Inplane Displacement of a Stressed Membrane witha Hole Measured by Holographic Interferometry

A. D. Wilson

The application of holographic interferometry techniques to measure the deformation parallel to thesurface of a membrane stressed parallel to its surface is described. The holographic technique used isconventional and is not insensitive to motions orthogonal to the applied stress; however, the object, arubber membrane, is such that the principal object deformation is in the plane of the membrane and isthus revealed in the holographic image without any data reduction. The holographic results are com-pared with computed displacement distributions obtained from integrations of classical solutions of thestress equation. The agreement ofgood.

the holographic and theoretical stress equation solution is very

Introduction

Holographic interferometry ' 6 has generally begun toemerge from the optical laboratories. Yet, in the areaof plane stress analysis, it has not replaced more con-ventional techniques such as photoelasticity. 7 Twopublications-by Ennos and by Aleksandrov andBonch-Bruevich-have illustrated the application ofholographic interferometry to a plane stress problem. 8' 9

This paper discusses the application of conventionalholographic interferometry to measure the deformationparallel to the surface of a membrane with a small holeand stressed parallel to its surface. From the holo-graphic images obtained, the inplane displacements areevident without any data reductions. This is notbecause the holographic technique employed is insen-sitive to motions orthogonal to the applied stress, butrather, because the object chosen has major deforma-tions parallel to the direction of applied stress. It isshown by comparison with displacement distributionscomputed by integration of classical solutions to thestress equation that the observed holographic fringesare in fact the inplane displacements.

The purpose of this paper is: (1) to show a simpleinplane stress example where holographic interferometrycan be useful without performing tedious computationsto get inplane displacements, (2) to present a com-parison of observed and expected fringes, and (3) tocontribute to the increasing number of papers dealing

The author is with IBM Systems Development Division,

Endicott, New York 13760.Received 11 September 1970.

with a theoretical and/or experimental verification ofholographic interferometry fringes.'10

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Analysis

Analytical Solution

A membrane of thickness t (Fig. 1), length L, andwidth 2b with a hole of radius a is placed under stressby the application of a force S acting in the x direction.The theoretical displacement distribution resultingfrom the force S is obtained by a plane stress analysis. 15

A stress function of the form

D = f(r) cos2o, (1)

which satisfies the compatibility equation

(a2 +1 a + 1 a0a6 1 as , 1 aft _( r' r Zr r' a c2/\ 1r' r or 1.2 b2 /(2)

is sought. The solution is

(r = (S/2)[(1 -p2) + (1 + 3p4-4p2) cos2O],

as= (S/2)[(1 + p2) - (1 + 3p4) cos2O],

T= -(S/2)[1 - 3p4 + 2p2] sin2G,

p = a/r, (3)

where or and o- are the radial and circumferential com-ponents of the stress (in polar coordinates), and rao isthe shearing stress component. Saint-Venant's prin-ciple is employed to obtain this solution-the hole hasnegligible effect on the stresses at distances that arelarge with respect to the hole size a.

The strains in the radial and theta directions areobtained from the stresses [Eq. (3)] by Hooke's laws:

e, = (l/E)(ar- ao)

908 APPLIED OPTICS / Vol. 10, No. 4 / April 1971

L points formed by dividing the radial coordinate of_ ~ length (b - a) into an integer number of equal size

increments and performing the same operation for thetheta coordinate between 7r/2 and 0. A typical number

_ v _ of points used is 400 per quad. Having computed Uand Vo, a transformation to cartesian coordinates is

s r ) made to obtain U and V. Contour intervals areselected, and lines of constant displacement in the x ory directions are computer plotted.

Holographic TechniquesIn the technique of holographic interferometry the

_ fringe system can be understood effectively as an inter-x ference between two or more wavefronts emanating

(a) from the same identifiable point on the object, but withan induced change in phase between the two or morewavefronts. A change in phase could be due to objectmotion, deformation, or optical path changes in general.To determine the optical data change due to object defor-imation, refer to Fig. 2, where a general diffuse surface is

2b shown in two states Q and Q2. An identifiable pointP is shown in the two states P1 and P2. Vector dp joins

t Xt a + . 1 P1 and P2. The optical path difference between wave-fronts traversing AP 1Bi and A2P2B2 is nŽ.l, where n is

Y U t y / the refractive index and Al the physical path difference.

s d x n

/~~~~~~~~~~~~~~~~~~~~ /c |\ oA 2SOURCE AND VIEWER B

Recver

(b)

Fig. 1. (a) Coordinate definitions. (b) Membrane and opticalgeometry. Q2

and Q,e = (1/E)( 0 - vo-r), (4) Fig. 2. Geometry of surface deformation and fringe formation.

where E is the modulus of elasticity and v is Poisson'sratio. The radial- and theta-directed displacementsare then obtained from a

Ur = fedr + h(o) 7 /and V = f(o - U)do + g(r). (5) &

The constants of integration h(o) and g(r) are found to \ \be zero by considering the solution as p - 0. Thus, the \Kdisplacements in polar coordinates are e dUr = (Sr/2E){(1 + p2) - V(1 - p2))/

+ [ - p4 + 2p + (1 - p4 )] cos2O},V= -(Sr/E)[(1 + p + p) + (1 + p4

- 2p2)] sin2O. (6) Fig. 3. Experimental holographic interferometer: a, lasersource; bmembrane; c, imaging lens; d, emulsion; and e,Equation (6) is evaluated numerically for a grid of reference wave.

April 1971 / Vol. 10, No. 4 / APPLIED OPTICS 909

Fig. 4. Test apparatus: A, membrane; B, moveable x stage;C, micrometer; D, mount; and E, stationary membrane end.

The physical path is given by the projection of d, ontothe directions i and r, which are unit vectors directedtoward the source illuminator and receiver, respectively.Interference between another pair of points, for ex-ample O1 and P2 or P1 and 02 does not occur withsufficient regularity for the formation of a visiblemacroscopic fringe. Hence, for the visible fringes,there exists pointwise a one-to-one correspondence forinterference between the surface in states Qi and Q2.The optical phase difference A4 corresponding to d, is

AO = (27 r/Xo)nAl,

where

Al d.(i + r)

and No is the vacuum wavelength of the illuminatingsource. Hence,

X = (27rn/Xo)d.(i + r). (7)

Fringes, regions of constant phase difference A4, aredependent not only upon the magnitude of the dis-placement vector do, but also upon its direction relativeto the illuminator and receiver directions. If the unitvectors i and r are at an angle 0 with respect to thedisplacement vector d, as shown in Fig. 1(b), then

AO = (4vn/Xo)1dp| cosO. (8)

For a stressed membrane, d lies mostly in the plane ofthe membrane. Thus, by selecting 0 to be less than7r/2 rad (ideally 0), the inplane displacement can beeasily observed.

Experimental

Fresnel carrier-beam holograms of the membrane arerecorded on Eastman Kodak 649F emulsion, using theoptical arrangement of Fig. 3. Coherent 632.8-nmlight is used. The holographic exposure is made in twosteps: the first half of the exposure is made of theobject, a membrane, with an arbitrary force applied asillustrated in Fig. 1(b); the second half of the holo-graphic exposure is made after the membrane isstretched a known increment by the application of anadditional force increment. Hereafter, this procedurewill be termed double exposure. The photographicemulsion is developed, stopped, fixed, washed, anddried. After the emulsion has dried, it is reilluminatedwith a wavefront similar to the reference wave ofFig. 3. The reconstructed wavefront is then observedand/or photographed with a conventional camera.

The object is a thin (0.006-cm) membrane of RTV(room temperature volcanized) white silastic rubbermounted on a fixture (Fig. 4) which can apply a dis-placement at one end of the membrane in increments assmall as W.'s of the wavelength of the light illuminatingthe membrane. The only motion imparted to themembrane is in the x direction (see Fig. 1). Theilluminator and receiver are in the xz plane at an angleof 0 with respect to the imparted displacement. Thesurface texture of the silastic material, when undertensile stress, is such that it acts as a diffuse reflectorwithout the addition of any retroreflective (or other)paints or surface treatment such as sandblasting. Themembrane is placed initially in the fixture with a slighttension on it. This prevents any bowing of the mem-brane and also tends to convert the smooth cast surfacesof the silastic from a specular to a diffuse reflector.The membranes are cast between two flat pieces of glassplates spaced apart by equal thickness shims. Themembranes, approximately 1.2 cm wide by 2.0 cm long,have either no holes, one hole, or two holes punched inthem..'Image holography, a lens between the object and theglass backed emulsion, is used to increase the object'sradiance at the emulsion by forming an image of the

S

Fig. 5. Theoretical x direction displacement distribution.

910 APPLIED OPTICS / Vol. 10, No. 4 / April 1971

object near the emulsion. For the 649F emulsion,3-mW He-Ne laser, and optical arrangement of Fig. 3,the total holographic exposure is 20 sec.

Results

A theoretical x direction displacement distributioncomputed by use of Eq. (6) is shown in Fig. 5. Pois-son's ratio and the modulus of elasticity are equal to0.5 and 1000, respectively. The applied stress is 2,and the hole diameter is 0.2 of the membrane width.The total number of computation points is 1600. Con-tours are drawn every 1/30 of the total displacement.

The stretched membrane with a hole and a singleexposure [Fig. 6(a) ] shows no fringes; rather, a uniformmembrane radiance is evident. Double-exposure holo-grams with nominal displacements between exposuresof four, eight, and twelve wavelengths (100, 200, and300 in.), reveal a displacement concentration [Figs.6(b)-6(d), respectively] in the neighborhood of thesingle hole. A membrane with no holes and stretchedsix wavelengths between exposures, Fig. 7(a), has auniform displacement distribution. Figure 7(b) shows amembrane with two holes and stretched eight wave-lengths between exposures. The amount of differentialdisplacement in the plane of the membrane corre-

sponding to one fringe (bright to bright or dark to dark)is approximately 0.8 of a wavelength (20 in.).

Discussion

The model (Fig. 2) for interpretation of the fringesdue to inplane displacements, while relatively simple,does indeed suffice for a first order analysis of the fringes.For example, for Fig. 6(b) the applied displacement isfour wavelengths and the number of differential dis-placement fringes (bright) is five in agreement with theexpected number. Moreover, the distribution of dis-placement in the neighborhood of the hole [e.g., Fig.6(d)] is that expected (Fig. 5). Note: Fig. 5 hasthirty contour intervals, and by letting alternate con-tours be bright fringes the similarity to Fig. 6(d) be-comes apparent. The fringes do not and should notnecessarily intersect normally the hole boundary. Thehole in the membrane is slightly oval because of theinitial stretching of the membrane. Fringe visibilitytends to decrease as the total applied displacementincreased.

While the applied stress is in the x direction theresultant displacement of a point P (Fig. 2) is notentirely in the x direction because the membrane shrinksin the y and z directions when it is stretched and thus

Fig. 6. Stretched membrane with hole: A-single exposure; B-double exposure, four wave-lenths of displacement between exposures; C-double exposure, eight wavelengths of displace-ment between exposures; and D-double exposure, twelve wavelengths of displacement

between exposures.

April 1971 / Vol. 10, No. 4 / APPLIED OPTICS 911

Fig. 7. Stretched membrane: A-no hole, six wavelengths ofdisplacement between exposures; and B-two holes, eight wave-

lengths of displacement between exposures.

d, (Fig. 2) has a component directed in the y and zdirections. The magnitude of the z component isapproximately 0.01 of the x direction component.Therefore, the maximum z component of d, is about0.12 wave and thus insufficient to contribute to thefringe pattern. We verified this by viewing the mem-brane normally [0 = 900 in Fig. 1 (b) ] and observed nofringes for 12X of displacement in the plane of the mem-brane. Due to a shrinking of the membrane in the ydirection, fringes are not observed with the illuminatingand viewing arrangement of Fig. 1(b). These fringesare observed by illuminating and viewing in the y-zplane instead of the x-z plane.

The author gratefully acknowledges sample prepara-tion by W. C. Hamm, model making by C. Fetcinko,and suggestions offered by P. A. Engel, J. P. Kirk,C. H. Lee, P. J. Magill, B. D. Martin, and T. Young.

References1. M. H. Horman, Appl. Opt. 4,333 (1965).2. R. J. Collier, E. T. Doherty, and K. S. Pennington, Appl.

Phys. Lett. 7, 223 (1965).3. R. E. Brooks, L. 0. Heflinger, and R. F. Wuerker, Appl.

Phys. Lett. 7, 248 (1965).4. R. L. Powell and K. A. Stetson, J. Opt. Soc. Amer. 55, 1593

(1965).5. B. P. Hildebrand and K. A. Haines, Appl. Opt. 5, 172 (1966).

6. L. 0. Heflinger, R. F. Wuerker, and R. E. Brooks, J. Appl.Phys. 37, 642 (1966).

7. M. M. Frocht, Photoelasticity (Wiley, New York, 1940).8. A. E. Ennos, J. Sci. Instrum. 1, 731 (1968).9. E. B. Aleksandrov and A. M. Bonch-Bruevich, Sov. Phys.

Tech. Phys. 12,258 (1967).10. J. E. Solid, Appl. Opt. 8, 1587 (1969).11. G. M. Brown, R. M. Grant, and G. W. Stoke, J. Acoust.

Soc. Amer. 45, 1166 (1969).12. K. A. Stetson, Optik 29, 386 (1969).13. J. Tsujiuchi, N. Takeya, and K. Matsuda, Opt. Acta 16,

709 (1969).14. T. Tsuruta, N. Shiotake, and Y. Itoh, Opt. Acta 16, 723

(1969).15. S. Timoshenko and J. N. Goodier, Theory of Elasticity (Maple

Press, York, Pa., 1951).16. P. J. Magill and A. D. Wilson, J. Appl. Phys. 39, 4717

(1968).

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