Optics 078 · Polarization, anisotropy, and modulation

Photoelastic Stress Viewer

How do colorful polarization fringes map the principal-stress difference inside a loaded object? Under load, a transparent material turns birefringent in proportion to the difference of its principal stresses. Between polarizers that delay appears as colored fringes, each a line of equal stress difference. A circular polariscope removes the dark bands tied to stress direction. Fringe orders still have to be counted, and equal principal stresses, such as uniform pressure, give no fringes.

Load three specimens with analytic stress solutions in a 3D polariscope. Compare fringes in plane and circular polarization, probe the stress tensor at any point and record exposures under LED light.

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 $\text{valid model regime}$

Model checked against analytic results

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 turn into fringes on the screen. Rotate the crossed linear polarizers, then insert the circular-polarizer stack: the dark bands that depend on orientation disappear, but the stress itself has not changed. Drag the probe, compare with the stress-tensor reference and record an exposure.

The stress fields come from linear elasticity for reference geometries, with an assumed constant stress-optic coefficient. The plate is a crop of an infinite plate, and the regions at the disk's point contacts are excluded. The LED channels are ideal. White-light color rendering and recovering stress from the fringes are not included.

Physics tutorial

Audit the physics of Photoelastic Stress Viewer

BackgroundUnder load, a transparent polymer acts as a retarder that varies from point to point. Its optical axes line up with the two principal stress directions. The stress-optic law is $\delta=\frac{2\pi C t}{\lambda}\Delta\sigma$ . The thickness is the actual path of light through the specimen at normal incidence, and stresses are in megapascals. The material coefficient is an assumed constant, not fitted to a real sample. Every specimen is drawn in its undeformed reference shape.

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 the extinction that depends on orientation, leaving $I_{\rm circ}/I_0=\sin^2(\delta/2)$ . A plane crossed pair adds a factor that depends on orientation. Rotating that pair moves the dark isoclinics, while the stress field stays the same. Neither image gives the hydrostatic stress or a unique fringe order directly.

Core mathematical model

Principal stresses 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 stated teaching assumption. When the two principal stresses are equal, the principal direction is undefined, and no false horizontal axis is drawn there. 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 to the light leaving the preparation polarizer. In circular mode, two ideal quarter-wave plates sit on either side of the sample, with opposite axes relative to the crossed pair. The screen receives the normal rays one to one, as in geometrical optics, and diffraction is neglected.

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 is loaded by opposing couples at its ends. The hole uses the Kirsch field for an infinite plate. The disk uses equal and opposite point loads plus a uniform correction that keeps its rim free. The regions within two millimetres of the singular contact points lie outside the model. There is no finite-element solver, deformation, failure prediction or stress inversion.

Common difficulties

Brighter means more stress?

Typical misconceptionA bright pixel uniquely determines both principal stresses.

Better mental modelIntensity repeats as the principal-stress difference grows, so each whole fringe order is dark again. In plane optics, a point also goes dark when its optical axis lines up with the polarizer. Adding the same hydrostatic stress to both principal stresses changes neither image. The reference stress tensor here comes from analytic mechanics, not 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 across the axis, but the stress difference and the circular intensity are symmetric.
What to observe: Light travels along positive X. Looking downstream from in front of the screen, right is positive Z and up is positive Y. The specimen coordinates follow these directions, and positive angles turn from right toward up.
02
Scene 2: Infinite plate around a hole
Choose the hole window and drag the probe to the top of the hole. There the tangential stress is three times the remote tension. The edges of the rectangular window are a crop, not free edges of the plate.
What to observe: The intensity image appears only on the physical receiving screen; no copy floats in space. The specimen thickness really changes, but no mechanical deflection is shown.
03
Scene 3: Diametric disk compression
Choose diametric compression and compare the center with the two excluded contact patches. The solution uses opposed point forces; it is not a Hertz contact calculation.
What to observe: Three independent narrow-band LEDs feed ideal red, green and blue channels. This is not a white-light spectrum, calibrated camera color or a real achromatic waveplate design.

Photoelastic Stress Viewer

Audit the physics of Photoelastic Stress Viewer

Core mathematical model

Principal stresses and optical phase

Plane versus circular dark field

Three different boundary loads

Common difficulties

Brighter means more stress?

Run the experiment

Scene 1: Pure bending beam

Scene 2: Infinite plate around a hole

Scene 3: Diametric disk compression

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