separation of isochromatics and isopachics using a faraday rotator in dynamic-holographic...

Separation of Isochromatics and Isopachics Using a Faraday Rotator in Dynamic.holographic Photoelasticity The method reported herein yields the simultaneous recordings of the whole-order isochromatic- and isopachic-fringe patterns for a plane-dynamic problem

by J.P. Lallemand and A. Lagarde

ABSTRACT--A method is presented that allows the simultaneous separation of isochromatic- and isopachic-fringe patterns for transient-plane stress problems. Isopachic fringes are obtained by means of holography with a Faraday cell and a pulsed ruby laser flashing dual pulses. As usual isochromatic whole-order fringes are recorded in a circular-light polariscope. The shock generator (air-gun) and its synchronizing system with the ruby laser is described. The procedure is applied to the recording of the isochromatic- and isopachic- fringe patterns in a disk under radial dynamic loads.

Introduction It is well known that holographic interferometry ~.~

applied to a photoelastic model produces two inter- mingled fringe patterns. The patterns which are not always distinguishable from each other contain information relative to the sum (isopachic fringes) and the difference (isochromatic fringes) of the principal stresses everywhere in the model. In an ordinary holographic experiment wi th circularly polarized light beams, the first-order intensity is given by the following expression:

I = lo [1 + 2 cos - - - 7 cos + cos 2 ]

(1) where 4, and 42 are the phase shifts of light vibrations in the principal directions xl and x2 between the loaded and unloaded state.

The phase shifts are related to the principal stresses oi and a2 by the well known relations

2 we 4, + 45 = ~- C ' ( , r l+ , r~ ) (2)

27re 4 , - 4~ = ~ c ( ~ , - ~ ) (3)

where

e = model's thickness X = light source wavelength

C and C ' = optic-mechanical constants of the photoelastic material

The difficulty in understanding the fringe pattern resulting from eq (1) induced experimenters to look for

J.P. Lallemand is Assistant Professor and A. Lagarde is Professor, Laboratory o f Mechanics of Solids, University of Poitiers, 40 avenue du Recteur Pineau, 86022 Poitiers Cedex, France. Original manuscript submitted: Janaury 20, 1981. Final version received: May 21, 1981.

methods producing separated isochromatic and isopachic fringe patterns allowing quick and easy computation of the principal stresses) -5 One means of achieving separation is for the light to pass through the model twice. Just before going through the stressed model for a second time, the polarization of the light beam is rotated 90 deg. The isochromatics are thereby eliminated.

The first experiments based on this principle were performed in 1971 by O'Regan and Dudde ra : for static stress fields and by Holloway 7 for dynamic stress fields. They used a quartz crystal to rotate the light vector. In their procedure, light passed through the model once, was reflected and allowed to pass through the model a second time. The transmitted and reflected beam were slightly misaligned such that the reflected beam could pass through the quartz crystal prior to passing through Lthe model the second time. However, the positioning of the crystal is quite critical and the reflected beam is no longer normal to the model. Other methods have been developed to remedy this drawback:

A passive 90-deg rotator composed of a set of two half- wave retarders 9,'~ can be used instead of a quartz rotator.

A technique based on a Faraday's cell was used in 1968 by Chau, H in 1971 by Chatelain, ~~ and later in 1976 by Cadoret. '2 This method takes advantage of the fact that some materials are optically inactive in their natural state and gain a nonreciprocal rotatory power due to a magnetic field (Faraday's effect). The effect is given by the relation

R = 0 e H (4)

where R is the rotatory power, t is the length of the glass rod, H is the magnetic field and 0 is the Verdet's constant. The latter is connected with wavelength k by the relation 0 = A / k 2 (A = constant). If the beam goes through the material a second time, but in the opposite direction, it acquires the same rotatory power as on the first pass. The magnetic field must be such that it gives the beam a rotatory power of 45 deg.

In each case, the isochromatic fringes are obtained by means of classical photoelastic analysis of the light beam having gone through the model only once.

Another method used by Holloway 7 and Holloway, Ranson and Taylor '3 used a photoelastic model whose front surface has been made semireflective. The transmitted beam allows the classical recording of the isochromatic fringes while the reflected rays are stored in a double-exposed holographic plate. The holographic reconstruction gives the phase changes between the model 's normal state and its loaded state contained in the reflected

174 �9 May 1982

beam. They account for the perpendicular movement of any point on the metallic-coated surface. The fringes observed on the reconstructed image are actually isopachic fringes only if the middle plane of the model remains plane after loading. This restriction is removed by the method described here which allows some bending of the model.

Separation of Fringe Patterns by Means of a Light Rotator

Consider the equivalent holography configuration shown in Fig. 1. Circularly polarized light is used in both the object (r.) and reference (~R) beams. It is required that the two circularly polarized beams have the same sense of rotation. The light intensities are k2a 2 and k2aR 2 respectively in the first exposure and a 2 and aR 2 in the second exposure. The reference beam does not pass through the optical-light rotator but the rotator acts on the object beam during both the first exposure and the second exposure.

The vectorial-complex amplitude of the object beam emerging from the model in the unstressed state is given by

* kaD/2 [ ( c o s R + j s i n R ) ~ q + ( s i n R - j c o s R ) ~ c 2 ] :40 = 2

Upon emerging from the stressed model it is given by

= ~ [(cos RoeZi'h + j sin R .e ic~1§ L

(5)

+ (sin R .e j(*'+*2) - j cos R.eZi'~2)~c2] (6)

The vectorial-complex amplitude of the reference beam during the first and the second exposure is respectively:

~ 1 . o _ k a . ~ [~c,-jx2]* (7) 2

- I x , - J x~ ] = 2 k (8)

The vectorial-complex amplitude reaching the film at the first exposure is

~x fao+ ARo

where only the portions of the reference beam and object beam that have the same polarization will interfere.

Thus the light energy at the hologram plate is given by

eo = (~4o+" ( A o + . a . o ) ) t . o ) . * ' *

where ( )* denotes the complex conjugate of () . In the same way the light energy at the hologram plate

for the loaded exposure is given by

E = (.~ +~).(.~ +,~.)* The holographic plate has been subjected to two ex-

posures, and consequently the total exposure of the film is

E, = Eo + E

If the holographic reconstruction is made with a d~ t reference beam whose light vector is denoted by ARand

CIRCULARY POLARIZED LIGHT

(z) ~ x l \ ~xl HOLOGRAM

MODEL ' ' MODEL / ~ I R O T A T O ~

CI RCUI~RY~(~) POLAR IZE~D ~ R' LIGHT

Fig. 1--Principle of the method of separating the isochromatic-isopachic patterns by use of an optical-light rotator

neglecting any nonlinear effects of the film or recording process, the vectorial-complex amplitude of the transmitted light in the virtual image may be written as

-~ 4, $

and by using eq (8)

. ~ r = m[(k Ao + ~ ) . .~ .1~1~ (9)

where m is the slope of the transmittance curve. Equations (5), (6), and (7) combined with (9) allow the

light vector ~lr to be expressed:

1 ~lr = -~- m a aR[cos R ( 2 k ~ + eZi*~ + eZi~)

+ 2j sin R ( k 2 + e~(~,+~))]~l~

The light intensity I in the virtual image is obtained by the well known relation

i = ~ . ~*

and with

1 12

we have finally

k[~ 4 l _ k 2 I = Io " 2 + -~- sin2R sin R-sin(q~l +q~2-R)

+ k 2 cos R .cos(C, - q~2) �9 cos(q~, + ~b2 - R)

1 + ~- cos 2 R �9 cos2(th, - ~b2)] (10)

When R equals 90 deg, the intensity is given by

: go [ L _ ~ + k s cos(~, + ,~2)] I (11)

Equation (11) shows that the isochromatic term (~1- ~2) disappears from the relation (10) when R = 90 deg. Therefore, the resulting fringes are only isopachics and their contrast 3' is given by

2k 2 3 " - l + k 4

How the contrast, 7, varies with k is clearly indicated by the following tabulation,

Expe r imen ta l Mechan i cs �9 175

k 0.5 0.7 0.8 1 1.2 1.4 2

3" 0.47 0.79 0.91 1 0.93 0.81 0.47

Results show that the equality condition of light intensity of both the first and second exposure is not a critical condition. So if a value k = 1 is used, the light intensity I may be written:

I = Io[1 + cos(4h + qh)] (12)

For this condition, the intensity will be a minimum when

0t+4~2 = ~r+2nTr

where n is an integer.

General Principles of Separation by a Faraday Rotator in Dynamics

The procedure to be described is a whole-field method allowing simultaneous but separate recordings of isochromatic- and isopachic-fringe patterns for transient- plane stress problems. This is achieved at any given time during the propagation of the disturbance through the model. The method requires good reproducibility of the propagation phenomena (boundary conditions, stable material properties, etc.), as well as a perfect synchronizing of the recording apparatus and the shock generator. Successive experiments thus may yield the pattern of shock propagation for several instants of time.

The experimental setup is composed of a light source (a Q-switched ruby laser giving dual short pulses), a shock generator (air gun), a magnetic-optical rotator (Faraday's cell) and an optical bench.

The optical bench, at least the elements giving the isopachic fringes, has been used for static cases, but to date has apparently never been applied to dynamic problems. The system uses equipment necessary for holographic interferometry with a Faraday's cell to eliminate the isochromatic fringes from the interferometric recording.

The need to provide a large field for studying wave propagation requires an optical bench with large dimensions. Therefore, it was not poSsible for the optical elements, including the laser and the model, to be located on the same rigid structure isolated from the surrounding vibrations. A very short double-exposure time was necessary for the isopachic fringes which in turn required the position of the elements to be absolutely unchanged. This severe constraint prohibits the use of a single impulse laser for which the flashing rate is measured in tens of seconds. Instead, we use a laser giving two light pulses separated by an adjustable delay of 5 to 220 microseconds. The first pulse lights up the model just before being shot by the projectile and allows the first exposure of the unloaded model. A second light pulse (the equality in intensity of the two light pulses is not critical as has been shown in the previous section) is emitted while the disturbance wave is traveling through the model and allows a second' exposure of the loaded model. The procedure does not allow visualizing the wave more than 220 microseconds after the beginning of the shock. This is not a significant limitation in studying wave propagation in common materials.

As usual, the isochromatic fringes are obtained in a circular-light polariscope. However it should be noted that if a light-field polariscope is used, the first light pulse

will evenly expose the film while the isochromatic fringes exposed during the second pulse will have little contrast (ratio �89 For this reason it is preferable to use a dark- field arrangement so that the first pulse going through the unloaded model will not expose the negative plate. The fringes of whole-order obtained from the loaded model with the second pulse will then have the best contrast.

Description of the Experimental Device The apparatus is shown on the diagram in Fig. 2.

Light Source The short-duration light pulses of the ruby laser (RL),

about 20 nanoseconds, stops the dynamic event such that large fringe gradients are captured without loss of contrast due to the movement of the wave during the exposure time. The ruby laser has its own oscillator and an am- plifying stage in order to induce an energy of 1 joule for both pulses. The pulses are triggered by a Pockels' cell.

Shock Generator The method requires a reproducible shock-generation

procedure. Explosive loading methods such as lead azide or PETN on the model 's edge were determined to be un- suitable for several reasons. The shock might last longer than 10 microseconds, is destructive in the area of the explosion, and is not accurately reproducible. For these reasons, explosives are best used with a cinematographic recording system (an ultrahigh-speed framing camera or a multiple spark-gap camera).

A 4.5~mm bore air gun (AG) was developed and con- structed which used compressed air up to 10 bars in pressure that could propel steel projectiles (balls or cylinders) at speeds ranging from 20 to 100 meters per second with a reproducibility of about 1 or 2 percent. The shock duration in the photoelastic material (Araldite B or Hysol 4290) varied from 20 to 50 microseconds.

Optical-measurement Apparatus The main incident beam emitted by the ruby laser (RL,

Fig. 2) lights up the model (M) in parallel rays 210 mm in diameter at normal incidence. The beam is reduced to a 25-mm diameter going through the Faraday's cell (FC). A beam splitter (BS2) is set up exactly at the focal point of the image of the model M formed by the lenses L3 and L4 and the cell (FC). The latter condition is a requirement because for a noninterferometric quality model, an incident beam emerging from the model should return to exactly the same point on the way back after reflection from BS2. The dioptric effect of the model (the thickness of which may vary no more than about 1/10 mm) is then well balanced. The beam having gone back through the model is reflected from beam splitter BS1 and arrives at the ground glass (GG) set in the model 's image plane defined by lenses L1, L2 and L5. The model 's image is thus collected on the ground glass which in turn diffuses light towards the holographic plate (H). In this way every portion of the hologram includes information for the image of the whole model. This is a widely used procedure which allows for better holographic reconstruction com- pared with a recording made in direct light (without ground glass).

A reference beam (RB), following a path (not shown in Fig. 2) of equal length to the working beam, impinges on

176 �9 May1982

.

�9 TS ~AG RL F~< I I I

"2 " L2 !L3 I BS1 L1 ~ L4

21 �88 H

L7 L8

HE-NE I

/_.A~S2 , A L6 PP

Fc

Fig. 2--Experimental setup to simultaneously obtain separate isochromatic and isopachic patterns for dynamic loading using a Faraday rotator

I/4 QUARTER WAVE PLATES M MODEL

A ANALYSER FC FARADAY CELL

BS BEAM-SPLITTERS RL RUBY LASER

GG GROUND GLASS AG AIR GUN

H HOLOGRAM HE-NE HELIUM-NEON LASER

RB REFERENCE BEAM PC PHOTOCELL

L LENS TS TRIGGERING SYSTEM

PP PHOTOGRAPHIC PLATE ~ MIRRORS

the holographic plate (H). The holographic reconstruction is achieved by a 2 mW Helium-Neon laser.

A portion of the working beam is circularly polarized before passing through the model, then proceeds through beam splitter BS2, is directed through a quarter-wave plate (M4), a rectilinear analyzer (A) and yields the whole-order isochromatic fringes on a photographic plate (PP). Of course (PP) stands in the model's image plane given by the optical elements within (M) and (PP).

The photographic emulsions are: --holographic plates AGFA 10E75 for the holographic

recordings --flat films AGFA PAN 400 for the recording of the

isochromatics and of the holographic photographs.

The mirrors and beam splitters are glass plates 10-ram thick, coated with multidielectric layers of interferometric quality for the specific wavelength: X = 6943 ,~. Several negative lenses (L1, L 4 . . . ) are used to avoid the problems caused by ionizing the atmosphere at points of beam focus.

Faraday Rotator The rotator is a solenoid supplied by a d-c generator

of 80 V and 100 amperes with a tolerance of one percent in which is positioned a glass rod SF 57 of 35-mm diameter and 180-ram length. In order to obtain 90 deg of rotation, a magnetic field of 3200 Gauss is produced by a current intensity of 70 amperes. Even though the system is water cooled, the working time must be limited to one minute.

The adjustment of current intensity is achieved by placing a statically loaded model in the field of the experimental setup. However, the optical elements are somewhat rearranged for that purpose. A quarter-wave plate and an analyzer are located between BS1 and L5 on the return beam and a flat film is used instead of the ground glass GG. By this procedure a circular-light polariscope in light field is produced with a beam going twice through the model. The solenoid current intensity is adjusted so that the half-order isochromatic fringes disappear from the model's image when recorded on the film. The schematic diagram of this arrangement is shown in Fig. 3.

POLARIZER xz x2 , ~ .~xl f4~o ANALYSER

X MODEL MODEL X - FARADAY CELL - 4 4

Fig. 3--Schematic diagram to adjust the current intensity of the solenoid

Experimental Mechanics �9 177

Y

1

0 . 1 _ ~ ,

0 90 ~ 180 ~

Fig. 4--Contrast of the half-order fringes vs. rotatory power R

For the arrangement of Fig. 3, the light intensity of the beam impinging on the film is given by

I ' = a2[sin 2 R + cos 2 R o cos2(4,, - 4'2)1 (13)

where a 2 is the energy of the incident light, R, 4,1 and 4,2 are defined in previous sections. Let 7 ' be the contrast of fringes defined by:

z '~ 7 t - -

! i l,,,,~x + lmin

Using eq (13), 7 ' may be written as

1 - sin 2 R 7 ' -- 1 + sin 2 R

(14)

The curve of 7 'versus R is drawn in Fig. 4. Examination of this curve shows that it is not necessary

to adjust the current intensity of the solenoid in order to obtain exactly a 90-deg rotation. It has been found that a current intensity ranging from about 50 to 80 amperes allows the isochromatic fringes to be eliminated.

Synchronization of the Air-Gun with the Ruby Laser The recording of transient phenomena traveling at a

high speed presents a difficult problem of synchronization. The projectile of the air gun triggers the action of the

ruby laser. Also, the projectile interrupts the light beam of a Helium-Neon laser twice (Fig. 2) and allows the velocity of the projectile to be measured. The signal given by the photocel ! (PC) is transmitted to triggering system (TS) which is a high-speed computer made with integrated circuits in T.T.L. logic. In this manner the trigger impulse is controlled by the projectile velocity. This setup allows the wave to be accurately captured at a single position in the model at the moment of film exposure and gives good reproducibility in spite of the fluctuation of the projectile velocity from one shot to the other.

Further information concerning the synchronizing problem will be found in other works of the author. 14-'5

Results The experiments were conducted to visualize the iso-

chromatic- and isopachic-fringe patterns in a dynamic problem. The model was a disk of araldite B (Hysol 4290), 180-mm diameter and 5-mm thick, tested by radial impact. The projectile was a ball 4.5-mm diameter with a velocity of 54 m/s. In order to avoid destruction of the model, a I-ram thick piece of metal made from an aluminum alloy was glued to the edge of the model. A new piece of metal was used for each shot. This operation did not significantly alter the boundary conditions.

Figures 5a and 5b are examples of exposures showing isopachic fringes and whole-order isochromatic fringes recorded 44 microseconds after impact. The resulting isochromatic- and isopachic-fringe patterns are shown together in Fig. 6. The principal stresses al and a2 may be computed using eqs (2) and (3) providing that the optic- mechanical constants C and C ' are known.

The ratio C ' / C can easily be obtained by observing the isochromatic- and isopachic-fringe patterns on the free edge of the model where the stress state is one dimen- sional. At the point where the isochromatics are of k- order and the isopachics are of k ' -order we have,

Fig. 5--Simultaneous isochromatic (a) and isopacmc (o) range patterns in a disk subjected to a radial impact for a delay of 44 microseconds, recorded by the Faraday rotator technique

178 �9 May 1982

.... ISOPACHIC FRINGES

...WHOLE-ORDER ISOCHROMATIC FRINGES

C

i / ' / / I I I , "

I I , I 1 '

l lk ' I7; ,' ',', fhl / t I I / r / , ,s;, i, i f . ; I i i 1 ! 11 I, I { , I ! ( t I I I

\ \ ~ \ \ \ \ \ \ 1

7~, \ \ N

\ ~- x �9

~)180

MPACT Fig. 6--1sochromatic-isopachic fringe patterns derived from Figs. 5(a) and (13)

2 ire ~ - 4 ~ 2 = ~ C e s = 2 k T r

2 r e 4~ + 0~ = - ~ - C ' as = 2 k ' Tr

A

where trs is the circumferential stress. The ratio of the above two equations leads to

k ' C ' k C

Study of Fig. 6 indicates the ratio C ' / C is about 4. With C = 50o10 -59 Pa- ' (a value elsewhere defined ~4 and corresponding to what is currently used for araldite B), we find C ' = 200.10 -'2 Pa-L

Conclusion The results reported herein represent the first successful

attempt in separating isochromatic and isopachic fringes for dynamic problems using a Faraday rotator in holographic interferometry. The method used yields the simultaneous recordings of the whole-order isochromatic- and isopachic-fringe patterns for a plane dynamic problem.

The arrangement of optical elements for a holographic bench of large dimensions would not have been possible without the use of a ruby laser which generated dual light pulses a very short time apart. Furthermore, the short duration of these pulses allows the dynamic process to be captured and allows the recording of large fringe gradients without contrast loss due to the wave displace- ment during the exposure time. it is important to note that the division of the light beam into three parts and the loss of energy in the beam splitters required the use of a ruby laser equipped with a n amplification stage for delivering high energy levels.

The visualization of a propagating wave was based upon the ability of performing repeated experiments. For this reason it was necessary to develop a shock generator

with good reproducibility of the boundary conditions. For the same reason, it was necessary to have a high-perfor- mance triggering system for the light source and the air gun.

References 1. Fourney, M.E., "Application of Holography to Photoelasticity, ""

E X P E R I M E N T A L M E C H A N I C S , 8 (1), 33-37 (1968). 2. Hovanesian, J.D., Brcic, V. and Powell, R.L., "A New Experi-

mental Stress-optic Method." Stress-Holo-lnterferometry, " " EXPERI- MENTAL MECHANICS, 8 (8), 362-368 (1968).

3. Sanford, R.J. and Durelli, A.J., "'Interpretation of Fringes in Stress-Holo-Interferometry, '" EXPERIMENTAL MECHANICS, 11 (4), 161-166 (1971).

4. Holloway, D.C. and Johnson, R.H., "Advancement in Holographic Photoelasticity, "" EXPERIMENTAL MECHANICS, 11 (2), 57-63 (1971).

5. Sciammarella, C.A. and Quintanilla, G., "'Techniques for the Determination of Absolute Retardation in Photoelasticity,'" EXPERI- MENTAL MECHANICS, 12 (2), 57-66 (1972).

6. O'Regan, R. and Dudderar, T.D., ",4 New Holographic lnterferometer for Stress Analysis,'" EXPERIMENTAL MECHANICS, 11 (6), 241- 247 (1971).

7. Holloway, D.C., "'Simultaneous Determination of the lsopachic and Isochromatic Fringe Patterns for Dynamic Loadings by Holographic Photoelasticity, "' T. & A.M. Report (349) (Aug. 1971).

8. Hovanesian, J.D., "Elimination of lsochromaties in Photo- elasticity, "" Strain, 7 (4), 151 (1971).

9. Gasvik, K., "'Separation of the lsochromatics-Isopachics Patterns by Use of Retarders in Holographic Photoelasticity, "" EXPERIMENTAL MECHANICS, 16 (4), 146-150 (1976).

10. Chatelain, B., "'Etude expErimentale de deux m~thodes d'observation simultande ou inddpendante des rdseaux d'isochromes et d'isopaches relatifs b u n moddle unique soumis b une seule sollicitation; '" Thdse de 3~me Cycle, Besangon (1972).

11. Chau, W.M., "'Holographic lnterferometer for lsopachic Stress Analysis, "Rev. Sci. Inst., 39 (12), 1789-1792 (1968).

12. Cadoret, G., "'Contribution Zi l'utilisation de m~thodes de l'optique cohErente clans l'analyse exp~rimentale des contraintes,'" Th#se de Docteur-Inge~nieur, Paris VI (1976).

13. Holloway, D.C., Ranson, W.F. and Taylor, C.E., "A Neoteric Interferometer for Use in Holographic Photoelasticity, "" EXPERIMENTAL MECHANICS, 12 (10). 461-465 (1972).

14. Lallemand, J.P., Interferometrie Holographique et Determination des Lois de Comportement Mdeanique et Optico-M~canique, en Rbgime Transitoire, "' Thdse de Doctorat bs Sciences, Pottiers (1980).

15. Lallemand, J.P., "Synchronization Problems in Dynamic Holo- Ph'otoelasticity," published in EXPERIMENTAL MECHANICS, 21 (12). 477- 480 (198I).

Experimental Mechanics �9 179

separation of isochromatics and isopachics using a faraday rotator in dynamic-holographic...

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