Abstract

The equations of caustics have been derived based on the exact theory of geometrical optics, and they have been adapted to the problem of a slanting internal crack in a disk under biaxial loading. A comparison of the exact caustics with those derived from the far-field theory used in the past in the applications showed negligible difference between the two theories. Thus, it is shown that the approximate theory of caustics used before for determining singular fields in mechanics is sufficiently accurate for engineering applications.

© 1983 Optical Society of America

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References

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  1. P. Manogg, “Anwendung der Schattenoptik zur Untersuchung des Zerreissvorgangs von Platten,” Dis. 4/64, U. Freiburg, 1964.
  2. P. Manogg, Int. J. Fract. Mech. 2, 604 (1966).
  3. P. S. Theocaris, J. Appl. Mech. 37, 409 (1970).
    [CrossRef]
  4. P. S. Theocaris, Appl. Opt. 10, 2240 (1971).
    [CrossRef] [PubMed]
  5. H. Favre, Rev. Opt. 8, 193 (1929).
  6. P. S. Theocaris, J. G. Michopoulos, “The Exact Form of Caustics in Mixed-Mode Fracture: A Comparison with Approximate Solutions,” Acta Mech.45, (1982), to appear.
  7. P. S. Theocaris, J. G. Michopoulos, Appl. Opt. 21, 1080 (1982).
    [CrossRef] [PubMed]
  8. D. L. Shealy, D. G. Burkhard, Opt. Acta 20, 287 (1973).
    [CrossRef]
  9. D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
    [CrossRef]
  10. D. G. Burkhard, D. L. Shealy, Appl. Opt. 20, 897 (1981).
    [CrossRef] [PubMed]
  11. D. G. Burkhard, Appl. Opt. 19, 3682 (1980).
    [CrossRef] [PubMed]
  12. P. S. Theocaris, J. Strain Anal. 8, 267 (1973).
    [CrossRef]
  13. P. S. Theocaris, J. G. Michopoulos, “A Closed-Form Solution of a Slant Crack Under Biaxial Loading,” Eng. Fract. Mech.16, (1982) to appear.
  14. J. Eftis, N. Subramonian, Eng. Fract. Mech. 10, 43 (1978).
    [CrossRef]

1982 (1)

1981 (1)

1980 (1)

1978 (1)

J. Eftis, N. Subramonian, Eng. Fract. Mech. 10, 43 (1978).
[CrossRef]

1973 (3)

D. L. Shealy, D. G. Burkhard, Opt. Acta 20, 287 (1973).
[CrossRef]

P. S. Theocaris, J. Strain Anal. 8, 267 (1973).
[CrossRef]

D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
[CrossRef]

1971 (1)

1970 (1)

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

1966 (1)

P. Manogg, Int. J. Fract. Mech. 2, 604 (1966).

1929 (1)

H. Favre, Rev. Opt. 8, 193 (1929).

Burkhard, D. G.

Eftis, J.

J. Eftis, N. Subramonian, Eng. Fract. Mech. 10, 43 (1978).
[CrossRef]

Favre, H.

H. Favre, Rev. Opt. 8, 193 (1929).

Manogg, P.

P. Manogg, Int. J. Fract. Mech. 2, 604 (1966).

P. Manogg, “Anwendung der Schattenoptik zur Untersuchung des Zerreissvorgangs von Platten,” Dis. 4/64, U. Freiburg, 1964.

Michopoulos, J. G.

P. S. Theocaris, J. G. Michopoulos, Appl. Opt. 21, 1080 (1982).
[CrossRef] [PubMed]

P. S. Theocaris, J. G. Michopoulos, “The Exact Form of Caustics in Mixed-Mode Fracture: A Comparison with Approximate Solutions,” Acta Mech.45, (1982), to appear.

P. S. Theocaris, J. G. Michopoulos, “A Closed-Form Solution of a Slant Crack Under Biaxial Loading,” Eng. Fract. Mech.16, (1982) to appear.

Shealy, D. L.

Subramonian, N.

J. Eftis, N. Subramonian, Eng. Fract. Mech. 10, 43 (1978).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris, J. G. Michopoulos, Appl. Opt. 21, 1080 (1982).
[CrossRef] [PubMed]

P. S. Theocaris, J. Strain Anal. 8, 267 (1973).
[CrossRef]

P. S. Theocaris, Appl. Opt. 10, 2240 (1971).
[CrossRef] [PubMed]

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

P. S. Theocaris, J. G. Michopoulos, “The Exact Form of Caustics in Mixed-Mode Fracture: A Comparison with Approximate Solutions,” Acta Mech.45, (1982), to appear.

P. S. Theocaris, J. G. Michopoulos, “A Closed-Form Solution of a Slant Crack Under Biaxial Loading,” Eng. Fract. Mech.16, (1982) to appear.

Appl. Opt. (4)

Eng. Fract. Mech. (1)

J. Eftis, N. Subramonian, Eng. Fract. Mech. 10, 43 (1978).
[CrossRef]

Int. J. Fract. Mech. (1)

P. Manogg, Int. J. Fract. Mech. 2, 604 (1966).

J. Appl. Mech. (1)

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

J. Opt. Soc. Am. (1)

J. Strain Anal. (1)

P. S. Theocaris, J. Strain Anal. 8, 267 (1973).
[CrossRef]

Opt. Acta (1)

D. L. Shealy, D. G. Burkhard, Opt. Acta 20, 287 (1973).
[CrossRef]

Rev. Opt. (1)

H. Favre, Rev. Opt. 8, 193 (1929).

Other (3)

P. S. Theocaris, J. G. Michopoulos, “The Exact Form of Caustics in Mixed-Mode Fracture: A Comparison with Approximate Solutions,” Acta Mech.45, (1982), to appear.

P. Manogg, “Anwendung der Schattenoptik zur Untersuchung des Zerreissvorgangs von Platten,” Dis. 4/64, U. Freiburg, 1964.

P. S. Theocaris, J. G. Michopoulos, “A Closed-Form Solution of a Slant Crack Under Biaxial Loading,” Eng. Fract. Mech.16, (1982) to appear.

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Figures (8)

Fig. 1
Fig. 1

Geometrical configuration of source, deflector, and receiver in their general form.

Fig. 2
Fig. 2

Optical setup for the method of caustics as well as the loading and geometry of the loaded infinite plate containing a slant crack.

Fig. 3
Fig. 3

Variation of the ratio ( r c e r c a ) / r c e vs the polar angle ϑ for various parametric values of the distances zO = 0.1, 0.3, 0.5, and 0.7 m and for zi = 0.1m, k = −1.0, and β = 90°.

Fig. 4
Fig. 4

Variation of the ratio ( r c e r c a ) / r c e at the angle ϑ = 360° vs the distance zO and the magnification factor λm for zi = 0.1m, k = −1, and β = 90°.

Fig. 5
Fig. 5

Variation of the ratio ( r c e r c a ) / r c e vs polar angle ϑ for the parametric values of angle of inclination β = 15°, 30°, 45°, 60°, 75°, and 90° and for biaxiality ratio k = −1.0 at zi = 0.1 m and zO = 0.7m.

Fig. 6
Fig. 6

Variation of the ratio ( r c e r c a ) / r c e vs polar angle ϑ for the parametric values of angle of inclination β = 15°, 30°, 45°, 60°, 75°, and 90° and for biaxiality ratio k = −0.5 at zi = 0.1 m and zO = 0.7m.

Fig. 7
Fig. 7

Variation of the ratio ( r c e r c a ) / r c e vs polar angle ϑ for the parametric values of angle of inclination β = 15°, 30°, 45°, 60°, 75°, and 90° and for biaxiality ratio k = 0 at zi = 0.1 m and zO = 0.7m.

Fig. 8
Fig. 8

Variation of the ratio ( r c e r c a ) / r c e vs polar angle ϑ for the parametric values of angle of inclination β = 15°, 30°, 45°, 60°, 75°, and 90° and for biaxiality ratio k = 0.5 at zi = 0.1 m and zO = 0.7m.

Equations (45)

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W x = x ( u 1 , u 2 ) + r 2 ξ ( u 1 , u 2 ) ,
W y = y ( u 1 , u 2 ) + r 2 η ( u 1 , u 2 ) ,
W z = z ( u 1 , u 2 ) + r 2 ζ ( u 1 , u 2 ) ,
r = r ( u 1 , u 2 ) = i x ( u 1 , u 2 ) + j y ( u 1 , u 2 ) + k z ( u 1 , u 2 ) ,
W = W ( υ 1 , υ 2 ) = i W x ( υ 1 , υ 2 ) + j W y ( υ 1 , υ 2 ) + k W z ( υ 1 , υ 2 ) ,
r 2 = | W ( υ 1 , υ 2 ) r ( u 1 , u 2 ) | ,
A = γ a + Ω N .
Ω = γ cos φ i ± ( 1 γ 2 sin 2 φ i ) 1 / 2 ,
sin φ i = 1 γ sin φ s .
Ω r = 2 cos φ i .
cos φ i = N · a .
Ω = γ N · a ± [ 1 γ 2 + γ 2 ( N · a ) 2 ] 1 / 2 ,
z = f ( x , y ) = d 2 ν E ( σ x + σ y ) ,
r = i x + j y + k f ( x , y ) .
W = i x + j y + k z O .
A x r = x D ( x , y ) 2 f ( x , y ) / x N 2 ( x , y ) D ( x , y ) × [ x f ( x , y ) x + y f y f ( x , y ) + z i ] ,
A y r = y D ( x , y ) 2 f ( x , y ) / y N 2 ( x , y ) D ( x , y ) × [ x f ( x , y ) x + y f y f ( x , y ) + z i ] ,
A z r = f ( x , y ) z i D ( x , y ) + 2 N 2 ( x , y ) D ( x , y ) × [ x f ( x , y ) x + y f y f ( x , y ) + z i ] ,
N 2 ( x , y ) = 1 + [ f ( x , y ) x ] 2 + [ f ( x , y ) y ] 2 ,
D ( x , y ) = { x 2 + y 2 + [ f ( x , y ) z i ] 2 } ,
ξ = A x ( A x 2 + A y 2 + A z 2 ) 1 / 2 , η = A y ( A x 2 + A y 2 + A z 2 ) 1 / 2 , ζ = A z ( A x 2 + A y 2 + A z 2 ) 1 / 2 ,
r 2 = z O f ( x , y ) ζ .
W x = x + [ z O + f ( x , y ) ] x N 2 2 f x ( x f x + y f y f + z i ) 2 ( x f x + y f y ) + ( f z i ) ( N 2 2 ) ,
W y = x + [ z O + f ( x , y ) ] y N 2 2 f x ( x f x + y f y f + z i ) 2 ( x f x + y f y ) + ( f z i ) ( N 2 2 ) ,
W z = z O .
W x = λ m x 2 z O f x = λ m x + z O d ν E x ( σ x + σ y ) ,
W y = λ m y 2 z O f y = λ m y + z O d ν E y ( σ x + σ y ) ,
W z = z O ,
λ m = ( z O + z i ) / z i .
Δ s = 2 f ( x , y ) = d ν E ( σ x + σ y ) ,
I 0 + r 2 I 1 + ( r 2 ) 2 I 2 = 0 ,
I 0 = A · ( f x × f f ) ,
I 1 = A · [ ( r x ) × ( A y ) + ( r y ) × ( A x ) ] ,
I 2 = A · ( A x × A y ) .
I 0 = A x f x A y f x + A z ,
I 1 = A x [ A z x f x A y y f y A y x ] + A y [ A z y + f y A z x ] + A z [ A y y A x x ] ,
I 2 = A x [ A y x A z y A z x A y y ] + A y [ A z x A x y A x x A z y ] + A z [ A x x A y y A y x A x y ] .
J ( W x , W y ) = ( W x , W y ) ( x , y ) = 0.
Δ s = C d ( σ x + σ y ) ,
C f = c f = ν / E ,
C r = 2 c r ,
C t = c t ,
( σ 1 + σ 2 ) = ( σ x + σ y ) = 2 K 1 ( 2 π r ) 1 / 2 cos ϑ 2 + 2 K I I ( 2 π r ) 1 / 2 × sin ϑ 2 + σ ( 1 k ) cos 2 β ,
K I = σ ( π a ) 1 / 2 2 [ ( 1 + k ) ( 1 k ) cos 2 β ] ,
K I I = σ ( π a ) 1 / 2 2 ( 1 k ) sin 2 β .

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