Abstract

Measurements of birefringence induced in fused-silica specimens by a crack produced by a 351-nm/500-ps Nd:glass laser as a function of laser fluence F L and of number of laser shots N are presented. The varying dimensional parameter is found to be the crack depth a and can be put in the form a(mm) = (0.0096 ± 0.0021)N[(F L/F exit/th) - 1]2/3 with F LF exit/th(F exit/th is the exit-surface damage threshold). The retardance data are converted into units of stress, thus permitting the estimation of residual stress near the crack. The results of the measured residual stress can be cast in the form σr(MPa) ≈ (0.0386 ± 0.0051)[(F L/F exit/th) - 1]1/2 N2/3 with F LF exit/th. A theoretical model giving the stress field around a crack is developed for comparison and shows reasonable agreement with the experiment. Good agreement with experimental data of others is also obtained. The effect of residual stresses on fracture strength is pointed out. The results obtained show that the presence of birefringence/residual stress in a fused-silica specimen with a crack on its surface has a strong effect on fracture and should be taken into account in any formulation that involves the failure strength of optical components used in inertial-confinement-fusion experiments.

© 1998 Optical Society of America

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    [CrossRef]
  3. Y. Z. Li, M. P. Harmer, Y. T. Chou, “Fracture behavior of fused quartz with laser-induced internal flaws,” J. Mater. Res. 9, 1780–1788 (1994).
    [CrossRef]
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    [CrossRef]

1996 (1)

J. C. Lambropoulos, S. Xu, T. Fang, “Constitutive law for the densification of fused silica, with applications in polishing and microgrinding,” J. Am. Ceram. Soc. 79, 1441–1452 (1996).
[CrossRef]

1995 (1)

P. J. Dwivedi, D. J. Green, “Determination of subcritical crack growth parameters by in situ observation of indentation cracks,” J. Am. Ceram. Soc. 78, 2122–2128 (1995).
[CrossRef]

1994 (4)

Y. Z. Li, M. P. Harmer, Y. T. Chou, “Fracture behavior of fused quartz with laser-induced internal flaws,” J. Mater. Res. 9, 1780–1788 (1994).
[CrossRef]

K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

J. E. Logan, N. A. Robertson, J. Hough, “Measurements of birefringence in a suspended sample of fused silica,” Opt. Commun. 107, 342–346 (1994).
[CrossRef]

D. Albagli, M. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

1988 (1)

1987 (1)

C. Meade, R. Jeanloz, “Frequency-dependent equation of state of fused silica to 10 GPa,” Phys. Rev. B 35, 236–244 (1987).
[CrossRef]

1985 (1)

D. H. Roach, A. R. Cooper, “Effect of contact residual stress relaxation of fracture strength of indented soda-lime glass,” J. Am. Ceram. Soc. 68, 632–636 (1985).
[CrossRef]

1982 (1)

K. Tanaka, T. Mura, “A theory of fatigue crack initiation at inclusions,” Metall. Trans. A 13A, 117–123 (1982).

1981 (1)

J. C. Newman, I. S. Raju, “An empirical stress-intensity factor equation for the surface crack,” Eng. Fract. Mech. 15, 185–192 (1981).
[CrossRef]

1977 (1)

M. Malin, K. Vedam, “Ellipsometric studies of environment-sensitive polish layers of glass,” J. Appl. Phys. 48, 1155–1157 (1977).
[CrossRef]

1974 (1)

N. Ingelstrom, H. Nordberg, “The fracture toughness of cemented tungsten carbides,” Eng. Fract. Mech. 6, 597–607 (1974).
[CrossRef]

1972 (1)

M. D. Crisp, N. L. Boling, G. Dubé, “Importance of Fresnel reflections in laser surface damage of transparent dielectrics,” Appl. Phys. Lett. 21, 364–366 (1972).
[CrossRef]

1969 (1)

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, “Ellipsometric study of polished glass surfaces,” Surf. Sci. 16, 265–274 (1969).
[CrossRef]

1965 (1)

H. M. Cohen, R. Roy, “Densification of glass at very high pressure,” Phys. Chem. Glasses 6, 149–161 (1965).

1953 (1)

P. W. Bridgman, I. Simon, “Effects of very high pressures on glass,” J. Appl. Phys. 24, 405–413 (1953).
[CrossRef]

Ai, C.

Albagli, D.

D. Albagli, M. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

D. Albagli, J. A. Izatt, G. B. Hayes, B. Banish, G. S. Janes, I. Itzkan, M. Feld, “Time dependence of laser-induced surface breakdown in fused silica at 355 nm in the nanosecond regime,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 146–153 (1991).
[CrossRef]

Banish, B.

D. Albagli, J. A. Izatt, G. B. Hayes, B. Banish, G. S. Janes, I. Itzkan, M. Feld, “Time dependence of laser-induced surface breakdown in fused silica at 355 nm in the nanosecond regime,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 146–153 (1991).
[CrossRef]

Basuray, A.

K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Bhattacharya, K.

K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Boling, N. L.

M. D. Crisp, N. L. Boling, G. Dubé, “Importance of Fresnel reflections in laser surface damage of transparent dielectrics,” Appl. Phys. Lett. 21, 364–366 (1972).
[CrossRef]

Bridgman, P. W.

P. W. Bridgman, I. Simon, “Effects of very high pressures on glass,” J. Appl. Phys. 24, 405–413 (1953).
[CrossRef]

Brimacombe, R. K.

Broek, D.

D. Broek, Elementary Engineering Fracture Mechanics, 3rd rev. ed. (Nijhoff, The Hague, 1982), Chap. 3.
[CrossRef]

Bumpas, S. E.

J. H. Campbell, P. A. Hurst, D. D. Heggins, W. A. Steele, S. E. Bumpas, “Laser-induced damage and fracture in fused silica vacuum windows,” in Laser-Induced Damage in Optical Materials: 1996, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, M. J. Soileau, eds. (Proc. SPIE2966, 106–125 (1997).
[CrossRef]

Burns, S.

F. Dahmani, J. C. Lambropoulos, A. W. Schmid, S. Burns, S. Papernov, “Fracture of fused silica with 351-nm-laser-generated surface cracks,” J. Mater. Res. (to be published).

Campbell, J. H.

J. H. Campbell, P. A. Hurst, D. D. Heggins, W. A. Steele, S. E. Bumpas, “Laser-induced damage and fracture in fused silica vacuum windows,” in Laser-Induced Damage in Optical Materials: 1996, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, M. J. Soileau, eds. (Proc. SPIE2966, 106–125 (1997).
[CrossRef]

J. H. Campbell, G. J. Edwards, J. E. Marion, “Damage and fracture in large aperture, fused silica, vacuum spatial filter lenses,” in Solid State Lasers for Application to Inertial Confinement Fusion, M. André, H. T. Powell, eds., Proc. SPIE2633, 522–534 (1995).
[CrossRef]

Chakraborty, A. K.

K. Bhattacharya, A. Basuray, A. K. Chakraborty, “Photoelastic testing using a birefringence-sensitive interferometer,” Opt. Commun. 109, 380–386 (1994).
[CrossRef]

Chou, Y. T.

Y. Z. Li, M. P. Harmer, Y. T. Chou, “Fracture behavior of fused quartz with laser-induced internal flaws,” J. Mater. Res. 9, 1780–1788 (1994).
[CrossRef]

Cochran, E. R.

Cohen, H. M.

H. M. Cohen, R. Roy, “Densification of glass at very high pressure,” Phys. Chem. Glasses 6, 149–161 (1965).

Cooper, A. R.

D. H. Roach, A. R. Cooper, “Effect of contact residual stress relaxation of fracture strength of indented soda-lime glass,” J. Am. Ceram. Soc. 68, 632–636 (1985).
[CrossRef]

Crisp, M. D.

M. D. Crisp, N. L. Boling, G. Dubé, “Importance of Fresnel reflections in laser surface damage of transparent dielectrics,” Appl. Phys. Lett. 21, 364–366 (1972).
[CrossRef]

M. D. Crisp, “Some aspects of surface damage that can be explained with linear optics,” in Laser Induced Damage in Optical Materials: 1973, Natl. Bur. Stand. (U.S.), Spec. Publ.387, 80–83 (1973).

Dahmani, F.

F. Dahmani, J. C. Lambropoulos, A. W. Schmid, S. Burns, S. Papernov, “Fracture of fused silica with 351-nm-laser-generated surface cracks,” J. Mater. Res. (to be published).

Dark, M.

Dubé, G.

M. D. Crisp, N. L. Boling, G. Dubé, “Importance of Fresnel reflections in laser surface damage of transparent dielectrics,” Appl. Phys. Lett. 21, 364–366 (1972).
[CrossRef]

Dwivedi, P. J.

P. J. Dwivedi, D. J. Green, “Determination of subcritical crack growth parameters by in situ observation of indentation cracks,” J. Am. Ceram. Soc. 78, 2122–2128 (1995).
[CrossRef]

Edwards, G. J.

J. H. Campbell, G. J. Edwards, J. E. Marion, “Damage and fracture in large aperture, fused silica, vacuum spatial filter lenses,” in Solid State Lasers for Application to Inertial Confinement Fusion, M. André, H. T. Powell, eds., Proc. SPIE2633, 522–534 (1995).
[CrossRef]

Eshelby, J. D.

J. D. Eshelby, “Elastic inclusions and inhomogeneities,” in Progress in Solid Mechanics, I. N. Sneddon, R. Hill, eds. (North-Holland, Amsterdam, 1961), Vol. II, pp. 89–140.

Fang, T.

J. C. Lambropoulos, S. Xu, T. Fang, “Constitutive law for the densification of fused silica, with applications in polishing and microgrinding,” J. Am. Ceram. Soc. 79, 1441–1452 (1996).
[CrossRef]

Feld, M.

D. Albagli, J. A. Izatt, G. B. Hayes, B. Banish, G. S. Janes, I. Itzkan, M. Feld, “Time dependence of laser-induced surface breakdown in fused silica at 355 nm in the nanosecond regime,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 146–153 (1991).
[CrossRef]

Feld, M. S.

Graff, K. F.

K. F. Graff, Wave Motion in Elastic Solids (Ohio State U. Press, Columbus, Ohio, 1975).

Green, D. J.

P. J. Dwivedi, D. J. Green, “Determination of subcritical crack growth parameters by in situ observation of indentation cracks,” J. Am. Ceram. Soc. 78, 2122–2128 (1995).
[CrossRef]

Harmer, M. P.

Y. Z. Li, M. P. Harmer, Y. T. Chou, “Fracture behavior of fused quartz with laser-induced internal flaws,” J. Mater. Res. 9, 1780–1788 (1994).
[CrossRef]

Hayden, J. E.

Hayes, G. B.

D. Albagli, J. A. Izatt, G. B. Hayes, B. Banish, G. S. Janes, I. Itzkan, M. Feld, “Time dependence of laser-induced surface breakdown in fused silica at 355 nm in the nanosecond regime,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 146–153 (1991).
[CrossRef]

Heggins, D. D.

J. H. Campbell, P. A. Hurst, D. D. Heggins, W. A. Steele, S. E. Bumpas, “Laser-induced damage and fracture in fused silica vacuum windows,” in Laser-Induced Damage in Optical Materials: 1996, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, M. J. Soileau, eds. (Proc. SPIE2966, 106–125 (1997).
[CrossRef]

Hough, J.

J. E. Logan, N. A. Robertson, J. Hough, “Measurements of birefringence in a suspended sample of fused silica,” Opt. Commun. 107, 342–346 (1994).
[CrossRef]

Hurst, P. A.

J. H. Campbell, P. A. Hurst, D. D. Heggins, W. A. Steele, S. E. Bumpas, “Laser-induced damage and fracture in fused silica vacuum windows,” in Laser-Induced Damage in Optical Materials: 1996, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, M. J. Soileau, eds. (Proc. SPIE2966, 106–125 (1997).
[CrossRef]

Ingelstrom, N.

N. Ingelstrom, H. Nordberg, “The fracture toughness of cemented tungsten carbides,” Eng. Fract. Mech. 6, 597–607 (1974).
[CrossRef]

Itzkan, I.

D. Albagli, M. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

D. Albagli, J. A. Izatt, G. B. Hayes, B. Banish, G. S. Janes, I. Itzkan, M. Feld, “Time dependence of laser-induced surface breakdown in fused silica at 355 nm in the nanosecond regime,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 146–153 (1991).
[CrossRef]

Izatt, J. A.

D. Albagli, J. A. Izatt, G. B. Hayes, B. Banish, G. S. Janes, I. Itzkan, M. Feld, “Time dependence of laser-induced surface breakdown in fused silica at 355 nm in the nanosecond regime,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 146–153 (1991).
[CrossRef]

Jacobs, S. D.

Janes, G. S.

D. Albagli, J. A. Izatt, G. B. Hayes, B. Banish, G. S. Janes, I. Itzkan, M. Feld, “Time dependence of laser-induced surface breakdown in fused silica at 355 nm in the nanosecond regime,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1441, 146–153 (1991).
[CrossRef]

Jeanloz, R.

C. Meade, R. Jeanloz, “Frequency-dependent equation of state of fused silica to 10 GPa,” Phys. Rev. B 35, 236–244 (1987).
[CrossRef]

Kinosita, K.

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, “Ellipsometric study of polished glass surfaces,” Surf. Sci. 16, 265–274 (1969).
[CrossRef]

Kuske, A.

A. Kuske, G. Robertson, Photoelastic Stress Analysis (Wiley, London, 1974).

Lambropoulos, J. C.

J. C. Lambropoulos, S. Xu, T. Fang, “Constitutive law for the densification of fused silica, with applications in polishing and microgrinding,” J. Am. Ceram. Soc. 79, 1441–1452 (1996).
[CrossRef]

F. Dahmani, J. C. Lambropoulos, A. W. Schmid, S. Burns, S. Papernov, “Fracture of fused silica with 351-nm-laser-generated surface cracks,” J. Mater. Res. (to be published).

Lawn, B.

B. Lawn, Fracture of Brittle Solids, 2nd ed., Cambridge Solid State Science Series, (Cambridge U. Press, Cambridge, UK, 1993), Chap. 2.
[CrossRef]

Leopold, K. E.

Li, Y. Z.

Y. Z. Li, M. P. Harmer, Y. T. Chou, “Fracture behavior of fused quartz with laser-induced internal flaws,” J. Mater. Res. 9, 1780–1788 (1994).
[CrossRef]

Logan, J. E.

J. E. Logan, N. A. Robertson, J. Hough, “Measurements of birefringence in a suspended sample of fused silica,” Opt. Commun. 107, 342–346 (1994).
[CrossRef]

Malin, M.

M. Malin, K. Vedam, “Ellipsometric studies of environment-sensitive polish layers of glass,” J. Appl. Phys. 48, 1155–1157 (1977).
[CrossRef]

Marion, J. E.

J. H. Campbell, G. J. Edwards, J. E. Marion, “Damage and fracture in large aperture, fused silica, vacuum spatial filter lenses,” in Solid State Lasers for Application to Inertial Confinement Fusion, M. André, H. T. Powell, eds., Proc. SPIE2633, 522–534 (1995).
[CrossRef]

Meade, C.

C. Meade, R. Jeanloz, “Frequency-dependent equation of state of fused silica to 10 GPa,” Phys. Rev. B 35, 236–244 (1987).
[CrossRef]

Mihailov, S.

Mura, T.

K. Tanaka, T. Mura, “A theory of fatigue crack initiation at inclusions,” Metall. Trans. A 13A, 117–123 (1982).

T. Mura, Micromechanics of Defects in Solids, 2nd rev. ed. (Nijhoff, Dordrecht, The Netherlands, 1987).
[CrossRef]

Newman, J. C.

J. C. Newman, I. S. Raju, “An empirical stress-intensity factor equation for the surface crack,” Eng. Fract. Mech. 15, 185–192 (1981).
[CrossRef]

Nishibori, M.

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, “Ellipsometric study of polished glass surfaces,” Surf. Sci. 16, 265–274 (1969).
[CrossRef]

Nordberg, H.

N. Ingelstrom, H. Nordberg, “The fracture toughness of cemented tungsten carbides,” Eng. Fract. Mech. 6, 597–607 (1974).
[CrossRef]

Papernov, S.

F. Dahmani, J. C. Lambropoulos, A. W. Schmid, S. Burns, S. Papernov, “Fracture of fused silica with 351-nm-laser-generated surface cracks,” J. Mater. Res. (to be published).

Perelman, L. T.

Raju, I. S.

J. C. Newman, I. S. Raju, “An empirical stress-intensity factor equation for the surface crack,” Eng. Fract. Mech. 15, 185–192 (1981).
[CrossRef]

Roach, D. H.

D. H. Roach, A. R. Cooper, “Effect of contact residual stress relaxation of fracture strength of indented soda-lime glass,” J. Am. Ceram. Soc. 68, 632–636 (1985).
[CrossRef]

Robertson, G.

A. Kuske, G. Robertson, Photoelastic Stress Analysis (Wiley, London, 1974).

Robertson, N. A.

J. E. Logan, N. A. Robertson, J. Hough, “Measurements of birefringence in a suspended sample of fused silica,” Opt. Commun. 107, 342–346 (1994).
[CrossRef]

Roy, R.

H. M. Cohen, R. Roy, “Densification of glass at very high pressure,” Phys. Chem. Glasses 6, 149–161 (1965).

Sakata, H.

H. Yokota, H. Sakata, M. Nishibori, K. Kinosita, “Ellipsometric study of polished glass surfaces,” Surf. Sci. 16, 265–274 (1969).
[CrossRef]

Schmid, A. W.

F. Dahmani, J. C. Lambropoulos, A. W. Schmid, S. Burns, S. Papernov, “Fracture of fused silica with 351-nm-laser-generated surface cracks,” J. Mater. Res. (to be published).

Simon, I.

P. W. Bridgman, I. Simon, “Effects of very high pressures on glass,” J. Appl. Phys. 24, 405–413 (1953).
[CrossRef]

Steele, W. A.

J. H. Campbell, P. A. Hurst, D. D. Heggins, W. A. Steele, S. E. Bumpas, “Laser-induced damage and fracture in fused silica vacuum windows,” in Laser-Induced Damage in Optical Materials: 1996, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, M. J. Soileau, eds. (Proc. SPIE2966, 106–125 (1997).
[CrossRef]

Tanaka, K.

K. Tanaka, T. Mura, “A theory of fatigue crack initiation at inclusions,” Metall. Trans. A 13A, 117–123 (1982).

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Figures (19)

Fig. 1
Fig. 1

Index ellipsoid for a point in a uniaxial stress field.

Fig. 2
Fig. 2

Experimental setup for measuring birefringence with a Soleil compensator (SC). The linearly polarized light from the He–Ne laser is focused by a lens (L 1) onto the fused-silica sample. The second lens (L 2) is used to diverge the beam on the photodetector (photodiode). The polarizer (P) is oriented perpendicular to the input polarization for maximum extinction in the absence of the sample and the compensator.

Fig. 3
Fig. 3

Cross-sectional view of a laser-induced crack after 520 laser shots at laser fluence of 1.8 × F exit/th. Threshold for exit-surface damage was F exit/th = 10 J cm-2.

Fig. 4
Fig. 4

Top view of cracks produced with N = 50 at (a) F L = 3.3 × F exit/th and (b) F L = 3.7 × F exit/th. The crack dimensions are for (a) c = 0.49 mm, a = 0.47 mm, and (b) c = 0.54 mm, a = 0.91 mm, respectively.

Fig. 5
Fig. 5

Crack depth versus the number of laser shots at laser-fluence levels of F L = 2.2 × F exit/th and 1.8 × F exit/th, respectively.

Fig. 6
Fig. 6

Birefringence [≥0 filled circles); <0 (open circles)] around a laser-induced crack of dimensions a = 1.66 mm and c = 0.49 mm produced with N = 100 laser shots at a laser-fluence level of F L = 2.2 × F exit/th. Filled and open circles indicate the approximate position of the input He–Ne light (spot diameter = 1 mm) (±0.5 mm) in x and y directions.

Fig. 7
Fig. 7

Residual stress around cracks versus the number of laser shots N at laser-fluence levels of F L = 1.8 × F exit/th and F L = 2.2 × F exit/th, respectively. Solid curves are Eq. (6).

Fig. 8
Fig. 8

Residual stress around cracks versus the laser fluence F L for N = 50 laser shots. Solid curve is Eq. (6).

Fig. 9
Fig. 9

Schematic view of laser-induced crack in bending.

Fig. 10
Fig. 10

Failure strength of laser-induced cracks in fused silica at laser fluences F L = 1.8 × F exit/th and F L = 2.2 × F exit/th as a function of crack depth a. Data points represent crack-depth measurements from cross-sectional views.

Fig. 11
Fig. 11

Superposition method for calculating the stress field of material with inhomogeneous inclusion (laser-induced crack). The stress (a) is equal to the sum of the stress field (b) at x = ∞ of the flawless specimen, denoted by σin, and to the stress (c) at infinity around a void with identical shape (as the crack) under remote applied pure shear -σin.

Fig. 12
Fig. 12

Circular hole with the same dimensions as the inclusion shown in Fig. 11(c), under pure shear -σin.

Fig. 13
Fig. 13

Comparison between the stresses (in unit of retardance) around a laser-induced crack in fused silica, determined photoelastically (birefringence measurements), and those calculated from the inclusion model [Eq. (20)] for a crack of dimensions 2c = 0.98 mm and a = 1.66 mm, produced by N = 100 laser shots at laser fluence of F L = 2.2 × F exit/th. Here σ r = 15.6 nm.

Fig. 14
Fig. 14

Comparison between the stresses (in unit of retardance) around a laser-induced crack in fused silica, determined photoelastically (birefringence measurements), and those calculated from the inclusion model [Eq. (20)] for a crack of dimensions 2c = 1.18 mm and a = 2.14 mm, produced by N = 200 laser shots at laser fluence of F L = 2.2 × F exit/th. Here σ r = 22 nm.

Fig. 15
Fig. 15

Comparison between the stresses (in unit of retardance) around a laser-induced crack in fused silica, determined photoelastically (birefringence measurements), and those calculated from the inclusion model [Eq. (20)] for a crack of dimensions 2c = 0.98 mm and a = 0.47 mm, produced by N = 50 laser shots at laser fluence of F L = 3.3 × F exit/th. Here σ r = 17.5 nm.

Fig. 16
Fig. 16

Comparison between the stresses (in unit of retardance) around a laser-induced crack in fused silica, determined photoelastically (birefringence measurements), and those calculated from the inclusion model [Eq. (20)] for a crack of dimensions 2c = 1.08 mm and a = 0.91 mm, produced by N = 50 laser shots at laser fluence of F L = 3.7 × F exit/th. Here σ r = 19.7 nm.

Fig. 17
Fig. 17

Comparison between residual stresses from Ref. 29 and the present work as a function of the geometrical factor Y. Data from Ref. 29 are for soda-lime glass and are obtained from several specimens. Data from our work (fused silica) represent many specimens irradiated at different laser fluences and numbers of laser shots. The Y in the current work is taken as Y = M b / Q b .

Fig. 18
Fig. 18

Fracture strength and retardance as a function of crack depth a for laser-induced cracks produced by different numbers of laser shots at laser fluences of (a) F L = 1.8 × F exit/th and (b) F L = 2.2 × F exit/th, respectively.

Fig. 19
Fig. 19

Fracture strength versus retardation with incorporation of results from all samples included in this work.

Equations (24)

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n x = n 0 + C 1 σ x + C 2 σ y + σ z , n y = n 0 + C 1 σ y + C 2 σ x + σ z , n z = n 0 + C 1 σ z + C 2 σ x + σ y ,
B = n y - n x n z - n y n x - n z = C σ x - σ y σ y - σ z σ z - σ x
n y = n z ,
n x - n y = C σ x .
Γ = n y - n x t = C σ x t ,
σ r MPa = 1 C nm / cm / MPa × Γ nm t cm .
a mm 0.0096 ± 0.0021 × N F L F exit / th - 1 2 / 3 , F L > F exit / th .
σ r MPa 0.0386 ± 0.0051 × F L F exit / th - 1 1 / 2 N 2 / 3 ,
F L F exit / th 1 .
K IC = σ F M b π a / Q b 1 / 2 ,
M b = 1.12274 - 1.39355 a t + 7.3 a t 2 - 13.0723 a t 3 + 13.992 a t 4 ,
ϕ = 0 π / 2 1 - c 2 - a 2 c 2 sin 2   θ 1 / 2 d θ
ϕ = π 8 3 + c 2 a 2 ,   for   1 < a / c <   2 ;
ϕ = 1.56 + 0.37 a / c 1.4 ,   a / c > 2 .
σ rr = - σ in 1 + 3 c 4 r 4 - 4 c 2 r 2 sin   2 θ ,
σ θ θ = σ in 1 + 3 c 4 r 4 sin   2 θ ,
σ r θ = - σ in 1 - 3 c 4 r 4 + 2 c 2 r 2 cos   2 θ ,
σ rr σ θ θ σ r θ = cos 2   θ sin 2   θ sin   2 θ sin 2   θ cos 2   θ - sin   2 θ - sin   2 θ 2 sin   2 θ 2 cos   2 θ M × σ xx σ yy τ xy .
σ xx σ yy σ xy = cos 2   θ sin 2   θ - sin   2 θ sin 2   θ cos 2   θ sin   2 θ sin   2 θ 2 -   sin   2 θ 2 cos   2 θ σ rr σ θ θ τ r θ .
σ xy = σ rr - σ θ θ sin   2 θ 2   + τ r θ   cos   2 θ .
σ xy = - σ in 2 xy r 2 2 + 6   c 4 r 4 - 4   c 2 r 2 + x 2 - y 2 r 2 2 × 1 - 3   c 4 r 4 + 2   c 2 r 2 .
τ xy = σ in 1 - 2 xy r 2 2 2 + 6   c 4 r 4 - 4   c 2 r 2 + x 2 - y 2 r 2 2 × 1 - 3   c 4 r 4 + 2   c 2 r 2 .
τ xy = - 3 σ r 1 - 2 xy r 2 2 2 + 6   c 4 r 4 - 4   c 2 r 2 + x 2 - y 2 r 2 2 1 - 3   c 4 r 4 + 2   c 2 r 2 .
σ r = K r / 2 π r 1 / 2 ,

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