Holographic photoelasticity: Determination of absolute retardations by a single hologram

In this paper, real-time holographic interferometry is applied to determine the absolute-retardation fringes in the photoelastic analysis of plane models. The method requires only one hologram recorded with the model unloaded. In the reconstruction process, the mcdel is loaded and the polarizers on the reference and object beams are rotated through the same angle. At the points of the model where the polarization direction is parallel to one of the principal stresses, only the corresponding family of absolute retardations is obtained. The technique is demonstrated experimentally using a deep beam centrally loaded.     

Interpretation of fringes in stress-holo-interferometry

Previous studies had concluded that stressholo-interferometry patterns consist of the independent superposition of the isopachic family (with half-order fringe shifts) and the isochromatic family. It is shown here that this interpretation is not always valid and can result in serious errors in some cases. In particular, it is demonstrated that the position, and even the existence of the fringes, are affected by the interaction of the isopachics and isochromatics. This effect is most pronounced when the two families of fringes are nearly parallel and of approximately the same spatial frequency. The independent superposition interpretation is most accurate when the two families of fringes are orthogonal, whatever the ratio of spatial frequencies might be. These properties are illustrated using computer-generated holographic interference patterns.     

Circularly polarized plane waves in a deformed Hadamard material

Small-amplitude inhomogeneous plane waves propagating in any direction in a homogeneously deformed Hadamard material are considered. Conditions for circular polarization are established. The analysis relies on the use of complex vectors (or bivectors) to describe the slowness and the polarization of the waves.

Generally, homogeneous circularly polarized plane waves may propagate in only two directions, the directions of the acoustic axes, in a homogeneously deformed Hadamard material. For inhomogeneous circularly polarized plane waves, the number of possibilities is far greater. They include an infinity of ‘transverse waves’, as well as ‘longitudinal waves’, and the superposition of transverse waves and longitudinal waves, where ‘transverse’ and ‘longitudinal’ are used in the bivector sense.

Each and every possibility of circular polarization is examined in turn, and explicit examples of solutions are given in every case.     

Determination of absolute retardations by vectorial reconstruction

Summary In this paper is described the application of the vectorial reconstruction to the holographic determination of absolute retardation fringes. The method requires a double-exposure hologram recorded with two reference beams. During the reconstruction process the image emerging from the hologram is viewed through a linear polarizer. Where the polarization direction is parallel to one of the principal stresses, only the corresponding family of absolute retardations is observed. By rotating the polarizer it is possible to observe the fringes over the whole model. The method is demonstrated experimentally using a deep beam centrally loaded.

---

Sommario In questa memoria si applica la ricostruzione vettoriale alla determinazione delle frange dei ritardi assoluti. Il metodo richiede l'esecuzione di un ologramma ad esposizione doppia registrato con due raggi di riferimento. In fase di ricostruzione l'immagine emergente dall'ologramma viene osservata attraverso un polarizzatore lineare. Nei punti del modello in cui la direzione di polarizzazione è parallela ad una tensione principale, si osserva la corrispondente famiglia dei ritardi assoluti. Ruotando il polarizzatore è quindi possibile osservare le frange dei ritardi assoluti in tutto il modello. Quale esempio di applicazione si riporta la determinazione delle frange dei ritardi assoluti in una trave corta caricata in mezzeria.

---

---

Paper presented at IV AIAS Conference held in Roma on September 30 – October 2, 1976.     

Full-field surface-strain and displacement analysis of three-dimensional objects by speckle interferometry

A method of full-field measurement of displacements as well as strain on arbitrarily curved surfaces is introduced. The speckle effect of coherent light is utilized to produce fringes due to displacements. Unlike the fringes produced by holographic interferometry, these fringes have a unique interpretation in relating to displacements and they localize on the surface. Three measurements are required to determine the three components of displacement; and, knowing the geometry of the object, its surface strains can be deduced. Three ways of recording displacement fringes, namely, real time, double exposure and superposition, are described.     

A neoteric interferometer for use in holographic photoelasticity

A shortcoming of the conventional holographic interferometer used in photoelasticity is that, for the double-exposure cases, the resulting fringe patterns are a complex combination of conventional isochromatic and isopachic fringes. This paper describes a holographic interferometer that may be used to obtain separate but simultaneous isochromatic- and isopachic-fringe patterns for photoelastic models in states of plane stress. The method requires a model with a partially reflecting front surface. Isopachics, which are proportional to the thickness change, are recorded using holographic interferometry from the transmitted light. The isochromatics are obtained from the transmitted light by conventional means. General equations relating the surface displacement of the specimen to the observed fringe patterns are developed, and examples of static and dynamic loadings are shown.     

On the effect of quarter-wave-plate errors in stress-holo-interferometry

The purpose of this paper is to analyze the influence of imperfect quarter-wave plates on the fringe configuration which is obtained in holographic interferometry. It has been found that isopachics may be substantially shifted by very small errors in the retardation of the quarter-wave plates. An alternative multiexposure technique using plane-polarized light is proposed. This technique eliminates the quarter-wave-plate error and preserves the full-field character of the interferometic method. The technique is proven analytically and experimentally.     

Further applications of holography to photoelasticity

A holographic method for reconstructing the polarization of light emitted by a photoelastic model is presented. Isochromatic and the entire family of isoclinic fringe patterns may be obtained by examining the holographic image, a posteriori. This is accomplished by using two orthogonally polarized reference beams so arranged that the reconstructed crosstalk images do not overlap. An analysis is preseted crosstalk images do not overlap. An analysis is presented which shows that, to reconstruct polarization, the phase relationship between the two reference beams must be maintained. This requirement is more stringent than that normally required in holography. Experimental verification is presented.The use of double-exposure holography to obtain isopachic-isochromatic fringe patterns in the reconstructed image is discussed. An analytic treatment of the method is presented. Expressions relating the isopachic-isochromatic fringe pattern in the reconstructed virtual image to the principal stresses are developed. Differences between this analysis and that presented in a recent paper are discussed. Experimental results are obtained and compared to theory and to other experimental results with good agreement.Paper was presented at 1969 SESA Spring Meeting held in Philadelphia, Pa., on May 13–16.The study reported herein was supported in part by NASA Grant No. NsG-401.     

Computer aided photoelasticity by an optimum phase stepping method

In this paper an automated photoelastic method based on the phase stepping technique is described. It provides full-field maps of the isoclinic parameter and the relative retardation. The technique is based on processing six images of a photoelastic specimen acquired using plane and circularly polarized light. The number of acquisitions and the type of polariscope used in this approach have been chosen with the aim at reducing the influence of quarter wave plate errors and obtaining raw photoelastic data in a periodic form suitable for easy applications of automatic unwrapping routines.     

A new holographic interferometer for stress analysis

In holographic interferometry with a photoelastic model, two families of fringes are generated simultaneously when the model is stressed. One family represents the isochromatic-fringe pattern normally associated with photoelasticity which yields the difference between the principal stresses. The other family represents the isopachic-fringe pattern associated with interferometry which yields the sum of the principal stresses. From these complementary patterns, the magnitudes of the principal stresses can readily be determined throughout the field of observation. Unfortunately, these fringe patterns are not completely independent but interact in such a way as to make interpretation difficult in critical regions of the model. A new system has been developed which readily permits simultaneous acquisition of these fringe patterns without their undesirable mutual interaction, as well as providing increased sensitivity. This new interferometer uses a double-pass object beam and an optical rotator to eliminate the isochromatic-fringe pattern and its effect from the isopachic interferogram. Such a system has considerable value in experimental mechanics for applications to both static and dynamic model studies and to materials investigation.     

On the limitations of interferometric methods in three-dimensional photoelasticity

The photoelastic response of tinree different epoxy materials was measured. Tensile strips, beams, disks and spheres were loaded and the deformations “frozen in.” Slices from these models were used to calibrate and analyze the absolute retardation of the materials. It was found that the absolute retardation of light vibrating in a given principal direction was directly proportional to the principal strain in that direction and to the deviatoric components of stress in that direction. This result was a consequence of a unique value of the ratio of the absolute-retardation constants and is contrary to published suggestions that absolute-retardation methods could be used to separate normal stresses in the interior of stress-frozen models.     

Rectilinear and circular analysis of a plane slice optically isolated in a three-dimensional photoelastic model

Because of the coherence of scattered light, it is possible to produce a speckle image from a plane beam of light passing through a transparent model. When two plane parallel beams of light are transmitted through the model the slice between the beams is then optically isolated. The two speckle patterns corresponding to the two beams are superposed and provide optical data relative to the slice (principal stress directions, birefrengence), the data being collected on high contrast photographic plates or by optical filtering to obtain the square of the contrast. The isoclinic and isochromatic fringes are shown to exist. The concepts of rectilinear or circular analysis are extended to the observation of a plane slice in a three-dimensional model without freezing or cutting the model.     

Limitation of fourier transform photoelasticity: Influence of isoclinics

The application of the Fourier transform to photoelasticity was used in the evaluation of the retardation using a carrier system of fringes. In photoelasticity, the light intensity from the analyzer in a circular polariscope depends on both the retardation (isochromatics) and the isoclinic parameter. The theoretical analysis shows that the angle between the principal stresses in the model and in the carrier system of fringes influences the evaluation of the retardation (isochromatics), as occurs when misaligned compensators (namely, Babinet) are used. As a consequence, this method may not be applied as a full-field technique, although the error is small if the angle between the principal stresses in the model and in the carrier is less than 25 deg. Numerical simulations and experimental tests were conducted to corroborate this prediction.     

Variable isochromatic/isopachic-fringe visibility in photoholoelasticity

This paper presents a new “hybrid” method whereby the ratio of the isochromatic-fringe visibility/isopachic-fringe visibility may be easily and continuously varied. This simple procedure merely combines a conventional polariscope with a holographic system. A variable beam splitter permits an incoherent superposition of the reconstruction of a doubly exposed hologram with real-time isochromatics, either dark or light field. By varying the ratios of the above two, in the image plane, numerous interesting results may be obtained including:

1. Isochromatics only, without errors in position
2. Isochromatics-isopachic fringes identical to those obtained through classical interferometry
3. Isochromatic-isopachic fringes whereby the amplitude modulation between the two may be minimized
4. Continuously variable isopachic/isochromatic-fringe visibility.

     

Moiré-fringe multiplication by means of filtering and a wave-front reconstruction process

Two principal problems associated with the practical application of moiré fringes are to obtain sufficient sensitivity for measuring small strains and to develop a simple and inexpensive technique for engraving lines on the surface of a model. This paper deals with simple solutions to both problems. It is shown that the maximum number of fringes that can be observed for a given model is independent of the gratings utilized and depend only on the geometry of the employed optical system. Examples of moiré patterns corresponding to the equivalent of 6000 lines per inch and patterns of the derivatives of the displacements corresponding to the same number of lines illustrate the paper.     

Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing

The authors show a nondestructive method for obtaining the isochromatic and isoclinic fringes in a three-dimensional photoelastic specimen. The basic idea is to delimit a slice between two plane laser beams. The properties of polarization of the scattered light (Rayleigh's law) and the interference possibilities of the diffused beams are used. By introducing speckle pattern properties, the correlation factor of the two scattered beams is similar to the illumination given in a plane polariscope for the investigation of a slice (in a classical frozen-stress technique). The authors use a monochromatic laser beam, a CCD camera and a personal computer. Because they cannot obtain the correlation factor directly, they do a statistical analysis of the speckle patterns. The variance (function of the correlation factor) is computed from the light intensities of three images corresponding to the speckle pattern for plane 1 alone, plane 2 alone, and both planes together.     

A new application of differential interferometry for stress analysis

In the past, differential interferometry has found interesting applications in gas dynamics. The gradients of density could be measured in gas flows. Now, a first trial is made to extend this method to the experimental treatment of stress problems. A Wollaston prism with polarizing elements is used in the optical arrangement. This prism combines two beams of light which have penetrated the model at locally separated points. A field of interference fringes can be produced behind the Wollaston prism. The deflections of the different conjugated light beams, which are caused by the deformed elements of the model, lead to a shifting of the interference fringes. A Stress Differential-interferometer Law is derived theoretically in order to interpret the optical data According to this theory, the optical effect caused by the deflection in this arrangement is proportional to the gradient of the sum of principal stresses. A calibration test is performed by using a circular disk, this method is applied to a circular ring for measuring the stress gradients. Under special conditions, interference fringes could be produced which represent the loci of equal stress gradient. Plexiglas plane models are loaded diametrically by single loads. The experimental results verify the statements of the developed theory.     

Superposition of finite-amplitude waves propagating in a principal plane of a deformed Mooney–Rivlin material

Here we consider finite-amplitude wave motions in Mooney–Rivlin elastic materials which are first subjected to a static homogeneous deformation (prestrain). We assume that the time-dependent displacement superimposed on the prestrain is along a principal axis of the prestrain and depends on two spatial variables in the principal plane orthogonal to this axis. Thus all waves considered here are linearly polarized along this axis. After retrieving known results for a single homogeneous plane wave propagating in a principal plane, a superposition of an arbitrary number of sinusoidal homogeneous plane waves is shown to be a solution of the equations of motion. Also, inhomogeneous plane wave solutions with complex wave vector in a principal plane and complex frequency are obtained. Moreover, appropriate superpositions of such inhomogeneous waves are also shown to be solutions. In each case, expressions are obtained for the energy density and energy flux associated with the wave motion.     

混合光载波法分离动静态主应力

本文将等差载波位相调制原理与夹层全息术结合形成的混合光载波法,可同时获得等差载波条纹图与混合载波条纹图.对混合载波纹图进行光学付里叶变换,利用空间滤波技术可将等和载波条纹从中分离出来.利用图象处理技术实现了对两种载波条纹图从预处理到应力计算的自动化处理.将混合光载波法应用于动态全息光弹性,分离了动态主应力. 文中还提出了一个经济实用的大尺寸载波片制作技术.