Magali Jaoul-Grammare
email: jaoulgrammare@beta-cnrs.unistra.fr

Anne Stenger
email: anne.stenger-letheux@inrae.fr

Documents de travail

Keywords: Education, Environmental concerns, socio demographic characteristics JEL codes C38, I20, Q53, Q54

In the Unesco"s report (2014), an essential role is given to education in the preservation of the environment, by improving understanding on environment deterioration or modifying individuals" behaviors. Indeed, many papers analyzed the importance of education on environmental sensitivity. However, they generally focus on only one environmental concern.

The originality of our study is that it takes into account a large range of green concerns. We lead an exploratory analysis in order to try to answer to the following problematic: what are the effects of educational level and socio demographic characteristics on various green concerns? Preliminary results tend to confirm and highlight some relationships between education and environmental concerns. We especially underline that the more educated, the more open to global issues like biodiversity or climate change issues.

Introduction

In its report entitled "Sustainable development begins with education", [START_REF] Unesco | Feuille de route pour la mise en oeuvre du programme d'action global pour l'éducation en vue du développement durable[END_REF] gave a central role to education in the objectives of environmental preservation in a society, looking at pro environmental behaviors and real efforts accomplished towards environment. This report was based on the results of various international studies (PISA, global study of values; see the World Value Surveys conducted between 2005 and 2012), which showed that a higher level of education tends to increase concern regarding environmental issues. Indeed, education has direct and indirect effects on individual behaviors (Commission Bruntland, 1987;[START_REF] Unesco | Feuille de route pour la mise en oeuvre du programme d'action global pour l'éducation en vue du développement durable[END_REF][START_REF] Granon | Environnement : comment changer nos comportements ?[END_REF], but it also improves understanding of the risks linked to environmental deterioration1 . For example, educated individuals have a higher chance to better grasp the consequences of climate change and then to take the right decisions and to adapt by reducing their vulnerability.

If many studies attest the importance of the level of education in the environmental concerns, the real effects of education on pro-environmental attitudes are not always significant and may sometimes be negative [START_REF] Torgler | The determinants of individuals' attitudes towards preventing environmental damage[END_REF][START_REF] Meyer | Does education increase pro-environmental behavior? Evidence from Europe[END_REF]. Moreover, the causal effect is sometimes unclear because of omitted variables in the analysis [START_REF] Meyer | Does education increase pro-environmental behavior? Evidence from Europe[END_REF]. In Meyer"s paper (2015), many papers are identified giving a positive correlation between pro environmental behaviors in a specified context (recycling, purchase of eco labelled food, water or energy saving, and higher education. Most papers analysed the behaviour or the attitude an individual faced to only one environmental concern: green electricity, eco-food consumption, energy consumption, recycling, water savings, global warming and climate change (Ajaps & McLellan, 2015). Even if individuals are sensitive to many environmental issues, they can express more or less concerns depending on the type of environmental concern they are more confronted or sensitive to, without ignoring the other ones but being relatively less committed to change their behaviors in every issue.

One important point is to define "education to environment": is it only giving information and knowledge on environmental concerns and issues or is it more than that? According to Ajaps and McLellan (2015), to promote pro environmental behaviors and to observe desired environmental attitude, three theories have to be taken together, none being able alone the basis for PEB: learning theory, the theory of planned behaviour and the value-belief-norm theory. Another important question is linked to a lasting change in behaviour which is important to consider as the duration of education matters as strongly as the duration of changes.

Instead of focusing on only one environmental concern, this note aims at studying a large range of green concerns. Using a multiple factorial analysis, we analyse the effect of education and others demographic factors on environmental sensitivity: what are the main environmental concerns and how can we explain them taking into account that we exploit one database not specifically dedicated to that question? We tend to determine the various socio demographic profiles accordingly to different green concerns. We use a specific part of a French survey ("Generation 2013" provided by the Cereq), which is devoted to sustainable development. This note is organised as follows: in the next part, we detail the database and the methodology we used; then, we present our estimations and we discuss the preliminary results. The last part concludes.

Data and methodology

Data and descriptive statistics

Our data come from the survey "Generation 2013" provided by the Cereq (Centre d"Etudes et de Recherche sur les Qualifications), a French public establishment which depends upon the French Ministry of National Education, the Ministry of Economy, Industry and Employment and the Ministry of Labor, Social Relations, Family, Solidarity and Towns. It gives advices in educational policies and is an expert in the production of statistical series, at regional and national levels, as well as for quantitative research on education, insertion and employment.

Among the usual statistics produced by the Cereq, we find investigations called « Generation ». These "Generation" investigations are surveys conducted every three years on individuals who left all levels of the educational system three years before. The first investigation was "Generation 1992", conducted in 1995, and the last one, conducted in 2019, is "Generation 2016". They provide information on schooling, on professional insertion and about the socio-demographic characteristics of individuals. These surveys are devoted to the analysis of the schooling and professional insertion. They are not specifically devoted to environmental analysis. However, we use here Generation 2013, conducted in 2016 on 22,000 individuals who left education in 2013. Indeed, among these 22,000 individuals, a sample of 3533 individuals answered additional questions on sustainable development. We lead our analysis on this particular sample.

In this particular sample, individuals were especially asked to inform about their essential green concern. The question was: "Among these various (identified) problems linked to the environment deterioration, according to you, which one is the most worrying?". It was a closed-question with seven items proposed: Noise pollution, Biodiversity, Natural disaster, Households waste increase, Water pollution (lakes, rivers), Air pollution, climate change and greenhouse effect. 2We study the effect of various determinants like socio demographic factors on these seven concerns: age, gender, highest diploma obtained by an individual, environmental studies, parents" level of education, living area during the sixth grade, actual living area, professional position, to have a chronic/lasting disease, number and type of protected areas in the living region. The highest diploma indicates the highest diploma owned by an individual. We during his secondary studies), Higher environmental studies (the individual received environmental studies only during higher education) and Secondary + Higher environmental studies (the individual received environmental studies during secondary and higher education). The parents" level of education is the highest level of education reached by the father (if the information is not available, we took into account the mother"s one). We distinguish two levels: <baccalaureate and >baccalaureate. The living area during the sixth grade and the actual living area are estimated with three modalities: rural, urban area and high urban area (more than 500,000 inhabitants). Three modalities constitute the professional position: executive, worker and others professions. We also use variables giving information on the actual position of the individual. First of all, we take into account if a chronic/lasting disease affects an individual. Indeed, a sick individual may be more sensitive to environmental problem than a healthy one. Finally, we are interested in the natural context around the living area: the number and the type of protected area in the living region (National park, Marine Natural Reserve, No protected area)3 .

The analysis is lead on these 3533 individuals (25,378 with weighting) aged from 16 to 35 years. The sample is not gender balanced (28% of women). All levels of education are represented both at the individual and parents" level. Most people live in an urban area (76%) but in a lower size than during the childhood (84%). Three essential green concerns can be emphasized, grouping more than 70% of the individuals of the sample: Climate change and greenhouse effect (26%), biodiversity (24%) and water pollution (20%) (Tables 1 and2). In order to analyse the impact of personal characteristics on environmental sensitivity, we use a Multiple Factorial Analysis to draw the various individual profiles of green concerns. We proceed in three steps: in a first time, we determine three individual profiles: the socio educational profile, the socio economic profile and the geographical profile. In a second time,

we bring together each profile and the various green concerns. Finally, we group all the profiles and the green concerns.

The various individual profiles of green concern: a MFA

The Multiple Factorial analysis (MFA) is a method of data analysis for m qualitative variables which exhaustively describes the phenomenon of study. Proposed in the 1960s by Benzécri, it has become the favoured method of data analysis, especially in sociology. It is based on the fact that the existence of common occurrences can highlight, in an inductive manner, and without any hypothesis, certain structures of dependence between variables. The objective is to transit from m dimensions to 2 synthetic dimensions while minimizing the loss of information. We represent all the modalities of all the variables on the same graph (Figure 1), in order to underline the role played by all modalities. Although the graph represents the essential result, one must take into account both the amount of information contained in the data (Relative contribution) and the contribution of the various modalities to the study (Absolute Contribution). The MFA takes place in four steps:

Eigenvalue analysis: Eigenvalues represent the amount of information contained in the data. It is advisable to select a number of eigenvalues implying the smallest loss of information. We use the Kaiser criterion: this consists in selecting the eigenvalues representing up to (1/P) % of the information, P being the number of variables. The number of the selected eigenvalues also represents the number of axes used for the graphical representation: if two eigenvalues are selected, this means that two axes (called factorial axes), that is to say one system, contain the bulk of the data information.

Contributions study: There are two contributions: the absolute contribution (CTA), which represents the weight of the modality of the variable in the definition of the factorial axis, and the relative contribution (CTR), which is the quality of representation of the modality along an axis. The critical values are respectively equal to 0.1 for the CTA (the definition of the axis, that is to say the information contained in this axis is due to less than 10% of the modality), and 0.3 for the CTR (the quality of representation of the modality upon the axis is smaller than 30%). If modalities present CTA and CTR as below the critical values, they are removed from the analysis. These values are weaker than those used in Simple Factorial

Analysis because the MFA is a cautious method.

Graphical Analysis: We interpret the groups of modalities which appear on the system of axes. This part of the analysis is the most interesting, because it allows us to highlight the various profiles.

Projection of additional modalities: The analysis is carried out on a sample of the variables, called active variables, on which previous criteria are calculated (eigenvalues, contributions) and which determine the profile; the non-selected variables are called additional variables.

They allow to refine the profile and to make the interpretation deeper; they are only taken into account in the graphical analysis and their selection is based on the test-value [START_REF] Lebart | Statistique exploratoire multidimentionnelle[END_REF].

Here, we try to determine three individual profiles on sensitivity to environmental problems.

For each profile, we consider different active and additional variables (Table 3).

Socio economic

Age, Gender, Professional position, Chronic disease

Green concern

Geographical

Actual living area, Number of protected areas in the living region, Type of protected area

Green concern

The socio educational profile

The analysis leads us to consider two eigenvalues which represent 76.6% of the initial information; the transition from m to 2 dimensions implies a loss of 23.4% of information.

Within these two synthetic dimensions, the modality "secondary and higher environmental studies" is not well represented and must be removed from the analysis. The test value also implies the removal of the additional modality "Green concern = waste increase".

The horizontal axis opposes on the left the lowest diplomas and on the right the highest diplomas whereas the vertical axis opposes at the bottom education without environmental studies and at the top, environmental studies (Figure 1). They allow us to emphasize three different profiles:

-Individuals with the highest level of education and who took environmental higher studies seem to be concerned with Water pollution and Climate change and natural disaster. They usually lived in rural areas during childhood and come from high social background.

-Individuals with lowest diplomas who never took environmental studies. They generally come from a lower social background. They appear to be concerned with Air

The socio economic profile

In this analysis, two eigenvalues represent 62.4% of the initial information; it implies a loss of 37.6% of information. The test value leads us to remove from the analysis the additional modality "Green concern = waste increase".

The horizontal axis opposes on the left the youngest and the men and on the right the oldest and the women whereas the vertical axis opposes at the bottom, people in employment and at the top, people who are no employed (Figure 2). Profiles are less apparent than in the previous profile. However, we can observe that people concerned with climate change and water pollution are those with a better professional status whereas those with a lower social position are concerned with Air pollution and noise pollution.

The geographical profile

In this analysis, 74.7% of the initial information appears with two eigenvalues that is to say a loss of 25.3% of information. Within these two synthetic dimensions, the modalities "Actual living area=rural" and "Type of protected area = Marine natural reserve" are not well represented and must be removed from the analysis. The test value also implies the removal of the additional modality "Green concern = natural disaster".

The horizontal axis opposes on the left a little number of protected areas and on the right a high number of protected areas whereas the vertical axis opposes at the bottom high urban areas and at the top little urban areas (Figure 3). This analysis shows that individuals who live in high urban areas and near protected areas are more sensitive to natural disaster, noise and air pollution. These results seem somehow consistent with some existing surveys on environmental concerns in France (CREDOC 2015; OCDE 2014) : "The declared sensitivity to the environment is linked, in particular, to the fact that the French reside, for nearly half of them (49%) in a particular geographical area: i.e.near the coast (23%), either near a mountain (12%) or near a natural area that is protected (33%). This proximity to a remarkable natural area increases interest in the environmental issue" (Enquête CREDOC pour la future agence pour la biodiversité, 2015). In environmental sensitivity, the place of residence is important: the more sensitive to environmental issues are in urban areas (OECD, 2014).

Global profile

We group here in the same analysis all the variables. Only the green concern is considered as an additional variable. In this analysis three eigenvalues can be considered. So we take into account two graphical systems: the first one is constituted with the first and the second factorial axis and the second graphical analysis is formed with the first and the third factorial axis. The table 4 summarizes these two studies. It appears that the educational profile seems to be essential in the environmental sensitivity (Figures 4 and5). Individuals with the highest diplomas and who pursue environmental higher studies are more sensitive to global concerns as climate change and water pollution.

Individuals with lowest diplomas and who did not take environmental studies are more sensitive to local concerns as waste increase, noise pollution, air pollution.

The socio economic profile is quite similar to the educational profile: individuals with better professional status are those who are concerned with global issues. On the other hand, we note that the geographical profile has a significant influence with the local problematics.

Lastly, the various analysis allows us to establish a very particular profile of individuals who are sensitive to biodiversity. They usually possess agricultural baccalaureate and live more frequently in a rural area. have to be improved getting more information on detailed questions regarding environmental concerns, this first analysis confirms that the more educated, the more open to global issues but education plays together with other variables that seem as important such as having a professional position, the living area or even some health events like asthma.

  distinguish three levels: Without diploma and <baccalaureate, Baccalaureate, > baccalaureate. Environmental studies indicates if an individual received training on environmental concerns and at which moment of his training he received it: No environmental studies, Secondary environmental studies (the individual received environmental training only

  pollution and noise pollution.-A third group highlights individuals concerned with the biodiversity. They usually possess baccalaureate and pursued secondary environmental studies. This group consists of individuals who possess agricultural baccalaureate.

Figure 1 .

 1 Figure 1. Socio educational profile

Figure 2 .

 2 Figure 2. Socio economic profile

Figure 3 .

 3 Figure 3. The geographical profile

Figure 4 .

 4 Figure 4. Global profile (Axis 1 and 2)

Table 1 .

 1 Qualitative variables

	Variable			Modalities	Nb (with	%
					weighting)
	Green concern			Air pollution		11,29
				Biodiversity		24,07
			Climate change		26,45
			Natural disaster		5,82
				Noise	423	1,66
			Waste increase		10,39
			Water pollution		20,33
	Environmental Studies		No		39,20
			Higher Environmental Studies		24,42
			Secondary +higher environmental studies		4,08
			Secondary environmental Studies		32,30
	Highest diploma		= baccalaureate		28,32
			> baccalaureate	10766	42,24
			Without diploma and < baccalaureate		29,45
	Gender			Female		28,48
				Male	18231	71,52
	Living area at the sixth grade		Rural area		15,68
				Urban area	11898	48,32
			Urban Area >500000 inhabitants		36,00
	Parents' level of education	< baccalaureate	14524	63,41
			> or = baccalaureate		36,59
	Living area at the survey time		Rural		24,51
			Urban < 500 000 inhabitants		37,08
			Urban > or = 500 000 inhabitants		38,41
	Chronic/lasting desease		No	21441	86,02
				Yes		13,98
	Type of protected area		No	18722	73,77
			Marine Natural Reserve		10,03
				National Park		16,20
	Professional position		Others		8,38
				Employment	18531	72,70
			Unemployment		18,92
	Tables 2. Quantitative variable
	Statistics	Age	Nb years of	Nb years of	Nb parks in the
			education	environmental studies	living region
	Min	16	10	0	1
	Max	35	20	8	12
	1st Quartile	19	12	0	3
	Median	21	12	2	4
	3rd Quartile	23	15	3	6
	Mean	20.97	13.58	2.09	4.71

Table 3 .

 3 Variables used in each profile

	Profile	Active variables		Additional variables
	Socio Educational	Gender,	Highest	diploma,	Living area at the sixth grade 4
		Environmental studies, Parents'	Green concern
		level of education

Table 4 .

 4 The two systems of the global analysis

	System	Axis 1 and 2	Axis 1 and 3
	% of information	58.6%	49.4%
	Modalities	Childhood living area = rural	Childhood living area (3 modalities)
	removed	No environmental studies	Secondary + higher environmental studies
		Secondary + higher environmental studies	Chronic disease
		Chronic disease	Age [20-22[
		Age [20-22[	Professional position = others
		Professional position (3 modalities)	Actual living area (3 modalities)
		Type of protected area (3 modalities)	Number of parks (3 modalities)
			Type of protected area (3 modalities)
		Waste increase
			Noise

«Un des rôles clés de l'éducation consiste à améliorer la compréhension scientifique du changement climatique et des autres problèmes environnementaux » (Unesco,

2014, p.11).

We take these green concerns as they were given in the questionnaire. However, these categories can have some interactions as for example, natural disasters like windstorms are more and more correlated to climate change; or air pollution can impact climate change.

We don"t take into account the natural regional park because almost all regions are concerned and so, it"s not a discriminant variable.

The living area during the childhood is considered as an additional variable because if we consider it as an active one, it was not significant in the analysis and must be removed.


Xavier Colin

Jacques Verdu

AGING OF ORGANIC MATRIX COMPOSITE MATERIALS

come

INTRODUCTION

Among the wide variety of organic matrix composite (OMC) applications, very scarce are those for which durability is not a key property. This is the reason why OMC aging raised a relatively abundant literature in the past half century.

Aging can be defined as a slow (e.g., needing the recourse to accelerated aging tests) and irreversible (in use conditions) evolution of the chemical structure, morphology (in its widest sense), and/or composition of the material. It is a technically and economically important problem when evolution leads to a deleterious change of use properties. This problem can be crucial in highly demanding domains, for instance in aerospace applications, which explains why a great part of the research on OMC aging was induced and supported by the aerospace industry.

Ideally, for an user, lifetime could be defined as follows: in a given structure, used in the given conditions, the lifetime is the age at which the probability of failure reaches an arbitrarily defined value. The main objective of research on material aging is to predict the lifetime. This prediction always involves an extrapolation. Empirical models, which can be efficient in interpolations, are inadequate in extrapolations. The objective of the research is thus to build nonempirical lifetime prediction models that need the knowledge of aging mechanisms. Two categories of aging mechanism can be distinguished.

1. Mechanical aging, in which failure results only from thermomechanical processes (creep, fatigue). This case is often treated with more or less sophisticated thermodynamical tools. Prominent contributors are Refs 1-4. These phenomena, which are relevant to composite mechanics, are not reported here. 2. Physicochemical aging resulting essentially from the interaction of the material with environmental agents such as radiation, oxygen, or water. This article focuses on this category of aging process.

Among the environmental degradation modes, some are set out here. Photochemical aging affects only a very thin superficial layer of a few dozens of micrometers [START_REF] Bellenger | [END_REF] and thus concerns coatings rather than composites. Radiochemical aging can penetrate deeply in composites and modify their mechanical properties. Results on usual thermoset matrices [START_REF] Wilski | Influence of ionizing radiation on thermoset plastics[END_REF] and on industrial composites [START_REF] Van De Voorde | Selection guide to organic materials in nuclear engineering[END_REF] are available, but the scarceness of the scientific literature and the restricted range of applications led us to sacrifice this topic for the benefit of more important ones. Biochemical aging is also negligible, in common industrial composites, although it could become important if materials issued from the ''green chemistry'' would replace, in the future, those coming from the organic synthesis. Physical aging resulting from structural relaxation in glassy state leads to an increase in the matrix yield stress and a decrease in its creep compliance and ductility [START_REF] Struik | Physical aging in amorphous polymers and other materials[END_REF]. Many papers have been published on the physical aging of composite matrices, especially epoxies [START_REF] Kong | [END_REF], but there is no clear evidence on the practical importance of physical aging in composite durability. As a matter of fact, it must be independent of the atmosphere, whereas degradation is considerably faster in air than in neutral atmosphere, in most cases.

Finally, we focus on two main aging modes, which cover the great majority of industrial aging problems: ''humid aging'' resulting from the material interaction with water (in liquid or vapor state) and ''thermo-oxidative aging'' resulting from the material interaction with atmospheric oxygen. Each aging mode is the subject of a section. The first section is devoted to problems common to both aging modes.

GENERAL PROBLEMS IN COMPOSITE AGING Interface/Interphase

Common OMCs can be considered as triphasic materials, as schematized in Fig. 1.

Except in very scarce cases, reinforcing agents, for instance, glass or carbon fibers, are stable in use conditions. OMCs thus perish by their weakest organic component: sometimes matrix, sometimes interface/interphase. The role of the latter is illustrated by the example of short glass fiber-thermoplastic composites exposed to boiling water [START_REF] Theberge | -D[END_REF] in Table 1.

The fact that for these three polymers, composite samples degrade more rapidly than the corresponding unfilled samples can be explained only by the occurrence of interfacial degradation. The literature reporting experimental evidences of the interfacial degradation, in humid as well as in thermo-oxidative aging, is relatively abundant, but the transformation of these results into nonempirical kinetic models for lifetime prediction appears extremely difficult for, at least, two reasons:

1. The triphasic model of Fig. 1 is often an oversimplification. Precise investigations reveal a multilayer structure and/or the presence of structural gradients in the interphase [START_REF] Ishida | [END_REF][START_REF] Ishida | Structural gradient in the silane coupling agent layers and its influence on the mechanical and physical properties of composites[END_REF]. How to localize exactly the decisive environmental effects? How to write the corresponding kinetic equations? 2. Even in the ideal case where the interfacial mechanisms are well identified, how to establish quantitative relationships between their conversion and the mechanical properties of practical interest?

Matrix

Interphase

Reinforcing agent Here also, the problem remains partially unresolved [START_REF] Thomason | [END_REF]14] despite the considerable amount of research done in this field in the past 30 years.

From the point of view of nonempirical kinetic modeling for lifetime prediction, interface/interphase degradation remains, in the first decade of the twenty-first century, a widely open research domain.

Transport Processes

Early theoretical developments on water or oxygen transport in composites were inspired by the analogy between gas diffusion and heat, electricity or magnetism conduction in heterogeneous materials, profiting the work of prestigious pioneers such as Maxwell and Rayleigh [START_REF] Barrer | Diffusion and permeation in heterogeneous media[END_REF]. The discipline received a strong impulse in the 1970s when problems of humid aging emerged in aeronautics. It was soon recognized that water transport kinetic modeling could be the key for lifetime prediction and that diffusion anisotropy, linked to the fibers spatial arrangement in laminates, is an important characteristic of fiber composites.

Examples of values of oxygen diffusivity [START_REF] Barjasteh | Compos Part A[END_REF] determined on thin composite samples by permeation using the half-time method [17] are reported in Table 2. As expected, oxygen diffusion is largely favored in the longitudinal direction of fibers (//) than in the transverse one (⊥). The epoxy matrix is an anhydride-cross-linked epoxy. The volume fractions of glass and carbon are 64 and 69%, respectively [START_REF] Barjasteh | Compos Part A[END_REF].

Simplified homogenized models, based only on geometric considerations, were proposed to evaluate the effective water diffusivities in the different principal directions of unidirectional laminates, knowing the diffusivities in the fiber and the polymer matrix [START_REF] Tsai | Composite design[END_REF]. As an example, in the fiber direction, one can write

D // = V f D f + (1 -V f )D m , ( 1 )
where D f , D m , and D // are the respective diffusion coefficients in the fiber, the polymer matrix, and the laminate and V f is the volume fraction of fibers in the laminate. These models make full account of the fact that water diffusivity is privileged in the fiber direction than in the transverse one [START_REF] Kondo | [END_REF]:

D // = 1 -V f 1 -2 V f π D ⊥ , ( 2 )
where D // and D ⊥ are the respective diffusion coefficients in the longitudinal and transverse directions of the fibers.

Colin et al. [20] showed that such models can also be used to predict oxygen diffusivity in composites. More recently, Roy and Singh [21] showed that these models can be improved to take into account physical discontinuities such as highly permeable fiber-matrix interface or fiber-matrix debonding due to oxidative shrinkage and erosion.

Mechanodiffusive Coupling

Even when no external loads are applied during aging, a stress state develops in OMCs as a result of diffusion-induced gradients of the volumetric properties, for example, swelling in humid aging and shrinkage in oxidative aging (see corresponding sections). Stresses are expected to influence gas or vapor diffusion. The simplest approach to the problem is to start from free volume considerations. Tensile stresses create free volume:

f S = dV V = ε(1 -2ν), (3)
where f S is the free volume fraction linked to the elastic strain ε. ν is the Poisson ratio. Conversely, compression decreases free volume. It remains to relate the free volume fraction to the gas or vapor diffusion coefficient. Despite the fact that the free volume theory is essentially applicable to the rubbery state [START_REF] Kumins | Free volume and other theories[END_REF], there are extensions to the glassy state [START_REF] Benedetto | [END_REF], but this approach seems questionable to us, at least, in the case of water [24]. A more general approach based on thermodynamics has been recently proposed. The key quantity, here, is the chemical potential that depends on temperature and stress state. This would influence gas solubility [START_REF] Derrien | Interaction between stress and diffusion in polymers[END_REF] as well as diffusion [START_REF] Rambert | Compos Part A[END_REF]. According to these authors, the local gas concentration [R] would be given by the following equation:

d[R] dt = D R div(grad [R]) + β R div(grad (tr ε)), ( 4
)
where D R is the coefficient of diffusion, ε is the deformation tensor, and β R is a coupling parameter to be determined experimentally.

Coupling parameters are difficult to determine in practical cases because (i) diffusion measurements under stress are not easy and (ii) coupling parameters are generally low and stress levels cannot be very high, otherwise creep and damages complicate the interpretation of results. When the gas reacts with the polymer (hydrolysis and oxidation), there is a way to determine β R from the thickness degradation profiles (see below).

Reaction-Diffusion Coupling

Let us consider a polymer sample of thickness L that is small relative to its other dimensions and a molecular reagent Y (water or oxygen) coming from the environment and diffusing through the polymer according to Fick's law, with a diffusion coefficient D. To simplify, one will consider the case where Y is consumed by its reaction with the polymer according to a first-order process:

d[Y] dt = -K[Y]. (5)
One can define a characteristic time t D for diffusion and t R for reaction:

t D = L 2 D and t R = 1 K . ( 6
)
The physical meaning of these quantities is very simple: at times when one order of magnitude lower than the characteristic time, the phenomenon's progression is negligible, and at times when one order of magnitude higher than the characteristic time, the phenomenon is almost totally achieved. Thus, here, if t D t R , it means that diffusion is fast enough to continuously saturate the whole sample volume. The reaction will occur homogeneously at its maximum rate (Fig. 2a). If t D t R , it means that reaction occurs at an appreciable conversion before sorption equilibrium is reached. Y is consumed in the superficial layers of the sample and cannot reach the inner layers (Fig. 2b).

An example of oxidized layer in a carbon fiber/epoxy matrix unidirectional composite is reported in Fig. 3. The thickness of the oxidized layer is more in the longitudinal direction of fibers (Fig. 3b) than in the transverse one (Fig. 3a) as a result of anisotropy of oxygen diffusion (Eq. 2).

One can define critical conditions from the equality of both above characteristic times that lead to define a critical thickness L C under the given exposure conditions:

L C = 2 D K 1/2 . ( 7
)
The factor 2 comes from the fact that in the majority of cases, the sample is hydrated or oxygenated by both faces. The thickness of the oxidized layer can be defined as:

= D K 1/2 = L C 2 . ( 8
)
This very simple scaling law, which can be derived from a more general approach [28], displays very good heuristic qualities. It allows to predict accurately the order of magnitude of the thickness of the oxidized layer, the sense of its variation with temperature, oxygen pressure, polymer morphology, and so on. A first step toward generalization is to write the balance equation in the case of unidirectional diffusion and reaction in steady state:

∂C ∂t = D ∂ 2 C ∂z 2 -r(C), (9)
where r(C) is the reagent consumption rate.

Making the assumption of steady state (δC/δt = 0), one obtains a second-order differential equation of which the resolution gives the concentration profile C = f (z), from which it is possible to determine the consumption rate profile r(C) = r[f (z)] and the profile of the number of reagent moles consumed: Q = r × t = r[f (z)] × t. In the case of hydrolysis, it is generally licit to use first-order kinetics:

r(C) = KC. ( 10
)
In the simplest case of oxidation, an hyperbolic equation can be used in a first approximation: 1/β. In certain cases of hydrolysis (equilibrated hydrolysis), and in most cases of oxidation, the assumption of steady state is not valid, the reaction rate [r(C)] is a function of time, and the reagent reaction with the polymer is one elementary reaction among many others. In these cases, a new step toward generalization is to incorporate the diffusion term [D × (δ 2 C/δz 2 )] in the whole kinetic scheme and to solve the latter in time (t) and space (z) using finite difference or finite element methods. This approach was used for the first time at Arts et Metiers ParisTech (e.g., [29]). The last step toward generalization is to take into account the coupling phenomena, for instance, mechanodiffusive ones [START_REF] Olivier | Prediction of thermal oxidative aging of organic matrix composites dedicated to aeronautics[END_REF]:

r(C) = αC 1 + βC . ( 11
) Q Y 1 0 L 0 L Q Y 2 3
dC dt = -r(C) + D div(grad (C)) + β div(grad(tr ε)), ( 12
)
where r(C) is the rate of reagent consumption derived from the chemical kinetic scheme.

HUMID AGING Introduction

The term humid aging covers all the aspects of aging due to the interaction of water with the material. When the weakest element of the OMC is the matrix, it is relatively easy to distinguish between physical and chemical humid aging processes. This distinction is not so easy to make in the case where the weakest element is the fiber-matrix interfacial region. As quoted in the section titled ''Interface/Interphase,'' relatively little is known on mechanisms and kinetics of humid aging at interfaces. This is the reason why most of this section is focused on aging processes linked to matrix interaction with water. Physical water-matrix interactions fall under the category of aging (e.g., irreversible) processes when they are at the origin of the following causal chain: water sorption/desorption → swelling/deswelling → stress state → damage Here, aging appears reversible at the molecular scale (no change of chemical structure) but irreversible at the macroscopic scale.

In the case of water-matrix chemical (e.g., never totally reversible) interaction, one would rather have the following causal chain: water absorption → hydrolysis → chain scission → embrittlement or water absorption → hydrolysis → chain scission → osmotic cracking ''Pure'' physical aging can occur when the polymer is nonreactive with water, for instance, in most cases of amine-cross-linked epoxies. In contrast, pure chemical aging cannot exist because to react, water must penetrate the material, inducing thus physical aging, although this can be negligible.

An example of combining chemical and physical effects of water absorption on a glass fiber/polymer composite is shown in Fig. 4. The strength decrease in dry state is irreversible and due to matrix hydrolysis. The strength decrease in wet state is mainly due to matrix plasticization by water. Plasticization effects are reversible.

Physical Processes of Humid Aging

The most common way to study physical interactions between water and a material is to make sorptiondesorption tests in air, at controlled temperature, and at a relative hygrometry (RH). A typical curve is presented in Fig. 5. Two time domains can be clearly distinguished:

1. The transient, during which the quantity of absorbed water increases continuously. During this period, the water concentration is inhomogeneously distributed in the sample thickness. 2. The equilibrium, in which the water concentration reaches its maximum value (in the chosen exposure conditions) in all the points of the sample volume.

m ∞ m ∞ (I) 0 0 (b) (a) L (II) t z m 0 t 1 t 1 t 2 t 2 m
It is thus homogeneously distributed in an initially homogeneous matrix.

Equilibrium Properties. The equilibrium water mass uptake m ∞ , mass fraction μ ∞ , concentration in the matrix C ∞ , and concentration in the composite C ∞ are linked by the following relationships:

μ ∞ = m ∞ 1 + m ∞ , C ∞ = μ ∞ 18 (mol/g), and C ∞ = C ∞ (1 -f ), (13)
f being the fiber mass fraction in the composite. Small deviations from this equation can be observed, especially in the case of interface porosity and interphase contribution to hydrophilicity. In most cases of industrial interest, m ∞ 0.1, so that in a first approximation, μ ∞ , C ∞ , and C ∞ are proportional to m ∞ .

Effect of Relative Hygrometry (RH).

Since water is far below its critical point (T CR = 647 K), the pertinent environmental parameter is the water activity a.

For humid atmospheres, with water partial pressure p:

a = p p S0 = RH 100 , ( 14
)
p S0 being the saturated vapor pressure of water. And in aqueous solutions:

a = p S p S0 , ( 15
)
where p S is the saturated vapor pressure above the solution.

In other words, exposure to a water solution is equivalent to exposure to a humid atmosphere of the same activity. The curve C ∞ = f (a) is called sorption isotherm. In most cases, sorption isotherms can display three different types of shapes (Fig. 6).

Type A is linear in the whole activity range and corresponds to the case where sorption obeys Henry's law:

C = Ha, ( 16
)
where H depends only on temperature. In Fig. 6, polyamide 11 (PA11), polycarbonate (PC), and poly(bisphenol A sulfone) (PSU) isotherms are of type A.

Type B is also linear at low activities but displays a positive concavity at high activities. This type of isotherm can generally be represented by the sum of two terms:

C = Ha + ba m . ( 17
)
The existence of this positive curvature can be attributed to cluster formation, and the above equation can be mathematically derived to determine cluster characteristics using the Zimm-Lundberg theory [START_REF] Zimm | [END_REF]. , poly(bisphenol A sulfone) (PSU); and ♦, poly(bisphenol A carbonate) (PC). Source: experimental data from Ref. 32 except for PADEG [START_REF] Coquillat | Interaction of aliphatic polyesters with water[END_REF].

In Fig. 6, alcohol-terminated poly(ethylene adipate) (PADEG) isotherm is of type B.

Type C corresponds to type B to which has been added a negative curvature at low activities. It can be represented by the following equation:

C = αC H a 1 + βa + Ha + ba m . ( 18
)
The first term is generally attributed to Langmuir's sorption mechanism. In Fig. 6, polyamide 6 PA6 isotherm is of type C.

Despite an impressive amount of research work since the 1950s, with, for instance, extensive reviews in the 1960s [START_REF] Barrie | Water in polymers[END_REF], there is no theory, to our knowledge, to predict the type of sorption isotherm from the polymer structure. Molecular dynamics simulations are used more and more to study polymer-water interactions. In a recent paper, Marque et al. [START_REF] Marque | [END_REF] quoted more than 40 articles on this subject. One can hope that such methods will give a new impulse in this research field, but we still are, apparently, far from the above type of prediction.

Effect of Temperature. It can be observed that the equilibrium water concentration in saturated vapor (C S = C ∞ at a = 1) depends only slightly on temperature, at least in moderately polar polymers. Detailed investigations show, however, that C S tends to increase with temperature for polymers of relatively low polarity, for instance, polyesters, and tends to decrease with temperature for polymers of relatively high polarity, for instance, hydroxyl-rich epoxies. An explanation, valid for Henry's regime, but which can be extended to other sorption modes, at least in a first approximation, was proposed by Merdas et al. [37]. According to Henry's law:

C S = Sp S , ( 19
)
where S is the solubility coefficient. The temperature variation of S and p S can be represented, at least in a first approach, by the Arrhenius law:

S = S 0 exp - H S RT and p S = p S0 exp - H W RT , (20)
where H W ≈ 43 kJ/mol. So that

C S = C S0 exp - H C RT , (21)
with

C S0 = S 0 p S0 and H C = H S + H W .
In fact, water-polymer interactions (through hydrogen bonds) are highly exothermic, so that H S is negative and its absolute value is close to H W . In polymers of relatively low polarity, |H S | < H W and H C > 0, so the equilibrium water concentration is an increasing function of temperature. In polymers of relatively high polarity, |H S | > H W and H C < 0, so C S is a decreasing function of temperature.

Effect of Structure.

Many authors, for instance, [38], have tried to relate hydrophilicity to the available free volume, but such theories fail to explain why free-volume-rich substances, for instance, liquid hydrocarbons of silicone rubbers, are hydrophobic. It seems clear that hydrophilicity is essentially linked to molecular interactions between water molecules and polar groups in the polymer. In a first approach, it has been tried to assume that water equilibrium concentration is a molar additive function [START_REF] Van Krevelen | Properties of polymers: their estimation and correlation with chemical structure[END_REF], for example, that each chemical group is characterized by a molar contribution independent of its structural neighboring. Schematically, one could define three categories of groups. Polymers containing only groups of type A, for instance, polyolefins, fluorinated polymers, silicones, polystyrene, and poly(phenylene sulfide), absorb generally <0.5 wt% of water. Polymers containing only groups of types A and B absorb generally <3 wt% of water, for instance, linear and cross-linked polyesters, polycarbonate, poly(etheretherketone), polyoxymethylene, and poly(methyl methacrylate). Polymers containing groups of type C can be water soluble if these groups are of high concentration, for instance, poly(vinyl alcohol) and poly(acrylic acid). They can absorb up to 7% water in industrial composite matrices such as amine-cross-linked tetraglycidyl methylene dianiline (TGMDA) or triglycidyl derivative of p-aminophenol (TGAP) types and polyamides.

Predictions of C S from the molar additive approach can work in certain homogeneous polymer families [START_REF] Bellenger | [END_REF] provided that ''elementary'' groups are sufficiently large to take into account eventual intramolecular interactions, for instance, intrasegmental hydrogen bonds. Anyhow, these theories fail to predict hydrophilicities in large families where the polar groups' concentrations vary extensively. It has been found that the molar contribution of these groups, for instance, hydroxyls in epoxies or sulfones in polysulfones, decreases and tends to vanish when the group concentration decreases [24,41]. Gaudichet-Maurin et al. [41] proposed a theory according to which an hydrophilic site is not a single group but a pair of polar groups that must satisfy certain geometric requirements to establish a double hydrogen bond with a given water molecule. This theory explains why the group molar contribution and the heat of dissolution H S are increasing functions of the groups' concentration.

Consequences of Water Absorption.

Water absorption induces polymer plasticization, for example, glass transition and temperature depletion. This can be estimated using various physical approaches, among which the free volume theory gives generally good results. According to this theory, T g of the wet polymer is linked to the volume fraction v of water by the following relationship [42]:

T g T gP = (1 -v)α P T gP + vα W T gW (1 -v)α P + vα W , ( 22
)
where T gP and T gW are the respective glass transition temperatures of the polymer in dry state and water (T gW ≈ 130 K). α P and α W are the respective dilatation coefficients of free volume defined as:

α = α l -α g , (23)
where α l and α g are the respective dilatation coefficients in liquid (or rubbery) and glassy states.

A simplified version of this theory gives:

1 T g = 1 T gP + Av, ( 24
)
where

A = 1 T gW - 1 T gP .
One sees that the T g depletion is an increasing function of the water concentration and of T gP . A, which can be considered as a parameter of plasticization efficiency, ranges from about 5 × 10 -3 /K for T gP = 100 • C to 6 × 10 -3 /K for T gP = 327 • C. Thus the T g depletion ranges between 7 and 20 K per percent water absorbed, which corresponds well to the experimental data.

Water absorption also induces polymer swelling. A swelling ratio λ can be defined as follows:

λ = v -v P v P × 100 %, ( 25
)
where v = 1/ρ and v P = 1/ρ P are the respective specific volumes of the polymer in wet and dry states. In polymer matrices, λ is generally positive but lower than the relative mass uptake m. An example given by Belan et al. [43] is shown in Table 3. Similar observations made, for instance, on polysulfones [START_REF] Marque | Water absorption in polymers[END_REF] show that the trends illustrated by Table 3 are general. According to this author, the polymer would swell in order to keep a constant packing density ρ * defined by ρ * = van der Waals volume/total volume. Diffusion. Water diffusion in matrices obeys Fick's law in most cases, but it can sometimes display more complex behaviors, often more or less abusively ranged in the category of Langmuir's processes (Fig. 7).

In both cases, the curves m = f (t 1/2 ) are linear in their initial part. In Langmuir's process, diffusion coexists with a ''trapping-detrapping'' process at peculiar sites able to establish strong interactions with water molecules. The kinetic model of [START_REF] Carter | [END_REF] is commonly used in these cases. In amine-cross-linked epoxies, for instance, the trapping-detrapping process is simply the reversible epoxide hydrolysis:

CH CH 2 O + H 2 O CH CH 2 OH OH
In these networks, the second stage of sorption, characteristic of Langmuir's mechanism, is clearly associated to the presence of unreacted epoxide groups [24].

The relationship between polymer structure and water diffusivity is far from being totally elucidated. Various theories have been proposed [46]. Thominette et al. [START_REF] Thominette | Effect of structure on water diffusion in moderately hydrophilic polymers[END_REF] observed that in various structural series, the diffusivity is a decreasing function of hydrophilicity:

D ≈ B m ∞ (26)
where B ≈ (25 ± 5) × 10 -12 m 2 /s for amine-cross-linked epoxies at 100 shows that the diffusion rate is slowed down by the water-polymer interactions.

The diffusion coefficient D is generally independent of water activity except in the case of high hydrophilicities where plasticization or clustering effects are important. It increases with temperature:

D = D 0 exp - E D RT . ( 27
)
Typical values of apparent activation energy E D are 30-40 kJ/mol for moderately polar thermoplastics and polyester networks, 30-75 kJ/mol for epoxy networks, and 20-40 kJ/mol for polyimides. The Arrhenius parameters D 0 and E D obey a compensation law (Fig. 8).

The slope of the compensation straightline (0.30) is not very different from the one reported by van Krevelen [START_REF] Van Krevelen | Properties of polymers: their estimation and correlation with chemical structure[END_REF] for permanent gases (0.276). In contrast, the preexponential coefficient (Ln D 0 = -23.5) is noticeably different (Ln D 0 = -11.5 for permanent gases).

Swelling Stresses.

From the knowledge of the diffusion law, established from sorption measurements, it is possible to measure the local water concentration in every point at every time, C = f (x, t). In principle, volumetric measurements allow to establish the relationship between water concentration and swelling. This relationship can be used to determine the local swelling ratio λ(x, t), which can be decomposed into strain components along trirectangular coordinates. If one assumes that swelling does not affect water transport properties (i.e., the water concentration has been fixed by solving the diffusion law), then the resulting local stresses can be deduced by solving a classical thermoelastic problem [START_REF] Gurtin | On the linear theory of diffusion through an elastic solid[END_REF][START_REF] Weitsman | [END_REF][50].

But, in general, stresses affect water equilibrium concentration and diffusivity, as evidenced, for example, by Weitsman in epoxy matrix composites [START_REF] Weitsman | [END_REF].

Chemical Processes of Humid Aging

Hydrolysis Kinetic Aspects. Hydrolytic processes are especially important in two main polymer families containing ester or amide groups in the chain. In these cases, each hydrolysis event is a chain scission.

dS dt = k H C(E 0 -S) -k R (b 0 + S) 2 , ( 28
)
where

C
dS dt = - dE dt = k H E 0 C = K, (29)
K is a pseudo zero-order constant, of which some typical values are given in Table 4.

Polymers containing ester groups (linear or cross-linked polyesters, anhydride-cured epoxies, urethane-cross-linked polyesters, polycarbonate, etc.) belong to this category. 2. Equilibrium occurs at low conversions (typical case of PA11). Then, the reverse reaction cannot be neglected, and the kinetic model is somewhat more complicated [52], especially when acids are present [53].

In composites, interphase hydrolysis can occur, for instance, in the case of certain silane coupling agents [54][55][56]. The level of knowledge, in this field, remains, however, far from what would be needed to predict the lifetime from mechanical criteria.

Consequences of Hydrolysis.

Since hydrolysis is a chain scission process, it always induces embrittlement. In initially ductile linear polymers, such as PET or PA11, the toughness fails of two or three decades when the weight average molar mass reaches a critical value M F of the order of 10-20 mol/kg corresponding to a small multiple of the entanglement molar mass [57]. In the frame of second-order kinetics (rate constant k H ), it is possible to determine a lifetime value, in fact, the time to embrittlement:

t F = 2 E 0 1 M F - 1 M W0 1 k H 1 C . ( 30
)
This nonempirical quantity can be decomposed in three almost independent factors: the first one linked to the polymer structure; the second one to temperature, through Arrhenius law; and the third one to water activity, through sorption isotherm equation.

NB: The third factor can also slightly depend on temperature.

In thermosets, which are often initially brittle, chain scission induces also a decrease in fracture properties, but the structure-property relationships are not well established in this domain.

In the most economically important class of hydrolyzable thermosets, for example, unsaturated polyesters (boats, swimming pools, tanks, etc.), failure comes generally from a specific consequence of hydrolysis, osmotic cracking. In laminates, subcutaneous cracks propagate preferentially parallel to the surface, giving blisters. This phenomenon was catastrophic to the composite boat industry in the 1970s-1980s. It was soon understood why cracks propagate. As a matter of fact, they contain water in which solutes, coming from the polyester matrix, create an osmotic pressure that increases until stress concentration at crack tips induces propagation. Then, the pressure decreases until the crack development stops, but new solutes are released by the polymer, the osmotic pressure increases again, and so on [58,59]. The mechanism of crack initiation was elucidated by Gautier et al. [60]: it is due to the accumulation of small highly hydrophilic molecules (diacids and dialcohols) resulting from hydrolysis events near the end of dangling chains (which are preexisting or being formed by hydrolysis events on elastically active chains). The composite resistance to osmotic cracking would be thus linked to three factors having additive effects on the initiation rate: the initial presence of solutes linked to the polymerization catalysts, the initial presence of dangling chains (which is a decreasing function of the prepolymer molar mass), and the polyester hydrolysis rate. If, by a proper optimization of the above factors, the rate of small molecules release is lowered enough, then they can eventually disappear (at least partially) by diffusion, and the time to cracking increases considerably or, even, becomes infinite.

In the case of glass fibers/unsaturated polyester matrix composites (boat hulls, swimming pools, tanks etc.), the kinetic curve of weight changes can present the peculiar shape, revealing the presence of an osmotic cracking 100 200 300 400 500 700 800 (Fig. 9). Such a behavior can be explained as follows: at t 1 , sorption equilibrium is reached; at t 2 , cracks initiate and the sorption capacity of the sample increases; and at t 3 , the cracks coalesce and the organic molecules, resulting from the polymer hydrolysis and dissolved into the water contained in the cracks, disappear in the bath. The weight begins to decrease.

1 2 0 -1 -2 -3 t 1 t 2 t 3 m(%) t(h)
NB: Both phenomena of sorption and cracking are distinguishable because the sample thickness L is low enough to have t D < t C , where t D = L 2 /D is the characteristic time of diffusion, D is the diffusion coefficient of water in the material, and t C is the characteristic time of osmotic cracking, mainly linked to the hydrolysis rate and thickness independent [60].

THERMO-OXIDATIVE AGING

A Brief History

The modern vision of oxidation kinetics was first elaborated by Semenov (received Nobel Prize in 1956), which recognized, for the first time, the branched radical chain character of the oxidation processes in any substrate [START_REF] Semenov | Chemical kinetics and chain reactions[END_REF]. Semenov created and animated a very powerful research team at the Institute of Chemical Physics in Moscow. The very well-known names of Emanuel or Denisov, authors of several books and monographs on the subject, belong to this school. Applications of these concepts to polymers emerged essentially after the Second World War.

In the Western countries, the pioneers are British workers [START_REF] Bolland | [END_REF]. In the United States, Tobolsky and coworkers [64] had a decisive contribution to kinetic modeling. In the 1960s-1970s, there was a fast development of the discipline in the United States where Ingold, Mayo, Russel, Shelton, Uri, Reich, and Stivala, among the authors, were prominent contributors [START_REF] Lundberg | Volumes 1 and 2, Autoxidation and antioxidants[END_REF][START_REF] Reich | Autoxidation of hydrocarbons and polyolefins: kinetics and mechanisms[END_REF]. During the same period, the research remained very active in former USSR [START_REF] Emanuel | Liquid phase oxidation of hydrocarbons[END_REF][START_REF] Emanuel | Chemical physics of polymer degradation and stabilization[END_REF]. Problems of reaction-diffusion coupling emerged in the 1980s [START_REF] Furneaux | [END_REF][70][71] as mentioned in the section titled ''General Problems in Composite Aging.'' The fact that thermal aging of composite materials results from a diffusion-controlled reaction did not become apparently a research objective before the beginning of 1990s [72][73][START_REF] Cunningham | Coupled diffusion-reaction models for predicting the distribution of degradation in polymer matrix composites[END_REF]. Moreover, the results obtained by the polymer (and radical oxidation) community were totally ignored by the composites community until the turn of the century. The history of the past decade is included in the following sections.

The Standard Mechanistic Scheme (SMS)

Analytical studies of the polymer thermal oxidation can reveal a great diversity of oxidation products (several dozens in certain cases), which is usually indicative of a very complex mechanistic scheme, and thus, to a not less complex kinetic scheme. However, generally, only a small number of elementary chemical events ''dictate their rhythm'' to the whole process (Table 5).

The main characteristics of thermal oxidation (at moderate temperatures where the substrate would be stable in neutral atmosphere) are the following ones:

1. Propagation involves two elementary steps: addition of oxygen to radicals P • (II) and hydrogen abstraction to the substrate by PO 2

• radicals, which regenerates a radical P • (III). Generally, k 2 /k 3 10 6 (Table 5). 2. Initiation results from the decomposition of hydroperoxides. The reaction generates its own initiator, which explains the eventual existence of an induction period in kinetics and the autoaccelerated character of the reaction. Initiation can be unimolecular

(δ = 1, α = 2, β = 0) or bimolecular (δ = 2, α = 1, β = 1)
. It can also begin unimolecular and turn to bimolecular when the hydroperoxide concentration reaches a critical value.

Tobolsky et al. [64] elaborated kinetic models that are able to simulate the main features of polymer thermal oxidation. Unfortunately, these models were almost ignored by the community until the 1990s, when they were rediscovered [75], investigated in detail [START_REF] Audouin | Modelling of hydrocarbon polymer oxidation[END_REF], and improved [START_REF] Colin | [END_REF]. In this model, the kinetic behavior depends only on three factors: the initial hydroperoxide concentration [POOH] 0 , the initiation rate constant k 1 , and a composite rate constant r = (k 2 3 [PH] 2 )/k 6 . This model starts from the hypothesis that oxygen is in excess, which allows to neglect termination processes IV and V (Table 5). However, there is no way, in this case, to simulate reaction-diffusion coupling. A solution elaborated in the 1980s [START_REF] Furneaux | [END_REF][70][71] was available for this, but it was based on a set of questionable simplified assumptions, for instance, the constancy of initiation rate, or the existence of a relationship between termination rate constants k 2 5 = 4k 4 × k 6 . Improvements were proposed in the frame of the study of polyimide-carbon composites [78]. At this time, the expression of oxygen consumption rate remains analytical. The following expressions give the steady concentrations of the main reactive species in the case of unimolecular initiation:

[P • ] = k 3 PH k 5 1 1 + βC , [POOH] = k 2 3 PH 2 k 1 k 6 βC 1 + βC , and [PO • 2 ] = k 3 PH k 6 βC 1 + βC , (31)
where

β = (k 2 k 6 /2k 5 k 3 [PH]
) and C is the oxygen concentration.

The rate of oxygen consumption is given in steady state by:

r ox = 2 k 2 3 [PH] 2 k 6 βC 1 + βC 1 - βC 2(1 + βC) (32)
Possible experimental checks are carbonyl concentrations from Infrared (local) measurements, mass variations, and glass transition temperature T g (local) measurements. These data can be connected to the kinetic model based on the following principle: propagation does not involve, in the cases under study, any secondary processes. The only physical consequence of propagation is a weight increase due to oxygen incorporation in reaction (II) (Table 5). All the other physical consequences of oxidation result essentially from fast secondary reactions accompanying hydroperoxide decomposition. Rearrangements of alkoxy radicals are important because they can generate volatile compounds and induce chain scission (Fig. 10).

The reactions accompanying unimolecular hydroperoxide decomposition and the corresponding balance equation are: where V is a hypothetical volatile molecule, the molar mass of which would be the number average of volatile molar masses; γ V is the yield of volatile formation in the PO • rearrangement; γ 1 is the yield of PO • rearrangement among PO • reactions; and V = γ 1 × γ V is the yield of volatiles per initiation event.

POOH -→ PO • + • OH HO • + P -→ H 2 O + P • PO • + P -→ POH + P • PO • -→ P = O + P • + S + γ V V POOH -→ 2P • + H 2 O + γ 1 P = O + γ S S + v 1 V + (1 -γ 1 )POH (balance equation)
Initiation Nonradical species→ radicals I δPOOH → αP • + βPO 2 • (k 1 ) Propagation One radical → one radical II P • + O 2 → PO 2 • (k 2 ) III PO 2 • + PH → POOH + P • (k 3 ) Termination Two radicals→ nonradical species IV P • + P • → inactive products (k 4 ) V P • + PO 2 • → inactive products (k 5 ) VI PO 2 • + PO 2 • → inactive products + O 2 ( k
Water is formed only in POOH decomposition. PO • radicals can also result from bimolecular peroxyl combinations, so that, in the general case, it can be written:

d H 2 O dt = k 1 [POOH], d [ P=O ] dt = γ 1 k 1 [POOH] + γ 6 k 6 [PO • 2 ] 2 , dV dt = ν 1 k 1 [POOH] + ν 6 k 6 [PO • 2 ]
2 , and so on [START_REF] Coquillat | Interaction of aliphatic polyesters with water[END_REF] In general, however, the kinetics is not far from the steady state, which means that initiation and termination rates are close. In these cases, one can suppress the termination term (k 6 [PO 2

• ] 2 ) in the above equations and modify the coefficient γ 1 to take into account both the initiation and termination rates. The mass variation is thus the algebraic sum of two members:

dm dt = 1 ρ 32k 2 C[P • ] -(18 + ν 1 M V )k 1 [POOH (34)
The first term (32k 2 C[P • ]) is linked to the propagation rate, whereas the second term ((18 + ν 1 M V )k 1 [POOH]) is linked to the initiation rate. A peculiarity of the hydroperoxide-initiated oxidation is that the kinetic chain length (the propagation rate/initiation rate ratio) decreases continuously during the initial period of exposure to tend asymptotically toward unity. One can thus find cases where, in an initial period, [POOH] so that the mass increases, whereas, at long term, the inequality is inversed and the mass decreases. Poly(bismaleimides) display this nonmonotonous behavior, whereas epoxy networks, in which hydroperoxides are highly unstable, loose mass continuously. The ability of the model to simulate these variety of behaviors [79] can be considered as a positive argument in its favor.

32k 2 C[P • ] > (18 + v 1 M V )k 1

Structure-Stability Relationships

The polymer oxidizability depends on many factors, and the most important ones are listed in Table 6.

Concerning first chemical factors, it is clear that propagation by hydrogen abstraction (k 3 ) plays a key role. Structure-k 3 relationships are relatively well known [80]. Taking polyethylene (methylene in a polymethylenic sequence) as a reference, one can distinguish two families: Polyetheretherketone (PEEK), poly(ether sulfones), polypyromellitimide, poly(dimethyl siloxane), poly (bisphenol sulfones), poly(bisphenol A carbonate), and so on belong to the first category. Polypropylene, polydienes, polyesters, polyamides, amine-cross-linked epoxies, and so on belong to the second category.

Indeed, oxygen transport properties are very important. If its permeability is very low (polymer glasses), oxidation will be restricted to a very thin superficial layer and will not have mechanical consequences.

The initial fracture behavior is also important. In initially ductile polymers, a very small number of chain scissions are able to induce a deep embrittlement. In contrast, in initially brittle polymers, a big number of chain scission events are needed to have significant consequences on the fracture behavior.

Kinetic Control by Oxygen Diffusion. New Developments

The classical approach of the reaction-diffusion coupling involves two steps:

1. The determination of the function linking the oxygen consumption rate to the oxygen concentration r(C). 2. The resolution of Equation ( 9):

∂C ∂t = D ∂ 2 C ∂z 2 -r(C).
As previously quoted in the section titled ''Reaction-Diffusion Coupling,'' a new approach was proposed in the beginning of 2000s (e.g., [29]). It was to integrate diffusion-reaction coupling in the set of differential equations constituting the kinetic scheme. Oxygen is always involved in two elementary steps, and it is consumed in step II (Table 5):

P • + O 2 → PO • 2 (k 2 )
and restituted in step VI:

PO 2 • + PO 2 • → inactive products + O 2 (k 6 )
Thus, in all cases:

∂C ∂t = D ∂ 2 C ∂z 2 -k 2 C[P • ] + k 6 [PO 2 • ] 2 (35)
Many authors, in the past, have neglected the termination term (k 6 [PO 2

• ] 2 ), but, since the kinetic chain length tends toward unity, this can carry noticeable errors.

The predictive value of this approach is clearly attested by experimental determinations of oxidation profiles [81,82]. Recent developments take into account fast diffusion in the interfacial region, role of cracks, and so on [83]. For these last authors, thermo-oxidation remains diffusion controlled as long as the material keeps its integrity. When cracks appear, they become the preferential way for oxygen penetration into deep layers. Homogenization approaches then allow to determine an effective oxygen diffusivity, including all the above-mentioned factors (interphase, cracks, etc.) [21].

Consequences of Oxidation

Shrinkage. Oxidation leads practically always to density increase and volatile evolution. Density increase results from the replacement of light atoms (C, H) by heavy ones (O). For a constitutive repeat unit, an ''average atomic mass'' M A can be defined as:

M A = Molar mass Number of atoms . ( 36
)
M A can be derived from the kinetic scheme, making simple assumptions about the nature of oxidation products [27]. The density can be calculated from an empirical relationship [START_REF] Pascault | Thermosetting polymers[END_REF]:

ρ = 350 + 120M A , ( 37
)
where ρ is expressed in kilograms per cubic millimeters and M A in grams per moles. The relative volume change is then given by:

ν ν = m m - ρ ρ . ( 38
)
ρ/ρ is always positive. m/m can be positive in the early period of exposure, but becomes negative at long term. Thus, oxidation generates shrinkage that induces a stress state, which, in turn, can induce damage, for instance, matrix-fiber debonding [START_REF] Rouquier | [END_REF].

Change of Glass Transition Temperature T g . In scarce cases of unsaturated matrices, for instance, epoxies cured by MTHPA (methyltetrahydrophthalic anhydride), radical addition to double bonds induces predominant cross-linking [86]. In most cases, however, chain scission predominates largely over cross-linking owing to the importance of PO • β scissions (The Standard Mechanistic Scheme).

Cross-linking and chain scission events can be associated to an elementary step of the mechanistic scheme, so that their respective numbers X and S can be calculated by the kinetic model. In linear polymers, the number (M n ) and weight (M W ) of the average molar masses are linked to S and X by the following equations [87]:

1 M n - 1 M n0 = S -X, ( 39
) 1 M W - 1 M W0 = S 2 -2X. ( 40
)
When the chain scission predominates (S > 4X), both molar masses decrease, and the glass transition temperature decreases. According to the Fox-Flory law [88], if cross-linking is negligible:

T g = T g∞ - K FF M n , (41)
so that

T g0 -T g = K FF S, (42)
where T g0 and T g are the respective glass transition temperatures before and after degradation and T g∞ and K FF are material constants. According to Bicerano [START_REF] Bicerano | Prediction of polymer properties[END_REF]:

K FF = 2.715 × 10 -3 T 3 g∞ . ( 43
)
When cross-linking largely predominates, gelation occurs at:

X = 1 2M W0 . ( 44
)
T g increases with X, but the slope dT g /dX is considerably lower than the absolute value of dT g /dS.

The relationship between the cross-linking density, the number of chain scissions, and/or cross-linking events can be relatively complex in nonideal networks (e.g., networks with dangling chains). In almost ideal networks, in the case of a predominating chain scission, one can write [START_REF] Pascault | Thermosetting polymers[END_REF]:

ν = ν 0 -ψS, ( 45
)
where ν 0 and ν are the concentrations of elastically active chains before and after aging, respectively. ψ = 3 for networks with trifunctional cross-links (amine-cross-linked epoxies), and ψ = 1 for all the other networks. When cross-linking largely predominates:

ν = ν 0 + 2X, in all the networks. ( 46
)
T g is an increasing function of ν. There are many relationships in the literature, the better one being, to our opinion [START_REF] Bellenger | [END_REF], Di Marzio's equation [91]:

T g = T gl 1 -2Fν . ( 47
)
T gl and F being characteristics of the network chains (essentially their dynamic stiffness). Thus, here also, chain scission leads to a decrease in T g , and cross-linking leads to an increase in T g . One can remark that:

T g dν = 2 FT 2 g T gl . ( 48
)
In other words, the sensitivity of T g to a change in cross-link density is an increasing function of T g .

Change of Elastic Properties. Elastic properties depend essentially on two factors: the cohesive energy density and the local mobility (responsible for subglass transitions at temperatures lower than the temperature at which elastic properties are measured). It can be written, schematically, in glassy state:

K = K 0 1 -α K T T g (bulk modulus), (49)
E = E 0 1 -α E T T g - i E i (Young's modulus), (50)
G = G 0 1 -α G T T g - i G i (shear modulus), (51)
μ = 0.5 -α μ 1 - T T g + i μ i (Poisson's ratio), (52)
where α K , α E , α G , and α μ are coefficients of the order of 0.3-0.5. E i , G i , and μ i are the respective gaps of Young's modulus, the shear modulus, and Poisson's ratio at the ith secondary transition below the measurement temperature T. The bulk modulus is not affected by the secondary transitions. K 0 , E 0 , G 0 , and μ 0 depend only on the cohesive energy density. E i , G i , and μ i are interrelated and depend on the activity of secondary transitions.

In linear polymers, it seems that a deep embrittlement occurs far before these properties have significantly changed. In certain networks displaying weak secondary transitions, such as styrene-cross-linked polyesters or vinyl esters, no significant changes of elastic properties (in a reasonable range of conversions) are expected. In networks displaying secondary transitions of noticeable activity, such as amine-cross-linked epoxies, the changes resulting from chain scissions are illustrated by Fig. 11.

In these networks, chain scission induces a decrease in the activity of β transition and thus an increase in the shear or Young's modulus and the concomitant decrease of Poisson's ratio, with the bulk modulus being not affected. Rasoldier et al. [92] have shown that, in the case of amine-cross-linked epoxies, this ''antiplasticization'' phenomenon is also displayed in virgin networks having an epoxide excess. These networks can thus be used as model networks. In polymerization of monomer reactants (PMR-15), the modulus increases also in the oxidized layer based, presumably, on the same mechanism [93].

There is no physical theory, to our knowledge, to establish a relationship between the network structure and elastic properties. This step of the lifetime prediction approach will remain empirical for a certain time.

Change in Fracture Properties. The trends described in the case of hydrolysis (see the section titled ''Consequences of Hydrolysis'') remain valid here. In the case of networks, it is observed that both cross-linking and chain scission lead to embrittlement for distinct reasons: cross-linking presumably shifts ductile-brittle transition toward high temperatures, and chain scission reduces the number of covalent bonds per volume unit, which presumably affects the brittle fracture behavior. These research domains are far from being systematically explored and would need noticeable research efforts.

An example of changes in fracture properties during thermal aging is reported in Fig. 12. Anhydride-cured epoxy matrix and its corresponding composites filled with 60 wt% of silica or 50 wt% of trihydrated alumina were exposed to air at various temperatures ranging from 140 to 200 • C, and their fracture properties were determined in flexural mode. The curves are rupture envelopes, that is, stress at break σ R against ultimate deflection ε R . In three cases, they coincide with initial stress-deflection curves. Here, oxidative aging does not modify the sample's mechanical behavior but creates superficial defects making the rupture occur sooner and sooner [94].

NB: The matrix is initially semiductile, and the maximum on the curves corresponds to the yield.

CONCLUSIONS

In common use conditions, OMCs perish mainly because of humid or thermal oxidative aging. Humid aging can be physical or chemical. Physical aging results mainly from damage induced by swelling stresses. There are now sophisticated tools for the mechanical analysis of this process, but some questions, for instance, relative to the effect of structure on water diffusivity and swelling ratio, remain largely open. Swelling mechanisms are, to our opinion, far from being fully understood. Chemical aging results from hydrolysis that includes osmotic cracking, especially in unsaturated polyester composites. Here, mechanisms and kinetics are almost fully elucidated in the case of matrix hydrolysis. However, interface/interphase can play a key role. We dispose of a great number of experimental observations in this field, but they are difficult to translate into nonempirical degradation kinetics laws. This domain will need noticeable research efforts in the future to reach the level required by kinetic modeling and lifetime prediction.

The thermal oxidative aging domain suffered a long stagnation period, partially because of the lack of communication between the polymer and composite communities. Joining these communities and introduction of numerical methods in the resolution of kinetic schemes recently created a strong impulse. Models having good predictive properties are now available, but they can still be considerably improved, especially in the following directions: (i) measurement of oxygen transport properties in composites, (ii) relationships between chemical changes and use properties (especially fracture properties), and (iii) role of interface/interphase.

In both humid and thermal oxidative aging domains, stress effects on diffusion and chemical reaction, which have been almost totally ignored in the past, will need also to be systematically investigated.

In the future, the turn of the century will probably appear as a transition period during which empiricism has been progressively eradicated and replaced fully by scientific approach of lifetime prediction for OMCs.

Figure 1 .

 1 Figure 1. Schematization of the triphasic structure of composites.

Figure 2 .

 2 Figure 2. Schematic shape of reaction product thickness profile in the absence of kinetic control by diffusion (a) and in the case of diffusion-controlled reaction (b). The numbers on the figures indicate increasing exposure times.

Figure 3 .

 3 Figure 3. Oxidized layer in a carbon fibers/epoxy matrix composite having 65 vol% of fibers (a) parallel and (b) perpendicular to the free surface after 200 h of exposure to air at 200 • C [27]. Examination of polished cross sections by optical microscopy using an interferential contrast. The epoxy matrix results from the reaction of a mixture of triglycidyl derivative of p-aminophenol (TGAP) and diglycidyl ether of bisphenol F (DGEBF) by diaminodiphenylsulfone (DDS) and is filled by about 30 wt% of thermoplastic polysulfone (PES). It reduces to first order r(C) = C at low concentration values, C 1/β, and to zero order r(C) = α/β at high concentration values, C1/β. In certain cases of hydrolysis (equilibrated hydrolysis), and in most cases of oxidation, the assumption of steady state is not valid, the reaction rate [r(C)] is a function of time, and the reagent reaction with the polymer is one elementary reaction among many others. In these cases, a new step toward generalization is to incorporate the diffusion term [D × (δ 2 C/δz 2 )] in the whole kinetic scheme and to solve the latter in time (t) and space (z) using finite difference or finite element methods. This approach was used for the first time at Arts et Metiers ParisTech (e.g.,[29]). The last step toward generalization is to take into account the coupling phenomena, for instance, mechanodiffusive ones[START_REF] Olivier | Prediction of thermal oxidative aging of organic matrix composites dedicated to aeronautics[END_REF]:

1 Figure 4 .

 14 Figure 4. Hydrolytic aging of an E-glass fibers/polyester laminate after immersion in water at 60 • C. The polyester is a fumarate/isophthalate of propylene glycol cross-linked by 45 wt% styrene. The weight fraction of glass is 50%. , shear strength

Figure 5 .

 5 Figure 5. (a) Typical shape of a sorption curve. (b) Concentration thickness profile at two different times of exposure. m, mass uptake; and m ∞ , equilibrium mass uptake.

Figure 6 .

 6 Figure 6. Sorption isotherms at 50 • C of several polymers: , polyamide 6 (PA6); , alcohol-terminated poly(ethylene adipate) (PADEG); •, polyethersulfone (PES); , polyamide 11 (PA11);

Figure 7 .

 7 Figure 7. Shape of sorption curves in the case of (a) Fick's and (b) Langmuir's diffusion processes.

Figure 8 .

 8 Figure 8. Logarithm of the preexponential coefficient D 0 against apparent activation energy E D of water diffusion for several polymers including thermoplastics and thermosets (polyesters and epoxies).

  -O-(polyesters) or -NH-(polyamides). Taking [-CO -X-] = E, [-COOH] = [HX-] = b, and S = number of chain scissions (mole per mass unit), one can write:

Figure 9 .

 9 Figure 9. Osmotic cracking in a polyester for boat hulls, as revealed by the kinetic curve of weight changes after immersion in boiling water. The resin is made of maleate/isophthalate (46/54) of propylene glycol cross-linked by 45 wt% styrene. The samples are sheets of 3-mm thickness. Source: Ref. 61.

Figure 10 .

 10 Sequence of reactions leading to chain scission by alkoxy radical rearrangement (β scission).

TEFigure 11 .

 11 Figure 11. Shape of dynamic mechanical analysis (DMA) curves for a network before (full line) and after degradation (dashed line). The horizontal arrow indicates the decrease of T g resulting from a predominating chain scission. The vertical arrow indicates the increase of modulus on the glassy plateau above T β . The oblique arrow indicates the change in amplitude and temperature of the β transition.

Figure 12 .

 12 Figure 12. Rupture envelopes for an anhydride-cured epoxy matrix ( , M) and its corresponding composites filled with 60 wt% of silica (•, Si) or 50 wt% of trihydrated alumina ( , Al) after thermal aging in air at various temperatures ranging from 140 to 200 • C. Fracture properties were determined in flexural mode. The arrows indicate the sense of aging effect. Source: Ref. 94.

Table 1 . Strength Retention (Percent of the Initial Value) After Immersion in Boiling Water

 1

	Polymer	Glass Fiber (%) Strength Retention (%)
			After 100 h After 1000 h
	Polycarbonate (PC)	30	51	28
	Polycarbonate (PC)	0	100+ a	28
	Polyoxymethylene	30	71	57
	(POM)
	Polyoxymethylene	0	100+ a	98
	(POM)
	Modified Polyphenylene	30	84	65
	oxide (PPO)
	Modified Polyphenylene	0	100+ a	100
	oxide (PPO)

Source: Adapted from Ref.

[START_REF] Theberge | -D[END_REF]

. a Strength increase after exposure.

Table 2 . Values of Oxygen Diffusivity for Glass and Carbon Fibers/Epoxy Matrix in the Longitudinal (D // ) and Transverse Directions (D ⊥ ) of the Fibers

 2

	Temperature	D for Glass/	D for Carbon/
	( • C)	Epoxy (m 2 /s)	Epoxy (m 2 /s 1 )
		D // ×10 -12 D ⊥ × 10 -12 D // ×10 -12 D ⊥ × 10 -12
	180	3.75	0.70	2.59	0.70
	200	5.36	1.10	3.70	1.10

  × 10 -12 m 2 /s for aromatic polysulfones at 50 • C, B ≈ (2.5 ± 1.0) × 10 -12 m 2 /s for styrene-cross-linked polyesters at 50 • C, and B ≈ 2 × 10 -12 m 2 /s for polyethylenes differing by the concentration of oxygen containing groups at 50 • C. This dependence

• C, B ≈ (18 ± 4)

Table 3 . Equilibrium Water Sorption Characteristics at 30 • C and 100% RH for Styrene-Cross-Linked Poly- esters [43]

 3

				Sample
		A	B	C	D
	ρ P (kg/m 3 )	1200	1184	1221	1153
	m (%)	1.55	1.52	5.00	2.80
	λ (%)	0.20	0.27	0.68	0.65
	λ/m	0.13	0.18	0.14	0.23

A and B are based on a maleate-phthalate copolymer. C and D are based on a maleate homopolymer.

  is the water concentration and k H and k R are rate constants depending only on temperature.

	Two cases can be distinguished,
	1. Equilibrium occurs at high conversions [(S ∞ /E) ≈
	1]. Since embrittlement occurs at low conversions,
	far from equilibrium, one can neglect the reverse
	reaction. Then, the rate of chain scission is well
	approximated by:

Table 4 . Approximate Value of the Pseudo Zero-Order Rate Constant of Hydrolysis for Ester-Containing Poly- mers [51]

 4

	Polymer	Temperature ( • C)	K × 10 8 (mol/L/s)	E act (kJ/mol)
	PET	99	6	107
	PET	60	0.08	107
	PC	85	0.2	75
	PC	100	0.7	75
	Unsaturated polyesters	100	20-150	70 ± 10
	Vinyl esters	100	0.2-1.0	-

Table 5 . Basic Radical Chain Mechanism Scheme and Standard Thermal Oxidation Mechanistic Scheme (For a Saturated Substrate)

 5

Table 6 . Main Factors Influencing Polymer Oxidizability

 6

	Factors			Effect on Oxidation Rate
	Substrate concentration ([PH])				+
	Initiation rate constant (k 1 )				-
	Propagation rate constant (k 3 )				++
	Termination rate constant (k 6 )				+
	Composite rate constant			+ + +
	[(k 3	2 [PH] 2 )/k 6 ]
	Oxygen solubility (S)				+
	Oxygen diffusivity (D)				+
	Presence of antioxidants				-
	• More stable polymers: Those containing no CH bonds
		(e.g., PTFE) or containing only
					or	CH 3
	• Less stable polymers: Those containing
		CH	, CH C C	,	CH O	,	CH N
		CH 3