Abstract

In this paper, the property of panda polarization maintaining fiber Bragg gratings (PPM-FBGs) embedded in a composite laminate system (CLS) under a transversal force from 1 × 105 Pa to 5 × 105 Pa is explored. Both the wavelengths shift and the rotation angle of the principal axes of the PPM-FBGs are surveyed theoretically. We investigate the corresponding relation between the direction of external force and the rotation angle of principal axes of the PPM-FBGs and prove that the magnitude and direction of the strain distribution in the CLS can be monitored simultaneously, which can realize the detection of interlaminar damage of the CLS. We find that the strain, which corresponds to the shift of wavelengths of the PPM-FBGs, has a different angular dependence, and therefore the potential failure forms of the CLS can be differentiated.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. Y. Wang, M. Wang, W. Xia, and X. Ni, “High-resolution fiber Bragg grating based transverse load sensor using microwave photonics filtering technique,” Opt. Express 24(16), 17960–17967 (2016).
    [Crossref] [PubMed]
  2. T. Yun and S. T. Tu, “Fabrication and characterization of a metal-packaged regenerated fiber Bragg grating strain sensor for structural integrity monitoring of high-temperature components,” Smart Mater. Struct. 23(3), 035001 (2014).
    [Crossref]
  3. Y. N. Tan, Y. Zhang, L. Jin, and B. O. Guan, “Simultaneous strain and temperature fiber grating laser sensor based on radio-frequency measurement,” Opt. Express 19(21), 20650–20656 (2011).
    [Crossref] [PubMed]
  4. J. Degrieck, W. De Waele, and P. Verleysen, “Monitoring of fiber reinforced composites with embedded optical fibre Bragg sensors, with application to filament wound pressure vessels,” NDT Int. 34(4), 289–296 (2001).
    [Crossref]
  5. G. Pereira, C. Frias, H. Faria, O. Frazão, and A. T. Marques, “On the improvement of strain measurements with FBG sensors embedded in unidirectional composites,” Polym. Test. 32(1), 99–105 (2013).
    [Crossref]
  6. L. Polz, B. Hopf, A. Jarsen, M. Eitzenberger, M. Lindner, H. Bartelt, and J. Roths, “Regenerated Bragg Gratings in Panda Fibers for Simultaneous Temperature and Force Measurements at High Temperatures,” J. Lightwave Technol. 34(19), 4550–4556 (2016).
    [Crossref]
  7. O. Rifaie-Graham, E. A. Apebende, L. K. Bast, and N. Bruns, “Self-Reporting Fiber-Reinforced Composites That Mimic the Ability of Biological Materials to Sense and Report Damage,” Adv. Mater. 30(19), e1705483 (2018).
    [Crossref] [PubMed]
  8. G. Luyckx, E. Voet, N. Lammens, W. De Waele, and J. Degrieck, “Residual strain-induced birefringent FBGs for multi-axial strain monitoring of CFRP composite laminates,” NDT Int. 54, 142–150 (2013).
    [Crossref]
  9. N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
    [Crossref]
  10. S. H. Yoo, M. G. Han, J. H. Hong, and S. H. Chang, “Simulation of curing process of carbon epoxy composite during autoclave degassing molding by considering phase changes of epoxy resin,” Compos., Part B Eng. 77, 257–267 (2015).
    [Crossref]
  11. P. P. Camanho, C. G. Davila, and M. F. De Moura, “Numerical simulation of mixed-mode progressive delamination in composite materials,” J. Compos. Mater. 37(16), 1415–1438 (2003).
    [Crossref]
  12. S. R. Lavoie, R. Long, and T. Tang, “Rate dependent fracture of a double cantilever beam with combined bulk and interfacial dissipation,” Int. J. Solids Struct. 75, 277–286 (2015).
    [Crossref]
  13. J. R. Reeder and J. H. Rews, “Mixed-mode bending method for delamination testing,” AIAA J. 28(7), 1270–1276 (1990).
    [Crossref]
  14. B. Hopf, A. W. Koch, and J. Roths, “Temperature dependence of glue-induced birefringence in surface-attached FBG strain sensors,” J. Lightwave Technol. 34(4), 1220–1227 (2016).
    [Crossref]
  15. W. Zhang, W. Chen, Y. Shu, X. Lei, and X. Liu, “Effects of bonding layer on the available strain measuring range of fiber Bragg gratings,” Appl. Opt. 53(5), 885–891 (2014).
    [Crossref] [PubMed]
  16. C. E. Banks, J. Grant, S. Russell, and S. Arnett, “Strain measurement during stress rupture of composite over-wrapped pressure vessel with fiber Bragg gratings sensors,” Proc. SPIE 6933, 69330O (2008).
    [Crossref]
  17. G. F. Pereira, L. P. Mikkelsen, and M. McGugan, “Crack detection in fibre reinforced plastic structures using embedded fibre bragg grating sensors: Theory, model development and experimental validation,” PLoS One 10(10), e0141495 (2015).
    [Crossref] [PubMed]
  18. G. Pereira, M. McGugan, and L. P. Mikkelsen, “FBG SiMul V1. 0: Fibre Bragg grating signal simulation tool for finite element method models,” SoftwareX 5, 163–170 (2016).
    [Crossref]
  19. G. Pereira, L. Mikkelsen, and M. McGugan, “Crack growth monitoring by embedded optical Fibre Bragg Grating sensors: Fibre reinforced plastic crack growing detection,” 2015 International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS) (IEEE, 2015), pp. 133–139.
    [Crossref]
  20. G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
    [Crossref]
  21. H. Z. Yan, S. Y. Li, Z. Y. Xie, X. P. Zheng, H. Y. Zhang, and B. K. Zhou, “Design of PANDA ring-core fiber with 10 polarization-maintaining modes,” Photon. Res. 5(1), 1–5 (2017).
    [Crossref]
  22. R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11(3), 240–254 (2005).
    [Crossref]
  23. E. Udd, “Review of multi-parameter fiber grating sensors,” Proc. SPIE 6770, 677002 (2007).
    [Crossref]
  24. M. G. Xu, J.-L. Archambault, L. Reekie, and J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
    [Crossref]
  25. S. E. Kanellopoulos, V. A. Handerek, and A. J. Rogers, “Simultaneous strain and temperature sensing with photogenerated in-fiber gratings,” Opt. Lett. 20(3), 333–335 (1995).
    [Crossref] [PubMed]
  26. L. A. Ferreira, F. M. Araujo, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using interferometrically interrogated fiber Bragg grating sensors,” Opt. Eng. 39(8), 2226–2234 (2000).
    [Crossref]
  27. F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathé, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003).
    [Crossref]
  28. J. F. Botero-Cadavid, J. D. Causado-Buelvas, and P. Torres, “Spectral properties of locally pressed fiber Bragg gratings written in polarization maintaining fibers,” J. Lightwave Technol. 28(9), 1291–1297 (2010).
    [Crossref]
  29. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed. New York: McGraw-Hill, 1970.
  30. S. Sulejmani, C. Sonnenfeld, T. Geernaert, G. Luyckx, D. Van Hemelrijck, P. Mergo, W. Urbanczyk, K. Chah, C. Caucheteur, P. Mégret, H. Thienpont, and F. Berghmans, “Shear stress sensing with Bragg grating-based sensors in microstructured optical fibers,” Opt. Express 21(17), 20404–20416 (2013).
    [Crossref] [PubMed]
  31. S. L. A. Carrara, B. Y. Kim, and H. J. Shaw, “Elasto-optic alignment of birefringent axes in polarization-holding optical fiber,” Opt. Lett. 11(7), 470–472 (1986).
    [Crossref] [PubMed]
  32. M. Lindner, B. Hopf, A. W. Koch, and J. Roths, “Three dimensional FEM model of FBGs in PANDA fibers with experimentally determined model parameters,” Optical Fiber Sensors Conference (OFS) (IEEE, 2017), pp. 1–4.

2018 (1)

O. Rifaie-Graham, E. A. Apebende, L. K. Bast, and N. Bruns, “Self-Reporting Fiber-Reinforced Composites That Mimic the Ability of Biological Materials to Sense and Report Damage,” Adv. Mater. 30(19), e1705483 (2018).
[Crossref] [PubMed]

2017 (1)

2016 (4)

2015 (3)

G. F. Pereira, L. P. Mikkelsen, and M. McGugan, “Crack detection in fibre reinforced plastic structures using embedded fibre bragg grating sensors: Theory, model development and experimental validation,” PLoS One 10(10), e0141495 (2015).
[Crossref] [PubMed]

S. H. Yoo, M. G. Han, J. H. Hong, and S. H. Chang, “Simulation of curing process of carbon epoxy composite during autoclave degassing molding by considering phase changes of epoxy resin,” Compos., Part B Eng. 77, 257–267 (2015).
[Crossref]

S. R. Lavoie, R. Long, and T. Tang, “Rate dependent fracture of a double cantilever beam with combined bulk and interfacial dissipation,” Int. J. Solids Struct. 75, 277–286 (2015).
[Crossref]

2014 (2)

T. Yun and S. T. Tu, “Fabrication and characterization of a metal-packaged regenerated fiber Bragg grating strain sensor for structural integrity monitoring of high-temperature components,” Smart Mater. Struct. 23(3), 035001 (2014).
[Crossref]

W. Zhang, W. Chen, Y. Shu, X. Lei, and X. Liu, “Effects of bonding layer on the available strain measuring range of fiber Bragg gratings,” Appl. Opt. 53(5), 885–891 (2014).
[Crossref] [PubMed]

2013 (4)

S. Sulejmani, C. Sonnenfeld, T. Geernaert, G. Luyckx, D. Van Hemelrijck, P. Mergo, W. Urbanczyk, K. Chah, C. Caucheteur, P. Mégret, H. Thienpont, and F. Berghmans, “Shear stress sensing with Bragg grating-based sensors in microstructured optical fibers,” Opt. Express 21(17), 20404–20416 (2013).
[Crossref] [PubMed]

G. Luyckx, E. Voet, N. Lammens, W. De Waele, and J. Degrieck, “Residual strain-induced birefringent FBGs for multi-axial strain monitoring of CFRP composite laminates,” NDT Int. 54, 142–150 (2013).
[Crossref]

N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
[Crossref]

G. Pereira, C. Frias, H. Faria, O. Frazão, and A. T. Marques, “On the improvement of strain measurements with FBG sensors embedded in unidirectional composites,” Polym. Test. 32(1), 99–105 (2013).
[Crossref]

2011 (1)

2010 (1)

2008 (1)

C. E. Banks, J. Grant, S. Russell, and S. Arnett, “Strain measurement during stress rupture of composite over-wrapped pressure vessel with fiber Bragg gratings sensors,” Proc. SPIE 6933, 69330O (2008).
[Crossref]

2007 (1)

E. Udd, “Review of multi-parameter fiber grating sensors,” Proc. SPIE 6770, 677002 (2007).
[Crossref]

2005 (1)

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11(3), 240–254 (2005).
[Crossref]

2004 (1)

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

2003 (2)

P. P. Camanho, C. G. Davila, and M. F. De Moura, “Numerical simulation of mixed-mode progressive delamination in composite materials,” J. Compos. Mater. 37(16), 1415–1438 (2003).
[Crossref]

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathé, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003).
[Crossref]

2001 (1)

J. Degrieck, W. De Waele, and P. Verleysen, “Monitoring of fiber reinforced composites with embedded optical fibre Bragg sensors, with application to filament wound pressure vessels,” NDT Int. 34(4), 289–296 (2001).
[Crossref]

2000 (1)

L. A. Ferreira, F. M. Araujo, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using interferometrically interrogated fiber Bragg grating sensors,” Opt. Eng. 39(8), 2226–2234 (2000).
[Crossref]

1995 (1)

1994 (1)

M. G. Xu, J.-L. Archambault, L. Reekie, and J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

1990 (1)

J. R. Reeder and J. H. Rews, “Mixed-mode bending method for delamination testing,” AIAA J. 28(7), 1270–1276 (1990).
[Crossref]

1986 (1)

Apebende, E. A.

O. Rifaie-Graham, E. A. Apebende, L. K. Bast, and N. Bruns, “Self-Reporting Fiber-Reinforced Composites That Mimic the Ability of Biological Materials to Sense and Report Damage,” Adv. Mater. 30(19), e1705483 (2018).
[Crossref] [PubMed]

Araujo, F. M.

L. A. Ferreira, F. M. Araujo, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using interferometrically interrogated fiber Bragg grating sensors,” Opt. Eng. 39(8), 2226–2234 (2000).
[Crossref]

Archambault, J.-L.

M. G. Xu, J.-L. Archambault, L. Reekie, and J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

Arnett, S.

C. E. Banks, J. Grant, S. Russell, and S. Arnett, “Strain measurement during stress rupture of composite over-wrapped pressure vessel with fiber Bragg gratings sensors,” Proc. SPIE 6933, 69330O (2008).
[Crossref]

Banks, C. E.

C. E. Banks, J. Grant, S. Russell, and S. Arnett, “Strain measurement during stress rupture of composite over-wrapped pressure vessel with fiber Bragg gratings sensors,” Proc. SPIE 6933, 69330O (2008).
[Crossref]

Bartelt, H.

Bast, L. K.

O. Rifaie-Graham, E. A. Apebende, L. K. Bast, and N. Bruns, “Self-Reporting Fiber-Reinforced Composites That Mimic the Ability of Biological Materials to Sense and Report Damage,” Adv. Mater. 30(19), e1705483 (2018).
[Crossref] [PubMed]

Berghmans, F.

Bosia, F.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathé, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003).
[Crossref]

Botero-Cadavid, J. F.

Botsis, J.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathé, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003).
[Crossref]

Bruns, N.

O. Rifaie-Graham, E. A. Apebende, L. K. Bast, and N. Bruns, “Self-Reporting Fiber-Reinforced Composites That Mimic the Ability of Biological Materials to Sense and Report Damage,” Adv. Mater. 30(19), e1705483 (2018).
[Crossref] [PubMed]

Camanho, P. P.

P. P. Camanho, C. G. Davila, and M. F. De Moura, “Numerical simulation of mixed-mode progressive delamination in composite materials,” J. Compos. Mater. 37(16), 1415–1438 (2003).
[Crossref]

Carrara, S. L. A.

Caucheteur, C.

S. Sulejmani, C. Sonnenfeld, T. Geernaert, G. Luyckx, D. Van Hemelrijck, P. Mergo, W. Urbanczyk, K. Chah, C. Caucheteur, P. Mégret, H. Thienpont, and F. Berghmans, “Shear stress sensing with Bragg grating-based sensors in microstructured optical fibers,” Opt. Express 21(17), 20404–20416 (2013).
[Crossref] [PubMed]

N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
[Crossref]

Causado-Buelvas, J. D.

Chah, K.

S. Sulejmani, C. Sonnenfeld, T. Geernaert, G. Luyckx, D. Van Hemelrijck, P. Mergo, W. Urbanczyk, K. Chah, C. Caucheteur, P. Mégret, H. Thienpont, and F. Berghmans, “Shear stress sensing with Bragg grating-based sensors in microstructured optical fibers,” Opt. Express 21(17), 20404–20416 (2013).
[Crossref] [PubMed]

N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
[Crossref]

Chang, S. H.

S. H. Yoo, M. G. Han, J. H. Hong, and S. H. Chang, “Simulation of curing process of carbon epoxy composite during autoclave degassing molding by considering phase changes of epoxy resin,” Compos., Part B Eng. 77, 257–267 (2015).
[Crossref]

Chen, G.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Chen, W.

Dakin, J. P.

M. G. Xu, J.-L. Archambault, L. Reekie, and J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

Davila, C. G.

P. P. Camanho, C. G. Davila, and M. F. De Moura, “Numerical simulation of mixed-mode progressive delamination in composite materials,” J. Compos. Mater. 37(16), 1415–1438 (2003).
[Crossref]

De Moura, M. F.

P. P. Camanho, C. G. Davila, and M. F. De Moura, “Numerical simulation of mixed-mode progressive delamination in composite materials,” J. Compos. Mater. 37(16), 1415–1438 (2003).
[Crossref]

De Waele, W.

G. Luyckx, E. Voet, N. Lammens, W. De Waele, and J. Degrieck, “Residual strain-induced birefringent FBGs for multi-axial strain monitoring of CFRP composite laminates,” NDT Int. 54, 142–150 (2013).
[Crossref]

J. Degrieck, W. De Waele, and P. Verleysen, “Monitoring of fiber reinforced composites with embedded optical fibre Bragg sensors, with application to filament wound pressure vessels,” NDT Int. 34(4), 289–296 (2001).
[Crossref]

Degrieck, J.

G. Luyckx, E. Voet, N. Lammens, W. De Waele, and J. Degrieck, “Residual strain-induced birefringent FBGs for multi-axial strain monitoring of CFRP composite laminates,” NDT Int. 54, 142–150 (2013).
[Crossref]

N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
[Crossref]

J. Degrieck, W. De Waele, and P. Verleysen, “Monitoring of fiber reinforced composites with embedded optical fibre Bragg sensors, with application to filament wound pressure vessels,” NDT Int. 34(4), 289–296 (2001).
[Crossref]

Eitzenberger, M.

Facchini, M.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathé, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003).
[Crossref]

Farahi, F.

L. A. Ferreira, F. M. Araujo, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using interferometrically interrogated fiber Bragg grating sensors,” Opt. Eng. 39(8), 2226–2234 (2000).
[Crossref]

Faria, H.

G. Pereira, C. Frias, H. Faria, O. Frazão, and A. T. Marques, “On the improvement of strain measurements with FBG sensors embedded in unidirectional composites,” Polym. Test. 32(1), 99–105 (2013).
[Crossref]

Ferreira, L. A.

L. A. Ferreira, F. M. Araujo, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using interferometrically interrogated fiber Bragg grating sensors,” Opt. Eng. 39(8), 2226–2234 (2000).
[Crossref]

Frazão, O.

G. Pereira, C. Frias, H. Faria, O. Frazão, and A. T. Marques, “On the improvement of strain measurements with FBG sensors embedded in unidirectional composites,” Polym. Test. 32(1), 99–105 (2013).
[Crossref]

Frias, C.

G. Pereira, C. Frias, H. Faria, O. Frazão, and A. T. Marques, “On the improvement of strain measurements with FBG sensors embedded in unidirectional composites,” Polym. Test. 32(1), 99–105 (2013).
[Crossref]

Gan, Z.

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11(3), 240–254 (2005).
[Crossref]

Geernaert, T.

Giaccari, P.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathé, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003).
[Crossref]

Grant, J.

C. E. Banks, J. Grant, S. Russell, and S. Arnett, “Strain measurement during stress rupture of composite over-wrapped pressure vessel with fiber Bragg gratings sensors,” Proc. SPIE 6933, 69330O (2008).
[Crossref]

Guan, B. O.

Guan, R.

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11(3), 240–254 (2005).
[Crossref]

Han, M. G.

S. H. Yoo, M. G. Han, J. H. Hong, and S. H. Chang, “Simulation of curing process of carbon epoxy composite during autoclave degassing molding by considering phase changes of epoxy resin,” Compos., Part B Eng. 77, 257–267 (2015).
[Crossref]

Handerek, V. A.

Hong, J. H.

S. H. Yoo, M. G. Han, J. H. Hong, and S. H. Chang, “Simulation of curing process of carbon epoxy composite during autoclave degassing molding by considering phase changes of epoxy resin,” Compos., Part B Eng. 77, 257–267 (2015).
[Crossref]

Hopf, B.

Huang, D.

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11(3), 240–254 (2005).
[Crossref]

Jarsen, A.

Jia, H.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Jin, L.

Kanellopoulos, S. E.

Kim, B. Y.

Kinet, D.

N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
[Crossref]

Koch, A. W.

B. Hopf, A. W. Koch, and J. Roths, “Temperature dependence of glue-induced birefringence in surface-attached FBG strain sensors,” J. Lightwave Technol. 34(4), 1220–1227 (2016).
[Crossref]

M. Lindner, B. Hopf, A. W. Koch, and J. Roths, “Three dimensional FEM model of FBGs in PANDA fibers with experimentally determined model parameters,” Optical Fiber Sensors Conference (OFS) (IEEE, 2017), pp. 1–4.

Lammens, N.

N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
[Crossref]

G. Luyckx, E. Voet, N. Lammens, W. De Waele, and J. Degrieck, “Residual strain-induced birefringent FBGs for multi-axial strain monitoring of CFRP composite laminates,” NDT Int. 54, 142–150 (2013).
[Crossref]

Lavoie, S. R.

S. R. Lavoie, R. Long, and T. Tang, “Rate dependent fracture of a double cantilever beam with combined bulk and interfacial dissipation,” Int. J. Solids Struct. 75, 277–286 (2015).
[Crossref]

Lei, X.

Li, S. Y.

Limberger, H. G.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathé, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003).
[Crossref]

Lindner, M.

L. Polz, B. Hopf, A. Jarsen, M. Eitzenberger, M. Lindner, H. Bartelt, and J. Roths, “Regenerated Bragg Gratings in Panda Fibers for Simultaneous Temperature and Force Measurements at High Temperatures,” J. Lightwave Technol. 34(19), 4550–4556 (2016).
[Crossref]

M. Lindner, B. Hopf, A. W. Koch, and J. Roths, “Three dimensional FEM model of FBGs in PANDA fibers with experimentally determined model parameters,” Optical Fiber Sensors Conference (OFS) (IEEE, 2017), pp. 1–4.

Liu, L.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Liu, S.

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11(3), 240–254 (2005).
[Crossref]

Liu, X.

Long, R.

S. R. Lavoie, R. Long, and T. Tang, “Rate dependent fracture of a double cantilever beam with combined bulk and interfacial dissipation,” Int. J. Solids Struct. 75, 277–286 (2015).
[Crossref]

Luyckx, G.

N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
[Crossref]

G. Luyckx, E. Voet, N. Lammens, W. De Waele, and J. Degrieck, “Residual strain-induced birefringent FBGs for multi-axial strain monitoring of CFRP composite laminates,” NDT Int. 54, 142–150 (2013).
[Crossref]

S. Sulejmani, C. Sonnenfeld, T. Geernaert, G. Luyckx, D. Van Hemelrijck, P. Mergo, W. Urbanczyk, K. Chah, C. Caucheteur, P. Mégret, H. Thienpont, and F. Berghmans, “Shear stress sensing with Bragg grating-based sensors in microstructured optical fibers,” Opt. Express 21(17), 20404–20416 (2013).
[Crossref] [PubMed]

Marques, A. T.

G. Pereira, C. Frias, H. Faria, O. Frazão, and A. T. Marques, “On the improvement of strain measurements with FBG sensors embedded in unidirectional composites,” Polym. Test. 32(1), 99–105 (2013).
[Crossref]

McGugan, M.

G. Pereira, M. McGugan, and L. P. Mikkelsen, “FBG SiMul V1. 0: Fibre Bragg grating signal simulation tool for finite element method models,” SoftwareX 5, 163–170 (2016).
[Crossref]

G. F. Pereira, L. P. Mikkelsen, and M. McGugan, “Crack detection in fibre reinforced plastic structures using embedded fibre bragg grating sensors: Theory, model development and experimental validation,” PLoS One 10(10), e0141495 (2015).
[Crossref] [PubMed]

G. Pereira, L. Mikkelsen, and M. McGugan, “Crack growth monitoring by embedded optical Fibre Bragg Grating sensors: Fibre reinforced plastic crack growing detection,” 2015 International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS) (IEEE, 2015), pp. 133–139.
[Crossref]

Mégret, P.

N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
[Crossref]

S. Sulejmani, C. Sonnenfeld, T. Geernaert, G. Luyckx, D. Van Hemelrijck, P. Mergo, W. Urbanczyk, K. Chah, C. Caucheteur, P. Mégret, H. Thienpont, and F. Berghmans, “Shear stress sensing with Bragg grating-based sensors in microstructured optical fibers,” Opt. Express 21(17), 20404–20416 (2013).
[Crossref] [PubMed]

Mergo, P.

Mikkelsen, L.

G. Pereira, L. Mikkelsen, and M. McGugan, “Crack growth monitoring by embedded optical Fibre Bragg Grating sensors: Fibre reinforced plastic crack growing detection,” 2015 International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS) (IEEE, 2015), pp. 133–139.
[Crossref]

Mikkelsen, L. P.

G. Pereira, M. McGugan, and L. P. Mikkelsen, “FBG SiMul V1. 0: Fibre Bragg grating signal simulation tool for finite element method models,” SoftwareX 5, 163–170 (2016).
[Crossref]

G. F. Pereira, L. P. Mikkelsen, and M. McGugan, “Crack detection in fibre reinforced plastic structures using embedded fibre bragg grating sensors: Theory, model development and experimental validation,” PLoS One 10(10), e0141495 (2015).
[Crossref] [PubMed]

Ni, X.

Pereira, G.

G. Pereira, M. McGugan, and L. P. Mikkelsen, “FBG SiMul V1. 0: Fibre Bragg grating signal simulation tool for finite element method models,” SoftwareX 5, 163–170 (2016).
[Crossref]

G. Pereira, C. Frias, H. Faria, O. Frazão, and A. T. Marques, “On the improvement of strain measurements with FBG sensors embedded in unidirectional composites,” Polym. Test. 32(1), 99–105 (2013).
[Crossref]

G. Pereira, L. Mikkelsen, and M. McGugan, “Crack growth monitoring by embedded optical Fibre Bragg Grating sensors: Fibre reinforced plastic crack growing detection,” 2015 International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS) (IEEE, 2015), pp. 133–139.
[Crossref]

Pereira, G. F.

G. F. Pereira, L. P. Mikkelsen, and M. McGugan, “Crack detection in fibre reinforced plastic structures using embedded fibre bragg grating sensors: Theory, model development and experimental validation,” PLoS One 10(10), e0141495 (2015).
[Crossref] [PubMed]

Polz, L.

Reeder, J. R.

J. R. Reeder and J. H. Rews, “Mixed-mode bending method for delamination testing,” AIAA J. 28(7), 1270–1276 (1990).
[Crossref]

Reekie, L.

M. G. Xu, J.-L. Archambault, L. Reekie, and J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

Rews, J. H.

J. R. Reeder and J. H. Rews, “Mixed-mode bending method for delamination testing,” AIAA J. 28(7), 1270–1276 (1990).
[Crossref]

Rifaie-Graham, O.

O. Rifaie-Graham, E. A. Apebende, L. K. Bast, and N. Bruns, “Self-Reporting Fiber-Reinforced Composites That Mimic the Ability of Biological Materials to Sense and Report Damage,” Adv. Mater. 30(19), e1705483 (2018).
[Crossref] [PubMed]

Rogers, A. J.

Roths, J.

Russell, S.

C. E. Banks, J. Grant, S. Russell, and S. Arnett, “Strain measurement during stress rupture of composite over-wrapped pressure vessel with fiber Bragg gratings sensors,” Proc. SPIE 6933, 69330O (2008).
[Crossref]

Salathé, R. P.

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathé, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003).
[Crossref]

Santos, J. L.

L. A. Ferreira, F. M. Araujo, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using interferometrically interrogated fiber Bragg grating sensors,” Opt. Eng. 39(8), 2226–2234 (2000).
[Crossref]

Shaw, H. J.

Shu, Y.

Sonnenfeld, C.

Sulejmani, S.

Tan, Y. N.

Tang, T.

S. R. Lavoie, R. Long, and T. Tang, “Rate dependent fracture of a double cantilever beam with combined bulk and interfacial dissipation,” Int. J. Solids Struct. 75, 277–286 (2015).
[Crossref]

Thienpont, H.

Torres, P.

Tu, S. T.

T. Yun and S. T. Tu, “Fabrication and characterization of a metal-packaged regenerated fiber Bragg grating strain sensor for structural integrity monitoring of high-temperature components,” Smart Mater. Struct. 23(3), 035001 (2014).
[Crossref]

Udd, E.

E. Udd, “Review of multi-parameter fiber grating sensors,” Proc. SPIE 6770, 677002 (2007).
[Crossref]

Urbanczyk, W.

Van Hemelrijck, D.

Verleysen, P.

J. Degrieck, W. De Waele, and P. Verleysen, “Monitoring of fiber reinforced composites with embedded optical fibre Bragg sensors, with application to filament wound pressure vessels,” NDT Int. 34(4), 289–296 (2001).
[Crossref]

Voet, E.

G. Luyckx, E. Voet, N. Lammens, W. De Waele, and J. Degrieck, “Residual strain-induced birefringent FBGs for multi-axial strain monitoring of CFRP composite laminates,” NDT Int. 54, 142–150 (2013).
[Crossref]

Wang, M.

Wang, W.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Wang, Y.

Xia, W.

Xie, Z. Y.

Xu, L.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Xu, M. G.

M. G. Xu, J.-L. Archambault, L. Reekie, and J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

Yan, H. Z.

Yoo, S. H.

S. H. Yoo, M. G. Han, J. H. Hong, and S. H. Chang, “Simulation of curing process of carbon epoxy composite during autoclave degassing molding by considering phase changes of epoxy resin,” Compos., Part B Eng. 77, 257–267 (2015).
[Crossref]

Yu, J.

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Yun, T.

T. Yun and S. T. Tu, “Fabrication and characterization of a metal-packaged regenerated fiber Bragg grating strain sensor for structural integrity monitoring of high-temperature components,” Smart Mater. Struct. 23(3), 035001 (2014).
[Crossref]

Zhang, H. Y.

Zhang, W.

Zhang, Y.

Zheng, X. P.

Zhou, B. K.

Zhu, F.

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11(3), 240–254 (2005).
[Crossref]

Adv. Mater. (1)

O. Rifaie-Graham, E. A. Apebende, L. K. Bast, and N. Bruns, “Self-Reporting Fiber-Reinforced Composites That Mimic the Ability of Biological Materials to Sense and Report Damage,” Adv. Mater. 30(19), e1705483 (2018).
[Crossref] [PubMed]

AIAA J. (1)

J. R. Reeder and J. H. Rews, “Mixed-mode bending method for delamination testing,” AIAA J. 28(7), 1270–1276 (1990).
[Crossref]

Appl. Opt. (1)

Compos., Part A Appl. Sci. Manuf. (1)

N. Lammens, D. Kinet, K. Chah, G. Luyckx, C. Caucheteur, J. Degrieck, and P. Mégret, “Residual strain monitoring of out-of-autoclave cured parts by use of polarization dependent loss measurements in embedded optical fiber Bragg gratings,” Compos., Part A Appl. Sci. Manuf. 52, 38–44 (2013).
[Crossref]

Compos., Part B Eng. (1)

S. H. Yoo, M. G. Han, J. H. Hong, and S. H. Chang, “Simulation of curing process of carbon epoxy composite during autoclave degassing molding by considering phase changes of epoxy resin,” Compos., Part B Eng. 77, 257–267 (2015).
[Crossref]

Electron. Lett. (1)

M. G. Xu, J.-L. Archambault, L. Reekie, and J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensors,” Electron. Lett. 30(13), 1085–1087 (1994).
[Crossref]

IEEE Photonics Technol. Lett. (1)

G. Chen, L. Liu, H. Jia, J. Yu, L. Xu, and W. Wang, “Simultaneous strain and temperature measurements with fiber Bragg grating written in novel Hi-Bi optical fiber,” IEEE Photonics Technol. Lett. 16(1), 221–223 (2004).
[Crossref]

Int. J. Solids Struct. (1)

S. R. Lavoie, R. Long, and T. Tang, “Rate dependent fracture of a double cantilever beam with combined bulk and interfacial dissipation,” Int. J. Solids Struct. 75, 277–286 (2015).
[Crossref]

J. Compos. Mater. (1)

P. P. Camanho, C. G. Davila, and M. F. De Moura, “Numerical simulation of mixed-mode progressive delamination in composite materials,” J. Compos. Mater. 37(16), 1415–1438 (2003).
[Crossref]

J. Lightwave Technol. (3)

NDT Int. (2)

G. Luyckx, E. Voet, N. Lammens, W. De Waele, and J. Degrieck, “Residual strain-induced birefringent FBGs for multi-axial strain monitoring of CFRP composite laminates,” NDT Int. 54, 142–150 (2013).
[Crossref]

J. Degrieck, W. De Waele, and P. Verleysen, “Monitoring of fiber reinforced composites with embedded optical fibre Bragg sensors, with application to filament wound pressure vessels,” NDT Int. 34(4), 289–296 (2001).
[Crossref]

Opt. Eng. (1)

L. A. Ferreira, F. M. Araujo, J. L. Santos, and F. Farahi, “Simultaneous measurement of strain and temperature using interferometrically interrogated fiber Bragg grating sensors,” Opt. Eng. 39(8), 2226–2234 (2000).
[Crossref]

Opt. Express (3)

Opt. Fiber Technol. (1)

R. Guan, F. Zhu, Z. Gan, D. Huang, and S. Liu, “Stress birefringence analysis of polarization maintaining optical fibers,” Opt. Fiber Technol. 11(3), 240–254 (2005).
[Crossref]

Opt. Lett. (2)

Photon. Res. (1)

PLoS One (1)

G. F. Pereira, L. P. Mikkelsen, and M. McGugan, “Crack detection in fibre reinforced plastic structures using embedded fibre bragg grating sensors: Theory, model development and experimental validation,” PLoS One 10(10), e0141495 (2015).
[Crossref] [PubMed]

Polym. Test. (1)

G. Pereira, C. Frias, H. Faria, O. Frazão, and A. T. Marques, “On the improvement of strain measurements with FBG sensors embedded in unidirectional composites,” Polym. Test. 32(1), 99–105 (2013).
[Crossref]

Proc. SPIE (2)

C. E. Banks, J. Grant, S. Russell, and S. Arnett, “Strain measurement during stress rupture of composite over-wrapped pressure vessel with fiber Bragg gratings sensors,” Proc. SPIE 6933, 69330O (2008).
[Crossref]

E. Udd, “Review of multi-parameter fiber grating sensors,” Proc. SPIE 6770, 677002 (2007).
[Crossref]

Smart Mater. Struct. (2)

F. Bosia, P. Giaccari, J. Botsis, M. Facchini, H. G. Limberger, and R. P. Salathé, “Characterization of the response of fibre Bragg grating sensors subjected to a two-dimensional strain field,” Smart Mater. Struct. 12(6), 925–934 (2003).
[Crossref]

T. Yun and S. T. Tu, “Fabrication and characterization of a metal-packaged regenerated fiber Bragg grating strain sensor for structural integrity monitoring of high-temperature components,” Smart Mater. Struct. 23(3), 035001 (2014).
[Crossref]

SoftwareX (1)

G. Pereira, M. McGugan, and L. P. Mikkelsen, “FBG SiMul V1. 0: Fibre Bragg grating signal simulation tool for finite element method models,” SoftwareX 5, 163–170 (2016).
[Crossref]

Other (3)

G. Pereira, L. Mikkelsen, and M. McGugan, “Crack growth monitoring by embedded optical Fibre Bragg Grating sensors: Fibre reinforced plastic crack growing detection,” 2015 International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS) (IEEE, 2015), pp. 133–139.
[Crossref]

M. Lindner, B. Hopf, A. W. Koch, and J. Roths, “Three dimensional FEM model of FBGs in PANDA fibers with experimentally determined model parameters,” Optical Fiber Sensors Conference (OFS) (IEEE, 2017), pp. 1–4.

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed. New York: McGraw-Hill, 1970.

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Figures (6)

Fig. 1
Fig. 1 (a) Configuration of a PPM-FBG embedded in a CLS, and θ is the angle between x-axis and the slow axis of the PPM-FBG; (b) The cross section of the PPM-FBG, and an external force L applied on the top of the CLS with an angle of α.
Fig. 2
Fig. 2 The wavelengths shift and rotation angle of principal axes of the PPM-FBG under different external forces L when α = 90°. The cross-section of the embedded PPM-FBG is shown as inset. (a) θ = 0°; (b) θ = 45°; (c) θ = 90°; (d) θ = 135°. The inset picture in Fig. (c) shows the magnified image of Δ λ x ' .The solid lines fit to the data.
Fig. 3
Fig. 3 The wavelengths shift and rotation angle of principal axes of the PPM-FBG under different L when α = 0°. The cross-section of the embedded PPM-FBG is shown as insets. (a) θ = 0°; (b) θ = 45°; (c) θ = 90°; (d) θ = 135°. The solid lines are fit to the data.
Fig. 4
Fig. 4 The difference between Δ λ x ' and Δ λ y ' under different θ and L. (a) α = 90°; (b) α = 0°.
Fig. 5
Fig. 5 The wavelengths shift and rotation angle of the principal axes of the PPM-FBG under different α with the stress L = 5 × 105 Pa. The cross-section of the embedded PPM-FBG is shown as inset. (a) θ = 0°; (b) θ = 45°; (c) θ = 90°; (d) θ = 135°. The solid line shows a sinusoidal fit.
Fig. 6
Fig. 6 (a) The difference between Δ λ x ' and Δ λ y ' dependence of the angle of α under different θ; (b) The direction of the new principal strain ϕ with the changes of α. The solid lines fit to the data.

Tables (2)

Tables Icon

Table 1 Parameters of the PPM-FBG

Tables Icon

Table 2 The strain sensitivities of the PPM-FBG when α = 90° and α = 0°

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

B=( C 2 C 1 )( σ 0,max σ 0,min )= n 0,F n 0,S .
( ε x ε y )= R 1 1 ( θ )( ε 0,F ε 0,S ) R 1 ( θ )+ R 2 1 ( ϕ )( ε L,x ε L,x ) R 2 ( ϕ ),
ε x,y ' = ε x + ε y 2 ± ( ε x ε y 2 ) 2 + ( γ xy 2 ) 2 ,
ϕ=tan [ γ xy / ( ε x ε y ) ] -1 .
( Δ λ x ' Δ λ y ' )=( n 0,1 2 p 11 n 0,1 2 p 12 1 n 0,1 2 p 12 n 0,2 2 p 12 n 0,2 2 p 11 1 n 0,2 2 p 12 )( ε x ' ε y ' ε z ' ),
Δ λ x ' =1.42L0.577 Δ λ y ' =7.32L0.595.
ϕ={ 15.874× e L 109481 87.068 ( θ= 45 ) 15.873× e L 109482 +87.068 ( θ= 135 ) .
Δ λ x ' =4.28L0.5685 Δ λ y ' =4.28L0.6012.
ϕ={ 21.116× e L 111647 +47.242 ( θ= 0 ) 21.199× e L 111705 +42.749 ( θ= 45 ) .