Photoelastic Stress Polariscope

What a polariscope shows

Stressed transparent plastics become birefringent: the refractive-index difference between the two principal-stress directions is proportional to the principal stress difference, n1 - n2 = C (σ1 - σ2) (stress-optic law, C in Brewsters = 10^-12 Pa^-1). Light crossing a plate of thickness t picks up a retardation of N wavelengths:

N = C t (σ1 - σ2) / λ = t (σ1 - σ2) / fσ, with material fringe value fσ = λ/C.

Intensities

Circular polariscope, dark field: I = I0 sin²(π N) (black where N is an integer: isochromatic fringes; light field is the complement). Plane polariscope adds the isoclinic factor sin²(2(θ - α)) where θ is the local principal direction and α the polariser angle: black bands wherever a principal axis lines up with the polariser. In white light each colour has its own N, so low orders show vivid colours and high orders wash out to pink and green.

Disc under diametral load (plane stress)

Superposing two point loads P at (0, ±R) (Flamant solutions) gives σy = -(2P/πt) [ (R-y)³/r14 + (R+y)³/r24 - 1/D ] and the analogous σx, τxy. At the centre σ1 - σ2 = 8P/(π D t) (tension 2P/πDt across the load line, compression 6P/πDt along it). This is why a disc is the classic calibration specimen for the fringe value.

Honest simplifications

• Point loads give infinite stress at the contacts; real platens flatten them (Hertz contact), so fringes cluster there but stay finite. The picture is trustworthy away from the contact patches (more than about 0.1 R).
• Plane stress, linear-elastic, constant stress-optic coefficient; no residual stress, no dispersion of C with wavelength, no creep.
• White light is nine spectral lines added in intensity, not a continuous spectrum.
• Material constants are typical textbook values; real batches differ by 10-20 %.
• The 3D scene is a schematic bench: the loading frame, screw and dial are drawn, not simulated structurally.

• Raise the load slowly in white light: new colour bands are born at the load points and march to the centre, one fringe order each time N passes an integer.
• Switch to sodium light and count black fringes along the load axis; compare with the fringe-order graph and the readout at the centre.
• Rotate the polariser in the plane polariscope and watch the black isoclinic bands sweep around the disc.
• Swap polycarbonate for acrylic: the stress is the same but almost no fringes appear because C is 15 times smaller.