__Two Dimensional Photo Elasticity:__

Photoelasticity is an experimental method to determine the stress distribution in a material. The method is mostly used in cases where mathematical methods become quite cumbersome. Unlike the analytical methods of stress determination, photo elasticity gives a fairly accurate picture of stress distribution, even around abrupt discontinuities in a material. The method is an important tool for determining critical stress points in a material, and is used for determining stress concentration in irregular geometries.

__Concepts of Light:__

The theory of photo elasticity is based on the wave nature of light. Light is regarded as a sinusoidal electromagnetic wave having transverse amplitude *a *and longitudinal wavelength, propagating in the *z *direction with velocity *v *(Fig. 3).

A wave propagating in the +*z *direction may be represented in trigonometric notation as

where the quantity

is called the *phase *of the wave. Terms related to the wavelength and speed are the ordinary frequency *f *(in Hz), the angular frequency (in rad/s), and the wave number *k*, as follows:

Thus the phase has the alternative representations

The speed of light *v *in a vacuum is approximately 299.79 Mm/s, independent of its wavelength or amplitude. From the last of the expressions in Eqns. (1), it will be seen that the frequency of a given light wave must depend on its wavelength:

Notice that the visible spectrum covers a nearly 2-to-1 ratio of wavelengths, the blue–violet wavelengths being much shorter than the orange–red ones. An equivalent way to express the trigonometric form of a wave is in complex notation as

Since the complex notation is much easier to manipulate when phases and amplitudes undergo changes, we shall use it in these notes. The operator Re{ } will be omitted for convenience, with the understanding that, if at any time a quantity is to be evaluated explicitly, the real part of the expression will be taken. A typical expression for a light wave will therefore be simply

__Photo elastic Effects:__

The occurrence of optical anisotropy in initially isotropic solids including polymers when the solids are subjected to mechanical stresses.

The photoelastic effect is a consequence of the strain dependence of the dielectric constant of a substance and is manifested as double refraction, or birefringence, and dichroism, which occur when a substance is mechanically loaded. Under uniaxial tension or compression, an isotropic solid takes on the properties of a uniaxial crystal with the optical axis parallel to the axis of tension or compression. Under more complex strain—for example, under bilateral tension—a specimen, or model, becomes biaxial.

Photoelasticity is caused by the deformation of the electron shells of atoms and molecules and by the orientation of optically anisotropic molecules or components of such molecules; in polymers, it is caused by the uncoiling and orientation of polymer chains. For a small uniaxial tension or compression, Brewster’s law is satisfied: Δn = kP, where Δn is the magnitude of the birefringence (that is, the difference between the refractive indexes for the ordinary and extraordinary waves), P is the stress, and k is the engineering stress-optical coefficient. For glasses, k = 10–1310–12 cm2/dyne; for celluloid, k = –10–12–10–11cm2/dyne.

The photoelastic effect is used in the study of stresses in mechanical structures for which calculation of the stresses is too complicated. The investigation of the birefringence due to loading in a transparent model—usually a small-scale model—of a structure being studied makes it possible to determine the nature and distribution of the stresses in the structure. The photoelastic effect underlies the interaction of light and ultrasound in solids.

__Stress Optic Law:__

__Interpretation of Fringe Pattern:__

The two waves are then brought together in a polariscope. The phenomena of optical interference take place and we get a fringe pattern, which depends on relative retardation.

Thus studying the fringe pattern one can determine the state of stress at various points in the material (Fig. 9.)