Madina Kilardj

Ghania Ikhenazen

Tanguy Messager

Toufik

Toufik Kanit

Linear and nonlinear buckling analysis of a locally stretched plate

Keywords: Buckling, FEM, Linear and nonlinear analysis, Local stretching, Thin plate

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Linear and nonlinear buckling analysis of a locally stretched plate

Introduction

Thin plates are structural elements that are widely used in many engineering fields, such as aerospace, shipbuilding and civil engineering. Ribbed and stiffened ship plate, offshore, and aerospace panels are commonly utilized and always subjected to partial edge traction on their own plane. This type of loading can induce buckling, which negatively affects the function of the structural elements concerned.

Plate instability is generated globally or locally through inplane compression and shear loading. The corrugations that result from plate instability may cause permanent deformation and reduce the efficiency of the entire structure. Thin plate buckling under compression and shear was studied by Timoshenko [START_REF] Timoshenko | Theory of elastic stability[END_REF] and Bulson [START_REF] Bulson | The Stability of Flat Plates[END_REF], and analytical solutions were presented. Chen and Kim [START_REF] Chen | An equivalent plate model for corrugated-core sandwich panels[END_REF] utilized a combined equivalent energy method and proposed a stretched corrugated plate model. Numerical methods, such as finite element method, can solve particular structures (e.g., crack propagation in a bearing ring [START_REF] Deng | A study on the effect of subsurface crack propagation on rolling contact fatigue in a bearing ring[END_REF], blast loaded corroded plates [START_REF] Eslami-Majd | Blast response of corroded steel plates[END_REF] and stretched corrugated beam elements [START_REF] Cartraud | Computational homogenization of periodic beam-like structures[END_REF]) and have provided solutions to highly complex problems encountered in plate buckling, such as laminated composite plates under compression [START_REF] Ghorbanpour Arani | Buckling analysis of laminated composite rectangular plates reinforced by swcnts using analytical and finite element methods[END_REF], partially compressed plate [START_REF] Ikhenazen | Finite element analysis of linear plates buckling under in-plane patch loading[END_REF], and defective plate under shear [START_REF] Brighenti | Buckling and fracture behavior of cracked thin plates under shear loading[END_REF], uniform compression [START_REF] Narayana | Fem buckling analysis of quasi-isotropic symmetrically laminated rectangular composite plates with a square/rectangular cutout[END_REF][START_REF] Khedmati | A comparative computational investigation on the effects of randomly distributed general corrosion on the post-buckling behavior of uniaxially loaded plates[END_REF] and local compression [START_REF] Seifi | Experimental and numerical studies on buckling of cracked thin-plates under full and partial compression edge loading[END_REF].

The buckling phenomenon appears as a result of the appli-cation of compression or shear load; it occurs in less usual situations, in which tensile loading induces buckling instability and the formation of buckling configurations. Despite the fact that no compressive component exists in the applied tension, local compressive stresses resulting from the presence of a cut, crack, or hole induce stability loss. Datta and Carlson [START_REF] Datta | Buckling and vibration of a thin tensioned sheet with an elliptical hole[END_REF] presented the results of an analytical and experimental study on the buckling of a thin sheet with a hole in the tensile field. The analytical method used in stress distribution around the opening was based on the elastic solution introduced by Timoshenko and Goodier [START_REF] Timoshenko | Theory of elasticity[END_REF].

Shaw [START_REF] Shaw | Buckling behavior of a central cracked thin plate under tension[END_REF] and Brighenti [START_REF] Brighenti | Buckling of cracked thin-plates under tension or compression[END_REF][START_REF] Brighenti | Numerical buckling analysis of compressed or tensioned cracked thin plates[END_REF][START_REF] Brighenti | Buckling sensitivity analysis of cracked thin plates under membrane tension or compression loading[END_REF] utilized the finite element method to investigate the buckling of a defective uniformly tensioned plate. A few authors studied the effect of point loading. Tomita et al. [START_REF] Tomita | Onset and growth of wrinkles in thin square plates subjected to diagonal tension[END_REF], Woo and Jenkins [START_REF] Woo | Analysis of crease-wrinkle interaction for thin sheets[END_REF], and Kremer [START_REF] Kremer | Buckling of tension-loaded thin-walled composite plates with cut-outs[END_REF] examined the case of concentrated tensile load applied to defective thin plates.

The experimental studies of Gilabert et al. [START_REF] Gilabert | Buckling instability and pattern around holes or cracks in thin plates under a tensile load[END_REF] and Seifi [START_REF] Seifi | Lateral load effects on buckling of cracked plates under tensile loading[END_REF] also revealed that local buckling occurs around defects on a stretched plate. The waves of the buckling undulation are perpendicular to the applied tensile stress.

These studies have shown that perfectly flat plates subjected to tensile load may buckle locally in the presence of defects. However, buckling occurs even in the absence of any geometric discontinuity. Such a phenomenon was well described by Bryan [START_REF] Bryan | On the stability of a plane plate under thrusts in its own plane, with applications to the "buckling" of the sides of a ship[END_REF], who compared a tensioned plate in one direction to a stretched sheet of paper wet in the middle. Moisture, which illustrates paper elasticity rather than weakness, causes the surface of the paper to expend and wrinkle. The wrinkles become fine and close as the tensile forces increase. This issue was discussed by Bryan [START_REF] Bryan | On the stability of a plane plate under thrusts in its own plane, with applications to the "buckling" of the sides of a ship[END_REF] through an analytic method. Friedl et al. [START_REF] Friedl | Buckling of stretched strips[END_REF] presented a numerical solution to this problem for special boundary conditions. However, only a few studies have been conducted on partially tensioned plates that exhibit no geometric discontinuity or imperfection. Moreover, only a few authors have addressed the problem of nonlinear buckling of thin stretched plates; even when an analysis on such plates had been conducted, the partially tensioned plate was not considered.

In the present study, the linear and nonlinear buckling of a thin, partially stretched, non-defective plate at its one end in the lateral direction, as depicted in Fig. 1, was investigated with the Finite element method (FEM). The aim of this study is to provide representative linear and nonlinear buckling load results for a simply supported plate. The effect of applied load length and plate length ( / l b ) ratio on the critical buckling load was investigated. The obtained numerical results in both linear and nonlinear fields were graphically summarized, and several interesting conclusions were obtained. To the authors' best knowledge, no experimental study is currently available to validate the results presented in this numerical investigation.

Stability problem formulation

Analytical estimations should be performed prior to numerical buckling analyses. However, no analytical solution exists for the present extremely complex buckling problem. The governing differential equation for the buckled plate is [START_REF] Timoshenko | Theory of elastic stability[END_REF]

ae ö ¶ ¶ ¶ ¶ + + = ç ÷ ¶ ¶ ¶ ¶ ¶ è ø , ( 1
)
where w is the plate deflection and D is the plate flexural rigidity. An analogous problem with a uniformly distributed compressive load applied over the width b of a simply supported thin plate was treated analytically in a previous study and has led to the well-known expression of the compressive load critical value as follows [START_REF] Timoshenko | Theory of elastic stability[END_REF]:

2 2 x D N k b p = for / 1 a b ³ , ( 2
)
where k is the plate buckling coefficient. However, Eq. ( 2) is inapplicable to the case of patch tensile loading, and numerical methods, such as FEM, should be utilized. The linear tensile buckling problem is formulated as an eigenvalue problem leading to the equation

( ){ } { } 0 i i K S l + Y = é ù é ù ë û ë û , (3)
where The Von-Mises' stress is computed as .

K
( ) ( ) ( ) ( ) 1
(5)

Results and discussions

Description and modelling of the analyzed plate

Description of the analyzed plate

The present investigation aims to examine the buckling behavior of a mild steel plate subjected to localized lateral tension (see Fig. 1). The plate is simply supported at its short sides and free at its lateral loaded edges. Only the displacement in the z direction is prevented on the AB side (Fig. 1), whereas the prevented displacements on the opposite side, CD, are along the z and y axes. A middle node is also restrained in the x direction on this last edge. Aspect ratio / a b , where a and b are the width and length of the examined plate, respectively, is equal to 1 / 3 . Aspect ratio / l b , where l is the distance over which the localized load extends on the plate length, provides different load cases to be analyzed. Given the importance of the plate thickness effect on the buckling load, the / a h ratio, where h is the plate thickness, is equal to 100. Poisson's ratio ν 0.

Finite element mesh density

The plate in the numerical tests performed with ANSYS 14.5 software academic version was modeled by an eightnodded element (depicted in Fig. 2). This element with six degrees of freedom at each node is well-suitable for linear and nonlinear applications.

To obtain reasonable results, several successive linear analyses were conducted to determine the optimum mesh density to be used. Three different types of meshes were designed for the plate described in Sec. 3.1.1 to perform a convergence analysis. The grids are depicted in Fig. 3 by their quarters. The Fig. 3(a) type mesh is uniformly gridded with 50 ´150 elements. The Fig. 3(b) type mesh that contains 70 ´150 elements is refined only at the neighborhood of the loaded edge. The Fig. 3(c) type mesh comprising 100 ´300 elements is uniformly and tightly refined. The subdivisions refer to the entire plate in the x and y directions. Table 1 presents a summary of the results obtained in the conducted numerical experiments. The figures in Table 1 show that the use of a mesh particularly refined in the vicinity of the local stretching (b) or entirely and tightly gridded (c) does not cause important changes in the results. Similar results were obtained by Deolassi [START_REF] Deolasi | Buckling and vibration of rectangular plates subjected to partial edge loading (compression or tension)[END_REF]. Consequently, the mapped discretization of type (a) depicted in Fig. 3(a) was adopted in the buckling analysis.

Linear and nonlinear plate buckling analyses

The numerical experiments show that the examined plates in their different mode shapes always buckle in the direction perpendicular to the applied tensile load. Thus, only stresses in this direction were analyzed. Several quantitative results were recorded for the plate described in Sec. ; these results are presented in Table 2. The finite element results suggest that a limited number of mode shapes is sufficient to describe the plate buckling behavior correctly [START_REF] Ikhenazen | Finite element analysis of linear plates buckling under in-plane patch loading[END_REF]. The first 50 mode shapes were derived, and the first 5 mode shapes in the linear and nonlinear fields were recorded and are shown in Fig. 4.

Linear longitudinal stress analysis

Fig. 4 shows the iso-stress areas for the linear longitudinal stress component on the entire plate. The drawing in the figure shows that negative stresses, which are compressive stresses perpendicular to the direction of the applied tensile loading, occurred in different parts of the plate. These stresses are evidently responsible for the occurrence of plate buckling because of the Poisson's effect, which is associated with the present boundary conditions. The maximum longitudinal stresses, both compressive and tensile, are located on the plate edges where in-plane loading was directly applied. The stress value decreased toward the longitudinal center line region and eventually reached its minimum. The maximum compressive stress showed a significant value of 20% higher than the maximum tensile stress although the loading was tensile (see Table 2). However, the minimum compressive stress value was almost nine times greater than the minimum tensile stress value. Fig. 5 shows that compressive stress covers the plate surface more than tensile stress does. The compressive zone is about five times larger than the tensile zone.

Nonlinear longitudinal stress analysis

The described plate was also analyzed in the nonlinear field. Fig. 6 shows the iso-stress areas for the nonlinear longitudinal stress component on the entire plate. The iso-stress areas show that compressive stresses perpendicular to the direction of the applied tensile loading occurred but over a much smaller surface area. The obtained nonlinear values show that the compressive stresses are exclusively located on the plate edge and in its nearby neighborhood, whereas the tensile stress spreads all over the plate and reaches its maximal value in the centerline region in the applied load vicinity.

The maximum compressive stress shows a significant value of 5.5 times the maximum tensile stress. Moreover, when the maximum compressive stress value in the linear analysis is smaller than that obtained in the nonlinear analysis, the maximum tensile stress value is greater than that obtained in the nonlinear field. The maximum stress values in both analyses are less than the yielding strength. Therefore, plate instability occurred in the elastic field.

Von-Mises' stress analysis

A Von-Mises' stress analysis in both linear and nonlinear fields was performed to provide further insight into the elastoplastic behavior of the plate. The obtained results are presented in Figs. 7 and8.

Comparison of the two stress distributions reveals that in the nonlinear analysis, plasticization occurred in the area bounded by the loading application points and the width of the simply supported plate. However, in the linear analysis where the obtained failure area is much smaller, this phenomenon appeared partially in the loaded zone. Furthermore, in the linear field, failure was reached with the stress value varying between 6% and 20% of the yielding strength, whereas in the nonlinear field, failure was obtained with a stress value equal to the yielding stress. Yielding may not occur in this last plate configuration. Thus, the strength of the whole material is utilized to the best advantage.

Critical buckling load for different load cases

The previous described plate was studied for different load cases to investigate the influence of partial loading expanse on critical tensile buckling load . cr P The / l b ratio value, which is the ratio between the distance over which the tensile loading extends and the plate length, was varied from 0.033 to 1.000 with an increase of one-thirtieth in each numerical test. Fig. 9 shows critical tensile buckling load cr P versus the / l b ratio. Given that it is used in plate stability, the critical buckling load resulting from numerous localized tensile load cases may be derived from this graph for a simply supported plate with / in both linear and nonlinear fields when loading is concentrated on one unique point.

Comparison of the two curves obtained in the linear and nonlinear buckling analyses shows three common significant sections as follows.

( . Given that / l b is small in this section, critical loading cr P can be likened to concentrated load, the value of which is the resultant loading cr cr R l P = ´.

Table 3 shows several cr R derived values obtained in the linear and nonlinear analyses. The figures in Table 3 show that when / critical buckling load of about rates reached in the present investigation. Nonlinearities tend to prevent "ideal" structural elements from achieving their theoretical elastic buckling strength. Thus, the eigenvalue buckling load is somewhat overestimated. The numerical tests conducted in this study show that when the analyzed plates tended to be uniformly stretched, they developed material nonlinearities.

Conclusions

The present paper considered the buckling phenomenon of thin, simply supported, partially stretched plates. Numerical FEM analyses were performed to examine the buckled plate compressive and tensile stresses and determine the critical buckling loads for various load cases from uniform to point loading tension in linear and nonlinear domains.

The main purpose of the present study was achieved. Local tensile loading induces buckling instability in thin plates with no defects. Owing to the Poisson's effect, which is associated with the present boundary conditions, compressive stresses are obviously responsible for the occurrence of flat plate buckling.

In determining the plate buckling shapes, the direction of the buckling shape half waves is always perpendicular to that of the applied tensile load, and no buckling exists under uniform tension.

The linear and nonlinear numerical analyses showed that plate instability in specific boundary conditions occurs in the elastic field. Furthermore, the Von-Mises' nonlinear analysis showed that yielding may not occur, and the strength of the entire material is used to the best advantage, which is not the case in the linear analysis.

Finally, diagrams with linear and nonlinear critical buckling loads as a function of the ratio / l b were introduced. The diagrams allow for the direct reading of the tensile critical load values for a plate with a ratio of / 1 / 3 a b = for conventional plate buckling problems. These diagrams show the same global tendency; generally, in the linear analysis, a greater cr P value is obtained, and no buckling occurs when the loading is uniform.

Further research is required to investigate the experimental buckling load of partially stretched plates so that the present numerical results can be compared with those from experimental work.

  é ù ë û : Stiffness matrix, S é ù ë û : Stress stiffness matrix, i l : ith eigenvalue used to multiply the loads that are generated S é ù ë û , and i Y : ith eigenvector of displacements. Hook's law is used to relate the strain vector { } e to the resultant stress vector { } s .

Fig. 1 .

 1 Fig. 1. Partially stretched thin plate.

Fig. 3 .

 3 Fig. 3. Meshes studied in the present convergence analysis.

Fig. 4 .Fig. 6 .Fig. 5 .

 465 Fig. 4. Buckling mode shape configurations obtained in the present investigation for a plate with / 1 / 3 a b = and / 1 / 6 l b = .

  of scale for / l b values up to 0.200 is shown in Fig.9. The continuous line represents the results of linear buckling analysis, and the dashed line represents those of nonlinear analysis.Fig.9shows that in the linear and nonlinear analyses, the graphs show the same global tendency, but linear analysis generally has a larger cr P value. Furthermore, in the numerical tests conducted for / 1 l b = , the plate shows no buckling when the loading is uniform. The obtained critical buckling load value is

  ) The first section is delimited by 0.033 / 0.200 l b £ £ values. The nonlinear curve reduces the overlap in the linear one up to about / 0.067. l b = The linear curve increases when 0.067 / 0.200 l b £ £

.Fig. 7 .

 7 Fig. 7. Linear Von-Mises' iso-stress areas (in MPa) on a plate with / 1 / 3 a b = and / 1 / 6 l b = .

Fig. 8 .

 8 Fig. 8. Nonlinear Von-Mises' iso-stress areas (in MPa) on a plate with / 1 / 3 a b = and / 1 / 6 l b = .

Fig. 9 .

 9 Fig. 9. Critical buckling load cr P versus / l b ratio for a plate with / 1 / 3 a b = .

Table 1 .

 1 Critical linear buckling loads obtained in the present convergence analysis using (a), (b) and (c) meshes for different loading cases.

	/ l b	cr P	é ë	/ MN ml	ù û	Error rate % é ù ë û
		(a)		(b)	(c)	(b)	(c)
	0.1667	7.7810	7.7820	7.7815	10 -2	10 -2
	0.3333	8.4427	8.4460	8.4407	4x10 -2	2x10 -2
	0.5000	8.4793	8.4783	8.4781	10 -2	10 -2
	0.6667	8.8732	8.8724	8.8740	10 -2	10 -2
	0.8333	9.9793	9.9791	9.9789	0	0
	Fig. 2. Isoparametric quadrilateral element with eight nodes.

Table 2 .

 2 Maximal and minimal buckling stress values ( c s = compressive stress value, t s = tensile stress value) in the present work.

	---	s	c max	[MPa]	s	c min	[MPa]	s	t max	[MPa]	s	t min	[MPa]	s	c max	/	s	t max	s	c min	/	s	t min
	Linear analysis	-210.6000		-38.6382		176.3140		4.3522		1.194		8.878
	Nonlinear analysis	-351.9490		-28.4486		63.9802		17.7658		5.501		1.601

Table 3

 3

	. Critical resultant buckling load cr R values reached in the present investigation for 0 / l b £	£	0.20	.
	/ l b	/ P MN ml cr é ë	ù û	cr R MN é ë	ù û		cr R relative increase
		Linear analysis	Nonlinear analysis	Linear analysis	Nonlinear analysis		Linear analysis	Nonlinear analysis
	0		0.41			0.41			---
	0.03		7.82			0.78			1.90 2 @	( P l b / cr	=	0	)
	0.07	6.34		6.00	1.27		1.20

  The obtained nonlinear values are always lower than the linear results and are closer to those obtained from linear analysis because the loading tends to cover the entire plate length. This tendency is illustrated in Table4, which contains the cr P

							8.14	/ MN ml , whereas nonlin-
	ear analysis results in cr P with an average value of
	4.69	/ MN ml , which is the minimum critical nonlinear buck-
	ling load.
	(3) With 0.900 / l b £	£	0.967	, both curves in the last sec-
	tion exhibit a dramatic increase in tensile buckling load with a
	small increase in the / l b ratio. For / l b =	0.967	in nonlin-
	ear analysis,	cr P	=	28.80	/ MN ml	; in linear analysis, cr P
	reaches the value of 40.13	/ MN ml .

Table 4 .

 4 Nonlinear to linear cr P rate values reached in the present investigation for 0.20 /

		£	l b	£	0.97	.
		/ P MN ml cr é ë	ù û	Nonlinear to linear cr P
	/ l b	Linear			Nonlinear	rate % é ù ë û
		analysis			analysis
	0.20	8.14				4.69	58
	0.90	12.50				7.50	60
	0.97	40.13			28.80	72


Denis Guiot

Maître De Conférences

ANTECEDENTS OF SUBJECTIVE AGE BIASES AMONG SENIOR WOMEN

From a conceptual framework based on developmental psychology and mechanisms of social comparison, this research proposes testing an explanatory model of the trend to see oneself as younger which is characteristic of seniors. The identification of antecedent variables of this tendency suggests a new approach to segmenting the feminine over fifties market in France.

INTRODUCTION

The concept of subjective age was introduced in marketing to palliate the limits of chronological, or real age as a variable for segmenting seniors' markets. A substantial body of research on people over 50 (Wolfe, 1992;Moschis, 1994) has emphasized how little pertinence chronological age has. As an alternative, research suggests studying the influence of the subjective dimension of age which complements chronological age insofar as it characterizes the way in which a person experiences his own age. Subjective age was conceptualized as a manifestation of a self-image defense and was initially defined as "the individual's selfperception in terms of reference age groups." [START_REF] Blau | Changes in Status and Age Identification[END_REF]. To this end, most studies have shown that seniors have a tendency to see themselves as 10 to 15 years younger than they actually are. Yet very few investigations have been carried out to reveal the cause of these differences. There are several reasons for the restricted amount of research in this field.

First it should be stressed that there are no formal theoretical frameworks likely to enable the identification of antecedent variables of discrepancies between the chronological age and subjective age of seniors. In addition, apart from Wilkes model (1992), most links established with subjective age are bivariable and do not always take into account the effect of chronological age on these relations. [START_REF] Henderson | Demographic Characteristics of Subjective Age[END_REF] reported that when chronological age is controlled, correlations with certain assumed demographic causes are no longer significant. Their work confirms the contradictory nature of results that have been published. The presumed causes seem to be correlates rather than real antecedents [START_REF] Barak | Subjective Age Correlates : A Research Note[END_REF]Barak, 1987;[START_REF] Barak | Inner-Ages of Middle-Aged Prime-Lifers[END_REF]. If these observations are correct, it is necessary to look deeper into the identification of antecedent variables of this characteristic in seniors to see themselves as younger than their chronological age (youthful bias). In particular, research should be orientated towards discovering psychographic determinants [START_REF] Chua | The Antecedents of Cognitive Age[END_REF][START_REF] Hubley | The Relationship of Personality Trait Variables to Subjective Age Identity in Older Adults[END_REF]. This research should help to profile the process leading to these discrepancies. Thereby, needs that are under or not satisfied are likely to be identified.

After having developed a conceptual framework based on developmental psychology and the mechanisms of social comparison, we propose testing an explanatory model of the formation of this youthful bias in seniors. Later, by using the antecedents that have been identified for this trend, we propose a new approach to segmenting the senior female population. Barak and Schiffman introduced the concept of subjective age in marketing in 1981, using hypotheses drawn from the cognitive theory of aging (Thomas, 1970). Different frameworks proposed by research led to the development of many measuring tools, probably as a consequence of the variety of approaches developed in gerontology and in developmental psychology [START_REF] Van Auken | An Assessment of the Trait Validity of Cognitive Age Measures[END_REF]. In the broadest sense, subjective age includes all cognitive and affective representations associated with age and with aging [START_REF] Gana | Concept de soi et vieillissement[END_REF].

THE CONCEPTUAL FRAMEWORK OF THE RESEARCH

Consumer behavior defines this in the narrowest sense as cognitive age, or self-perceived age, a component of self-concept. [START_REF] Barak | Cognitive Age: A Non-Chronological Age Variable[END_REF] proposed a scale for cognitive age. This scale and the gap between chronological and cognitive age, can be used interchangeably to measure the same concept [START_REF] Chua | The Antecedents of Cognitive Age[END_REF][START_REF] Hubley | The Relationship of Personality Trait Variables to Subjective Age Identity in Older Adults[END_REF][START_REF] Johnson | Cognitive Age: Understanding Consumer Alienation in the Mature Market[END_REF]. The first measurement lets us characterize the age to which individuals identify, and the second determines the direction and the size of the subjective age bias, the tendency to see oneself as either older or younger than one's chronological age [START_REF] Steitz | Subjective Age, Age Identity, and Middle-Age Adults[END_REF]McClary, 1985, 1988;[START_REF] Staats | Youthful and Older Biases as Special Cases of A Self Age Optimization Bias[END_REF]. This ambiguity has managerial and methodological repercussions: should the researcher identify the causes of cognitive age or the subjective age bias? The second approach was preferred for several reasons. Conceptually speaking, the self-discrepancy theory [START_REF] Higgins | Self-Discrepancy: A Theory Relating Self and Affect[END_REF] states that the difference between the objective self and the subjective self influence behavior more than any single element. [START_REF] Moschis | Marketing to Older Consumers[END_REF] concurs, suggesting that the gap between chronological and cognitive age influences the consumption of products destined to defend self-image because of the self-consistency motive. Practically speaking, cognitive age taken alone does not allow us to identify the presence or absence of an eventual gap between chronological and cognitive age. The correlation coefficients between these two variables, observed in female populations between 20 and 90 years of age, are between .75 and .90 [START_REF] Barak | Women's Age in Advertising: An Examination of Two Age Profiles[END_REF][START_REF] Henderson | Demographic Characteristics of Subjective Age[END_REF]. These coefficients let us suppose a lack of discriminant validity of cognitive age because of the strong relationship. One explanation is the higher proportion of individuals between 20 and 50 years of age with no age bias, for whom cognitive age is the same as chronological age. (Barak and Gould, 1985;[START_REF] Henderson | Demographic Characteristics of Subjective Age[END_REF]. This proportion seems much lower for senior females the majority of whom reveal a youthful bias [START_REF] Staats | Youthful and Older Biases as Special Cases of A Self Age Optimization Bias[END_REF]. [START_REF] Heckhausen | Developmental Expectations for the Self and Most Other People: Age Grading in Three Functions of Social Comparison[END_REF] claim that normative conceptions of human development cause the gap between chronological and subjective age. These normative conceptions are temporal references that allow an individual to evaluate his or her position in terms of physical and social development, and future prospects with respect to life cycle. They include two types of interdependent cognition: those linked to the self and those linked to others. For selfevaluation, these norms use comparison mechanisms [START_REF] Festinger | A Theory of Social Comparison Processes[END_REF] and reference groups are therefore extremely important. A considerable amount of research has shown that age constitutes an identifiable reference group. A person belongs or wants to belong to an age group, is affected by a group or sees the group as an ideal insofar as it is a means of social categorization (Montepare and Zebrowitz, 1998). Age is also the subject of labeling due to associated social roles and stereotypes [START_REF] Tepper | The Role of Labeling Processes in Elderly Consumers' Responses to Age Segmentation Cues[END_REF][START_REF] Alaphilippe | Evolution de la représentation de soi chez les personnes âgées[END_REF].

Subjective age bias concepts fall into the life-span theory of control developed by Heckhausen and Shulz (1995). Age biases are defined as the manifestation of a compensatory regulation process of self elements linked to age. These control mechanisms are particularly important at stages in life when self-image is being questioned frequently (Erikson, 1963;[START_REF] Neugarten | The Awareness of Middle Age, Middle Age and Aging, a Reactor in Social Psychology[END_REF][START_REF] Neugarten | Time, Age and the Life Cycle[END_REF]. The latter seem more frequent for women than for men according to the double-standard theory [START_REF] Sontag | The Double Standard of Aging, No Longer Young: The Older Woman in America[END_REF][START_REF] Hagestad | Age and the Life Course, Handbook of Aging and the Social Sciences[END_REF][START_REF] Hansson | Old Age: Testing the Parameters of Social Psychological Assumptions, The Social Psychology of Aging[END_REF].

These reasons led us to be more interested in senior females in this research.

Through certain characteristic, repeated events in life, they become conscious of aging which inevitably happens. Heckhausen and Shulz underlined the importance of social comparison mechanisms in the regulation process. This process results in a confrontation between an individual's questioning of his identity and his idea of others' views of normal human development. From this confrontation he sets developmental aspirations. These are feelings, expectations or beliefs about his own physical, psychological and social development.

Depending on the degree of difference between these elements and evocations associated with chronological age, a difference between chronological age and subjective age is likely to be revealed. If there is no difference in these elements, individuals tend to show no age bias.

Developmental aspirations can be considered as one of the main factors determining subjective age biases. Becoming aware of aging activates several defense mechanisms of self-image which simultaneously affect the formation of developmental aspirations. Two of them may be distinguished in seniors.

The first, manifestation of a process of social comparison, reflects the desire to maintain social roles perceived as characteristic of younger people (aspiration to be younger). By their development, individuals exceeding 50 occupy or have occupied roles that they consider enhancing. Thus they wish to remain in these roles for as long as possible. Seniors are very unlikely to have a tendency in the opposite direction (aspiration to be older) in the aim of selfimprovment. This is due to negative stereotypes associated with aging.

The second corresponds to a desire to be detached from chronological age: the feeling of remaining young. Exposed by the theory of continuity [START_REF] Atchley | A Continuity Theory of Normal Aging[END_REF][START_REF] Atchley | Continuity Theory and the Evolution of Activity in Later Adulthood, Activity and Aging[END_REF], this can be defined for seniors as the feeling of conserving the same aptitudes and physical, mental or social capacities as when the person was younger. The discrepancy between chronological age and subjective age will be bigger when this feeling is stronger. In general, the stronger the developmental aspirations the more pronounced the discrepancy between the two types of age.

We have formulated the following hypotheses: Recently published research in developmental psychology and consumer behavior suggest that when there is negative self-esteem, the defense process and the process of improved self-image are more likely to be activated (Brandtstadter, 1989). Physical factors seem to be affected first because they are visible. By their very nature, the formation of developmental aspirations should be influenced by variables of appearance.

In Western societies, cultural norms dictate the importance of being physically attractive and in particular, thin. This preoccupation evokes agility. It symbolizes a young appearance for seniors [START_REF] Barak | Women's Age in Advertising: An Examination of Two Age Profiles[END_REF]. All these factors affect self-assessment due to a stereotype of physical attractiveness. It is a process which consists of attributing favorable characteristics or types of personality to physically attractive people and unfavorable characteristics to unattractive people. The existence of this stereotype was shown by a meta-analysis based on the implicit theory of personality [START_REF] Eagly | What Is Beautiful Is Good, But …: A Meta-Analytic Review of Research on the Physical Attractiveness Stereotype[END_REF]. Most studies have shown that a positive relation exists between perceived physical attractiveness and self-esteem [START_REF] Krantz | Physical Attractiveness and Popularity: The Mediating Role of Self Perception[END_REF][START_REF] Rozin | Body Image, Attitudes to Weight and Misperceptions of Figure Preferences of the Opposite Sex: A Comparison of Men and Women in Two Generations[END_REF][START_REF] Franzoi | The Body as Object Versus the Body as Process: Gender Differences in the Experience of Body Awareness: An Experiential Sampling Study[END_REF].

Moreover, the same research theme suggests that double stereotypes of aging and physical attractiveness associate youth and beauty [START_REF] Vacker | Beauty And The Beholder : The Pursuit of Beauty Through Commodities[END_REF]. For this reason, seniors whose perceived level of attractiveness is high should have a stronger feeling of remaining youthful.

The following hypothesis may be made: [START_REF] Vacker | Beauty And The Beholder : The Pursuit of Beauty Through Commodities[END_REF]. They particularly influence persons who place more importance on physical features as these people are more inclined to improve their image in situations of social comparison [START_REF] Richins | Social Comparison and the Idealized Images of Advertising[END_REF].

When seniors are specifically targeted (e.g. anti-wrinkle creams), standards of beauty tend to be increasingly projected by older models who look young or whose apparent age is under fifty (Engel, Blackwell and Miniard, 1995).

With the aim of self-improvement, we have assumed there is a relation between the concern for physical appearance and the desire to do something to remain young amongst seniors. Another important factor stems from the nature of seniors' stage in life. Seniors are more or less preoccupied with the physical and/or social consequences of their advanced age. Those persons that are very frightened of aging, are more likely to take steps to preserve and conserve attributes that are typical of younger people (physical appearance, social roles etc.). We have called this compensatory desire "aspiration to be younger". We shall study the interactions of these variables on the formation of subjective age biases in seniors.

Self-esteem must exercise an indirect influence on seniors. People over fifty are conscious of their aging. The first physical signs of this are likely to appear as early as twenty years old but are only perceptible much later. Observing such signs may provoke self-image defense reactions. The aspiration to be younger in seniors has been defined as one of these reactions. This desire should only be activated by negative self-esteem if the individual is frightened of aging. Consequently, the link between self-esteem and the aspiration to be younger should depend on the mediating role of the fear of aging.

We have made the following hypothesis: In the above argument, the presence of a mediating variable leads us to question the relationship between, on the one hand, self-esteem and fear of aging, and on the other hand, the fear of aging and the aspiration to be younger. The second link must be stronger for persons who have the feeling they can alter their physical and/or social appearance. [START_REF] Heckhausen | Developmental Expectations for the Self and Most Other People: Age Grading in Three Functions of Social Comparison[END_REF] studied defense mechanisms of self-image in seniors. They underlined the importance of the locus of control. This is defined as the representation of instances which reinforce or inhibit the behavior of a person. Research on this personality trait distinguish and oppose external and internal instances of determination. External ones are situations, powerful others or chance situations. Internal instances are the person's willpower, competencies, desires, pleasures and the satisfaction that he feels. [START_REF] Lefcourt | Research With the Locus of Control Construct[END_REF] and [START_REF] Gana | Concept de soi et vieillissement[END_REF] reported that individuals with a representation of external control often have the impression of being subjected to events that they see as being out of their control. However, those with an internal representation of control, feel they have the power to affect their existence. We propose testing the following hypothesis:

RESEARCH METHODOLOGY

The choice of measuring tools of subjective age biases and their antecedents

Most scales used were developed in the USA. In order to adapt them to a French cultural and social context, they have been subjected to cross translations and a validation procedure.

Subjective age biases scores were obtained by computing the discrepancies expressed in years between chronological age and each of the items of the two subjective age scales: cognitive age and desired age. They were developed successively by [START_REF] Barak | Cognitive Age: A Non-Chronological Age Variable[END_REF] and by [START_REF] Barak | Ideal Age Concepts: An Exploration[END_REF]. The uni-dimensional structure of the cognitive age scale has been established on several occasions [START_REF] Wilkes | A Structural Modelling Approach to the Measurement and Meaning of Cognitive Age[END_REF][START_REF] Van Auken | An Assessment of the Trait Validity of Cognitive Age Measures[END_REF]. It was also used successfully in a French social context [START_REF] Gana | Concept de soi et vieillissement[END_REF][START_REF] Guiot | Age subjectif et segmentation des seniors[END_REF]. Adapted from this measure, the scale of desired age is used to measure ideal age-role self-concept (the age that an individual would like to be). The two scales of subjective age were administered at the end of the questionnaire.

Two scales from Self-Perception Profile by [START_REF] Harter | Manual for the Self-Perception Profile for Adults[END_REF] enabled the measurement of selfesteem and perceived physical attractiveness. They include items whose format was designed to minimize the phenomena of social desirability due to the sensitive nature of this part of the inquiry. These measures were further validated by [START_REF] Martin | Advertising and Social Comparison : Consequences for Female Preadolescents and Adolescents[END_REF]. The aspiration to be younger was put into operation by adapting a measuring scale for the tendency to social comparison proposed by these authors using Harter's format. It was also justified due to the very marginal proportion of seniors who present an older bias. Only 1.5% of a pre-tested sample of senior women (N = 128) showed a cognitive older bias (difference between chronological age and cognitive age). However, no senior female questioned showed a desired older bias, which is not surprising.

Items of the degree of feeling of staying young were developed from two sources: articles on the sense of continuity [START_REF] Mischel | Continuity and Change in Personality[END_REF][START_REF] Kaufman | The Ageless Self Sources of Meaning in Late Life[END_REF][START_REF] Gana | Concept de soi et vieillissement[END_REF] and the scale of selfconsciousness (Fenigtsein et al., 1975) from which certain questions were adapted. In total, 17 items (Likert in 7 points) were developed after a series of pretests.

Klemmack, Durand and Roff's scale (1980) was used to measure the concept of fear of aging which was defined as the collection of personal preoccupations of an individual towards the consequences of aging.

Alaphilippe and Chasseigne proposed a French version of Rotter's scale (1966) which was validated on seniors. This contained 10 items, 3 were kept after a purification phase.

Finally the scale of concern for physical appearance (Likert in 7 points) was developed from works by [START_REF] Netemeyer | Trait Aspects of Vanity : Measurement and Relevance to Consumer Behavior[END_REF].

TABLE I. -Sample Measurement Items

Really true for me

Sort of true for me

Aspiration to be younger (3 items : α = 0.72)

Sort of true for me

Really true for me

Some women do everything to remain young

BUT

Other women hardly care

Self esteem (3 items : α = 0.70) Some women like the kind of person they are

BUT

Other women often wish they were someone else

Perceived physical appearance (3 items : α = 0.71) Some women are not happy with the way they look

BUT

Other women are happy with the way they look

Implementation of the final study

As the main problem is essentially theoretical, the achievement of research goals should be facilitated by recourse to a homogenous sample and by controlling exogenous sources of variance. In order to minimize the influence of differences in regions and living standards on the formation of subjective age biases (Attias-Donfut, 1991), the respondents recruited lived in the same city. This was checked by the postal code which was given at the end of the questionnaire. To this end, a self-administered questionnaire was circulated to a convenience sample of 225 women aged between 50 to 83 years (average age = 62.73) living in Melun (Seine et Marne department). Its composition is given in detail in appendix 1.

RESULTS

Validation of constructs

It was necessary to pre-test the scales because certain concepts included here have never been used in France.

Initially, measures of cognitive age and desired age underwent a validation process. 1 . An exploratory analysis of all eight subjective age items on an initial sample (N=128) let us identify a factorial structure corresponding to the 2 assumed dimensions. The factorial analysis (oblimin rotation) let us restore 82.7% of the variance, explained with the 2 correlated factors (r=0.44) with eigenvalues higher than 1 and where each of the items strongly load their original factor (loading mean = 0.89). The reliability of cognitive and ideal age scales are satisfactory (α=0.93 and α=0.91 respectively).

Next, two variables corresponding to cognitive and desired age biases were built by calculating the difference between chronological age and each of the items of the cognitive and desired age scales. The average differences are 11.97 years for cognitive age bias and 20.16 years for the desired age bias. Then, a confirmatory model (EQS 5.7) was established for the final data 1 The item « I look like a person who is ___ years old » (LOOK AGE ) was eliminated from the statistical analysis. It does not correspond well to the concept of cognitive age which is more an inner belief [START_REF] Wilkes | A Structural Modelling Approach to the Measurement and Meaning of Cognitive Age[END_REF][START_REF] Barak | Inner-Ages of Middle-Aged Prime-Lifers[END_REF].

A confirmatory factorial analysis confirmed this decision.

(N=225). The estimation was carried out by the maximum likelihood method with robust statistics given the non-multinormality of the data (critical Mardia ratio = 41.620). The fit was satisfactory (Satorra-Bentler χ 2 =9.968; p=0.696; robust CFI=1.000; AGFI=0.950; RMSEA=0.04). Loadings varied from 0.728 to 0.922 with an average of 0.824. The factors are correlated (r = 0.51).

In a complete model of structural equations, the reliability of the two new variables obtained was estimated by calculating Joreskorg's rho (ρ=0.91 and ρ=0.93 respectively). The composite reliability coefficients of the other constructs varied between 0.75 and 0.91. The convergent and discriminant validities of the constructs were verified using the method proposed by Fornell and Larker (1981). In general, psychometric qualities of measures used proved to be enough. These first results enabled us to empirically test the integral conceptual framework.

The causal model

The hypotheses were tested using a structural equation model estimated with the help of partial least squares (PLS) estimation. This method is preferred to a classical analysis of covariance structures for several reasons. First the PLS approach is particularly well suited to the basic analysis of heterogeneous data because of the different types of variables (particularly nominals and ordinals) and the measurement formats associated with them. Collecting data relative to subjective age bias (ratio scales), self-esteem (scale using Harter format) and other constructs evaluated using the Likert scales makes the estimation difficult because of the maximum of likelihood properties. This one is based on hypotheses of multinormality and of sufficient sample sizes. Some of the variables showed a non-multinormal distribution (subjective age biases). In addition, the test of a moderating effect of the locus of control required an analysis of two groups of senior women. Those with an internal locus of control and those with an external locus of control. The samples formed were too small for a LISREL type model to be applied without biasing the estimation of the parameters. Several authors have shown that the PLS method correctly manages constraints [START_REF] Fornell | Two Structural Equation Models: LISREL and PLS applied to Consumer Exit-Voice Theory[END_REF][START_REF] Falk | A Primer for Soft Modeling[END_REF].

In order to facilitate the presentation of results, we shall begin by examining the role of the fear of aging in the relation between self-esteem and the aspiration to being younger. Then all causal links of the model are studied.

The mediating effect of the fear of aging on the relationship between self-esteem and the aspiration to be younger in seniors.

On a theoretical level, the mediating nature of the variable "fear of aging" will be established if the following three conditions are respected [START_REF] Baron | The Moderator-Mediator Variable Distinction in Social Psychological Research: Conceptual, Strategic and Statistical Considerations[END_REF]: (1) self-esteem significantly affects the fear of aging (2) self-esteem exercises a significant direct influence on the aspiration to be younger and (3), when the relationship between the fear of aging and the aspiration to be younger is controlled, the intensity of the relationship in the first condition falls. The explanatory capacity of the indirect model obtained increases significantly compared to the direct relation. Each condition was examined and subjected to a PLS model. A bootstrap replication procedure was applied using PLS-Graph software (version 2.91.03.04) which enabled the significance of the relations to be checked [START_REF] Chin | The Partial Least Squares Approach for Structural Equation Modeling, Modern Methods for Business Research[END_REF][START_REF] Guiot | Age subjectif et segmentation des seniors[END_REF]. The following table

summarizes the results.

Table II -The mediating role of the fear of aging in the relationship between self-esteem and an aspiration to be younger

(1) Effect of self-esteem on the fear of aging: R²_ = 9.8 % -0.31* self-esteem = fear of aging (t = -5.48)

(2) Direct effect of self-esteem on the aspiration to be younger: R²_ = 10.8 % -0.33* self-esteem = aspiration to be younger (t = -5.29)

(3) Indirect effect of self-esteem on the aspiration to be younger: R²_ = 15,2 % -0,26* self-esteem+ 0.21* fear of aging = aspiration to be younger (t = -3,89) (t = 2,86)

Self-esteem has an impact on the fear of aging. It also has a direct influence (t = -5.29) on the aspiration to be younger. However, this effect falls from 0.33 to 0.26 when the fear of aging is included in the model. Simultaneously, the predictive capacity of the model obtained significantly increases according to the results of an increment test (F = 11.52; p = 0.00) using the procedure described by [START_REF] Jaccard | Interaction Effects in Multiple Regression, Sage University Paper[END_REF]. Introduced into the model, the fear of aging has an indirect effect of 21.2% (0.33 -0.26 / 0.33) between self-esteem on the aspiration to be younger. It is a partial mediator in this relationship. This is stronger when individuals are very scared of getting older. If the direct link is significant, the impact on other variables should be identified that are likely to affect this relation. This leads us to examine the nature of the influence of locus of control.

Locus of control: a moderating variable of the relation between fear of aging and aspiration to be younger

We postulated there would be a stronger link in persons with internal locus of control than those with external locus of control. Unfortunately PLS-Graph software does not directly enable several groups to be analysed. However, it automatically provides correlations between latent variables. This possibility facilitates the comparison of the intensity of effects on two independent samples with the help of a difference of correlations test. In order to test the hypothesis of the moderating role of the control representation, we split the seniors' sample in two, at the median of the control representation variable. Then on each group we carried out a difference of correlation test 1 between the fear of aging and the aspiration to be younger (table III).

1 The results of this procedure are similar to those obtained by the method of multi-group analysis developed by Keil et al. (1999) and recommended by Chin. It is a technique which is compatible with PLS. ** significant at 0.01 NS non significant

Tableau III -The moderating role of locus of control

Locus

Although the coefficients are relatively low, the results show that the relation between the fear of aging and the aspiration to be younger is only significant for "internal" persons. These people have a higher feeling of control over events in their life. They are more inclined to react when faced with signs of aging.

Thus hypotheses H6 and H7 are validated.

The causal model

The hypotheses were tested using a PLS approach with a bootstrap simulation. The results of the structural model are presented in figure I.

On the whole, the relations tested were significant (table IV). The intensity of the link between perceived physical appearance and self-esteem was modest with 14.6% explained variance.

Hypothesis H3 is supported for seniors.

CONCLUSION

From the theory of social comparison, subjective age biases were seen as the effects of developmental aspirations.

The results from seniors show that the cognitive age bias seems more to result from a feeling of remaining young than an aspiration to be younger. This observation underlines the interest of the theory of continuity to explain the formation of subjective age biases in seniors. It reveals feelings of permanence and stability of individuals over 50 despite physical and social changes confronting them.

In the framework of a multivariate approach, contrasting most other studies, other determinants were identified. Concern for physical appearance and self-esteem are linked to developmental aspirations. The latter particularly affect the desired age bias. Seen as manifestations of selfimage defense mechanisms, the desired age bias is more likely to be activated when self-esteem is negative.

Moreover, relations established with elements of physical self (appearance) show their importance in the formation of youthful biases.

The causal model identified determinants of subjective age biases. Large individual differences may be revealed on these variables. Combining them enables different groups of seniors to be characterized according to the degree of cognitive and desired age biases. For the over fifties, the breakdown obtained could be used as targets for marketing anti-aging products. Women over fifty show distinct attitudes towards aging and have varied expectations in terms of consuming anti-aging products. The possibilities of market segmentation should be similar to the groups defined above. They are the starting point for marketing operations as they propose a targeting tool. Such a tool is based on the elaboration of typologies of subjective age biases prepared using their antecedent variables. These typologies may be considered as a partial response to strategic and operational imperatives of companies specialized in the marketing of cosmetic companies such as, for example Beiersdorf (Nivea Vital) (Allérès, 1986). They are also of interest to individuals in other age brackets. Indeed, according to managerial journals, anti-aging products are used at an increasingly young age. Typologies could find a wider application to include all anti-age products, not just anti-aging treatments. Such products include many personal hygiene products.

The limits of this research must also be presented. First, the existence of these antecedent variables must be verified on a more representative sample of senior French females.

Moreover, other variables must be integrated in order to fine-tune the typologies presented.

This would improve the possibilities of market segmentation. Thus a more in-depth study of antecedent variables should be carried out in order to improve the prospecting or even the loyalty of segments most susceptible to manifest subjective age biases.

  H1 : FOR SENIOR WOMEN, THE STRONGER THE ASPIRATION TO BE YOUNGER THEN THE BIGGER THE DISCREPANCY BETWEEN CHRONOLOGICAL AND SUBJECTIVE AGE. H2 : FOR SENIOR WOMEN, THE STRONGER THE FEELING OF REMAINING YOUNG THEN THE BIGGER THE DISCREPANCY BETWEEN CHRONOLOGICAL AND SUBJECTIVE AGE.

  H3 : FOR FEMALE SENIORS THE MORE THEY ARE SATISFIED WITH THEIR PHYSICAL APPEARANCE THEN THE HIGHER THEIR SELF-ESTEEM H4 : FOR FEMALE SENIORS, THE MORE THEY ARE SATISFIED WITH THEIR PHYSICAL APPEARANCE THEN THE STRONGER THEIR FEELING OF REMAINING YOUNG Several authors have shown the impact of standards of appearance in the context of social comparison. They play the role of indicators or guides of the potential of improving physical features

  H5 : FOR SENIOR WOMEN, THE GREATER THE CONCERN FOR PHYSICAL APPEARANCE THEN THE HIGHER THE ASPIRATION TO BE YOUNGER

  H6 : FOR SENIOR WOMEN, THE FEAR OF AGING IS A MEDIATING VARIABLE OF THE RELATIONBETWEEN SELF-ESTEEM AND THE ASPIRATION TO BE YOUNGER.

  H7 : THE RELATIONSHIP BETWEEN THE FEAR OF AGING AND THE ASPIRATIONS TO BE YOUNGER IS STRONGER IN SENIOR WOMEN WHO HAVE AN INTERNAL VS. EXTERNAL LOCUS OF CONTROL

˙

  Cognitive age (3 items: α = 0.93) I feel as though I were ___ years old ˙ Desired age (4 items: α = 0.91) I would like to look as though I were ___ years old LIKERT SCALES: ˙ Fear of aging (3 items : α = 0.85) I feel that people will ignore me when I am old ˙ Feeling of remaining young (3 items : α = 0.73) Even though I'm getting older, I feel as if I'm still young ˙ Concern for physical appearance (4 items : α = 0.72) The way I look is extremely important to me ˙ Locus of control (3 items : α = 0.73) I'm capable of almost everything if I really want it SCALES USING HARTER'S FORMAT:

Table IV -

 IV The estimation of parameters of the causal model by the bootstrap method: Determinants of subjective age bias in women seniors The antecedents of subjective age biases 1 chronological age as explanatory variables in a second discriminant analysis of groups obtained from the cluster analysis (lambda of Wilks = 0.665; F = 8.01; p = 0.00) Four determinants explain where individuals are classed, the clusters defined in terms of subjective age bias: the aspiration to be younger, concern for physical appearance, the locus of control and self-esteem. The correct classification percentage is 47.1 % with 4 clusters.Although quite low, the result is not insufficient. It is linked to poorer prediction in one cluster compared to the others (cluster 3, with 26.7% of individuals classed correctly). This is not due to hazard, because in this case the classification percentage was only approximately 25%. A graph showing the averages of variables of each cluster enables their profile to be determined (figure II).Clusters 3 and 4 have radically opposing profiles. With 13.3% of numbers, cluster 3 is characterized by average age seniors (in their sixties) with a high subjective age bias on both dimensions, an average desire to be younger, rather negative self-esteem and an internal locus of control. Cluster 4 is large. It accounts for 34.2% of the sample. It unites women in their fifties with low subjective age biases. These individuals have relatively low concern for physical appearance. Their self-esteem is on the positive side and they manifest only a low desire to be younger.With a third of individuals, category 1 has a profile similar to category 4. The main differences are that it has an average desired age bias and lower self-esteem.Finally, 19.1% of senior women make up cluster 2. These have an average chronological age (62.3 years), they manifest a desire to become younger and place great concern for physical appearance. Their cognitive age bias is rather high whilst their desired age bias is average.

	Results of hypothèses H2b : Aspiration to be younger Cognitive age bias REJECTED H1b : Aspiration to be younger Desired age bias VALIDATED H3 : Perceived physical appearance Self-esteem VALIDATED H5 : Concern for physical appearance Aspiration to be younger FIGURE I -FIGURE II -Profiles of women seniors obtained from hierarchical cluster analysis B1 Para-meter B2 Para-meter Standar deviation t Test sig 0.15 0.13 0.09 1.69 0.09 0.23 0.24 0.06 3.85 0.00 0.38 0.39 0.07 5.61 0.00 0.13 0.15 0.07 1.83 0.07
			REJECTED
	H6 : Self-esteem Fear of aging 2.0		-0.31	-0.31	0.07	-4.68	0.00
	Fear of aging	Aspiration
	1.5		to be younger		0.19	0.19	0.07	2.59	0.01
	Self-esteem H4 : Perceived physical appearance Aspiration to be younger VALIDATED Feeling of staying young VALIDATED H2a : Feeling of staying young Cognitive age bias VALIDATED H2b : Feeling of staying young Desired age bias REJECTED -0.5 0.0 0.5 Standardized latent variables scores 1.0	-0.27 0.36 0.29 0.02	-0.27 0.37 0.30 0.02	0.07 0.06 0.06 0.08 SES ASPI BAC	-3.65 5.92 0.00 0.00 4.83 0.00 0.03 0.98
	B1: Parameter from total sample B2 : Average parameters from bootstrap simulations -1.0				IMP BAI
	-1.5	G_1:1	G_2:2	G_3:3	G_4:4		LOCUS AGECHRO
			4 CLASSES
		SES	= Self-Esteem
		ASPI	= Aspiration au rajeunissement
		BAC	= Cognitive age bias
		BAI	= Desired age bias
		IMP	= Concern for physical appearance
		LOCUS	= Locus of control
		AGECHRO = Chronological age

This group along with cluster 3 could form the most interesting target for marketing anti-aging products or other products designed to increase youthfulness. Such products meet compensatory expectations and/or consolidate tendencies of younger self-perception. The potential of the two categories is 31.4% in our sample. This classification was carried out for descriptive and operational purposes. It validates a large part of the PLS model. The results further underline the interest of the aspiration to younger variable for segmenting and targeting prospects.

In the measurement model, figures correspond to loadings and residual variances

Aspiration to be younger in seniors is influenced by low self-esteem and fear of aging.

However the relation between concern for physical appearance and aspiration to be younger is not very significant (B = 0.13; p = 0.07).

The structure of the model founded on hypotheses H5 and H6 enables 15.0% of the variance of the aspiration to be younger to be reconstituted.

The feeling of remaining young is only linked to the cognitive age bias (B = 0.29; t = 4.83; p = 0.00). It explains 15.8% of the variance. Hypothesis H2 is therefore validated. However, the relation between this variable and the desired age bias variable is not significant.

The feeling of remaining young seems to be one of the antecedent variables of cognitive age bias. Based on the theory of continuity, it seems to be induced by perceived physical appearance (B = 0.36) with 12.6 % of explained variance (hypothesis H4). The influence of the aspiration to be younger is very low on the desired age bias (5.1 % of explained variance).

It appears that the cognitive age bias is more forceful for women over fifty.

In order to identify groups of senior women characterized by subjective age biases a cluster analysis was carryied out.

PROFILES OF SENIOR WOMEN ACCORDING TO THEIR SUBJECTIVE AGE BIASES

Individual latent variable scores taken from the PLS model were clustered using Ward's method. A dendrogramm revealed a division into 4 classes of individuals in groups of approximately the same size. An initial discriminant analysis enables individuals to be placed in the clusters determined in the preceding stage. The results (lambda of Wilks = 0.12; F = 138.00 ; p = 0.00) validates the profile carried out on active variables with a good percentage of classification of 95.1 % .

Typology analysis is important to determine group profiles of seniors from subjective age biases and their determinants. To this end, the determinants were first introduced with


X Chapeleau

P Casari

D Leduc

Y Scudeller

C Lupi

R Le Ny

C Boisrobert

Determination of strain distribution and temperature gradient profiles from phase measurements of embedded fibre Bragg gratings

Keywords:

In this paper, a powerful method for measuring non-uniform strain is reviewed and extended to the measurement of temperature gradients inside an epoxy part. The sensor is an embedded fibre Bragg grating and the method is based on the measurements of the grating phase obtained by combining optical low coherence interferometry and an inverse scattering algorithm. This method has a great potential and for the first time experimental results obtained for both strain distribution and temperature gradient are presented. By applying a tensile test, strain distribution measurements are first compared with finite element simulations to validate previous published results. Second, original temperature gradient measurements in a solid body are shown. The results are in good accordance with reference measurements achieved with thermocouples incorporated into the body.

Introduction

Since the 1990s, research in sensors based on optical fibre Bragg gratings (FBGs) has developed new sensing systems [START_REF] Rao | In-fibre Bragg grating sensor[END_REF][START_REF] Kersey | A review of recent developments in fibre optic sensor technology[END_REF][START_REF] Kersey | Fibre grating sensors[END_REF]. The main advantages of this technology are small size, light weight, high sensitivity, easily multiplexability, immunity to electromagnetic perturbation, and resistance to corrosion and fatigue. Systems based on FBG sensors have already been installed and are commercially available in large scale practical applications [START_REF] Rao | Recent progress in applications of in-fibre Bragg grating sensors[END_REF] such as bridges, mines, marine vessels, vehicles, aircraft, . . ..

An FBG is a segment of single-mode or multimode fibre in which a periodic modulation of the core refractive index is written by exposure to a spatial pattern of ultraviolet light.

According to Bragg's law, when an FBG is illuminated by a broadband source, only one wavelength (called the Bragg wavelength, λ B ) is preferentially reflected:

λ B = 2n co (1)
where n co is the effective index of the fundamental mode and the period of the index modulation. The sensing principle is related to the optical properties of the FBG. When an FBG is subjected to strain and/or temperature changes, n co and are modified and so the Bragg wavelength is shifted. Thus, for a temperature change of T , the corresponding Bragg wavelength shift is given by

λ BT = λ B (α + ξ ) T ( 2
)
where α is the coefficient of thermal expansion and ξ is the fibre thermo-optic coefficient. For an applied longitudinal strain , the Bragg wavelength shift is given by

λ B = λ B 1 - n 2 co 2 [P 12 -ν(P 11 + P 12 )] ( 3
)
where P i j are the components of the fibre-optic strain tensor and ν is Poisson's ratio. According to (2) and (3), strain or temperature changes can then be deduced from the Bragg wavelength shift measurement.

To achieve good sensor performances, it is crucial to determine with high precision the wavelength shift. A wavelength resolution of about 1 pm is required to resolve a temperature change of 0.1 • C or a strain change of 1 µ . Investigation on high resolution interrogation methods led to the development of a certain number of techniques. The most common are based on the use of either an edge filter or a tuneable filter [START_REF] Zhao | Discrimination methods and demodulation techniques for fibre Bragg grating sensors[END_REF]. An interferometric scanning method [START_REF] Kersey | High-resolution fibre-grating based strain sensor with interferometric wavelength-shift detection[END_REF] has also been reported and its high dynamic resolution demonstrated. In spite of the high precision of these wavelength shift detection techniques, they can only be used if the strain is uniform or if there is no temperature gradient along the grating. In the opposite case, the narrow band spectrum of the reflection coefficient becomes relatively broader and comprises multiple peaks. The FBG spectral shape is then more complicated to interpret so that it cannot be directly related to the strain or the temperature gradient along the grating. This problem becomes crucial for the so-called 'fibre-optic smart structures' [START_REF] Peters | Embedded optical fibre Bragg grating sensor in a nonuniform strain field: measurements and simulations[END_REF] where FBGs are embedded into a complex structure to monitor its strain or temperature distribution and to check the structural integrity. This is particularly obvious for long gratings embedded into complex structures such as textile-reinforced composites in which the manufacturing process may induce non-uniform micro-mechanical deformation.

Several studies have focused on the determination of nonuniform strain along an FBG. Two first approaches based on the reflection spectrum [START_REF] Leblanc | Distributed strain measurement based on a fibre Bragg grating and its reflection spectrum analysis[END_REF] and the phase spectrum [START_REF] Huang | Phase-based Bragg intragrating distributed strain sensor[END_REF] of the FBG were proposed near the middle of the 1990s. However, they are valid only when the strain profile is monotonic and the sign of its slope is known. To overcome these difficulties, a method [START_REF] Huang | Continuous arbitrary strain profile measurements with fibre Bragg gratings Smart[END_REF] based on the Fourier transform of the complex reflection coefficient of FBG was proposed in 1998. However, this technique also presents two drawbacks: the reflectivity level cannot be higher than 30% and the method is very sensitive to system noise. In 2000, a time-frequency signalanalysis technique [START_REF] Azaña | Reconstructing arbitrary strain distributions within fibre gratings by time-frequency signal analysis[END_REF] demonstrated its efficiency, and it is free from the limitations mentioned above and can be applied without any restrictions on the strain profile. Nevertheless, this method does not work properly for FBGs in which different zones along the length of the grating couple with the same local Bragg frequency when the coupling strength is too strong. In 2002, a new approach based on a genetic algorithm was developed [START_REF] Casagrande | From the reflected spectrum to the properties of a fibre Bragg grating: a genetic algorithm approach with application to distributed strain sensing[END_REF]. This method gives good results but its performance depends on the number of unknown parameters and on the choice of their range. Since this algorithm executes a multidirectional search that simultaneously analyses many possible solutions, the computation time is generally very long. In 2005, a new method which allows a direct reconstruction of the strain without any assumptions and a computational time of only of a few seconds was proposed [START_REF] Giaccari | On a direct determination of non-uniform internal strain fields using fibre Bragg gratings Smart[END_REF]. It is based on the measurement of the complex impulse response of FBG by optical low-coherence interferometry (LCI) and the retrieval of the local Bragg wavelength by the layer-peeling (LP) technique. Axial strain distribution measurements using uniform FBG sensors can be measured accurately using this method. However, it was never applied to temperature gradient profile measurement.

In this paper, we propose to show the great potential of this method while presenting both experimental results obtained for strain distribution and temperature gradient using uniform or chirped gratings. Indeed, we extend the method for the first time to temperature distribution measurements and demonstrate its efficiency. For this, we have developed a slightly different method, which is based on the measurement of the complex reflection coefficient of FBG by LCI and the retrieval of the phase of the grating by LP. In section 2, we carefully describe our method and then we will show how a non-uniform strain distribution or temperature gradient can be extracted from the phase of the grating. In sections 3 and 4, we will present and discuss the experimental results obtained respectively for a non-uniform strain distribution and a temperature gradient in a steady state condition.

Method for strain or temperature distribution measurements

Description of the LCI technique

The interferometer is a Michelson-type interferometer assembled around a 'two-by-two' single-mode fibre coupler (figure 1). The test arm is connected to the grating sample to be tested. The wave, which propagates along the reference arm, is bent on a sliding corner cube reflector and oriented toward a fixed plane mirror. The He-Ne laser fringe counting system output is used to sample the infrared interferogram every 80 nm (λ He-Ne /8).

The signal received by the detector is the superposition of the DC signal corresponding to the intensities reflected by each arm and the AC signal created by the interferences of the two reflected waves. The receiver is AC-coupled and amplifies only the oscillations. Theoretically, the variable part of the intensity is

I (x) = +∞ -∞ S(σ )r (σ ) exp(-8iπσ x) dσ ( 4
)
where x is the displacement of the moving carriage, σ the wavenumber, S(σ ) the source spectral power density and r (σ ) the complex amplitude reflection coefficient. Relation [START_REF] Rao | Recent progress in applications of in-fibre Bragg grating sensors[END_REF] shows that r (σ ) can easily be computed from the Fourier transform of the interferogram when the power spectral density of the source is known. From the square of the modulus of r (σ ) the spectrum of the FBG is obtained, and from the phase of r (σ ) the group delay and the chromatic dispersion of the FBG can be calculated. This reflectometric method has already been thoroughly tested and reported [START_REF] Chapeleau | Experimental synthesis of fibre Bragg gratings using optical low coherence reflectometry[END_REF][START_REF] Leduc | Accurate low-coherence interferometric relative group delay and reflectance measurements; characterization of a free space optics multiplexer/demultiplexer[END_REF]. We have demonstrated the repeatability of fibre Bragg grating reflectance and group delay LCI measurements and showed their accuracy by comparison with the results of the NIST Telecom Round Robin. Accuracy in these measurements is absolutely required to determine the index profile modulation and the phase of FBG using the layer-peeling method.

Description of layer-peeling algorithm

The layer-peeling inverse method has been reported in many recent publications [START_REF] Feced | An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings[END_REF][START_REF] Poladian | Simple grating synthesis algorithm[END_REF][START_REF] Skaar | On the synthesis of fibre Bragg gratings by layer peeling[END_REF].

It is a powerful tool for the design of the index profile from a theoretical complex reflection coefficient. Indeed, the LP algorithm takes into account multiple reflections into the FBG so that it can reconstruct correctly the index profile of gratings whose reflection coefficient is high but lower than 90%. We point out the fact that we could also use other inverse scattering algorithms [START_REF] Leduc | Comparison of two inverse scattering algorithms for the experimental synthesis of fiber Bragg gratings[END_REF] based on integral formulations which give similar results but they are greedy in computational time.

The LP algorithm is based on the assumption that the index change (caused by UV exposure) is assumed to be quasisinusoidal:

n(z) = n co + n DC (z) + n AC (z) × cos 2π z + 2π 2 z 0 (z ) - dz (5)
where n AC is the index modulation amplitude, n DC the variation of the refractive index averaged over a single period of the grating, n co the refractive index of the core of the fibre and (z) is the grating's period. Then, FBG is discretized with N layers, so that the index change can be considered as constant. We also assume that the fibre is lossless and guides one single mode. The quantity obtained with the LP algorithm is the complex coupling coefficient q(z). The modulus of q(z) is proportional to the index modulation amplitude n AC and the phase of q(z) is linked to the variation of the refractive index n DC and to the variation of the grating's period:

(z) = - 2π n co z 0 η n DC (z ) dz - 2π 2 z 0 (z ) - dz (6)
where η is the overlap factor of the fibre. In this paper, we call it the phase of the grating.

With the combination of our LCI technique and LP algorithm the phase of the grating is measured with a spatial resolution of 20 µm, and from this quantity the strain distribution or the temperature gradient along the grating can be deduced as shown in the following section.

Description of strain or temperature distribution measurements

Let us call 0 (z) the phase of the free grating. If an external constraint, either a temperature gradient T (z) or an axial tension which induces a strain (z) supposed as non-uniform, is applied to the FBG, then both the period of the grating and the core refractive index are affected. In the linear limit, the variations of the grating's period are related to T (z) or to

(z) by δ (z) = α T (z) δ (z) = (z). ( 7
)
By the photo-elastic effect or photo-thermal effect, the variations of the refractive core index are

δn DC (z) = n co ξ T (z) δn DC (z) = - n 3 co 2 [P 12 -ν (P 11 + P 12 )] (z). ( 8
)
Taking into account these two effects, the temperature gradient or the variation of the strain along the FBG can be retrieved from the measurement of the phase of the grating with and without external constraints:

T (z) = K T d [ (z) -0 (z)] dz ( 9
) (z) = K d [ (z) -0 (z)] dz (10)
where K T is a constant which depends on the thermo-optic and thermal expansion coefficients of the FBG and K is a constant which depends on the photoelastic coefficients and Poisson's ratio of the FBG.

To summarize, the method is based on the measurements of the phase of the grating. These quantities are obtained using the combination of the LCI technique and the LP algorithm previously discussed in sections 2.1 and 2.2. To retrieve the strain or the temperature gradients along the grating, the scheme of the method is as follows.

(i) The phase 0 (z) of the grating without strain distribution or temperature gradient is determined. (ii) The phase (z) of the grating with strain distribution or temperature gradient is measured. (iii) The strain distribution or the temperature gradient are obtained by using respectively the formulae ( 9) and ( 10).

This last step requires us to calculate a discrete derivative which is not straightforward due to the noise in the experimental data. To reduce it, we choose to take the average of ten measurements of the phase. Nevertheless, the result of the derivative still shows some ripples which have no significant physical meaning. So, the results of strain or the gradient of temperature measurements presented in this paper are obtained by applying a smoothing filter based on the least squares.

The method presented previously cannot be applied if there are both an unknown strain distribution and an unknown temperature gradient. Indeed, these two effects are added, so that it is impossible to distinguish them. To validate our method we then studied them separately, and in the next section results of strain distribution measurements will be first shown.

Strain distribution measurement

3.1. The experimental procedure and the finite element simulations 3.1.1. The experimental procedure and calibration. To test the method of strain distribution measurement proposed in this paper, an FBG without fibre coating was embedded in a rectangular block of epoxy with the following dimensions: 200 × 32 × 8 mm 3 . The FBG is 11 mm long and is located in the middle of the specimen and oriented in the direction of the length axis. It was fabricated in a SMF28 fibre by means of the phase mask technique and it is a uniform grating.

The epoxy resin is Axson Epolam 2020, which cures at room temperature. The choice of this resin is because of a good compatibility with the FBG regarding the bonding and the easiness of insertion. During filling and polymerization, the FBG is maintained under slight tension inside the mould in order to ensure its location in the middle of the sample. The fittings at the ends of the sample allow the fibre to emerge straight.

After removal from the mould, the FBG sensor was calibrated. Indeed, it is necessary to determine the coefficient K (10) which is characteristic of the grating used. For this, a preliminary tensile test was performed. The tensile test set-up (see figure 2) is an aluminium frame with a ball joint on each side, a load cell placed at one end for the load measurement and a loading screw able to apply the load. The FBG is then stretched with the sample. This small device was easily placed along the interferometer during measurements. By determining the phase of the FBG for different strains measured with an extensometer placed on the sample, a coefficient K equal to about 1.1×10 2 µ mm rad -1 was found. After this calibration, the sample was drilled with one hole on each side of the FBG (see figure 3), which induces a symmetrical strain gradient in the area of the FBG sensor. Then, new tensile tests were performed.

The finite element simulation.

A numerical mechanical model has been implemented with the finite element method in the code CAST3M [20]. It consists in representing the epoxy sample as a plate under the hypothesis of plane stresses. The FBG is not included in the model because of its small diameter and its negligible contribution to the stiffness of the sample. The geometry is the one of the sample but two symmetries are used in order to simplify the problem. One quarter of the sample is represented in figure 3.

Results and discussions

Figure 4 shows the amplitude reflection coefficient of the FBG before and after it was embedded. The polymerization of the resin causes a shift of the Bragg wavelength of about 4.45 nm and the spectrum becomes broader. This widening can be explained by the shrinkage of the resin and the parttool interactions. They imply deformation of the FBG which changes its properties, thus causing a birefringent effect. This effect leads to a change in the refractive index so that two plane-polarized waves will propagate along the grating. The response of the FBG is then composed of two peaks. In our case, we cannot distinguish them in figure 4, so the birefringent effect due to the embedment and the polymerization process is small and we can neglect it. Figure 5(a) shows the amplitude reflection coefficient of the FBG loaded under different tensile loads. These measurements have been obtained by the optical low coherence interferometer described in section 2.1. The higher the force, the more the spectrum is shifted toward higher wavelengths and the more it is distorted. This distortion is obviously due to a non-uniform strain distribution along the grating.

Before measuring the strain distribution by using the method described in section 2.3, we verified that the birefringence induced by the axial load is insignificant. When the sample is loaded, the presence of the holes creates two transverse stresses σ xx and σ yy , respectively perpendicular and parallel to the axes of the holes, which are not equal. As a consequence, the refractive index will change by the photoelastic effect and become n x and n y for the xand y-polarization respectively. A numerical simulation by the finite element method was made to evaluate the transverse stress σ xx (figure 6) with a maximum of 3 MPa, while the zz longitudinal stress average on the sample is set to 50 MPa. The transverse stress σ yy was set to zero through the plane stress assumption. This approach is widely accepted in the field of structural mechanics when the thickness of the structure is small compared to the width. Effective change in refractive index in both planes could be derived using photoelasticity theory [START_REF] Gafsi | Analyse of induced-birefringence effects on fibre Bragg gratings[END_REF]:

n x (z) -n y (z) = n 3 co 2E K σ xx -σ yy ( 11
)
where E is Young's modulus for the fibre and K is a constant which depends on the photoelastic coefficients and Poisson's coefficient. Using K = 0.1743, E = 74 GPa and n co = 1.45, the value of the birefringence due to the load of the sample cannot exceed 1 × 10 -5 , in the worst case. This value is sufficiently small that we can neglect the birefringence effect in strain distribution measurements. From the complex reflection coefficients, the phase of the FBG is then determined by using the layer-peeling algorithm (see section 2.2). Figure 5(b) shows them for different tensile loads. The strain distribution was then calculated using [START_REF] Huang | Continuous arbitrary strain profile measurements with fibre Bragg gratings Smart[END_REF] and the results are plotted in figure 7. The strain curves are not monotonic and correspond to a non-uniform strain distribution as expected. With a difference below 5%, a good agreement is observed between simulated (dashed curves) and experimental distributions.

Temperature gradient measurement

Experimental procedure and calibration

The experimental set-up shown in figure 8 has been designed in order to create a carefully controlled temperature gradient in a solid body where the FBG sensor is incorporated. The body is formed from polymethyl methacrylate-based organic material (PMMA). As a preliminary, a cylindrical polymethyl methacrylate block (diameter of 80 mm and thickness of 20 mm) was cut in half and a micro-groove with a cross-section of about 200 µm × 200 was machined in the middle of one half. An FBG sensor was then placed in this micro-groove and the two half spherical blocks were bonded. A temperature gradient was created by placing the PMMA cylinder between a heater assembly and a brass cold plate controlled at fixed temperature by a circulator (see figure 8). The heater was composed of an electrical resistor brazed on a brass plate. In steady state conditions, the thermal gradient becomes uniform because of the high thermal conductivity of brass and the size of the cylinder.

To demonstrate that our method can be applied with any type of gratings, the FBG used in this temperature gradient experiment is a chirped fibre Bragg grating. It is 10 mm long and its chirp is equal to 1.5 nm cm -1 . Before embedding the FBG sensor in the block of polymethyl methacrylate, it is necessary to calibrate it. The goal of this calibration is to determine the coefficient K T [START_REF] Huang | Phase-based Bragg intragrating distributed strain sensor[END_REF] which is characteristic of the FBG sensor used. The grating was embedded with a thermocouple in an isothermal brass block. In order to calibrate the FBG sensor, the block was carefully maintained at different temperatures by using a heater. The dimensions of the block are sufficient to ensure a uniform temperature around the grating and the reference thermocouple. By determining the phase of the FBG sensor for different temperatures measured with the thermocouple, we found a coefficient K T equal to 11.8 • C mm rad -1 .

Wavelength [nm]

Results and discussions

In figure 9(a), the solid curve shows the spectrum of the FBG without any temperature gradient. In contrast to a uniform grating, a chirp grating has a broader band-width. The full width at half maximum (FWHM) is equal to 1.4 nm. With the temperature gradient, the dashed curve shows that the spectrum is slightly shifted and the FWHM decreases by about 160 pm. These effects are due to the modifications of the chirp of the grating and of the refractive index distribution by the temperature gradient. These modifications depend on the direction of the grating period compared to the temperature gradient. In other words, if the grating had been turned in the other direction, the FWHM would have probably increased.

The solid curve of figure 9(b) shows the phase of the grating without the temperature gradient. With the temperature gradient (the dotted curve) the curve of the phase becomes less rounded and we can interpret it like a reduction of the chirp of the grating. It is in agreement with the decrease of band-width observed in figure 9(a).

From these phase measurements, the temperature gradient was calculated with the help of ( 9) and the result, the solid curve, is shown in figure 10. The measured temperature gradient is linear with a slope of 2 • C mm -1 with a spatial resolution of 20 µm. The thickness of the PMMA cylinder is 20 mm and the length of the FBG sensor is about 10 mm, so the grating used is too short to measure the entire temperature gradient. We chose to place the FBG sensor close to the heated plate but it is very difficult to locate it accurately. In order to be able to compare the gradient measurement with reference measurement achieved with thermocouples (in figure 10, each dot corresponds to a thermocouple), the curve was shifted only on the Z -coordinate. The agreement between the two results is very good and it demonstrates the validity of the LCI-LP method.

Conclusion

In this paper, we have described a powerful method for measuring non-uniform strain and temperature gradient with fibre Bragg gratings. It is based on the measurement of the phase of the grating which can be obtained by combining the optical low coherence interferometry and the layer-peeling algorithm. Two measurements of the phase of the grating (with and without strain distribution or temperature gradient) are required and from the derivative of the difference of these phases, the strain or the gradient temperature along the grating can be retrieved.

By this method, we performed non-uniform strain distribution and temperature gradient measurements using uniform or chirped FBG sensors. The measured strain distribution was compared to finite element simulation and temperature gradient was compared with thermocouples. In each case, the agreement was excellent. This demonstrates the efficiency of the method.

So, it is a promising technique to quantify strain or temperature distributions inside composite parts. These measurements of gradients would lead to an accurate characterization of both the mechanical and thermal behaviours where classical techniques are only able to provide surface measurements.

Figure 1 .

 1 Figure 1. Optical low-coherence interferometer.

Figure 2 .

 2 Figure 2. Tensile test set-up.

Figure 3 .Figure 4 .

 34 Figure 3. Scheme of the sample and quarter of the finite element mesh.

Figure 5 .

 5 Figure 5. Amplitude reflection coefficient and phase of FBG for several tensile loads.

Figure 6 .

 6 Figure 6. Transverse stress σ x x obtained by the finite element simulation.

Figure 7 .

 7 Figure 7. Strain distribution measurements from the phase grating (solid curves) and from the finite element simulations (dashed curves).

Figure 8 .

 8 Figure 8. Experimental set-up of temperature gradient.

Figure 9 .

 9 Figure 9. Amplitude reflection coefficient and phase of FBG without (solid curves) and with (dotted curve) temperature gradient.

Figure 10 .

 10 Figure10. Measurement of the temperature gradient from the phase grating (solid curves) and from the thermocouples (dashed curve).

Acknowledgments

We thank Patrick Derval and Michel Roche for their technical support.