The use of phase-stepping for the measurement of characteristic parameters in integrated photoelasticity

Abstract

ABSTRACT--A three-dimensional photoelastic body can be represented by an optically equivalent model, which consists of a linear retarder, 5, at a certain angle, 0, and a pure rotator, X. These have been described as the characteristic retardation, ~, and the primary and secondary characteristic directions, 0 and 0+X. Until now these characteristic parameters have only been determined using manual, point-by-point collection methods which are involved and time consuming. Therefore an automated phase-stepping method has been developed to enable the determination of the three characteristic parameters for three-dimensional or integrated photoelasticity. Expressions have been derived to obtain 3, 0 and 0 4-7, from six phase-stepped images. These images are collected using a CCD camera and the full-field data is processed using a standard personal computer. This novel method allows accurate, full-field maps of all three characteristic parameters to be obtained in a relatively short time, which makes full-field tomographic reconstruction of photoelastic data a real possibility.

Characteristic parameters in integrated photoelasticity: an application of Poincaré's equivalence theorem

“…If one of the parameter values (29) is taken, the right-hand side of eqs. (25,27) becomes the ill-defined expression …”

“…If the replacements (32) are used in (25) we obtain (27), again under the conditions (29). -It is easy to check that, whenever one of the values in (29) is taken, the right-hand sides in both (25) and (27) are ill-defined.…”

“…From the proof of theorem 4.1 we know that the decompositions (52, 53) are unique provided that the parameters (θ, φ, χ) in the parametrisation (22) of U are restricted according to (29).…”

“…parameters (θ, φ, χ) satisfy (29), or equivalently, the phase lag ∆ = −2δ of the equivalent linear retarder is not a multiple of π. In this case the linear retarder has precisely two preferred directions, namely the fast and slow optical axis.…”

“…Although references to the equivalence theorem, or applications thereof, are made frequently in the literature [20,22,23,24,28,29], its precise content is apparently not widely published; a rigorous proof of the theorem, or some equivalent statement thereof, seems even less readily available. This observation constitutes the starting point of our paper: We rigorously prove a matrix decomposition theorem for unitary unimodular (2, 2) matrices [section 4], from which the Poincaré Equivalence Theorem follows as a corollary [section 5].…”