Fractional Moir6 Strain Analysis Using Digital Imaging Techniques by A.S. Voloshin, C.P. Burger, R.E. Rowlands and T.G. Richard ABSTRACT--A new method for analyzing low-order moir6- fringe patterns of displacement fields is presented. This method adapts the techniques of half-fringe photoelasticity to molt6 and extracts continuous displacement information in the regions between integral fringes. The effectiveness of the technique is illustrated with three examples: a uniform uniaxial field, a tapered specimen in tension, and a disk in diametral compression. Introduction Moir~ patterns were first interpreted by Rayleigh more than a century ago in 1874. Developments since then have resulted in many applications of moir6 fringes to static and dynamic displacement and strain analysis. '2 The conventional method for two-dimensional displacement measurement usually compares the distortions of either a stretchable grating bonded to a specimen, or a photo- graphically reproduced grating on the surface of the speci- men, with an undeformed reference grating. Under applied load the specimen grating deforms and optically interferes with the undeformed reference, 'master' or analyzing grating. Moir~ fringes result from this interference. Each fringe corresponds to the locus of points of equal displace- ment in a direction normal to that of the reference ruling. If before deformation the two gratings had identical pitches, a displacement equal to the pitch produces one fringe. Thus, if there is no rigid-body displacement or rotation between the two gratings, a simple count of the obtained fringes gives the displacement u = Np (1) where u = displacement normal to the moir~ gratings, N = fringe count from any reference fringe, and p = pitch of the undeformed gratings. If a Cartesian coordinate system is chosen such that u coincides with x, then consecutive differentials with respect to x result in the strain cx normal to the grating lines. e= = U,x (2) A.S. Voloshin (SEM Member) is Associate Professor, Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015. C.P. Burger (SEM Fellow) is Professor, Department of En- gineering Science and Mechanics and Engineering Research Institute, Iowa State University, Ames, IA 50011. R.E. Rowlands (SEM Fellow) and T.G. Richard (SEM Member) are Professors, Department of Engineering Mechanics, University of Wisconsin-Madison, Madison, W153706. Original manuscript submitted: July 31, 1984. Final manuscript received: March 25, 1986. For a uniaxial field this means that the distance between successive fringes becomes the effective gage length and the pitch of the grating becomes the resolution. The value for ex is an average over the gage length. In the absence of rigid-body motions, this need for spatial rate of change of fringe density is the Achilles heel of the moirg method. For small strains it is not possible to obtain a sufficient number of moir6 fringes for accurate differentiation. Until now, attempts to reduce the seriousness of this limitation have concentrated on increasing the density of initial gratings or on increasing the number of obtained fringes from a specific grating by sharpening and/or multiplication of the moir~ fringes? .3.' Probably the most significant advance in this direction was Post's intro- duction of a practically useful method for generating and using very high density gratings (2400 e/ram). '.5 An alternative method which uses fractional moir~ fringes was introduced and developed by Sciammarella in 1965. 6 9 The theory utilizesa light intensity versus displacement relationship and was applied to several examples by using a microdensitometer to record the light intensity. Un- fortunately, this approach did not catch on, mainly because of the relative inconvenience of using the micro- densitometer and the cumbersome way in which the data had to be processed. A renewed interest in using the fractional moir~ system has been shown in recent work by Hunter. ~~ Hunter, however, still utilizes fairly large numbers of full moir~ fringes in the field. Recent developments in the field of digital image analysis have led to significant progress~ in automated measure- ments of light intensity in photoelasticity over fairly large fields and with high accuracy. ~2." When this general approach is modified to apply to molt6 f!elds , it leads to a greatly improved capability for interpolating between relatively widely spaced moir6 fringes. This paper describes how it can be done. It can be shown 6 that for a grating with known pitch the relationship between displacement and light intensity is I(x) = Io + l, cos 27r O(x) + /2 cos 67r O(x) + ... where [Fig. l(b)] (3) Io = the average background intensity I,,Is,etc. = the intensity amplitudes of corresponding harmonics [Figs. l(a) and l(b)] Q(x) = relative displacement of the two grids (master and deformed) The relative displacement 0(x) can be expressed through 254 " September 1986