Internal strain measurement by fiber Bragg grating sensors in textile composites

Internal strain measurement by ®ber Bragg grating sensorsin textile composites

Xiaoming Taoa,*, Liqun Tanga,c, Wei-chong Dua, Chung-loong Choyb

aInstitute of Textiles and Clothing, The Hong Kong Polytechnic University, Hong KongbMaterials Research Centre, The Hong Kong Polytechnic University, Hong Kong

cDepartment of Engineering Mechanics, South China University of Technology, PR China

Received 10 March 1998; received in revised form 25 May 1999; accepted 2 November 1999

Abstract

This paper is concerned with a study of internal strain measurement by ®ber Bragg grating sensors (FBGSs) embedded insidetextile composites. First, on the basis of the measurement principle of FBGSs, the e�ects of transverse strain and temperature on

the measurement results are discussed. A composite model comprising an optical ®ber, coating and resin was developed to deter-mine the measurement e�ectiveness of an embedded optical ®ber sensor by analyzing the strain ®eld of the system under a uniformthermal load. Factors in¯uencing the measurement e�ectiveness were considered including the elastic modulus and Poisson's ratio

of coating material, tension sti�ness ratio and the length of the host. Secondly, an experimental investigation was carried out todetermine the reliability of FBGSs embedded in textile composites with di�erent interfaces of ®ber/coating and coating/resin andwith two con®gurations, that is, single-ended and dual-ended. The measurement errors induced by the deviation of position and

direction of the sensors were estimated. Finally, Volanthen's low-coherence technique was applied to measure the internal straindistribution along the length of FBGSs which were embedded into textile composite. # 2000 Elsevier Science Ltd. All rightsreserved.

Keywords: Textile composites; Smart materials; Non-destructuve testing; Thermomechanical properties

1. Introduction

Since the successful introduction of permanent Bragggrating in the core of an optical ®ber by the UnitedTechnologies Research Center [1,2], a new type of opti-cal ®ber sensor called a ®ber Bragg grating sensor(FBGS) has been developed rapidly [3±7]. The FBGShave all the inherent advantages of optical ®bers such aslow density, small size, ¯exibility, ease of embeddmentinto a structure, and immunity to electromagnetic ®elds.In addition, they are capable of localized, distributiveand absolute measurement with good linearity, havehigher potentials for mass production comparing withother kinds of optical ®ber sensors such as FP (Fabry-Perot), TM (two-mode), P (polarimetric) sensors. A

detailed review and analysis of the sensors have beengiven by Udd [5] and Measures [6].Because of all the above-mentioned advantages,

FBGSs have been applied as embedded sensors tomonitor or measure the internal strain of compositestructures [8±16]. Friebele [9,10] reported internal dis-tributed strain sensing with ®ber grating arrays embeddedin continuous resin-transfer-molded (CRTMTM) compo-sites; Bullock and his colleagues [11] used a trans-lami-naminar embedded ®ber grating sensor system to monitorthe structural integrity of composites; Du and Tao [16]embedded ®ber Bragg gratings in a glass-®ber-cloth lami-nated structure for internal strain measurements.The coupling problems between the embedding host

and the sensor appear inevitably. In some special casesthe optical ®ber embedded into the composite hosts mayhave signi®cant e�ects on the strength, sti�ness, evenfailure properties of hosts [17±31], and these e�ects canin¯uence the reliability of the measurement results of

0266-3538/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

PI I : S0266-3538(99 )00163-3

Composites Science and Technology 60 (2000) 657±669

* Corresponding author.

®ber optic sensor [17,18]. Many people investigatedthese problems: Sirk and Dasgupta [19], Pak [20], Madsen[21], Hadjiprocopiou [22,23], and Levin [18,24] studied thestress concentration induced by embedded optical ®bersand the optimized parameters to minimize the stressconcentration; the strength and sti�ness of compositesa�ected by embedded ®ber optical sensor were studiedby Jensen [25], Sirkis [26] and Mall [27].The debonding at the interfaces of ®ber/coating and

coating/resin is a phenomenon which cannot be neglec-ted. Micrographs of test specimens containing embed-ded sensors by Haasksma [28] showed de-laminationoccurring at the coating/optical ®ber interface in pre-ference to the coating/epoxy resin interface; Henkel [20]provided a crack growth mechanism for debonding ofthe optical ®ber and loss of sensor performance; LeBlanc [14] indicated that high interfacial stresses at thetip of the sensor can result in debonding of the sensorand, consequently, the loss of the sensor's function;Zhong [30,31] discovered that the fracture process wassensitive to the cohesive properties of the polymer aswell as the adhesive bond strength of the interface.The principle of measurement by embedded FBGS

has been based on two assumptions made by Butter andHocker [32]: (1) the optical ®ber's axial strain is equal tothat of host in the optical ®ber direction; (2) there is notransverse strain from the host to be coupled to theoptical ®ber. Then the shift of the re¯ective peak wave-length has a linear relation with the strain in the host inthe optical ®ber direction. However, many problemsrequire further investigation. For instance, the couplingof strain and temperature, the e�ect of embedded FBGSon the strain ®eld of host and to what an extent theoptical ®ber's axial strain can represent the host strain,even in the optic ®ber direction. In relation to measure-ment reliability, the uniformity of sensors, interfaces aswell as deviation of position and direction of optical®ber may a�ect the measurement results.This paper is mainly concerned with the problems

mentioned above. First the measurement principle ofFBGS will be analyzed to quantify the e�ects of trans-verse strain and temperature on the measurementresults. Secondly, the strain ®eld of a host embeddedwith a coated optical ®ber will be determined under auniform thermal load. This will illustrate how much theembedded ®ber a�ects the initial strain ®eld of the hostand whether the axial strain of optical ®ber can repre-sent the strain of the surrounding host. Thirdly, aninvestigation will be presented to determine the relativemeasurement error of FBGSs embedded in textile com-posites with various types of interface and sensor con-®guration, as well as the error induced by the deviationof position and direction of the sensors. Finally, a tech-nique to measure the internal strain by distributed grat-ing sensors will be introduced and discussed with someexperimental results.

2. Principle analysis of FBGS

The sensing element of FBGS sensor consists of amodulation of the index of refraction along a shortlength of the core in a germania-doped silica ®ber [2±4],normally by means of exposure to an interference pat-tern of intense ultraviolet pulses. The optical re¯ectivespectrum comprises a very narrow spike in which thepeak wavelength is the Bragg wavelength l; which hasthe following relation to the linear core index of refrac-tion n for a guided mode, and the period of the indexmodulation �:

l � 2n� �1�

Any change in either n or � may cause a shift of theBragg wavelength l; which will be served to determinethe strain or the temperature environment to which theoptical ®ber is subjected.If we assume that the change in the index modulation

period is independent of the state of polarization of theinterrogating light and only dependent on the ®ber axialstrain, di�erentiating the Bragg wavelength in Eq. (1)yields:

�ll� "1 ��n

n�2�

where "1 is the total axial strain of optical ®ber. Gen-erally speaking, l and n will have di�erent values in thedirections of polarization. We will denote subscript i=1, 2, 3 for l and n as their values in the de®nedpolarized direction. A local Cartesian coordinate systemwill be set-up, with 1, 2, 3 representing the three princi-pal directions respectively. Eq. (2) can be rewritten as:

�lili� "1 ��ni

ni�3�

For strain, we use subscript � j �=1, 2, 3, 4, 5, 6. The®rst three represent the normal strains in the 1st (®beraxis), 2nd, 3rd direction respectively, the latter threebeing the three shear strains respectively. The strain " ofan optical ®ber may be contributed by either thermalexpansion or stress, hence the symbol "� is used for theoptical ®ber strain induced by stress only. The refrac-tion index n is related to both temperature T and strain"�; therefore:

�nini� 1

ni

Xj

@ni@"�j

"�j �@ni@T

�T �4�

According to the strain optic theory [33]:

1

ni

Xj

@ni@"�j

"�j � ÿn2i2

Xj

Pij"�j �5�

658 X. Tao et al. / Composites Science and Technology 60 (2000) 657±669

where Pij is the strain-optical coe�cient matrix. For ahomogeneous isotropic medium,

Pij �

P11 P12 P12

P12 P11 P12 0P12 P12 P11

P44

0 P44

P44

26666664

37777775 �6�

where P44 � �P11 ÿ P12�=2:For a homogeneous isotropic medium, it can be

assumed that the index of refraction n has the linearrelation with temperature T:

� � @ni@T

�7�

where � is regarded as a thermo-optic constant.Because lights are the transverse waves, only the

transverse (2, 3 directions) deviations of the re¯ectiveindex can cause the shift of the Bragg wavelength. Sub-stitute (5), (6) and (7) into (4), the peak wavelengthshifts for the light linearly polarized in the 2nd and 3rddirection are given as below:

�l2l2� "1 ÿ n22

2�P11"

�2 � P12�"�1 � "�3�� T �8�

and

�l3l3� "1 ÿ n23

2P11"

�3 � P12�"�1 � "�2�

� �� T �9�

In the general case, the wavelength shift for the Bragggrating sensor observed for each polarization eigen-mode of the optical ®ber depends on all the three prin-cipal strain components within the optical ®ber. Sirkisand Haslach [34] extended the Butter and Hocher'smodel [32] and have shown that their results are closerto those observed in transverse loading experiments [35]for the interferometric optical ®ber sensor. Here we willonly discuss the axisymmetric problem where "�2 � "�3: Ifthe optical ®ber is a thermal isotropic material with aconstant expansion coe�cient �; then "�j � "j ÿ ��T( j=1, 2, 3). Eqs. (8) and (9) can be written into the sameform:

�ll� "1 1ÿ n2

2P12 � �P11 � P12� "2

"1

� �� n

2

2�P11 � 2P12��T� ��T � f"1 � ��T

�10�

where

f � 1ÿ n2

2P12 � �P11 � P12� "2

"1

� ��11�

and

�� n2

2�P11 � 2P12� �12�

f is de®ned as the sensitivity factor, �� as the revisedoptical±thermal constant.

2.1. Sensitivity factor

Let us de®ne �� ÿ"2="1� as the e�ective Poisson'sratio (EPR) of optic ®ber. From Eq. (12), it is obviousthat the sensitivity factor f is not a constant but afunction of ��;

f � 1ÿ n2

2�P12 ÿ �P11 � P12�� 13�

Fig. 1 shows a typical curve of the sensitivity factor as afunction of the e�ective Poisson's ratio, calculated byusing the material parameters of optic ®ber, provided byBerholds and Dandiliker [36] in Table 1.Let us consider the following special cases:

1. �� 0:17; f � 0:798: implies that the EPR is equalto the ®ber material Poisson's ratio and meets therequirement of the Butter and Hocher [32]assumption. The value of sensitivity factor,f=0.798, is recommended by many FBGS manu-facturers.

2. �� ÿ1; f � 0:344: implies that the strains in threeprincipal directions of the ®ber are equal, whichcorresponds to the case of static uniform stress orthe case of thermal expansion;

3. �� 0:0; f � 0:732; implies that there is no trans-verse deformation.

Therefore, if a FBGS is used as an embedded sensor,it is necessary to make a correction of the sensitivityfactor with respect to the transverse principal strain. Or

Fig. 1. Sensitivity factor plotted against e�ective Poisson's ratio.

X. Tao et al. / Composites Science and Technology 60 (2000) 657±669 659

only when the transverse principal strain of the optical®ber is not sensitive to the host's strain ®eld, the sensi-tivity factor can be regarded as a constant.

2.2. Temperature compensation

As an embedded strain sensor, ideally, the measuredstrain "1 in (10) should represent the host strain in theoptical ®ber direction. The temperature compensationcan be made simply by:

"1 � ��l=lÿ ��T�=f �14�

For the germania doped silica core [37], the thermal-optical coe�cient � is approximately equal to 8:3�10ÿ6; then the revised constant �� is 8:96� 10ÿ6: If themeasured strain is greater than 0.001 and the variationof the temperature is smaller than 10�C, comparativelythe term ��T in Eq. (14) is one order of magnitudesmaller than that of the strain, then the temperaturecompensation would be unnecessary in some cases.

2.3. Coupling of temperature and strain

If the internal temperature of the host is unknownand its contribution to the shift of the wavelength iscompatible to that of the strain, it is impossible todetermine strain and temperature from Eq. (10) only.Additional relation between physical paramters has tobe used in order to distinguish the intrinsic strain andtemperature. For Fabry±Perot sensors, the birefringencemethod was introduced by Leilabady and his colleagues[38]. A novel ®ber Bragg grating Fabry±Perot sensorhas been conceived [39] for simultaneous measurementof temperature and strain by measuring the opticalphase shift or power change and wavelength shift of thelight re¯ected from the sensor.

2.4. Temperature sensor only

If a stand-alone FBGS is subjected to a temperaturechange, then:

"1 � "2 � ��T �15�

where � is the isothermal expansion coe�cient of optic®ber. Substituting (15) into (10), we can derive thefollowing:

�ll� f��1ÿ n2

2�2P12 � P11�� g�T

� �� T �16�

Usually � is over 10 times greater than � [3] (for silica,� � 0:55� 10ÿ6; � � 8:3� 10ÿ6), therefore the e�ect ofthermal expansion on the measurement result is negli-gible for a stand-alone FBGS.

3. Measurement e�ectiveness of FBGS

For FBGS to be an ideal embedded strain sensor in atextile composite, the following conditions will have tobe met:

1. The integrated optical ®ber has little e�ect on thehost strain ®eld;

2. The axial strain of optical ®ber "1 can representthe nearby host strain in the optical ®ber direction;

3. The e�ective Poisson's ratio �� is constant duringthe measurement period.

In the succeeding section, we will examine the e�ec-tiveness of an embedded FBGS with respect to theseconditions. A special case will be considered, in which athermal load is applied to a host embedded with anoptical ®ber, then the strain ®elds of both host andoptical ®ber will be studied. The factors in¯uencing themeasurement e�ectiveness will be discussed.

3.1. Three-layered composite and numerical analysis

The optical-®ber±host system can be illustrated as acomposite comprising a cylindrical ®ber and two con-centric shell layers, with ®ber, coating and host from thecenter to the out surface (Fig. 2). The height of thecomposite cylinder is 2H, the radius of ®ber is R1; which

Table 1

Material parameters of optical ®ber

Strain-optical

coe�cient

Index of

refraction

Elastic

modulus

Poisson's

ratio

P11 P12 n E (GPa) �0.113 0.252 1.458 70 0.17

Fig. 2. Three-layer composite model of host, coating and ®ber.


is also the inter radius of coating, R2 is the outer radiusof coating and the inter radius of host, R3 is the outradius of the host. The following basic assumptionshave been made so that the case can be simpli®ed asaxial- symmetric problem:

. The optical ®ber, coating of ®ber and host are lin-ear elastic;

. The thermal expansion coe�cients of ®ber, coatingand host are constants;

. There is no discontinuity in displacement at theinterfaces of ®ber and coating, coating and hostunder loading;

. Thermal load is uniform in the whole compositecylinder.

The ®nite element (FE) analysis was applied by usingthe program ABAQUS. Four node bilinear elements(CAX4) were used to generate meshes. Because of thesymmetry to the r axis, only half of the cylinder wasconsidered for the strain analysis. The boundary condi-tions are ujr�0� 0 and vjz�0� 0; where u; v are the dis-placements along the r; z direction, respectively. The twoends (z � �H) are assumed to be free from any externalforce and no constraint on displacement. The materialconstants and dimensions are listed in Table 2.

3.2. Numerical results and discussions

A thermal load of �T=25�C was applied to the sys-tem. Without the embedded ®ber, the strain ®eld shouldbe uniform in the host and equal to:

"0 � �h ��T � 0:000325 �17�

As the strain in ®ber axial direction "z is the mostimportant term in our discussion, if without specialnoti®cation, the normalized strain we mention in the

rest of this section means the normalized strain in the®ber axial direction e given by:

e � "z="0 �18�

3.2.1. General views of the normalized strain distributionFig. 3 illustrate the normalized strain distribution in

the ®ber, host and coating, respectively, along the ®beraxial direction with various values of radius r. Fig. 3(a)shows that the normalized strain of ®ber declines withthe increasing length z. Because the thermal expansioncoe�cient of optical ®ber is more than one order ofmagnitude smaller than that of host, the strain level ofoptical ®ber depends largely on the restriction of host.As the middle part of optical ®ber is restricted morethan the parts near two ends, the strain of middle optic

Table 2

The material constants and dimensions used for computationa

Parameter Fiber Coating Host

E (GPa) 72.9 0.045±72.9, 0.045� 49.

� 0.17 0.1±0.45, 0.34� 0.288

� (10ÿ6/�C) 0.45 10 13.38

R1 (mm) 0.063

R2 (mm) 0.123

R3 (mm) 0.323±9.123, 1.0�

H (mm) 10±90, 10�

a E; � and � are the elastic modulus, Poisson's ratio and thermal

expansion coe�cient respectivelt, subscripts ``h'', ``c'', ``f'' represent

®bre, coating and host respectively. The host parameters are those of

solidi®ed polyester. The values with ``*'' are the default values for the

commercial material system and the experimental set-up used in the

present investigation. Fig. 3. Distribution of ®ber's axial strain at various radial positions.


®ber is closer to that of the host without embedded ®bersensor (e � 1). The strain distribution curves are iden-tical regardless the radial position in the ®ber, thus thestrain of ®ber can be regarded as a function of z only.Fig. 3(c) shows the reverse trends of strain distribu-

tion in the host. The strain level of the middle host islower than that of the host near the boundary, which isbecause the middle host is restricted more by the ®berand the host near boundary can expanse more freelyunder a thermal load. The normalized strain values ofthe host are very close to 1 (0.995<e<1), which impliesthat the host with embedded ®ber sensor has a strain®eld very close to that without embedded optical ®ber.Because of its lower elastic modulus, the strain dis-

tribution of the coating is signi®cantly a�ected by boththe ®ber and host, as shown in Fig. 3(b). The straindistribution along the z is similar with that of the ®berwhen r approaches R1; and the strain near the outersurface of coating varies little along z like that of host.

3.2.2. E�ects of parameters on e�ectiveness coe�cient3.2.2.1. Measurement effectiveness coefficient of FBGS.Fig. 3(a) shows that the normalized strain e declinesmonotonously with z. A term H95 was introduced, atwhich e has a value of 0.95. The physical meaning ofH95 is that only when zj j of optical ®ber grating is

smaller than H95; the measurement result of ®ber ise�ective. The length of the zone represents the limits ofe�ective measurement of host strain by an embeddedFBGS. The longer the zone length, the more e�ective aFBGS in a host. Thus the relative length of the e�ectivezone is de®ned as the e�ectiveness coe�cient � by:

� � H95

H�19�

3.2.2.2. Coating's elastic modulus Ec. Fig. 4 shows thatincreasing Ec leads to a sharp increment in � when Ec

varies from 0.045 to 1 GPa. Then the curve reaches aplateau and the e�ect on � becomes very small. ThusEc=1 GPa can be regarded as a threshold value withthe conditions under our current investigation. Appli-cation of coating materials with an appropriate elasticmodulus is a very desirable method to increase mea-surement e�ectiveness, especially in the case of shorthosts.

3.2.2.3. Poisson's ratio of coating. The e�ect of coatingPoisson's ratio �c on � is demonstrated in Fig. 5. In thepermissible range, � declines approximately linearlywith the increasing Poisson's ratio of coating.

3.2.2.4. Tension stiffness ratio. Variation in the outerradius of the host means the change in the thickness andthe tension sti�ness of the host as well. A tension sti�-ness ratio R was de®ned as the ratio of the host tensionsti�ness to that of optical ®ber:

R � EhAh

EfAf�20�

where Ah; Af are the cross-sectional area of the host andoptical ®ber, respectively.Fig. 6 shows that � increases with increasing R: Simi-

lar to the e�ect of the coating elastic modulus, there isa threshold value of log�R�=2.5 with our current

Fig. 4. E�ect of elastic modulus of coating on the measurement e�ec-

tiveness of optical ®ber.

Fig. 5. E�ect of Poisson's ratio of coating on the measurement e�ec-

tiveness of optical ®ber.

Fig. 6. E�ect of tension sti�ness ratio of host to ®ber on the mea-

surement e�ectiveness of optical ®ber.


calculation conditions, i.e. the tension sti�ness of host isover 300 times more than that of optical ®ber. In mostcases, this threshold can be reached easily if the host'sradius is 20 times more than that of optic ®ber. How-ever the increment of � is very limited by increasing,R; � < 0:3, as shown in Fig. 6, where the default valuesof a soft coating are used (see the default value in Table2).

3.2.2.5. Length of the cylinder. The e�ectiveness, �;increases with the increment of the half length of thecylinder H; and approaches unity, as shown in Fig. 7.However the incremental rate becomes smaller gradually.The measurement e�ectiveness needs to be consideredwhen the host dimension is small in the measurementdirection.

3.2.3. E�ective Poisson's ratio of optical ®berFig. 8 plots the e�ective Poisson's ratio of optical

®ber against the radial position r:Within a range from 0to 7 mm, the e�ective Poisson's ratio declines slightlywith the increment of z: Thus it can be regarded as aconstant along the z axis except for the short portionnear the boundary. The value of �� near z � 0 is 0.13which is not equal to the Poisson's ratio of the ®ber(0.17). This indicates that the transverse strain of optical®ber is not dominated by the host strain (otherwise it

should be close to ÿ1) but the Poisson's ratio of the®ber. By substituting this value into Eq. (11), the sensi-tivity factor f � 0:783 which is close to the sensitivityfactor f � 0:798: Fig. 8 also illustrates that the e�ectivePoisson's ratio of the optical ®ber is a function of zonly, independent of the radial position, �r�.It was suggested in the previous discussion that to

increase the elastic modulus of coating Ec properly is avery desirable means for the improvement of the e�ec-tiveness of optical ®ber. In the particular case under ourinvestigation, the elastic modulus of the coating shouldbe equal to or greater than the threshold value Ec � 1GPa, if the measurement e�ectiveness is concerned.However, the increment of Ec will a�ect the value of thee�ective Poisson's ratio, ��; then the sensitivity factor, f.Fig. 9 plots �� as a function of Ec; which exhibits thereversed trend compared with that of the e�ectiveness �:When the coating modulus is smaller than or equal tothe threshold value of 1 GPa, the variation of �� israther small. By considering the e�ects of coating elasticmodulus on both measurement e�ectiveness and e�ec-tive Poisson's ratio, the optimal coating modulus shouldbe chosen as the threshold value Ec � 1 GPa.

4. Reliability of FBGS

The analysis on measurement e�ectiveness analysis inthe previous section is based on the composite modelassuming perfect interfaces between the coating and®ber as well as the coating and host. Another conditionis the strain measured by the optic ®ber sensor is at apredetermined position and in a predetermined direc-tion. Hence, this section will investigate their e�ectswhen these assumed conditions are not met.

4.1. Experimental design

4.1.1. FBGS and textile compositeThree types of interfaces were introduced in the sen-

sor/composite system. The ®rst is the control with nor-mal coating conditions. The second and third types were

Fig. 7. E�ect of height of host on the measurement e�ectiveness of

optical ®ber.

Fig. 8. Distribution of ®ber's e�ective Poisson's ratio along the ®ber

length at various radial positions.

Fig. 9. E�ect of elastic modulus of coating on ®ber's e�ective

Poisson's ratio.


made by applying PVA between the ®ber and coating,and the coating and resin, respectively. The details arelisted in Table 3.Plain-woven glass fabrics and polyester resin were

used to fabricate two laminar composite specimen. Asshown in Fig. 10, two con®gurations of the optical ®ber,that is, single-ended and dual-ended optical ®ber, wereembedded into the two composite specimens, respec-tively, each with 36 layers of fabrics (Fig. 11). In eachcomposite specimen, there were six dual-ended or sin-gle-ended FBGSs with the three classes of interfacestructures. The sensors were placed inside the skin lay-ers from the top and bottom sides of the compositespecimen.

4.1.2. Three-point bending testThree-point bending tests of the composite beam were

carried out using an apparatus as shown in Fig. 12. Thespecimen was simply supported on its two ends, thegrating position was in the middle of the two supportedpoints, the distance l between the two supported pointswas 350 mm. A sti� bar was applied across the beam toproduce the transverse load at the middle point of thebeam. The vertical displacement � of the sti� bar wascontrolled and recorded by a micrometer.

4.2. Error analysis of normal strain induced by deviationof the position and direction

With the assumption that the Yong's modulus in thex-direction of specimen is a constant, the strain of thespecimen can be deduced as:

"�x; y� �3�xy

a2�lÿ a� �0 < x < a�3��lÿ x�ya�lÿ a�2 �a < x < l�

8>><>>: �21�

4.2.1. E�ects of positional and directional deviationIdeally, all the FBGS should be laid along the x-axis

as shown in Figs.11 and 12. If there are small deviationsdx, dy in the x; y direction, respectively, and let thedirectional cosines of the new deviated direction o� thex-axis be:

cos�d�1�; cos��2ÿ d�2�; cos��

2ÿ d�3�

� �or cos�d�1�; sin�d�2�; sin�d�3��

�22�

The relative error Re can be deduced as:

Re � Rp � Rd � dy

y

��

�dx

x

�� dx

lÿ x

�� d�1�2 �

"d�2

�� 0 < x < a�

�a < x < l�

8>><>>:�23�

Table 3

The classes of the interfaces of sensors

Class Interface construction

1 Fiber+coating+resin

2 Fiber+PVA+coating+resin

3 Fiber+coating+PVA+resin

Fig. 10. Two sensor con®gurations: dual-ended and single-ended ®ber Bragg grating sensors.

Fig. 11. Textile composite samples with dual-ended and single-ended optical ®ber sensors, respectively.


where Rp; Rd are the relative errors induced by thedeviations of position and direction, respectively. Com-paratively, the relative error induced by the deviation ofposition is much greater than the term by the deviationof direction.When x � a and a � l=2; the relative error induced by

the positional deviation in the x axis has the smallestvalue. When y � h=2; the relative error induced bypositional deviation in the y-axis reaches the smallestvalue, and the relative error induced by the directionaldeviation in the y-direction is also the minimum underthe condition � 0 and " � "max:

4.2.2. Estimations of the maximum relative errorThe maximum relative error can be estimated based

on the parameters used in experiment as shown in Figs.11 and 12:

Re � Rp < 7:02% �24�

In our experiments, the sensors were embedded in theoptimal position of the minimized relative error. If thesensors were embedded in a large gradient strain ®eldsuch as nearby the tip of a crack, the relative errorinduced would be enormous.

4.3. Experimental results and discussion

4.3.1. Calibration of FBGSBefore being embedded into composites, all 12 FBGS

were calibrated by a simple tensile experiment. Thesensitivity factors determined are listed in Table 4,showing a small variation between individual FBGS.

During the calibration, some sensors broke when thestrain was far less than their critical strain, whichimplies that these sensors may be damaged and losttheir strength. Hence, the calibration procedure can alsohelp one to choose the good sensors to be embeddedinto the composite.

4.3.2. Dual-ended FBGSFig. 13 illustrates the measured strains of all dual-

ended FBGSs plotted against the loading displacement.From Eq. (21), the strain and displacement should havea linear relation, and all the measurement lines shouldbe identical. The experimental results in Fig. 13 showthat all the dual-ended FBGS worked well and measuredvalues were consistent with the theoretical predictions.Though the sensors had di�erent interface structure andmight debond at the interfaces during testing, butbecause debonding occurred in a local area, other partsof the sensor remained in coherence, which kept thedeformation of the ®ber near the grating with that ofhost. This result implies that the interface debondinghas little e�ect on the strain values measured by dual-ended FBGS.Fig. 14 shows that the maximum average relative

error between the di�erent sensors was approximately7%, which was in the same range of the maximumrelative error induced by the location.

4.4. Single-ended FBGS

The single-ended FBGS with Class 1 interface struc-ture (no PVA) worked well during the experiment. Figs.15 and 16 show the measured strain values plottedagainst the loading displacement of FGBSs with Class 2and Class 3 interface respectively. It was obvious that

Fig. 12. Set-up of three-point bending experiment.

Fig. 13. Experimental results of the dual-ended sensors.

Table 4

The corrected sensitive factor

Sensor 1_1 1_2 1_3 1_4 1_5 1_6

Sensitivity factor 0.7697 0.7856 0.8048 0.7964 0.8030 0.8060

Sensor 2_1 2_2 2_3 2_4 2_5 2_6

Sensitivity factor 0.8066 0.7946 0.8066 0.8041 0.7922 0.8001

Fig. 14. Relative error of measured values of strain.


these two sensors lost their function. Because thesesensors were covered PVA between ®ber/coating andcoating/resin, respectively, the loss in function is mostlikely contributed by the debonding at the interfaces.Single-ended FBGS are very sensitive to the interfaceconditions.

5. Measurement of strain distribution

5.1. Introduction

By the means of measuring the shift of the peakwavelength of the re¯ective spectrum, the average strainover the whole grating length is measured rather thanthe distribution of strain. In many applications, the dis-tribution of strain is of the major interest, thus a num-ber of methods can be implemented to serve thispurpose. The grating length of FBGS can be made veryshort (e.g. 2 mm), thus the strain measured by the shiftof the peak wavelength of the re¯ective spectrum can beregarded as a localized strain. Multiplexing of suchFBGS arrays can be used to obtain pseudo distributedmeasurement [39].If it is monotonic, the strain distribution can be

determined along the grating by analyzing the re¯ectionspectrum [41±44]. Volanthen and his colleagues [45±47]

developed a new measuring system of distributed grat-ing sensor to measure the wavelength and re¯ectivity ofgratings as functions of time delay. They employed thelow-coherence re¯ectometry to measure strain witharbitrary distribution. This technique was applied toour three-point bending experiments of a textile com-posite embedded with FBGS.

5.2. Measurement system and materials

Fig. 17 shows the Volthanen's measuring system. Thelocation in the grating to be interrogated is selected bybalancing its path length with that of mirror, while theinterrogation wavelength is determined by the tunable®lter. The wavelength of the ®lter is tuned by a voltagecontrolled oscillator (VCO). When the wavelength ofthe ®lter is close to that at the location under inter-rogation, a coherent optical interference is observed atthe received end. The coherence path-length can beadjusted by a micro-meter to observe several inter-ference fringes. When the amplitude of the fringes ismaximized, the peak ®lter wavelength matches the localBragg wavelength. To summarize, the distance is mea-sured from the variable delay, the wavelength is mea-sured from the wavelength of the tunable ®lter and there¯ectivity is measured from the amplitude of the inter-ference fringes, or the optical output power.The specimen under investigation was a laminate tex-

tile composite comprising 18 layers of fabrics, similarwith that shown in Fig. 11, and a distributed gratingoptic ®ber. The three-point bending experiment wascarried out using the same set-up as shown in Fig. 12.The length of grating is 2.5 cm, one end of the grating isat the center of the two supporting points. The strain inthis composite beam has a distribution as described byEq. (21).

6. Results and discussions

Fig. 18 plots the theoretical and experimental dis-tributions of strain along the x-direction (starting from

Fig. 16. Experimental results of FBGS with PVA between the coating

and resin.

Fig. 15. Experimental results of FBGS with PVA between the ®ber

and coating.Fig. 17. The Volanthen's low-coherence measurement system.


the mid-point) while the specimen under given trans-verse load in three-point bending experiment. It is evi-dent that the trends of the distribution were quitesimilar, especially the slopes of the two lines were thesame. The relative error between the experimentalresults and theoretical prediction is about 9%. Possiblecauses of this discrepancy in the absolute strain may beexplained by the following: (1) The theoretical predic-tion was based on the estimated height of the ®berinside the composite beam. Our previous analysis onreliability has indicated that the deviation of verticalposition will cause signi®cant amount of relative error.In particular the thickness of the specimen was only ahalf of the specimen used for the reliability test. (2) Inthis experiment, the adjusted length between lens andmirror was shorter than that of the grating, i.e. themeasurement did not cover the whole length of thegrating, and there was no reference to indicate the realposition in the grating. The estimated x values mayinclude error. (3) In the experiment we did not use PZT(Fig. 17) to additionally sinusoidally modulate theinterference path to enhance the interference fringes.The detection of the peaks of the interference fringeswas rather di�cult and may add some operationalerror.Hence the establishment of a reference point for the

distributed strain measurement is very important. Toembed a supplementary FBGS near a distributed grat-ing sensor would be desirable, for the supplementaryFBGS can detect strain of a given point whose strainvalue measured by the supplementary FBGS can beused as a reference for the measurement results of adistributed grating sensor, if there are not too manypoints along the distributed grating to have the samestrains.

7. Conclusions

Our investigation of the principle, e�ectiveness andreliability of embedded FBGS and Volanthen's dis-tribute grating sensors in textile composites havebrought the following conclusions:

1. The sensitivity factor f of FBGS is not only afunction of the axial strain but also a function ofthe ratio of the transverse strain over the axialstrain or e�ective Poisson's ratio (EPR). When aFBGS is embedded in a composite, the EPR ofoptic ®ber is no longer equal to the Poisson's ratioof optical ®ber but some other values dependingon the property combination of the ®ber, coatingand resin system as well as the type of loading.Therefore the sensitivity factor may not be a con-stant and may have a few values.

2. The temperature compensation for a FBGS isnecessary if the variation of the temperature islarge during the measurement. However whenthe variation of temperature is small (<10�C)and the measured strain is greater than 0.1%,the temperature e�ect on the measurement isnegligible. In a pure temperature measurement,the e�ect of thermal expansion of the ®ber isnegligible.

3. The embedded optical ®ber sensor has little e�ecton the initial strain ®eld of host under our currentinvestigation, because of its small size.

4. When the length of a grating is compatible withthe dimension of a composite, considerationshould be given to the measurement e�ective-ness as the strain of the FBGS may not repre-sent that of the composite host. Selection ofappropriate coating is very important to obtaina high level of measurement e�ectiveness. Themost e�ective way to improve the measuremente�ectiveness is by increasing the coating elasticmodulus properly, which also can serve thepurpose to ensure a stable sensitivity factor ofoptical ®ber. The increment of the Poisson'sratio of coating may reduce the e�ectiveness ofoptical ®ber;

5. The local debonding at the interfaces of ®ber/coating and coating/resin has great e�ect onthe reliability of the single-ended FBGS, but ithas little e�ect on that of the dual-ended FBGS.Hence, it is recommended to use dual-endedFBGS wherever possible or let the embeddedend of the FBGS be long enough from the grating.To reduce the measurement error, determinationof real location of the grating of FBGS inside thecomposite may be a key issue for future work,especially when the sensors are embedded in alarge gradient strain ®eld such as near the tip of acrack.

6. Apart from multi-plexing FBGS arrays which canachieve a pseudo distributed measurement ofstrain, the Volanthen's method is another feasibleway to measure the internal strain distribution ofthe host, but a reference for determining the realposition of the measured strain is necessary.

Fig. 18. The comparison between strain distribution measured by the

distributed grating sensor with that of theoretical prediction.


Acknowledgements

We would like to acknowledge the Hong KongResearch Grant Council for supporting this work(Grant No. Polyu123/96E). We also appreciate thetechnical assistance by Mr. Yu Jian Ming, in preparingthe composite samples.

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Internal strain measurement by fiber Bragg grating sensors in textile composites