Photoelasticity
= Photoelasticity
{wiki}
<Photoelasticity> is the dependence of a material’s optical response on <stress>, used to reveal otherwise invisible mechanical patterns. A transparent material that is optically isotropic before loading can become <birefringent> under load: orthogonal <polarizations> acquire different <refractive indices> and therefore a relative <phase retardation>.<Experimental Strain Analysis by David Roylance>[Photoelasticity subsection, stress-optical response]{ref}
For a thin, initially isotropic specimen in a suitable linear regime, the <stress-optic law> relates induced birefringence to the difference of in-plane <principal stresses>:
$$
\Delta n=C(\sigma_1-\sigma_2),\qquad
\delta=\frac{2\pi t\Delta n}{\lambda}.
$$
Here $C$ is a material coefficient with units $\mathrm{Pa^{-1}}$, $t$ is the optical thickness, $\lambda$ the vacuum wavelength and $\delta$ the phase delay. This simple form assumes the relevant stress directions and difference do not change along the light path. A general three-dimensional specimen needs a more careful optical and mechanical model.
With ideal crossed <polarizers>, intensity normalized to the light after the first <polarizer> is
$$
\frac I{I_0}=\sin^2(2\theta)\sin^2(\delta/2),
$$
where $\theta$ is the angle between the first polarization direction and a principal optical axis. A dark region can therefore arise from axis alignment or from an integer multiple of $2\pi$ retardation. Darkness alone does not prove that <stress> is zero.
For illustrative $C=3\times10^{-12}\,\mathrm{Pa^{-1}}$, $t=5\,\mathrm{mm}$ and $\lambda=500\,\mathrm{nm}$, one full phase cycle corresponds to
$$
|\sigma_1-\sigma_2|=\frac{\lambda}{|C|t}\approx33.3\,\mathrm{MPa}.
$$
These chosen values illustrate conversion from optical delay to a stress difference, not a calibration for every plastic. The method does not directly recover a hydrostatic stress shared equally by the two in-plane directions.
A <polariscope> makes the measurement visible. The Illinois teaching apparatus places a specimen between a <polarizer> and analyzer, applies load with a manual leadscrew, measures it with a load cell and records the pattern with a camera. Monochromatic illumination and adjustable polarization optics separate information about stress directions and differences.<Polariscope and Photoelasticity by the University of Illinois Mechanical Testing Instructional Laboratory>[Apparatus description and direction/magnitude settings]{ref} Comparing patterns as load changes is a motivating experiment in connecting an optical observable to a mechanical model.