State-of-strain evaluation with fiber Bragg grating rosettes: application to discrimination between strain and temperature effects in fiber sensors

State-of-strain evaluation with fiber Bragggrating rosettes: application to discriminationbetween strain and temperature effects in fiber sensors

Sylvain Magne, Stephane Rougeault, Manuel Vilela, and Pierre Ferdinand

An optical rosette that incorporates fiber Bragg gratings as strain gauges has been designed, fabricated,and tested. We investigated it by measuring the state of strain of a thin plate as the test structuresubmitted to an increasing load in a four-point bending configuration and for various angular orienta-tions. This device has also been successfully investigated as a self-temperature-compensated in situuniaxial strain sensor without any angular dependence and with high accuracy in recovery analysis,leading us to expect many industrial applications. Printed circuit processes or integrated optics onpolymers would provide a means for accuracy, reproducibility, and integration in a mass-producedprocess. © 1997 Optical Society of America

Key words: Fiber Bragg grating, rosette, optical fiber sensor, smart material and structure, strainand stress analysis, temperature compensation.

1. Introduction

Fiber Bragg gratings ~FBG’s! are now recognized asimportant optoelectronic components for guided-wave optics, owing to the large number of device func-tions in which they can be used.1,2 They are usedextensively in telecommunications ~wavelength-division multiplexing, gain flattening of erbium-doped fiber amplifiers, and dispersion compensation!,in instrumentation ~wavelength-selective reflectorsfor fiber lasers and, to a lesser extent, semiconductorlasers!, and in sensors for the measurement of strain,temperature, and hydrostatic pressure.3

The first reported Bragg grating ~the so-called Hillgrating! dates back to 1978 when Hill and co-workerslaunched light at 488 nm from an argon-ion laser intoa germanosilicate optical fiber.4 In those experi-ments the grating was formed by the standing-waveinterference pattern set up by counterpropagatingbeams inducing periodic perturbations of the refrac-tive index along its length, and the center Bragg

The authors are with the Department d’Electronique etd’Instrumentation Nucleaire, Service de Physique Electronique,Laboratoire d’Electronique de Technologie et d’Instrumentation,Commissariat a l’Energie Atomique ~Technologies Avancees!, Cen-tre d’Etudes de Saclay, 91191 Gif-sur-Yvette Cedex, France.

Received 1 April 1997; revised manuscript received 16 July 1997.0003-6935y97y369437-11$10.00y0© 1997 Optical Society of America

wavelength equaled the laser wavelength. Becauseof the long interaction length, narrow-band Braggreflection filters were formed with reflectivities ap-proaching 100%. After that, in spite of the signifi-cant interest raised at the time, few studies werepursued until 1989 when Meltz et al.5 reported thatgratings could be written by two-beam holographicexposure through the side of the fiber by use of UVradiation. This method dramatically improved thewriting efficiency because of the direct photoioniza-tion process induced by UV light ~in contrast with theHill experiment, in which photoionization resultedfrom a two-photon absorption process!. Moreover,this method allowed the possibility of producing grat-ings with an arbitrarily selected Bragg wavelengthsimply by adjusting the angle between the exposedbeams with respect to the fiber axis. Although thefree-space holographic method ~Meltz’s method! al-lowed great flexibility in fabrication of FBG’s, ahigher reproducibility was achieved by a methodbased on near-contact exposure through a phasemask.6 In this case the grating was photowritten byinterference of 21 and 11 diffraction orders with aperiod equal to half of that of the mask independentof the exposure wavelength. Since then activity inphoto-induced Bragg gratings has increased rapidly,leading to the demonstration of many types of grating~moire, chirped, phase-shifted, etc.!1,2 in fibers or onsubstrates and in many materials, such as germano-silicate glasses with or without hydrogen loading;7

20 December 1997 y Vol. 36, No. 36 y APPLIED OPTICS 9437

cerium-, terbium-, or tin-doped silicate glasses;8,9

cerium-doped fluorozirconate glasses;10 and poly-mers.11

FBG’s are known to be well suited for measuringstrain and temperature, e.g., in smart structures,12–14

and have unique advantages over classical electricalstrain gauges ~metal-foil resistance and piezoelectricceramic strain gauges!. These advantages are atfirst conveyed by intrinsic features of optical fibersensors, such as electromagnetic interference immu-nity, light weight and small size, high temperatureand radiation tolerance, flexibility, stability, and du-rability against harsh environments. FBG’s havethe advantages of being absolute, linear in response,interrupt immune, and of low insertion loss so thatthey can be multiplexed in a series on a single-modefiber. The spectral signature renders the measure-ment free of intensity fluctuations. This is a greatasset for long-term reproducibility. Moreover,FBG’s can be easily embedded into composite ma-terials and can provide local damage detection aswell as internal strain field mapping with high res-olution in strain and localization and a large mea-surement range. The FBG is therefore a majorcomponent in the development of fiber-optic smart-structure technology. It offers the promise of un-dertaking real-time structural measurements withbuilt-in sensor systems expected to be cost effectivewhen the number of sensors that can be multi-plexed is large.

In the scope of making strain measurements usingBragg gratings, one can design FBG rosettes follow-ing the example of electric strain rosettes. Rosettesare made of two or three noncollinear strain gaugesmounted on a common substrate, typically arrangedat 45° or 120° with respect to one another to formrectangular or delta rosettes, respectively. They areused extensively in experimental stress analysis tomeasure the two principal strains ~and stresses! andthe orientation of the principal axis whenever it is notknown a priori. Such a concept has already beendemonstrated with fiber interferometer straingauges15–17 and fiber Fabry–Perot strain gaugesplaced onto common substrates18 or embedded inneat resin and graphite–epoxy composite laminatesfor mapping of two-dimensional strain fields.19 Toour knowledge rosettes have not yet been demon-strated with FBG’s as strain gauges. FBG rosetteshave several advantages over interferometric strainrosettes, including the fact that they are potentiallyeasy to manufacture ~amenable to batch-processmanufacturing! and the multiplexing capability ofBragg grating-based rosettes and gauges in a seriesalong a single monomode fiber.

However, whatever the method used, temperaturesensitivity of the fiber sensor can complicate its ap-plication as a strain gauge. Let us focus on Bragggrating-based optical sensors: many solutions havebeen proposed to eliminate their thermal-apparentstrain. The most obvious solution lies in compensat-ing for the temperature influence by using anotherFBG ~reference grating! shielded from strain ~e.g., by

9438 APPLIED OPTICS y Vol. 36, No. 36 y 20 December 1997

Teflon sleeving! that measures only temperature.The Bragg wavelength of this grating is then sub-tracted from that of the FBG-measuring strain andtemperature.20 This correction is fully efficient ifboth transducers are at the same temperature ~i.e.,they are close enough! or if the difference in temper-ature remains constant.

We now attempt to make a survey of in situtemperature-compensating methods for Bragg grat-ings. They can be classified according to whetherthey are intrinsic ~i.e., make use of the fiber proper-ties themselves! or extrinsic ~i.e., require an extramaterial!. At least eight intrinsic methods havebeen proposed. The first method involves the use ofa saturated chirped grating21 of which the spectralbandwidth increases with strain, thus increasing thereturn signal. The second method involves the useof two superimposed Bragg gratings of different cen-ter wavelengths.22,23 A third method relies on theuse of a Bragg grating photowritten in a highly bire-fringent fiber.24 A fourth method involves the use oftwo Bragg gratings of close center wavelengths pho-towritten on either side of a splice between two fibersof different cladding diameters ~80 and 120 mm!.25 Afifth method relies on the primary and second ordersof diffraction from a single Bragg grating.26 A sixthmethod makes use of the dependence of rocking fil-ters on strain and temperature.27,28 A seventhmethod involves the use of two Bragg gratings and along-period grating,29 and an eighth method relies onthe use of a FBG and a Fabry–Perot interferometer.30

The first method relies on an intensity measurement~4-mε resolution in strain!, and temperature is notmeasured. For any of the seven latter methods tem-perature and strain information can be recovered,provided that two linear equations, both functions oftemperature and strain, can be solved.31

Conversely, at least two extrinsic methods havebeen proposed. The first makes use of a cantileverwith two FBG’s surface mounted on opposite surfaces~top and bottom!.32 One grating is stretched whilethe other is compressed. The difference in Bragggrating center wavelengths is temperature indepen-dent because both Bragg gratings have the samethermal sensitivity. The second method consists ofa passive temperature-compensating package andmakes use of a proper choice of materials and di-mensions to nullify the temperature-to-wavelengthcoefficient.33,34 These two methods involve a me-chanical compensation and do not provide a temper-ature measurement.

Here we describe the design, fabrication, and testof a Bragg grating rosette. This device was testedin two ways. The first test was dedicated to themeasurement of the state of strain of a plate as thetest structure. The second test was more unex-pected, as it provided a rigorous measurement of auniaxial strain ~or stress! independent of both tem-perature and orientation of the rosette onto the teststructure.

2. Bragg Grating Properties

The Bragg center wavelength of a grating lg is givenby the Bragg phase-matching condition

lg 5 2neffL, (1)

where L is the fringe spacing of the grating and neff isthe effective refractive index of the LP01 mode ~de-pending on the V value of the fiber!. The relativechange in the center Bragg wavelength is therefore

Dlg

lg5

Dneff

neff1

DL

L. (2)

Both refractive index and spacing depend on temper-ature and strain, as follows:

Dlg

lg5 εax 2

ncore2

2@εr~P11 1 P12! 1 P12εax# 1 ~as 1 zs!DT,

(3)

where the first term refers to changes in the gratingspacing caused by axial strain, the second term inbrackets refers to photoelastically induced changes inthe refractive index ncore of the fiber core ~;1.46 forgermanosilicate fibers!, and the third and fourthterms refer to thermal expansion ~;5 3 1027 K21!and thermo-optic ~;7 3 1026 K21! coefficients of thefiber material, respectively ~i.e., change of shape andrefractive index with temperature!. P11 ; 0.113and P12 ; 0.252 are the photoelastic constants ofsilica,35 and εax and εr are the axial and radial strains,respectively.

When the Bragg grating is axially strained, theradial strain can be determined by Poisson’s law,

εr 5 2nsεax, (4)

where ns is the Poisson ratio of the fiber material ~ns; 0.17 for silica!. Then the strain–temperature-to-wavelength relation reduces to

DlB

lB5 ~1 2 pe!εax 1 ~as 1 zs!DT, (5)

where pe 5 ~ncore2y2!@P12 2 ns~P11 1 P12!# ' 0.22 for

a germanosilicate fiber. At room temperature theexperimental relative change in the Bragg centerwavelength is then

Dlg

lg5 0.78εax 1 7.5 3 1026DT~K!. (6)

At the wavelength of 1.3 mm the temperature-to-wavelength coefficient is approximately 0.1 Kypm atroom temperature, and the strain-to-wavelength co-efficient is approximately 1 mεypm ~1 mε 5 1mstrain 5 1026!. Therefore a change in temperatureof 0.1 K induces the same wavelength shift as thatinduced by 1 mstrain. It should not be overlookedthat both strain- and temperature-to-wavelength co-efficients depend on temperature. For example,Meltz et al.5 reported slopes of relative Bragg gratingwavelength shift versus temperature that increased

from 6.7 3 1026 K21 ~from room temperature to150 °C! to 9.57 3 1026 K21 ~at 450 °C!.36 Thereforerecalibrations are needed every time there is achange in the temperature range of use. ~This isparticularly true for all intrinsic temperature-compensation methods previously described.!

Once the FBG is bonded onto or embedded into thetest structure, the thermal expansion of the structurematerial causes a change in the grating period. Letastructure be the thermal-expansion coefficient of thetest material. Then Eq. ~5! becomes

DlB

lB5 ~1 2 pe!εax 1 @zs 1 as 1 ~1 2 pe!

3 ~astructure 2 as!#DT.

For astruc .. as ~for example, astructure 5 23 3 1026

K21 for aluminum!, it reduces to

DlB

lB5 ~1 2 pe!εax 1 @zs 1 ~1 2 pe!astructure#DT. (7)

The experimental relative change in the center wave-length of a FBG bonded on aluminum is then

Dlg

lg5 0.78εax 1 25 3 1026DT~K!. (8)

3. Two-dimensional Stress and Strain Analysis

In practical strain gauging the two principal strainsare unknown, and the goal is to find their magnitudesand directions from a number of strain measure-ments taken at various angles. Therefore three in-dependent strain measurements in three differentdirections along the plane ~arranged at convenientangles in relation to each other! are necessary andsufficient. A fourth gauge may be added for temper-ature compensation.

In the following we assume that the test structureis isotropic, homogeneous, and elastic. In practice,the test structure was weakly loaded so that Hooke’slaw held ~strain did not exceed 0.1%!. Let x and y bethe coordinates of the plane structure and z be theoutward normal to the surface. Whatever the stateof strain, a reference frame ~X, Y, Z! always exists inwhich shear stresses vanish and only normal stressesremain ~i.e., principal stresses sX and sY!.37 Now letus assume that the test structure behaves under astate of plane stress ~i.e., sz 5 sZ 5 0! and thatprincipal axes are well known. Let txy and sy be,respectively, the shear and normal stresses acting ona plane of outward normal y, and let sx be the normalstress acting on a plane of outward normal x @Fig.1~a!#. The plane surface makes an angle a with re-spect to the principal axis X. sX and sY give rise to,respectively, two orthogonal stresses CX 5 sX sin aand CY 5 sY cos a acting on the tilted plane. Agauge aligned along the y axis measures

sy 5 CX sin a 1 CY cos a 5 d9 1 r9 cos~2a!, (9)

txy 5 r9 sin~2a!, (10)


where the parameters d9 and r9 are called the meanstress and the deviatoric stress ~or stress deviator!,respectively, and are expressed as follows:

d9 5 ~sX 1 sY!y2, (11a)

r9 5 ~sY 2 sX!y2. (11b)

Local strains are deduced from local stresses in Eq.~9! with Hooke’s law:

εy 5 syyE 5 d 1 r cos~2a!, (12)

where E is Young’s modulus of the test material andd and r are the mean and deviatoric strains, respec-tively, given by

d 5 ~εX 1 εY!y2 , (13a)

r 5 ~εY 2 εX!y2 . (13b)

The normal and shear stresses are the two coordi-nates of a state of stress that can be drawn on a circlecalled the Mohr stress circle37 @Fig. 1~b!#, described bythe following equation:

~sy 2 d9!2 1 txy2 5 r92. (14)

If the test structure no longer behaves under a stateof plane stress, the three normal stresses are drawnon three circles called the Mohr stress tricircle. In

Fig. 1. Normal and shear stresses acting on a surface plane of the~a! outward normal y and ~b! corresponding Mohr stress circle.

the case of a state of plane stress, the principalstrains εX and εY are then

εX 5 ~1yE!~sX 2 nsY!,

εY 5 ~1yE!~sY 2 nsX!. (15)

The principal stresses sX and sY can be derived fromthe principal strains

sX 5 @Ey~1 2 n2!#~εX 1 nεY!,

sY 5 @Ey~1 2 n2!#~εY 1 nεX!, (16)

where n is the Poisson ratio of the test material.The local strain equation ~12! can then be rewritten

for each gauge of the rosette:

ε1 5 d 1 r cos@2a# ~gauge 1!,

ε2 5 d 1 r cos@2~a 1 b!# ~gauge 2!,

ε3 5 d 1 r cos@2~a 1 2b!# ~gauge 3!, (17)

where a is the angular orientation of the principalaxis Y with respect to FBG 1. Analytical solutionsfor Eqs. ~17! can be obtained for two types of rosettes:rectangular ~b 5 45°! or delta ~b 5 120°!. Knowingb and measuring ε1, ε2, and ε3, one can determine thestrain parameters d, r, and a ~see Table 1!. Rosettesof arbitrary angles ~different from 45° or 120°! areconceivable, but their practical use requires numeri-cal resolution of Eq. ~17!.

An analytical form for the strain deviator can begiven for both rosette types. The deviatoric formulafor the delta rosette is given by

r120° 513

~ε1 2 ε3!H3 1 F2~ε2 2 ε3!

~ε1 2 ε3!2 1G2J1y2

, (18)

whereas the deviatoric formula for the rectangularrosette is given by

r45° 512

~ε1 2 ε3!H1 1 F2~ε2 2 ε3!

~ε1 2 ε3!2 1G2J1y2

. (19)

The principal strains and stresses, as well as theangular orientation a, can thus be computed from themeasured strains ε1, ε2, and ε3 by use of Eq. ~18! or

Table 1. Parameters for Rosette Evaluation of Principal Strains «X, «Y, and Orientation a

Three-GaugeRosette Types Delta Rosette ~b 5 120°! Rectangular Rosette ~b 5 45°!

Mean strain ~d!ε1 1 ε2 1 ε3

3ε1 1 ε3

2Deviatoric strain ~r! ε2 1 ε3 2 2ε1

3 cos~2a!

ε1 2 ε3

2 cos~2a!

13

~ε1 2 ε3!H3 1 F2~ε2 2 ε3!

~ε1 2 ε3!2 1G 2J 1y2 1

2~ε1 2 ε3!H1 1 F2

~ε2 2 ε3!

~ε1 2 ε3!2 1G 2J 1y2

Principal axis ori-entation @tg~2a!# Î3

ε2 2 ε3

2ε1 2 ε2 2 ε3

ε3 1 ε1 2 2ε2

ε1 2 ε3


Eq. ~19!, Table 1, and the following derived from Eqs.~13a! and ~13b!:

εX 5 d 2 r, (20a)

εY 5 d 1 r. (20b)

4. Bragg Grating Rosette as aSelf-Temperature-Compensated Uniaxial Strain Gauge

Under hydrostatic pressure or temperature changes,the Mohr stress circle remains unchanged ~same ra-dius and angular orientation 2a! but shifts along thesX axis. Let us assume that the gradient in temper-ature is negligible and that all gauges are at the sametemperature. Therefore temperature induces thesame thermal-apparent strain in any direction ofspace, and so does hydrostatic pressure. As can beseen in Table 1, only the d parameter ~mean strain!depends on temperature and hydrostatic pressure,whereas r ~strain deviator! and a ~angular orienta-tion! do not because they both involve differencesbetween local strains. As a consequence, tempera-ture and hydrostatic effects rigorously cancel out.In the case of a uniaxial strain applied to the teststructure along the X direction ~i.e., sY 5 0; sX 5EεX!, Eqs. ~16! reduce merely to Hooke’s and Pois-son’s laws, and the strain deviator can be written as

uru 5uεY 2 εXu

25

~1 1 n!

2uεXu. (21)

Therefore the uniaxial strain can be written as

uεXu 5 S 21 1 nDuru, (22)

where r is given by Eq. ~18! or Eq. ~19! for the delta orrectangular rosette, respectively. In the particularcase of the delta rosette, the principal strain is then

εX 52

1 1 n

13

~ε1 2 ε3!H3 1 F2~ε2 2 ε3!

~ε1 2 ε3!2 1G2J1y2

. (23)

As can be seen, the rosette evaluation of the principalstrain εX is temperature independent.

Note that two-gauge rosettes can also be usedwhenever the principal axis orientation a is known.In the case of a 90° tee rosette, the rosette strainrelations are then

ε1 5 d 1 r cos@2a# ~gauge 1!,

ε2 5 d 1 r cos@2~a 1 py2!# 5 d 2 r cos@2a# ~gauge 2!.

(24)

Therefore the deviatoric strain as measured by a two-gauge tee rosette type is

r 5ε1 2 ε2

2 cos@2a#. (25)

The principal strain εX can then be deduced with Eq.~22!, and one can see that it is absolutely temperatureindependent as well.

The accuracy of the measurement of the state ofstrain is partly limited by spectral resolution. Infact, since three noncollinear gratings are requiredfor one to perform the measurement, one can easilyfind @by differentiating Eq. ~23!# that the error de-pends on a and is approximately twice to three timesthat of a single grating ~owing to the vectorial natureof the strain information!. Therefore a maximumerror of approximately 3 mεypm is anticipated for therosette-based uniaxial strain sensor. However, agreat inaccuracy arises from angular dispersion inthe positioning of gauges with respect to one another.Let us suppose that one gauge is badly positionedwith respect to the two other gauges. There is asmall change in local strain Dε in comparison withthat measured by the perfectly aligned gauge thatcan be estimated by a first-order Taylor series ap-proximation of Eqs. ~17! as a function of a small de-parture Db from the exact angle. Then this newstrain ε 1 Dε is inserted into Eq. ~18!, and the corre-sponding change in the deviator value is estimated bya first-order Taylor-series approximation as a func-tion of Dε. After simplification the relative error indeviator estimation with respect to the departure an-gle Db is then found to be

Drr

589

sin2 b sin@4~a 1 b!#Db (26)

and can be found to be, at worst 60.12% for a 0.1°angular misalignment. In actual fact, two gaugescan be badly positioned with respect to the third one.Therefore the relative error of the rosette sensor isactually, at worst, 60.24% for a 0.1° angular mis-alignment.

Once εX and a are known ~Table 1!, pure localstrains ε1°, ε2°, and ε3° can be calculated by use of thefollowing deduced from Eqs. ~17!:

ε1°εX

5 sin2~a! 2 n cos2~a! 5 S1 2 n

2 D2 S1 1 n

2 Dcos~2a!,

ε2°εX

5 sin2~a 1 b! 2 n cos2~a 1 b! 5 S1 2 n

2 D2 S1 1 n

2 D3 cos~2a 1 2b!,

ε3°εX

5 sin2~a 1 2b! 2 n cos2~a 1 2b! 5 S1 2 n

2 D2 S1 1 n

2 D3 cos~2a 1 4b!. (27)

The temperature information T 2 T0 can be calcu-lated with Eq. ~7! for any grating:

T 2 T0 5 FDli

li2 ~1 2 pe!εi°G 1

@zs 1 ~1 2 pe!astructure#.

(28)

The absolute temperature T can then be estimated,provided that an accurate calibration of the referencetemperature T0 has been made beforehand. Sincethe strain εi° is a function of εX and a, the relative


error in temperature with respect to spectral resolu-tion can be estimated. The calculation is lengthybut does not create any difficulty. It yields

DTDl

~Kypm! 5 F1l

1 ~1 2 pe!S1 2 n

2 DUdεX

dlU 1

43

1lG

31

@zs 1 ~1 2 pe!astructure#. (29)

The third term refers to the error in angular orien-tation and is maximum for a 5 45°.

5. Strain-to-Deflection Governing Equations for Platesunder Bending

A four-point bending setup was chosen for a proof-of-principle experiment. It is depicted in Fig. 2. Itconsists of two support bars fixed on an optical table,two loading bars applying a force set by a screw, anda micrometer to measure the center deflection of thetest structure. Four-point bending induces a stateof plane strain. Such a setup enables one to performstatic measurements with great stability over longperiods of time to achieve a good spectral resolution.20

Moreover, the curvature radius of the test structureis constant at the center so that the bending-inducedstrain is constant over the grating length ~i.e., thereis no spectral chirp of the grating!, and there is noneed for accurate positioning of the test structure onthe bench. Finally, the test structure can be turnedat any angle between 630°, and the principal axisorientation is determined by the support bars.

It can be shown from the theory of elasticity37,38

that the bending profile w~x! of a beam is ruled by thewell-known differential equation of equilibrium:

d2w~x!

dx2 5M~x!

EI(30a)

or

d4w~x!

dx4 5p~x!

EI, (30b)

Fig. 2. Four-point bending setup: SLD, superluminescent diode.


where M~x! is the bending moment applied to thebeam with respect to the abscissa x, p~x! is the ap-plied force per unit area, and I is the moment ofinertia of the beam cross section with respect to the xand y axes, which is considered constant over itslength. By integrating twice and solving for bound-ary conditions, one can easily find the deflection W atthe center of the beam:

W 5FaEI Sa2

62

L2

8 D , (31)

where L is the distance between support bars and a isthe distance between a support bar and the nearestloading bar. The upper part of the structure is un-der compression, whereas its lower part is under trac-tion. The local strain at the surface of the structureincreases linearly with distance from the neutral axisof the beam according to

εX 5 2~h 1 F!

2d2w~x!

dx2 Ux5Ly2

, (32)

where F is the diameter of the polymer-coated fiberand h is the thickness of the plate. Substitution ofEq. ~30a! into Eq. ~32! gives

εX 5 2~h 1 F!

2M~Ly2!

EI. (33)

Since the bending moment at the center of the beamis M~Ly2! 5 Fa, substitution of Eq. ~31! into Eq. ~33!finally gives the following relation between centerdeflection and local strain:

εX 5 212~h 1 F!W

~4a2 2 3L2!. (34)

The beam-bending theory has the merits of both sim-plicity and ease of use because most bending prob-lems can be solved analytically. However, owing tothe overall dimensions of our Bragg grating rosettewith respect to our bending setup, the test structureon which it is bonded is better described by the thinplate-bending model rather than by the beam-bending one. The differential equation of equilib-rium is then written in two-dimensional coordinatesas38

d4w~x, y!

dx4 1 2d4w~x, y!

dx2dy2 1d4w~x, y!

dy4

512~1 2 n2!

Eh3 p~x, y!. (35)

Solutions for Eq. ~35! can be obtained by the inversemethod that relies on assumed solutions for w~x, y!that satisfy the governing equation and the boundaryconditions. A powerful method is the Fourier series~Navier or Levy solutions!, but manageable solutionsrequire one to fulfill particular boundary conditionsthat are not generally met in practical experiments.On the other hand, finite-element methods have

proved to be the most flexible for determining themechanical behavior of arbitrary test structures andmapping their state of strain. We performed plate-bending numerical analysis using a finite-elementsoftware ~resistance des materiaux!. Under a dis-placement imposed by the loading bars ~as boundaryconditions!, we obtained principal strains and princi-pal axis orientation of arbitrary points on the surfaceas a function of plate deflection for several orienta-tions of the plate on the test bench.

Plate-bending behavior differs considerably fromthat of beam bending because of edge effects in theway that a transverse stress sY takes place in platebending but is not noticeable in beam bending.Moreover, the plate takes the shape of a saddle.This behavior has been reported by several au-thors38,39 and can be analytically described in simplecases. As a consequence of this, the ratio 2εYyεX isinevitably less than the Poisson ratio of the test ma-terial. This apparent Poisson ratio is given by

εY

εX5 2napp 5

sY 2 nsX

sX 2 nsY. (36)

The apparent Poisson ratio was confirmed by finite-element analysis and was found to be nearly constantwith respect to load. Indeed, the theory of elasticityalways predicts a linear relation between transverseand longitudinal strains ~i.e., Poisson’s law alwaysholds but with an effective Poisson ratio differentfrom that of the test material!. At any rate, as aproof of principle, the experiment is as conclusive asif it were performed on a beam ~i.e., small width-to-length ratio!, except that further calculations werecarried out on the basis of an apparent Poisson ratio.

Moreover, numerical analysis shows that the de-flection of the plate at 0° orientation is identical tothat of its beam equivalent @i.e., Eq. ~34! holds for 0°plate bending#, owing to the symmetry of the distrib-uted applied load. The experimental deflection dif-fers from that determined by the beam-bendingtheory at other orientations, and results obtained bynumerical analysis of plate bending must be consid-ered instead.

6. Experiment and Discussion

An optical Bragg grating delta rosette was fabricatedand is shown in Fig. 3~a!. Each side was 35 mmlong. The three Bragg gratings arranged at 60°from one another were bonded onto an aluminumplate as the test structure @Fig. 3~b!# on the reverseside ~tensile face!. Seen from the top, the Bragggratings were arranged in a counterclockwise config-uration. The thickness of the plate was h 5 4.28mm, and its Young’s modulus and Poisson ratio wereE 5 67.5 GPa and n 5 0.34, respectively. The opti-cal instrumentation has already been described in aprevious publication.20 Light from a broadband su-perluminescent diode was launched into a single-mode fiber, in which the three Bragg gratings werephotowritten at 1294.8, 1296.7, and 1297.4 nm. Theouter diameter of the polymer-coated fiber was 280

mm. The spectral linewidths of the Bragg gratinggauges were 0.2 nm. Bragg spectra were recordedwith a 64-cm-focal-length scanning monochromatorequipped with a 600-lineymm grating. The spectralresolution was approximately 65 pm, yielding astrain resolution of 65 mε.

In a first approach we investigated the use of thisrosette to determine the state of strain. Althoughtheory @Eq. ~23! and Table 1# predicts that the rosettecalculation of principal strains and stresses is inde-pendent of the rosette orientation with respect to theprincipal axis, we intended to verify its reproducibil-ity. With this aim in mind, we performed severalmeasurements at various angular orientations andcompared rosette evaluations of the principal strain.The plate was turned along the bench at 0°, 14°, and30°. For each angular orientation the plate was pro-gressively loaded so that the deflection increased insteps of 0.5 mm. Spectra of the three Bragg gratingswere recorded for each deflection. Figures 4 and 5show the Bragg grating wavelengths with respect tothe center deflection of the plate for 0° and 30°, re-spectively. Also shown are the experimental princi-

Fig. 3. Bragg grating ~a! delta rosette and ~b! epoxy bonding ontoa metallic test plate.

Fig. 4. Experimental and theoretical evolutions of the Bragg grat-ing wavelengths under an increased load for 0° orientation ~FBG 1is aligned along the Y principal axis! and experimental and theo-retical evaluations of the principal strain εX.


pal strain εX as calculated from individual Bragggrating strains with Eq. ~23! and the theoretical oneas obtained by numerical analysis. The theoreticallocal strains ε1°, ε2°, and ε3° were then deduced by useof Eq. ~27!, and the corresponding individual Bragggrating center wavelengths were obtained with Eq.~5!. They are also drawn in dotted lines for compar-ison, and it is obvious that agreement between theexperiment and the numerical analysis is good. Asmentioned above, finite-element results are identicalto those obtained with beam-bending calculation for0° orientation. Experimental values for εX fit boththe calibration ~metal-foil resistance strain gauge!and the plate bending well whatever the orientation.For 0° and 14° orientation the dimensions of the testbench were L 5 400 mm and a 5 111 mm. At 30°orientation the dimensions were L 5 295 mm and a 566 mm. We estimated strain-to-deflection coeffi-cients using Eq. ~34! and finite-element modeling aswell. For a given displacement imposed by loadingbars, the principal strain εX and deflection W werereported for each angular orientation of the plate ~0°,14°, and 30°!. Principal axis angular orientations awere 0°, 11°, and 26.5°. Finite-element numericalanalysis gave the principal stresses sX and sY, andprincipal strains could be deduced with Eq. ~15!.With Eq. ~36! we obtained the apparent Poisson ratiobetween the two principal strains, which was 0.17.We compared the theoretical results with the Bragggrating rosette experimental evaluation of the prin-cipal strain εX, taking into account a strain-to-wavelength coefficient of 1.01 pmymε. The resultsare outlined in Table 2.

A quite good agreement is found between experi-mental evaluation of the principal strain and finite-element numerical analysis, whereas a greatdifference exists when we used the beam-bendingmodel ~except for the special case of 0° orientation!.Figures 4 and 5 show a nonlinearity of the strainresponse of grating 1 ~mostly submitted to the Pois-

Fig. 5. Experimental and theoretical evolutions of the Bragg grat-ing wavelengths under an increased load for 30° orientation ~FBG1 is at 30° with respect to the Y principal axis! and experimentaland theoretical evaluations of the principal strain εX.


son strain!. We believe this is due to a slight changein the Poisson ratio during plate bending ~see Section5!. Since the relative change in the Poisson ratio isless than 10%, it leads to a small inaccuracy in theprincipal strain measurement ~approximately 1%!.As already described, this problem arises because ofplate behavior under bending and is expected to van-ish for linear structures ~rods, tubular members! forwhich the Poisson ratio is exactly that of the struc-ture material.

In a second approach we investigated an innova-tive way of using this rosette in combination withuniaxial strain to make a self-temperature-compensated optical fiber Bragg grating-based straingauge. Once again the plate was progressivelyloaded, but, as the deflection reached 1 mm, we im-posed a sudden change in temperature by heating theplate. After temperature stabilization the loadingcontinued until the deflection reached 3.5 mm ~Fig.6!. The corresponding strain εX was approximately435 mε. As expected, every Bragg grating centerwavelength exhibited a step owing to temperaturechange. Heating the plate manifested itself in athermal-apparent strain added to the real strain~shown by the dotted line! as it affected each Bragggrating wavelength. On the other hand, the exper-imental value of the real strain εX @as calculated byEq. ~23!# remained unaffected by the change in tem-

Table 2. Principal Strain-to-Plate Deflection Coefficients «XyW~m«ymm! for Various Angular Orientations of the Platea

PlateOrientation

~deg!Beam-Bending

ModelNumericalAnalysis

ExperimentalEvaluation

0 127 127 128 6 214 127 120 116 6 230 225 200 193 6 2

aFBG 1, FBG 2, and FBG 3 are arranged in a counterclockwiseconfiguration.

Fig. 6. Experimental and theoretical evolutions of the Bragg grat-ing wavelengths under an increased load for 0° orientation anddemonstration of the self-temperature-compensated measurementof the uniaxial strain εX.

perature. Since the rosette evaluation of the princi-pal strain was independent of its angular orientation~as shown in Fig. 5!, we therefore realized a uniaxialstrain sensor absolutely independent of temperatureas well as its positioning onto the test structure.The temperature information can be recovered aswell with Eq. ~28!, and in the case of Fig. 6 it wasfound to be approximately 7 6 0.7 °C.

Let us assume that the Poisson ratio is accuratelyknown. In addition to spectral resolution, a majorinaccuracy stems from the fabrication tolerance of theBragg grating rosette ~by means of angular disper-sion!. In our experiment the rosette was accuratelypositioned to better than 0.1° so that the relativeerror in uniaxial strain was expected to be less than0.24%, as estimated with Eq. ~26!. This means that,for strains ranging from several to 1000 mε, the pre-cision in strain was limited mainly by the spectralresolution of our instrumentation ~65 pm!. There-fore the error caused by spectral resolution was 615mε, whereas the error caused by fabrication tolerancewas 62.5 mε. On the other hand, for strains higherthan 1000 mε, the precision in strain was determinedby the angular inaccuracy ~fabrication tolerance! aswell. For example, for most smart-structure appli-cations ~strain range to as high as 1%!, this errorwould be approximately 625 mε. As a consequenceof that, the real error of our rosette-based uniaxialstrain sensor was estimated to be, at worst, 65.5 mεfor a strain of 1000 mε and a given spectral resolutionof 61 pm. Using Eq. ~29!, we found the correspond-ing precision in temperature to be approximately 0.14Kypm.

7. Conclusion

For the first time to our knowledge, a Bragg gratingrosette has been designed, fabricated, and tested.We first investigated this rosette by measuring thestate of strain of a thin plate as the test structure ina four-point bending setup ~i.e., measuring the prin-cipal strains and orientation of the principal axis!.This rosette competed with its electrical counterpartsin terms of precision and conveyed all intrinsic opticalfiber sensor advantages. Satisfactory results wereobtained that allowed the use of this rosette on planestructures under different loading conditions, possi-bly with an extra grating for temperature compensa-tion. Other structures, such as tie rods or membersof a truss ~either plane or tubular!, might be equippedas well.

In addition to this classical use, we have describedan innovative application of this device as a self-temperature-compensated uniaxial strain sensorwith no angular dependence. The uniaxial strain,the angular orientation of its principal axis, and thetemperature can be accurately recovered by calcula-tion. In support of this conception, note that mostextensometric measurements involve only uniaxialstrains. In-plane determination of principal axisorientation and principal strains are seldom encoun-tered in industrial applications and are mostly re-stricted to engineering and laboratory experiments.

The basic idea of this self-compensation scheme isthen to add more information than strictly required~i.e., three gauges instead of one! and to use the ex-cess information to get rid of temperature, pressure,and orientation influence. This concept is, of course,of great relevance to industrial applications, sincemost in situ strain sensors are inherently tempera-ture dependent and often require accurate position-ing.

In contrast with other in situ temperature-compensation methods, the FBG rosette evaluation ofthe uniaxial strain makes use of a rigorous calcula-tion ~i.e., deviatoric calculation! that does not rely onmatrix inversion. A great advantage over alterna-tive methods is that calculation of the uniaxial strainis inherently independent of temperature. Since itdoes not involve the temperature sensitivity of thetransducer ~e.g., FBG’s!, the rosette-based sensor canbe used over any arbitrary temperature range with-out requiring frequent recalibrations because ofchanges in the temperature-to-wavelength coefficientof FBG’s.

Finally, one can use this sensor for any isotropictest material simply by adjusting the Poisson ratio.~One must use an effective Poisson ratio instead ofthe structure material when making plate-bendingtests.! The deviatoric equation can be calculated inreal time, or the corresponding signal can be shapedby a dedicated electronic circuit. In addition to spec-tral resolution, a major source of inaccuracy stemsfrom angular fabrication tolerances. The precisionsof our rosette-based uniaxial strain sensor are ap-proximately 3 mεypm 1 2.5 mε for a strain of 1000 mεand 0.14 Kypm, and maximum errors of approxi-mately 3 mεypm and 0.11 Kypm are anticipated for aperfectly arranged rosette-based uniaxial strain sen-sor. Typical strain- and temperature-to-spectral-resolution relative errors are summarized in Table 3~only for methods involving spectral measurements!.Strain and temperature precisions are limited by themeasurement accuracy, i.e., 1 mεypm and 0.1 Kypm,respectively. The great limitation of most methodsrequiring matrix inversion is that the inversion isoften poorly conditioned because of a small determi-nant and, as a consequence, gives large errors andlow accuracy.

The satisfactory performances of this compensat-ing scheme allow us to be confident about its manytechnical applications. Clear advantages over alter-native methods are angular independence, high ac-curacy, and self-temperature compensation over abroad temperature range without recalibration. Al-though it has been demonstrated with FBG’s asstrain gauges, other transducers ~e.g., interferomet-ric! can be implemented in a similar manner.

However, as a proof of principle, this rosette hasbeen realized in fiber form. This implementationturned out to be difficult to handle, and the angularpositioning required a meticulous design. As a con-sequence, a fiber-based device would preferably berealized with printed circuit processes on a flexiblesubstrate ~e.g., bonded on an epoxy film!. Such a


Table 3. Comparison of Recovery Analysis Performances of Several Temperature-Compensation Methods of FBG’s Involving SpectralMeasurements in Terms of Relative Error to Spectral Resolution

Compensating Method

Relative ErrorsExtrinsic Versus

Intrinsic ReferenceStrain Temperature

Dual overlaid FBG’s at different centerwavelengths

17 mεypm 1.7 Kypm Intrinsic 22, 23

Bragg grating in a highly birefringent fiber Not indicated Not indicated Intrinsic 24Two FBG’s of different cladding diameters 17 mεypm 1 Kypm Intrinsic 25Two diffraction orders of a FBG 17 mεypm 1.7 Kypm Intrinsic 26One rocking filter and one FBG 40 mεy0.1 nm 0.25 Ky0.1

nmIntrinsic 27, 28

Two FBG’s and a long-period grating Not indicated Not indicated Intrinsic 29Fabry–Perot interferometer and FBG 1.25 mεypm 0.35 Kypm Extrinsic 30FBG rosette 3 mεypm 1

2.5 mε0.14 Kypm Intrinsic This

studyTwo FBG’s mounted on opposite surfaces of

a cantilever;1 mεypm Not measured Extrinsic 32

Passive temperature-compensating package 70 mε erroron 120 °C

range

Not measured Extrinsic 33, 34

substrate of small thickness is required to allow forlarge elongations ~several 10,000 mε! without signif-icant alteration of the strain distribution in the struc-ture under test. An even more efficient technologycould be integrated optics on polymers as a means ofachieving high accuracy in fabrication tolerances, re-producibility, and integration in a mass-producedprocess. Indeed, polymer substrates have beenshown to be photosensitive, and channel waveguidesand gratings have been photowritten in such sub-strates.11 Fibers or channel waveguides of great in-dex difference between the core and the cladding~1022! do not suffer significant curvature losses for abending radius as low as several millimeters. Con-sequently, we can anticipate that the design andelaboration of integrated rosettes of overall dimen-sions of approximately 10 3 10 mm2 or less may bepossible. This compactness is comparable with oreven smaller than that of their resistive foil counter-parts.

As a strain transducer the Bragg grating can beused to sense many other physical parameters bymeans of an appropriate strain-transducing mecha-nism. For example, one might use magnetostrictivesubstrates to measure the amplitude of a magnetic-field vector40 or, alternatively, an electric current ~forexample, in a transformer!. Such a Bragg gratingrosette-based sensor would be self-temperature com-pensated and independent of its angular orientationwith respect to the field vector. One might use pi-ezoelectric substrates or coatings as well to measurean electric-field vector. A section of fiber containinga Bragg grating can be bonded onto or embedded intomaterials that change size or shape in the presence ofthe electric field. One could also use FBG’s coatedwith a piezoelectric polymer jacket @e.g., poly-~vinylidene fluoride!# to perform such a task.

We thank C. Prioul and M. Pasquet of Ecole Cen-trale des Arts et Manufactures, Chatenay-Malabry,


France, for technical support with the finite-elementnumerical analysis resistance des materiaux soft-ware obtained from Universite du Mans, France! andW. W. Morey of 3M Bragg grating technology ~spe-cialty optical fibers! for fruitful collaborations and forproviding us with Bragg gratings.

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Documents

State-of-strain evaluation with fiber Bragg grating rosettes: application to discrimination between strain and temperature effects in fiber sensors