Damage Assessment in Composites using Fiber Bragg Grating Sensors. Mohanraj Prabhugoud

Transcription

1 ABSTRACT PRABHUGOUD MOHANRAJ. Damage Assessmen in Composies using Fibe Bagg Gaing Sensos. (Unde he diecion of Assisan Pofesso Kaa J. Pees). This disseaion develops a mehodology o assess damage in composies using fibe Bagg gaing (FBG) sain sensos. Fis, a sain-ansfe model using he finie elemen (FE) mehod is developed o simulae he esponse of an embedded FBG o he applied loading. This FE model is also able o calculae biefingence in he FBG due o applied ansvese loads. The model is validaed consideing he wo-dimensional poblem of diameical compession of polaizaion-mainaining fibes. A modified T-maix model is hen fomulaed o simulae he esponse of an embedded FBG due o an applied axial sain field. The esponse of FBGs suface mouned on PMMA and wo-dimensional woven composies subjeced o muliple low velociy impacs is expeimenally invesigaed. The complex specal esponse is elaed o he esidual sains afe impac in he PMMA specimens and he suface sain o failue in he wo-dimensional woven composies. The feasibiliy of using FBGs o measue inenal sain in woven composies duing damage pogession is finally consideed.

2 Damage Assessmen in Composies using Fibe Bagg Gaing Sensos by Mohanaj Pabhugoud A disseaion submied o he Gaduae Faculy of Noh Caolina Sae Univesiy in paial saisfacion of he equiemens fo he Degee of Doco of Philosophy Mechanical Engineeing Raleigh 25 Appoved By: D. Eic C. Klang D. Tasnim Hassan D. Kaa J. Pees Chai of Advisoy Commiee D. Mohammed A. Ziky

3 To my Paens ii

4 iii Biogaphy MOHANRAJ PRABHUGOUD was bon in Bangaloe, India. He was bough up in Bangaloe, whee he aended Cambidge School fo his seconday educaion. He majoed in Science a S. Nijalingappa College. He oped fo a caee in Engineeing and aended M. S. Ramaiah Insiue of Technology, Bangaloe, fom Sepembe 995 o Sepembe 999. He gaduaed wih a Bachelo of Technology degee in Mechanical Engineeing. In Januay 2 he moved o he Unied Saes fo his gaduae sudies a Noh Caolina Sae Univesiy, Raleigh, NC. He obained a Mase of Science in Mechanical Engineeing in 22. In Sping 23, he saed woking owads his Ph.D. in Mechanical Engineeing.

5 iv Acknowledgemens I would like o hank D. Kaa Pees fo he suppo, encouagemen and invaluable guidance houghou he couse of his wok. She was one of he fis people o fomally inoduce me o he wondeful wold of fibe opics and has since moivaed me o pusue a eseach caee in his field. I am exemely hankful fo he financial suppo povided by Naional Science Foundaion, which enabled me o pusue his docoal pogam. I would like o hank Advanced Composies fo donaing he composie will and Coning Inc., fo donaing he gaings used fo he expeimens. I would like o specially hank D. Mohammed Ziky fo his valuable inpus fo he plexiglass expeimens. I would like o hank James D Peason fo being available a all imes (wih uniing enhusiasm) and helping me ou wih he expeimens especially using he dop owe. I would like o hank my commiee membes, D. Eic Klang, and D. Tasnim Hassan fo seving on my commiee. I would like o exend my hanks o Shaon Kiesel, Mike Sia, Luke Davis and Saah Wilson fo hei help wih he expeimens. I would also like o hank Rufus (Skip) L. Richadson and Mike Beedlove fo hei help in machining componens equied fo conducing he expeimens. Gaduae sudy a NCSU has been a wondeful expeience, duing which I made a numbe of dea fiends. I would like o hank all my fiends fo hei suppo and help. I would like o hank my family membes fo hei moal suppo and encouagemen. My deep gaiude o my fahe fo eaching me he high values of educaion and life.

6 v Conens Lis of Figues Lis of Tables vii xiv Inoducion 2 Backgound 3 2. FIBERBRAGGGRATINGS APPLICATION OF BRAGG GRATING STRAIN SENSORS TO COM- POSITES MODELING OF EMBEDDED BRAGG GRATING STRAIN SENSORS MOTIVATION... 3 Finie Elemen Model fo Embedded Fibe Bagg Gaing Senso 2 3. FINITEELEMENTFORMULATION Oveview Calculaionofindicesofefacionfoanelemen Calculaion of popagaion consans fo a senso segmen Calculaionofhesensoesponse NUMERICALEXAMPLES Ciculacoe,sep-indexfibe PMfibes SUMMARY Modified Tansfe Maix Fomulaion fo Bagg Gaing Sain Sensos INTRODUCTION REVIEW OF TWO-MODE COUPLING IN BRAGG GRATINGS Coupledmodeheoy Tansfemaixappoximaion T-MATRIX FORMULATION BASED ON MODIFIED PERIOD FUNCTION Unifomgaings Chipedgaings... 55

7 vi Calculaionofappliedsain Discussion EXPERIMENTALVALIDATION APPLICATIONTOPMFIBER Addiionalcommens EllipicalSAPfibe SUMMARY Expeimenal Invesigaion of FBG Sain Sensos INTRODUCTION BENCHMARKPMMASTUDIES Resuls Discussion WOVENCOMPOSITES SufaceMounedSensos EmbeddedSensos Discussion SUMMARY Conclusions 3 Refeences 5 Appendices 22 A Expeimenal Daa fo PMMA Specimens 23 B Expeimenal Daa fo Composie Specimens 49

8 vii Lis of Figues 2. Classificaionofopicalfibes FBGwieninoaPMfibe,subjecedomuli-axisloading Typical opical fibe coss-secions: (a) cicula coe, sep-index (no biefingence), (b) ellipical coe, sep-index (geomeical biefingence), (c) bow-ie fibe wih pe-sessed egions (geomeical and esidual sess biefingence). 3. Schemaic of he pocedue fo calculaion of FBG specal esponse fo a sensoembeddedinahosmaeialsysem Disceizaion of opical fibe ino FBG senso elemens. Also shown ae definiion of local polaizaion axes p and q, global polaizaion axes, X and Y, and local pinciple sain axes, 2 3. z is he diecion of popagaion alongheopicalfibe Definiionofiangulaelemenwihnodalcoodinaes Global index ellipse showing vaiaion of n eff fo a given mode wih oaion of global axes X Y. X Y coesponds o global popagaion axes of opicalfibe Coss-secion of cicula coe, sep-index fibe wih applied ansvese load. Coesizeisexaggeaedoshowdimensions Vaiaion of effecive index of efacion wih nomalized fequency fo cicula coss-secion, sep-index opical fibe. Cicles epesen exac soluion. Tiangles epesen esuls of coase mesh simulaions and squaes epesen esuls of fine mesh (27 coe elemens, 452 cladding elemens) simulaions Vaiaion of Bagg wavelengh wih applied diameical load fo FBG in cicula coe, sep-index opical fibe. Squaes epesen esul of cene sainappoximaion.feesulisploedasasolidline Geomey of PM fibe ypes consideed: (a) ellipical coe fibe; (b) D-fibe; (c) ellipical coe SAP fibe; (d) Bow-Tie fibe; (e) Panda fibe. The slow and fas axes ae also indicaed. All dimensions shown in µm Vaiaion of Bagg wavelengh wih applied load fo FBG in ellipical coe fibe of figue 3.8 (a). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =

9 viii 3. Vaiaion of Bagg wavelengh wih applied load fo FBG in D-fibe of figue 3.8 (b). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ = Vaiaion of Bagg wavelengh wih applied load fo FBG in ellipical coe SAP fibe of figue 3.8 (c) wih oue diamee d = 25 µm. Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ = Vaiaion of Bagg wavelengh wih applied load fo FBG in ellipical coe SAP fibe of figue 3.8 (c) wih oue diamee d = 8 µm. Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9 ;(c)γ =36 ; and (d) γ = Vaiaion of Bagg wavelengh wih applied load fo FBG in bow-ie fibe of figue 3.8 (d). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9 ;(c)γ =36 ; and (d) γ = Vaiaion of Bagg wavelengh wih applied load fo FBG in panda fibe of figue 3.8 (e). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9 ;(c)γ =36 ; and (d) γ = Vaiaion of he oienaion of global index ellipse wih applied load fo FBG in panda fibe of figue 3.8 (e). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9 ;(c)γ =36 ;and (d) γ = Vaiaion of Bagg wavelengh wih applied load fo muliple oienaions of applied loading angle, γ, elaive o geomeical axes of Panda fibe. Resuls fom boh he finie elemen and cene sain soluions ae ploed Vaiaion of Bagg wavelengh wih applied empeaue (a) ellipical coe fibe (d = 25 µm); (b) bow-ie fibe; (c) panda fibe. Resuls fo each axis calculaed using he cene sain appoximaion (CSA) ae also ploed Response of a unifom gaing subjeced o an applied (a) linea sain field. (b) Refleced speca due o he applied linea sain field along he gaing lengh: dashed line is Runge-Kua simulaion of he coupled mode equaions; solid line is T-maix simulaion using Λ(z); doed line is T-maix simulaion using Λ(z) Response of a unifom gaing subjeced o an applied (a) quadaic sain field. (b) Refleced speca due o he applied quadaic sain field along he gaing lengh: dashed line is Runge-Kua simulaion of he coupled mode equaions; solid line is T-maix simulaion using Λ(z); doed line is T-maix simulaion using Λ(z) Response of a unifom gaing subjeced o an applied (a) linea sain field wih high sain gadien. (b) Refleced speca due o he applied linea sain field along he gaing lengh: dashed line is Runge-Kua simulaion of he coupled mode equaions; solid line is T-maix simulaion using Λ(z); doed line is T-maix simulaion using Λ(z)... 54

10 ix 4.4 Response of a unifom gaing subjeced o an applied (a) exponenially vaying sain field wih high sain gadien. (b) Refleced speca due o he applied exponenially vaying sain field along he gaing lengh: dashed line is Runge-Kua simulaion of he coupled mode equaions; solid line is T-maix simulaion using Λ(z); doed line is T-maix simulaion using Λ(z) Response of a chiped gaing: (a) ɛ (z) = 6 and ɛ(z) =( /mm)z; (b) ɛ (z) = 6 and ɛ(z) =( 6 /mm 2 )z 2 ;and (c) ɛ (z) =( /mm)z and ɛ(z) =( 6 /mm 2 )z 2. Runge-Kua simulaions ae ploed as dashed lines and T-maix simulaions using Λ(z) aeploedasdoedlines Reflecion speca fo a gaing suface mouned on an aluminum diaphagm. (a) Expeimenally measued speca (fom Chang and Voha [84]), (b) Speca simulaed including sain gadien in he effecive peiod fomulaion, and (c) Speca simulaed wihou sain gadien in he effecive peiod fomulaion Coss-secion of PM fibe wih applied ansvese load. Coe size is exaggeaedoshowdimensions Azimuhal vaiaion of global effecive efacive index fo P =N/mmand γ =54 (squaes). Ellipse oiened a θ wih majo and mino axes n max eff and n min eff isploedasasolidline (a) Vaiaion of Bagg wavelengh wih applied diameical load a γ =36 fo PM fibe. Squaes epesen esul of cene sain appoximaion, while FE esuls ae ploed as solid line. (b) Vaiaion of he oienaion of global index ellipse wih he applied load. The vaiaion of θ is ploed ove a geae ange of P odemonsaeasympoicbehavio (a) Vaiaion of Bagg wavelengh wih applied diameical load a γ =72 fo PM fibe. Squaes epesen esul of cene sain appoximaion, while FE esuls ae ploed as solid line. (b) Vaiaion of he oienaion of global index ellipse wih he applied load. The vaiaion of θ is ploed ove a geae ange of P odemonsaeasympoicbehavio (a) Linea vaiaion of applied diameical load along he lengh of he FBG. FBG speca obained befoe (dashed line) and afe (solid line) applicaion of ansvese load a (b) γ =36 and (c) γ = (a) PMMA specimen wih suface mouned FBG. (b) Schemaic epesenaion of PMMA specimen. Also indicaed ae he locaion of he opical fibes and FBGs fo each specimen and he impac poin. All dimensions in mm InsumenaionfoFBGsenso Dopowefoimpacloadingofhespecimens Vaiaion of sain wih sike # fo FBG suface mouned on specimen A Vaiaion of sain wih sike # fo FBG suface mouned on specimen A Vaiaion of sain wih sike # fo FBG suface mouned on specimen A Vaiaion of sain wih sike # fo FBG suface mouned on specimen A Vaiaion of sain wih sike # fo FBG suface mouned on specimen A Vaiaion of sain wih sike # fo FBG suface mouned on specimen A75. 78

11 x 5. Vaiaion of sain wih sike # fo FBG suface mouned on specimen A Measued ansmied speca of a FBG suface mouned on specimen A8 a wo diffeen load levels. (a) 3 lbs, maximum load; (b) 3 lbs, zeo load; (c) lbs, maximum load; (d) lbs, zeo load Vaiaion of sain wih load fo an FBG suface mouned on specimen A Vaiaion of sain wih load fo an FBG suface mouned on specimen A Vaiaion of he maximum esidual sain wih he adius fo 9 and 45 oienaions. Solid and dashed lines ae he quadaic fi fo 9 and 45 oienaionsespecively Cuing cycle fo he fabicaion of 2D woven composies Phoogaph of wo-dimensional woven composie specimen (a) befoe and (b) afeimpac Coss-secion of wo-dimensional woven composie specimen unde a opical micoscope(a)befoeand(b)afeimpac Insumenaion fo coninuous FBG sensing. Also shown is he 5 Hz ineogao fom Micon Opics Schemaic epesenaion of specimen C5 along wih opical fibe and impac locaion.alldimensionsinmm Global measuemens fo specimen C5. (a) Vaiaion of peak acceleaion wih sike #. (b) Vaiaion of dissipaed enegy wih sike # Vaiaion of sain wih sike # fo an FBG suface mouned on specimen C5. Sains compued fom boh he eflecion and ansmission speca ae ploed Schemaic epesenaion of composie specimens. Also shown ae he locaion of FBGs fo C6,C7,C8 and C5 along wih impac locaion. All dimensionsinmm Global measuemens fo C6. (a) Vaiaion of peak acceleaion wih sike #.(b)vaiaionofdissipaedenegywihsike# Vaiaion of sain wih sike # fo an FBG suface mouned on C6. Sains compued fom boh he eflecion and ansmission speca ae ploed Global measuemens fo C7. (a) Vaiaion of peak acceleaion wih sike #.(b)vaiaionofdissipaedenegywihsike# Vaiaion of sain wih sike # fo an FBG suface mouned on C7. Sains compued fom he eflecion speca of (a) End and (b) End 2 ae ploed Measued eflecion speca fom End fo a FBG suface mouned on specimenc7afesikes#:(a),(b)4,(c)7,and(d) Measued eflecion speca fom End 2 fo a FBG suface mouned on specimenc7afesikes#:(a),(b)4,(c)6,and(d) Global measuemens fo specimen C8. (a) Vaiaion of peak acceleaion wih sike #. (b) Vaiaion of dissipaed enegy wih sike # Vaiaion of sain wih sike # fo an FBG suface mouned on specimen C8. Sains compued fom boh he eflecion and ansmission speca ae ploed

12 xi 5.3 Measued specal esponse of FBG suface mouned on specimen specimen C8 afe sikes #: (a) 6, (b) 9, (c), (d) 4, (e) 5, (f) 6, and (g) Global measuemens fo C5. (a) Vaiaion of peak acceleaion wih sike #.(b)vaiaionofdissipaedenegywihsike# Coss-secion of wo dimensional woven composie specimen embedded wih opical fibe unde opical micoscope Schemaic epesenaion of specimen C49 along wih opical fibes and impaclocaion.alldimensionsinmm Schemaic epesenaion of specimen C5 along wih opical fibes and impaclocaion.alldimensionsinmm Measued speca fo FBG-A embedded in specimen C49 (a) afe embedding, afesikes#:(b),(c)6,and(d) Measued speca fo FBG-B embedded in specimen C49 (a) afe embedding, afesikes#:(b),(c)8,and(d) Measued speca fo FBG-C embedded in specimen C5 (a) afe embedding, afesikes#:(b),(c)4,and(d) Measued speca fo FBG-D embedded in specimen C5 (a) afe embedding, afesikes#:(b),(c)4,and(d) Coss-secion of wo dimensional woven composie specimens unde opical micoscope: (a) specimen C29, (b) specimen C3, (c) specimen C32, and (d) specimenc A. Measued specal esponse of FBG suface mouned on A58 fo impac sikes A.2 Measued specal esponse of FBG suface mouned on A59 fo impac sikes A.3 Measued specal esponse of FBG suface mouned on A59 fo impac sikes A.4 Measued specal esponse of FBG suface mouned on A59 fo impac sikes A.5 Measued specal esponse of FBG suface mouned on A69 fo impac sikes A.6 Measued specal esponse of FBG suface mouned on A69 fo impac sikes A.7 Measued specal esponse of FBG suface mouned on A69 fo impac sikes A.8 Measued specal esponse of FBG suface mouned on A69 fo impac sikes A.9 Measued specal esponse of FBG suface mouned on A69 a he end of impacloading A. Measued specal esponse of FBG suface mouned on A7 fo impac sikes A. Measued specal esponse of FBG suface mouned on A73 fo impac sikes

13 xii A.2 Measued specal esponse of FBG suface mouned on A73 fo impac sikes A.3 Measued specal esponse of FBG suface mouned on A73 fo impac sikes A.4 Measued specal esponse of FBG suface mouned on A73 fo impac sikes A.5 Measued specal esponse of FBG suface mouned on A73 fo impac sikes A.6 Measued specal esponse of FBG suface mouned on A75 fo impac sikes A.7 Measued specal esponse of FBG suface mouned on A79 fo impac sikes A.8 Measued specal esponse of FBG suface mouned on A79 fo impac sikes A.9 Measued specal esponse of FBG suface mouned on A79 fo impac sikes A.2PMMAspecimenssufacemounedwihFBGs A.2 Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom -7 lbs A.22 Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom 8-3 lbs A.23 Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom -4L lbs. Noaion L and U sand fo loading and unloading especively A.24 Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom 4U-8U lbs. Noaion L and U sand fo loading and unloading especively A.25 Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom 9L-U lbs. Noaion L and U sand fo loading and unloading especively B. Dimensions of he mold used o fabicae he composie specimens. All dimensionsininches... 5 B.2 Pogession of damage in a ypical wo-dimensional woven composie specimen.5 B.3 Measued specal esponse of FBG suface mouned on C7 fo impac sikes B.4 Measued specal esponse of FBG suface mouned on C7 fo impac sikes B.5 Measued specal esponse of FBG suface mouned on C7 fo impac sikes B.6 Measued specal esponse of FBG suface mouned on C7 fo impac sikes B.7 Measued specal esponse of FBG suface mouned on C8 fo impac sikes

14 B.8 Measued specal esponse of FBG suface mouned on C8 fo impac sikes B.9 Measued specal esponse of FBG suface mouned on C8 fo impac sikes B. Measued specal esponse of FBG embedded in C5 fo impac sikes B. Measued specal esponse of FBG embedded in C5 fo impac sikes B.2 Measued specal esponse of FBG-A embedded in C49 fo impac sikes -4.6 B.3 Measued specal esponse of FBG-A embedded in C49 fo impac sikes B.4 Measued specal esponse of FBG-B embedded in C49 fo impac sikes B.5 Measued specal esponse of FBG-B embedded in C49 fo impac sikes B.6 Measued specal esponse of FBG-B embedded in C49 fo impac sikes B.7 Measued specal esponse of FBG-C embedded in C5 fo impac sikes B.8 Measued specal esponse of FBG-C embedded in C5 fo impac sikes B.9 Measued specal esponse of FBG-C embedded in C5 fo impac sikes B.2 Measued specal esponse of FBG-C embedded in C5 fo impac sikes B.2 Measued specal esponse of FBG-D embedded in C5 fo impac sikes -4.7 B.22 Measued specal esponse of FBG-D embedded in C5 fo impac sikes B.23 Measued specal esponse of FBG-D embedded in C5 fo impac sikes B.24 Measued specal esponse of FBG-D embedded in C5 fo impac sikes xiii

15 xiv Lis of Tables 3. Paamees of scala wave equaion fo fundamenal LP modes Paameesofciculacoe,sep-indexfibe MaeialPopeiesofModeledOpicalFibes NumbeofElemensMeshedfoPMFibeTypes Compaison of Sensiiviy o Applied Load fo PM Fibe Types Compaison of Sensiiviy o Applied Tempeaue fo PM Fibe Types PaameesofPMfibe ConfiguaionsofPMMASpecimens Configuaionsof2Dwovencomposiespecimens Configuaions of 2D woven composie embedded wih polyimide coaed fibeswihoufbgs Configuaions of 2D woven composies embedded wih polyimide coaed fibeswihoufbgs... 3

16 Chape Inoducion The goal of a damage assessmen sysem is o measue failue iniiaion and pogession in a sucue. Fom his one esimaes he emaining life ime of he damaged sucue. This subsysem heefoe foms an inegal pa of long-em sucual healh monioing sysem. The concep of damage assessmen is paiculaly impoan fo fibe einfoced composies subjeced o low velociy impacs due o he diffeen failue mechanisms involved. The ineacion of hese vaious mechanisms duing he pogession of damage poses a challenge o he assessmen sysem. The fis sep owads building an effecive damage assessmen sysem is he choice of he senso(s). Due o is abiliy o measue sain disibuions as well as wih muliplexing capabiliies fibe Bagg gaing (FBG) senso is an effecive ool fo a damage assessmen sysem. In addiion, FBGs can be embedded ino many hos sucues o measue inenal damage. Afe selecing he senso, he nex sep is o model he sain ansfe fom he hos maeial o he senso and he senso esponse due o he physical change in he hos sucue incopoaing his ansfe. One of he challenges posed by a damage assessmen sysem is o be able o disinguish beween senso failues and hos maeial failues fom he complex senso esponse. This disseaion eliminaes he need fo a educed sain ansfe model by using a finie elemen (FE) fomulaion o model he esponse of an embedded FBG senso. A linea sain-opic effec is applied o model he local opo-mechanical esponse of he opical fibe. Also a fis aemp is made o analyze he complex senso esponse when

17 2 FBG sensos ae sufaced mouned and embedded in wo-dimensional woven composies. The developed FE model fo he senso esponse and he analysis of expeimenal daa is impoan owads (i) bee undesanding he senso esponse o be able o disinguish beween senso failue and hos maeial failue and (ii) inegaing ino exising FE models fo sucual behavio. This hesis is oganized as follows: Chape 2 pesens a lieaue eview of exising models fo sain ansfe and moivaions fo developing he FE fomulaion. Chape 3 pesens he FE fomulaion fo he complex sain ansfe using a linea sain-opic effec. The model is applied o a wo-dimensional poblem of diameical compession of FBGs wien in PM fibes fo validaion. Chape 4 pesens he specal esponse of he FBG senso using a modified T-maix model including he effecs of sain gadiens. Chape 5 pesens expeimenal esuls and analysis of he FBG senso esponse when suface mouned and embedded in wo-dimensional woven composies. We also pesen he esuls of FBG s suface mouned on PMMA specimens. Finally Chape 6 pesens conclusions and fuue wok.

18 3 Chape 2 Backgound 2. FIBER BRAGG GRATINGS Figue 2. shows a classificaion of opical fibes based on he diamee of he coe ino single mode o muli mode fibes. Fo small coe diamees (4- µm) he fibe suppos only he fis mode. The diamee of he claddding is ypically 25 µm. Singlemodefibes ae fuhe classified ino cicula coe and polaizaion mainaining fibes. In polaizaion mainaining fibes he fis linealy polaized mode, LP, is spli ino wo modes each having a specific popagaion consan. Thus wo polaizaion axes ae esablished. PM fibes ae fuhe classified as low biefingen and high biefingen fibes based on whehe he biefingence is due o geomey o a sess applying pa (SAP) [8]. The fis fibe opic senso was inoduced by Bue and Hocke in 978 []. They used an inefeomeic echnique o measue he sain on he suface of a canileve beam. They elaed he change in phase of a lighwave popagaing hough he opical fibe, φ, o he axial sain applied o he fibe by ={β 2 βn2 [( ν)p 2 νp ]}ɛl (2.) whee β = nk is he popagaion consan, n is he effecive efacive index, k =2π/λ is he wavenumbe, λ is he wavelengh of ligh, L is he sensing lengh of he opical fibe, ν is he Poisson s aio of he fibe, and ɛ is he applied axial sain. Assuming a linea

19 4 Opical fibes Single-mode Muli-mode Cicula coe PM fibe Low biefingen Ellipical coe D-fibe High biefingen Panda fibe Figue 2.: Classificaion of opical fibes. sain-opic effec one aives a he Bue and Hocke fomulaion. The coefficiens p and p 2 ae he sain opic coefficiens fo he opical fibe. Lae Hocke, showed ha he sensiiviy o empeaue is highe han he pessue sensiiviy [2]. In 988, Beholds and Dandlike developed a echnique o expeimenally deemine he sain opic coefficiens fo an isoopic, mechanically homogeneous opical fibe [3]. They applied axial sain o he fibe and used (2.) o compue he diffeence in sain opic coefficiens [5]. They applied osional sain o he fibe and elaed he change in polaizaion of ligh o he diffeence in sain opic coefficiens [4]. Using hese wo independen measuemens hey compued he individual sain opic coefficiens. The values of sain-opic coefficiens compued ae p =.3 and p 2 =.252. These values have been univesally acceped and widely used, including in his hesis. The main disadvanages o applying wih he inefeomeic echniques o measue sain ae he high vibaion sensiiviy and he need o coun finges which becomes laboious a high sains. The fomaion of pemanen gaings in an opical fibe was fis demonsaed by Hill e al. in 978 a he Canadian Communicaions Reseach Cene (CRC) [8]. This achievemen was an ougowh of eseach on he nonlinea popeies of gemania-doped silica fibes. Cuenly, hese gaings ae used exensively as files fo elecommunicaion puposes and as sain sensos. A Bagg gaing is a peiodic modulaion of he coe index of efacion along a segmen of opical fibe. I is fomed usually by means of exposue o an inefeence paen of inense ulaviole ligh ( 245 nm). The opical back-efleced specum of such a Bagg gaing compises a vey naow spike. This song backeflecion

20 5 occus a he Bagg wavelengh λ B, which can be elaed o he effecive coe efacive index, n, and he peiod of he index modulaion, Λ, by he elaion [6]: λ B =2nΛ (2.2) This Bagg wavelengh will shif wih changes o eihe n o Λ, hus measuing he wavelengh of his naow-band specum will deemine he sain o empeaue o which he opical fibe is subjeced. The applicaion of hese gaings o measue axial sain was demonsaed by Measues in 992 [6]. Measues fomulaed he change in Bagg wavelengh, λ B, o he applied axial sain, ɛ, by he elaion λ B =( p e )ɛ (2.3) λ B whee p e =.22 is he effecive phooelasic consan. This linea elaion has been widely used o measue axial sain. Since hen, Bagg gaings have been used fo many applicaions including aicaf healh monioing [9], and spacecaf [2]. Bagg gaings can be classified based on hei lengh as sho peiod ( 5 nm) and long peiod gaings. Fom he modeling poin of view, in a sho peiod gaing he coupling is beween a fowad popagaing coe mode and a backwad popagaing coe mode. In a long peiod gaing he coupling is beween a fowad popagaing coe mode and a fowad popagaing cladding mode [2, 2]. Fo a eview of he fomaion, modeling and applicaions of fibe Bagg gaings see Kashyap [9] and Ohonos e al. []. Fo moe specific eviews, Hill e al. fom CRC pesen an excellen eview on he fomaion of Bagg gaings in [7]; Kesey e al. fom he Naval Reseach Laboaoy eview he inumenaion fo Bagg gaings sensos in [7]; and Edogan eviews he modeling of Bagg gaing senso esponse in [3]. To model he Bagg gaing senso esponse, Edogan fis assumes ligh o be an elecomagneic wave popagaing hough a weakly guiding waveguide (opical fibe) [3]. One noes ha his appoximaion is valid fo all opical fibes wih a diffeence in coe and cladding efacive index of less han 2%. Using he above appoximaion, one solves he wave equaion o obain he guided elecomagneic fields in he opical fibe. An impoan poin o noe is ha he exac soluion o he wave equaion exiss in ems of modified Bessel funcions due o he geomeical symmey of he fibe. The elecomagneic fields ae defined in ems of modified Bessel funcion of fis kind, and in he cladding in ems of modified Bessel funcion of second kind. The fibe Bagg gaing can hen be descibed

21 6 as a peubaion o he effecive efacive index, n of he guided mode(s) as [4]: δn eff = δn eff (z){+νcos[ 2π z + φ(z)]} (2.4) Λ whee δn eff is he dc index change spaially aveaged ove a gaing peiod, ν is he finge visibiliy of he index change, and φ(z) descibes he gaing chip. This peubaion causes he fowad popagaing coe mode o couple o he backwad popagaing coe mode. This coupling is maximum a he Bagg wavelengh whee in enegy fom he fowad popagaing coe mode is ansfeed o he backwad popagaing coe mode. The coupling beween he modes is well descibed by he coupled-mode heoy [5, 6]. The specal esponse of he Bagg gaing senso can hen be deived using coupled-mode heoy [3]. This modeling appoach allows one o conside he esponse of he Bagg gaing o moe geneal sain disibuions han he consan axial sain assumed in (2.3). 2.2 APPLICATION OF BRAGG GRATING STRAIN SEN- SORS TO COMPOSITES McKenzie e al. used opical fibe sensos o sudy he feasibiliy of opical fibe sensos fo healh monioing of bonded epai sysems [25]. They suface mouned FBGs o an aluminum-boon epoxy pach o monio faigue cack gowh and used he finie elemen mehod o deemine he opimal placemen of sensos. This wok demonsaes ha FBGs can be used o deec cack gowh. The poblems encouneed wee debonding of he fibes and a disoion in he esponse of FBGs due o song sain gadiens. Muukeshan e al. used fibe Bagg gaings fo cue monioing of composies [26]. They consideed diffeen composie maeial sysems embedded wih FBGs. The esuls demonsaed he use of FBGs o measue esidual sain duing cuing and also he linea esponse of he senso o bending. An aveage sain was compued fom he specal esponse of he FBG, howeve an aemp o analyze senso esponse was no made. Takeda e al. embedded Bagg gaings in Cabon Fibe Reinfoced Plasic (CFRP) laminaes o deec ansvese cacks [27]. They pefomed ensile ess on laminaes embedded wih FBG s o induce ansvese cacks. They pefomed boh heoeical and expeimenal specal analyses o monio he cack densiy. They also sudied he effec of

22 7 fibe coaing on sain ansfe fom he laminae o he embedded opical fibe by applying he widely used shea lag heoy fo he sain ansfe o numeically simulae he specal esponse of he embedded gaing [3]. The auhos also demonsaed he use of small diamee fibe Bagg gaing sensos o deec ansvese cack and delaminaion in CFRP laminaes [33, 34]. Kuang e al. embedded Bagg gaings in advanced composie maeials and fibe meal laminaes (FML) [28]. They measued he esidual sess duing fabicaion and aibued he spli in he Bagg eflecion peak o ansvese loading. Afewads hey subjeced he FML s o ensile loading and obseved an incease in inensiy of one Bagg peak wih espec o one anohe. They called his wavelengh hopping phenomenon. This wavelengh hopping phenomenon could be due o sess elief duing loading. One obseves high sain gadiens induced duing loading fom he specal esponse. Howeve, a deailed analysis is lacking in [28]. They also sudied he specal esponse of FBG s, embedded in FML s, subjeced o low velociy impacs [29, 32]. They obseved specum disoion duing impac loading and also a dop in inensiy. They measued he S-shape end in he shif in Bagg peak wih sike numbe. A simila obsevaion was made by Guemes and Menéndez [3]. Kuang e al. also demonsaed he use of FBG s, embedded in FML s, o monio damage induced duing cyclic loading [35]. Fibe Bagg gaings have also been used o measue inenal sain in exile composies [36, 37, 38]. 2.3 MODELING OF EMBEDDED BRAGG GRATING STRAIN SENSORS As menioned above, opical fibe Bagg gaing (FBG) sensos have been widely embedded in composie maeial sysems fo he measuemen of cuing sesses, inelamina sesses, delaminaion, cack gowh, and ohe phenomena [22]. Thei unique abiliy o be embedded wihin fibe-einfoced composies wih a minimum peubaion o he suounding hos maeial makes hem aacive fo he above applicaions. One paicula field of cuen inees is long-em healh monioing of composie sucues, fo example aicaf o FRP einfoced concee sucues. Theefoe, a clea undesanding of he specal esponse of he FBG as a funcion of he behavio of he sucue is equied.

23 8 A vaiey of models have been developed fo he sess ansfe in isoopic maeials embedded wih opical fibes [39, 4, 4]. The goal of such models is o calculae he sain in he opical fibe, and hence he senso esponse, due o loading applied o he hos maeial sysem. Kollá and Van Seenkise developed a model of he sain ansfe beween a laminaed composie and an opical fibe embedded beween wo laminae [42]. The opical fibe is eaed as an ellipical inclusion in he composie laminae coss-secion and he aveage fibe sain due o he suounding sain in he laminae is calculaed analyically. Lae, Pabhugoud and Pees modeled a unidiecional composie by applying a combinaion of he finie elemen mehod and opimal shea-lag heoy [43]. While applying vaying levels of deail o he hos sucue, each of hese models consides he opical fibe as an isoopic o ohoopic, homogeneous fibe wih consan sain acoss he coss-secion of he fibe. Alhough his assumpion poduces excellen esuls fo he mechanical esponse of embedded fibes [44], i is no eviden ha he same is ue fo he opical esponse of he FBG due o he opical non-homogeneiies of he opical fibe. Once he sain field applied o he opical fibe is known, he shif in Bagg wavelengh of he FBG, λ B, is calculaed as [6], λ B = (2Λn eff ) = 2Λ n eff +2n eff Λ (2.5) whee Λ is he peiod of he gaing and n eff is he change in effecive efacive index of he fibe coss-secion due o he applied sain. Fo example, in he case of an unconsained fibe subjeced o an axial load, λ B = λ B ɛ z [ n2 ] eff 2 {p ν(p + p 2 )} (2.6) whee ɛ z is he axial sain in he fibe, p and p 2 ae phooelasic consans fo silica, and ν is he Poisson s aio of he fibe []. Alhough he em Λ in (2.5) can be diecly elaed o he esuling axial sain in he FBG, he em n eff is moe difficul o calculae. This change is due boh o geomeical changes in he coss-secion and he sain-opic effec. In addiion, induced biefingence in he fibe sepaaes he single Bagg eflecion peak ino wo peaks [45]. This biefingence is highly dependen on he index of efacion and maeial disibuion houghou he fibe coss-secion. Examples of ypical opical fibe coss-secions ae shown in figue 2.3. The cicula and ellipical sep-index coss-secions shown in figue 2.3(a) and (b) ae mechanically isoopic wih an elasic modulus E coe = E clad and poisson s

24 9 P 2 P P 3 Figue 2.2: FBG wien ino a PM fibe, subjeced o muli-axis loading. aio ν coe = ν clad. Opically, hey ae non-homogeneous wih wo index of efacion egions: n coe in he coe and n clad in he cladding. Wheeas he cicula fibe popagaes ligh a one popagaion consan, β, pe mode, he ellipical fibe popagaes modes along wo ohogonal axes wih wo diffeen popagaion consans, β and β 2. The hid opical fibe, efeed o as he polaizaion mainaining (PM) fibe, shown in figue 2.3(c), includes egions of a sepaae maeial (and is heefoe boh mechanically and opically non-homogeneous) called he sess applying pa (SAP). The pupose of hese egions is o povide sess o he fibe coe as he fibe is dawn, inducing esidual biefingence due o he hemal expansion mismach of he silica and SAP. Seveal ohe configuaions fo he PM fibe exis ohe han he bow-ie fom shown in figue 2.3(c). To calculae n eff due o an applied sain field, Kim e al. consideed he fibes shown in figue 2.3 o be opically and mechanically homogeneous wih isoopic o ansvesly isoopic maeial popeies [46]. Based on he model of an inclusion in a composie laminae, he emoe sains wee analyically linked o he pinciple sains in he fibe coe and β in he diecion of popagaion calculaed. As in lae models, he assumpion is made ha mos of he enegy of he fundamenal mode popagaing in he fibe is conained in he coe, heefoe he pinciple sains a he cene of he fibe ae sufficien o esimae n eff. In addiion, none of he above models accoun fo he biefingence due o he change in he geomey of he fibe alhough his effec is consideably smalle han

25 n clad, E clad n clad, E clad n clad, E clad n coe, E coe n coe, E coe n E coe coe n E SAP SAP (a) (b) (c) Figue 2.3: Typical opical fibe coss-secions: (a) cicula coe, sep-index (no biefingence), (b) ellipical coe, sep-index (geomeical biefingence), (c) bow-ie fibe wih pe-sessed egions (geomeical and esidual sess biefingence). he sain-opic effec. Sikis elaed he change in Bagg wavelenghs, λ B, and λ B,2, o he pinciple sains a he cene of he coe, ɛ, ɛ 2,andɛ 3, as [47] λ B, λ B, λ B,2 λ B,2 = ɛ n2 o 2 (p ɛ 2 + p 2 ɛ 3 + p 2 ɛ ) (2.7) = ɛ n2 o 2 (p 2ɛ 2 + p ɛ 3 + p 2 ɛ ) whee ɛ 2 and ɛ 3 ae in he plane of he fibe coss-secion, ɛ is in he axial diecion and n o is he effecive efacive index of he fundamenal mode. Wageich e al. demonsaed he linea dependence of λ B, and λ B,2 on he applied load fo a cicula coe fibe (figue 2.3(a)) unde diameical compession [45]. Lawence e al. [48] modeled he mechanical non-homogeneiies in a PM fibe wih ellipical SAP using a finie elemen analysis o calculae he sains a he cene of he fibe due o applied ansvese loading. Thei main goal was o calculae he calibaion maix fo he ansvese sensiiviy of a FBG senso in a PM fibe. Lae Bosia e al. [49] also modeled he mechanical non-homogenieies in a bowie ype PM fibe using finie elemen analysis o calculae he pinciple sains a he cene of he coe and hence he shif in Bagg wavelengh due o applied ansvese loading using (2.7). Boh he expeimenal and numeical sudies of [48] and [49] demonsaed ha fo a PM fibe he shif in Bagg wavelengh is nonlinea wih ansvese load fo ceain loading angles. Gafsi and El-Sheif expanded he cene sain fomulaion o include vaiaions of efacive indices along he axis of he fibe by inoducing (2.7) ino he coupled mode equaions descibing he specal esponse of he FBG [5]. Howeve, in all he above mehods, hee is sill a discepancy beween he expeimenally measued

26 and heoeical sensiiviy o applied sain. 2.4 MOTIVATION As descibed in he above secion hee exiss handful of models o simulae he esponse of a FBG subjeced o abiay loading. One should noe ha all he models use he cene sain appoximaion (CSA). None of he models descibed in he above secion can heefoe accoun fo he vaiaion of he efacive index in he coss-secion of he fibe due o he applied sain. The goal of his hesis is o pesen a mehodology o calculae he esponse of a FBG subjeced o abiay loading as shown in figue 2.2. We fomulae a finie elemen (FE) model o calculae he biefingence effec due o he applied ansvese load. The linea sain-opic law is assumed o calculae he change in efacive index due o he applied ansvese load. One can inegae his model ino exising FE models fo sucual behavio. We will conside a diameical compession of PM fibes o validae he model. As descibed in he above secion PM fibes ae boh mechanically and opically nonhomogeneous. We will also fomulae a modified T-maix model o simulae he esponse of FBG subjeced o longiudinal sain. This model consides he effec of sain gadiens in he esponse and is compuaionally efficien. Afewads, he esponse of FBGs suface mouned on PMMA and wo-dimensional woven composies subjeced o low velociy impacs ae analyzed. Finally, FBGs ae embedded in wo-dimensional woven composies o sudy he feasibiliy of using FBG s o measue inenal sains.

27 2 Chape 3 Finie Elemen Model fo Embedded Fibe Bagg Gaing Senso The goal of his chape is o deive a finie elemen (FE) fomulaion o pedic he opical esponse of an embedded FBG senso as a funcion of he loading applied o he hos sucue. The fomulaion incopoaes boh he mechanical and opical nonhomogeneiies of he opical fibe. Fisly, he FE fomulaion calculaes he change in index of efacion disibuion houghou he coss-secion of he fibe due o he esuling mechanical sesses. Fom he updaed index of efacion disibuion, he popagaion consans of he fundamenal modes, as well as he popagaion axes, ae obained. Pevious wok by Huang has appoached a simila poblem fo plana waveguides analyically [5]. As can be obseved fom [5], analyical soluions ae only obainable fo a few loading condiions. In he cuen fomulaion, he popagaion consans ae hen inoduced ino a disceized vesion of he coupled mode equaions o deemine he specal esponse of he FBG. The FE model can be applied o FBGs embedded in a vaiey of hos maeial

28 3 sysems fo which exensive FE modeling has aleady been pefomed. Examples include fibe einfoced composies and concee sucues. The cuen model also allows one o accuaely calculae he sensiiviy of he FBG o ansvese sains and is applicable o vaious fibe ypes including bowie and panda PM fibes. The fomulaion also includes he effec of oaing polaizaion axes due o significan sain ampliudes especially when he fibe is embedded nea a sess concenaion o failue locaion. 3. FINITE ELEMENT FORMULATION The popagaion of a given guided mode hough an opical fibe can be chaaceized hough he mode disibuion in he coss-secion of he opical fibe and he popagaion consan, β, fo a given fequency. The mode popagaion consan is elaed o he effecive index of efacion, n eff, fo he paicula mode hough β = 2π λ n eff (3.) whee λ is he popagaing wavelengh [52]. Exac soluions fo he popagaion chaaceisics of opical fibes, obained by solving wave equaions, ae limied o elaively simple geomeies (e.g., cicula o ellipical coss-secions) wih an axisymmeic index of efacion disibuion in he coe. Thus, o calculae he popagaion chaaceisics of an opical fibe wih an abiay coss-secional shape o abiay vaiaion of efacive index in he coe, cladding, and SAP, one needs o adop a numeical mehod such as he finie elemen mehod [53]. Cuen finie elemen mehods fo opical fibe waveguides can be classified ino veco mehods and scala mehods. Veco finie elemen mehods ae applicable o all values of efacive index diffeence beween he coe and he cladding. The main disadvanages of hese mehods ae he lage compuaional effo equied and he appeaance of spuious modes in he soluion. The spuious modes can be eliminaed using a penaly appoach [53]. Diffeen vaiaions of he veco fomulaion ae based on he componens of he elecic field, E, o magneic field, H, consideed. Fo example, in he fomulaion of Yeh e al., he axial componens of E and H field ae consideed [54]. All ohe componens ae hen expessed in ems of hese axial componens using Maxwell s equaions. A minimizing funcional is obained by applying he veco wave equaion along wih a coninuiy condiion a he coe-cladding

29 4 ineface. In he wok of Koshiba, he minimizing funcional is obained fom he complee E o H field saisfying he veco wave equaion [53]. Diffeen appoaches have also been poposed o addess he open bounday poblem fo example, applying he FE fomulaion o he coe and appopiae bounday condiion o he coe-cladding ineface. This educes he numbe of elemens equied in he cladding o obain an accuae soluion [55, 56]. Scala finie elemen mehods, on he ohe hand, ae only applicable o weakly guiding fibes, i.e. fo which he vaiaion of he efacive index is negligible ove a disance of one wavelengh [52]. Howeve, such an assumpion is easonable fo mos fibes ino which FBGs ae wien, wihin hei elasic sain limi. The advanages of a scala mehod ae ha no spuious soluions appea (since only linealy polaized modes ae capued) and only one componen of he E o H field is consideed, educing he size of he equied sysem of equaions o solve fo he popagaion consan. Fo his eason, in his chape we deive a senso elemen based on a scala fomulaion wihou imposing he assumpion of axisymmey. 3.. Oveview In he cuen analysis, he pedicion of he FBG specal esponse is pefomed hough he following seps (see figue 3.): The suounding hos composie maeial and opical fibe senso ae meshed using a commecial FE package (e.g., ANSYS fo he cuen wok). The chosen senso mesh is shown in figue 3.2, whee he fibe is divided ino segmens of lengh z in he axial diecion and each coss-secion is meshed using 2D plane sess iangula elemens. An example elemen is also shown in figue 3.2. Fo he pupose of lae calculaions, his 3D popagaion elemen is chaaceized by is siffness popeies, indices of efacion, and lengh, z. Axis is along he popagaion diecion and coincides wih he global popagaion axis, z. Axes p and q ae he local opical axes of popagaion. Using he hemo-mechanical FE model, he nodal displacemens ae obained due o he exenal applied loads. Fom he nodal displacemens, sain componens in each elemen ae also calculaed. Fo each 2D iangula elemen (as shown in figue 3.2), he index of efacion change

30 5 due o he applied sain field is calculaed in local opical axes using a linea sainopic law (see secion 3..2). The updaed indices of efacions ae hen ansfomed fom he local axes o he global sucual axes. Fo each senso segmen, he popagaion consan fo he opical fibe abou he global sucual axis is calculaed using he opical FE fomulaion including he updaed index of efacion disibuion and he nodal displacemens. The popagaion consan/effecive index of efacion a wo ohe angles wih espec o he global sucual axis is calculaed. Fom hese values he maximum and mininum popagaion consans ae calculaed fo he coss-secion as well as he global opical axes coesponding o hese exema. The FBG specal esponse is calculaed fom he local axial sain, effecive indices of efacion, and cuvaue of each segmen using he modified T-maix mehod (see secion 3..4). The deails of some of hese calculaions ae given below Calculaion of indices of efacion fo an elemen Each elemen is assumed o be opically isoopic wih an index of efacion in he unsessed sae of n o e. The displacemen field veco fo ligh popagaing hough he elemen in he -diecion is given as [46], [ ] [ ] {D} = A p {s p } sin ω 2πnp e λ x + A q {s q } sin ω 2πnq e λ x (3.2) whee s p and s q ae ohogonal uni vecos in he 2-3 plane in he diecion of he pinciple opical axes, n p e and n q e ae he elemen index of efacions abou hese axes, ω is he angula fequency of he wave, and A p and A q ae he ampliudes of he displacemen veco componens. We can wie he wave equaion fo his displacemen field veco as [46], {s} ({s} [B]{D})+ {D} = (3.3) (n e ) 2

31 6 Sucual Model (geomey, maeial popeies, loading) ANSYS Nodal Displacemens Sain - Opic Law Segmen Popagaion Consans a Oienaions α, β, and γ Finie Elemen Fomulaion Elemen Indices of Refacion and Local Opical Axes Global Index Ellipse Segmen Global Opical Axes and Pinciple Popagaion Consans Tansfe Maix Fomulaion FBG Specal Response Figue 3.: Schemaic of he pocedue fo calculaion of FBG specal esponse fo a senso embedded in a hos maeial sysem. whee n e = n p e o n q e, {s} is he uni veco in he popagaion diecion, and [B] is he maeial dielecic impemeabiliy enso, [B] = B B 6 B 6 B 2 B 5 B 4 (3.4) B 5 B 4 B 3 Wiing {D} in ems of is componens {D} =(,D 2,D 3 ), {s} =(,, ), and evaluaing (3.3) yields he maix equaion, B 2 /n 2 e B 4 B 4 B 3 /n 2 e The non-ivial soluions o (3.5) n p e and n q e ae, D 2 D 3 = (3.5) (n p,q e ) 2 = (B 2 + B 3 ) ± (B2 B 3 ) 2 +4B (3.6)

32 7 diecion of popagaion z Y q 3 p 2 X Z Figue 3.2: Disceizaion of opical fibe ino FBG senso elemens. Also shown ae definiion of local polaizaion axes p and q, global polaizaion axes, X and Y,andlocal pinciple sain axes, 2 3. z is he diecion of popagaion along he opical fibe. These soluions coespond o he indices of efacion abou he pinciple opical axes p and q in he 2 3 plane shown in figue 3.2. These axes will be deemined lae. Fo an opically isoopic maeial, B = B 2 = B 3 =/(n o e) 2, B 4 = B 5 = B 6 =. Theefoe, n p e = n q e = n o e. Once sain is applied o he elemen, he dielecic impemeabiliy enso change is defined by he linea sain-opic equaion, B i = 6 p ij ɛ j (3.7) j= whee [p] is he sain-opic enso and he compac noaion is used fo he sain componens (ɛ = ɛ,ɛ 2 = ɛ 22,ɛ 3 = ɛ 33,ɛ 4 = γ 23,ɛ 5 = γ 3,ɛ 6 = γ 2 ) [46]. Expanding he soluion of (3.6), wiing B i = Bi o + B i, and applying he isoopic popeies o Bi o, we find, (n p,q e ) 2 = (n o e) + ( B 2 + B 3 ) ± ( B 2 B 3 ) B4 2 (3.8)

33 8 Fo an opically isoopic maeial, he sain-opic enso, [p], educes o, p p 2 p 2 p 2 p p 2 p 2 p 2 p [p] = (p p 22 )/2 (p p 22 )/2 (p p 22 )/2 (3.9) Subsiuing (3.9) ino (3.7) ino (3.8) yields n p e and n q e fo an elemen in he sessed sae. (n p,q e ) 2 = (n o e) 2 + p 2ɛ + (p + p 2 ) (ɛ 2 + ɛ 3 ) 2 ± (p p 2 ) 2 (ɛ 2 ɛ 3 ) 2 + ɛ 2 4 (3.) Kim e al. consideed he same fomulaion fo he opical fibe as a single homogeneous elemen and deived a lineaized fom of (3.) o calculae he sensiiviy of he FBG o ansvese sain [46]. To model a polaizaion mainaining fibe (such as figue 2.3(c)) hey consideed he fibe o be iniially opically ohoopic. This appoach poduces idenical esuls o he cene sain appoximaion of (2.7). Fo FBG senso poblems including hemal loading, (3.6) can be expanded o include a linea hemo-opic effec [46], B i = W i T + p ij (ɛ j α j T ) (3.) whee {α} ae he coefficiens of hemal expansion of he senso in he local opical coodinaes. Fo an isoopic senso (α = α 2 = α 3 = α and α 4 = α 5 = α 6 = ). The coefficiens W i,definedas, ( ) B i W i = (3.2) T σ=cons. ae measued duing iso-sess condiions. Fo an opically isoopic maeial, he non-zeo coefficiens ae hus evaluaed as, ( ) ( ) W = W 2 = W 3 = T (n o e) 2 = 2 n o e (n o e) 3 (3.3) T A ypical value fo he hemoopic coefficien ( n o e)/( T) fo silica is given by Kim e al. as.2 5 / o C [46]. Applying (3.), he local pinciple indices of efacion (fomely

34 9 (3.)) would hus be modified o (n p,q eff )2 = (n o e) 2 + p 2ɛ + (p + p 2 ) 2 ± (p p 2 ) 2 ( (ɛ 2 + ɛ 3 ) 2 (n o e) 3 n o e T ɛ 2 4 +(ɛ 2 ɛ 3 ) 2 (3.4) The angle of oienaion of he elemen pinciple opical axes ae idenical o he pinciple sain diecions [57]. Alhough he index of efacion is no a ue enso quaniy, i can be epesened by an ellipse in he 2-3 plane wih he majo and mino axes of lengh n p e and n q e in he pinciple sain diecions [57]. Theefoe o calculae n e abou he global axes X and Y,wefind n X e = n Y e = n p en q e (n p e cos ψ e ) 2 +(n q e sin ψ e ) 2 n p en q e (n p e sin ψ e ) 2 +(n q e cos ψ e ) 2 (3.5) whee ψ e is he angle equied o oae he p axis o he X axis. ) 3..3 Calculaion of popagaion consans fo a senso segmen Once he index of efacion fo each elemen is known abou boh he X and Y axes, he popagaion consans β max and β min and he oienaion of he pinciple opical axes fo he complee coss-secion mus be calculaed. The popagaion chaaceisics fo linealy polaized (LP) modes popagaing hough a waveguide of abiay coss-secion and abiay vaiaion of efacive index ae deemined by solving he scala wave equaion ove he coss-secion of he fibe, hee defined as he egion Ω, [ ] p zx x x (p xφ(x, y)) + [ ] p zy y y (p yφ(x, y)) + (qk 2 β 2 )Φ(x, y) = (3.6) whee he field Φ(x, y) and he coefficiens p x, p zx, p y, p zy,andq ae defined in able 3. fo he fundamenal LPx and LPy modes. The scala wave equaion defined in (3.6) is deived as follows: Maxwell s equaions fo souce fee, ime hamonic fields ae [52] E = jωµ H (3.7)

35 2 Table 3.: Paamees of scala wave equaion fo fundamenal LP modes. Mode Φ p x p zx p y p zy q LP x E x n 2 x /n 2 z n 2 x LP y E y n 2 y /n 2 z n 2 y H = jωɛ [ɛ] E (3.8) D = ɛ ([ɛ ] E) = (3.9) H = (3.2) whee E(x, y, z) = E(x, y)e jβz and H(x, y, z) = H(x, y)e jβz ae he elecic field and magneic fields especively, ɛ and µ ae he fee space pemiiviy and pemeabiliy consans, and [ɛ ] is he maeial pemiiviy enso given by n 2 x [ɛ ]= n 2 y (3.2) Taking he cul of (3.7) and subsiuing (3.8) we obain 2 E y x y 2 E x y 2 2 E x x y 2 E y x 2 2 E x z 2 2 E y z 2 n 2 z + 2 E z x z = k2 n 2 xe x (3.22) + 2 E z y z = k2 n 2 ye y (3.23) 2 E x x z 2 E z x 2 2 E z y E y y z = k2 n 2 ze z (3.24) whee k =2π/λ. Noing ha fo he LPx mode, E y = and using (3.9) yields E z = j βn 2 z x (n2 xe x ) (3.25) Subsiuing (3.25) ino (3.22) we obain ] x[ n 2 z x (n2 xe x ) + 2 E x y 2 + k2 n 2 xe x β 2 E x = (3.26) Similaly fo he LPy mode, E x = and ] y[ n 2 z y (n2 ye y ) + 2 E y x 2 + k2 n 2 ye y β 2 E y = (3.27)

36 2 Thus, he scala wave equaion allows us o solve fo he fields E x and E y independenly. Alhough, he scala wave equaion deived above is simila o he one deived by Koshiba [53], hee we include he gadien of he efacive index in he fomulaion. The funcional fo (3.6) is given by, { [ F = δφ p zx (x, y) ] x x (p x(x, y)φ(x, y)) + [ p zy (x, y) y y (p y(x, y)φ(x, y)) Ω ] } +(q(x, y)k 2 β 2 )Φ(x, y) dxdy (3.28) Taking he fis vaiaion of (3.28) and educing, one obains [ δf = p zx x (p xφ) x (δφ) + p zy y (p yφ) y (δφ) Ω ] +(β 2 qk)(δφ)φ 2 dxdy { [ δφ p zx Γ x (p ]} xφ) + p zy y (p yφ) dγ = (3.29) whee Γ is he bounday of he egion Ω. Since we ae only concened wih popagaed modes, i.e. modes ha ae fully conained in he opical fibe, we apply he bounday condiion Φ = on Γ. Disceizing he egion ino iangula elemens as shown in figue 3.2 and noing ha he coefficiens p x, p zx, p y, p zy,andq ae consan fo each elemen, (3.29) educes o [ p e zxp e Φ x x x (δφ) + pe zyp e Φ y y y (δφ) e Ω e ] +(β 2 q e k)(δφ)φ 2 dxdy = (3.3) whee he supescip e efes o he values fo a given elemen. elemen, Φ e = [N N 2 N 3 ] Φ Φ 2 Φ 3 We expand Φ in each = {N}T {Φ} (3.3) whee N, N 2,andN 3 ae elemen shape funcions and Φ,Φ 2,andΦ 3 ae nodal values of Φ. Subsiuing (3.3) ino (3.3), we obain he global maix equaion, [K]{Φ} β 2 [M]{Φ} = (3.32)

37 22 wih [K] = e [ q e k 2 {N}{N} T p e zxp e x{n x }{N x } T Ω e ] p e zyp e y{n y }{N y } T dxdy [M] = [ ] {N}{N} T dxdy (3.33) e Ω e The equied deivaives of he shape funcion fo he iangula elemen defined in figue 3.3 ae [53]: 2 {N}{N} T dxdy = Ae 2 2 Ω e 2 b 2 b b 2 b b 3 {N x }{N x } T dxdy = 4A e b b 2 b 2 2 b 2 b 3 Ω e b b 3 b 2 b 3 b 2 3 whee, A e = 2 {N y }{N y } T dxdy = 4A e Ω e x x 2 x 3 y y 2 y 3 b = y 2 y 3 b 2 = y 3 y b 3 = y y 2 c 2 c c 2 c c 3 c c 2 c 2 2 c 2 c 3 c c 3 c 2 c 3 c 2 3 (3.34) c = x 3 x 2 c 2 = x x 3 (3.35) c 3 = x 2 x Afe calculaing he maices [K] and[m], (3.32) is solved fo he eigenvalues, β, a a fixed wavelengh. Knowing ha he effecive efacive indices fo he LP modes lie beween he efacive index of he coe and cladding, i is compuaionally efficien o sep he effecive efacive index fom he cladding value o he coe value and solve fo values of β saisfying De{[K] β 2 [M]} = (3.36) The value of β fo he fundamenal LP mode is hus he lowes value of β obained. This pocedue is epeaed ove he ange of wavelenghs equied. The popagaion consans compued coespond o he wo ohogonal axes X and Y chosen as he opical fibe global axes in figue 3.2. Howeve hese axes do no

38 23 (x,y ) 2 (x,y ) (x,y ) 3 Figue 3.3: Definiion of iangula elemen wih nodal coodinaes. necessaily coespond o he opical popagaion axes fo he fibe coss-secion. The popagaion consan β, o similaly he effecive index of efacion n eff via (3.), vaies as an ellipse wih he oienaion of he axes X and Y, as shown in figue 3.4. The ohogonal opical popagaion axes, X and Y, coespond o he maximum and minimum values of β o n eff on he global index ellipse, as also indicaed in figue 3.4. θ is defined as he angle of oaion fom he X Y sysem o he X Y sysem. In ode o calculae n max eff and n min eff, one could oae he axes X and Y, pefom he FE calculaions of his secion and seach fo n max eff o n min eff. Howeve, as his mus be pefomed fo evey segmen of he FBG, a moe efficien mehod is o calculae n eff abou hee diffeen axes (no co-linea) and solve fo he hee unknowns n max eff, nmin eff,andθ. As veified by he esuls of he numeical example consideed in secion 3.2, he vaiaion of effecive index of efacion fo each segmen is an ellipse. The magniude and oienaion of he index ellipse is no known apioi, bu mus be deemined fom he hee indices calculaed abou he X, Y, and M (a an angle γ o he X axis) axes, as shown in figue 3.4. These hee known indices ae efeed o hee as n eff, n9 eff,andnγ eff. The indices o be obained ae labelled n max eff and n min eff and ae he maximum (majo axis) and minimum (mino axis) effecive efacive indices, shown in figue 3.4. Thus, he following mehod is used o solve fo he hee unknowns (θ, n max eff,andnmin eff ): We define X and Y o be he global popagaion axes fo he coss-secion fo which n max eff and n min eff ae calculaed. The

39 24 Y X Y C θ M 9 n eff γ n eff B γ O n eff A X max n eff min n eff Figue 3.4: Global index ellipse showing vaiaion of n eff fo a given mode wih oaion of global axes X Y. X Y coesponds o global popagaion axes of opical fibe. index ellipse defined in he opical popagaion coodinae sysem is given by: ( X n max eff ) 2 + ( Y n min eff ) 2 = (3.37) Poins A, B and C lie on he ellipse as shown in figue 3.4. Defining m = an(θ), m 2 = an(9 θ), and m 3 = an( γ θ ), he lenghs OA, OB and OC ae given by, ( ) 2 n eff =(x ) 2 +(y ) 2 =(+m 2 ) ( ( n 9 eff ) 2 =(x 2) 2 +(y 2) 2 =(+m 2 2) ( ( ) 2 n γ eff =(x 3) 2 +(y 3) 2 =(+m 2 3) ( ( n min eff n min eff n min eff ) 2 n max eff nmin eff ) 2 ( ) 2 (3.38) + m n max eff ( ) 2 n max eff nmin eff ) 2 ( ) 2 (3.39) + m 2 n max eff ( ) 2 n max eff nmin eff ) 2 + ( m 3 n max eff ) 2 (3.4) Equaions ae a coupled algebaic sysem of equaions fo he hee unknowns.

40 25 Rewiing (3.38) fo n max eff, subsiuing ino (3.39), and solving fo nmin eff, one obains ) 2 ( 2[m ( ) 2 (n n min eff n 9 2 eff) 2 m 2 ] eff = ( 2[ ( 2[ (3.4) neff) + m 2 2 ] neff) 9 + m 2 ] Equaion (3.4) gives he elaionship beween θ and n min eff. Subsiuing (3.4) ino (3.38) one obains a simila elaionship beween θ and n max eff, ) 2 ( 2[m ( ) 2 (n eff neff) m 2 ] = n max eff ( n 9 eff ) 2m 2 2 [ + m 2 ] ( n eff ) 2m 2 [ + m 2 2 ] (3.42) Thus, subsiuing (3.4) and (3.42) ino (3.4) one obains an equaion fo θ, which can be solved ieaively Calculaion of he senso esponse Once he popagaion consans and opical axes ae known fo each fibe segmen, he specal esponse of he FBG mus be calculaed. The T-maix appoximaion, fis inoduced by Yamada and Sakuda, is widely used o calculae he specal esponse of Bagg gaings wih non-consan popeies [58]. The pimay advanage of he T-maix mehod is ha i is compuaionally efficien as compaed o diec numeical inegaion of he coupled mode equaions fo he FBG. A second mehod would be o expand he scala wave equaion of (3.6) o include vaiaions in he z-diecion and fomulae a FE soluion fo he full 3D poblem fo β(z). Howeve, he T-maix mehod has been shown o convege apidly o he soluion of he coupled mode equaions and a FE soluion o he 3D poblem would equie disceizaion in he z-diecion anyway. Thus, no fuhe accuacy would be gained using 3D elemens a he cos of consideably moe compuaional effos. Due o he song sain gadiens expeced along he opical fibe, howeve, a modified T-maix mehod was deived incopoaing hese gadiens. This mehod will be descibed in deail in chape NUMERICAL EXAMPLES In ode o check he validiy of he FE appoach pesened in secion 3., a benchmak case modeled was consideed: a cicula, sep-index fibe fo which biefingence

41 26 is due o he applied loading. Afewads PM fibes fo which he biefingence is due o he geomey of he fibe, he esidual sesses in he fibe, and he applied loading wee analyzed. Fo each example, he loading is hough diameical compession. We have assumed a consan peiod of he gaing of Λ = 53 nm Cicula coe, sep-index fibe The geomey of he cicula coe sep-index fibe coss-secion, along wih he applied ansvese load is shown in figue 3.5. Table 3.2 liss he maeial and geomeic paamees of he fibe fo his simulaion. As obseved fom able 3.2, he fibe is mechanically isoopic. This paicula example was chosen since analyical soluions ae known fo boh he popagaion consan a zeo applied load [52] and he sain field houghou he fibe coss-secion due o diameical compession [6]. Table 3.2: Paamees of cicula coe, sep-index fibe. Paamee Value n coe.46 n clad.44 coe 4 µm clad 62.5 µm E coe = E clad 72 GPa ν coe = ν clad.7 p, p 2.3,.252 Duing he soluion pocess, he calculaed index of efacion fo each segmen (fo diameical compession, all segmens ae he same), n max eff and n min eff, ae ploed as a funcion of he nomalized fequency V = 2π λ coe n 2 coe n 2 clad (3.43) fo a fixed value of he applied load P. The obained soluions ae ploed in figue 3.6. One obseves ha muliple mode soluions appea fo lage values of V. To check he convegence of he FE soluion o he exac soluion, wo diffeen mesh sizes wee consideed as shown in figue 3.6. The FE soluion was found o convege o he analyical, exac soluion fo he fine mesh size which was used fo all fuhe simulaions. One excepion is he LP mode

42 27 P clad 2 coe Coe Cladding P Figue 3.5: Coss-secion of cicula coe, sep-index fibe wih applied ansvese load. Coe size is exaggeaed o show dimensions. which is no found by he fine mesh simulaion. This is pobably due o he axisymmeic bounday condiions, Φ =, ha wee applied on he oue edge of he fibe, since he LP mode is no axisymmeic. Howeve, fo FBG sensos he fibe diamee is chosen o be small such ha only he fis LP mode popagaes and conibues o he Bagg eflecion heefoe he highe modes will no even be consideed. Fo he following examples, we heefoe conside he case V = 2. Simila plos o he one shown in figue 3.6 ae hen geneaed fo he ange of applied loads equied. Fom he vaiaion of effecive indices of efacion wih diameical load, we calculae he new Bagg wavelenghs as λ max B λ min B =2n max eff Λ =2n min eff Λ (3.44) Figue 3.7 plos he obained linea dependence of each Bagg wavelengh wih applied ansvese load fo he cicula, sep-index fibe. The cene sain appoximaion of (2.7) is also ploed in figue 3.7 fo compaison. As obseved, he FE soluion maches well wih he soluion obained fom he cene sain appoximaion and coecly evaluaes he induced biefingence due o he applied loading.

43 LP n eff V LP LP LP LP LP LP LP Figue 3.6: Vaiaion of effecive index of efacion wih nomalized fequency fo cicula coss-secion, sep-index opical fibe. Cicles epesen exac soluion. Tiangles epesen esuls of coase mesh simulaions and squaes epesen esuls of fine mesh (27 coe elemens, 452 cladding elemens) simulaions PM fibes The pedicion of he sensiiviy of a fibe Bagg gaing (FBG) senso o ansvese loading is a complex poblem when he FBG is wien ino a polaizaion-mainaining (PM) fibe. This complexiy is due o he mechanical and opical non-homogeneiies of PM fibes. In geneal, FBGs wien in PM fibes exhibi diffeen sensiiviies o muliple loading componens (as seen in figue 2.2) due o geomeical and sess induced biefingence in he PM fibe. The sess induced poion of he biefingence can be due o esidual sesses and/o he applied ansvese loading componens. This biefingence effec has been exploied fo he independen measuemen of muliple sain componens and/o empeaue changes using single o muliple Bagg gaings [48]. Fuhemoe, his capabiliy makes he FBG senso ideal fo he monioing of esidual sesses duing cuing of laminaed composies as well as fuhe monioing of inenal sesses duing loading of he laminaes [26, 63, 64, 66, 68, 7]. Geneally, he elaive sensiiviies of he FBG wien in a PM fibe ae cali-

44 CSA FE (nm) B λ Slow axis Fas axis P (N/mm) Figue 3.7: Vaiaion of Bagg wavelengh wih applied diameical load fo FBG in cicula coe, sep-index opical fibe. Squaes epesen esul of cene sain appoximaion. FE esul is ploed as a solid line. baed (as discussed in Lawence e al. [48]), due o he complexiy of pedicing hese sensiiviies and he lack of deailed maeial infomaion fo he paicula PM fibes used. Uanczyk, Chmielewska and Bock [7], Chehua e al. [62], Lawence e al. [48] and Bosia e al. [49] pefomed expeimenal measuemens of he wavelengh shifs in FBGs due o muli-axis loading in vaious PM fibe ypes. Chehua e al. [62] pefomed a compaaive expeimenal sudy of vaious PM fibe ypes o deemine which fibe ype was opimal fo muli-axis sain sensing and empeaue monioing. The fibe ypes consideed in Chehua e al. [62] ae shown in figue 3.8. The auhos deemined ha amongs he specific, commecially available PM fibes sudied he ellipical coe SAP fibe povided he maximum sensiiviy o ansvese loading, while he Panda fibe povided he maximum sensiiviy o empeaue loading. Howeve, he paicula ellipical coe SAP fibe used had a significanly smalle cladding diamee han he ohe fibes, which he auhos noe was he cause of he inceased ansvese load sensiiviy. An independen evaluaion of he sandad ellipical coe SAP fibe was no pefomed. As poposed by Uanczyk, Chmielewska and Bock [7], he ansvese load and empeaue sensiiviies of he FBG

45 3 (a) (b) R62.5 P γ R62.5 P γ Slow axis Coe SAP Cladding P 5 P 5 Fas axis P R24 P P γ γ R8 γ 53 R7 28 R2.5 R4 R2.5 R62.5 R2.5 R62.5 P 22 P 6 P (c) (d) (e) Figue 3.8: Geomey of PM fibe ypes consideed: (a) ellipical coe fibe; (b) D-fibe; (c) ellipical coe SAP fibe; (d) Bow-Tie fibe; (e) Panda fibe. The slow and fas axes ae also indicaed. All dimensions shown in µm. can be inceased hough he simulaneous measuemen of he esponse of boh he fundamenal LP modes and he LP modes, howeve popagaing he LP modes in he FBG pesens subsanial challenges. The fibe ypes shown in figue 3.8 can be divided ino wo caegoies. The fis goup shown in figue 3.8 (a) and (b), he ellipical coe fibe and D-fibe especively, ae PM fibes due o he fac ha he fibes ae no axisymmeic. Fo he puposes of his chape, hese fibes ae consideed mechanically homogeneous, ye opically inhomogeneous wih a sep index disibuion in he index of efacion. The second goup of fibes shown in figue 3.8 (c)-(e) achieve biefingence due he lack of axisymmey as well as esidual sesses induced by he sess applying pa (SAP) suounding he coe. This SAP has diffeen mechanical and hemal expansion popeies han he silica, inducing lage esidual sesses on he fibe coe duing cooling of he fibe afe i is dawn. Due o he pesence of he SAP, hese fibes ae consideed boh opically and mechanically inhomogeneous.

46 3 In his secion, he above fomulaion is applied o he five PM fibe ypes shown in figue 3.8, subjeced o diameical compession, o calculae he Bagg wavelengh shifs fo an FBG wien ino each ype. Fo seveal of he fibe ypes, he vaiaion of he wavelengh sensiiviy wih he oienaion of applied sain axes elaive o he global axes of he fibe was also consideed. Since he ellipical coe SAP fibe consideed in he expeimenal esuls of Chehua e al. [62] was a small diamee fibe (d = 8 µm) he sensiiviy of his fibe compaed o a simila geomey in a sandad diamee fibe (d = 25 µm) was also pefomed. Finally, he sensiiviy of seveal of he PM fibes o empeaue loading was analyzed since he muli-axis FBG sensos ae ofen used fo simulaneous measuemens of empeaue and sain. The maeial popeies of he coe, cladding and SAP egions fo each of he opical fibes consideed ae lised in able 3.3. To sepaae he effec of he vaious PM fibe configuaions, he maeial popeies and coe sizes emained he same fo all fibes. Obviously vaying each of hese paamees would also significanly affec he sensiiviy of each fibe o ansvese sain, howeve opimizing he maeial popeies was no wihin he scope of his hesis and would also need o ake ino consideaion he fabicaion pocesses of he opical fibes hemselves. The paicula values used wee chosen fom he lieaue as ypical fo PM fibes [49, 53, 65]. In addiion, he simulaions wee pefomed a he same nomalized fequency value fo each fibe, V =.73. Table 3.3: Maeial Popeies of Modeled Opical Fibes. Paamee Value E coe = E clad 7 GPa E SAP 5 GPa ν coe = ν clad.9 ν SAP.2 α coe = α clad.55 6 / C α SAP /C p.3 p n coe.46 n SAP = n clad.45 T iniial 8 C To simulae he ole of he SAP fo he fibes in figue 3.8 (c)-(e) duing solidi-

47 32 ficaion of he opical fibes, he fibe geomey was modeled as shown in figue 3.8, and afewads a hemal loading of T iniial = 8 C was applied. The effec of he index of efacion change due o empeaue, δn o e/δt, was no included in equaion (3) since he final empeaue was assumed o be oom empeaue a which empeaue he index values of able 3.3 wee oiginally measued. The Bagg wavelenghs, λ B, a zeo applied load wee calculaed afe T iniial was applied. Since n eff fo he LPx and LPy modes vaied fom fibe o fibe due o he esidual sess condiions, i will be seen in he lae calculaions ha λ slow B and λ fas B a P = ae no he same beween he vaious fibe ypes. Howeve, his does no affec he compaison of he FBG sensiiviies since hese ae based on λ B / P. Table 3.4: Numbe of Elemens Meshed fo PM Fibe Types. Fibe Type Numbe of Elemens Coe Cladding SAP Ellipical coe NA D-fibe NA Ellipical coe SAP (25 µm) Ellipical coe SAP (8 µm) Bow-ie Panda ELLIPTICAL CORE FIBER The fis PM fibe ype consideed is he ellipical coe fibe of figue 3.8 (a). This fibe is a PM fibe due o he geomeic biefingence of he opical fibe. As he fibe is mechanically homogeneous, sess induced biefingence does no appea duing he dawing of he opical fibe. The numbe of elemens meshed fo his and all ohe fibes ae lised in able 3.4. As demonsaed in secion 3.2., hese wee sufficien fo convegence of he soluion. Figue 3.9 plos he calculaed Bagg wavelengh fo boh he fas and slow axes of he fibe as a funcion of applied load P, applied a an angle of γ = and γ =9 o he global X-axis. A maximum load of P = N/mm was applied o each fibe. Fo compaison, he esuls of cene sain appoximaion of Sikis [47] ae also ploed fo each simulaion. One can also see fom figue 3.9 ha his appoximaion is a good appoach fo hese loading

48 λ B (nm) 54 FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis λ B (nm) 54 FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis P (N/mm) P (N/mm) Figue 3.9: Vaiaion of Bagg wavelengh wih applied load fo FBG in ellipical coe fibe of figue 3.8 (a). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9. cases. D-FIBER The second fibe ype consideed is he D-fibe of figue 3.8 (b). As fo he ellipical coe fibe, his fibe is mechanically homogeneous and heefoe sess induced biefingence does no appea pio o loading of he fibe. Figue 5 plos he calculaed Bagg wavelengh shifs of he fibe fo γ = and γ =9 as a funcion of applied load, P. As can be seen fom figue 3., he sensiiviy of he D-fibe o he ansvese load is highe han ha of he pevious ellipical coe fibe. As fo he pevious case, he cene sain appoximaion is a good appoximaion fo hese loading cases. Fuhe numeical simulaions wee pefomed o deemine he sensiiviy of he D-fibe and he ellipical coe fibe o he applied loading angle γ. Howeve, he biefingence effec due he induced sesses is significanly lage han ha due o he oiginal geomeical effec [67]. Theefoe, even a he minimum loading P consideed, he fas and slow axes wee aligned wih he pinciple sain diecions which wee in he diecion of and pependicula o he applied loading. Theefoe, he sensiiviy of he fas and slow axes o P was independen of he loading angle γ. This was no he case fo lae simulaions of fibes wih SAP egions and heefoe high iniial esidual sesses.

49 FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis λ B (nm) 542 FE Fas Axis λ B (nm) FE Slow Axis CSA Fas Axis CSA Slow Axis P (N/mm) P (N/mm) Figue 3.: Vaiaion of Bagg wavelengh wih applied load fo FBG in D-fibe of figue 3.8 (b). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9. ELLIPTICAL SAP FIBER The hid PM fibe consideed is he ellipical SAP fibe which diffes fom he ellipical coe fibe due o he esidual sess biefingence induced by he SAP. Two sepaae cladding diamees wee consideed fo his fibe: he 25 µm sandad cladding diamee o be consisen wih he ohe PM fibes consideed and he 8 µm cladding diamee o sudy he effecs of cladding diamee on he FBG sensiiviies. In addiion, he fas and slow axis sensiiviies of he FBG wien ino he ellipical coe SAP fibe wee highly dependen upon he angle of applied loading γ. Theefoe, fou diffeen loading angles wee consideed: γ = and 9 o coincide wih he geomeical axes of he fibe, and γ =36 and 72. The simulaion esuls fo he 25 µm diamee fibe ae ploed in figue 3. and he esuls fo he 8 µm diamee fibe in figue 3.2. Fo boh fibes, when γ =9 and 72 he Bagg wavelenghs of he fas and slow axes convege owads one anohe as he load P is inceased. This effec due o he fac ha he applied load is in he invese sense han he iniial esidual sess in he opical fibe. A he value of P a which he applied load ovecomes he iniial esidual sesses in he fibe, he fas and slow axes coss one anohe and coninue o divege as he applied load inceases. The 8 µm diamee fibe demonsaed a significanly highe sensiiviy o applied load han he sandad 25 µm diamee fibe. This inceased sensiiviy is due o he

50 FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis λ B (nm) 54.2 λ B (nm) FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis P (N/mm) P (N/mm) (a) (b) Figue 3.: Vaiaion of Bagg wavelengh wih applied load fo FBG in ellipical coe SAP fibe of figue 3.8 (c) wih oue diamee d = 25 µm. Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9. educion in coss-secional aea of he fibe, and heefoe he inceased sain pe uni load. I is o be expeced ha his end would apply o he ohe PM fibe ypes. Geneally, fo each of he cases fo he ellipical SAP fibe, he cene sain appoximaion pedics he sensiiviy of he fas axis well, howeve i unde pedics he sensiiviy of he slow axis. BOW-TIE FIBER The fouh fibe modeled is he Bow-Tie fibe PM shown in figue 3.8 (d). As fo he pevious, ellipical coe SAP fibe, biefingence in he Bow-Tie fibe is achieved by he lage esidual sesses induced duing cooling of he fibe. This fibe was loaded a fou diffeen angles, γ =,9,36 and 72. The esuls of he FE simulaions ae ploed in figue 3.3 fo he fou loading angles. The ends obseved in figue 3.3 ae simila o hose of he ellipical coe SAP fibe. Howeve he sensiiviies o ansvese loading of he FBG wien in he Bow-Tie fibe ae less han hose of he ellipical coe SAP fibe.

51 FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis λ B (nm) 54.2 λ B (nm) P (N/mm) P (N/mm) (a) (b) FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis λ B (nm) 54.2 λ B (nm) P (N/mm) P (N/mm) (c) (d) Figue 3.2: Vaiaion of Bagg wavelengh wih applied load fo FBG in ellipical coe SAP fibe of figue 3.8 (c) wih oue diamee d = 8 µm. Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9 ;(c)γ =36 ;and (d) γ =72.

52 FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis 54.2 λ B (nm) λ B (nm) P (N/mm) P (N/mm) (a) (b) 54.6 FE Fas Axis 54.6 FE Fas Axis 54.4 FE Slow Axis CSA Fas Axis CSA Slow Axis 54.4 FE Slow Axis CSA Fas Axis CSA Slow Axis λ B (nm) 54 λ B (nm) P (N/mm) P (N/mm) (c) (d) Figue 3.3: Vaiaion of Bagg wavelengh wih applied load fo FBG in bow-ie fibe of figue 3.8 (d). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9 ;(c)γ =36 ; and (d) γ =72.

53 38 PANDA FIBER The final fibe ype consideed is he Panda Fibe of figue 3.8 (e), also a fibe wih lage induced esidual sesses. The esuls of he FE simulaions fo he Panda fibe ae ploed in figue 3.4 fo he fou loading angles γ =,9,36 and 72. In addiion he angle of oienaion of he slow and fas axes wih espec o he X axis fo each loading angle consideed is ploed in figue 3.5. When loading is applied along one of he fibe geomeic axes (γ =,9 ), he slow and fas axes emain oiened along hese diecions, alhough hey do swich diecions fo he case of γ =9. Fo he ohe wo loading angles, he fas and slow axes asympoe owads γ and γ ± 9 demonsaing he song dependence of he pinciple sain oienaions discussed in Bosia e al. [49]. The cene sain appoximaion pedics he angle of oienaion of he fas and slow axes vey well, which is o be expeced since i is based on he pinciple sain diecions. Finally, figue 3.6 plos he Bagg wavelengh as a funcion of applied load fo seveal loading angles simulaneously fo compaison. To compae he sensiiviies of he six fibe ypes consideed o ansvese loading, he maximum and minimum sensiiviies fo each fibe ype ae colleced in able 3.5. Since he maximum and minimum sensiiviies occued a he oienaions of γ = and 9 simulaions of which wee pefomed fo all fibes, he esuls ae compaable. Table 3.5 demonsaes ha amongs he 25 µm sandad cladding diamee fibes, he D-fibe had he highes sensiiviy fo boh he slow and fas axes. The ellipical coe fibe had he lowes sensiiviy fo he slow axis, while he ellipical coe SAP fibe had he lowes sensiiviy fo he fas axis. Consisen wih he expeimenal esuls of Chehua e al. [62], he small diamee (8 µm) ellipical coe SAP fibe had a significanly highe sensiiviy o boh he slow and fas axes han any ohe fibe consideed. Clealy, educing he oue cladding diamee of a PM fibe will incease he sensiiviy of a FBG senso wien ino he fibe, pesumably due o he educed coss-secional aea of he fibe and heefoe inceased sain fo a fixed load. Such an appoach could also be applied o he ohe fibe ypes of able 3.6. THERMAL LOADING Fo he final se of numeical simulaions, he hee PM fibes wih SAP egions wee subjeced o a unifom empeaue change T. These hee fibes will be moe

54 FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis λ B (nm) λ B (nm) P (N/mm) P (N/mm) (a) (b) FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis λ B (nm) λ B (nm) P (N/mm) P (N/mm) (c) (d) Figue 3.4: Vaiaion of Bagg wavelengh wih applied load fo FBG in panda fibe of figue 3.8 (e). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9 ;(c)γ =36 ; and (d) γ =72.

55 4 8 8 θ (degees) 6 4 θ (degees) FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis 2 FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis P (N/mm) P (N/mm) (a) (b) 2 γ + 9 ο 2 θ (degees) FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis γ θ (degees) FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis γ P (N/mm) P (N/mm) γ 9 ο (c) (d) Figue 3.5: Vaiaion of he oienaion of global index ellipse wih applied load fo FBG in panda fibe of figue 3.8 (e). Resuls fom boh he finie elemen and cene sain soluions ae ploed. (a) γ = ; (b) γ =9 ;(c)γ =36 ; and (d) γ =72.

56 FE Fas Axis FE Slow Axis 45, λ B (nm) , , , P (N/mm) Figue 3.6: Vaiaion of Bagg wavelengh wih applied load fo muliple oienaions of applied loading angle, γ, elaive o geomeical axes of Panda fibe. Resuls fom boh he finie elemen and cene sain soluions ae ploed. sensiive o hemal loading due o he hemal expansion coefficien mismach beween he SAP and silica, esuling no only in hemal sain, bu also in a educion in he iniial esidual sesses. The esuls of hese simulaions ae ploed in figue 3.7. The fas and slowaxesfoeachfibehavealineavaiaionwih T, howeve he diffeence in slopes beween he wo axes is small. These esuls ae consisen wih he expeimenal esuls of Chehua e al. [62]. The sensiiviies fo each of he hee opical fibes ae summaized in able 3.6. Fo his sudy, he ellipical coe SAP fibe demonsaed he highes sensiiviies o hemal loading. The expeimenal esuls of Chehua e al. [62] found he panda fibe o have he highes sensiiviy. Howeve, he values in able 3.6 ae much close o one anohe han hose fo he case of he applied load P and heefoe one expecs he paicula maeial popeies of he fibe (and in paicula he SAP maeial) o have a much songe influence on he ode of he hee fibes. One can also deemine fom figue 3.7 ha he cene sain appoximaion (hee including he addiional hemal em inoduced by Kim, Kollá, and Spinge [65] ahe han he oiginal fomulaion of [47]) pedics he hemal sensiiviies of he FBG well. Theefoe, he full opical/mechanical

57 42 Table 3.5: Compaison of Sensiiviy o Applied Load fo PM Fibe Types. Fibe Type Slow Axis Sensiiviy Fas Axis Sensiiviy [nm/(n/mm)] [nm/(n/mm)] Maximum Minimum Maximum Minimum Ellipical coe D-fibe Ellipical coe SAP (25 µm) Ellipical coe SAP (8 µm) Bow-ie Panda FE mehod is no necessaily equied o pedic hemal sensiiviies. Table 3.6: Compaison of Sensiiviy o Applied Tempeaue fo PM Fibe Types. Fibe Type Slow Axis Sensiiviy Fas Axis Sensiiviy (pm/c) (pm/c) Ellipical coe SAP (25 µm) Bow-ie Panda SUMMARY To summaize, we have pesened a mehodology o calculae he esponse of he embedded FBG due o applied load. We fomulaed a FE model o calculae he change in effecive efacive index due o he applied ansvese load. We consideed he case of diameical compession o validae he model. We fis consideed a benchmak case of cicula coe. Nex we modeled PM fibes and compued he sensiiviies of diffeen fibes due o applied diameical load. We found ha he D-fibe demonsaes he highes sensiiviy o ansvese loading. I is also shown ha educing he fibe cladding diamee significanly impoves he sensiiviies of he FBG senso o ansvese load. All fibe ypes exhibi appoximaely he same sensiiviy o hemal loading, wih he ellipical coe SAP fibe slighly moe sensiive. In he nex chape, we conside he effec of longiudinal load

58 λ B (nm) λ B (nm) FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis o T ( C) o T ( C) (a) (b) λ B (nm) FE Fas Axis FE Slow Axis CSA Fas Axis CSA Slow Axis o T ( C) (c) Figue 3.7: Vaiaion of Bagg wavelengh wih applied empeaue (a) ellipical coe fibe (d = 25 µm); (b) bow-ie fibe; (c) panda fibe. Resuls fo each axis calculaed using he cene sain appoximaion (CSA) ae also ploed.

59 44 on he FBG senso esponse. We fomulae a modified T-maix model consideing he effec of sain gadien in he specal esponse.

60 45 Chape 4 Modified Tansfe Maix Fomulaion fo Bagg Gaing Sain Sensos 4. INTRODUCTION The use of fibe Bagg gaings o obain local sain disibuions has been peviously exploied fo he measuemen of cack bidging acions [72], cack locaions [73], and cack densiies [34, 74] in laminaed composies. The specal esponse of a unifom, chiped, o apodized Bagg gaing is easily calculaed hough numeical soluion of he coupled mode equaions o hough appoximaion mehods such as he ansfe maix (T-maix) mehod [58]. The T-maix appoximaion models he non-unifom gaing popeies as piecewise consan funcions and educes he calculaion of he gaing specal esponse o a single esponse maix. The T-maix mehod is hus an exemely fas echnique fo he calculaion of Bagg gaing file popeies and has been demonsaed by Edogan [3] o model chiped files well using elaively few segmens. Fo axial sain sensing applicaions, he T-maix mehod was fis applied o a

61 46 Bagg gaing subjeced o a non-unifom sain disibuion by Huang e al. [75]. Huang e al. applied he T-maix fomulaion by appoximaing he applied sain as a piecewise coninuous funcion, calculaing he aveage peiod in each gaing segmen due o he applied sain, and subsiuing his local peiod ino he coupling coefficien fo he T-maix mehod. This T-maix model has since been used o model Bagg gaing sain sensos fo cack deecion in composies, fibe pullou ess, and fo he design of opical devices incopoaing mechanically loaded gaings [3, 33, 72, 76, 77]. Huang e al. consideed boh a unifom and an iniially chiped gaing subjeced o a linea sain disibuion along he lengh of he gaing [75]. They compaed simulaed gaing speca wih expeimenal esuls fom a gaing suface mouned on a beam and fi he speca o back calculae a linea sain pofile along he gaing, as expeced. Some discepancies beween he simulaed eflecion speca and he measued speca wee obseved a highe applied loading o he beam. Such discepancies will be shown in his chape o be due o he pesence of he sain gadien. This chape demonsaes ha in he pesence of a significan sain gadien he fomulaion of Huang e al. [75] does no convege o he soluion of he coupled mode equaions fo a lage numbe of gaing segmens. Theefoe, he cuen wok poposes a modified local peiod funcion o be used fo he T-maix mehod. The deived peiod funcion is also exended o he case of oiginally chiped gaings subjeced o boh consan and non-unifom sain fields. The convegence of each case o he coupled mode equaion soluions is demonsaed hough simulaions. Finally, he modified ansfe maix mehod is applied o peviously published expeimenal daa fo independen validaion. 4.2 REVIEW OF TWO-MODE COUPLING IN BRAGG GRATINGS This secion eviews he wo pimay analysis mehods fo wo-mode coupling in Bagg gaings. The fis mehod, diec numeical inegaion of he coupled mode equaions, will be used as he exac soluion fo evaluaion of he deived appoximaions.

62 Coupled mode heoy In he axial diecion, he Bagg gaing can defined as a peiodical peubaion o he effecive index of efacion, n eff, of he opical fibe coe descibed by [3], [ ]} 2π δn eff (z) =δn eff {+νcos z + φ(z) Λ whee ν is he finge visibiliy, Λ he nominal peiod, φ(z) he gaing chip, and δn eff (4.) he dc index change spaially aveaged ove a gaing peiod. This peubaion δn eff is assumed o be small. The applicaion of he gaing popeies calculaed in he pevious chape o (4.) will be discussed lae in secion 4.5, afe we deive he T-maix fomulaion. Coupled mode heoy is an excellen ool fo obaining quaniaive infomaion abou he specal esponse of fibe Bagg gaings [6, 78]. Hee we conside a Bagg eflecion gaing in a single mode opical fibe. The assumpions ae made ha he opical fibe is weakly guiding and no enegy is coupled o adiaion modes [9]. Using coupled mode heoy wih he above assumpions, one obains he following coupled fis ode diffeenial equaions descibing mode popagaion hough he gaing (z is he coodinae in he diecion of popagaion) [3]: dr(z) = iˆσr(z)+iκs(z) dz ds(z) = iˆσs(z) iκr(z) (4.2) dz whee R(z) ands(z) ae he ampliudes of he fowad and backwad popagaing modes especively. ˆσ is he geneal dc self-coupling coefficien as a funcion of he popagaing wavelengh, λ, definedas ( ) ˆσ =2πn eff λ + 2π λ D λ δn eff 2 φ (z) (4.3) whee φ (z) =dφ/dz, andλ D =2n eff Λ is he design wavelengh. κ is he ac coupling coefficien defined as κ = π λ νδn eff. (4.4) We define he limis of he gaing o be ( L/2) z (L/2). The acual bounday condiions fo he Bagg gaing ae R(L/2) = and S(L/2) =. Howeve,

63 48 an exac soluion fo he unifom gaing (i.e. Λ (z) =φ (z) = ) exiss o (4.2) fo he bounday condiions R( L/2) = and S(L/2) =. I can be shown ha he soluion of he fome bounday condiions yields he same efleciviy as he lae, heefoe we use he lae bounday condiions fo simpliciy [3]. When used as a sain senso, he Bagg gaing descibed in (4.) is consideed axially non-unifom in one of he hee cases lised below: Case : An iniially unifom gaing (consan peiod) is subjeced o a non-consan sain field; Case 2: An iniially chiped gaing (non-consan peiod) is subjeced o a consan sain field; Case 3: An iniially chiped gaing is subjeced o a non-consan sain field. Fo he above hee cases (4.2) can be solved numeically using diffeen echniques [9]. Fo example, (4.2) can be ansfomed ino a Ricai diffeenial equaion by subsiuing ρ(z) = S(z)/R(z). Diffeeniaing ρ wih espec o z and subsiuing ino (4.2) yields dρ(z) = iκ 2iˆσρ(z) iκρ(z) 2. (4.5) dz The bounday condiions ae hen ansfomed o ρ(l/2) =. A fouh ode Runge-Kua numeical inegaion wih adapive sep size can hen be used o solve (4.2) o (4.5) by inegaing backwads fom z = L/2 oz = L/2. This numeical soluion is used as he exac soluion houghou his chape. The efleciviy (λ) ofhebagg gaing is calculaed as a funcion of wavelengh as S( L/2) 2 (λ) = = ρ( L/2) 2. (4.6) R( L/2) Diec numeical inegaion of (4.2) o (4.5) is compuaionally inensive and heefoe no pacical fo analyzing lage amouns of expeimenal daa. Fo his eason, pevious auhos have modeled he iniially unifom gaing subjeced o a non-consan sain field (case ) as an equivalen iniially chiped gaing subjeced o a consan sain field (case 2) applying he T-maix appoximaion [33, 75, 79]. Edogan demonsaed ha he piecewise unifom appoximaion can be used o accuaely model chiped gaings (i.e. case 2) [3]. The pupose of his chape is o popose a modified T-maix o model he non-consan applied sain o be used fo cases and 3. I will also be shown ha his

64 49 modified model conveges o he exac soluion of (4.2) in he limi of a lage numbe of gaing segmens fo cases and Tansfe maix appoximaion The T-maix appoximaion, fis inoduced by Yamada and Sakuda, is widely used o model Bagg gaings wih non-consan popeies [58]. This appoach divides he gaing ino M smalle secions each wih unifom coupling popeies. Defining R i and S i o be he field ampliudes afe avesing he i h secion, he popagaion hough his unifom secion is descibed by, [ ] [ ] Ri Ri = F i. (4.7) S i S i F i = [ cosh(γb z) i ˆσ γ B sinh(γ B z) i κ γ B sinh(γ B z) ] i κ γ B sinh(γ B z) cosh(γ B z)+i ˆσ γ B sinh(γ B z) (4.8) whee F i is he opical ansfe maix, γ B = κ 2 ˆσ 2 and z is he lengh of he secion [3]. Fo he enie gaing, he T-maix, F can be wien, [ ] R( L/2) [ ] R(L/2) S( L/2) = F S(L/2) (4.9) whee, F = F M F M... F. The efleciviy as a funcion of wavelengh can hen be calculaed using (4.6). The pimay advanage of he T-maix mehod is ha i is compuaionally efficien as compaed o diec numeical inegaion of (4.2). Typically, M is moe han sufficien o accuaely model chiped gaings. A limiaion of he T-maix appoximaion howeve is ha he numbe of secions M canno be abiaily lage since seveal gaing peiods ae equied fo complee coupling. Theefoe, we equie ha [9], M 2n effl. (4.) λ D Howeve, fo sensing applicaions his limiaion is no ypically eached.

65 5 4.3 T-MATRIX FORMULATION BASED ON MODIFIED PERIOD FUNCTION As menioned peviously, Edogan demonsaed ha he T-maix appoximaion can be applied successfully o iniially chiped gaings [3]. Fo sain sensing applicaions, when he gaing is subjeced o a sain gadien, is eflecive specum will no only be shifed bu also disoed due o non-unifom changes in boh he local index of efacion and he gaing peiod. Fo axial sain, ɛ(z), applied o an unchiped gaing hese wo effecs can be combined hough an effecive gaing peiod defined by [8], Λ(z) =Λ [ + ( p e )ɛ(z)] (4.) whee Λ is he peiod of he gaing befoe sain is applied and he phooelasic consan p e akes ino accoun he change in effecive index of efacion of he opical fibe wih sain. Though simulaion and compaison wih he diec soluion of (4.2), one can show ha his fomulaion coecly models he gaing efleced specum fo a consan applied sain field (i.e. ɛ (z) = ) Unifom gaings In his chape, we fis conside he case of an iniially unifom gaing subjeced o an axial sain disibuion along he gaing, ɛ(z). Fo his and all fuhe cases, only small defomaions of he senso ae consideed (i.e ɛ(z) ), heefoe non-linea gaing effecs ae negleced. To numeically solve (4.2) using he Runge-Kua mehod, an effecive chip funcion, φ(z), is fis obained by focing he peubaion δn eff of (4.) o have a peiod vaiaion equivalen o ha of (4.), φ(z)+ 2πz Λ = 2πz Λ [ + ( p e )ɛ(z)]. (4.2) Solving fo φ(z), we find he effecive gaing chip funcion, [ ] φ(z) = 2π ( p e )ɛ(z) z (4.3) Λ +( p e )ɛ(z) The deivaive of his chip funcion is hen used in he dc coupling coefficien of (4.3) when numeically inegaing (4.2). Examples of such simulaions fo an iniially unifom

66 5 gaing subjeced o a linea and quadaic sain field ae shown in figues 4. and 4.2 especively. To apply he T-maix appoximaion, Huang e al. calculaed he aveage peiod, Λ fo each gaing segmen due o he applied sain using (4.) and subsiued Λ(z) = Λ diecly ino he dc coupling coefficien of (4.3) assuming φ (z) = [75]. ˆσ = 2π λ (n eff + δn eff ) 2πn eff λ D = 2π λ (n eff + δn eff ) π Λ(z). (4.4) As will be shown lae, his T-maix appoximaion does no mach well wih numeical soluion of he coupled mode equaions fo non-consan sain fields. In his chape, we popose a modified effecive gaing peiod Λ(z) obeusedinˆσ, definedas, Λ(z) Λ [ + ( p e )ɛ(z)+( p e )zɛ (z)]. (4.5) The appopiae dc coupling coefficien fo he T-maix mehod, hee enamed σ modf, hen becomes, ˆσ modf = 2π λ (n eff + δn eff ) π Λ(z). (4.6) Alhough (4.5) does no epesen he physical gaing peiod due o he applied sain, we show ha he modified coupling coefficien ˆσ modf is equivalen o (4.3) including he gaing chip of (4.3). Assuming small defomaions, i.e. ɛ, we can wie (4.3) as, Diffeeniaing (4.7) wih espec o z gives φ(z) = 2π Λ ( p e )zɛ(z). (4.7) 2 φ (z) = π Λ [( p e )ɛ(z)+( p e )zɛ (z)]. (4.8) Now calculaing he em (/2) φ (z) by equaing he wo coupling coefficiens ˆσ modf in (4.6), and ˆσ in (4.3) yields π Λ [ + ( p e )ɛ(z)+( p e )zɛ (z)] = π Λ + 2 φ (z) (4.9)

67 ε(z) (µ sain) Refleciviy z (mm) (a) (b) Figue 4.: Response of a unifom gaing subjeced o an applied (a) linea sain field. (b) Refleced speca due o he applied linea sain field along he gaing lengh: dashed line is Runge-Kua simulaion of he coupled mode equaions; solid line is T-maix simulaion using Λ(z); doed line is T-maix simulaion using Λ(z). o, [ ] 2 φ (z) = π ( p e )ɛ(z)+( p e )zɛ (z) Λ +( p e )ɛ(z)+( p e )zɛ. (4.2) (z) Making he above assumpion of ɛ yields 2 φ (z) = π Λ [( p e )ɛ(z)+( p e )zɛ (z)] (4.2) which is equivalen o (4.8). Theefoe he coupling coefficiens ˆσ modf in (4.6) and ˆσ in (4.3) ae equivalen. To apply Λ(z) in he T-maix assumpion, he aveage value of Λ(z) is hen calculaed fo each gaing segmen of lengh z, accoding o Λ i = z z i + z/2 z i z/2 Λ(z)dz (4.22) whee z i is he midpoin of he i h segmen. This value is hen used in he calculaion of F i (in (4.8)).

68 53 ε(z) (µ sain) Refleciviy z (mm) (a) (b) Figue 4.2: Response of a unifom gaing subjeced o an applied (a) quadaic sain field. (b) Refleced speca due o he applied quadaic sain field along he gaing lengh: dashed line is Runge-Kua simulaion of he coupled mode equaions; solid line is T-maix simulaion using Λ(z); doed line is T-maix simulaion using Λ(z). One noes ha Λ(z) is a funcion of he applied sain ɛ(z) and sain gadien ɛ (z). Theefoe, in he pesence of a significan sain gadien one expecs a diffeence beween he gaing efleced specum simulaed using he T-maix mehod incopoaing ˆσ modf in (4.6) as compaed o ˆσ in (4.4). This diffeence can be clealy seen in figues 4. and 4.2 whee simulaions of a gaing subjeced o linea and quadaic sain fields ae ploed using boh he Λ(z) and Λ(z) fomulaions. The modelled gaing has he following popeies: n eff =.46, λ D = 557 nm, L = 4 mm, δn eff =2.5 4, p e =.26. Fo each of he T-maix simulaions 2 gaing segmens wee used in ode o check he convegence of he T-maix appoximaion o he Runge-Kua soluion of (4.2). I can be seen fom he gaphs of figues 4. and 4.2 ha he T-maix fomulaion using Λ(z) does no convege o he diec soluion of he coupled mode equaions. The fomulaion using he modified peiod funcion Λ(z) does convege coecly howeve. This is o be expeced since he coupling coefficien conveges o he coec value as shown above. An impoan poin o be gained fom figues 4. and 4.2 is ha fo hese examples he sain gadien em in Λ(z) is of he same ode as he sain em and heefoe canno be negleced in he simulaions. In conas, highe ode sain deivaive ems,

69 54 ε(z) (µ sain) z (mm) Refleciviy (a) (b) Figue 4.3: Response of a unifom gaing subjeced o an applied (a) linea sain field wih high sain gadien. (b) Refleced speca due o he applied linea sain field along he gaing lengh: dashed line is Runge-Kua simulaion of he coupled mode equaions; solid line is T-maix simulaion using Λ(z); doed line is T-maix simulaion using Λ(z). fo example ɛ (z), do no appea in (4.5) even hough hey may be non-zeo as in he case of figue 4.2(a), howeve he modified T-maix fomulaion sill conveges o he exac soluion. As an exeme case, we also conside a linea sain field wih a elaively high sain gadien applied o a unifom gaing wih he popeies as above. Figue 4.3 plos he efleced specum due o he applied linea sain field. Noe ha he maximum efleciviy of he gaing has been educed o appoximaely.23. Once again, he specum obained fom he Runge-Kua mehod maches well wih he specum obained fom he T-maix mehod using he modified peiod funcion Λ(z). Also ploed is he specum obained fom he T-maix mehod using Λ(z) which diffes significanly fom he ohe wo cuves. So fa we have consideed he cases of linea and quadaic sain fields applied along he gaing lengh. These cases could be consideed as specialized examples since he gadien em in Λ(z) is a scala muliple of ɛ(z) (i.e. zɛ (z) =C ɛ(z)). Theefoe, we also pesen he example of an applied sain field ha is exponenially vaying along one half

70 55 ε(z) (µ sain) z (mm) Refleciviy (a) (b) Figue 4.4: Response of a unifom gaing subjeced o an applied (a) exponenially vaying sain field wih high sain gadien. (b) Refleced speca due o he applied exponenially vaying sain field along he gaing lengh: dashed line is Runge-Kua simulaion of he coupled mode equaions; solid line is T-maix simulaion using Λ(z); doed line is T-maix simulaion using Λ(z). of he gaing lengh, as shown in figue 4.4(a). Figue 4.4(b) plos he efleced speca obained fom he Runge-Kua mehod, he T-maix mehod using he modified peiod funcion Λ(z), and he T-maix mehod using he peiod funcion Λ(z). One obseves ha he specum obained fom he T-maix mehod using Λ(z) maches well wih he speca obained fom he Runge-Kua mehod Chiped gaings We now conside he case of a chiped gaing (i.e. φ (z) ) subjeced o a non-unifom sain field (case 3). The case of an apodized gaing (i.e. δn eff consan) is no consideed explicily in his chape since axial sain does no significanly affec he apodizaion of he gaing. As menioned peviously, a chiped gaing subjeced o a unifom sain field is equivalen o a unifom gaing subjeced o a non-unifom sain field. Howeve, o simulae a non-unifom gaing subjeced o a non-unifom sain field, we fis find an equivalen sain, ɛ (z), o be applied o a non-chiped gaing o simulae he iniial chip. Then one applies he non-unifom sain field ɛ(z), esuling in a modified peiod funcion which is a funcion of boh ɛ (z) andɛ(z).

71 56 We find he equivalen sain ɛ (z) due o he gaing chip fom (4.3), ɛ (z) = ( p e ) φ (z) [ 2πz Λ + φ (z) ]. (4.23) To apply he Runge-Kua mehod, we hen find φ(z), given in (4.24), o be he equivalen gaing chip such ha he esuling peiod due o he applied non-unifom sain field maches ha of (4.), i.e. 2πz Λ [ + ( p e )ɛ (z)] + φ(z) = φ(z) = φ (z) 2πz Λ [ + ( p e )ɛ (z)][ + ( p e )ɛ(z)] [ ] ( p e )ɛ(z)z +( p e )ɛ(z) 2π Λ [ + ( p e )ɛ (z)] (4.24) To apply he T-maix mehod, he modified effecive gaing peiod is solved by seing he coupling coefficiens ˆσ and ˆσ modf o be equivalen, in he same manne as in he calculaions fo a unifom gaing. The esuling modified peiod funcion Λ(z) is given by (4.25). Λ(z) =Λ [ + ( p e )ɛ (z)+( p e )zɛ (z)][ + ( p e )ɛ(z)+( p e )zɛ (z)] (4.25) As an example, we conside a unifom gaing wih popeies as given in he above secion. Figue 4.5 plos seveal examples of he efleced specum fo he gaing subjeced o boh an iniial chip and a non-unifom sain field. Simulaions using boh he Runge-Kua mehod and he T-maix mehod using Λ(z) ae ploed. Fo figues 4.5 (a) and (b) he gaing is iniially subjeced o a consan chip of he fom ɛ (z) =C whee C is a consan. Afewads, a linea and quadaic sain field ɛ(z) ae applied o he gaing fo he wo gaphs especively. The esuling Λ(z) is calculaed using (4.25). Fo figue 4.5(c) he gaing is iniially subjeced o a chip of he fom ɛ (z) =C z. Afewads, a quadaic sain field ɛ(z) is applied. Fo each case, he modified T-maix fomulaion and he coupled mode equaions esuls ae idenical, demonsaing ha he fomulaion of (4.25) conveges o he soluion of he coupled mode equaions Calculaion of applied sain When he Bagg gaing is used as a sain senso, he effecive peiod funcion Λ(z) would be deemined fom measued eflecion specum daa, fo example a geneic al-

72 Refleciviy Refleciviy (a) (b) Refleciviy (c) Figue 4.5: Response of a chiped gaing: (a) ɛ (z) = 6 and ɛ(z) = ( /mm)z; (b) ɛ (z) = 6 and ɛ(z) =( 6 /mm 2 )z 2 ; and (c) ɛ (z) = ( /mm)z and ɛ(z) =( 6 /mm 2 )z 2. Runge-Kua simulaions ae ploed as dashed lines and T-maix simulaions using Λ(z) ae ploed as doed lines.

73 58 goihm o mach he measued efleced specum o simulaed ones [8, 82]. Once he bes fi specum has been obained, he funcion Λ(z) is known, fom which he applied sain ɛ(z) mus be calculaed using (4.5). The soluion of his odinay diffeenial equaion yields [ ][ ɛ(z) = Λ(z)dz Λ o ]. (4.26) ( p e )Λ o z Thee is also a homogeneous soluion ha saisfies (4.5), given in (4.27) whee C is an abiay consan. This soluion is pesen due o he small sain appoximaion in (4.5). Howeve, his soluion is no physically ealisable, since i is infinie a he midpoin of he gaing, z =. Theefoe, his em has been dopped fom he soluion in (4.26). [ ] C ɛ h (z) = (4.27) ( p e ) zλ o Discussion A quick noe should be menioned as o why he diffeence beween he T-maix fomulaion using Λ(z) and expeimenal measuemens of gaing speca have no been menioned in he pevious lieaue. As an example, Huang e al. suface mouned a Bagg gaing on a specially designed canileveed beam [75]. They assumed ha he sain disibuion along he gaing was linea and calculaed he slope such ha he simulaed gaing esponses using ˆσ in (4.4) mached he measued eflecion speca. Fo confimaion, he simulaed speca wee compaed o he soluion of Kogelnik [83]. Since he sain disibuion was linea hee is simply a diffeence in scaling faco beween he peiod funcion Λ(z) and he modified peiod funcion Λ(z) used in he modified T-maix mehod, as menioned in secion secion Thus he diffeence beween he T-maix mehod and modified T-maix mehod would no be appaen in a cuve fi o he gaing esponse. This appoach was also applied by Pees e al. [76] wih he same esul. 4.4 EXPERIMENTAL VALIDATION Finally, his secion demonsaes he impoance of including he sain gadien in he effecive peiod fomulaion, while validaing he modified T-maix fomulaion

74 59 hough independen expeimenal esuls. We conside expeimenal specal daa peviously obained by Chang and Voha [84]. To consuc a ansduce, Chang and Voha suface mouned a gaing adially on an aluminum diaphagm. The diaphagm was hen subjeced o a unifom pessue P. The non-linea axial sain along he fibe diecion was calculaed by Chang and Voha o be ɛ() = 3PZ f ( ν 2 ) 4E 3 (R ) (4.28) whee Z f is he disance fom he fibe coe o he cene plane of he diaphagm, E, ν, and R ae he Young s modulus, Poisson s aio, hickness, and adius of he diaphagm, especively, and is he posiion along he diaphagm. The following gaing popeies wee obained fom he iniial specum: n eff =.46, L = 8 mm, λ B =(+δn eff /n eff )λ D = 55 nm, p e =.26, Z f =62.5 µ, andδn eff = e 8.6(z/L)2. The diaphagm popeies ae: R = 9 mm, =.3 mm, ν =.3, and E =72.5 GPa. Figue 4.6(a) shows he efleced specum of he gaing measued a vaious pessues published by Chang and Voha. As seen, his is an excellen example of a Bagg sain senso subjeced o a non-unifom sain field. Fo he same pessue values figue 4.6(b) plos he simulaed efleced specum fo he gaing due o he applied sain of (4.28) using he modified peiod funcion Λ(z) in he T-maix mehod. Figue 4.6(c) plos he simulaed efleced specum using he effecive peiod funcion Λ(z). Alhough a qualiaive compaison, one obseves ha he simulaed efleced specum including he sain gadien in he effecive peiod funcion maches well wih he measued specum. On he ohe hand, wihou he sain gadien in he effecive peiod funcion, he T-maix simulaion does no closely mach he expeimenal daa. 4.5 APPLICATION TO PM FIBER To apply he T-maix appoach using he esuls obained in Chape 3, he maix [F i ] is evaluaed fo each segmen of he gaing. I is assumed ha he ae of oaion of he global opical axes (dθ/dz) (see figue 3.4) is no oo lage such ha he T-maix fomulaion assuming does no hold. In ode o incopoae he popagaion

75 6 λ/λ B (a) Refleciviy psi λ/λ B (b) Refleciviy psi λ/λ B (c) Figue 4.6: Reflecion speca fo a gaing suface mouned on an aluminum diaphagm. (a) Expeimenally measued speca (fom Chang and Voha [84]), (b) Speca simulaed including sain gadien in he effecive peiod fomulaion, and (c) Speca simulaed wihou sain gadien in he effecive peiod fomulaion.

76 6 consan of each segmen, he coupling coefficien ˆσ is evaluaed as, ˆσ = β + 2π λ δn eff π Λ(z) (4.29) whee δn eff is he mean mode effecive index of efacion vaiaion and β = β max o β min depending on he mode consideed. Fo ease of calculaion, i is geneally assumed ha he vaiaion of β is small ove he bandwidh of he FBG, heefoe, he value of β coesponding o he Bagg wavelengh is used. Λ(z) is he modified peiod funcion used o mach he coupling popeies of he gaing (wih one change), Λ(z) Λ [ + ɛ (z)+zɛ (z)] (4.3) The soluion of Λ(z) in (4.5) includes ems muliplied by he effecive sain-opic coefficien p e. Howeve, hese ems ake ino accoun he poisson conacion of he fibe due o an axial load and he change in index of efacion of he maeial wih sain. Since boh of hese effecs ae aleady included in he FE fomulaion, hey ae dopped fom he funcion Λ(z) in (4.3). Fo his fomulaion, he value of Λ is calculaed fo each gaing segmen using he aveage sain ɛ and sain gadien ɛ / z beween he wo faces of he elemen shown in figue Addiional commens Some addiional commens ae equied a his poin o discuss effecs negleced in he cuen FE/T-maix fomulaion. Fisly, he effec of bending on he FBG, alhough only ponounced once he cuvaue is significan, can easily be included in he FE fomulaion. Such a high cuvaue in he FBG would be expeced fo some damage deecion applicaions. The fibe bending acs o educe he efleciviy of he gaing, paiculaly in egions of maximum efleciviy [6]. Pabhugoud and Pees modeled he loss of efleciviy hough he bending powe loss em, P (λ, z), as a funcion of he localized cuvaue, ρ [43]. The fom of P (λ, z) was deemined expeimenally in [43]. Since his em acs o educe he powe of all modes popagaing hough he gaing in eihe diecion, he P can be evaluaed fo each senso segmen and he opical ansfe maix educed accoding o, [F ]=( P M )[F M ]( P M )[F M ]...( P )[F ] (4.3)

77 62 Secondly, by educing he Maxwell s equaions o he 2D wave popagaion poblem of (3.6), he effec of he shea sains γ 2 and γ 3 have been eliminaed. Howeve, i has been demonsaed by seveal auhos ha unless hese shea componens ae exemely lage, as in he case of a iled gaing, his effec is negligible (see [43]). In addiion, he pesence of a coaing on he fibe would educe his effec even fuhe. Should i be desied o include his effec, howeve one can modify he coupling coefficien κ of (4.4) fo each segmen accoding o κ = π λ ν(γ)δn eff (4.32) whee ν(γ) is he finge visibiliy of he gaing [43]. Fo small shea sains, ν(γ) cos γ. Finally, alhough n eff is assumed o change houghou he opical fibe due o he applied sess, he ampliude of he index modulaion δn eff is assumed o emain consan along he gaing due o he fac ha δn eff n eff. This assumpion is consisen wih pevious soluions [5] Ellipical SAP fibe Fo he final numeical simulaion, we conside a FBG wien ino a PM fibe incopoaing SAP egions as shown in figue 4.7. The coe and cladding egions ae cicula, wheeas he SAP egion is ellipical. This example demonsaes he abiliy of he FE model o capue biefingence due o he fibe geomey, esidual sesses due o he hemal expansion coefficien mismach beween he SAP and he silica, and he applied loading. The PM fibe was meshed wih 272 coe elemens, 64 inne cladding elemens, 376 SAP elemens, and 49 oue cladding elemens. The maeial and geomeical paamees of he PM fibe used fo he simulaions ae given in able 4.. Since exac opical popeies of he SAP ae difficul o obain, n, p,andp 2 fo he SAP wee assumed o be same as fo he cladding. Simulaions wee pefomed fo seveal values of hese paamees, howeve no noiceable diffeences wee obained fo he LP modes, pesumably due o he low enegy densiy of hese modes in he SAP. Theefoe, only he mechanical popeies of he SAP ae of impoance when calculaing he Bagg wavelenghs. To simulae he ole of he SAP duing solidificaion of he opical fibe, he geomey of figue 4.7 was modeled and afewads a hemal loading of T = 8 C was applied. The effec of he index of efacion change due o empeaue was no included since he final empeaue was assumed o be oom empeaue a which he index values of able 4. wee oiginally

78 63 measued. P o clad γ 2b P i 2 clad 2a Coe SAP Cladding Figue 4.7: Coss-secion of PM fibe wih applied ansvese load. Coe size is exaggeaed o show dimensions. Figue 4.8 demonsaes ha he azimuhal vaiaion of effecive efacive index, obained fom he FE simulaion, is an ellipse. Also ploed fo compaison is an ellipse oiened a angle θ o he global X axis wih n max eff and n min eff as majo and mino axes lenghs. Figues 4.9(a) and 4.(a) plo he vaiaion of Bagg wavelenghs wih applied ansvese load fo wo diffeen loading angles. One noes he nonlinea vaiaion of Bagg wavelengh wih applied ansvese load as also obseved in [49]. Fo compaison, he esuls using he cene sain appoximaion of (2.7) ae also ploed. Fo his example of a PM fibe in diameical compession hee is no a significan diffeence in he calculaions using he cuen FE fomulaion and he cene sain appoximaion. Even fo he case whee significan sain gadiens exis acoss he fibe coss-secion he diffeence beween he CSA and FE soluion is small. To demonsae he impoance of calculaing he global opical axes fo each segmen, he global slow axis is also ploed vesus applied load in figues 4.9(b) and 4.(b). Fo he fis case, γ =36, he slow axis is oiginally a θ = due o he esidual sesses. As he applied loading inceases, he applied sain ovecomes he esidual sesses and θ asympoes o he loading angle γ. Similaly, fo he second case, γ =72, he slow axis sas a θ = and inceases owads he applied loading angle. Howeve, once γ =45 he esidual sesses ae effecively balanced by he applied loading and fo highe loads he fas and he slow axes ae evesed (as seen a P = 2 N/mm in figue 4.(b)).

79 64 Table 4.: Paamees of PM fibe. Paamee Value n coe.46 n clad = n SAP.44 coe 2.5 µm clad i, o clad 7.5 µm, 62.5 µm a, b 3.75 µm, 4.25 µm E coe = E clad 7 GPa E SAP 5 GPa ν coe = ν clad.9 ν SAP.2 α coe = α clad.55 6 /C α SAP /C p, p 2.3,.252 To demonsae he complee pocedue oulined in his chape, we also simulaed he specal esponse of a FBG, in he PM fibe consideed above, o vaying diameical load along he lengh of he gaing. The following unifom gaing paamees wee assumed L = 8 mm, δn eff =.5 4, Λ = 53 nm, and z =.5 mm. Figue 4.(a) plos he linea vaiaion of applied diameical load along he lengh of he gaing. Two diffeen loading angles, γ =36 and γ =72 wee consideed. Figues 4.(b) and 4.(c) plo he speca obained befoe and afe loading is applied. One can obseve he spli peak a zeo load due o he esidual sain. We have assumed 5% powe coupling ino each of he LPx and LPy modes. As seen in figues 4.(b) and 4.(c), a γ =36 he peak spliing is due o he biefingence is inceased by he specal bandwidh incease due o he load disibuion along he gaing. Howeve, fo γ =72 he induced biefingence is in he opposie sense o ha of he load disibuion heefoe he wo peaks appea o have coalesced. 4.6 SUMMARY We pesen a mehodology o apply he T-maix appoximaion o a FBG subjeced o a highly nonunifom axial sain field. A modified effecive peiod funcion is deived by maching he coupling coefficien of he T-maix appoximaion o ha in he

80 Y X 65 Y θ n eff ϕ X n min eff n max eff Figue 4.8: Azimuhal vaiaion of global effecive efacive index fo P = N/mm and γ =54 (squaes). Ellipse oiened a θ wih majo and mino axes n max eff and n min eff is ploed as a solid line. coupled-mode equaions. Simulaions fo vaious applied sain disibuions compaing he deived fomulaion wih diec inegaion of he coupled-mode equaions ae pesened o validae he modified peiod funcion. In all cases, he T-maix appoximaion using he poposed peiod funcion conveges o he soluion of he coupled-mode equaions fo a lage numbe of gaing segmens. In he nex chape we measue he esidual sain afe impac in a PMMA specimen suface mouned wih a FBG senso. We analyze he complex senso esponse o low velociy impac in a 2D woven composie. We coelae he global measuemens o local sain measuemens, o sudy he evoluion of damage in composies due o low velociy impac loading.

81 λ B (nm) CSA FE Slow axis θ (degees) Fas axis P (N/mm) P (N/mm) (a) (b) Figue 4.9: (a) Vaiaion of Bagg wavelengh wih applied diameical load a γ =36 fo PM fibe. Squaes epesen esul of cene sain appoximaion, while FE esuls ae ploed as solid line. (b) Vaiaion of he oienaion of global index ellipse wih he applied load. The vaiaion of θ is ploed ove a geae ange of P o demonsae asympoic behavio CSA FE λ B (nm) Slow axis Fas axis θ (degees) P (N/mm) P (N/mm) (a) (b) Figue 4.: (a) Vaiaion of Bagg wavelengh wih applied diameical load a γ =72 fo PM fibe. Squaes epesen esul of cene sain appoximaion, while FE esuls ae ploed as solid line. (b) Vaiaion of he oienaion of global index ellipse wih he applied load. The vaiaion of θ is ploed ove a geae ange of P o demonsae asympoic behavio.

82 67 P (N/mm) z (mm) (a) (b) (c) Figue 4.: (a) Linea vaiaion of applied diameical load along he lengh of he FBG. FBG speca obained befoe (dashed line) and afe (solid line) applicaion of ansvese load a (b) γ =36 and (c) γ =72.

83 68 Chape 5 Expeimenal Invesigaion of FBG Sain Sensos 5. INTRODUCTION The goal of his chape is o sudy he feasibiliy of using FBG s o measue suface and inenal sains in composies and PMMA subjeced o low velociy impac loading; coelae global measuemens o local senso measuemens; and sudy he effec of diffeen coaings on FBG fo suvivabiliy and sain ansfe fom he hos maeial o he senso. A fis aemp is also made o bee inepe he specal esponse so as o disinguish qualiaively senso failue fom he hos maeial failue. Fisly, a homogeneous, isoopic PMMA sysem suface mouned wih FBG is subjeced o muliple low velociy impac loadings. The effec of he adhesive laye on sain ansfe fom he PMMA o he opical fibe is sudied. Fom he measued specal esponse, he suface sain fields equied fo failue of he PMMA ae compued. Secondly, wo-dimensional woven composies ae suface mouned wih FBGs and subjeced o low velociy impacs o coelae he global measuemens (conac foce and dissipaed enegy) o local senso measuemens. The global measuemens indicae he accumulaion of damage afe evey sike. Finally, wo-dimensional woven composies embedded wih

84 69 FBGs coaed wih polyimide ae subjeced o low velociy impac loadings. The effec of coaing on he sain ansfe fom he composie o he fibe is also sudied. 5.2 BENCHMARK PMMA STUDIES In he fis seies of expeimens, FBG sensos wee applied o a homogeneous, isoopic maeial sysem, PolyMehyl-MehAcylae (PMMA). The elasic modulus of PMMA as povided by he manufacue is.3 GPa. A fis, specimens wih a 45 V- noch was consideed. A small sli was cu a he ip of he noch o aid he cack gowh. Seveal in-plane low velociy impacs a 2 m/sec wee pefomed. The oal mass of he impaco was fixed a 5.5 kg, esuling in an available kineic enegy of J. Due o he poblems of specimen oaion in he esing fame and unsable cack gowh, his geomey was no consideed fo fuhe impacs. Addiionally ou of plane impac loading was aemped on solid plae specimens, bu he specimens failed afe he fis sike. Thus, specimens wih a single hole geomey, as shown in figue 5., wee consideed. A seies of es specimens wee impaced a 2 m/sec yielding a sable cack gowh. As shown in figue 5., he opical fibes wee mouned on he face opposie o he impac suface. This is o measue he high sain induced duing impac. The daa acquisiion sysem used o measue he efleced and ansmied speca of he FBG is shown in figue 5.2. The FBGs wee ineogaed using a unable lase (Phooneics) wih a esoluion of. nm. Measuemens wee made befoe and afe evey sike. The specimen suface was cleaned wih alcohol befoe mouning he FBG senso. Befoe mouning he opical fibe wih FBG, he acylae coaing is emoved by immesing he fibe in aceone fo a few seconds. M-Bond 2 adhesive fom Vishay Measuemens Goup Inc., was used o adhee he opical fibe o he specimen. The amoun of adhesive used o adhee he opical fibe o he specimen was lae deemined o play an impoan ole in he suface sain measuemen. Theefoe, a minimum quaniy of M-Bond was used o ensue a line conac beween he opical fibe and he specimen. The FBGs used fo he expeimens wee povided by Coning Inc., and wee wien ino SMF-28 singlemode opical fibes. The ansmission speca of each FBG was measued befoe suface mouning on he PMMA specimen. Fo each gaing, he Bagg wavelengh was hen compued fom he measued ansmission specum and acs

85 7 as a efeence wavelengh fo fuhe calculaions. All he gaings used fo he PMMA sudies wee iniially chiped gaing. A soldeing ion was used o locae he cene of he FBG along he fibe by seaching fo he locaion of maximum senso esponse. Using his echnique he locaion of he FBG was measued o wihin 25.4 mm. The appoximae lengh of he FBG was deemined o be -2 mm using he T-maix mehod descibed in Chape 4 o simulae he ansmission specum of each FBG. The specialized insumened dop-owe used fo all he impac expeimens is shown in figue 5.3. The posiion and acceleaion of he impaco wee coninuously monioed houghou impac using a magneosicive posiion senso and piezoelecic acceleomee especively. Assuming he impaco o be igid, he acceleomee signal povides a measue of impac-foce ime hisoy. The oal enegy dissipaed duing each impac was deemined by a balance of anslaional kineic enegy of he impaco. The geomey of he impaco chosen fo his invesigaion was cicula-cylindical, wih a 9 mm diamee and a hemispheical nose. The squae specimens wee clamped beween wo hick aluminum annula plaes, wih a 76 mm inne diamee and a 52 mm oue diamee. The enie sysem was fixed o a igid base. Six diffeen senso locaions and oienaions wee seleced o map he sain field in he PMMA specimen. These ae lised in able 5.. Addiionally, a final specimen, A79, was impac esed o sudy he effec of educing he velociy fom 2 m/sec o.5 m/sec. We have defined he nea, inemediae and fa fields o be, 2.7 and 25.4 mm fom he poin of impac especively as seen in figue 5.. Table 5.: Configuaions of PMMA Specimens No. Disance fom Oienaion Impac velociy Sikes o Impac failue A58 Nea 45 2 m/sec 5 A59 Fa 45 2 m/sec 9 A69 Inemediae 45 2 m/sec 27 A7 Inemediae 9 2 m/sec 4 A73 Fa 9 2 m/sec 36 A75 Nea 9 2 m/sec 4 A79 Nea 9.5 m/sec 2

86 Resuls As lised in able 5., six PMMA specimens each mouned wih a single FBG wee impaced a 2 m/sec o measue he close, inemediae, and fa field sains a a 9 and 45 oienaion induced duing failue of he specimen. Thee was no vaiaion in he conac foce and dissipaed enegy afe evey sike. This can be aibued o he fac ha hee was no significan accumulaion of damage in he specimen befoe failue. The vaiaions in he numbe of sikes equied o fail can be aibued o he maeial vaiabiliy in each specimen. Howeve, he assumpion was made ha once failue occued i was he same fo all specimens. The specal esponse of each FBG measued duing he muliple low velociy impacs ae given in Appendix A. Figue 5.4 plos he vaiaion of he measued nea field sain value fo specimen A58 afe evey sike. Due o he mismach in he maeial popeies of he PMMA, he opical fibe, and he adhesive, a sain elaxaion was obseved afe each sike. Due o his hee is a small vaiaion in he measued ansmission specum afe each sike. Thus he sain value ploed in figue 5.4 is he aveage of five ansmission speca measued afe evey sike. This pocedue was applied o all he ohe PMMA specimens as well. One should noe ha he decease in sain afe sike #2 is due o he debonding of he opical fibe fom he PMMA. Figue 5.5 plos he vaiaion of he measued fa field sain value fo specimen A59 afe evey sike. Due o line conac beween he bae opical fibe and PMMA, he fibe saed o debond afe sike #5. The fibe was heefoe ebonded afe sike #, esuling in he muliple sain values afe sike # seen in figue 5.5. Afe sike #5 he fibe boke fa fom he gaing. Thus only he eflecion specum was measued afe sike #5. Compaed o figue 5.4, he maximum fa field sain value a 45 is one ode of magniude less han he nea field value. Figue 5.6 plos he vaiaion of he measued inemediae field sain value fo specimen A69 afe evey sike. As fo he pevious specimen, he opical fibe was ebonded o he PMMA afe sike #4. As seen in Appendix A, he dop in he ansmied inensiy afe sike #5 can be cedied o he low level beaks in he opical fibe as shown in figue 5.(a). These low level beaks wee idenified by pobing he opical fibe wih visible ligh afe sike #5. The widening of he ansmied specum afe sike #5 is due o he nonunifom esidual sain field along he lengh of he gaing. Since he Bagg

87 72 gaing is he weakes secion of he fibe, he opical fibe boke fis a he locaion of he gaing afe sike #22. Thus he sains wee compued fom he eflecion speca measued fom each end of he opical fibe. An ineesing poin o be noed is ha hee was sill ansmission hough he fibe alhough he specimen failed a sike #27. This is indicaed a sike #3 in figue 5.6. Alhough he fibe boke afe sike #22, he failue of he PMMA specimen aligned he wo secions of he gaing esuling in ansmission hough he fibe. This demonsaes he influence of hos maeial failue (PMMA specimen) on he sensing chaaceisics of he FBG. One should noe ha due o he non-epeaabiliy of he bile cack fomaion, his demonsaion is no epeaable. The fom of he eflecion specum measued fom boh ends of he fibe wee no same. The sain values compued fom he eflecion speca ploed in figue 5.6 coespond o he esidual sains ha exis in he wo secions of he gaing. This infomaion is no evealed fom he measued ansmission specum. Figues 5.7, 5.8 and 5.9 plo he vaiaion of measued sain values fo specimens A7, A73 and A75 especively afe evey sike. Fo hese specimens, he sain gage was oiened a 9 o he locaion of he hole as seen in figue 5.. The decease in sain magniude afe sike #3 in specimen A7 can be aibued o he debonding of he fibe fom he PMMA specimen. The high sain induced in A75 is due o he locaion of he opical fibe igh unde he poin of impac. Figue 5. plos he vaiaion of measued nea field sain value fo specimen, A79 fo which he impac velociy was.5 m/sec. We obseve ha educing he impac velociy by 25% has he effec of educing he maximum sain induced by 5%. Close obsevaion of he ansmission specum of specimen A75 afe sike # (see Appendix A) eveals ha a compessive sain of high magniude is applied o one secion of he gaing while he ohe secion is a zeo sain. As obseved fom figue 5.9 he fibe sas o debond afe sike #2. Alhough he measued esidual sain afe sike # and #3 ae close o each ohe, he fom of he measued ansmied speca ae diffeen. This may be due o he change in he suounding medium fom adhesive o ai and is influence on he ansmied specum. The widening of he ansmied specum afe sike #7 fo specimen, A79 is due o he nonunifom axial sain induced by he impac. Again he dop in he ansmied inensiy afe sike #2 is due o micocacking in he fibe. The oscillaions in he ansmied specum afe sike #2 ae due o a high magniude of nonunifom sain

88 induced by he impac in a iniially chiped gaing 73

89 Y X' Clamp locaion 45 X Opical fibe Low level beaks (a) A7 A75 A73 A59 A69 A Hole R 2.7 Clamp Locaion of FBG Impac head D9 Opical fibes (b) Figue 5.: (a) PMMA specimen wih suface mouned FBG. (b) Schemaic epesenaion of PMMA specimen. Also indicaed ae he locaion of he opical fibes and FBGs fo each specimen and he impac poin. All dimensions in mm.

90 75 unable Lase daa acquisiion sysem/ conolle phoodeeco efleced signal 3-way ciulao ansmied signal Bagg gaing Figue 5.2: Insumenaion fo FBG senso. Acceleomee Posiion senso Impac head Specimen Clamp Figue 5.3: Dop owe fo impac loading of he specimens

91 76 sain) -2 Sain (mico Sike # Figue 5.4: Vaiaion of sain wih sike # fo FBG suface mouned on specimen A58. 2 Fibe Rebonded Reflecion sain) Sain (mico Sike # Figue 5.5: Vaiaion of sain wih sike # fo FBG suface mouned on specimen A59.

92 77 Fibe Rebonded Fibe Respliced Reflecion sain) - Sain (mico Tansmission End End 2 Failue of Specimen A Sike # Figue 5.6: Vaiaion of sain wih sike # fo FBG suface mouned on specimen A69. - sain) -2 Sain (mico Sike # Figue 5.7: Vaiaion of sain wih sike # fo FBG suface mouned on specimen A7.

93 78 5 Sain (mico sain) Sike # Figue 5.8: Vaiaion of sain wih sike # fo FBG suface mouned on specimen A73. sain) -5 Sain (mico Sike # Figue 5.9: Vaiaion of sain wih sike # fo FBG suface mouned on specimen A75.

94 79-2 sain) Sain (mico Sike # Figue 5.: Vaiaion of sain wih sike # fo FBG suface mouned on specimen A79. INSTRON TESTS In all he impac expeimens conduced on PMMA specimens, he opical fibe was mouned on he face opposie o he impac suface. Alhough his face expeiences ension, he sains measued wee compessive. This can be explained as follows: () Duing impac loading he boom suface expeiences ensile sain and compessive sain duing he ebound. (2) This apid change debonds secions of fibe. (3) In addiion, his phenomenon is enhanced by he popey mismach beween he fibe and PMMA. To bee undesand his phenomenon, saic bending ess wee pefomed on PMMA specimens wih mouned gages. The Inson ensile esing machine used fo hese ess is a displacemen based conol. Thus, a small decease in load was obseved when he machine is sopped o measue he FBG esponse. Fo each specimen, an opical fibe wih FBG was mouned on he face opposie o he loading suface, and diecly unde he poin of load. A minimum amoun of adhesive was used o ensue line conac beween he fibe and PMMA. Two diffeen cases wee consideed. In he fis case he specimen, A8, was coninuously loaded o he poin of failue. FBG Specums wee measued in incemens of lbs load. In he second case a cyclic loading was applied o he specimen, A8. Evey

95 8 cycle he load was inceased by lbs. FBG speca wee measued a maximum and minimum load fo evey cycle Inefeence (a) (b) Inefeence (c).4.2 Inefeence (d) Figue 5.: Measued ansmied speca of a FBG suface mouned on specimen A8 a wo diffeen load levels. (a) 3 lbs, maximum load; (b) 3 lbs, zeo load; (c) lbs, maximum load; (d) lbs, zeo load Figue 5. plos he measued ansmied speca of a FBG suface mouned on specimen A8. The specal esponse is ploed fo wo load levels, 3 and lbs. Boh he speca measued a maximum load and a zeo load ae ploed. The Bagg peak is indicaed by a dashed line in each specum. The oscillaions in he specum measued a 3 lbs ae due o he faby-peo like inefeence and a nonunifom axial sain in he

96 8 Bagg Peak Cenoid Reflecion 8 Sain (mico) Load (lbs) Figue 5.2: Vaiaion of sain wih load fo an FBG suface mouned on specimen A8. iniially chiped gaing. The fomaion of his inefeence is due o he back eflecions beween he wo ends of he boken fibe. These oscillaions ae evesible, i.e. a smooh ansmied specum was obained when he specimen was unloaded o lbs indicaing ha he sain is no pemanen. The fequency of oscillaion and he span inceased a a highe load of lbs. One obseves he oscillaions in he measued ansmied specum wee sill pesen when he specimen was unloaded fom lbs o lbs (see figue 5.(d)). The oscillaions ae on he igh hand side of he Bagg peak in figue 5.(d) as compaed o he lef hand side of he Bagg peak in figue 5.(d). This is due o he flip in he applied non unifom axial sain fom ension o compession. Figue 5.2 plos he vaiaion of measued sain wih load. Due o he disoed specum wih epeaable oscillaions, sains compued using boh he maximum peak inensiy (Bagg peak) and cenoid of he ansmied specum ae ploed in figue 5.2. The diffeence beween he wo lines indicaes he pesence of nonunifom sain disibuions along he FBG. Figue 5.3 plos he vaiaion of measued sain wih load fo each specimen. The unloading sain is he esidual sain compued fom he specum measued a lbs. In figue 5.3(a) he unloading sain is ploed agains he load, bu coesponds o lbs. In figue 5.3(b) plos he esidual sain induced a zeo load agains he maximum ensile sain in each cycle. Figue 5.3(b) indicaes he end of inceasing esidual compessive

97 82 2 Loading Unloading Sain (mico sain) Residual Sain (mico sain) Load (lbs) -2 5 Maximum Tensile Sain (mico sain) (a) (b) Figue 5.3: Vaiaion of sain wih load fo an FBG suface mouned on specimen A8. sain wih inceasing load. Thus, while in all he impac expeimens wih PMMA we measued he compessive sain afe each impac, we can elae his qualiaively o he maximum ensile sain in he specimen Discussion Iniially he PMMA specimens wee seleced o benchmak he maximum sain o failue in a homogeneous maeial. Howeve, due o he non-epeaabiliy of he bile failue, and maeial popey mismach beween he fibe, PMMA, and M-Bond, we could measue only he maximum esidual sain. Figue 5.4 plos he vaiaion of maximum esidual sain measued by he FBG a each oienaion. A quadaic fi o he expeimenal daa poins is also ploed. Due o he pesence of hole, he sain field is no axisymmeic. Thus, hee is a diffeence in he magniude of maximum esidual sain field beween 9 and 45 oienaions. One should noe ha he vaiaion in he maximum sain induced in he PMMA specimens is he same as he vaiaion of he maximum esidual sain measued by he FBG. Thus alhough we could no measue he sain o failue, he maximum sain o failue scales

98 Inemediae field Fa field Residual Sain (mico sain) Nea field (mm) Figue 5.4: Vaiaion of he maximum esidual sain wih he adius fo 9 and 45 oienaions. Solid and dashed lines ae he quadaic fi fo 9 and 45 oienaions especively. wih he esidual sain measued. As menioned in he pevious secion, he amoun of adhesive used also played an impoan ole in he sain ansfe beween he PMMA and he opical fibe. Low velociy impacs on PMMA specimens suface mouned wih FBGs whee an abundan amoun of adhesive was applied o adhee he fibe o he PMMA specimen wee also pefomed. We obseved ha afe evey sike he esponse was consan. The sain ansfe fom he he suface of PMMA specimen o he fibe hough he adhesive is pimaily due o shea sain. Due o he maeial popey mismach beween he PMMA, adhesive, and fibe he maximum sain o failue was no ansfeed when a significan amoun of adhesive was used. Thus he minimum amoun of adhesive was used fo all specimens o ensue ha a line conac was esablished beween he fibe and PMMA specimen. 5.3 WOVEN COMPOSITES The goal of his secion is o sudy he feasibiliy of using FBGs o measue suface and inenal sains in 2D woven composies. The effec of he ype of coaing on he suvivabiliy and sain ansfe is also sudied.

99 Suface Mouned Sensos Fou diffeen composies specimens wee impaced o measue he suface sains as lised in able 5.2. The FBGs wee locaed in he opical fibe using he same pocedue as adoped fo he PMMA expeimens. The bae opical fibes wih FBGs wee adheed o he composie sufaces using he M-Bond 2 adhesive ki fom Vishay. An abundan amoun of M-Bond was used o ensue ha he fibes wee well adheed o he composie suface. The FBG was mouned on he face opposie o he impac suface. Table 5.2: Configuaions of 2D woven composie specimens No. Coaing Mouning Sikes o failue C5 Bae Suface 5 C6 Bae Suface 6 C5 Acylae Embedded 2 C7 Bae Suface 2 C8 Bae Suface 23 C49 Polyimide Embedded 8 C5 Polyimide Embedded 25 Fabicaion A wo-dimensional woven cabon fibe-epoxy will pe-peg fom Advanced Composies was used o fabicae he specimens. The esuling composie has a fibe volume facion of.6. The will was efigeaed o avoid cuing a oom empeaue. To fabicae he composie specimens, he will was fis cu ino 4.75 inches squae coupons. A oal of 24 coupons was used fo each specimen. The and 9 weave can be seen in figue 5.6. The layes wee consideed o be isoopic due o he geomeical symmey of he weave as seen in figue 5.6. Nex, he coupons wee sacked one ove anohe in a mold. The coupons wee sacked such ha all he weaves in one coupon wee paallel o he weaves in is adjacen coupons. The coupons could also be sacked such ha all he weaves wee aligned o he weaves in he adjacen coupons i.e., he coupons sacked such ha he ces and ough of each weave is aligned o he ces and ough of he weave in he adjacen coupons. Howeve, his sacking is hade o achieve. Small vaiaions in

100 85 he sacking pocedue, could esul in a vaiaion in he impac sengh of he fabicaed composies. Afe sacking he coupons, he mold was pepaed fo he ho-pess. The mold consised of hee pas: base plae, plunge and side ails. The side ails ae used o peven epoxy loss duing pessing. The dimensions of he mold ae given in figue B.. The mold was waxed wih wo coas of Canuba wax. Nex, wo coas of mold elease (dy film silicone lubican fom LPS) wee spayed. A oal of 2 elease shees and 2 vacuum shees wee used fo each specimen. The elease shees faciliaed he easy emoval of he composie specimen afe fabicaion. The vacuum shees pevened he epoxy fom adheing o he mold. The sacked specimen was placed in he mold and he plunge added o he op. The mold wih he specimen is vacuumed o avoid ai bubbles in he fabicaed composie. The vacuumed mold was hen placed in he ho-pess and a cue cycle defined in figue 5.5, povided by he manufacue, was applied. A he end of he cue cycle, he specimen was emoved fom he mold and he shap edges sanded. A seies of specimens wee fabicaed as descibed above and impac esed o ensue he epeaabiliy of he cack gowh. Figue 5.6 shows a ypical fabicaed composie specimen befoe and afe he impac es. The aows indicae he diecion of cack gowh. To ensue no aigaps wee pesen afe fabicaion, he woven specimen was cu using a ilesaw ino 38. mm by 38. mm coupons. The coupons wee hen polished and obseved wih an opical micoscope. Figue 5.7 shows a micoscope image of he coss-secion of a wo-dimensional woven composie specimen befoe and afe failue of he composie. Fom figue 5.7(a) we can obseve a unifom layup and he absence of aigaps afe fabicaion. Diffeen ypes of ypical composie failues: maix cacking, delaminaion and fibe beakage can be seen in figue 5.7(b). The Si72 opical specum analyze fom Micon Opics was used o measue he esponse of he FBG suface mouned on specimens C5 and C6 as shown in figue 5.8. Fo all he ohe composie specimens, he unable lase seup shown in figue 5.2 was used. Due o he pesence of wo phoodeecos, boh he ansmission and eflecion specums wee measued simulaneously. The analyze scans he wavelengh ange (52-57 nm) a a 5 Hz fequency. Since he fequency of he impaco was geae han ha of he analyze, he impac expeimens wee consideed o be semi-dynamic. The oscillaions in he speca afe each sike wee obseved. These wee due o he oscillaion of he suface opposie o he impac face fom ension o compession because of he impac loading.

101 86 8 Tempeaue ( C) o Time (hs) 8 Pessue (psi) Time (hs) Figue 5.5: Cuing cycle fo he fabicaion of 2D woven composies. A schemaic epesenaion of specimen C5 is shown in figue 5.9. One should noe he offse in he impac poin. Figue 5.2 plos he global measuemens obained fom specimen C5. One obseves a geneal end of decease in conac foce and incease in dissipaed enegy wih inceasing sike numbe. This end was also obseved fo all he lae composies impaced. These plos indicae he accumulaion of damage wih inceasing sike numbe. Figue 5.2 plos he sain compued fom boh he ansmission and eflecion speca. Due o he lage defomaion as seen in figue 5.6, he FBG failed befoe he failue of he composie. A schemaic epesenaion of specimens C6, C7 and C8 is shown in figue Figue 5.23 plos he global measuemens obained fom specimen C6. Figue 5.24 plos

102 87 he sain compued fom boh he ansmission and eflecion speca. Since he FBG is he weak secion of he fibe, he fibe boke a he locaion of he FBG afe sike #29. One secion of he gaing is debonded while he ohe secion is in compession as can be seen fom figue Figue 5.25 plos he global measuemens obained fom specimen C7 demonsaing accumulaion of damage wih muliple sikes. The fibe boke a he gaing afe sike #. Figues 5.26(a) and (b) plo he sain compued fom End and End 2 of he FBG especively. Figues 5.27 and 5.28 plo he measued eflecion speca measued fom End and End 2 of he fibe especively afe seleced impac sikes. The complee specal esponses ae ploed in Appendix B. The pogessive widening of he ansmied specum afe sike #4 coesponds o a nonunifom axial sain induced due o impac. The spli in he Bagg peak afe sike #8 indicaes ha one secion of he gaing is in ension while no sain is applied o he emaining secion. One should noe ha simulaing he ansmission specum o compue he nonunifom sain disibuion is beyond he scope of his hesis. The complex loading occuing due o pogession of damage in he composie hus splis a single Bagg peak ino muliple peaks. The sains ploed in figue 5.26(a) ae compued fom he Bagg peaks of he eflecion specum measued afe sikes #, 4, 7, and 2 and ae indicaed by dashed lines in figue One obseves ha a secion of he gaing is debonded while he ohe secion is in compession as seen in figue 5.26(a). The sains ploed in figue 5.26(b) ae compued fom he Bagg peaks of he eflecion specum measued afe sikes #, 4, 6, and 8 and ae indicaed by dashed lines in figue One obseves ha secions of he gaing ae a zeo, ensile and compessive sains as seen in figue 5.26(b). Thee is also a change in ampliude of eflecion in he wo Bagg peaks duing he pogession of damage. This is a simila effec o one obseved by Kuang e al. [28]. Fom figues 5.27 and 5.28 demonsae ha due o complex sain induced duing low velociy impacs, he single Bagg peak splis ino muliple peaks and hen ecombines duing he pogession of damage.

103 88 Impac locaion Clamp locaion Opical fibe Locaion of FBG o 9 weave o weave Opical fibe Figue 5.6: Phoogaph of wo-dimensional woven composie specimen (a) befoe and (b) afe impac.

104 89 weave 9 weave Fibe beak Maix cacking Delaminaion Figue 5.7: Coss-secion of wo-dimensional woven composie specimen unde a opical micoscope (a) befoe and (b) afe impac.

105 9 opical senso analyze (Micon Opics Si72) ch ch 2 lase oupu efleced signal 3-way ciculao Bagg gaing ansmied signal Figue 5.8: Insumenaion fo coninuous FBG sensing. Also shown is he 5 Hz ineogao fom Micon Opics. 4.6 Clamp Locaion of FBG D Impac head Opical fibe Figue 5.9: Schemaic epesenaion of specimen C5 along wih opical fibe and impac locaion. All dimensions in mm.

106 9 4 Conac Foce (N) 2 8 Enegy Dissipaed(J) Sike # Sike # Figue 5.2: Global measuemens fo specimen C5. (a) Vaiaion of peak acceleaion wih sike #. (b) Vaiaion of dissipaed enegy wih sike #. 3 Reflecion Tansmission 2 Sain (mico sain) Sike # Figue 5.2: Vaiaion of sain wih sike # fo an FBG suface mouned on specimen C5. Sains compued fom boh he eflecion and ansmission speca ae ploed.

107 Clamp D9 Locaion of FBG Impac head Opical fibe 27 C5 C6 C7 C8 Figue 5.22: Schemaic epesenaion of composie specimens. Also shown ae he locaion of FBGs fo C6,C7,C8 and C5 along wih impac locaion. All dimensions in mm Conac Foce (N) 8 Enegy Dissipaed (J) Sike # Sike # (a) (b) Figue 5.23: Global measuemens fo C6. (a) Vaiaion of peak acceleaion wih sike #. (b) Vaiaion of dissipaed enegy wih sike #.

108 Reflecion Tansmission 2 Sain (mico sain) Sike # Figue 5.24: Vaiaion of sain wih sike # fo an FBG suface mouned on C6. Sains compued fom boh he eflecion and ansmission speca ae ploed Conac Foce (N) 8 6 Enegy Dissipaed (J) Sike # Sike # (a) (b) Figue 5.25: Global measuemens fo C7. (a) Vaiaion of peak acceleaion wih sike #. (b) Vaiaion of dissipaed enegy wih sike #.

109 Maximum Minimum 2 x Maximum Minimum Sain (mico sain) Sain (mico sain) -2 Tansmission Reflecion fom End 2 x -4 Tansmission Reflecion fom End Sike # Sike # (a) (b) Figue 5.26: Vaiaion of sain wih sike # fo an FBG suface mouned on C7. Sains compued fom he eflecion speca of (a) End and (b) End 2 ae ploed.

110 95.5 λ B λ B2.5 λ B λ B (a) (b).5 λ B λ B2.5 λ B λ B (c) (d) Figue 5.27: Measued eflecion speca fom End fo a FBG suface mouned on specimen C7 afe sikes #: (a), (b) 4, (c) 7, and (d) 2.

111 96.5 λ B λ B2.5 λ B λ B (a) (b).6.5 λ B λ B2 λ B3 λ B λ B2 λ B (c) (d) Figue 5.28: Measued eflecion speca fom End 2 fo a FBG suface mouned on specimen C7 afe sikes #: (a), (b) 4, (c) 6, and (d) 8.

112 Conac Foce (N) Enegy dissipaed (J) Sike # Sike # (a) (b) Figue 5.29: Global measuemens fo specimen C8. (a) Vaiaion of peak acceleaion wih sike #. (b) Vaiaion of dissipaed enegy wih sike #. 2 Tansmission Maximum Minimum Reflecion fom End Sain (mico sain) Sike # Figue 5.3: Vaiaion of sain wih sike # fo an FBG suface mouned on specimen C8. Sains compued fom boh he eflecion and ansmission speca ae ploed.

113 98 Figue 5.29 plos he global measuemens obained fom specimen C8. Figue 5.3 plos he sain compued fom he eflecion and ansmission specums fo he same specimen. The pogessive widening of he ansmied specum afe sike # coesponds o a nonunifom axial sain induced due o impac. The sains shown in figue 5.3 compued fom he Bagg peaks of he specums measued afe sikes #6, 9,, 4, 5, 6, and 2 ae indicaed by dashed lines in figue 5.3. All he specal daa ae ploed in Appendix B. Simila o specimen C7, a single Bagg peak is spli ino muliple peaks indicaing nonunifom sain induced by impac duing he pogession of damage. The fibe boke afe sike #6. One secion of he gaing is debonded while he ohe secion is in compession as seen in figue 5.3. The esponse of he FBG suface mouned on he wo-dimensional woven composie specimens subjeced o low velociy impacs is complex as compaed o he esponse of FBG suface mouned on he PMMA specimens. The above analysis demonsaes a sysemaic appoach o compue esidual sains afe impac fom he complex specal esponse Embedded Sensos To sudy he effec of fibe coaing on he suvivabiliy and sain ansfe hee diffeen composie specimens as lised in able 5.2 wee consideed. ACRYLATE COATING An opical fibe coaed wih he sandad acylae coaing was embedded a he mid-plane in he specimen C5 as shown in figue Fis, wo eleven laye panel wee fabicaed sepaaely using he pocedue as descibed above. Nex, a single laye was sacked on op of he eleven laye panel. The opical fibe wih a FBG was mouned on his laye followed by anohe single laye and he ohe eleven laye panel. The complee assembly was cued using he cue cycle as descibed in figue 5.5. Seveal specimens embedded wih opical fibes wihou FBG wee fabicaed o make sue he above pocedue povided saisfacoy esuls. They wee also impac esed o ensue he conac foce is in he same ange as fo a single weny-fou laye panel. Figue 5.33 shows he micoscope image of he coss-secion of a wo dimensional woven composie embedded wih opical fibe. An ellipical esin ich egion called esin

114 99 eye is fomed suounding he opical fibe as seen in figue Since he specimen is fabicaed wih ++2 laye configuaion, he epoxy esin beween he panels is also shown in figue The esponse of he FBG embedded in specimen C5 is ploed in Appendix B. Due o high pessue applied duing fabicaion, he fibe boke and only eflecion fom one end was measued. The low inensiy in eflecion can be aibued o he bending loss due o he woven naue of he composie. Figue 5.32 plos he global measuemens obained fom specimen C5. POLYIMIDE COATING Afewads, composie specimens embedded wih polyimide coaed opical fibes wee fabicaed o ensue he fibes suvived he fabicaion pocedue, as lised in able 5.3. The polyimide coaing is siffe han he sandad acylae coaing and povides moe poecion o he fibe duing he specimen fabicaion. To fabicae composie specimens Temp I and II, wo ses each compising a en laye and a five laye panel wee fabicaed using he cue cycle ploed in figue 5.5. Fis en squae coupons wee sacked beween he elease shees and vacuum shees. Nex, five squae coupons wee also sacked beween he elease and vacuum shees. The sacked coupons wee pessed ogehe in he ho-pess such ha he en laye sack was a he boom of he mold. A he end of he cue cycle, i was obseved ha one of he sufaces fo each en laye and five laye panels was uneven due o he fomaion of idges. A simila pocedue was followed fo he las wo five laye panels equied o fabicae composie specimens Temp I and II. Fo he final pess, he configuaion lised in able 5.3 was fomed wih a pessue of 8 psi. Fo all he specimens lised in able 5.3, he final pess configuaion is lised in paenhesis. The fibes wee aligned along he 9 weave and wee 2.7 mm on eihe side of he impac poin o balance he applied pessue. Fo all specimens he pessue lised in able 5.3 is fo he final pess. As seen fom able 5.3, fo specimens Temp I and II, none of he fibes suvived he high pessue. Two composie specimens, Temp III and IV, embedded wih polyimide coaed fibes wee also fabicaed. The same configuaion and fabicaion pocedue as descibed above was followed. A educed pessue of 9 psi was used fo he final cycle. Again he fibes did no suvive he fabicaion. This was deemined o be due o he fomaion of

115 Table 5.3: Configuaions of 2D woven composie embedded wih polyimide coaed fibes wihou FBGs. No. Configuaion Pessue (psi) Fibe Suvivabiliy Temp I +(+fibe++5++fibe+)+5 8 No Temp II +(+fibe++5++fibe+)+5 8 No Temp III +(+fibe++5++fibe+)+5 9 No Temp IV +(+fibe++5++fibe+)+5 9 No Temp V (+fibe+) 9 Yes Temp VI (2+fibe+2) 9 Yes C49 +(2+fibes+2)+ 9 Yes C5 7+(2+fibes+5) 9 Yes idges when wo panels ae fabicaed ogehe. This is veified by fabicaing composie specimens V and VI. Hee a educed pessue was applied as lised in able 5.3. The fibes suvived he fabicaion. Addiionally, composie specimens wih configuaion simila o Temp V and VI wih a pessue of 8 psi wee fabicaed. The fibes did no suvive he fabicaion. Fom he visual inspecion of composie specimen Temp V, we obseved ha he diamee of he opical fibe coaed wih polymide is geae han he hickness of he specimen. Thus, a minimum of wo layes on eihe side of he opical fibe is neccessay as in specimen Temp VI o ensue ha he hickness of he specimen is geae han he diamee of he embedded fibe. Composie specimens wih configuaions simila o Temp VI bu wih fibes a an angle of 3 o 9 o he weave wee also fabicaed. Fo hese specimens, he fibe suvived he fabicaion pocess. Afe esablishing a fabicaion pocedue fo he suceessful embedmen of opical fibes coaed wih polyimide wo composie specimens C49 and C5 wih configuaions as lised in able 5.3 wee fabicaed. The FBGs wien in polyimide coaed opical fibes used fo he expeimens wee puchased fom Avensys. A schemaic epesenaion of specimen C49 along wih he locaion of he opical fibes is shown in figue Fo his specimen, he opical fibes wih FBGs wee embedded a he mid-plane, 2.7 mm and 9 mm on eihe side of he impac poin, A and B, especively as shown in figue The fibes wee aligned along he 9 weave diecion. The specal esponse of he FBGs, labelled A and B, subjeced o low velociy impac ae ploed in he Appendix B. A schemaic

116 epesenaion of specimen C5 along wih he locaion of he opical fibes is shown in figue The opical fibes wih FBGs fo his specimen wee embedded a 2.7 mm on eihe side of he impac poin, C and D, as shown in figue The fibes wee aligned along he 9 weave diecion. The specal esponse of he FBGs, labelled C and D, subjeced o low velociy impacs ae ploed in he Appendix B. Figue 5.36 plos he esponse of FBG-A embedded in specimen C49 subjeced o muliple low velociy impacs. The dashed line indicaes he efeence wavelengh (Bagg wavelengh befoe embedding). Due o high pessue applied duing fabicaion, he single Bagg peak spli ino wo as seen in figue 5.36(a). The vaiaion in he inensiy of each peak duing he pogession of damage is due o micobending losses inheen wih wo dimensional woven composies. The fibe boke afe sike #6 and afewads only eflecion speca wee measued. Fom figue 5.36(d), one noes ha a secion of gaing debonded while he ohe secion was in compession. Figue 5.37 plos he esponse of FBG-B embedded in specimen C49 subjeced o muliple low velociy impacs. The dashed line indicaes he efeence wavelengh (Bagg wavelengh befoe embedding). Due o high pessue applied duing fabicaion, he single Bagg peak spli ino wo as seen in figue 5.37(a), consisen wih FBG-A. One noes ha FBG-A is close o impac poin ha he FBG-B. Thee was sill ansmission hough he fibe afe he failue of he specimen C49 demonsaing he inceased duabiliy of he polyimide coaed opical fibe. Figue 5.38 plos he esponse of FBG-C embedded in specimen C5 subjeced o muliple low velociy impacs. The dashed line indicaes he efeence wavelengh (Bagg wavelengh befoe embedding). Due o high pessue applied duing fabicaion, he single Bagg peak spli ino wo as seen in figue 5.38(a), consisen wih FBG-A. Fom figue 5.38(d), one noes ha he fibe is debonded. Since he gaing is he weak secion of he fibe, he fibe boke a he gaing afe sike #7. One should noe ha he low eflecion inensiies and hei vaiaion ae due o he muliple fibe beaks caused by he successive impacs. They can also be aibued o bending losses due o he woven naue of specimen C5. Figue 5.39 plos he esponse of FBG-D embedded in specimen C5 subjeced o muliple low velociy impacs. The dashed line indicaes he efeence wavelengh (Bagg wavelengh befoe embedding). Due o high pessue applied duing fabicaion, he single Bagg peak spli ino wo as seen in figue 5.39(a), consisen wih FBG-C. Fom fig-

117 2 ue 5.38(d), one noes ha a secion of he gaing is in compession while he ohe secion is in ension. Again, he fibe boke a he gaing afe sike #8 consisen wih FBG-C Discussion In his secion an aemp is made o answe quesions ha aise while analyzing boh he global and local measuemens obained duing muliple low velociy impacs on wo dimensional woven composie specimens. Fom boh he suface mouned and embedded senso esuls one noes ha he fibe beakage was no always a he locaion of he gaing. Alhough he gaing is he weakes secion of he fibe, hee ae ohe facos ha goven he fibe beakage, such as micoscopic defecs in he opical fibe and damage in he composie, no necessaily nea he gaing. The inepeaion of he embedded FBG esponse is complex. Since he oienaion and alignmen of he opical fibe ae no known afe fabicaion, coelaing he senso esponse o physical damage is also difficul. Also he numbe of he weaves along he lengh of he fibe is no known afe fabicaion. The lengh of he FBG as compaed o he dimensions of he weave plays an impoan ole in he senso esponse. Thus, an aemp o calculae he inenal sain fom he specal esponse simila o specimens C7 and C8 was no made fo he embedded FBGs. The global and local senso esponse of he FBGs suface mouned on specimens C7 and C8 ae diffeen alhough he FBGs ae boh locaed 2.7 mm fom he impac poin. The vaiaion in he global measuemens is due o he vaiaion of he siffness of he specimens. Alhough all he specimens wee fabicaed using he same pocedue he vaiaion in siffness can be aibued o he woven naue each laye. The decease in peak conac foce wih successive impacs coelaes well o he pogession of damage. Thus, he ae of decease in conac foce is a good measue fo he global paamee ahe han he sike numbe fo each composie specimen. The oscillaions in he acceleomee daa fo each sike indicae he iniiaion of damage. Alhough FBGs C and D wee embedded, in specimen C5, symmeically on eihe side of he impac poin, he esponse o muliple low velociy impacs is no he same. This is because he failue of he composie was no symmeic. A ensile sain was induced duing fabicaion of specimen C49 as seen fom he ansmission speca of

118 3 FBGs A and B. Howeve, hee was no sain duing fabicaion of specimen C5 as seen fom ansmission speca of FBGs C and D. This is due o he vaiaion in he fabicaion pocedue of he specimens. Table 5.4: Configuaions of 2D woven composies embedded wih polyimide coaed fibes wihou FBGs. No. Configuaion No. of sikes Peak conac foce (N) C C C C To sudy his effec of vaiaion in he fabicaion pocedue on he damage iniiaion and pogession, fou composie specimens wee fabicaed as lised in able 5.4. The numbe of sikes lised in able 5.4 is he numbe of impac sikes befoe he specimen was cu fo opical micoscopy. Figue 5.4 shows opical micoscope images of he coss-secion of he composie specimens lised in able 5.4. As he failue in composies was due o shea, he damage iniiaed on he suface opposie o he impac suface. One should noe ha hee was no significan diffeence in damage iniiaion and pogession due o muliple low velociy impacs beween he +2+ and 24 laye configuaions. Due o he vaiaion in he siffness of each specimen, he conac foce afe each sike is no known a-pioi. Thus opical micoscopy a a paicula conac foce is had o achieve. This accouns fo he diffeence in conac foce alhough he numbe of sikes ae he same fo specimens C3 and C34 as seen in able 5.4. Afe he fis sike, hee was fibe beakage and delaminaions as seen in figue 5.4. Geneally in woven composies, due o he low siffness of he maix, maix cacking appeas fis. This is followed by high shea sess beween he layes esuling in delaminaions. Lasly, due o lage defomaions, fibe beakage occus. Since we obseve fibe beakage on he fis sike, all he enegy in he successive sikes is used o accumulae damage by pogession of cacks. One obseves fom figue 5.4 ha visible damage is highe fo lowe conac foces.

119 4 5.4 SUMMARY To summaize, a sysemaic appoach o compue sain fom he complex FBG senso esponse is pesened. Fis, FBGs wee suface mouned on PMMA specimens and subjeced o muliple low velociy impacs. The esidual sain field was calculaed fom he specal esponse. The effec of adhesive on he sain ansfe was also sudied. Nex, FBGs wee suface mouned on wo-dimensional woven composie specimens and subjeced o muliple low velociy impacs. The sain induced due o impac was calculaed afe evey sike fom he complex specal esponse. The global measuemens demonsae he accumulaion of damage duing he pogession of cack. Finally, FBGs wee embedded in wo-dimensional woven composie specimens and subjeced o muliple low velociy impacs o measue inenal sains.

120 .9.8 λ B.9 λ B.9 λ B λ B2 λ B2 λ B λ B λ B (a) (b) (c).8.9 λ B λ B2.9 λ B λ B λ B3.2.. λ B λ B (d) (e) (f) λ B λ B (g) Figue 5.3: Measued specal esponse of FBG suface mouned on specimen specimen C8 afe sikes #: (a) 6, (b) 9, (c), (d) 4, (e) 5, (f) 6, and (g) 2. 5

121 6 2 4 Conac Foce (N) Enegy Dissipaed(J) Sike # Sike # (a) (b) Figue 5.32: Global measuemens fo C5. (a) Vaiaion of peak acceleaion wih sike #. (b) Vaiaion of dissipaed enegy wih sike #. Resin (epoxy) Opical fibe Figue 5.33: Coss-secion of wo dimensional woven composie specimen embedded wih opical fibe unde opical micoscope.

122 7 B A Clamp D9 Locaion of FBG Impac head Opical fibe 9 27 Figue 5.34: Schemaic epesenaion of specimen C49 along wih opical fibes and impac locaion. All dimensions in mm. D C Clamp D9 Locaion of FBG Impac head Opical fibe Figue 5.35: Schemaic epesenaion of specimen C5 along wih opical fibes and impac locaion. All dimensions in mm.

123 (a) (b) (c) (d) Figue 5.36: Measued speca fo FBG-A embedded in specimen C49 (a) afe embedding, afe sikes #: (b), (c) 6, and (d) 8.

124 (a) (b) (c) (d) Figue 5.37: Measued speca fo FBG-B embedded in specimen C49 (a) afe embedding, afe sikes #: (b), (c) 8, and (d) 3.

125 (a) (b) (c) (d) Figue 5.38: Measued speca fo FBG-C embedded in specimen C5 (a) afe embedding, afe sikes #: (b), (c) 4, and (d) 7.

126 (a) (b) (c) (d) Figue 5.39: Measued speca fo FBG-D embedded in specimen C5 (a) afe embedding, afe sikes #: (b), (c) 4, and (d) 8.

127 (a) (b) (c) Figue 5.4: Coss-secion of wo dimensional woven composie specimens unde opical micoscope: (a) specimen C29, (b) specimen C3, (c) specimen C32, and (d) specimen C34. (d) 2

128 3 Chape 6 Conclusions This hesis pesens a mehodology o simulae he esponse of embedded FBG sensos o applied load as well as he expeimenal sudies o es he feasibiliy of using FBG sensos duing low velociy impacs. The simulaion mehod logically disinguishes he effec of ansvese and longiudinal sain on he specal esponse of he FBG. We fomulaed a FE model o calculae he change in effecive efacive index due o he applied ansvese sain. The change in efacive index fo each elemen along he local fas and slow axis was deived assuming a linea sain-opic effec. Using he global index ellipse concep he popagaion consan along he fas and slow axis was compued. The model was applied o a numbe of diffeen coss-secions subjeced o diameical compession o demonsae is applicaion o loading of he sensos. The examples consideed compaed he sensiiviies of diffeen PM fibes (all having same maeial popeies) o ansvese load. These sudies deemined ha D-fibe exhibied highes sensiiviy. I was also shown ha educing he fibe cladding diamee significanly impoves he sensiiviies of he FBG senso o ansvese load. In addiion, a numeical example of a PM fibe subjeced o diameical load was pesened ha demonsaes he limiaion of he cene sain appoximaion. Afewads, he T-maix appoximaion fo a FBG was exended o he case of a highly nonunifom axial sain field. A modified effecive peiod funcion was deived by maching he coupling coefficien of he T-maix appoximaion o ha in he coupledmode equaions. Simulaions fo vaious applied sain disibuions compaing he deived

129 4 fomulaion wih diec inegaion of he coupled-mode equaions validaed he modified peiod funcion. In all cases, he T-maix appoximaion using he poposed peiod funcion conveged o he soluion of he coupled-mode equaions fo a lage numbe of gaing segmens. Finally, he expeimenal invesigaions of he specal esponse of a FBG suface mouned on PMMA and wo-dimensional woven composie specimens demonsaed ha esidual sains due o impac could be obained. All he specimens wee subjeced o muliple low velociy impacs o monio he pogession of damage. Deailed analysis of he FBG specal esponse pemied moe infomaion fom he impac specimens han peviously obained. Fuhemoe, embedded FBGs in he woven composie specimen yielded some inenal sain infomaion afe each impac.

130 5 Refeences [] C. D. Bue and G. B. Hocke, Fibe opics sain gauge, Applied Opics, vol. 7, pp , 978. [2] G. B. Hocke, Fibe-opic sensing of pessue and empeaue, Applied Opics, vol. 8, pp , 979. [3] A. Beholds and R. Dandlike, Deeminaion of he individual sain-opic coefficiens in single-mode opical fibes, Jounal of Lighwave Technology, vol. 6, pp. 7-2, 988. [4] R. Ulich and A. Simon, Polaizaion opics of wised single-mode fibes, Applied Opics, vol. 8, pp , 979. [5] A. Beholds and R. Dandlike, Defomaion of single-mode opical fibes unde saic longiudinal sess, Jounal of Lighwave Technology, vol. LT-5, pp , 987. [6] R. M. Measues, Sma composie sucues wih embedded sensos, Composies Engineeing, vol. 2, pp , 992. [7] K. O. Hill and G. Melz, Fibe Bagg gaing echnology fundamenals and oveview, Jounal of Lighwave Technology, vol. 5, pp , 997. [8] K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, Phoosensiiviy in opical fibe waveguides: applicaion o eflecion file fabicaion, Applied Physics Lees, vol. 32, pp , 978. [9] R. Kashyap, Fibe Bagg gaings, Academic Pess, 999.

131 6 [] A. Ohonos, K. Kalli, Fibe Bagg gaings: fudamenals and applicaions in elecommunicaions and sensing, Acech House Inc., 999. [] R. M. Measues, Sucual monioing wih fibe opic echnology, Academic Pess, 999. [2] E. J. Fiebele C. G. Askins, A. B. Bosse, A. D. Kesey, H. J. Paick, W. R. Pogue, M. A. Punam, W. R. Simon, F. A. Taske, W. S. Vincen, and S. T. Voha, Opical fibe sensos fo spacecaf applicaions, Sma Maeials and Sucues, vol. 8, pp , 999. [3] T. Edogan, Fibe gaing speca, Jounal of Lighwave Technology, vol. 5, pp , 997. [4] V. Mizahi and J. E. Sipe, Opical popeies of phoosensiive fibe phase gaings, Jounal of Lighwave Technology, vol., pp , 993. [5] H. A. Haus and Weiping Huang, Coupled-mode heoy, Poceedings of he IEEE, vol. 79, pp , 99. [6] A. Yaiv, Coupled-mode heoy fo guided-wave opics, IEEE Jounal of Quanum Eleconics, vol. QE-9, pp , 973. [7] A. D. Kesey, M. A. Davis, H. J. Paick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Punam, and E. Joseph Fiebele, Fibe gaing sensos, Jounal of Lighwave Technology, vol. 5, pp , 997. [8] J. Noda, K. Okamoo, and Y. Sasaki, Polaizaion-mainaining fibes and hei applicaions, Jounal of Lighwave Technology, vol. LT-4, pp. 7-89, 986. [9] C. Bolle, Ways and opions fo aicaf sucual healh managemen, Sma Maeials and Sucues, vol., pp , 2. [2] Y. Sasaki, Long-lengh low-loss polaizaion-mainaining fibes, Jounal of Lighwave Technology, vol. LT-5, pp , 987. [2] S. W. James and R. P. Taam, Opical fibe long-peiod gaing sensos:chaaceisics and applicaion, Measuemen Science and Technology, vol. 4, pp. R49-R6, 23.

132 7 [22] G. Zhou and L. M. Sim, Damage deecion and assessmen in fibe-einfoced composie sucues wih embedded fibe opic sensos-eview, Sma Maeials and Sucues, vol., pp , 22. [23] J. R. Dunphy, G. Melz, F. P. Lamm, and W. W. Moey, Muli-funcion, disibued opical fibe senso fo composie cue and esponse monioing, SPIE Fibe Opic Sma Sucues and Skins III, vol. 37, pp. 6-8, 99. [24] W. W. Moey, G. Melz, and W. H. Glen, Fibe opic Bagg gaing sensos, SPIE Fibe Opic and Lase Sensos VII, vol. 69, pp. 98-7, 989. [25] I. McKenzie, R. Jones, I. H. Mashall, and S. Galea, Opical fibe sensos fo healh monioing of bonded epai sysems, Composie Sucues, vol. 5, pp , 2. [26] V. M. Muukeshan, P. Y. Chan, L. S. Ong, and L. K. Seah, Cue monioing of sma composies using fibe Bagg gaing based embedded sensos, Sensos and Acuaos, vol. 79, pp. 53-6, 2. [27] Y. Okabe, S. Yashio, T. Kosaka and N. Takeda, Deecion of ansvese cacks in CFRP composies using embedded fibe Bagg gaing sensos, Sma Maeials and Sucues, vol. 9, pp , 2. [28] K. S. C. Kuang, R. Kenny, M. P. Whelan, W. J. Canwell, and P. R. Chalke, Embedded fibe Bagg gaing sensos in advanced composie maeials, Composies Science and Technology, vol. 6, pp , 2. [29] K. S. C. Kuang, R. Kenny, M. P. Whelan, W. J. Canwell, and P. R. Chalke, Residual sain measuemen and impac esponse of opical fibe Bagg gaing sensos in fibe meal laminaes, Sma Maeials and Sucues, vol., pp , 2. [3] J. A. Guemes and J. M. Menendez, Response of Bagg gaing fibe-opic sensos when embedded in composie laminaes, Composies Science and Technology, vol. 62, pp , 22. [3] Y. Okabe, N. Tanaka, and N. Takeda, Effec of fibe coaing on cack deecion in cabon fibe einfoced plasic composies using fibe Bagg gaing sensos, Sma Maeials and Sucues, vol., pp , 22.

133 8 [32] K. S. C. Kuang and W. J. Canwell, Deecion of impac damage in hemoplasicbased fibe-meal laminaes using opical fibe sensos, Jounal of Maeial Science Lees, vol. 2, pp , 22. [33] S. Takeda, Y. Okabe and N. Takeda, Delaminaion deecion in CFRP laminaes wih embedded small-diamee fibe Bagg gaing sensos, Composies: Pa A, vol. 33, pp , 22. [34] T. Mizuani, Y. Okabe, and N. Takeda, Quaniaive evaluaion of ansvese cacks in cabon fibe einfoced plasic quasi-isoopic laminaes wih embedded small-diamee fibe Bagg gaing sensos, Sma Maeials and Sucues, vol. 2, pp , 23. [35] K. S. C. Kuang, W. J. Canwell, L. Zhang, I. Bennion, M. Maalej, and S. T. Quek, Damage monioing in aluminum-foam sandwich sucues based on hemoplasic fibe-meal laminaes using fibe Bagg gaings, Composies Science and Technology, vol. 65, pp. 8-87, 25. [36] X. Tao, L. Tang, W. Du, and C. Choy, Inenal sain measuemen by fibe Bagg gaing sensos in exile composies, Composies Science and Technology, vol. 6, pp , 2. [37] W. Du, X. M. Tao, H. Y. Tam, and C. L. Choy, Fundamenals and applicaions of opical fibe Bagg gaing sensos o exile sucual composies, Composie Sucues, vol. 42, pp , 998. [38] Y. Shenfang, H. Rui, L. Xianghua, and L. Xiaohui, Deeminaion of inenal sain in 3-D baided composies using opic fibe sain sensos, Aca Mechanica Solida Sinica, vol. 7, pp , 24. [39] Y. E. Pak, Longiudinal shea ansfe in fibe opic sensos, Sma Maeials and Sucues, vol., pp , 992. [4] H. Y. Yang and M. L. Wang, Opical fibe senso sysem embedded in a membe subjeced o elaively abiay loads, Sma Maeials and Sucues, vol.4, pp. 5-58, 994.

134 9 [4] Y. Libo, F. Ansai, and M. Qasim, Sess analysis of bond fo inegaed opical fibe sensos, Inelligen Civil Engineeing Maeials and Sucues, ed. F Ansai e al., ASCE New Yok, pp , 997. [42] L. P. Kollá and R. J. Van Seenkise, Calculaion of he sesses and sains in embedded fibe opic sensos, Jounal of Composie Maeials, vol. 32, pp , 998. [43] M. Pabhugoud and K. Pees, Efficien simulaion of Bagg gaings sensos fo implemenaion o damage idenificaion in composies, Sma Maeials and Sucues, vol. 2, pp , 23. [44] R. M. Jones, Mechanics of composie maeials, Taylo and Fancis Inc, Philadelphia PA, 999. [45] R. B. Wageich, W. A. Aia, and J. S. Sikis, Effecs of diameic load on fibe Bagg gaings fabicaed in low biefingen fibe, Eleconics Lees, vol. 32, pp , 996. [46] K. S. Kim, L. Kollá, and G. S. Spinge, A model of embedded fibe opic Faby- Peo empeaue and sain sensos, Jounal of Composie Maeials, vol. 27, pp , 993. [47] J. S. Sikis, Unified appoach o phase-sain-empeaue models fo sma sucue inefeomeic opical fibe sensos: pa, developmen, Opical Engineeing, vol. 32, pp , 993. [48] C. M. Lawence, D. V. Nelson, E. Udd, and T. Benne, A fibe opic senso fo ansvese sain measuemen, Expeimenal Mechanics, vol. 39, pp , 999. [49] F. Bosia, P. Giaccai, J. Bosis, M. Facchini, H. G. Limbege, and R. P. Salahé, Chaaceizaion of he esponse of fibe Bagg gaing sensos subjeced o a wodimensional sain field, Sma Maeials and Sucues, vol. 2, pp , 23. [5] R. Gafsi and M. A. El-Sheif, Analysis of induced-biefingence effecs on fibe Bagg gaings, Opical Fibe Technology, vol. 6, pp , 2. [5] M. Huang, Sess effecs on he pefomance of opical waveguides, Inenaional Jounal of Solids and Sucues, vol. 4, pp , 23.

135 2 [52] P. A. Bèlange, Opical fibe heoy - a supplemen o applied elecomagneism, Wold Scienific, Rive Edge, NJ, 993. [53] M. Koshiba, Opical waveguide heoy by he finie elemen mehod, Kluwe Academic Publishes, Boson, MA, 992. [54] C. Yeh, S. B. Dong, and W. Olive, Abiaily shaped inhomogenous opical fibe o inegaed opical waveguides, Jounal of Applied Physics, vol. 46, pp , 975. [55] C. Yeh, K. Ha, S. B. Dong, and W. P. Bown, Single-mode opical waveguides, Applied Opics, vol. 8, pp , 979. [56] K. S. Chiang, Finie elemen analysis of opical fibes wih ieaive eamen of he infinie 2-D Space, Opical and Quanum Eleconics, vol. 7, pp , 985. [57] J. F. Nye, Physical popeies of cysals, Oxfod Univesiy Pess, New Yok NY, 23. [58] M. Yamada and K. Sakuda, Analysis of almos-peiodic disibued feedback slab waveguides via a fundamenal maix appoach, Applied Opics, vol. 26, pp , 987. [59] M. Pabhugoud and K. Pees, Modified ansfe maix fomulaion fo Bagg gaing sain sensos, Jounal of Lighwave Technology, vol. 22, pp , 24. [6] H. G. Taylo, Bending effecs in opical fibes, Jounal of Lighwave Technology, vol. 2, pp , 984. [6] M. M. Foch, Phooelasiciy, John Wiley and Sons Inc., New Yok NY, 948. [62] E. Chehua, C. C. Ye, S. E. Saines, S. W. James, and R. P. Taam, Chaaceizaion of he esponse of fibe Bagg gaings fabicaed in sess and geomeically induced high biefingence fibes o empeaue and ansvese load, Sma Maeials and Sucues, vol. 3, pp , 24. [63] C. Doyle, A. Main, T. Liu, M. Wu, S. Hayes, P. A. Cosby, G. R. Powell, D. Books, and G. F. Fenando, In-siu pocess and condiioning monioing of advanced fibeeinfoced composie maeials using opical fibe sensos, Sma Maeials and Sucues, vol. 7, pp , 998.

136 2 [64] H. K. Kang, D. H. Kang, C. S. Hong, and C. G. Kim, Simulaneous monioing of sain and empeaue duing and afe cue of unsymmeic composie laminae using fibe-opic sensos, Sma Maeials and Sucues, vol. 2, pp , 23. [65] K. S. Kim, L. Kollá, and G. S. Spinge, A model of embedded fibe opic fabypeo empeaue and sain sensos, Jounal of Composie Maeials, vol. 27, pp , 993. [66] J. S. Leng and A. Asundi, Real-ime cue monioing of sma composie maeials using exinsic faby-peo inefeomee and fibe Bagg gaing sensos, Sma Maeials and Sucues, vol., pp , 22. [67] Y. Liu, B. M. A. Rahman, and K. T. V. Gaan, Themal-sess-induced biefingence in bow-ie opical fibes, Applied Opics, vol. 33, pp , 994. [68] M. J. O Dwye, G. M. Maisos, S. W. James, R. P. Taam, and I. K. Paidge, Relaing he sae of cue o he eal-ime inenal sain developmen in a cuing composie using in-fibe Bagg gaings and dielecic sensos, Measuemen Science and Technology, vol. 9, pp , 998. [69] M. Pabhugoud and K. Pees, Finie elemen model fo embedded fibe Bagg gaing senso, submied o Sma Maeials and Sucues, 25. [7] N. Takeda and Y. Okabe, Duabiliy analysis and sucual healh managemen of sma composie sucues using small-diamee fibe opic sensos, Science and Engineeing of Composie Maeials, vol. 2, pp. -2, 25. [7] W. Uanczyk, E. Chmielewska, and W. J. Bock, Measuemens of empeaue and sain sensiiviies of a wo-mode Bagg gaings impined in a bow-ie fibe, Measuemen Science and Technology, vol. 2, pp. 8-84, 2. [72] M. Sude, K. Pees, and J. Bosis, Mehod fo deeminaion of cack bidging paamees using long opical fibe Bagg gaing sensos, Composies Pa B - Engineeing, vol. 34, pp , 23. [73] Y. Okabe, R. Tsuji, and N. Takeda, Applicaion of chiped fibe Bagg gaings sensos fo idenificaion of cack locaions in composies, Composies Pa A - Applied Science and Manufacuing, vol. 35, pp , 24.

137 22 [74] Y. Okabe, T. Mizuani, S. Yashio, and N. Takeda, Deecion of micoscopic damages in composie laminaes wih embedded small-diamee fibe Bagg gaing sensos, Composies Science and Technology, vol. 62, pp , 22. [75] S. Huang, M. LeBlanc, M. M. Ohn, and R. M. Measues, Bagg inagaing sucual sensing, Applied Opics, vol. 34, pp , 995. [76] K. Pees, P. Pais, and J. Bosis, Novel echnique o measue axial sain disibuion along fibe duing pullou es, Jounal of Maeials Science Lees, vol. 2, pp , 22. [77] P. Toes and L. C. G. Valene, Specal esponse of locally pessed fibe Bagg gaing, Opics Communicaions, vol. 28, pp , 22. [78] H. Kogelnik, Theoy of opical waveguides, in Guided-Wave Opoeleconics, T. Tami, Ed. New Yok, Spinge-Velag, 99. [79] K. Pees, M. Sude, J. Bosis, A. Iocco, H. Limbege and R. Salahé, Embedded opical fibe Bagg gaing senso in a nonunifom sain field: measuemens and simulaions, Expeimenal Mechanics, vol. 4, pp. 9-28, 2. [8] S. Huang, M. M. Ohn, M. LeBlanc, and R. M. Measues, Coninuous abiay sain pofile measuemens wih fibe Bagg gaings, Sma Maeials and Sucues, vol. 7, pp , 998. [8] G. Comie, R. Boudeau, and S. Théiaul, Real-coded geneic algoihm fo Bagg gaing paamee synhesis, Jounal of Sociey of Ameica B, vol. 8, pp , 2. [82] A. Gill, K. Pees, and M. Sude, Geneic algoihm fo he econsucion of Bagg gaing senso disibuion, Poceedings of he 23 ASME Inenaional Mechanical Engineeing Congess and Exposiion, Washingon DC, 23. [83] H. Kogelnik, File esponse of nonunifom almos-peiodic sucues, AT & T Technical Jounal, vol. 55, pp. 9-26, 976. [84] C. Chang and S. T. Voha, Specal boadening due o non-unifom sain fields in fibe Bagg gaing based ansduces, Eleconics Lees, vol. 34, pp , 998.

138 23 Appendix A Expeimenal Daa fo PMMA Specimens This appendix plos he expeimenal daa fo PMMA specimens. The esponse of FBGs suface mouned on PMMA specimens subjeced o low velociy impacs ae ploed. The esponse of FBGs oiened a angles 45 (see figues A.-A.9) and 9 (see figues A.-A.6) ae included. The specal daa fo nea, inemediae and fa fields ae ploed fo boh he oienaions (efe o able 5.). Figues A.7-A.9 plo he specal esponse of FBGs suface mouned on specimen A79. One noes ha fo his specimen he impac velociy was educed o.5 m/sec. Phoogaphs of he PMMA specimens indicaing diffeen failue modes ae shown in figue A.2. Finally, he esponse of FBGs suface mouned on PMMA specimens subjeced o a saic bending load ae ploed (see figues A.2-A.25). The eponse fo boh coninuous loading (see figues A.2-A.22) and cyclic loading (see figues A.23-A.25) ae shown.

139 2 A58 5 Sikes Befoe mouning 2 Afe mouning 2 Sike # Wavelengh(nm) Sike #2 Sike #3 Sike # Sike # Figue A.: Measued specal esponse of FBG suface mouned on A58 fo impac sikes -5.

140 2.5 A59 9 Sikes Befoe mouning 2.5 Afe mouning 2.5 Sike # Wavelengh(nm) Sike #2 Sike #3 Sike # Sike #5 Sike #6 Sike # Wavelengh(nm) Wavelengh(nm) 25 Figue A.2: Measued specal esponse of FBG suface mouned on A59 fo impac sikes -7.

141 2.5 A59 9 Sikes Sike #8 2.5 Sike #9 2.5 Sike # Wavelengh(nm) Sike # Afe ebonding Sike # Sike #3 Sike #4 Sike # Wavelengh(nm) Wavelengh(nm) 26 Figue A.3: Measued specal esponse of FBG suface mouned on A59 fo impac sikes 8-5.

142 A59 9 Sikes Sike #6 Sike #7 Sike # Wavelengh(nm) Sike # Figue A.4: Measued specal esponse of FBG suface mouned on A59 fo impac sikes

143 2.5 A69 27 Sikes Befoe mouning 2.5 Afe mouning 2.5 Sike # Sike #2 Sike #3 Sike # Afe ebonding Sike #5.2 Sike # Figue A.5: Measued specal esponse of FBG suface mouned on A69 fo impac sikes -6.

144 A69 27 Sikes Sike #7.8 Sike #8.7 Sike # Sike #.5 Sike #.3 Sike # Sike #3.3 Sike #4.2 Resplicing Figue A.6: Measued specal esponse of FBG suface mouned on A69 fo impac sikes 7-4.

145 .4.2 A69 27 Sikes Sike #5.4.2 Sike #6.5 Sike # Sike #8.3 Sike #9.4 Sike # Sike #2.2.5 Sike # Sike #23 End Figue A.7: Measued specal esponse of FBG suface mouned on A69 fo impac sikes 5-23.

146 A69 27 Sikes Sike #23 End Sike #24 End Sike #24 End Sike #25 End.3 Sike #25 End 2 Sike #26 End Sike #26 End 2.5 Sike #27 End.3 Sike #27 End Figue A.8: Measued specal esponse of FBG suface mouned on A69 fo impac sikes

147 .2 A69 27 Sikes.5 End of expeimen End eflecion.5 Endof expeimen Tansmission Endof expeimen End 2 eflecion Figue A.9: Measued specal esponse of FBG suface mouned on A69 a he end of impac loading. 32

148 A7 4Sikes Befoe mouning Afe mouning Sike # Wavelengh(nm) Sike #2 Sike #3 Sike # Wavelengh(nm) Figue A.: Measued specal esponse of FBG suface mouned on A7 fo impac sikes

149 A73 36 Sikes Befoe mouning Afe mouning Sike # Wavelengh(nm).6 Sike #2.6 Sike #3.6 Sike # Wavelengh(nm) Sike #5.6 Sike #6.6 Sike # Wavelengh(nm) Figue A.: Measued specal esponse of FBG suface mouned on A73 fo impac sikes -7.

150 A73 36 Sikes Sike #8 Sike #9 Sike # Wavelengh(nm).6 Sike #.6 Sike #2.6 Sike # Wavelengh(nm) Sike #4.6 Sike #5.6 Sike # Wavelengh(nm) Figue A.2: Measued specal esponse of FBG suface mouned on A73 fo impac sikes 8-6.

151 A73 36 Sikes Sike #7 Sike #8 Sike # Wavelengh(nm).6 Sike #2.6 Sike #2.6 Sike # Wavelengh(nm) Sike #23.6 Sike #24.6 Sike # Wavelengh(nm) Figue A.3: Measued specal esponse of FBG suface mouned on A73 fo impac sikes 7-25.

152 A73 36 Sikes Sike #26 Sike #27 Sike # Wavelengh(nm) Sike # Sike # Sike # Wavelengh(nm) Sike #32.6 Sike #33.6 Sike # Wavelengh(nm) Figue A.4: Measued specal esponse of FBG suface mouned on A73 fo impac sikes

153 .5 A73 36 Sikes Sike #35.6 Sike # Figue A.5: Measued specal esponse of FBG suface mouned on A73 fo impac sikes

154 .5 A75 4 Sikes Befoe mouning.6 Afe mouning.6 Sike # Wavelengh(nm).6 Sike #2.6 Sike #3.6 Sike # Wavelengh(nm) Wavelengh(nm) Figue A.6: Measued specal esponse of FBG suface mouned on A75 fo impac sikes

155 A Sikes Befoe mouning.4 Afe mouning.4 Sike # Sike #2.4 Sike #3.4 Sike # Sike #5.4 Sike #6.4 Sike # Figue A.7: Measued specal esponse of FBG suface mouned on A79 fo impac sikes -7.

156 A79 Sike #8.4 2 Sikes.4 Sike #9.4 Sike # Wavelengh(nm) Wavelengh(nm).4 Sike #.4 Sike #2.4 Sike # Wavelengh(nm).4 Sike #4.4 Sike #5.4 Sike # Wavelengh(nm) 4 Figue A.8: Measued specal esponse of FBG suface mouned on A79 fo impac sikes 8-6.

157 A79 Sike #7.4 2 Sikes.4 Sike #8.4 Sike # Sike # Figue A.9: Measued specal esponse of FBG suface mouned on A79 fo impac sikes

158 A59 A59 Opical fibe Opical fibe A69 A7 Opical fibe Opical fibe A73 Opical fibe A75 Opical fibe Figue A.2: PMMA specimens suface mouned wih FBGs. 43

159 Befoe mouning Afe mouning lbs.8 A8 Coninuous Loading Wavelengh(nm) Wavelengh(nm) 2 lbs 3 lbs 4 lbs Wavelengh(nm) 5 lbs 6 lbs 7 lbs Wavelengh(nm) 44 Figue A.2: Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom -7 lbs.

160 8 lbs lbs End lbs End A8 Coninuous Loading lbs End lbs End 2 2 lbs End lbs End 2 3 lbs End.5 3 lbs End Figue A.22: Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom 8-3 lbs.

161 .6 A8 Loading and Unloading Befoe mouning.6 Afe mouning.6 lbs L Wavelengh(nm) Wavelengh(nm).6 lbs U.6 2 lbs L.6 2 lbs U Wavelengh(nm).6 3 lbs L.6 3 lbs U.6 4 lbs L Wavelengh(nm) 46 Figue A.23: Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom -4L lbs. Noaion L and U sand fo loading and unloading especively.

162 .6 A8 Loading and Unloading 4 lbs U.6 5 lbs L.6 5 lbs U lbs L.6 6 lbs U.6 7 lbs L lbs U.6 8 lbs L.6 8 lbs U Figue A.24: Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom 4U-8U lbs. Noaion L and U sand fo loading and unloading especively.

163 .6 A8 Loading and Unloading 9 lbs L.6 9 lbs U.6 lbs L lbs U lbs L End lbs U End Figue A.25: Measued specal esponse of FBG suface mouned on A8 fo a saic loading fom 9L-U lbs. Noaion L and U sand fo loading and unloading especively. 48

164 49 Appendix B Expeimenal Daa fo Composie Specimens This appendix pesens he expeimenal daa fo he wo dimensional woven composie specimens. The dawing of he mold used o fabicae he composie specimens is shown in figue B.. The pogession of damage in a ypical composie specimen subjeced o muliple low velociy impacs is shown in figue B.2. The esponse of FBGs suface mouned on he composie specimens subjeced o low velociy impacs fo specimen C7 (see figues B.3-B.6) and specimen C8 (see figues B.7-B.9) ae ploed. Finally, he esponse of he FBGs embedded in wo dimensional woven composie specimens subjeced o muliple low velociy impacs ae also ploed. The specal esponse of FBG embedded in specimen C5 ae ploed in figues B.-B.. One noes ha a bae opical fibe was embedded in specimen C5. The specal esponse of FBGs, coaed wih polyimide, embedded in specimen C49 (see figues B.2-B.6) and specimen C5 (see figues B.7-B.24) ae also ploed.

165 Base plae Plunge 2 Numbes Numbes Side ails Figue B.: Dimensions of he mold used o fabicae he composie specimens. All dimensions in inches.

166 Sike # Sike #4 Sike #6 Sike #9 Sike #2 Sike #5 Sike #7 Sike #2 5 Figue B.2: Pogession of damage in a ypical wo-dimensional woven composie specimen.