Abstract

A theoretical analysis of strain transfer of six-layer surface-bonded fiber Bragg gratings (FBGs) subjected to uniform axial stress is presented. The proposed six-layer structure consists of optical fiber, protective coating, adhesive layer, substrate layer, outer adhesive layer, and host material, which is different from the four-layer case of common acknowledgement. A theoretical formula of strain transfer rate from host material to optical fiber is established to provide an accurate theoretical prediction. On the basis of the theoretical analysis, influence parameters of the middle layers that affect the average strain transfer rate of the six-layer surface-bonded FBG are discussed. After the parametric study, a selection scheme of sensor parameters for numerical validation, which makes the average strain transfer rate approach unity, is determined. Good agreement is observed between numerical results and theoretical predictions. In the end, the six-layer model is extended to the general situation of multiple substrate layers, which lays a theoretical groundwork for the research and design of surface-bonded FBGs with substrate layers in the future.

© 2012 Optical Society of America

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  1. J. Gomez, J. Zubia, G. Aranguren, J. Arrue, H. Poisel, and I. Saez, “Comparing polymer optical fiber, fiber Bragg grating, and traditional strain gauge for aircraft structural health monitoring,” Appl. Opt. 48, 1436–1443 (2009).
    [CrossRef]
  2. C. Cheng, Y. Lo, and W. Li, “Accurate simulations of reflective wavelength spectrum of surface-bonded fiber Bragg grating,” Appl. Opt. 49, 3394–3402 (2010).
    [CrossRef]
  3. E. Udd, “Optical fiber smart structures,” Proc. IEEE 84, 884–894 (1996).
    [CrossRef]
  4. C. Galiotis, R. J. Young, P. H. J. Yeund, and D. N. Batchelder, “The study of model polydiacetylene/epoxy composites. Part 1: the axial strain in the fibre,” J. Mater. Sci. 19, 3640–3648 (1984).
    [CrossRef]
  5. A. Nanni, C. C. Yang, K. Pan, J. Wang, and R. R. Michael, “Fiber-optic sensor for concrete strain-stress measurement,” ACI Mater. J. 88, 257–264 (1991).
  6. Y. E. Pak, “Longitudinal shear transfer in fiber optic sensors,” Smart Mater. Struct. 1, 57–62 (1992).
    [CrossRef]
  7. F. Ansari and Y. Libo, “Mechanics of bond and interface shear transfer in optical fiber sensors,” J. Eng. Mech. 124, 385–394 (1998).
    [CrossRef]
  8. K. T. Lau, L. M. Yuan, L. Zhou, J. Wu, and C. H. Woo, “Strain monitoring in FRP laminates and concrete beams using FBG sensors,” Composite Struct. 51, 9–20 (2001).
    [CrossRef]
  9. Q. Li, G. Li, G. Wang, F. Ansari, and Q. Liu, “Elasto-plastic bonding of embedded optical fiber sensors in concrete,” J. Eng. Mech. 128, 471–478 (2002).
    [CrossRef]
  10. D. S. Li and H. N. Li, “Strain transferring analysis of embedded fiber Bragg grating sensors,” J. Theor. Appl. Mech. 37, 435–441 (2005).
  11. J. Zhou, Z. Zhou, and D. Zhang, “Study on strain transfer characteristics of fiber Bragg grating sensors,” J. Intell. Mater. Syst. Struct. 21, 1117–1122 (2010).
    [CrossRef]
  12. K. T. Wan, C. K. Y. Leung, and N. G. Olson, “Investigation of the strain transfer for surface-attached optical fiber strain sensors,” Smart Mater. Struct. 17, 035037 (2008).
    [CrossRef]
  13. W. Y. Li, C. C. Cheng, and Y. L. Lo, “Investigation of strain transmission of surface-bonded FBGs used as strain sensors,” Sens. Actuators A 149, 201–207 (2009).
    [CrossRef]
  14. S. C. Her and C. Y. Huang, “Effect of coating on strain transfer of optical fiber sensors,” Sensors 11, 6926–6941(2011).
    [CrossRef]
  15. D. S. Li, H. N. Li, L. Ren, and G. B. Song, “Strain transferring analysis of fiber Bragg grating sensors,” Opt. Eng. 45, 024402 (2006).
    [CrossRef]

2011 (1)

S. C. Her and C. Y. Huang, “Effect of coating on strain transfer of optical fiber sensors,” Sensors 11, 6926–6941(2011).
[CrossRef]

2010 (2)

J. Zhou, Z. Zhou, and D. Zhang, “Study on strain transfer characteristics of fiber Bragg grating sensors,” J. Intell. Mater. Syst. Struct. 21, 1117–1122 (2010).
[CrossRef]

C. Cheng, Y. Lo, and W. Li, “Accurate simulations of reflective wavelength spectrum of surface-bonded fiber Bragg grating,” Appl. Opt. 49, 3394–3402 (2010).
[CrossRef]

2009 (2)

2008 (1)

K. T. Wan, C. K. Y. Leung, and N. G. Olson, “Investigation of the strain transfer for surface-attached optical fiber strain sensors,” Smart Mater. Struct. 17, 035037 (2008).
[CrossRef]

2006 (1)

D. S. Li, H. N. Li, L. Ren, and G. B. Song, “Strain transferring analysis of fiber Bragg grating sensors,” Opt. Eng. 45, 024402 (2006).
[CrossRef]

2005 (1)

D. S. Li and H. N. Li, “Strain transferring analysis of embedded fiber Bragg grating sensors,” J. Theor. Appl. Mech. 37, 435–441 (2005).

2002 (1)

Q. Li, G. Li, G. Wang, F. Ansari, and Q. Liu, “Elasto-plastic bonding of embedded optical fiber sensors in concrete,” J. Eng. Mech. 128, 471–478 (2002).
[CrossRef]

2001 (1)

K. T. Lau, L. M. Yuan, L. Zhou, J. Wu, and C. H. Woo, “Strain monitoring in FRP laminates and concrete beams using FBG sensors,” Composite Struct. 51, 9–20 (2001).
[CrossRef]

1998 (1)

F. Ansari and Y. Libo, “Mechanics of bond and interface shear transfer in optical fiber sensors,” J. Eng. Mech. 124, 385–394 (1998).
[CrossRef]

1996 (1)

E. Udd, “Optical fiber smart structures,” Proc. IEEE 84, 884–894 (1996).
[CrossRef]

1992 (1)

Y. E. Pak, “Longitudinal shear transfer in fiber optic sensors,” Smart Mater. Struct. 1, 57–62 (1992).
[CrossRef]

1991 (1)

A. Nanni, C. C. Yang, K. Pan, J. Wang, and R. R. Michael, “Fiber-optic sensor for concrete strain-stress measurement,” ACI Mater. J. 88, 257–264 (1991).

1984 (1)

C. Galiotis, R. J. Young, P. H. J. Yeund, and D. N. Batchelder, “The study of model polydiacetylene/epoxy composites. Part 1: the axial strain in the fibre,” J. Mater. Sci. 19, 3640–3648 (1984).
[CrossRef]

Ansari, F.

Q. Li, G. Li, G. Wang, F. Ansari, and Q. Liu, “Elasto-plastic bonding of embedded optical fiber sensors in concrete,” J. Eng. Mech. 128, 471–478 (2002).
[CrossRef]

F. Ansari and Y. Libo, “Mechanics of bond and interface shear transfer in optical fiber sensors,” J. Eng. Mech. 124, 385–394 (1998).
[CrossRef]

Aranguren, G.

Arrue, J.

Batchelder, D. N.

C. Galiotis, R. J. Young, P. H. J. Yeund, and D. N. Batchelder, “The study of model polydiacetylene/epoxy composites. Part 1: the axial strain in the fibre,” J. Mater. Sci. 19, 3640–3648 (1984).
[CrossRef]

Cheng, C.

Cheng, C. C.

W. Y. Li, C. C. Cheng, and Y. L. Lo, “Investigation of strain transmission of surface-bonded FBGs used as strain sensors,” Sens. Actuators A 149, 201–207 (2009).
[CrossRef]

Galiotis, C.

C. Galiotis, R. J. Young, P. H. J. Yeund, and D. N. Batchelder, “The study of model polydiacetylene/epoxy composites. Part 1: the axial strain in the fibre,” J. Mater. Sci. 19, 3640–3648 (1984).
[CrossRef]

Gomez, J.

Her, S. C.

S. C. Her and C. Y. Huang, “Effect of coating on strain transfer of optical fiber sensors,” Sensors 11, 6926–6941(2011).
[CrossRef]

Huang, C. Y.

S. C. Her and C. Y. Huang, “Effect of coating on strain transfer of optical fiber sensors,” Sensors 11, 6926–6941(2011).
[CrossRef]

Lau, K. T.

K. T. Lau, L. M. Yuan, L. Zhou, J. Wu, and C. H. Woo, “Strain monitoring in FRP laminates and concrete beams using FBG sensors,” Composite Struct. 51, 9–20 (2001).
[CrossRef]

Leung, C. K. Y.

K. T. Wan, C. K. Y. Leung, and N. G. Olson, “Investigation of the strain transfer for surface-attached optical fiber strain sensors,” Smart Mater. Struct. 17, 035037 (2008).
[CrossRef]

Li, D. S.

D. S. Li, H. N. Li, L. Ren, and G. B. Song, “Strain transferring analysis of fiber Bragg grating sensors,” Opt. Eng. 45, 024402 (2006).
[CrossRef]

D. S. Li and H. N. Li, “Strain transferring analysis of embedded fiber Bragg grating sensors,” J. Theor. Appl. Mech. 37, 435–441 (2005).

Li, G.

Q. Li, G. Li, G. Wang, F. Ansari, and Q. Liu, “Elasto-plastic bonding of embedded optical fiber sensors in concrete,” J. Eng. Mech. 128, 471–478 (2002).
[CrossRef]

Li, H. N.

D. S. Li, H. N. Li, L. Ren, and G. B. Song, “Strain transferring analysis of fiber Bragg grating sensors,” Opt. Eng. 45, 024402 (2006).
[CrossRef]

D. S. Li and H. N. Li, “Strain transferring analysis of embedded fiber Bragg grating sensors,” J. Theor. Appl. Mech. 37, 435–441 (2005).

Li, Q.

Q. Li, G. Li, G. Wang, F. Ansari, and Q. Liu, “Elasto-plastic bonding of embedded optical fiber sensors in concrete,” J. Eng. Mech. 128, 471–478 (2002).
[CrossRef]

Li, W.

Li, W. Y.

W. Y. Li, C. C. Cheng, and Y. L. Lo, “Investigation of strain transmission of surface-bonded FBGs used as strain sensors,” Sens. Actuators A 149, 201–207 (2009).
[CrossRef]

Libo, Y.

F. Ansari and Y. Libo, “Mechanics of bond and interface shear transfer in optical fiber sensors,” J. Eng. Mech. 124, 385–394 (1998).
[CrossRef]

Liu, Q.

Q. Li, G. Li, G. Wang, F. Ansari, and Q. Liu, “Elasto-plastic bonding of embedded optical fiber sensors in concrete,” J. Eng. Mech. 128, 471–478 (2002).
[CrossRef]

Lo, Y.

Lo, Y. L.

W. Y. Li, C. C. Cheng, and Y. L. Lo, “Investigation of strain transmission of surface-bonded FBGs used as strain sensors,” Sens. Actuators A 149, 201–207 (2009).
[CrossRef]

Michael, R. R.

A. Nanni, C. C. Yang, K. Pan, J. Wang, and R. R. Michael, “Fiber-optic sensor for concrete strain-stress measurement,” ACI Mater. J. 88, 257–264 (1991).

Nanni, A.

A. Nanni, C. C. Yang, K. Pan, J. Wang, and R. R. Michael, “Fiber-optic sensor for concrete strain-stress measurement,” ACI Mater. J. 88, 257–264 (1991).

Olson, N. G.

K. T. Wan, C. K. Y. Leung, and N. G. Olson, “Investigation of the strain transfer for surface-attached optical fiber strain sensors,” Smart Mater. Struct. 17, 035037 (2008).
[CrossRef]

Pak, Y. E.

Y. E. Pak, “Longitudinal shear transfer in fiber optic sensors,” Smart Mater. Struct. 1, 57–62 (1992).
[CrossRef]

Pan, K.

A. Nanni, C. C. Yang, K. Pan, J. Wang, and R. R. Michael, “Fiber-optic sensor for concrete strain-stress measurement,” ACI Mater. J. 88, 257–264 (1991).

Poisel, H.

Ren, L.

D. S. Li, H. N. Li, L. Ren, and G. B. Song, “Strain transferring analysis of fiber Bragg grating sensors,” Opt. Eng. 45, 024402 (2006).
[CrossRef]

Saez, I.

Song, G. B.

D. S. Li, H. N. Li, L. Ren, and G. B. Song, “Strain transferring analysis of fiber Bragg grating sensors,” Opt. Eng. 45, 024402 (2006).
[CrossRef]

Udd, E.

E. Udd, “Optical fiber smart structures,” Proc. IEEE 84, 884–894 (1996).
[CrossRef]

Wan, K. T.

K. T. Wan, C. K. Y. Leung, and N. G. Olson, “Investigation of the strain transfer for surface-attached optical fiber strain sensors,” Smart Mater. Struct. 17, 035037 (2008).
[CrossRef]

Wang, G.

Q. Li, G. Li, G. Wang, F. Ansari, and Q. Liu, “Elasto-plastic bonding of embedded optical fiber sensors in concrete,” J. Eng. Mech. 128, 471–478 (2002).
[CrossRef]

Wang, J.

A. Nanni, C. C. Yang, K. Pan, J. Wang, and R. R. Michael, “Fiber-optic sensor for concrete strain-stress measurement,” ACI Mater. J. 88, 257–264 (1991).

Woo, C. H.

K. T. Lau, L. M. Yuan, L. Zhou, J. Wu, and C. H. Woo, “Strain monitoring in FRP laminates and concrete beams using FBG sensors,” Composite Struct. 51, 9–20 (2001).
[CrossRef]

Wu, J.

K. T. Lau, L. M. Yuan, L. Zhou, J. Wu, and C. H. Woo, “Strain monitoring in FRP laminates and concrete beams using FBG sensors,” Composite Struct. 51, 9–20 (2001).
[CrossRef]

Yang, C. C.

A. Nanni, C. C. Yang, K. Pan, J. Wang, and R. R. Michael, “Fiber-optic sensor for concrete strain-stress measurement,” ACI Mater. J. 88, 257–264 (1991).

Yeund, P. H. J.

C. Galiotis, R. J. Young, P. H. J. Yeund, and D. N. Batchelder, “The study of model polydiacetylene/epoxy composites. Part 1: the axial strain in the fibre,” J. Mater. Sci. 19, 3640–3648 (1984).
[CrossRef]

Young, R. J.

C. Galiotis, R. J. Young, P. H. J. Yeund, and D. N. Batchelder, “The study of model polydiacetylene/epoxy composites. Part 1: the axial strain in the fibre,” J. Mater. Sci. 19, 3640–3648 (1984).
[CrossRef]

Yuan, L. M.

K. T. Lau, L. M. Yuan, L. Zhou, J. Wu, and C. H. Woo, “Strain monitoring in FRP laminates and concrete beams using FBG sensors,” Composite Struct. 51, 9–20 (2001).
[CrossRef]

Zhang, D.

J. Zhou, Z. Zhou, and D. Zhang, “Study on strain transfer characteristics of fiber Bragg grating sensors,” J. Intell. Mater. Syst. Struct. 21, 1117–1122 (2010).
[CrossRef]

Zhou, J.

J. Zhou, Z. Zhou, and D. Zhang, “Study on strain transfer characteristics of fiber Bragg grating sensors,” J. Intell. Mater. Syst. Struct. 21, 1117–1122 (2010).
[CrossRef]

Zhou, L.

K. T. Lau, L. M. Yuan, L. Zhou, J. Wu, and C. H. Woo, “Strain monitoring in FRP laminates and concrete beams using FBG sensors,” Composite Struct. 51, 9–20 (2001).
[CrossRef]

Zhou, Z.

J. Zhou, Z. Zhou, and D. Zhang, “Study on strain transfer characteristics of fiber Bragg grating sensors,” J. Intell. Mater. Syst. Struct. 21, 1117–1122 (2010).
[CrossRef]

Zubia, J.

ACI Mater. J. (1)

A. Nanni, C. C. Yang, K. Pan, J. Wang, and R. R. Michael, “Fiber-optic sensor for concrete strain-stress measurement,” ACI Mater. J. 88, 257–264 (1991).

Appl. Opt. (2)

Composite Struct. (1)

K. T. Lau, L. M. Yuan, L. Zhou, J. Wu, and C. H. Woo, “Strain monitoring in FRP laminates and concrete beams using FBG sensors,” Composite Struct. 51, 9–20 (2001).
[CrossRef]

J. Eng. Mech. (2)

Q. Li, G. Li, G. Wang, F. Ansari, and Q. Liu, “Elasto-plastic bonding of embedded optical fiber sensors in concrete,” J. Eng. Mech. 128, 471–478 (2002).
[CrossRef]

F. Ansari and Y. Libo, “Mechanics of bond and interface shear transfer in optical fiber sensors,” J. Eng. Mech. 124, 385–394 (1998).
[CrossRef]

J. Intell. Mater. Syst. Struct. (1)

J. Zhou, Z. Zhou, and D. Zhang, “Study on strain transfer characteristics of fiber Bragg grating sensors,” J. Intell. Mater. Syst. Struct. 21, 1117–1122 (2010).
[CrossRef]

J. Mater. Sci. (1)

C. Galiotis, R. J. Young, P. H. J. Yeund, and D. N. Batchelder, “The study of model polydiacetylene/epoxy composites. Part 1: the axial strain in the fibre,” J. Mater. Sci. 19, 3640–3648 (1984).
[CrossRef]

J. Theor. Appl. Mech. (1)

D. S. Li and H. N. Li, “Strain transferring analysis of embedded fiber Bragg grating sensors,” J. Theor. Appl. Mech. 37, 435–441 (2005).

Opt. Eng. (1)

D. S. Li, H. N. Li, L. Ren, and G. B. Song, “Strain transferring analysis of fiber Bragg grating sensors,” Opt. Eng. 45, 024402 (2006).
[CrossRef]

Proc. IEEE (1)

E. Udd, “Optical fiber smart structures,” Proc. IEEE 84, 884–894 (1996).
[CrossRef]

Sens. Actuators A (1)

W. Y. Li, C. C. Cheng, and Y. L. Lo, “Investigation of strain transmission of surface-bonded FBGs used as strain sensors,” Sens. Actuators A 149, 201–207 (2009).
[CrossRef]

Sensors (1)

S. C. Her and C. Y. Huang, “Effect of coating on strain transfer of optical fiber sensors,” Sensors 11, 6926–6941(2011).
[CrossRef]

Smart Mater. Struct. (2)

K. T. Wan, C. K. Y. Leung, and N. G. Olson, “Investigation of the strain transfer for surface-attached optical fiber strain sensors,” Smart Mater. Struct. 17, 035037 (2008).
[CrossRef]

Y. E. Pak, “Longitudinal shear transfer in fiber optic sensors,” Smart Mater. Struct. 1, 57–62 (1992).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

(a) Cross section of a six-layer surface-bonded FBG; (b) longitudinal section of a six-layer surface-bonded FBG; (c) stress distribution of a six-layer surface-bonded FBG.

Fig. 2.
Fig. 2.

(a) Influence of the half bonding length Lf with the width of adhesive layer Da varying from 0.3 mm to 1.2 mm and the assigned parameter values ra=0.14mm, Ra=0.06mm, Eb=4×109Pa, λb=0.34, Et=0.1×109Pa, rt=0.17mm; (b) influence of the bottom thickness of adhesive layer rarb with the top thickness of adhesive layer Ra varying from 0.02 mm to 0.10 mm and the assigned parameter values Eb=4×109Pa, λb=0.3.4, Et=0.1×109Pa, rt=0.14mm, Da=0.6mm, Lf=25mm; (c) influence of the thickness of substrate layer rtra with the Young’s modulus Et varying from 0.05×109Pa to 0.5×109Pa and the assigned parameter values Ra=0.06mm, Eb=4×109Pa, λb=0.34, Da=0.6mm, Lf=25mm; (d) influence of the Young’s modulus of outer adhesive layer Eb with its Poisson’s ratio λb varying from 0.2 to 0.4 and the assigned parameter values Ra=0.06mm, rt=0.17mm, Et=0.1×109Pa, ra=0.14mm, Da=0.6mm, Lf=25mm.

Fig. 3.
Fig. 3.

Finite element meshes.

Fig. 4.
Fig. 4.

Comparison of the distribution of normal strain along the optical fiber achieved by FEM and Eq. (19).

Fig. 5.
Fig. 5.

Stress distribution of a multi-substrate-layer surface-bonded FBG.

Tables (1)

Tables Icon

Table 1. Known Quantities of FBG Sensors

Equations (26)

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πrf2σf=πrf2(σf+dσf)+2πrfdxτ(x,rf).
π(rp2rf2)σp+2πrfdxτ(x,rf)=π(rp2rf2)(σp+dσp)+2πrpdxτ(x,rp),
(DaRa+Daraπrp2)σa+2πrpdxτ(x,rp)=(DaRa+Daraπrp2)(σa+dσa)+τ(x,ra)Dadx,
Wt(rtra)σt+τ(x,ra)Wtdx=Wt(rtra)(σt+dσt)+τ(x,rt)Wtdx,
Wt(rrt)σb+τ(x,rt)Wtdx=Wt(rrt)(σb+dσb)+τ(x,r)Wtdx.
τ(x,r)=2πrf2Dadσfdx+π(rf2rp2)Dadσpdx(Raraπrp2Da)dσadx(rtra)dσtdx(rrt)dσbdx.
τ(x,r)=2πrf2EfDadεfdx+π(rf2rp2)EpDadεpdx(Raraπrp2Da)Eadεadx(rtra)Etdεtdx(rrt)Ebdεbdx=Ef[2πrf2Dadεfdx+π(rf2rp2)EpDaEfdεpdx(Raraπrp2Da)EaEfdεadx(rtra)EtEfdεtdx(rrt)EbEfdεbdx].
dεpdxdεadxdεtdxdεbdxdεfdx.
τ(x,r)=2πrf2Efπ(rf2rp2)Ep+[Da(Rara)πrp2)]Ea+(rtra)DaEt+(rrt)DaEbDadεfdx.
τ(x,r)=Gbγ(x,r)=Gb(ur+wx)Gbdudr.
rfrm(Gbdudr)dr=rfrm[2πrf2Efπ(rf2rp2)Ep+[Da(Rara)πrp2)]Ea+(rtra)DaEt+(rrt)DaEbDadεfdx]dr,
umuf=(rmrf){2πrf2Efπ(rf2rp2)Ep+[Da(Rara)πrp2]Ea+(rtra)DaEt+(12rm+12rt)DaEb}Dadεfdx,=1k2dεfdx.
k=DaGb(rmrf){2πrf2Efπ(rf2rp2)Ep+[Da(Rara)πrp2]Ea+(rtra)DaEt+(12rm+12rfrt)DaEb},
d2εf(x)dx2k2εf(x)=k2εm.
εf(x)=c1ekx+c2ekx+εm,
εf(Lf)=εf(Lf)=0.
c1=c2=εm2cosh(kLf).
εf(x)=εm[1cosh(kx)cosh(kLf)].
α(x)=εf(x)εm=1cosh(kx)cosh(kLf).
αm(0)=εf(0)εm=11cosh(kLf).
α¯=εf(x)¯εm=20Lfεf(x)dx2Lfεm=1sinh(kLf)kLfcosh(kLf).
τ(x,r)=2πrf2Efπ(rf2rp2)Ep+[Da(Rara)πrp2)]Ea+(rt1ra)DaEt1+(ra1rt1)DaEa1++(rtnra(n1))DaEtn+(rrtn)DaEanDadεfdx,
rfrm(Gandudr)dr=rfrm1Da{2πrf2Efπ(rf2rp2)Ep+[Da(Rara)πrp2]Ea+(rt1ra)DaEt1+(ra1rt1)DaEa1++(rtnra(n1))DaEtn+(rrtn)DaEan}dεfdxdr.
[rfrt1+rt1ra1+ra1rt2++rtnrm(Gandudr)]dr=[rfrt1+rt1ra1+ra1rt2++rtnrm(1Da{2πrf2Efπ(rf2rp2)Ep+[Da(Rara)πrp2]Ea+(rt1ra)DaEt1+(ra1rt1)DaEa1++(rtnra(n1))DaEtn+(rrtn)DaEan}dεfdx)]dr.
umutn=(rmrfDaGan{2πrf2Efπ(rf2rp2)Ep+[Da(Rara)πrp2]Ea+(rt1ra)DaEt1+(ra1rt1)DaEa1++(rtnra(n1))DaEtn}+Da(rtnrf12rtn212rf2)Ean)dεfdx=rmrfDaGan{2πrf2Efπ(rf2rp2)Ep+[Da(Rara)πrp2]Ea+Dai=1n(rtira(i1))Eti+Dai=1n1(rairti)Eai+Da(rtnrf12rtn212rf2)Ean}dεfdx=1K2dεfdx.
K=DaGan(rmrf){2πrf2Efπ(rf2rp2)Ep+[Da(Rar)πrp2]Ea+Dai=1n(rtira(i1))Eti+Dai=1n1(rairti)Eai+Da(rtnrf12rtn212rf2)Ean}.

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