Abstract

An optical method was developed for the study of constrained zones around cracks in transparent specimens made of birefringent materials under conditions of plane stress. The absolute retardations of light rays impinging normally at the plate, partly reflected from the back face, and twice refracted at the front face create a caustic when emerged from the plate, which has the shape of a generalized epicycloid, when it is projected on a screen far from the plate. The caustic defines the constrained zone surrounding a crack tip. The characteristic properties of the caustic were studied in relation to the straining mode of the plate.

© 1971 Optical Society of America

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References

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  1. E. G. Coker, L. N. G. Filon, A Treatise on Photoelasticity (Cambridge U. P., Cambridge, 1957), pp. 204–210.
  2. H. Favre, Rev. Opt. 8, 5 (1929).
  3. C. Fabry, Compt. Rend. 190, 457 (1930).
  4. A. Dose, R. Landwehr, Ing. Arch. 21, 73 (1953).
    [CrossRef]
  5. D. Post, Proc. Soc. Exp. Stress Anal. 12, 99 (1954).
  6. H. Favre, W. Schumann, “A Photoelastic-Interferometric Method to Determine Separately the Principal Stresses in Two-Dimensional States and Possible Applications to Surface and Thermal Stresses,” in Photoelasticity, Proc. Intern. Symp. Photoelasticity, M. M. Frocht, Ed. (Pergamon, New York, 1963), pp. 3–25.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 41–47, 110–113.
  8. P. C. Paris, G. C. Sih, “Stress Analysis of Cracks,” Fracture Toughness Testing and its Applications, ASTM Sp. Tech. Publ. 381 (1969), pp. 30–81.
  9. P. S. Theocaris, J. Appl. Mech. 37, 409 (1970).
    [CrossRef]
  10. H. Henschen, Interferenzoptische Untersuchung der Spannungsverteilung von dem laufenden Bruch in Kunststoff CR-39, doctoral thesis, University of Freiburg, 1962.
    [PubMed]

1970 (1)

P. S. Theocaris, J. Appl. Mech. 37, 409 (1970).
[CrossRef]

1969 (1)

P. C. Paris, G. C. Sih, “Stress Analysis of Cracks,” Fracture Toughness Testing and its Applications, ASTM Sp. Tech. Publ. 381 (1969), pp. 30–81.

1954 (1)

D. Post, Proc. Soc. Exp. Stress Anal. 12, 99 (1954).

1953 (1)

A. Dose, R. Landwehr, Ing. Arch. 21, 73 (1953).
[CrossRef]

1930 (1)

C. Fabry, Compt. Rend. 190, 457 (1930).

1929 (1)

H. Favre, Rev. Opt. 8, 5 (1929).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 41–47, 110–113.

Coker, E. G.

E. G. Coker, L. N. G. Filon, A Treatise on Photoelasticity (Cambridge U. P., Cambridge, 1957), pp. 204–210.

Dose, A.

A. Dose, R. Landwehr, Ing. Arch. 21, 73 (1953).
[CrossRef]

Fabry, C.

C. Fabry, Compt. Rend. 190, 457 (1930).

Favre, H.

H. Favre, Rev. Opt. 8, 5 (1929).

H. Favre, W. Schumann, “A Photoelastic-Interferometric Method to Determine Separately the Principal Stresses in Two-Dimensional States and Possible Applications to Surface and Thermal Stresses,” in Photoelasticity, Proc. Intern. Symp. Photoelasticity, M. M. Frocht, Ed. (Pergamon, New York, 1963), pp. 3–25.

Filon, L. N. G.

E. G. Coker, L. N. G. Filon, A Treatise on Photoelasticity (Cambridge U. P., Cambridge, 1957), pp. 204–210.

Henschen, H.

H. Henschen, Interferenzoptische Untersuchung der Spannungsverteilung von dem laufenden Bruch in Kunststoff CR-39, doctoral thesis, University of Freiburg, 1962.
[PubMed]

Landwehr, R.

A. Dose, R. Landwehr, Ing. Arch. 21, 73 (1953).
[CrossRef]

Paris, P. C.

P. C. Paris, G. C. Sih, “Stress Analysis of Cracks,” Fracture Toughness Testing and its Applications, ASTM Sp. Tech. Publ. 381 (1969), pp. 30–81.

Post, D.

D. Post, Proc. Soc. Exp. Stress Anal. 12, 99 (1954).

Schumann, W.

H. Favre, W. Schumann, “A Photoelastic-Interferometric Method to Determine Separately the Principal Stresses in Two-Dimensional States and Possible Applications to Surface and Thermal Stresses,” in Photoelasticity, Proc. Intern. Symp. Photoelasticity, M. M. Frocht, Ed. (Pergamon, New York, 1963), pp. 3–25.

Sih, G. C.

P. C. Paris, G. C. Sih, “Stress Analysis of Cracks,” Fracture Toughness Testing and its Applications, ASTM Sp. Tech. Publ. 381 (1969), pp. 30–81.

Theocaris, P. S.

P. S. Theocaris, J. Appl. Mech. 37, 409 (1970).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 41–47, 110–113.

Compt. Rend. (1)

C. Fabry, Compt. Rend. 190, 457 (1930).

Fracture Toughness Testing and its Applications (1)

P. C. Paris, G. C. Sih, “Stress Analysis of Cracks,” Fracture Toughness Testing and its Applications, ASTM Sp. Tech. Publ. 381 (1969), pp. 30–81.

Ing. Arch. (1)

A. Dose, R. Landwehr, Ing. Arch. 21, 73 (1953).
[CrossRef]

J. Appl. Mech. (1)

P. S. Theocaris, J. Appl. Mech. 37, 409 (1970).
[CrossRef]

Proc. Soc. Exp. Stress Anal. (1)

D. Post, Proc. Soc. Exp. Stress Anal. 12, 99 (1954).

Rev. Opt. (1)

H. Favre, Rev. Opt. 8, 5 (1929).

Other (4)

E. G. Coker, L. N. G. Filon, A Treatise on Photoelasticity (Cambridge U. P., Cambridge, 1957), pp. 204–210.

H. Henschen, Interferenzoptische Untersuchung der Spannungsverteilung von dem laufenden Bruch in Kunststoff CR-39, doctoral thesis, University of Freiburg, 1962.
[PubMed]

H. Favre, W. Schumann, “A Photoelastic-Interferometric Method to Determine Separately the Principal Stresses in Two-Dimensional States and Possible Applications to Surface and Thermal Stresses,” in Photoelasticity, Proc. Intern. Symp. Photoelasticity, M. M. Frocht, Ed. (Pergamon, New York, 1963), pp. 3–25.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 41–47, 110–113.

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of successive reflections of a ray normally impinging on a transparent plate.

Fig. 2
Fig. 2

Diagram of the formation of the generalized epicycloid.

Fig. 3
Fig. 3

Shape of the principal epicycloid {for r = [ ( 3 / 2 ) C p ] 2 5} and geometry of its formation.

Fig. 4
Fig. 4

Interference pattern and caustic for a transversely cracked plate. (The outside part of the caustic corresponds to a reflection at the back face, while the inside part to a reflection at the front face. It is easy to show that both parts belong to the same curve for Plexiglas.)

Equations (80)

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Δ s 3 = c 3 d ( σ 1 - σ 2 ) ,
Δ s 1 = d ( c 1 σ 1 + c 2 σ 2 ) , Δ s 2 = d ( c 2 σ 1 + c 1 σ 2 ) ,
I r ( k + l ) = R k ( 1 - R ) l I i ,
R = [ ( n - 1 ) 2 ] / [ ( n + 1 ) 2 ] .
I 1 = 2 R [ 1 - cos ( 4 π n d / λ ) ]
I 1 = 2 R [ 1 - cos ( 4 π n 1 d / λ ) ] ,
ϑ 1 ( 1 + 2 ) = 2 λ ( n d - n 1 d ) 2 λ [ d ( n - n 1 ) + n ( d - d ) ] .
I 1 = 2 R ( 1 - cos { 2 π [ 2 n d λ - ϑ 1 ( 1 + 2 ) ] } ) .
Δ s 1 , 2 = 2 [ ( n 1 , 2 - n ) d + ( n 1 , 2 - 1 2 ) Δ d ] ,
Δ n 1 = b 1 1 + b 2 ( 2 + 3 ) , Δ n 2 = b 1 2 + b 2 ( 1 + 3 ) ,
Δ s 1 = 2 d [ b 1 1 + b 2 2 + ( b 2 + n - 1 2 ) 3 ] , Δ s 2 = 2 d [ b 1 2 + b 2 1 + ( b 2 + n - 1 2 ) 3 ] .
Δ s 1 = 2 d ( α σ 1 + β σ 2 ) , Δ s 2 = 2 d ( β σ 1 + α σ 2 ) ,
α = ( 1 / E ) [ b 1 - 2 ν b 2 - ν ( n - 1 2 ) ] , β = ( 1 / E ) [ b 2 - ν ( b 1 + b 2 ) - ν ( n - 1 2 ) ] ,
Δ s 1 , 2 = 2 d c r [ ( σ 1 + σ 2 ) ± ξ ( σ 1 - σ 2 ) ] ,
c r = ( α + β ) / 2             and             ξ = ( α - β ) / ( α + β ) .
α = β = 1 / E [ b 1 ( 1 - 2 ν ) - ν ( n - 1 2 ) ] Δ s 1 = Δ s 2 = ( 2 d / E ) [ b 1 ( 1 - 2 ν ) - ν ( n - 1 2 ) ] ( σ 1 + σ 2 ) = 2 d c r ( σ 1 + σ 2 ) .
Δ s r ( 1 + 2 ) = ( 2 Δ s t ( 0 + 2 ) + Δ d )
Δ d = - ( ν d / E ) ( σ 1 + σ 2 ) and Δ s t ( 0 + 2 ) = c d ( σ 1 + σ 2 )
Δ s r ( 1 + 2 ) = d ( σ 1 + σ 2 ) [ 2 c - ( ν / E ) ] .
Δ s r ( 1 + 2 ) = c d ( σ 1 + σ 2 ) .
or             w = z 0 grad S ( x , y , z ) w = - z 0 grad s ( x , y ) = - z 0 ( s x i + s y j ) ,
w = - z 0 grad Δ s ( x , y ) .
and             z = x + i y Re σ ( z ) = ( σ x + σ y )
σ ( z ) = f ( z ) / z 1 2 ,
or             w = - z 0 grad Δ s = - z 0 c d grad ( σ x + σ y ) w = C grad ( σ x + σ y ) ,
C = - z 0 d c .
w = C ( u x i + u y j ) ,
w = C ( u x + i u y ) .
W = z + C ( u x + i u y )
W = ( x + C u x ) + i ( y + C u y ) .
x = ( x + C u x ) , y = ( y + C u y ) ,
W = ( x + i y ) .
D = ( x , y ) / ( r , ϑ ) = 0.
D = ( x , y ) ( r , ϑ ) = ( x , y ) ( x , y ) ( x , y ) ( r , ϑ ) = 0 ,
| ( 1 + C 2 u x 2 ) C 2 u x y C 2 u x y ( 1 + C 2 u y 2 ) | = 0.
1 + C ( 2 u x 2 + 2 u y 2 ) + C 2 [ 2 u x 2 2 u y 2 - ( 2 u x y ) 2 ] = 0.
( 2 u x 2 + 2 u y 2 ) = 0.
σ ( z ) = 2 u x 2 - i 2 u x y ,
σ ( z ) 2 = - 2 u x 2 2 u y 2 + ( 2 u x y ) 2 .
σ ( z ) = - C - 1 .
and             W = z + C ( u x - i ν x ) σ ( z ) = - C - 1 ,
W = z + C σ ( z ) ¯ , σ ( z ) = - C - 1 .
and             W = z - z 0 t c σ ( z ) ¯ σ ( z ) = - 1 / ( z 0 t c ) .
σ ( z ) = 2 K * / ( 2 π z ) 1 2 ,
W = z + ( C p / z / 2 3 )
r = [ ( 3 / 2 ) C p ] 2 5 ,
C p = ( z 0 d c ) K * ( 2 π ) 1 2 .
x = r cos ϑ + ζ K I r - / 2 3 cos 3 ϑ 2 - ζ K II r - / 2 3 sin 3 ϑ 2 , y = r sin ϑ + ζ K I r - / 2 3 sin 3 ϑ 2 + K II r - / 2 3 cos 3 ϑ 2 , }
ζ = C p / K * .
z = r e i ϑ + ζ r - / 2 3 K * ¯ exp ( i 3 ϑ 2 ) ,
and cos ( ϑ + m t ) = cos ϑ , sin ( ϑ + m t ) = sin ϑ , cos ( 3 ϑ + m t 2 ) = cos 3 ϑ 2 , sin ( 3 ϑ + m t 2 ) = sin 3 ϑ 2 ,
m t = 2 κ π
3 m t / 2 = 2 κ π ,
x = r [ cos ϑ + 2 3 cos ( 3 ϑ 2 + ω ) ] , y = r [ sin ϑ + 2 3 sin ( 3 ϑ 2 + ω ) ] .
z = ρ exp ( i φ ) = r { exp ( i ϑ ) + 2 3 exp [ i ( 3 ϑ / 2 + ω ) ] } ,
ρ = r { exp [ i ( ϑ - φ ) ] + 2 3 exp [ i ( 3 ϑ / 2 + ω - φ ) } .
ρ = r [ cos ( ϑ - φ ) + 2 3 cos ( 3 ϑ 2 + ω - φ ) ] ,
0 = r [ sin ( ϑ - φ ) + 2 3 sin ( 3 ϑ 2 + ω - φ ) ] .
F ( ϑ , φ ) = r [ cos ( ϑ - φ ) + 2 3 cos ( 3 ϑ 2 + ω - φ ) ] + λ r [ sin ( ϑ - φ ) + 2 3 sin ( 3 ϑ 2 + ω - φ ) ] ,
sin ( ϑ - φ ) = 0 , sin ( 3 ϑ 2 + ω - φ ) = 0 ,
cos ( ϑ - φ ) = ± 1 , cos ( 3 ϑ 2 + ω - φ ) = ± 1.
for ϑ = φ = - 2 ω             :             ρ max = 5 r / 3 ,
for ϑ = φ = 2 ( π - ω )             :             ρ min = r / 3.
tan φ = 3 sin ϑ + 2 sin ( 3 ϑ / 2 + ω ) 3 cos ϑ + 2 cos ( 3 ϑ / 2 + ω ) .
tan ( φ - 2 ω ) = 3 sin ( ϑ - 2 ω ) + 2 sin ( 3 ϑ / 2 - 2 ω ) 3 cos ( ϑ - 2 ω ) + 2 cos ( 3 ϑ / 2 - 2 ω ) .
3 sin ( ϑ - 2 ω ) + 2 sin ( 3 ϑ / 2 - 2 ω ) 3 cos ( ϑ - 2 ω ) + 2 cos ( 3 ϑ / 2 - 2 ω ) = - tan 2 ω .
or sin ( ϑ - φ ) + 2 3 sin ( 3 ϑ 2 - φ ) = 0 , sin [ ( - ϑ ) - ( - φ ) ] + 2 3 sin [ 3 2 ( - ϑ ) - ( - φ ) ] = 0.
5 r 2 18 ρ - ρ 2 + 2 r 3 cos [ ω + φ ± 3 cos - 1 ( 9 ρ 2 + 5 r 2 ) 18 ρ r 2 ] = 0.
x I = r [ cos ( ϑ + 2 ω ) + 2 3 cos 3 2 ( ϑ + 2 ω ) ] ,
y I = r [ sin ( ϑ + 2 ω ) + 2 3 sin 3 2 ( ϑ + 2 ω ) ] .
ρ mid = 4 r / 3 ,
( ϑ + 2 ω ) = 2 cos - 1 ( 1 4 )             or             ( ϑ + 2 ω ) = 151° 3 ,
( ϑ + 2 ω ) = 4 π - 2 cos - 1 ( 1 4 ) or ( ϑ + 2 ω ) = 568°57 .
D lon = ( 4 r / 3 ) + ( 5 r / 3 ) = 3 r .
ρ t r = 1.5817 r .
x = r cos ϑ + C p r - / 2 3 cos ( 3 ϑ 2 + ω ) , y = r sin ϑ + C p r - / 2 3 sin ( 3 ϑ 2 + ω ) .
E = 1 2 θ 1 θ 2 ( x y θ - y x θ ) d θ .
E = 1 2 0 4 π ( r 2 + 3 2 C p 2 r - 3 ) d θ .
d E d r = 2 π ( 2 r - 9 2 C p 2 r - 4 ) = 0.
r = [ ( 3 / 2 ) C p ] 2 5 .

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