Abstract

A full-field digital gradient sensing method is proposed for measuring small angular deflections of light rays due to local stresses in transparent planar solids. The working principle of the method is explained, and the governing equations are derived. The analysis shows that angular deflections of light rays can be linked to nonuniform changes in thickness and refractive index of the material. In mechanically loaded planar solids, the angular deflections can be further related to spatial gradients of first invariant of stresses under plane stress conditions. The proposed method is first demonstrated by capturing the angular deflection fields in two orthogonal directions for a thin plano-convex lens. The measured contours of constant angular deflection of light rays are in good agreement with the expected ones for a spherical wavefront. The method is also successfully implemented to study a stress concentration problem involving a line load acting on an edge of a large planar sheet. Again, the stress gradients, measured simultaneously along and perpendicular to the loading directions, are in good agreement with the analytical predictions. The measured stress gradients have also been used to estimate stresses in the load point vicinity where plane stress results hold.

© 2012 Optical Society of America

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References

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  1. W. N. Sharpe, ed., Handbook of Experimental Solid Mechanics (Springer, 2008).
  2. N. Shukla and J. W. Dally, Experimental Solid Mechanics(College House, 2010).
  3. M. A. Sutton, U. Orteu, and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements (Springer, 2009).
  4. E. Strassburger, “Ballistic testing of transparent armour ceramics,” J. Eur. Ceram. Soc. 29, 267–273 (2009).
    [CrossRef]
  5. P. Patel, G. A. Gilde, P. G. Dehmer, and J. W. McCauley, “Transparent armor,” AMPTIAC Newsletter 4(3), 1–9 (2000).
  6. S. Iwamoto, A. N. Nakagaito, H. Yano, and M. Nogi, “Optically transparent composites reinforced with plant fiber-based nanofibers,” Appl. Phys. A: Mater. Sci. Process. 81, 1109–1112 (2005).
    [CrossRef]
  7. E. J. A. Pope, M. Asami, and J. D. Mackenzie, “Transparent silica gel-PMMA composites,” J. Mater. Res. 4, 1018–1026 (1989).
    [CrossRef]
  8. S. Ravi, “Development of transparent composite for photoelastic studies,” Adv. Compos. Mater. 7, 73–81 (1998).
  9. H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
    [CrossRef]
  10. H. V. Tippur, “Coherent gradient sensing—a Fourier optics analysis and applications to fracture,” Appl. Opt. 31, 4428–4439 (1992).
    [CrossRef]
  11. H. V. Tippur, “Coherent gradient sensing (CGS) method for fracture mechanics: a review,” Fatigue Fract. Eng. Mater. Struct. 33, 832–858 (2010).
    [CrossRef]
  12. H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing—a study of crack tip K-dominance,” Int. J. Fract. 52, 91–117 (1991).
  13. H. V. Tippur and A. J. Rosakis, “Quasi-static and dynamic crack-growth along bimaterial interfaces—a note on crack-tip field-measurements using coherent gradient sensing,” Exp. Mech. 31, 243–251 (1991).
    [CrossRef]
  14. J. Kimberley and J. Lambros, “Dynamic crack kinking from a PMMA/homalite interface,” Exp. Mech. 44, 158–166 (2004).
    [CrossRef]
  15. J. J. Mason, J. Lambros, and A. J. Rosakis, “The use of a coherent gradient sensor in dynamic mixed-mode fracture-mechanics experiments,” J. Mech. Phys. Solids 40, 641–661 (1992).
    [CrossRef]
  16. S. Ramaswamy, H. V. Tippur, and L. Xu, “Mixed-mode crack-tip deformations studied using a modified flexural specimen and coherent gradient sensing,” Exp. Mech. 33, 218–227 (1993).
    [CrossRef]
  17. J. K. Sinha, H. V. Tippur, and L. M. Xu, “An interferometric and finite element investigation of interfacial crack tip fields: role of mode-mixity on 3-D stress variations,” Int. J. Solids Struct. 34, 741–754 (1997).
    [CrossRef]
  18. X. J. Dai, H. Yun, and Q. Pu, “Measuring thickness change of transparent plate by electronic speckle pattern interferometry and digital image correlation,” Opt. Commun. 283, 3481–3486 (2010).
    [CrossRef]
  19. J. W. Dally and W. F. Riley, Experimental Stress Analysis, 4th ed. (College House, 2005).
  20. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).
  21. M. S. Kirugulige, H. V. Tippur, and T. S. Denney, “Measurement of transient deformations using digital image correlation method and high-speed photography: application to dynamic fracture,” Appl. Opt. 46, 5083–5096 (2007).
    [CrossRef]
  22. M. S. Kirugulige and H. V. Tippur, “Measurement of surface deformations and fracture parameters for a mixed-mode crack driven by stress waves using image correlation technique and high-speed photography,” Strain 45, 108–122(2009).
    [CrossRef]
  23. H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements—analysis and experimental results,” Int. J. Fract. 48, 193–204 (1991).
    [CrossRef]
  24. R. G. Budynas, Advanced Strength and Applied Stress Analysis (McGraw-Hill, 1998).

2010 (2)

H. V. Tippur, “Coherent gradient sensing (CGS) method for fracture mechanics: a review,” Fatigue Fract. Eng. Mater. Struct. 33, 832–858 (2010).
[CrossRef]

X. J. Dai, H. Yun, and Q. Pu, “Measuring thickness change of transparent plate by electronic speckle pattern interferometry and digital image correlation,” Opt. Commun. 283, 3481–3486 (2010).
[CrossRef]

2009 (2)

M. S. Kirugulige and H. V. Tippur, “Measurement of surface deformations and fracture parameters for a mixed-mode crack driven by stress waves using image correlation technique and high-speed photography,” Strain 45, 108–122(2009).
[CrossRef]

E. Strassburger, “Ballistic testing of transparent armour ceramics,” J. Eur. Ceram. Soc. 29, 267–273 (2009).
[CrossRef]

2007 (1)

2005 (2)

S. Iwamoto, A. N. Nakagaito, H. Yano, and M. Nogi, “Optically transparent composites reinforced with plant fiber-based nanofibers,” Appl. Phys. A: Mater. Sci. Process. 81, 1109–1112 (2005).
[CrossRef]

H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
[CrossRef]

2004 (1)

J. Kimberley and J. Lambros, “Dynamic crack kinking from a PMMA/homalite interface,” Exp. Mech. 44, 158–166 (2004).
[CrossRef]

2000 (1)

P. Patel, G. A. Gilde, P. G. Dehmer, and J. W. McCauley, “Transparent armor,” AMPTIAC Newsletter 4(3), 1–9 (2000).

1998 (1)

S. Ravi, “Development of transparent composite for photoelastic studies,” Adv. Compos. Mater. 7, 73–81 (1998).

1997 (1)

J. K. Sinha, H. V. Tippur, and L. M. Xu, “An interferometric and finite element investigation of interfacial crack tip fields: role of mode-mixity on 3-D stress variations,” Int. J. Solids Struct. 34, 741–754 (1997).
[CrossRef]

1993 (1)

S. Ramaswamy, H. V. Tippur, and L. Xu, “Mixed-mode crack-tip deformations studied using a modified flexural specimen and coherent gradient sensing,” Exp. Mech. 33, 218–227 (1993).
[CrossRef]

1992 (2)

J. J. Mason, J. Lambros, and A. J. Rosakis, “The use of a coherent gradient sensor in dynamic mixed-mode fracture-mechanics experiments,” J. Mech. Phys. Solids 40, 641–661 (1992).
[CrossRef]

H. V. Tippur, “Coherent gradient sensing—a Fourier optics analysis and applications to fracture,” Appl. Opt. 31, 4428–4439 (1992).
[CrossRef]

1991 (3)

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing—a study of crack tip K-dominance,” Int. J. Fract. 52, 91–117 (1991).

H. V. Tippur and A. J. Rosakis, “Quasi-static and dynamic crack-growth along bimaterial interfaces—a note on crack-tip field-measurements using coherent gradient sensing,” Exp. Mech. 31, 243–251 (1991).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements—analysis and experimental results,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

1989 (1)

E. J. A. Pope, M. Asami, and J. D. Mackenzie, “Transparent silica gel-PMMA composites,” J. Mater. Res. 4, 1018–1026 (1989).
[CrossRef]

Asami, M.

E. J. A. Pope, M. Asami, and J. D. Mackenzie, “Transparent silica gel-PMMA composites,” J. Mater. Res. 4, 1018–1026 (1989).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

Budynas, R. G.

R. G. Budynas, Advanced Strength and Applied Stress Analysis (McGraw-Hill, 1998).

Dai, X. J.

X. J. Dai, H. Yun, and Q. Pu, “Measuring thickness change of transparent plate by electronic speckle pattern interferometry and digital image correlation,” Opt. Commun. 283, 3481–3486 (2010).
[CrossRef]

Dally, J. W.

J. W. Dally and W. F. Riley, Experimental Stress Analysis, 4th ed. (College House, 2005).

N. Shukla and J. W. Dally, Experimental Solid Mechanics(College House, 2010).

Dehmer, P. G.

P. Patel, G. A. Gilde, P. G. Dehmer, and J. W. McCauley, “Transparent armor,” AMPTIAC Newsletter 4(3), 1–9 (2000).

Denney, T. S.

Gilde, G. A.

P. Patel, G. A. Gilde, P. G. Dehmer, and J. W. McCauley, “Transparent armor,” AMPTIAC Newsletter 4(3), 1–9 (2000).

Handa, K.

H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
[CrossRef]

Hikita, M.

H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
[CrossRef]

Iwamoto, S.

S. Iwamoto, A. N. Nakagaito, H. Yano, and M. Nogi, “Optically transparent composites reinforced with plant fiber-based nanofibers,” Appl. Phys. A: Mater. Sci. Process. 81, 1109–1112 (2005).
[CrossRef]

Kimberley, J.

J. Kimberley and J. Lambros, “Dynamic crack kinking from a PMMA/homalite interface,” Exp. Mech. 44, 158–166 (2004).
[CrossRef]

Kirugulige, M. S.

M. S. Kirugulige and H. V. Tippur, “Measurement of surface deformations and fracture parameters for a mixed-mode crack driven by stress waves using image correlation technique and high-speed photography,” Strain 45, 108–122(2009).
[CrossRef]

M. S. Kirugulige, H. V. Tippur, and T. S. Denney, “Measurement of transient deformations using digital image correlation method and high-speed photography: application to dynamic fracture,” Appl. Opt. 46, 5083–5096 (2007).
[CrossRef]

Krishnaswamy, S.

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements—analysis and experimental results,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing—a study of crack tip K-dominance,” Int. J. Fract. 52, 91–117 (1991).

Lambros, J.

J. Kimberley and J. Lambros, “Dynamic crack kinking from a PMMA/homalite interface,” Exp. Mech. 44, 158–166 (2004).
[CrossRef]

J. J. Mason, J. Lambros, and A. J. Rosakis, “The use of a coherent gradient sensor in dynamic mixed-mode fracture-mechanics experiments,” J. Mech. Phys. Solids 40, 641–661 (1992).
[CrossRef]

Mackenzie, J. D.

E. J. A. Pope, M. Asami, and J. D. Mackenzie, “Transparent silica gel-PMMA composites,” J. Mater. Res. 4, 1018–1026 (1989).
[CrossRef]

Mason, J. J.

J. J. Mason, J. Lambros, and A. J. Rosakis, “The use of a coherent gradient sensor in dynamic mixed-mode fracture-mechanics experiments,” J. Mech. Phys. Solids 40, 641–661 (1992).
[CrossRef]

Matsuura, T.

H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
[CrossRef]

McCauley, J. W.

P. Patel, G. A. Gilde, P. G. Dehmer, and J. W. McCauley, “Transparent armor,” AMPTIAC Newsletter 4(3), 1–9 (2000).

Nakagaito, A. N.

S. Iwamoto, A. N. Nakagaito, H. Yano, and M. Nogi, “Optically transparent composites reinforced with plant fiber-based nanofibers,” Appl. Phys. A: Mater. Sci. Process. 81, 1109–1112 (2005).
[CrossRef]

H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
[CrossRef]

Nogi, M.

H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
[CrossRef]

S. Iwamoto, A. N. Nakagaito, H. Yano, and M. Nogi, “Optically transparent composites reinforced with plant fiber-based nanofibers,” Appl. Phys. A: Mater. Sci. Process. 81, 1109–1112 (2005).
[CrossRef]

Orteu, U.

M. A. Sutton, U. Orteu, and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements (Springer, 2009).

Patel, P.

P. Patel, G. A. Gilde, P. G. Dehmer, and J. W. McCauley, “Transparent armor,” AMPTIAC Newsletter 4(3), 1–9 (2000).

Pope, E. J. A.

E. J. A. Pope, M. Asami, and J. D. Mackenzie, “Transparent silica gel-PMMA composites,” J. Mater. Res. 4, 1018–1026 (1989).
[CrossRef]

Pu, Q.

X. J. Dai, H. Yun, and Q. Pu, “Measuring thickness change of transparent plate by electronic speckle pattern interferometry and digital image correlation,” Opt. Commun. 283, 3481–3486 (2010).
[CrossRef]

Ramaswamy, S.

S. Ramaswamy, H. V. Tippur, and L. Xu, “Mixed-mode crack-tip deformations studied using a modified flexural specimen and coherent gradient sensing,” Exp. Mech. 33, 218–227 (1993).
[CrossRef]

Ravi, S.

S. Ravi, “Development of transparent composite for photoelastic studies,” Adv. Compos. Mater. 7, 73–81 (1998).

Riley, W. F.

J. W. Dally and W. F. Riley, Experimental Stress Analysis, 4th ed. (College House, 2005).

Rosakis, A. J.

J. J. Mason, J. Lambros, and A. J. Rosakis, “The use of a coherent gradient sensor in dynamic mixed-mode fracture-mechanics experiments,” J. Mech. Phys. Solids 40, 641–661 (1992).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing—a study of crack tip K-dominance,” Int. J. Fract. 52, 91–117 (1991).

H. V. Tippur and A. J. Rosakis, “Quasi-static and dynamic crack-growth along bimaterial interfaces—a note on crack-tip field-measurements using coherent gradient sensing,” Exp. Mech. 31, 243–251 (1991).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements—analysis and experimental results,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

Schreier, H.

M. A. Sutton, U. Orteu, and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements (Springer, 2009).

Shukla, N.

N. Shukla and J. W. Dally, Experimental Solid Mechanics(College House, 2010).

Sinha, J. K.

J. K. Sinha, H. V. Tippur, and L. M. Xu, “An interferometric and finite element investigation of interfacial crack tip fields: role of mode-mixity on 3-D stress variations,” Int. J. Solids Struct. 34, 741–754 (1997).
[CrossRef]

Strassburger, E.

E. Strassburger, “Ballistic testing of transparent armour ceramics,” J. Eur. Ceram. Soc. 29, 267–273 (2009).
[CrossRef]

Sugiyama, J.

H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
[CrossRef]

Sutton, M. A.

M. A. Sutton, U. Orteu, and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements (Springer, 2009).

Tippur, H. V.

H. V. Tippur, “Coherent gradient sensing (CGS) method for fracture mechanics: a review,” Fatigue Fract. Eng. Mater. Struct. 33, 832–858 (2010).
[CrossRef]

M. S. Kirugulige and H. V. Tippur, “Measurement of surface deformations and fracture parameters for a mixed-mode crack driven by stress waves using image correlation technique and high-speed photography,” Strain 45, 108–122(2009).
[CrossRef]

M. S. Kirugulige, H. V. Tippur, and T. S. Denney, “Measurement of transient deformations using digital image correlation method and high-speed photography: application to dynamic fracture,” Appl. Opt. 46, 5083–5096 (2007).
[CrossRef]

J. K. Sinha, H. V. Tippur, and L. M. Xu, “An interferometric and finite element investigation of interfacial crack tip fields: role of mode-mixity on 3-D stress variations,” Int. J. Solids Struct. 34, 741–754 (1997).
[CrossRef]

S. Ramaswamy, H. V. Tippur, and L. Xu, “Mixed-mode crack-tip deformations studied using a modified flexural specimen and coherent gradient sensing,” Exp. Mech. 33, 218–227 (1993).
[CrossRef]

H. V. Tippur, “Coherent gradient sensing—a Fourier optics analysis and applications to fracture,” Appl. Opt. 31, 4428–4439 (1992).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing—a study of crack tip K-dominance,” Int. J. Fract. 52, 91–117 (1991).

H. V. Tippur and A. J. Rosakis, “Quasi-static and dynamic crack-growth along bimaterial interfaces—a note on crack-tip field-measurements using coherent gradient sensing,” Exp. Mech. 31, 243–251 (1991).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements—analysis and experimental results,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

Xu, L.

S. Ramaswamy, H. V. Tippur, and L. Xu, “Mixed-mode crack-tip deformations studied using a modified flexural specimen and coherent gradient sensing,” Exp. Mech. 33, 218–227 (1993).
[CrossRef]

Xu, L. M.

J. K. Sinha, H. V. Tippur, and L. M. Xu, “An interferometric and finite element investigation of interfacial crack tip fields: role of mode-mixity on 3-D stress variations,” Int. J. Solids Struct. 34, 741–754 (1997).
[CrossRef]

Yano, H.

H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
[CrossRef]

S. Iwamoto, A. N. Nakagaito, H. Yano, and M. Nogi, “Optically transparent composites reinforced with plant fiber-based nanofibers,” Appl. Phys. A: Mater. Sci. Process. 81, 1109–1112 (2005).
[CrossRef]

Yun, H.

X. J. Dai, H. Yun, and Q. Pu, “Measuring thickness change of transparent plate by electronic speckle pattern interferometry and digital image correlation,” Opt. Commun. 283, 3481–3486 (2010).
[CrossRef]

Adv. Compos. Mater. (1)

S. Ravi, “Development of transparent composite for photoelastic studies,” Adv. Compos. Mater. 7, 73–81 (1998).

Adv. Mater. (1)

H. Yano, J. Sugiyama, A. N. Nakagaito, M. Nogi, T. Matsuura, M. Hikita, and K. Handa, “Optically transparent composites reinforced with networks of bacterial nanofibers,” Adv. Mater. 17, 153–155 (2005).
[CrossRef]

AMPTIAC Newsletter (1)

P. Patel, G. A. Gilde, P. G. Dehmer, and J. W. McCauley, “Transparent armor,” AMPTIAC Newsletter 4(3), 1–9 (2000).

Appl. Opt. (2)

Appl. Phys. A: Mater. Sci. Process. (1)

S. Iwamoto, A. N. Nakagaito, H. Yano, and M. Nogi, “Optically transparent composites reinforced with plant fiber-based nanofibers,” Appl. Phys. A: Mater. Sci. Process. 81, 1109–1112 (2005).
[CrossRef]

Exp. Mech. (3)

H. V. Tippur and A. J. Rosakis, “Quasi-static and dynamic crack-growth along bimaterial interfaces—a note on crack-tip field-measurements using coherent gradient sensing,” Exp. Mech. 31, 243–251 (1991).
[CrossRef]

J. Kimberley and J. Lambros, “Dynamic crack kinking from a PMMA/homalite interface,” Exp. Mech. 44, 158–166 (2004).
[CrossRef]

S. Ramaswamy, H. V. Tippur, and L. Xu, “Mixed-mode crack-tip deformations studied using a modified flexural specimen and coherent gradient sensing,” Exp. Mech. 33, 218–227 (1993).
[CrossRef]

Fatigue Fract. Eng. Mater. Struct. (1)

H. V. Tippur, “Coherent gradient sensing (CGS) method for fracture mechanics: a review,” Fatigue Fract. Eng. Mater. Struct. 33, 832–858 (2010).
[CrossRef]

Int. J. Fract. (2)

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing—a study of crack tip K-dominance,” Int. J. Fract. 52, 91–117 (1991).

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements—analysis and experimental results,” Int. J. Fract. 48, 193–204 (1991).
[CrossRef]

Int. J. Solids Struct. (1)

J. K. Sinha, H. V. Tippur, and L. M. Xu, “An interferometric and finite element investigation of interfacial crack tip fields: role of mode-mixity on 3-D stress variations,” Int. J. Solids Struct. 34, 741–754 (1997).
[CrossRef]

J. Eur. Ceram. Soc. (1)

E. Strassburger, “Ballistic testing of transparent armour ceramics,” J. Eur. Ceram. Soc. 29, 267–273 (2009).
[CrossRef]

J. Mater. Res. (1)

E. J. A. Pope, M. Asami, and J. D. Mackenzie, “Transparent silica gel-PMMA composites,” J. Mater. Res. 4, 1018–1026 (1989).
[CrossRef]

J. Mech. Phys. Solids (1)

J. J. Mason, J. Lambros, and A. J. Rosakis, “The use of a coherent gradient sensor in dynamic mixed-mode fracture-mechanics experiments,” J. Mech. Phys. Solids 40, 641–661 (1992).
[CrossRef]

Opt. Commun. (1)

X. J. Dai, H. Yun, and Q. Pu, “Measuring thickness change of transparent plate by electronic speckle pattern interferometry and digital image correlation,” Opt. Commun. 283, 3481–3486 (2010).
[CrossRef]

Strain (1)

M. S. Kirugulige and H. V. Tippur, “Measurement of surface deformations and fracture parameters for a mixed-mode crack driven by stress waves using image correlation technique and high-speed photography,” Strain 45, 108–122(2009).
[CrossRef]

Other (6)

J. W. Dally and W. F. Riley, Experimental Stress Analysis, 4th ed. (College House, 2005).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

W. N. Sharpe, ed., Handbook of Experimental Solid Mechanics (Springer, 2008).

N. Shukla and J. W. Dally, Experimental Solid Mechanics(College House, 2010).

M. A. Sutton, U. Orteu, and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements (Springer, 2009).

R. G. Budynas, Advanced Strength and Applied Stress Analysis (McGraw-Hill, 1998).

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Figures (12)

Fig. 1.
Fig. 1.

Experimental setup for the DGS method to determine planar stress gradients in phase objects.

Fig. 2.
Fig. 2.

Working principle of DGS.

Fig. 3.
Fig. 3.

Contour plots of angular deflections (left to right) ϕx, ϕy and ϕ fields caused by a plano-convex spherical lens. Contour interval=2.5×103rads. (The heavy dot in the angular deflection field is due to a reference mark on the speckle plate.)

Fig. 4.
Fig. 4.

Mapping coordinates of specimen and target planes.

Fig. 5.
Fig. 5.

Contour plots of angular deflections of corrected (broken lines) and uncorrected (solid lines) for (left) ϕx, (middle) ϕy, and (right) ϕ fields caused by a long focal length plano-convex lens. Contour interval=2.5×103rads.

Fig. 6.
Fig. 6.

(top) Experimental setup used for studying stress concentration caused by a line load acting on the edge of a large PMMA sheet. (bottom) The close-up shows loading pin resting on the top edge of the transparent specimen and speckles on the target.

Fig. 7.
Fig. 7.

Line load acting on a (left) half-space and (right) an actual speckle image recorded. Note the blurred/distorted region adjacent to the loading pin in the enlarged speckle image.

Fig. 8.
Fig. 8.

Measured (left) ϕx and (right) ϕy contours near the loading point for different load levels. Contour interval=1×103rads. (The left vertical edge of each image corresponds to the loading edge where F acts at the origin.)

Fig. 9.
Fig. 9.

Comparison of experimental and analytical (top) ϕx and (bottom) ϕy contours for F=2022N.

Fig. 10.
Fig. 10.

Plot of experimentally extracted load (symbols) at various radial locations from the two angular deflection fields ϕx and ϕy along r, θ=0° and r, θ=45°, respectively. Deviation between the Flamant solution and measurement close to the loading point where plane stress approximations are violated can be seen.

Fig. 11.
Fig. 11.

Measured resultant angular deflection of light rays (left column) ϕand (right column) estimated radial stress σrr contours for various load levels. Contours are plotted every 1×103rads and 2 MPa, respectively. (The left vertical edge in each image is the edge where the line load acts horizontally at the origin.)

Fig. 12.
Fig. 12.

Comparison between experimental and analytical (top) ϕ and (bottom) radial stress (σrr) contours for F=2022N.

Equations (16)

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d^=αi^+βj^+γk^,
δS(x,y)=2B(n1)01/2εzzd(z/B)+2B01/2δnd(z/B).
δn(x,y)=D1(σxx+σyy+σzz),
δS=2B(D1υE(n1)){01/2(σxx+σyy)[1+D2(σzzυ(σxx+σyy))]}d(z/B),
δS(x,y)CσB(σxx+σyy),
d^(δS)xi^+(δS)yj^+k^
α=(δS)x=CσB(σxx+σyy)xandβ=(δS)y=CσB(σxx+σyy)y.
cosθx=δxR,cosθy=δyR,tanϕx=δxΔ,andtanϕy=δyΔ,
tanϕx=RΔcosθx=1+δx2+δy2Δ2cosθx,tanϕy=RΔcosθy=1+δx2+δy2Δ2cosθy.
ϕxα=CσB(σxx+σyy)x,ϕyβ=CσB(σxx+σyy)y,
ϕx=x(x2+y22fl)=xflandϕy=y(x2+y22fl)=yfl,
σxx+σyy=σrr=2FπBcos(θ)r=2FπBxr2,σθθ=0,σrθ=0,
ϕx=CσB(σxx+σyy)x=CσB(σrr)xandϕy=CσB(σxx+σyy)y=CσB(σrr)y.
ϕx=CσB2FπBcos(2θ)r2andϕy=CσB2FπBsin(2θ)r2.
ϕ=ϕx2+ϕy2=CσB2FπBr2.
(σxx+σyy)=σrr=ϕCσB(rcosθ).

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