Photoelastic Disc Fringes

What is modelled

A circular disc of radius R and thickness h is pressed by two equal and opposite line loads P at the ends of a vertical diameter. A transparent polymer becomes doubly refracting under stress: the retardation between the two polarised rays is proportional to the difference of the principal stresses. In a circular polariscope this shows as coloured isochromatic fringes.

Equations

• Stresses in the disc (loads at y = ±R, r_1,2² = x² + (R &mn; y)²) σx = −(2P / π h) [ x²(R − y)/r14 + x²(R + y)/r24 − 1/(2R) ], with the matching σ_y and τ_xy.
• Stress-optic law N = (σ1 − σ2) h / f, with f the material fringe value in force per length.
• Intensity per colour I = sin²(π N λref / λ).
• At the centre σx = P/(π R h) in tension and σy = −3P/(π R h), so Nc = 4P / (π R f).

Checks run in the page

• The vertical stress integrated across the horizontal diameter balances the load.
• The rim is traction-free away from the contacts.
• The sum of the principal stresses satisfies Laplace's equation.
• The centre fringe order against the closed form.

Honest simplifications

This is the plane-stress solution for a thin disc with ideal line loads, so the stress is infinite at the contact points and the fringes are capped near them. A real disc has finite contact patches, edge effects, residual stress and a fringe value that depends on wavelength, temperature and time. Colours are computed from three wavelengths, not a full spectrum.

• Raise the load and watch new fringes appear at the contacts and move inward.
• Switch to monochromatic light: the fringes become dark and bright bands that are easy to count.
• Lower the fringe value to mimic a more sensitive polymer: the same load gives more fringes.