8133 engineering & laboratory notes

High Resolution Moiré Photography: Extension to Variable Sensitivity Displacement Measurement and to the Determination of Direct Strains

Pramod K. Rastogi, Swiss Federal Institute of Technology, Stress Analysis Laboratory, Lausanne, Switzerland.

Abstract A method for obtaining a direct full field display of in-plane strain contours is demonstrated. On another front, the paper proposes the basis of a multi-sensitivity high resolution moiré photography system for in-plane displacement mea­surement.

High resolution moiré photography1-3 is an important tech­nique for the measurement of in-plane displacements of deformed objects. The method has many desirable features such as white-light object illumination, low sensitivity, whole field mapping of in-plane displacements, and the abil­ity to be applied to specimens of largely varying sizes. The method uses the unique imaging properties of a lens covered with a mask containing two parallel slits. A so-masked lens

Figure 1 . Schematic of high resolution moiré shearography for obtain­ing in-plane strain contours.

Figure 2. (a) Schematic of the two four-slot pupil arrangements cover­ing the split-lens device and (b) its corresponding monochromatic opti­cal transfer function along the x direction; b shows 2x magnified view.

system is used to image a periodic pattern, fixed on the object surface, onto a high resolution photographic film. This type of system has two main advantages. First, it serves to enhance the resolution over a narrow band of spatial fre­quencies in a direction parallel to the line joining the center of the two slits. Second, it increases the depth of focus by a significant amount.

In this paper, a novel scheme that combines the princi­ples of optical shearing and a slotted mask arrangement to obtain directly the whole-field mapping of in-plane strains of a deformed object is introduced. The second aim of the paper is to extend the method's capability to include multi­ple frequency channels. The multi-frequency transmission is made possible by the use of a novel aperture masking arrangement, which during reconstruction permits observa­tion in all bands. An immediate fall out of this proposal is the development of a multi-sensitivity high resolution moiré photography system for displacement measurement.

Strain measurement The specimen under investigation is imaged by an image-shearing device on to high-resolution photographic film (see Fig. 1). The image-shearing device consists of a split lens assembly with each half of the assembly covered by a mask

engineering & laboratory notes 8134

Figure 3. Setup for reconstructing moiré shearograms.

containing an appropriate pattern of slots. By incorporating the mask element into the lens assembly we are able to tailor the response of the imaging system to a select narrow fre­quency band1-3 centered on the frequency contained in the grid pattern attached to the object surface. The grid frequen­cy also determines the method's sensitivity. The device, in a horizontal shear configuration, is shown in Figure 2 a. The monochromatic optical transfer function of the pupil arrangement along the f axis is shown in Figure 2b. The two images produced by the two halves of the lens are laterally sheared with respect to each other. In other words, two neighboring points P(x, y, z) and P' (x + Δx, y, z) on the object surface are brought to contribute to a point Q in the image plane. The principle of measurement is based on the detection of the relative displacement between the two points. Let [u(P), v(P)] and [u(P'), v(P')] represent the hori­zontal and vertical components of the in-plane displacement vectors at the points P and P', respectively. The film after being exposed to the deformed object was processed and placed in an optical Fourier filtering system of the type shown in Figure 3. The field distribution in the focal plane contained a series of spots corresponding to the diffraction pattern of the grid. The strain related information was con­tained in the first order spots about the x and y axes in the focal plane. Using a photographic camera to image only the light associated with the first order spot in the horizontal direction generates a fringe system described by

or

where n1 is the fringe number. A spatial filter used to pass only the first order spot in the vertical direction generates a fringe system given by

where n2 is the fringe number and p0 is the grid pitch. A continuous change of the shear and hence the method's sen­sitivity can be obtained with the proposed configuration.

By adjusting the split lens assembly to give a linear shear of magnitude Δy along the y direction, the information cor­responding to ∂vl∂y and ∂u/∂y is now recorded on the pho­tographic film. This information can be reconstructed as in Equation 3 in a spatial filtering setup.

Figure 4a depicts the contours of ∂u/∂x for the case of a Plexiglas beam, with a hole, submitted to a three point bend-

Figure 4. Examples of moiré shearograms corresponding to in-plane strain derivatives (a) ∂u/∂x and (b, c) ∂v/∂x.

ing load. Figures 4b and 4c show the distributions of ∂v/∂x for the case of a thin sheet under tension. The amount of strain per fringe space in Figures 4a and 4c is equal to 3.125 × 10-3 and in Figure 4b is equal to 6.25 × 103.

Figure 5a corresponds to the contours of ∂u/∂y for the case of a thin Plexiglas sheet under tension. Figure 5b shows the distribution of ∂v/∂y for the case of a beam subjected to three point bending. The amount of strain per fringe space in Figures 5a and 5b is equal to 6.25 × 10-3

and 3.125 × 10-3, respectively.

Multi-sensitivity approach in displacement measurement Measuring the strain concentrations in bodies, especially near holes and cutouts, is of great interest in experimental and computational mechanics. These regions are character­ized by high displacement gradients. Hence, in these mea­surement problems, it is clearly attractive to work with a method offering a multi-sensitivity feature consisting of low-sensitivity observation of high stress concentration zones and high-sensitivity observation of less deformed zones. The multi-sensitivity approach provides a good dis­play of the whole field displacement map, since fringes could still be seen with clarity in high stress concentration regions, and the presence of more fringes would mean improved information mapping in low stress regions on the specimen.

The need for tuning an optical system to multiple nar­row frequency bands is, thus, well apparent. In its simplest form the photographic moiré method involves the double exposure recording of a grid pattern fixed to the surface of an object in two different states of stress. The camera lens is

8135 engineering & laboratory notes

Figure 5. Examples of moiré shearograms corresponding to in-plane strain derivatives (a) ∂u/∂y and (b) ∂v/∂y.

covered with a mask containing two slits. In the multi-sensi­tivity approach, the mask containing two slits is replaced by a mask containing four slits, as shown in Figure 6a. A com­parison between the optical transfer function of an imaging system with a square (see Fig. 6b) and four slit apertures is shown in Figures 6c and 6d. The polychromatic transfer function is found by averaging a small range of wavelengths. As is evident from Figure 6d, the response of the optical sys­tem with four slit apertures is tuned to resolve spatial fre­quencies centered at ±ƒi, ±3/4ƒi, ±1/2ƒi, and ±1/4ƒi in the out­put image, where ƒi is the maximum resolvable frequency that the device can resolve

s' is the distance between the lens and the film plane and d is the separation between the two outermost slits. The corre­sponding tuned spatial frequencies on the object are ƒ0, 3/4ƒo, 1/2ƒ0, and 1/4f0 , where ƒ0 is the maximum spatial frequency contained on the object surface

m is the magnification of the imaging system, s is the dis­tance between the object and the entrance pupil, and λ is the effective wavelength set by the film and illumination.

In the spatial frequency set-up the film diffracts light into different directions associated with the fine details recorded on it. The use of a filter is made to pass one fre­quency component of the Fourier spectrum. This results in the reconstruction of the displacement information carried by that particular frequency. The process of filtering is repeated for other frequency bands to extract the complete information contained in the film.

The schematic example (see Fig. 7a) is that of an object prepared by applying a periodic pattern of frequency ƒ0 lines/mm on the regions of supposedly low deformation gradient and a periodic pattern of 3/4ƒ0 lines/mm on the regions of supposedly high deformation gradient. The reconstructed image obtained by passing the higher frequen-

Rgure 6. Schematic of (a) four slit and (b) square aperture mask to cover the imaging lens. The monochromatic optical transfer function of the two pupil functions are are shown in (d) and (c) respectively; c and d show 2x magnified view.

Figure 7. (a) Periodic patterns of frequencies fσ and 3/4fσ are applied on the specimen surface, (b, c) Reconstructed images of the specimen obtained by the spatial filtering process.

cy spot in the focal plane is shown in Figure 7b. The central part of the object, in the form of a cross, is absent from the reconstructed image. Similarly the reconstructed image obtained by passing the lower frequency spot is shown in Figure 7c. The sensitivities of the fringe contours in Figures 7b and 7c are po and 3/4po μm/contour lines, respectively.

To summarize, this paper proposes a novel scheme for directly displaying the derivatives of in-plane displacement, and a basis for extending the use of high resolution moiré photography to obtain simultaneously multi-sensitivity dis­placement measurements.

Acknowledgments The author thanks Ours Vokinger for his help in experi­ments, and the Swiss National Science Foundation for finan­cial support.

References 1. J.M. Burch and C. Forno, "A high sensitivity moiré grid tech­

nique for studying deformation in large objects," Opt. Eng. 14, 178-185 (1975).

2. J.M. Burch and C. Forno, "High resolution moiré photography," Opt. Eng. 12, 602-614 (1982).

3. C. Forno, "Deformation measurement using high resolution moiré photography," Optics and Lasers in Engineering 8, 189-212 (1988).