse case. This explains why the robustness curve is vertical for α ≥ α1 meaning that increasing horizon of uncertainty no longer has an impact on the worst case maximum response. As such, the performance in this zone is seen to be robust to uncertainty.

In this simple case, it is can be concluded that if a response amplitude of y 1 is acceptable than it is shown to be robust to the uncertainty in the first eigenfrequency. Here the robustness curve obtained by optimization methods can be validated using a more basic approach. Since there is only one uncertain eigenparameter, namely, the first eigenfrequency, we can model this uncertainty as an interval with the worst case found at each boundary of the interval. The simulated validation data using this approach, indicated by the red points in Figure 4, yields the same maximum response levels at f = 30 Hz when the first eigenfrequency f 1 is in the interval This can be seen in the upper red curve for α 3 which represents the worst case at the boundary of the interval. It is evident here that the worst case is still found for α = α1 , as indicated in Figure 5.

The initial response corresponds to the amplitude of the response at the frequency f = 30 Hz when no uncertainty is taken into account (α = 0). To summarize, when the horizon of uncertainty increases there are two possibilities. The point 2 corresponds to the best possibility since the amplitude at f = 30 Hz is lower than the original one. The point 1 is the worst possibility when α = α 1 since the amplitude at f = 30 is larger than the original one.

This simple numerical example illustrates the approach that will be adopted in this study. However in real life applications the uncertainty in a set of eigenparameters must be taken into account, including eigenfrequencies, eigenvectors and modal damping factors.

EXPERIMENTAL VALIDATION

As with stochastic approaches, the results of an info-gap robustness analysis are sensitive to the underlying uncertainty model used in the analysis. The objective of this section is to detail the construction of an info-gap model of uncertainty based on a subset of test data. As an illustration, the methodology is applied to the experimental structure shown in Figure 6, which is the structure described in section 3. The system is instrumented with five accelerometers on the free edges of each plate and with an impedance head at the excitation point. An experimental modal analysis is carried out using a low level white noise excitation from 5 to 312 Hz.

Modal Identification

The objective of this section is to demonstrate that the dominant uncertainties in the system under study are only the consequences of the addition of lumped masses in the system. Hence, the modal identification should be as rigorous as possible in order to minimize the errors during the identification process. In the present study, the modal identification is performed in two steps:

1. Pole/residue model identification by using LSCF (Least Square Complex Frequency) method [START_REF] Guillaume | A poly-reference implementation of the least-squares complex frequency-domain estimator[END_REF]. The measured Frequency Response Functions are synthetized by

FRF id ij = N ν=1 R ij ν s -λ ν + R ij * ν s -λ * ν ∈ C, (10)
with s = ω: the Laplace variable;

N : the number of identified poles;

λ ν = -ξ ν ω ν + jω ν 1 -ξ 2 ν ∈ C: the ν-th identified pole 2 ;
R ij ν ∈ C: the identified residue corresponding to the mode ν between the input i and the output j. The complex mode ψ ν is derived from the residue matrix R ν by R ν = ψ ν (ψ * ν F )

where (ψ * ν F ) is the modal factor3 .

2. Normal mode approximations derived from the complex modes where the synthesis is rewritten using the normal eigenvectors φ ν :

FRF nm ij = N ν=1 φ i ν φ j ν T s 2 + 2ξ ν ω ν s + ω ν ∈ C. ( 11
)
As shown in Figure 8, the measured FRFs have an acceptable signal/noise ratio. Furthermore, due to the simplicity of the tested structure, the mode complexity is very low.

φ ν = ψ ν √ 2λ ν . ( 12
)
The quality of the normal mode model, as well as the pole/residu model, is directly related to their ability to represent the structural dynamic behavior. One way to assess the quality is to examine to what extent the models are able to synthesize the measured FRFs. Quantitative tools such as the FRAC (Frequency Response Assurance Criterion) can be used to assess quantitatively this quality. However, it should be noted that the superposition requires the mass normalized normal modes which can only be obtained when the colocated FRFs are available. Let µ ν ∈ R be the identified modal mass of the mode ν defined by

µ ν = φ i ν 2 2ω ν Im (R ii ν ) , (13)
the mass normalized normal modes φν are thus obtained by

φν = φ ν √ µ ν . ( 14
)
As observed in Figure 10, the pole/residue model as well as the mass normalized normal mode model properly match the measured FRFs. The same approach is performed for all experiments.

Variability in test data

Experimental modal analyses are performed in order to validate the proposed approach. The various cases are obtained by changing the added masses locations and numbers. Compared to the nominal structure, the mean mass added is around 6% whereas the standard deviation is around 0.6%. The variability is then evaluated on the results of 31 experimental modal bases.

Variability in mode shapes, eigenfrequencies and modal damping factors

The MAC (Modal Assurance Criterion) is a standard criterion to evaluate the correlation between two vectors [START_REF] Allemang | The modal assurance criterion: twenty years of use and abuse[END_REF] and is used to compare the mode shapes of the initial and modified systems. Figure 11 shows the mean (left) and standard deviation (right) of the MAC between the two systems. The number inside the MAC corresponds to the number of paired mode shapes where it is assumed that two modes are paired when their MAC is higher than 0.7. When two modes shapes are paired, we can compare their corresponding eigenfrequencies and modal damping factors (cf. Table 1 and2). It can clearly be observed that the variations corresponding to the perturbations of mass do not have the same impact on the different eigenproperties. In terms of frequencies, the first and fifth modes are the most affected in terms of mean, while modes 2 to 4 have almost the same mean as the nominal model. The relative standard deviation is of the same order of magnitude (7 to 11% of the mean) for the 5 modes except mode 4 whose sensitivity is much lower (about 2% of the mean). The variation of the mode shapes (Figure 11) is coherent with physical arguments, that is to say, the impact on both mean value and standard deviation is increasing with mode order. Globally, the highest variations are observed on the damping values.

For all modes except mode 5, the standard deviation on the damping factor is between 18% and 31% of the mean value. For mode 5, the standard deviation level is of the order of the mean,

Variability in test data

indicating a very large variability. The mean damping factor of this mode is also highly impacted by the mass changes, while for the other modes the impact on the mean damping factor is not significant.

Variability in the identified normal mode based FRFs

After having detailed the variations affecting the different eigenproperties, one can observe the global variation affecting the normal mode based FRFs associated to the various configurations.

Figure 12 shows the envelop FRFs for the 31 configurations (the initial system is in black) for 4 sensors. The previous conclusions are obviously still valid, in particular that the first three modes have similar properties in terms of frequency and damping dispersion, but also the fact that mode 5 is more dispersive both in frequency and damping. Moreover, the variability ranges of the first 3

modes are distinct from one to another.

Construction and calibration of the info-gap model of uncertainty based on a subset of identified modal model

The robustness function assesses the impact of eigensolution uncertainty on the performance requirement. It is therefore important that the model of uncertainty represents the correct idealization of the real uncertainty in the model. A well-known result of info-gap theory is that the more informative the info-gap model of uncertainty is, the more robust the decision making process is. This is analogous to the notion of informativeness as seen in Figure 13, where the info-gap model of uncertainty IGMU2 is more informative than the IGMU1. As a consequence, decisions based on the IGMU2 should be more robust than decision based on the IGMU1. For simplicity, it is assumed that there are no outliers in the experimental data. It is also assumed that no finite element model is available so that the best estimate is provided by the experimental modal model. Let X be the whole set of experimental data and X e a subset used only to construct and calibrate the info-gap model of uncertainty. Obviously, the larger the set X e is, the more informative the info-gap model of uncertainty will be. According to the notations introduced in section 2, x is uncertain, and is in the convex set denoted U w α, x 0 = U α, x 0 in this section, and expressed by equation [START_REF] Klir | Generalized information theory: aims, results and open problems[END_REF]. Some strategies for calibrating this uncertainty model can be found in the literature (see [START_REF] Ben-Haim | Uncertainty, probability and information-gaps[END_REF] or chap.8 in [START_REF] Ben-Haim | Information-Gap Theory: Decisions Under Severe Uncertainty[END_REF]). The calibration process may concern not only the parameters of the model of uncertainty (center or weighting coefficient) but also the structure (envelop, absolute error model, etc.). One quantitative way to assess the fidelity between experimental data and info-gap model of uncertainty is the gap-function Γ U w α, x 0 , w defined by

Γ U w α, x 0 , w = min{α : X e ⊆ U w α, x 0 }. (15)
Small (resp. large) values for Γ U w α, x 0 , w indicate a good (resp. low) agreement between the model of uncertainty U w α, x 0 and the subset of test-data X e . Nevertheless, the main question is: does the test data provide experimental evidence against the info-gap model? Here, confidence in the ability of the structure of the model of uncertainty to idealize uncertainties is high. The problem to be addressed is the fundamental choice of the weighting vector. It is obvious that the eigenfrequencies can not be considered as more uncertain than the modal damping factors or even than the mode shapes (section 4.2). Nevertheless, the weighting vector will be considered to be constant as the uncertainty increases, though an α-dependent or mode-dependent weighting vector can be defined. For the sake of clarity, we assume that the weighting vector which defines the uncertainty in the eigenfrequencies, denoted w ω , is the unit vector. The objective here is to assess w ξ and w φ based on the subset of test data. Hence, they are referred as the parameters to be calibrated in order to reduce the discrepancy between the info-gap model of uncertainty and the test data X e . The cost function used to calibrate the model of uncertainty is the empirical robustness. It is an estimate of the horizon of uncertainty within which the info-gap model captures some fraction of the measurements (ch. 8 in [START_REF] Ben-Haim | Information-Gap Theory: Decisions Under Severe Uncertainty[END_REF]). In our case, the empirical robustness, αe (w, r c ), is defined by

αe (w, r c ) = max{Γ (X e , w) : Γ (X i , w) ≤ Γ (X e , w) if max x∈Uw(Γ(X i ,w),x 0 ) s ≤ r c }. (16)
Figure 14 shows the repartition of the modal data before calibration: the starting values for the weighting vectors are w ξ = 1 and w φ = 1. Two test data are used to both assess and improve the model of uncertainty (depicted in red squares in Figure 14).

Minimal values

Fig. 15. Empirical Robustness (log) vs (w ξ , w φ ) and selected values for w ξ and w φ = 20.

Confrontation between the model of uncertainty and real experiments

The final objective of this paper is to illustrate that the calibrated model of uncertainty is consistent with real experimental data when the robustness of dynamic response levels to an uncertain modal model is evaluated. According to the approach described in section 2, a robustness analysis is performed to investigate the impact of lack of knowledge affecting the modal model on dynamic response levels. The dynamic response level is evaluated at f = 30 Hz. The robustness curve is plotted in Figure 16. Meanwhile, test data coming from the real experiments are plotted (black circle on the Figure 16). One can observe that the robustness curve maximizes well the response levels when uncertainty is increasing. However, the model of uncertainty tends to underestimate the robustness values, though a rigorous calibration of the model of uncertainty has been performed. Several explanations for this result are possible, including for instance the topology of the uncertainty model or the fact that the weighting factors remain constant for each mode and for each value of the horizon of uncertainty. However, the approach adopted here consists in using the information available, thought additional tests could be performed in order to obtain a best model of uncertainty that represents much better the reality.

Fig. 2 .

 2 Fig. 1. FE model

Fig. 3 .

 3 Fig. 3. Mode shapes used to assess the robustness of the system

2 .

 2 The objective function to be maximized is the displacement amplitude at the node 1 along the z-direction 3. The box constraints on the realizable eigenfrequency value depend directly on the current horizon of uncertainty 4. A local nonlinear optimization algorithm is used to maximize the objective function thus providing the largest displacement at α i .

Fig. 4 .

 4 Fig. 4. Robustness curve

Fig. 5 .

 5 Fig. 5. Real FRFs for checking the accuracy of the robustness curve

Fig. 7 .

 7 Fig. 6. Experimental set-up

Figure 9 plots 4 . 1 Fig. 9 .

 419 Fig. 8. Measured FRFs

Fig. 10 .

 10 Fig. 10. Results of the modal identification for the collocated FRF and one non-collocated FRF: -measured FRFs (-), Poles/residues model (--), Mass-normalized normal mode synthesis (--).

Fig. 11 .

 11 Fig. 11. Variation of the 5 first mode shapes: mean (left) and standard deviation (right).

4. 3 Fig. 12 .

 312 Fig. 12. Envelop mass normalized normal mode synthesis for the 31 configuration and associated to 4 sensors.

4. 3 Fig. 13 .

 313 Fig. 13. Notion of informativeness (2 uncertain parameters)

4. 3

 3 Construction and calibration of the info-gap model of uncertainty based on a subset of identified modal model The calibrated parameters, ŵ, are the ones that minimize the empirical robustness: ŵ = arg min w αe (w, r c ) .

Fig. 14 . 1 ) 25 4. 4

 141254 Fig. 14. Experimental data before calibration (w ξ = 1 and w φ = 1)

Fig. 16 . 5 CONCLUSION

 165 Fig. 16. Robustness curve (black curve) and test data (black circle). Zoom on the right.

Table 1 .

 1 Variation on the eigenfrequencies

	1	31.6	27	26.6	32.2	28.9	1.9
	2	45	29	36.9	59	44.3	5.3
	3	57.7	20	43.5	59.6	54.1	5.1
	4	61.9	17	58	62.1	60.5	1.4
	5	119.5	22	98.6	145.3	115.8	9.1

Table 2 .

 2 4.3 Construction and calibration of the info-gap model of uncertainty based on a subset of identified modal model Variations on the modal damping factors

	Mode Init. Damping (%) Matched (over 31) Min (%) Max (%) Mean Damping (%) Std. Dev. (%)
	1	0.25	27	0.14	0.55	0.29	0.09
	2	0.27	29	0.13	0.47	0.28	0.08
	3	0.33	20	0.25	0.53	0.34	0.06
	4	0.25	17	0.20	0.53	0.30	0.09
	5	0.11	22	0.06	2.6	0.58	0.6

Satisficing is the contraction of satisfactory and sufficient.

The modal damping assumption is supposed to be valid.

Here, (ψ * ν F ) ∈ R since a mono-excitation (punctual force) is applied on the structure.


Julio César

Livia Santos

Tulio Hallak Avila De Oliveira

Chrystel Panzera

Ian Remillat

Vincent Farrow

Fabrizio Placet

Scarpa

Julio César

Avila

Júlio César

Dos Santos

Lívia Ávila De Oliveira

Hallak Panzera

Chrystel D L Remillat

Ian Farrow

Vincent Placet

Fabrizio Scarpa

Ageing of autoclaved epoxy/flax composites: Effects on water absorption, porosity and flexural behaviour

Keywords: flax composite, prepreg, ageing process, fibre orientation, flexural properties, water absorption, apparent porosity

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Flax (Linum usitatissimum) is one of the most widely used types of biofibres for textile applications, the latter dating back to 5000 BC [START_REF] Baley | Compressive strength of flax fibre bundles within the stem and comparison with unidirectional flax/epoxy composites[END_REF][START_REF] Yan | Flax fibre and its composites -A review[END_REF]. Linen fabric represents a strong traditional niche among high quality household textiles, such as bedding, furnishing fabrics and interior decoration accessories [START_REF] Yan | Flax fibre and its composites -A review[END_REF]. In recent years, the use of flax fibres as reinforcement in composite materials has attracted the attention of the research community due to biodegradability, renewable aspects, low cost, low density and good mechanical performance [START_REF] Masseteau | An evaluation of the effects of moisture content on the modulus of elasticity of a unidirectional flax fiber composite[END_REF][START_REF] Zivkovic | Influence of moisture absorption on the impact properties of flax, basalt and hybrid flax/basalt fiber reinforced green composites[END_REF][START_REF] Flax | Linum usitatissimum L.) fire reinforced polymer composite materials: A review on preparation, properties and prospects[END_REF][START_REF] Hallonet | Durability and tensile characterization of wet lay-up flax/epoxy composites used for external strengthening of RC structures[END_REF]. The most promising applications of these composites are in the transport industries because of the lightweight characteristics, high specific strength and stiffness required in those domains [START_REF] Hu | Moisture absorption, tensile strength and microstructure evolution of short jute fiber/polylactide composite in hygrothermal environment[END_REF][START_REF] Assarar | Influence of water ageing on mechanical properties and damage events of two reinforced composite materials: Flax-fibres and glassfibres[END_REF].

Due to their high hydrophilicity, the use of plant fibre composites in humid or outdoor environments requires however a thorough evaluation to assess the impact on the mechanical behaviour [START_REF] Assarar | Influence of water ageing on mechanical properties and damage events of two reinforced composite materials: Flax-fibres and glassfibres[END_REF][START_REF] Regazzi | Modeling of hydrothermal aging of short flax fiber reinforced composites[END_REF]. Since the thermosetting polymer is less hydrophilic than natural fibres, water absorption is mainly governed by the characteristics of the fibres, and this aspect plays an important role in the ageing of biocomposites [START_REF] Moudood | Flax fiber and its composites: An overview of water and moisture absorption impact on their performance[END_REF][START_REF] Fiore | Aging resistance of bioepoxy jute-basalt hybrid composites as novel multilayer structures for cladding[END_REF][START_REF] Wang | A mechanical degradation model for bidirectional natural fiber reinforced composites under hydrothermal ageing and applying in buckling and vibration analysis[END_REF][START_REF] Fiore | Influence of sodium bicarbonate treatment on the aging resistance of natural fiber reinforced polymer composites under marine environment[END_REF]. Swelling, plasticizing and hydrolysis due to water absorption result in physical damage to the polymeric matrix. Interfacial damage between fibres and matrix and delamination of the layers are also present, and all these factors contribute to reduce the overall mechanical performance of the composites [START_REF] Wang | A mechanical degradation model for bidirectional natural fiber reinforced composites under hydrothermal ageing and applying in buckling and vibration analysis[END_REF].

Several studies have shown that biocomposites feature a reduced stiffness and strength in humid environments. Scida et al. [START_REF] Scida | Influence of hygrothermal ageing on the damage mechanisms of flax-fibre reinforced epoxy composite[END_REF] evaluated the influence of hygrothermal ageing on the mechanical performance and damage behaviour of quasi-unidirectional flax-fibre reinforced epoxy composites. The tensile properties were clearly affected by hygrothermal ageing, with an overall significant reduction, in particular for the Young's modulus. The reduction in terms of stiffness and strength was explained by the reorientation of the flax microfibrils and the plasticizing effect of the water on the matrix. Yan et al. [START_REF] Yan | Effect of water, seawater and alkaline solution ageing on mechanical properties of flax fabric/epoxy composites used for civil engineering applications[END_REF] evaluated the effect of water, seawater and alkaline (5% NaOH) solution ageing on the mechanical properties of flax fabric/epoxy composites used for civil engineering applications. All the ageing procedures used have led to a noticeable degradation of the tensile/flexural properties. Alkaline ageing was the most damaging, followed by the ones provided by seawater and water. Fiore et al. [START_REF] Fiore | Effects of aging in salt spray conditions on flax and flax/basalt reinforced composites: Wettability and dynamic mechanical properties[END_REF] have investigated the wettability and the dynamic mechanical properties of flax and flax/basalt reinforced composites subjected to long-term ageing under critical environmental conditions.

The results showed that the storage modulus of the two types of composites decreased after 15 days of ageing, mainly due to the progressive damage to the resin/fibre interface that induces formation of cracks. The hydrophilic nature of the fibres (with resultant water uptake) and the poor fibre-matrix bonding were also other significant factors that contributed to the reduction of the mechanical performance. Regazzi et al. [START_REF] Regazzi | Coupled hydro-mechanical aging of short flax fiber reinforced composites[END_REF] examined the influence of thermohydromechanical-based ageing on the elastic behaviour of flax/poly (lactic acid) composites. The results confirmed that the ageing temperature in wet conditions led to a loss of the elastic properties. In addition, the combination of mechanical loading with water immersion generated in a strong synergistic effect on the stiffness loss. Berges et al. [START_REF] Berges | Influence of moisture uptake on the static, cyclic and dynamic behaviour of unidirectional flax fibre-reinforced epoxy laminates[END_REF] studied the effect of moisture absorption on the mechanical properties of unidirectional flax fibre-reinforced epoxy composites through quasi-static, fatigue and vibration tests. The results revealed that the water vapour sorption promoted a significant change in the shape of the tensile stress-strain curve, a decrease in the dynamic elastic modulus of about 20% and a great increase in damping ratio of 50%. Contrary to all expectations, water saturation does not degrade the monotonic tensile strength of flax-epoxy composites and leads to an increase in fatigue strength over a large number of cycles. Jeannin et al. [START_REF] Jeannin | Inflence of hydrothermal ageing on the fatigue behaviour of a unidirectional flxepoxy laminate[END_REF] investigated the influence of hydrothermal ageing on the fatigue behaviour of the unidirectional flax-epoxy laminated composite, implementing fatigue tests in a water bath. The results showed that hydrothermal ageing, while inducing a decrease in stiffness and strength, leads to an improvement in fatigue resistance. The maximum stress level for an expected lifetime of 5.3 x 10 6 cycles is comparable to one of the unaged specimens.

Despite the high sensitivity of the composite to water, its resistance to ageing under very severe conditions is good. Cheour et al. [START_REF] Cheour | Effect of water ageing on the mechanical and damping properties of flax-fibre reinforced composite materials[END_REF] have evaluated the effect of water ageing on the mechanical and damping properties in quasi-unidirectional flax fibre reinforced epoxy composites. The results showed that the water uptake induces a decrease in flexural modulus and an increase of the loss factors in comparison with the ageing time. Shen et al. [START_REF] Shen | Effect of Temperature and Water Absorption on Low-Velocity Impact Damage of Composites with Multi-Layer Structured Flax Fiber[END_REF] have investigated the effect of the manufacturing and environment temperatures and water absorption on the low-velocity impact response and damage mechanisms of flax fibre reinforced epoxy plastic laminates at the onset of damage. The results showed that excessive temperature can cause chemical decomposition and structural damage to the flax fibres, resulting in a serious reduction of the impact damage threshold and of the resistance to damage. Shen et al. also observed that a small amount of water absorption can slightly improve the damage threshold load and the damage resistance, while more water uptake causes severe degradation at the composite interface and structural damage of the flax fibre, reducing the impact performance of flax fibre reinforced composites. Moudood et al. [START_REF] Moudood | Environmental effects on the durability and the mechanical performance of flax fiber/bio-epoxy composites[END_REF] have evaluated the durability and the mechanical performance of flax/bio-epoxy composites exposed to different environmental conditions. The results showed that exposure to warm and humid environments slightly reduced the mechanical properties of the bio-composites. Almost no detrimental effects were however observed regarding the performance of the composites after freeze/thaw cycles. The authors suggested that flax/bio epoxy composites can therefore be used in most environmental conditions, excluding underwater applications that cause severe damage to the materials properties. Recently, Koolen et al. [START_REF] Koolen | Interface modification and the influence on damage development of flax fibre -Epoxy composites when subjected to hygroscopic cycling[END_REF] investigated the development of damage in epoxy composites reinforced with unidirectional flax fibres during the hygroscopic cycle by inserting an elastomeric silicone in the fibre/matrix interphase. Hygroscopic cycling was performed by varying the relative humidity between 25 and 80% at 80 °C. The results showed that the degradation of the mechanical properties can be attributed to the fibre-matrix debonding and the cohesive failure within the fibre bundles. Contrary to the hypothesis, the insertion of the silicone interphase led to an accelerated decrease of the transverse strength.

Chaudhary et al. [START_REF] Chaudhary | Comparative study of mechanical properties of dry and water aged jute/flax/epoxy hybrid composite[END_REF] studied the properties of hybrid jute/flax epoxy composites in dry and one-year water aged condition; those composites were subjected to tensile, flexural, impact and hardness tests. The experimental findings showed reductions of 37.8% and 43.2% in tensile strength and modulus; 33.6% and 45.1% in flexural strength and modulus. The hardness and the impact resistance however decreased by only 2.1% and 4.3% respectively, compared to the dry composites. Cheng et al. [START_REF] Cheng | Environmental durability of carbon/flax fiber hybrid composites[END_REF] investigated the durability of carbon/flax fibre reinforced polypropylene composites exposed to water immersion ageing at 60 o C until saturation. The inclusion of carbon fibres reduced the water absorption and improved the mechanical properties of the composites when compared to pure flax/polypropylene composites. The carbon/flax interface is however vulnerable to both hygrothermal attack and delamination, reducing the tensile and flexural properties of the laminates while increasing the tensile failure strain due to plasticization. Wang and Petru [START_REF] Wang | Degradation of bending properties of flax fiber reinforced polymer after natural aging and accelerated aging[END_REF] studied the effect of natural and accelerated ageing on the flexural properties of flax fibre reinforced polymer composites. Natural ageing was carried out under daily temperature and humidity and lasted 180 days, while accelerated ageing was carried out at 60 o C, with 100% relative humidity. Although both ageing tests have shown a decrease in mechanical properties, they generated different degradation effects on the properties of composites, since only limited ageing factors (temperature and humidity) are considered.

Natural ageing is on the opposite caused by a combination of various environmental factors, such as rainwater, ultraviolet rays, oxygen and ozone. Panzera et al. [START_REF] Panzera | Static, fatigue and impact behaviour of an autoclaved flax fibre reinforced composite for aerospace engineering[END_REF] investigated the tensile, flexure and impact properties of autoclaved UD and cross-ply flax fibre composites impregnated with fire retardant epoxy polymer, following the recommendations for aerospace applications, as a previous study to the current work.

Although several studies in open literature describe the effect of critical environments on the mechanical properties of flax reinforced composites, this work is the first to report about the interaction between the ageing time, type of bending loading (with related sample size) and fibre orientations, through a statistical design, of autoclaved composites made with a commercialised unidirectional prepreg flaxtape, and the influence provided on the water absorption capability, porosity and flexural properties of those composites, meeting the requirements of AC 25-853a [START_REF]AC 25.853-1 standard: Flammability requirements for aircraft seat cushions, ANM-110[END_REF] in terms of self-extinguishing at 50% RH (relative humidity).

Moreover, this work evaluates composites machined via a laser cutting process. Laser cutting reduces the presence of microstructural defects, stress concentrations, premature swelling due to the coolant, or any other uncontrolled geometric parameter that affects the responses.

Materials and Methods

Materials

The unidirectional flax fibre reinforced composites are made of flax fibres preimpregnated with fire-retardant epoxy polymer (XB 3515 GB -Huntsman), combined with Aradur 1571 BD and Accelerator 1573 BD. The prepreg flaxtape is supplied by EchoTechnilin-Lineo (France). The prepreg has a nominal 50% flax weight fraction, and the matrix/fibre volume fraction is estimated at 43/56%.

Fabrication process

The composites have been fabricated in two processes: hand lay-up followed by autoclave curing. Twelve prepreg flax plies have been laid up along the unidirectional orientation. Layup [0]12 is autoclaved and cured at ~0.7 MPa (100 psi), with two dwell times of 100 minutes each at 80 o C and 140 o C. An aluminium plate is placed on the upper surface of the lay-up to obtain a surface finish similar to the one of the lower plane in contact with the tool plate. The aluminium plate also helps to obtain a final flat laminate. The final composites have an average thickness of 2.05 mm, corresponding to approximately 0.17 mm per layer.

Water saturation (50% and 100%)

A laser cutting machine has been used to obtain samples along the longitudinal and transverse directions with the required dimensions for the characterisation of the composites (Figure 1). The laser cutting avoids any swelling effect attributed to traditional coolant cutting and peripherical damages that may also affect the physical and mechanical properties of the composites. Prior to the ageing and the characterisation process, the samples have been weighed in a precision scale (0.001g) and placed in a climatic chamber (Solab climatic chamber SL -206) for saturation during 2, 4, 8 and 16 days. The humidity and temperature inside the chamber were adjusted to 50 ± 5% and 21 ± 5°C, respectively, following the ASTM D5229/D5229M standard [START_REF]Standard Test Method for Moisture Absorption Properties and Equilibrium Conditioning of Polymer Matrix Composite Materials[END_REF]; the sorption measurements have been however carried out here under a transient regime at each ageing time level, i.e., not corresponding to the water equilibrium state. Hundred percent humidity (100%RH) samples were immersed in water within the climatic chamber at 21 ± 5°C, for 2, 4 and 8 days. An additional immersion time level of 30 minutes is used to measure apparent porosity only. The results are compared with 50% saturated humidity samples for 16 days to identify the equivalent ageing time for a similar mechanical behaviour. This test has been performed with UD flax composite samples of 65 ´ 13 ´ 2 mm 3 and 80 ´ 13 ´ 2 mm 3 and oriented along the transverse and longitudinal fibre directions (Figure 1).

Physical and mechanical properties 2.4.1 Water absorption and apparent porosity

The water absorption and the apparent porosity are determined using the Archimedes principle. Water absorption is measured during every experimental condition considering the sample size (3PB, 4PB), the fibre orientation (longitudinal and transverse) and the ageing time Water absorption is measured according to the ASTM D570-98 standard [START_REF]Standard Test Method for Water Absorption of Plastics[END_REF], in which the relative weight uptake (Mt) of the specimens is calculated for each exposure time:

𝑀 ! = " ! #" " " " 𝑥 100% (1)
where Wo is the weight of the dry specimen and Wt is the weight of the wet specimen at time t.

The apparent porosity P is expressed as a percentage of the volume of open pores to the exterior volume, and may be calculated according to Eq. 2, following the recommendations of the ASTM C1039-85 standard [START_REF]Standard Test Methods for Apparent Porosity, Apparent Specific Gravity, and Bulk Density of Graphite Electrodes[END_REF]:

𝑃 = " ! #" " " ! #" # 𝑥 100% (2)
In ( 2), Wt, Wo and Ws are the weight of the wet specimen at the time t, the dry and suspended specimens, respectively.

Flexural Properties

Longitudinal and transverse UD flax composites have been tested under three-point bending (3 PB) according to the ASTM D790-17 standard [START_REF]Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials[END_REF], while for the four-point bending (4 PB) one the ASTM D7264/D7264M-15 protocols has been followed [START_REF]Standard Test Method for Flexural Properties of Polymer Matrix Composite Materials[END_REF]. The sizes of the samples for the two types of bending loading are 65 ´ 13 ´ 2 mm 3 and 80 ´ 13 ´ 2 mm 3 , respectively, as shown in Figure 1. A load span of 32 mm and a support span of 64 mm were used for the four-point bending test, while a support span of 52 mm was used for the three-point bending test. The experiments have been performed using a Shimadzu AG-X Plus equipped with 100 kN load cell at 2 mm/min. Flexural strength and modulus have been determined based on ASTM recommendations [START_REF]Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials[END_REF][START_REF]Standard Test Method for Flexural Properties of Polymer Matrix Composite Materials[END_REF].

Statistical Design

A Full Factorial Design (5 1 2 2 ) has been used to identify the effects of the type of bending loading (3 and 4-point bending), the fibre orientation (longitudinal and transverse) and ageing time (0, 2, 4, 8 and 16 days) on the water absorption and the flexural properties of the composites at 50% of humidity. The 'bending type' factor also relates to the effect of the 'sample size (3PB and 4PB)' when the water absorption is evaluated. The Design results in 20 experimental conditions (ECs) (Table 1). Three specimens have been fabricated for each experimental condition in two different runs (replicate) to estimate the experimental error [START_REF] Montgomery | Design and Analysis of Experiments[END_REF],

totalling 120 specimens. The Minitab software v. 18 has been used to perform the Design of Experiment (DoE) and the Analysis of Variance (ANOVA). ANOVA is used to establish the statistical significance of the effects of factors and their interactions on physical and mechanical responses, considering a 95% confidence interval [START_REF] Montgomery | Design and Analysis of Experiments[END_REF]. The interaction among factors occurs when the effect of one factor on a specific response depends on the level of other factors. The indicator of the significance of factors and interactions within the confidence interval is a P ≤ α. The parameter α (the so-called significance level, fixed at 0.05 in this study) represents the risk of stating that the effect of a factor (or interaction of factors) is significant when, in fact, it is not. The validity of ANOVA is based on the normality of the underlying probability distribution of the analysed data. The Anderson-Darling test is performed to verify this assumption. Given the construction of the Anderson-Darling hypothesis test, P ≥ 0.05.

Significant effects are interpreted through main or interaction plots. In addition to ANOVA, a statistical mean comparison test, such as Tukey, can be useful to compare the means between levels of the same factor. Tukey´s test attributes a letter to each mean; the means that do not share a letter are significantly different from each other [START_REF] Montgomery | Design and Analysis of Experiments[END_REF].

Results

Experimental data

Table 2 presents the mean values and standard deviation of the responses obtained for 50% moisture saturated samples. As expected, increases in water absorption do compromise the bending properties of plant fibre composites. Similar trends of those shown in the current data can be also observed in other works from open literature focusing on the effects of ageing on the mechanical performance of flax composites [START_REF] Shen | Effect of Temperature and Water Absorption on Low-Velocity Impact Damage of Composites with Multi-Layer Structured Flax Fiber[END_REF][START_REF] Moudood | Environmental effects on the durability and the mechanical performance of flax fiber/bio-epoxy composites[END_REF][START_REF] Li | Hydrothermal ageing mechanism s of unidirectional flax fabric reinforced epoxy composites[END_REF]. According to Kollia et al. [START_REF] Kollia | Effect of water aging on the mechanical properties of flax fiber/bio-based resin composites[END_REF] for example, the flexural modulus and strength are significantly reduced after hydrothermal ageing due to the degrading effect of the water on the fibre-matrix bonding. Other causes of this degradation are the presence of excessive cracks in the matrix, swelling of the fibre-matrix interface, plasticization of the material, degradation of the cellulose structure (i.e., dissolution)

and fibre sliding.

The data present in Table 2 will be better assessed in the statistical design described in the following section.

Statistical Design

Table 3 presents the DoE/ANOVA analysis for the 50% moisture saturated samples. All P-values are lower than 0.05, meaning that the main and the interaction effects are statistically significant within a 95% confidence level. This implies that change in the level of the factors produces a change in the response-variables [START_REF] Montgomery | Design and Analysis of Experiments[END_REF]. The results are interpreted by using effect plots (Figures 234567). The R Figure 3. Third order interaction effect plot for the mean (average) water absorption levels.

Flexural properties

Typical behaviours of longitudinal and transverse samples under three-and four-point bending tests are shown in Figure 4. Although similar behaviour is observed between 3PB and 4PB samples, the 4PB test leads to higher maximum force values (Figure 4b). Flax composites tested in the longitudinal direction provide greater force than the transverse ones. Substantial reductions in force are achieved by water-saturated composites after 16 days at 50%RH. The effects of each factor on flexural strength and modulus will be assessed in sections 3.2.2.1 and

3.2.2.2.
Fracture modes of saturated samples under 3PB and 4PB loading are similar to those of pristine specimens. The cracks present in the longitudinal composites propagate along the diagonal direction (Figure 5a), while the transverse samples show cracks along the direction of the fibre orientation (Figure 5b).

Flexural Strength

Flexural strength ranges from 25.5 MPa (3 PB, Transverse, 16 days) to 325 MPa (3 PB, longitudinal, 0 day), as shown in Table 2. Figure 6 shows the effect plots for the main factors.

Laminates tested under four-point bending show an increased strength (35%) compared to three-point bending samples (Figure 6a). The stress in the four-point bending loading is located between the two loading noses, which produces peak stresses over an extended region of the sample surface and thus more resistance to the applied load. On the other hand, the maximum stress, under three-point bending, is located under the loading roller, resulting in a non-uniform distributions and reduced stress elsewhere. Three-point bending also generates shear stress in the whole specimen, except on the vertical plane containing the central loading roller. In 4- point bending, the shear stress is zero in the region between the two loading rollers and has a non-zero value outside this central region (out of the two loading noses). The shear stress direction is almost parallel to the applied loading direction throughout. The flexural strength of the UD longitudinal laminates is nearly 8-fold higher than the transverse composites (Figure 6b). Fibres oriented along the longitudinal direction, especially those under the neutral line, increase their flexural stress due to their improved mechanical performance under tensile loading [START_REF] Santos | Eco-friendly sodium bicarbonate treatment and its effect on epoxy and polyester coir fibre composites[END_REF]. A 20% reduction in strength is observed after 16 days of ageing (Figure 6c). No noticeable difference can be discerned between the mechanical performance related to the 2 and 4 days of ageing, as evidenced by the same group B provided by Tukey's comparison test.

There is no significant variation in strength between the transverse 3PB and 4PB samples. A 38% increase for the 4P laminates UD oriented is however evident (Figure 7a). Figure 7. Third order interaction effect plot for the mean flexural strength.

Flexural Modulus

The flexural modulus ranges from 3.23 GPa (3 PB, Transverse, 16 days) to 30.58 GPa (4 PB, longitudinal, 0 day), as shown in Table 2. The main (Figure 8) and interaction (Figure 9) effects of the flexural modulus are similar to the flexural strength presented above. The highest flexural modulus value is provided by the 4PB samples (Figure 8a), and with the fibres longitudinally oriented (Figure 8b). A similar reduction in stiffness (22%) is obtained after 16 days of ageing. Samples after 4 and 8 days had however equivalent means (average) values, as also shown within the same group C.

The interaction effects reveal small variations between the 3PB and 4PB samples when transversely tested (Figure 9a), and also in pristine condition (Figure 9b). Similarly, to the flexural strength response, the stiffness of the transverse laminates is not affected by ageing (Figure 9c), since the mechanical performance is mainly dominated by the properties of the matrix.

Figure 8. Main effect plot for the mean flexural modulus.

Figure 9. Third order interaction effect plot for the mean flexural modulus.

Fully saturated water environment

Flax laminate composites are tested after a fully saturated environment (100% humidity) and then compared with those saturated for 16 days at a humidity level of 50%. Results are presented in Table 4 and Figures 101112. In general, the results presented here follow a trend similar to the one of the 50% saturated moisture samples, i.e. increases in water absorption and reduction of the flexural performance (3P and 4P bending) with ageing time and also along the transverse direction. After eight days in a fully saturated environment, the water absorption increases by 30

and 25 times for the longitudinal 3PB and 4PB samples, respectively (Figure 10a). Similar behaviour is also observed in the transverse composites (Figure 10b). The water absorption levels are greater for the 4PB samples due to the increase in the fibre length, which raises the level of the permeability of the composites as described by Habibi et al. [START_REF] Habibi | Effect of surface density and fiber length on the porosity and permeability of nonwoven flax reinforcement[END_REF]. For the same surface density, composites manufactured with longer fibres possess therefore higher water absorption and porosity. A general trend towards equilibrium in water absorption is shown after 4 days for the 3PB fully (Group A) and 50% saturated samples, respectively (Figure 10). It is worth mentioning that the 4PB fully saturated samples do not reach such equilibrium after 4 days, as also revealed by the different Tukey letters (Group A, B); this can be attributed to their larger sample size affecting the water diffusion (Fick´s law). It is emphasised that the determination of the hygro-equilibrium state is not the scope of this work, and further investigations must be carried out to reach reliable conclusions. compared to 50%. The laminates tested at four-point bending have however better mechanical properties. Samples saturated for 16 days at 50% humidity are equivalent to the ones with less than one day ageing at 100% humidity. Boris et al. [START_REF] Boris | Study of the diffusion behavior of water vapor sorption in natural fiber composite: Plaster/Rhecktophyllum camerunense[END_REF] showed that the water diffusion coefficient increases with the relative humidity. The water diffusing within the composite material creates hydrogen bonds with the fibres, which can lead to the reduction of the interactions between fibres and matrix. Capillarity could provoke the flow of water molecules along the fibre/matrix interfaces, as well as a diffusion process through the bulk matrix. This water is bound to the network by hydrogen bonds breaking the existing bonds between the hydroxyl groups of the matrix chain [START_REF] Boris | Study of the diffusion behavior of water vapor sorption in natural fiber composite: Plaster/Rhecktophyllum camerunense[END_REF]. This could result in interfacial debonding affecting the mechanical strength of the composite. Table 5 shows a comparison between the present results at 100% saturation with those obtained by Cheour et al. [START_REF] Cheour | Effect of water ageing on the mechanical and damping properties of flax-fibre reinforced composite materials[END_REF], who investigated the effect of the water ageing on the flexural modulus of composites reinforced with flax fibres. Although both studies show similar water absorption, the present work considers a transient sorption regime, while Cheour et al [START_REF] Cheour | Effect of water ageing on the mechanical and damping properties of flax-fibre reinforced composite materials[END_REF] performed the measurements in equilibrium state. In addition, the latter authors have investigated compacted flax composites made with a volume fraction of 35%, while the present study is related to a 43/56% epoxy/flax prepreg laminates manufactured with autoclave. The present work shows the presence of a larger flexural modulus only for dry composites (0 days); this is attributed to the larger amount of fibres along the loading direction (0 o ). In contrast, when the transverse matrix-dominated direction is considered (90 o ) the flexural modulus of the prepreg composites decreases due to its smaller matrix fraction (Table 5). The reduced flexural modulus featured by the prepreg flax composites compared to [START_REF] Cheour | Effect of water ageing on the mechanical and damping properties of flax-fibre reinforced composite materials[END_REF] after 4 days of ageing can be also attributed to the greater degradation due to the larger amount of fibres. The significant loss in modulus observed for the autoclaved prepreg epoxy/flax laminates and the composites shown in [START_REF] Cheour | Effect of water ageing on the mechanical and damping properties of flax-fibre reinforced composite materials[END_REF] may also be due to the different role that the interlaminar shear strength plays in the flexural behaviour of composites. The interlaminar shear strength in flexure is more dominant than in tensile failure; the compaction manufacturing process and lower fibre volume fraction in [START_REF] Cheour | Effect of water ageing on the mechanical and damping properties of flax-fibre reinforced composite materials[END_REF] can lead to different consolidation states between the plies and therefore generate different interlaminar shear strength in those composites compared to the case shown in this paper. As a consequence, the interlaminar shear strength could be more sensitive to the moisture sorption than in the case of the thermo-compressed specimens in [START_REF] Cheour | Effect of water ageing on the mechanical and damping properties of flax-fibre reinforced composite materials[END_REF]. The composites evaluated in the present work may also present heterogenous moisture contents due to the transient regime, in particular along the thickness of the specimens. This may lead to higher internal stresses and a consequent reduction of the stiffness and strength in the autoclaved samples. The apparent porosity ranges from 3.02 to 13.79% (Table 6). Figure 14 shows the apparent porosity results for UD flax composites (3PB samples) with up to 8 days of water saturation. Tukey´s test reveals that porosity levels for transverse and longitudinal samples are similar at 30 min (Group C) and 2 days (Group B) of saturation. Longitudinal samples have similar porosity levels at 2, 4 and 8 days, as indicated by the same letter group B. In contrast, the porosity of transverse samples increases up to 4 days, remaining constant after this period.

A 52% variation is obtained between the transverse and longitudinal composites after 4 days of saturation. The water penetration is easier along the direction of the cross-section of the fibres [START_REF] Boris | Study of the diffusion behavior of water vapor sorption in natural fiber composite: Plaster/Rhecktophyllum camerunense[END_REF][START_REF] Mbou | Modelling of the Water Absorption Kinetics and Determination of the Water Diffusion Coefficient in the Pith of Raffia vinifera of Bandjoun, Cameroon[END_REF] because more porosity is present there than along their peripherical surface. Transverse prismatic samples feature a better percolation of water because of the presence of a larger number of fibres cross-sections, in comparison to the longitudinal samples. Mbou et al. [START_REF] Mbou | Modelling of the Water Absorption Kinetics and Determination of the Water Diffusion Coefficient in the Pith of Raffia vinifera of Bandjoun, Cameroon[END_REF] investigated the water diffusion from the pith of Raffia vinifera by Fick's second law. The diffusion coefficients decrease from the centre to the periphery in the radial position, while the water absorption increases from the periphery to the centre in the radial position. It is noteworthy that the microstructural characteristics of the transverse and longitudinal composites are the same, as also indicated by the equal apparent density (1.38±0.01 g/cm 3 ).

However, the exposure of the fibre cross-sections facilitates the water diffusion along the length of fibres, leading to higher porosity values.

Conclusions

Unidirectional composites made with prepreg flaxtape have been here characterised to investigate the interaction between the ageing time, the type of bending loading and the fibre orientation and their influence on the water absorption, porosity and overall flexural properties.

The main conclusions are described below:

Figure 1 .

 1 Figure 1. Longitudinal and transverse (a) 3PB and (b) 4PB samples.

( 2 ,

 2 4, 8 and 16 days at 50%RH and 2, 4 and 8 days at 100%RH). The apparent porosity is measured considering 3PB samples (longitudinal and transverse) saturated at 100%RH for 30 min, 2, 4 and 8 days.

Figure 2 .

 2 Figure 2. Main effect plot for the mean (average) water absorption.

Figure 4 .Figure 5 .

 45 Figure 4. Typical bending behaviours of 3PB (a) and 4PB (b) samples.

Figure

  Figure7breveals a similar strength increase when the ageing time is reduced for all the sample types. Figure7cshows the interaction effect between fibre orientation and ageing time. A large reduction in strength over time is noted for longitudinal laminates, while no substantial change occurs for transverse samples. Although the transverse composites absorb more water (Figures2b, 7a and 7c), their strength is however slightly affected by ageing. The tensile properties of the UD transverse laminates are known to be dominated by the matrix properties, and the polymer is less affected by water saturation.

Figure 6 .

 6 Figure 6. Main effect plot for the mean flexural strength.

Figure 10 .

 10 Figure 10. Water absorption at 50% and 100% humidity for (a) longitudinal and (b) transverse fibre composites.

Figure 11 Figure 11 .

 1111 Figure11shows the typical flexural behaviours of longitudinal and transverse 3PB (a) and 4PB (b) samples saturated at 50% and 100% for 8 days. 4PB samples reach a higher maximum force than 3PB samples. Substantial reductions in maximum force are achieved when the water saturation level increases from 50% to 100%. The effects of each factor on the flexural strength and modulus will be discussed in the following paragraphs.

Figures 12 and 13

 13 Figures 12 and 13 show reductions in strength and stiffness over ageing and moisture levels for the longitudinal (a) and transverse (b) laminates. drastic drop in strength is observed after 8 days in 3PB and 4PB samples saturated at 100% humidity. A slight variation in mechanical properties is found between the 3PB and 4PB samples saturated at 100% humidity

Figure 12 .

 12 Figure 12. Flexural strength at 50% and 100% humidity for (a) longitudinal and (b) transverse fibre composites.

Figure 13 .

 13 Figure 13. Flexural modulus at 50% and 100% humidity for (a) longitudinal and (b) transverse fibre composites.

Figure 14 .

 14 Figure 14. Apparent porosity for longitudinal and transverse fibre composites (3PB samples) under 100% humidity.

  i.The main factors (bending type, fibre direction and ageing time), as well as their interactions, affect all the responses related to the 50% humidity composites, as evidenced by ANOVA;ii. Four-point bending test samples exhibit higher water absorption and flexural properties in the transient sorption regime, compared to three-point bending, being attributed to the dimensional effect of the sample; iii. Longitudinal flax laminates lead to lower water absorption in the transient sorption regime, compared to the transverse ones, due to their lower cross-sectional exposure to moisture. The samples with [0 o ]12 architecture also show improved flexural properties, as the bending behaviour is strongly dominated by the tensile stress under the neutral line; iv. The water absorption increases progressively with ageing time, compromising the flexural properties of the composites; v. The flexural properties of the flax composites at 50% humidity for 16 days are equivalent to less than one day in a fully saturated environment; vi. The water immersion time affects the apparent porosity of the flax composites, which increases until the fourth day, remaining constant thereafter.

Table 1 .

 1 Full Factorial Design (5 1 2 2 ) for 50% moisture saturated samples.

	E.C.	Bending Type	Fibre Orientation	Ageing Time (day)	E.C.	Bending Type	Fibre Orientation	Ageing Time (day)
	1	3 PB Longitudinal	0	11	4 PB Longitudinal	0
	2	3 PB Longitudinal	2	12	4 PB Longitudinal	2
	3	3 PB Longitudinal	4	13	4 PB Longitudinal	4
	4	3 PB Longitudinal	8	14	4 PB Longitudinal	8
	5	3 PB Longitudinal	16	15	4 PB Longitudinal	16
	6	3 PB	Transverse	0	16	4 PB	Transverse	0
	7	3 PB	Transverse	2	17	4 PB	Transverse	2
	8	3 PB	Transverse	4	18	4 PB	Transverse	4
	9	3 PB	Transverse	8	19	4 PB	Transverse	8
	10	3 PB	Transverse	16	20	4 PB	Transverse	16

Table 2 .

 2 Statistical descriptive for 50% moisture saturated samples.

	E.C.	Water Absorption (%) Replicate 1 Replicate 2	Flexural Strength (MPa) Replicate 1 Replicate 2	Flexural Modulus (GPa) Replicate 1 Replicate 2

Table 3 .

 3 Analysis of variance (ANOVA) for 50% moisture saturated samples.

	Darling normality test, showing P-values from 0.111 to 0.985. In this case, P-values ≥ 0.05
	indicate that the data follow a normal distribution.
	P-value ≤ 0.05
		Water	Flexural
	Experimental Factors	Absorption	Strength
		(%)	(MPa)

2

-adj values, ranging from 98.03% to 99.98%, indicate high predictability models, since they are close to 100%. The ANOVA is validated by the Anderson-

Table 4 .

 4 Physical and mechanical results for 100% moisture saturated samples.

	Bending Test	Fibre Orientation	Ageing Time (Days)	Water Absorption (%)	Flexural Strength (MPa)	Flexural Modulus (GPa)
	3 PB	Longitudinal	0	0.00	248.02 (±26.24)	30.19 (±1.16)
	3 PB	Longitudinal	2	5.76 (±0.86)	130.83 (±9.86)	6.98 (±0.93)
	3 PB	Longitudinal	4	8.97 (±1.00)	77.93 (±8.25)	5.67 (±0.39)
	3 PB	Longitudinal	8	9.83 (±0.95)	64.59 (±10.79)	4.60 (±1.51)
	3 PB	Transverse	0	0.00	28.64 (±2.85)	4.41 (±0.17)
	3 PB	Transverse	2	8.62 (±0.14)	10.30 (±2.14)	0.56 (±0.03)
	3 PB	Transverse	4	11.23 (±0.16)	7.53 (±0.41)	0.48 (±0.03)
	3 PB	Transverse	8	11.68 (±0.15)	5.90 (±1.57)	0.44 (±0.10)
	4 PB	Longitudinal	0	0.00	318.14 (±21.71)	30.75 (±10.71)
	4 PB	Longitudinal	2	7.48 (±0.37)	149.81 (±11.35)	9.77 (±0.56)
	4 PB	Longitudinal	4	12.96 (±0.63)	96.90 (±4.16)	6.41 (±1.11)
	4 PB	Longitudinal	8	16.05 (±2.03)	65.47 (±11.73)	5.59 (±1.42)
	4 PB	Transverse	0	0.00	34.07 (±2.68)	5.17 (±0.14)
	4 PB	Transverse	2	10.95 (±0.24)	10.01 (±1.13)	0.55 (±0.04)
	4 PB	Transverse	4	14.51 (±0.24)	9.64 (±0.67)	0.50 (±0.07)
	4 PB	Transverse	8	15.32 (±0.25)	10.87 (±1.43)	0.60 (±0.05)

Table 5 .

 5 Comparison of the results for 100% RH.

		Fibre volume fraction	Water absorption (%)	Flexural Modulus (GPa) 0 o 90 o
			0 o	90 o	0 days 4 days 0 days 4 days
	Present work (prepreg)	56%	9.83	11.68 30.19	5.67	4.41	0.48

Table 6 .

 6 Apparent porosity result for laminate composites under 100% humidity.

	Fibre	Ageing	Apparent
	Orientation	Time	Porosity (%)
	Longitudinal	30 min	3.36 (±0.36)
	Longitudinal	2 days	7.24 (±0.97)
	Longitudinal	4 days	9.05 (±1.02)
	Longitudinal	8 days	8.89 (±0.83)
	Transverse	30 min	3.02 (±0.58)
	Transverse	2 days	8.36 (±0.78)
	Transverse	4 days	13.79 (±0.28)
	Transverse	8 days	13.49 (±1.28)

Acknowledgement

This project has received funding from the Bio Based Industries Joint Undertaking under the European Union's Horizon 2020 research and innovation programme under grant agreement No 744349 (SSUCHY project). JCdS, THP and LAdO are also grateful for the support provided by the Bristol Composites Institute, and the Brazilian Research Agencies CAPES and CNPq (PQ -309885/2019-1) for the sandwich-doctoral and postdoctoral scholarships, respectively.

An increase of 42% in water absorption is observed for the 4 PB samples (Figure 2a), which is attributed to the larger volume of the 4 PB samples (2080 mm 3 ) compared to the 3PB specimens (1690 mm 3 ). The transverse orientation of the fibres also contributes to the increase water absorption by 23% (Figure 2b) due to the greater number of fibre cross sections along the length of the sample and the shorter size along the width direction. This facilitates the percolation of the water along the fibre direction, as demonstrated by the diffusion coefficients determined in the works of Cheour et al. [START_REF] Cheour | Effect of water ageing on the mechanical and damping properties of flax-fibre reinforced composite materials[END_REF]. Although flax composites show a slight absorption of water after 16 days of ageing, a significant increase (176%) is noticed when the ageing times change from 2 to 16 days (Figure 2c). This could be attributed to a greater exposure of the sample to a constant humidity environment.

Figure 3 shows the interaction effect plot for mean water absorption. Figure 3a shows that the increase in water absorption by samples with transverse fibre orientation [90 o ] is larger for the 4PB samples (69%). This is an indicator of the interaction between the size of the sample and the architecture of the stacking sequence, i.e., a larger sample size increases the number of transverse fibres. The interaction between the type of bending loading and the ageing time (Figure 3b) shows a significant increase in water absorption (242%) between levels from day 2 to day 16. In addition, an increase of 77% is observed between the samples of 3PB and 4Pb.

The interaction effect between fibre orientation and ageing time (Figure 3c) is quite similar to the previous one (Figure 3b), i.e. the sample size and the fibre direction factors similarly affect the water absorption of the flax laminates as a function of time.