Research in Automated Photoelasticity
Parmjit Singh Kanth
Mechanical and Industrial Engineering, University
of Toronto, 1997.
Photoelasticity, from an experimental perspective, is the study of stress and strain field distributions within a loaded photoelastic specimen, as viewed under a polariscope.
A polariscope is an optical instrument that utilizes the properties
of polarized light to analyze the stress distribution within a photoelastic
specimen. Two types of polariscope are commonly employed in stress analysis
work, the plane polariscope, and the circular polariscope.
The Plane polariscope is the simplest optical system used in photoelasticity; it consists of two linear polarizers (which transmit light only along their axis of polarization) and a light source. The linear polarizer nearest the light source is called the polarizer, while the second linear polarizer is known as the analyzer. In the plane polariscope, the two axes of polarization are always crossed; hence no light is transmitted through the analyzer.
Circular polariscope: This polariscope employs circularly polarized light (light which sweeps a circular helical trace through time as it passes through a wave plate, which basically has two perpendicular axes of polarization). The photoelastic apparatus contains four optical elements and a light source. Various configurations of the polarizer, 1st and 2nd wave plates, and analyzer produce light and dark bands beyond the analyzer.
THEORY OF PHOTOELASTICITY
The stress field at any point in a photoelastic specimen can be related
to its index of refraction through Maxwell’s stress optic laws. The light
emerging from the analyzer is subject to prior conditioning from the polarizer
and specimen, and can be described as follows: The intensity I diminishes
when either sin term goes to zero, and therefore we have two possible fringe
patterns of points where the light is extinguished, i.e.,
In a plane polarizer, isoclinic and isochromatic fringe patterns are superimposed upon each other. Therefore, in order to obtain the individual values of the stress, we must investigate separation techniques.
At interior regions of the model, stresses obtained from isochromatic and isoclinic patterns without using:
1: Methods based on the Equilibrium Equations: Shear Difference Method
The method described here is based solely on the equations of equilibrium (i.e., equilibrium of applied body forces, stresses, and shears), and as a result is independent of the elastic constants of the photoelastic model material. The equations of equilibrium when applied to the plane-stress problem can integrated in approximate form using the following finite difference expressions:
The benefit of the above method is that is can be readily visualized graphically, and applicable to arbitrary specimen geometry: Since the above procedure implements finite difference techniques, it leads way to the possibility of incorporating the shear difference method to automate the entire separation of isochromatic and isoclinic fringe patterns - see Current Research.
2: Methods based on the Compatibility Equations
The compatibility or continuity equations can be expressed in the form of Laplace’s equation, the solution of which is known as a harmonic function. There are many methods for modeling and solving using Laplace’s function, including superposition of analytical harmonic functions, finite element techniques, as well as physical analogy methods such as electrical circuits modeled to suit the geometry in question.
3: Methods based on Hooke’s Law
Separation methods based on Hooke’s law make use of the fact that the sum of principal stresses can be determined if the change in thickness of the model, as a result of the applied loads, can be measured accurately at the point of interest. Instruments developed for the measurements of these changes (which are in the order of a few thousandths of a cm) include lateral extensometers and interferometers.
4: Oblique Incidence Methods
Rather than having the light pass through the model at normal incidence, the model can be rotated in the polariscope so that the light passes through the model at some other angle, producing an oblique incidence fringe pattern. This oblique incidence fringe pattern provides additional data which can be employed to separate the principal stresses.