Illustrated Physics
physics, drawn
Concept

Bending stiffness — where it appears

The resistance of a beam, plate or fibre to being curved, the product of its material's stiffness and a factor fixed by the shape of its cross-section. For a plate it grows as the cube of the thickness, which is why thin sheets bend so easily.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

Restitution with nothing lossy in it. The coefficient of restitution of a perfectly elastic ball striking a large, perfectly elastic plate, against Zener's parameter λ, which grows as the square of the ball's radius over the plate's thickness and weakly with speed. At λ = 0 the ball comes back at its arrival speed. The rebound falls below half at λ ≈ 0.65, with no friction, plasticity or damping anywhere — the missing energy is travelling outward through the plate as bending waves. Marked: a steel ball 20 mm across at 1 m/s on steel plates 5 mm thick (λ = 0.94), 10 mm thick (λ = 0.23), 20 mm thick (λ = 0.06).

The bounce an elastic plate cannot give back

Drop a steel ball on a thick steel block and it bounces almost to where it started. Drop it on a steel plate five millimetres thick and it comes back at barely half its speed, although steel on steel is about as elastic as materials get and nothing in the impact is plastic, sticky or damped. The energy has not been turned into heat. It has been turned into waves, running out through the plate faster than the ball can take them back — and a ball striking a longer rod loses energy the same way, by leaving it ringing.

mechanics · Momentum The pull of a meniscus against the stiffness of a wall. For two neighbouring lines of a wet pattern leaning towards each other, the net Laplace pressure pulling them together (heavy curve) and the bending stiffness pushing them back (straight lines), against the lean as a fraction of the half-gap. The pull grows faster than linearly, because narrowing a gap raises its suction, and it diverges as the gap closes. The stiffness is a straight line whose slope falls as the fourth power of the line's height; the lines drawn are for 0.8 times, 1 times, 1.25 times the critical height. At 0.8 of the critical height the straight state is stable and an unstable balance lies at a lean of 0.77 of the half-gap, a barrier the lines must be pushed over; at the critical height the two start with the same slope and the straight state is marginal; at 1.25 times the straight state is unstable and the pair collapses.

The walls a rinse pulls together

Every result about capillary rise assumes the walls hold still. Make them thin enough and they do not. The water between two walls is below atmospheric pressure, so it pulls them together, and as the gap narrows the suction grows. Beyond a critical height or length the walls cannot push back and they snap shut. It is why wet hair clumps, why a paintbrush comes out of water as a point, and why a chip factory cannot rinse lines of resist taller than about three times their width without laying them flat.

fluids · Capillarity Every wave a plate can carry, at every frequency. The phase velocities of the guided waves in an aluminium plate (longitudinal speed 6.32 km/s, shear 3.13), against the product of frequency and thickness in MHz·mm, from the Rayleigh–Lamb equations: symmetric modes, in which the two faces bulge and thin together (light), and antisymmetric ones, in which they bend together (dark). At low frequency only two exist. The symmetric S₀ is a stretching wave at the plate velocity, 5.44 km/s; the antisymmetric A₀ is a bending wave whose speed falls towards zero, 0.96 km/s at 0.1 MHz·mm. At high frequency both approach the 2.92 km/s of a Rayleigh wave on each face. Higher modes enter at cut-off frequencies where the plate's thickness holds a whole number of half-wavelengths of one bulk wave — the first at 1.56 MHz·mm — arriving from infinite phase velocity and slowing as the frequency rises.

The plate that carries two kinds of sound

Tap the edge of a sheet of aluminium and the sound that runs along it is not one wave but two, and neither is the sound of the metal. One stretches the sheet and travels at 5.44 km/s, faster than a shear wave and slower than a compression wave in the same metal. The other bends it, travels at a speed that depends on how high the note is and how thick the sheet is, and at low frequency crawls at a few hundred metres a second. Both are made of the same two bulk waves bouncing between the faces, and the two faces together decide everything.

waves · Guided waves

Named alongside it

The objects these essays reach for when they reach for this one.

Elastic wavesCapillarityCoefficient of restitutionCollisionCutoffDispersionDispersion relationElastic instabilityElastocapillarityEnergyGroup velocityGuided waves

All concepts