Abstract

We evaluate the effect of the nonlinear stress–strain relationship on elastic stability, free vibrations, and bending of optical glass fibers. The analysis is carried out under an assumption that this relationship, obtained for the case of uniaxial tension is also valid in the case of compression, and is applicable to bending deformations as well. We examine low-temperature microbending of infinitely long dual-coated fibers, elastic stability of short bare fibers, free vibrations of long fused portions of light-wave couplers that are subjected to uniaxial tension, and bending deformations of optical fibers that experience large deflections. We conclude that the nonlinear stress–strain relationship in silica materials can have a significant effect on the mechanical behavior of optical fibers and that, since the experimental data were obtained for tensile strains not exceeding 5%, future experimental research should include evaluation of the nonlinear stress–strain relationship, both in tension and compression, for higher strains and for high-strength fibers (such as, for instance, fibers protected by metallic coatings).

© 1992 Optical Society of America

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References

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  1. F. P. Mallinder, B. A. Proctor, “Elastic constants of fused silica as a function of large tensile strain,” Phys. Chem. Glasses 5, 91–103 (1964).
  2. J. T. Krause, L. R. Testardi, R. N. Thurston, “Deviations from linearity in the dependence of elongation upon force for fibers of simple glass formers and of glass optical lightguides,” Phys. Chem. Glasses 20, 135–139 (1979).
  3. G. S. Glaesemann, S. T. Gulati, J. D. Helfinstine, “Effect of strain and surface composition on Young’s modulus of optical fibers,” in Optical Fiber Communications Conference, Vol. 1 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TUG5.
  4. D. Gloge, “Bending loss in multimode fibers,” Appl. Opt. 11, 2506–2513 (1972).
    [CrossRef] [PubMed]
  5. W. B. Gardner, “Microbending loss in optical fibers,” Bell Syst. Tech. J. 54, 457–465 (1975).
  6. E. Suhir, “Effect of initial curvature on low temperature microbending in optical fibers,” IEEE J. Lightwave Technol. 6, 1321–1327 (1988).
    [CrossRef]
  7. Y. Katsuyama, Y. Mitsunaga, Y. Ishida, K. Ishihara, “Transmission loss of coated single-mode fibers at low temperatures,” Appl. Opt. 19, 4200–4205 (1980).
    [CrossRef] [PubMed]
  8. E. Suhir, “Spring constant in the buckling of dual-coated optical fibers,” IEEE J. Lightwave Technol. 6, 1240–1244 (1988).
    [CrossRef]
  9. E. Suhir, “Stresses in dual-coated optical fibers,” J. Appl. Mech. 55, 822–830 (1988).
    [CrossRef]
  10. S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability (McGraw-Hill, New York, 1961).
  11. E. Suhir, “Structural analysis in microelectronic and fiber-optic systems,” Basic Principles of Engineering Elasticity and Fundamentals of Structural Analysis. (Van Nostrand, New York, 1991), Vol. 1.
  12. C. R. Kurkjian, AT&T Bell Laboratories, Murray Hill, N.J. 07974–0636 (personal communication).
  13. J. B. Murgatroyd, “The strength of glass fibers,” J. Soc. Glass Technol. 28, 388–405 (1944).
  14. P. W. France, M. J. Paradine, M. H. Reevee, G. R. Newns, “Liquid nitrogen strength of coated optical glass fibers,” J. Mater. Sci. 15, 825–830 (1980).
    [CrossRef]
  15. S. F. Cowap, S. D. Brown, “Static fatigue testing of a hermetically sealed optical fiber,” Am. Ceram. Soc. Bull. 63, 495–500 (1984).
  16. M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
    [CrossRef]

1988 (3)

E. Suhir, “Effect of initial curvature on low temperature microbending in optical fibers,” IEEE J. Lightwave Technol. 6, 1321–1327 (1988).
[CrossRef]

E. Suhir, “Spring constant in the buckling of dual-coated optical fibers,” IEEE J. Lightwave Technol. 6, 1240–1244 (1988).
[CrossRef]

E. Suhir, “Stresses in dual-coated optical fibers,” J. Appl. Mech. 55, 822–830 (1988).
[CrossRef]

1986 (1)

M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
[CrossRef]

1984 (1)

S. F. Cowap, S. D. Brown, “Static fatigue testing of a hermetically sealed optical fiber,” Am. Ceram. Soc. Bull. 63, 495–500 (1984).

1980 (2)

P. W. France, M. J. Paradine, M. H. Reevee, G. R. Newns, “Liquid nitrogen strength of coated optical glass fibers,” J. Mater. Sci. 15, 825–830 (1980).
[CrossRef]

Y. Katsuyama, Y. Mitsunaga, Y. Ishida, K. Ishihara, “Transmission loss of coated single-mode fibers at low temperatures,” Appl. Opt. 19, 4200–4205 (1980).
[CrossRef] [PubMed]

1979 (1)

J. T. Krause, L. R. Testardi, R. N. Thurston, “Deviations from linearity in the dependence of elongation upon force for fibers of simple glass formers and of glass optical lightguides,” Phys. Chem. Glasses 20, 135–139 (1979).

1975 (1)

W. B. Gardner, “Microbending loss in optical fibers,” Bell Syst. Tech. J. 54, 457–465 (1975).

1972 (1)

1964 (1)

F. P. Mallinder, B. A. Proctor, “Elastic constants of fused silica as a function of large tensile strain,” Phys. Chem. Glasses 5, 91–103 (1964).

1944 (1)

J. B. Murgatroyd, “The strength of glass fibers,” J. Soc. Glass Technol. 28, 388–405 (1944).

Brown, S. D.

S. F. Cowap, S. D. Brown, “Static fatigue testing of a hermetically sealed optical fiber,” Am. Ceram. Soc. Bull. 63, 495–500 (1984).

Cowap, S. F.

S. F. Cowap, S. D. Brown, “Static fatigue testing of a hermetically sealed optical fiber,” Am. Ceram. Soc. Bull. 63, 495–500 (1984).

France, P. W.

P. W. France, M. J. Paradine, M. H. Reevee, G. R. Newns, “Liquid nitrogen strength of coated optical glass fibers,” J. Mater. Sci. 15, 825–830 (1980).
[CrossRef]

Gardner, W. B.

W. B. Gardner, “Microbending loss in optical fibers,” Bell Syst. Tech. J. 54, 457–465 (1975).

Gere, J. M.

S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability (McGraw-Hill, New York, 1961).

Glaesemann, G. S.

G. S. Glaesemann, S. T. Gulati, J. D. Helfinstine, “Effect of strain and surface composition on Young’s modulus of optical fibers,” in Optical Fiber Communications Conference, Vol. 1 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TUG5.

Gloge, D.

Gulati, S. T.

M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
[CrossRef]

G. S. Glaesemann, S. T. Gulati, J. D. Helfinstine, “Effect of strain and surface composition on Young’s modulus of optical fibers,” in Optical Fiber Communications Conference, Vol. 1 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TUG5.

Helfinstine, J. D.

G. S. Glaesemann, S. T. Gulati, J. D. Helfinstine, “Effect of strain and surface composition on Young’s modulus of optical fibers,” in Optical Fiber Communications Conference, Vol. 1 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TUG5.

Ishida, Y.

Ishihara, K.

Katsuyama, Y.

Krause, J. T.

J. T. Krause, L. R. Testardi, R. N. Thurston, “Deviations from linearity in the dependence of elongation upon force for fibers of simple glass formers and of glass optical lightguides,” Phys. Chem. Glasses 20, 135–139 (1979).

Kurkjian, C. R.

M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
[CrossRef]

C. R. Kurkjian, AT&T Bell Laboratories, Murray Hill, N.J. 07974–0636 (personal communication).

Mallinder, F. P.

F. P. Mallinder, B. A. Proctor, “Elastic constants of fused silica as a function of large tensile strain,” Phys. Chem. Glasses 5, 91–103 (1964).

Matthewson, M. J.

M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
[CrossRef]

Mitsunaga, Y.

Murgatroyd, J. B.

J. B. Murgatroyd, “The strength of glass fibers,” J. Soc. Glass Technol. 28, 388–405 (1944).

Newns, G. R.

P. W. France, M. J. Paradine, M. H. Reevee, G. R. Newns, “Liquid nitrogen strength of coated optical glass fibers,” J. Mater. Sci. 15, 825–830 (1980).
[CrossRef]

Paradine, M. J.

P. W. France, M. J. Paradine, M. H. Reevee, G. R. Newns, “Liquid nitrogen strength of coated optical glass fibers,” J. Mater. Sci. 15, 825–830 (1980).
[CrossRef]

Proctor, B. A.

F. P. Mallinder, B. A. Proctor, “Elastic constants of fused silica as a function of large tensile strain,” Phys. Chem. Glasses 5, 91–103 (1964).

Reevee, M. H.

P. W. France, M. J. Paradine, M. H. Reevee, G. R. Newns, “Liquid nitrogen strength of coated optical glass fibers,” J. Mater. Sci. 15, 825–830 (1980).
[CrossRef]

Suhir, E.

E. Suhir, “Effect of initial curvature on low temperature microbending in optical fibers,” IEEE J. Lightwave Technol. 6, 1321–1327 (1988).
[CrossRef]

E. Suhir, “Spring constant in the buckling of dual-coated optical fibers,” IEEE J. Lightwave Technol. 6, 1240–1244 (1988).
[CrossRef]

E. Suhir, “Stresses in dual-coated optical fibers,” J. Appl. Mech. 55, 822–830 (1988).
[CrossRef]

E. Suhir, “Structural analysis in microelectronic and fiber-optic systems,” Basic Principles of Engineering Elasticity and Fundamentals of Structural Analysis. (Van Nostrand, New York, 1991), Vol. 1.

Testardi, L. R.

J. T. Krause, L. R. Testardi, R. N. Thurston, “Deviations from linearity in the dependence of elongation upon force for fibers of simple glass formers and of glass optical lightguides,” Phys. Chem. Glasses 20, 135–139 (1979).

Thurston, R. N.

J. T. Krause, L. R. Testardi, R. N. Thurston, “Deviations from linearity in the dependence of elongation upon force for fibers of simple glass formers and of glass optical lightguides,” Phys. Chem. Glasses 20, 135–139 (1979).

Timoshenko, S. P.

S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability (McGraw-Hill, New York, 1961).

Am. Ceram. Soc. Bull. (1)

S. F. Cowap, S. D. Brown, “Static fatigue testing of a hermetically sealed optical fiber,” Am. Ceram. Soc. Bull. 63, 495–500 (1984).

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

W. B. Gardner, “Microbending loss in optical fibers,” Bell Syst. Tech. J. 54, 457–465 (1975).

IEEE J. Lightwave Technol. (2)

E. Suhir, “Effect of initial curvature on low temperature microbending in optical fibers,” IEEE J. Lightwave Technol. 6, 1321–1327 (1988).
[CrossRef]

E. Suhir, “Spring constant in the buckling of dual-coated optical fibers,” IEEE J. Lightwave Technol. 6, 1240–1244 (1988).
[CrossRef]

J. Am. Ceram. Soc. (1)

M. J. Matthewson, C. R. Kurkjian, S. T. Gulati, “Strength measurement of optical fibers by bending,” J. Am. Ceram. Soc. 69, 815–821 (1986).
[CrossRef]

J. Appl. Mech. (1)

E. Suhir, “Stresses in dual-coated optical fibers,” J. Appl. Mech. 55, 822–830 (1988).
[CrossRef]

J. Mater. Sci. (1)

P. W. France, M. J. Paradine, M. H. Reevee, G. R. Newns, “Liquid nitrogen strength of coated optical glass fibers,” J. Mater. Sci. 15, 825–830 (1980).
[CrossRef]

J. Soc. Glass Technol. (1)

J. B. Murgatroyd, “The strength of glass fibers,” J. Soc. Glass Technol. 28, 388–405 (1944).

Phys. Chem. Glasses (2)

F. P. Mallinder, B. A. Proctor, “Elastic constants of fused silica as a function of large tensile strain,” Phys. Chem. Glasses 5, 91–103 (1964).

J. T. Krause, L. R. Testardi, R. N. Thurston, “Deviations from linearity in the dependence of elongation upon force for fibers of simple glass formers and of glass optical lightguides,” Phys. Chem. Glasses 20, 135–139 (1979).

Other (4)

G. S. Glaesemann, S. T. Gulati, J. D. Helfinstine, “Effect of strain and surface composition on Young’s modulus of optical fibers,” in Optical Fiber Communications Conference, Vol. 1 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TUG5.

S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability (McGraw-Hill, New York, 1961).

E. Suhir, “Structural analysis in microelectronic and fiber-optic systems,” Basic Principles of Engineering Elasticity and Fundamentals of Structural Analysis. (Van Nostrand, New York, 1991), Vol. 1.

C. R. Kurkjian, AT&T Bell Laboratories, Murray Hill, N.J. 07974–0636 (personal communication).

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Figures (3)

Fig. 1
Fig. 1

Effect of the nonlinear stress–strain relationship on the critical stress in coated (η1) and bare (η2) fibers.

Fig. 2
Fig. 2

Cross section of a fiber subjected to bending: zc, deviation of the neutral axis from the geometrical center of the fiber cross section.

Fig. 3
Fig. 3

Maximum stress in a fiber subjected to two-point bending versus distance between faceplates (tube diameter): σt, maximum tensile stress; σc, maximum compressive stress; σt0, maximum tensile stress without considering the shift in the centroid; σc0, maximum compressive stress without considering the shift in the centroid, E0, nominal (low strain) Young modulus.

Tables (2)

Tables Icon

Table 1 Stresses and Strains in Optical Fibers Subjected to Two-Point Bending With Consideration of the Nonlinear Stress–Strain Relationship

Tables Icon

Table 2 Ratios of Induced Stresses to the Nominal (Low Strain) Young Modulusa

Equations (45)

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σ = E 0 ( 1 + ½ α ) .
σ = E 0 ( 1 - ½ α ) .
T c = r 0 2 ( π K E ) 1 / 2 .
σ c = ( 1 π K E ) 1 / 2 ,
σ c = T c π r 0 2
E = d σ d = E 0 ( 1 - α ) .
σ c = η 1 σ 0 ,
σ 0 = ( 1 π K E 0 ) 1 / 2
η 1 = ( 1 - α ) 1 / 2
η 1 4 + 2 α ¯ η 1 - 1 = 0 ,
α ¯ = α σ 0 E 0 .
T c = π 2 E I ( μ l ) 2 .
σ c = ( π r 0 2 μ l ) 2 E ,
σ 0 = ( π r 0 2 μ l ) 2 E 0 ,
η 2 = 1 - α
η 2 = ( 1 + α ¯ 2 ) 1 / 2 - α ¯ ,
T 2 w x 2 - m 2 w t 2 = 0.
w = i = 1 θ i ( t ) sin i π x l ,
θ ¨ i + ω i 2 θ i = 0 , i = 1 , 2 ,
ω i = i π l ( T m ) 1 / 2 = i π l ( σ ρ ) 1 / 2 , i = 1 , 2 , ,
ω i = η 3 ω i 0 ,
ω i 0 = i π l ( E 0 ρ ) 1 / 2
η 3 = ( 1 + ½ α ) 1 / 2
A E z 1 d A = 0 ,
d A = [ r 0 2 - ( z 1 + z c ) 2 ] 1 / 2 d z 1 .
= z 1 R ,
E = E 0 ( 1 + α z 1 R ) .
- ( r 0 + z c ) r 0 - z c ( z 1 + α z 1 2 R ) [ r 0 2 - ( z 1 + z c ) 2 ] 1 / 2 d z 1 = 0.
z 1 + z c = r 0     sin ξ .
- z c ( 1 - α R z c ) - π / 2 π / 2 cos 2 ξ d ξ + α R r 0 2 - π / 2 π / 2 sin 2 ξ cos 2 ξ d ξ + r 0 ( 1 - 2 α R z c ) - π / 2 π / 2 sin ξ cos 2 ξ d ξ = 0.
z c 2 - R α z c + r 0 2 4 = 0.
z c = r 0 1 - [ 1 - ( α 0 ) 2 ] 1 / 2 2 α 0 ,
0 = r 0 R
z c = ¼ α 0 r 0 ,
z c r 0 = / 2 3 0 .
E I = A E z 1 2 d A = E 0 - ( r 0 + z c ) r 0 - z c ( z 1 2 + α z 1 3 R ) [ r 0 2 - ( z 1 + z c ) 2 ] 1 / 2 d z 1 = ( 2 E 0 r 0 4 - 3 α E 0 r 0 4 R z c ) - π / 2 π / 2 sin 2 ξ cos 2 ξ d ξ + ( 2 E 0 r 0 2 z c 2 - α E 0 r 0 2 R z c 3 ) - π / 2 π / 2 cos 2 ξ d ξ + α E 0 r 0 5 R - π / 2 π / 2 sin 3 ξ cos 2 ξ d ξ - ( 4 E 0 r 0 3 z c - 3 α E 0 r 0 3 R z c 2 ) - π / 2 π / 2 sin ξ cos 2 ξ d ξ .
E I = η E E 0 I 0 ,
E 0 I 0 = π 4 E 0 r 0 4
η E = 1 - / 2 3 α 0 z c r 0 + 4 ( z c r 0 ) 2 - 2 α 0 ( z c r 0 ) 3
η E = 2 - ( 2 - α 2 0 2 ) ( 1 - α 2 0 2 ) 1 / 2 2 α 2 0 2 .
η E 1 - ( α 0 ) 2 = 1 - / 2 9 0 2 .
t = r 0 - z c R = 0 [ 1 - 1 - ( α 2 0 2 ) 1 / 2 2 α 0 ] , c = r 0 + z c R = 0 [ 1 + 1 - ( 1 - α 2 0 2 ) 1 / 2 2 α 0 ] .
σ t = E 0 t ( 1 + ½ α t ) , σ c = E 0 c ( 1 - ½ α c ) .
R = ½ 1 / 2 2 E ( 1 / 2 ) - K ( 1 / 2 ) D = 0.4173 D .
0 = r 0 R = 2.3964 r 0 D = 2.4 r 0 D .

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