Application of Sharafutdinov's Ray Transform in Integrated Photoelasticity

We explain the main concepts centered around Sharafutdinov's ray transform, its kernel, and the extent to which it can be inverted. It is shown how the ray transform emerges naturally in any attempt to reconstruct optical and stress tensors within a photoelastic medium from measurements on the state of polarization of light beams passing through the strained medium. The problem of reconstruction of stress tensors is crucially related to the fact that the ray transform has a nontrivial kernel; the latter is described by a theorem for which we provide a new proof which is simpler and shorter as in Sharafutdinov's original work, as we limit our scope to tensors which are relevant to Photoelasticity. We explain how the kernel of the ray transform is related to the decomposition of tensor fields into longitudinal and transverse components. The merits of the ray transform as a tool for tensor reconstruction are studied by walking through an explicit example of reconstructing the $\sigma_{33}$-component of the stress tensor in a cylindrical photoelastic specimen. In order to make the paper self-contained we provide a derivation of the basic equations of Integrated Photoelasticity which describe how the presence of stress within a photoelastic medium influences the passage of polarized light through the material

2D and 3D separation of stresses using automated photoelasticity

A procedure for the separation of full-field photoelastic images for use with an automated polariscope is described. Regions of background in the image are identified thus producing the boundary of the model. The shear difference method is used to calculate the components of stress along all raster lines in the image using photoelastic parameters at the boundary points to calculate the initial values of stress. Algorithms were also used to evaluate the stress components along raster lines which did not contain boundary points. A plastic template was used to evaluate the efficiency of the boundary routine. It was found that it was able to identify edges to within approximately one pixel on screen. The complete procedure for stress separation was evaluated using a stress frozen disc in compression and a turbine slot. The values of stress found using the automated polariscope with the stress-separation procedure were found to agree well with theory and with results determined using the method of Tardy compensation and manual analysis. The automated polariscope was also used to analyze three-dimensional stress components along arbitrary lines of a 3D model. A two-model slicing regime was used to analyze a strut subjected to a vertical load. This work was compared to results obtained by Frocht and Guernsey on an identical model machined from Fosterite and subjected to a higher load. Good agreement was found between the results for points away from the region of loading. Significant differences were found near to the load point, however. A finite element analysis of the same problem suggested that this was due to the effects of plasticity

Evaluation of full field automated photoelastic analysis based on phase stepping

A full field automated polariscope for photoelastic analysis has been developed in the author's laboratory and has been described in detail elsewhere. Briefly, the system uses phase-stepping to determine both the fractional isochromatic fringes and isoclinic parameter at all points in the field of view independently of their neighbouring points. A wrapping algorithm is then employed to produce continuous ischromatic and isoclinic data, which can be subsequently be used in stress separation procedures. The idea of using phase stepping in photoelasticity is a fairly recent innovation and can be described as changing incremently the absolute phase of the reference wave by rotating the output elements of the polariscope and measuring the local light intensity after each step. In the apparatus described here, the output elements are rotated to six different positions providing six images of the specimen. Maps of the periodic values of the isoclinic and isochromatic parameters are subsequently obtained by combining, mathematically, these six images. A number of full field techniques have been developed. Poloshin and Redner have developed half fringe photoelasticity, and two laboratories in Japan are working on the technique of phase stepping. It appears, however, that no detailed evaluation has been made of the accuracy and reliability of the results generated by the technique. The objective of the work described in this paper has been provided such an evaluation. Five different models were selected for analysis using the automated system and manually using the Tardy compensation method: (a) a disk in diametral compression: (b) a constrained beam subject to a point load: (c) a tensile plate with a central hole: (d) a turbine blade; and (e) a turbine disk slot. These models provided a range of different fringe patterns, orders and stress gradients to test the performance of the system