Tempered Glass - 3D Tempering Line

How a quench locks stress into glass

Hot glass above its freezing (strain-point) temperature Tf flows and carries no stress. Air jets cool the faces first: the skin passes Tf and becomes elastic while the core is still soft. The core then cools and tries to shrink by alpha (Tf - Tair) but is held by the rigid skin, so the core ends in tension and the skin in compression. The through-thickness profile is close to a parabola, sigma(z) = sigma_c (1 - 3 (z/H)^2), whose force balance integral sigma dz = 0 forces the surface value to be about -2 sigma_c. Typical tempered glass: 90-150 MPa compression at the surface, 40-70 MPa tension in the core.

The model in this lab

The temperature field comes from the heat equation in a slab, rho c dT/dt = k d2T/dz2, with a surface loss -k dT/dz = (hc + h_rad(Ts)) (Ts - Tair) (convection plus linearised radiation, emissivity 0.84), solved by finite differences. Layers colder than Tf are elastic: each time step the frozen layers receive thermal-strain increments alpha dT, the common in-plane strain increment follows from sum(w d sigma) = 0, and d sigma = E' (d e - alpha dT) with E' = E/(1 - nu) (E = 70 GPa, nu = 0.22, alpha = 9e-6 /K). More cooling (larger Biot number Bi = hc H / k) means a larger temperature difference at the moment the core freezes, so more stress; thicker glass at the same jets also gains stress.

Polariscope

Photoelastic glass is birefringent in proportion to the stress: the path difference between the two polarisations is delta = C sigma L with the stress-optic coefficient C = 2.7e-12 /Pa and light path L. Between crossed polarisers at 45 degrees the intensity of wavelength lambda is sin^2(pi delta / lambda), so each colour has its own fringes. You look along the plate edge (the in-plane stress acts across the light path); the neutral lines, where stress changes sign, are dark.

Why it shatters into dice

Breaking the skin releases the stored elastic energy per unit area U = integral sigma^2 / (2 E') dz. A running crack keeps branching while that energy pays for new surface; with an effective crack energy Gamma (8 J/m2 here, including roughness) and half of U available, square dice of side d = 2 Gamma h / (0.5 U) result. Typical output for 6 mm glass: dice of 3-5 mm, and the standard EN 12150 asks for at least 40 fragments in any 50 x 50 mm window (d at most 7.9 mm). Annealed glass has almost no stored energy, so it breaks into a few large shards. The Prince Rupert drop is the extreme case: quenched in water, its surface compression is several hundred MPa; the head withstands a hammer, yet snapping the thin tail sends a fracture front through the whole drop at about 1.5 km/s, turning it to powder.

What is simplified

Instant freezing at one temperature replaces the full viscoelastic structural relaxation of glass (real tempered stress is somewhat different); constant glass properties; one-dimensional symmetric cooling, so edge and corner effects, quench marks and the roller-wave distortion are ignored; the drop is a slab of half-thickness equal to its head radius (a sphere has a different surface-to-core ratio, which is why the printed ratio differs); fragment size comes from an energy balance, not from crack-branching dynamics; the crack pattern is a Voronoi picture at the computed scale. Plate sizes and the drop are drawn for clarity; slow-motion factors are noted on screen.

• Furnace and rollers: the plate leaves the furnace glowing red; colour follows its surface temperature.
• Quench box: upper and lower plenums with a nozzle array; white streaks are the air jets.
• Punch table: a spring punch on one corner; the crack pattern spreads from the impact point.
• Polariscope bench and screen: the edge view between crossed polarisers; the green line is the fringe order.
• Graphs: temperature history, stress through the thickness (now, final, ideal parabola), fringes, and fragment size against cooling intensity.

• Press Quench with "Tempered 6 mm" and watch the stress column: first the skin goes into compression, then the core swings into tension as it cools. Open the Polariscope column to see the colour fringes grow.
• Then press "Punch / snap test": about 180 dice in the 50 x 50 mm window. Pick "Annealed (slow cool)" and punch again: a handful of large shards.
• Drag "Cooling intensity" up and down and watch the fragment-size graph cross the EN 12150 line.
• Compare 3 mm with strong jets against 12 mm: thicker plates reach higher stress at the same jets.
• Choose the Prince Rupert drop and press Snap: the fracture front runs from tail to head.