Optics 078 · Polarization, anisotropy, and modulation

Photoelastic Stress Viewer

Three analytic loaded specimens, a real 3D polariscope, plane/circular fringes, tensor probes and recorded LED exposures.

Probe principal-stress difference

\Delta\sigma

$0.50$

Probe green fringe order

N_{550}

$50\%$

Probe green transmission

I_{550}/I_0

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Photoelastic Stress Viewer

Boundary stress scale; definition below

q

4\,\mathrm{MPa}

Actual optical thickness

t

4\,\mathrm{mm}

Crossed pair angle from image right toward up

\theta_P

30^{\circ}

Circular optical stack inserted

j_C

1

Three narrow LEDs selected

j_L

1

Probe horizontal coordinate

u

0\,\mathrm{mm}

Probe vertical coordinate

v

10\,\mathrm{mm}

Specimen identity

j_S

0

Load a clear specimen and watch its invisible stress become fringes on the receiving screen. Rotate a plane crossed pair, then insert the circular stack: orientation-dark bands disappear, but the stress stays. Drag the probe, compare the tensor reference and record an exposure.

Linear-elastic reference geometry and assumed constant stress-optic coefficient. Infinite-plate crop and excluded disk point contacts; ideal LED channels, no white-light colorimetry or inverse stress recovery.

Physics tutorial

Audit the physics of Photoelastic Stress Viewer

BackgroundA loaded transparent polymer becomes a spatially varying retarder. Its two principal-stress directions are the optical eigenaxes. The stress-optic law is $\delta=\frac{2\pi C t}{\lambda}\Delta\sigma$ . Thickness is the real normal optical path; stresses use megapascals. Our constant material coefficient is assumed, not fitted to a real sample. All specimens are displayed in their undeformed reference geometry.

Why it mattersHow do colorful polarization fringes map the principal-stress difference inside a loaded object?

Start with the essentials

Focus question: How do colorful polarization fringes map the principal-stress difference inside a loaded object?
One-sentence intuition: A circular dark-field stack removes orientation-dependent extinction, leaving $I_{\rm circ}/I_0=\sin^2(\delta/2)$ . A plane crossed pair has an additional orientation factor. Turning that pair moves dark isoclinics without changing the stress field. Neither image directly gives hydrostatic stress or a unique fringe order.

Core mathematical model

Tensor eigenvalues and optical phase

\begin{aligned}\Delta\sigma&=\sqrt{(\sigma_{uu}-\sigma_{vv})^2+4\sigma_{uv}^2}\\N_\lambda&=\frac{Ct\Delta\sigma}{\lambda}\\C&=60\times10^{-12}\,\mathrm{m^2/N}\end{aligned}

The coefficient is a transparent teaching assumption. Equal eigenvalues make the principal-axis direction undefined, not a spurious horizontal axis. Stress and thickness are converted to pascals and metres exactly once.

Plane versus circular dark field

\begin{aligned}I_{\rm plane}/I_0&=\sin^2[2(\phi-\theta_P)]\sin^2(\pi N_\lambda)\\I_{\rm circ}/I_0&=\sin^2(\pi N_\lambda)\end{aligned}

Intensity is normalized after the preparation polarizer. In circular mode two ideal quarter-wave plates flank the sample, with opposite axes relative to the crossed pair. The screen receives a geometrical one-to-one normal-ray field, neglecting diffraction.

Three different boundary loads

\begin{aligned}\text{beam:}\quad q&=\frac{6M}{th^2},\quad\sigma_{uu}=-\frac{2qv}{h}\\\text{hole:}\quad q&=\sigma_\infty,\quad\sigma_{\theta\theta}(a,\pi/2)=3q\\\text{disk:}\quad q&=\frac{F}{Dt},\quad\Delta\sigma(0)=\frac{8q}{\pi}\end{aligned}

The beam uses opposing end couples; the hole uses the infinite-plate Kirsch field; the disk uses equal opposite point loads and a uniform correction enforcing the free rim. Two-millimetre neighborhoods of the singular contacts are outside the model. No finite-element solver, deformation, failure prediction or stress inversion is included.

Common difficulties

Brighter means more stress?

Typical misconceptionA bright pixel uniquely determines both principal stresses.

Better mental modelIntensity is periodic in principal-stress difference. Whole fringe orders can be dark again; plane optics also go dark when an eigenaxis aligns with the polarizer. Adding equal hydrostatic stress to both eigenvalues changes neither image. The reference tensor here is known from analytic mechanics, not recovered from a photograph.

Run the experiment

01
Scene 1: Pure bending beam
In pure bending, raise the stress scale and probe above and below the neutral axis. The stress changes sign; the stress difference and circular intensity are symmetric.
What to observe: The ray travels along positive X. In the screen-front view looking downstream, right is positive Z and up is positive Y. Specimen coordinates follow these directions; positive angles turn from right toward up.
02
Scene 2: Infinite plate around a hole
Choose the hole window. Drag to its top crown; the tangential stress is three times remote tension. The rectangular crop is not a free plate edge.
What to observe: Only the physical receiving screen shows the intensity image. No duplicate image floats in space. The specimen thickness changes physically; no mechanical deflection is claimed.
03
Scene 3: Diametric disk compression
Choose diametric compression. Compare the center with the two excluded contact patches. The opposed point-force solution is not a Hertz contact calculation.
What to observe: Three narrow independent LEDs feed ideal red, green and blue channels. This is not a white-light spectrum, calibrated camera color or a real achromatic waveplate design.