Abstract

The fringe-vector theory of holographic strain analysis was applied experimentally to determine strain and rotation tensors of rotated and deformed objects. One object was supported in a specially designed positioner, and the other was heated internally with a temperature controlled device. The rigid body motions and deformations were determined directly from the fringes that were observed on the surface of the object. This was done by taking photographs of the virtual image of the object from several different directions through the same hologram; the photographs were digitized manually, and the parameters obtained were, in turn, analyzed using a computer. The strains and rotations determined experimentally from holograms compared satisfactorily with the theoretical ones.

© 1978 Optical Society of America

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References

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  1. R. Dändliker, B. Ineichen, F. M. Mottier, Opt. Commun. 9, 412 (1973).
    [CrossRef]
  2. R. Dändliker, B. Ineichen, F. M. Mottier, in Proceedings of the International Optical Computing Conference, Zurich, Switzerland, April (1974), pp. 69–72.
  3. R. Dändliker, B. Eliasson, B. Ineichen, F. M. Mottier, in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., Cambridge, 1976), pp. 99–117.
  4. M. Dubas, W. Schumann, Opt. Acta 21, 547 (1974).
    [CrossRef]
  5. M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
    [CrossRef]
  6. K. A. Stetson, Appl. Opt. 14, 2256 (1975).
    [CrossRef] [PubMed]
  7. K. A. Stetson, J. Opt. Soc. Am. 64, 1 (1974).
    [CrossRef]
  8. R. Pryputniewicz, K. A. Stetson, Appl. Opt. 15, 725 (1976).
    [CrossRef] [PubMed]
  9. R. J. Pryputniewicz, Ph.D. Dissertation, U. Connecticut, Storrs (1976).
  10. S. K. Dhir, J. P. Sikora, Exp. Mech. 12, 323 (1972).
    [CrossRef]
  11. This fact becomes clear when one realizes that the illumination and observation perspectives as well as the linear displacements vary on the object’s surface from one point to another.

1976 (1)

1975 (2)

K. A. Stetson, Appl. Opt. 14, 2256 (1975).
[CrossRef] [PubMed]

M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
[CrossRef]

1974 (2)

M. Dubas, W. Schumann, Opt. Acta 21, 547 (1974).
[CrossRef]

K. A. Stetson, J. Opt. Soc. Am. 64, 1 (1974).
[CrossRef]

1973 (1)

R. Dändliker, B. Ineichen, F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

1972 (1)

S. K. Dhir, J. P. Sikora, Exp. Mech. 12, 323 (1972).
[CrossRef]

Dändliker, R.

R. Dändliker, B. Ineichen, F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

R. Dändliker, B. Ineichen, F. M. Mottier, in Proceedings of the International Optical Computing Conference, Zurich, Switzerland, April (1974), pp. 69–72.

R. Dändliker, B. Eliasson, B. Ineichen, F. M. Mottier, in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., Cambridge, 1976), pp. 99–117.

Dhir, S. K.

S. K. Dhir, J. P. Sikora, Exp. Mech. 12, 323 (1972).
[CrossRef]

Dubas, M.

M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
[CrossRef]

M. Dubas, W. Schumann, Opt. Acta 21, 547 (1974).
[CrossRef]

Eliasson, B.

R. Dändliker, B. Eliasson, B. Ineichen, F. M. Mottier, in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., Cambridge, 1976), pp. 99–117.

Ineichen, B.

R. Dändliker, B. Ineichen, F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

R. Dändliker, B. Ineichen, F. M. Mottier, in Proceedings of the International Optical Computing Conference, Zurich, Switzerland, April (1974), pp. 69–72.

R. Dändliker, B. Eliasson, B. Ineichen, F. M. Mottier, in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., Cambridge, 1976), pp. 99–117.

Mottier, F. M.

R. Dändliker, B. Ineichen, F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

R. Dändliker, B. Ineichen, F. M. Mottier, in Proceedings of the International Optical Computing Conference, Zurich, Switzerland, April (1974), pp. 69–72.

R. Dändliker, B. Eliasson, B. Ineichen, F. M. Mottier, in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., Cambridge, 1976), pp. 99–117.

Pryputniewicz, R.

Pryputniewicz, R. J.

R. J. Pryputniewicz, Ph.D. Dissertation, U. Connecticut, Storrs (1976).

Schumann, W.

M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
[CrossRef]

M. Dubas, W. Schumann, Opt. Acta 21, 547 (1974).
[CrossRef]

Sikora, J. P.

S. K. Dhir, J. P. Sikora, Exp. Mech. 12, 323 (1972).
[CrossRef]

Stetson, K. A.

Appl. Opt. (2)

Exp. Mech. (1)

S. K. Dhir, J. P. Sikora, Exp. Mech. 12, 323 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (2)

M. Dubas, W. Schumann, Opt. Acta 21, 547 (1974).
[CrossRef]

M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
[CrossRef]

Opt. Commun. (1)

R. Dändliker, B. Ineichen, F. M. Mottier, Opt. Commun. 9, 412 (1973).
[CrossRef]

Other (4)

R. Dändliker, B. Ineichen, F. M. Mottier, in Proceedings of the International Optical Computing Conference, Zurich, Switzerland, April (1974), pp. 69–72.

R. Dändliker, B. Eliasson, B. Ineichen, F. M. Mottier, in The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U.P., Cambridge, 1976), pp. 99–117.

R. J. Pryputniewicz, Ph.D. Dissertation, U. Connecticut, Storrs (1976).

This fact becomes clear when one realizes that the illumination and observation perspectives as well as the linear displacements vary on the object’s surface from one point to another.

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Figures (9)

Fig. 1
Fig. 1

A reconstruction from a typical double-exposure hologram of a rigid body rotation of an object. The fringes seen on the object’s surface seem to appear along the lines of intersection of the object’s surface with a set of parallel equally spaced fringe-locus planes.

Fig. 2
Fig. 2

Illumination and observation geometry in hologram interferometry. The hologram recording and reconstruction geometry is defined with respect to an arbitrarily assigned orthogonal coordinate system xyz. An object is illuminated with spherical wavefronts from a point defined by a space vector R1 (from the origin of the coordinate system to the point source of illumination) and observed with a spherical perspective from a point at R2. If the value of fringe-locus function Ω at a point P (located by RP) on the object is known, the Ω at a nearby point Q may be expressed in terms of the value of Ω(K,RP), and its derivatives as shown in Eq. (1). K1 and K2 are the illumination and observation vectors, respectively.

Fig. 3
Fig. 3

Multiple observations of the holographically reconstructed image. The direction of observation is changing from that along the vector K 2 1 to observations along vectors K 2 m (m = 2,3, …,9), one at a time, as shown in the figure. Each change in the direction of observation from K 2 1 to K 2 m is accompanied by determination of a corresponding fringe shift n1,m.

Fig. 4
Fig. 4

Rotation device. Test object is mounted on a precision ground steel plate. The plate is supported in such a way at three points that the object can be easily given known rotations by adjustments of the appropriate differential translators.

Fig. 5
Fig. 5

Experimental setup for a generation of homogeneous strains. A long thin-walled copper cylinder was suspended at its upper end, and its lower end was free to expand; the deformations were produced by internal heating with a temperature controlled device. The temperature of the cylinder was monitored by twenty thermocouples placed on the cylinder’s surface; their readings were recorded on a multichannel chart recorder.

Fig. 6
Fig. 6

Typical photographic recordings of a holographically reconstructed virtual image of a thermally deformed cylinder. Comparing both photographs it becomes obvious that the fringe patterns are slightly different; note the fringe location with respect to the grid pattern. Illumination and observation geometry was identical for both holograms, the only change was the temperature difference ΔT. The hologram in (a) records a cylinder at a higher ΔT than the hologram in (b).

Fig. 7
Fig. 7

Rotations of the object. The abscissa represents the actual rotations given to the object as read off the positioner’s differential translator; the rotations measured experimentally from holograms are plotted on the ordinate.

Fig. 8
Fig. 8

Normal strains of the object. The actual strains were determined from a temperature difference and coefficient of thermal expansion of the cylinder. The measured strains are those obtained directly from the holograms using the fringe-vector theory of strain analysis.

Fig. 9
Fig. 9

Formation of fringes in hologram interferometry: (a) object surface intersected by a set of unequally spaced curved fringe-locus surfaces (represented by spatially varying fringe vectors— K f 1 , K f 11 , , K f υ), and (b) object surface intersected by a set of equally spaced parallel fringe-locus planes (represented by a spatially constant fringe vector Kf).

Equations (10)

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Ω ( K , R Q ) = Ω ( K , R P + Δ R P Q ) = Ω ( K , R P ) + Δ R P Q · K f .
K f = K f x i ˆ + K f y j ˆ + K f z k ˆ = Ω x i ˆ + Ω y j ˆ + Ω z k ˆ ,
Ω = K · L ,
( K f ) = ( K ) [ f ] + ( L ) [ g ] .
[ f ] = [ e ] + [ θ ] = [ e x x e x y e x z e y x e y y e y z e z x e z y e z z ] + [ 0 θ z θ y θ z 0 θ x θ y θ x 0 ] .
( K ) [ f ] = ( K f ) ( L ) [ g ] .
[ K ] [ f ] = [ K f c ] ,
[ f ] = [ [ K ] T [ K ] ] 1 [ [ K ] T [ K f c ] ] .
e x x = e y y = e z z = α ( Δ T ) ,
e x y = e y x = e x z = e z x = e y z = e z y = 0.

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