Illustrated Physics
physics, drawn
Concept

Elasticity — where it appears

The property of a material that makes it return to its original shape after a stress is removed, storing energy rather than dissipating it. It is what a spring constant measures at small strain and what fails at the yield point, past which the deformation is permanent.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

Rough contact grows as W^1.000, and one smooth bump as W^0.667. Real contact area against load, both logarithmic, three ways. Plastic flow of the asperities gives A = W/H, a slope of exactly one, which is the account Bowden and Tabor's argument uses. A single elastic sphere gives Hertz's answer, A ∝ W^(2/3), and a friction coefficient falling as W^(−1/3) — Amontons' law would be false. A rough elastic surface, with many asperities spread exponentially in height and integrated here rather than quoted, gives a slope of 1.0000, because pressing harder mostly recruits new contacts rather than enlarging existing ones. The dotted curve repeats that integration with a Gaussian spread of heights and gives 0.9695 — the upper tail of a Gaussian is nearly exponential, so the answer is nearly and not exactly linear. Amontons' law does not need plasticity. It needs roughness, it is exact for one distribution and good to three per cent for another, and it survives whichever way the individual junctions deform.

The grip that is not a coefficient

A friction coefficient is independent of load and of area, and the reason is usually given as plastic flow of the asperities. It is not — a rough elastic surface gives the same law, integrated here to a slope of 1.0000, because pressing harder recruits new contacts rather than enlarging old ones. What breaks the law is a contact that is smooth, or a material that dissipates in its bulk — where μ passes one and stops being a coefficient at all.

mechanics · Friction Four supports, and a whole line of answers. The same top on four supports at the corners of a square, with the load in the same place. Five sets of reactions are drawn and every one of them satisfies all three equilibrium equations exactly — worst residual 1.1e-16 across the whole family — so statics does not prefer any of them. They differ by a multiple of the pattern plus, minus, plus, minus around the square: pressing one diagonal pair harder and the other pair less adds no net force and no moment about either axis, which is exactly what it means for the problem to have a fourth unknown and only three equations. The rigid-body idealisation has not been applied carelessly here; it has been applied correctly, and the answer it gives is that there is no answer. Every member drawn keeps all four reactions positive, so the requirement that a leg can only push narrows the family without closing it. What decides is left out of the model entirely — how much each leg gives under load.

The table statics cannot settle

A rigid top on three legs has one possible set of reactions and a rigid top on four has infinitely many, all of them balancing every force and every moment exactly. The extra leg does not make the problem harder; it makes it unanswerable, and the answer has to come from somewhere the model deliberately threw away.

mechanics · Free-body 5 balls whose contact force goes as the overlap to the three halves, touching. The velocity of every ball in a line of 5, against time in units of one binary contact, with the first arriving at unit speed. Nothing about collisions is assumed: neighbours push on each other with k times their overlap raised to the power 1.5, and the equations of motion are integrated. With the balls touching there is no such separation — several overlaps are non-zero at once and the disturbance crosses the line as a single compression wave. The far ball leaves at 0.989 of the striking speed and the others keep 0.011 between them, which is why a real cradle's balls do not quite come to rest. Momentum and energy are conserved to 4.4e-16 and 4.1e-8, so the difference between the two cases is the contact law and not the bookkeeping.

Five balls, and the law that does not choose

The usual account of a Newton's cradle says that momentum and energy conservation force one ball out at the striking speed. For three balls or more they do no such thing: the two laws leave a whole curve of possible outcomes, and what picks one is the shape of the force between two touching spheres.

mechanics · Momentum Force against slip, and the knee between them. The force a tyre delivers against how much faster its tread is going than the road, for a patch 120 mm long under 4000 N with a friction coefficient of 1. The curve is the integral over the bristles, and the dashed line is the cubic the brush model gives in closed form; they agree to 0.00 per cent of the sliding force. The first slope is 80 kN per unit slip, and it belongs entirely to the elasticity of the rubber — at vanishing slip nothing is sliding, so no friction coefficient can appear in it. Full sliding is reached at 15.0 per cent slip and not before. Everything a driver calls grip lives on the rising part of this curve, at a few per cent of slip, where the patch is partly stuck and partly sliding — and the quantity that decides handling in that region is the slope rather than the friction coefficient at the top.

The grip that needs a little slipping

A wheel that transmits any force at all is not rolling. Part of its contact patch is stuck to the road and part is already sliding, and the force it delivers is a measure of how much has given up. The useful part of the curve is a few per cent of slip, the peak is not the end of it, and everything past the peak is unstable.

mechanics · Friction The stress a temperature change puts into a bar that cannot move. The stress in a member held between supports that will not let it change length, against how much its temperature changes, for four materials. Each line is EαΔT, computed here from a free expansion and the force needed to undo it, and checked by evaluating it for a bar half a metre long and one thirty-seven metres long: the two agree to every figure carried, because neither the length nor the cross-section appears in the answer. steel develops 2.40 MPa for every kelvin and reaches yield at 104 K; aluminium develops 1.59 MPa for every kelvin and reaches yield at 151 K; concrete develops 0.30 MPa for every kelvin and reaches cracking at 10 K; invar develops 0.17 MPa for every kelvin and reaches yield at 1655 K. Concrete reaches its cracking stress after ten kelvin, which is less than a sunny afternoon, and is why every slab has movement joints in it. Invar is in the comparison because it was made to have a small product: it is as stiff as steel and develops a fourteenth of the stress, which is a statement about the expansion coefficient and nothing else.

The load nobody applied

A redundant structure develops forces with nothing on it. Change its temperature and the same extra constraint that made statics unanswerable also refuses the expansion — and a restrained steel member reaches its yield stress after a hundred and four kelvin, a figure that contains no length, no area and no load.

mechanics · Free-body Four stress states in one beam, and only one of them has no tension. Stress across the depth of a 300 by 600 millimetre concrete section at the middle of an eight-metre span, compression to the right, for four conditions. Under the load alone the bottom fibre is in tension at 11.1 MPa, which is four times what concrete can carry, so an ordinary reinforced beam cracks there and relies on steel to hold the crack together. With 1,500 kilonewtons of prestress 120 millimetres below the centroid and the full load applied, the section runs from 9.8 to 3.6 MPa and every fibre of it is in compression. The second case is the one that surprises: with the prestress applied and nothing whatever to oppose it, the top fibre is in tension at 1.7 MPa, because a force below the centroid bends the beam upwards. A prestressed beam is at its most vulnerable when nothing is on it, and what rescues it is its own weight: adding that alone brings the top back to 0.3 MPa of compression. Each stress block is checked by integrating it and recovering the force and the moment that produced it.

A state no load could reach

The free direction a redundant structure leaves open can be driven on purpose. Tighten a tendon through a concrete beam and its whole stress state moves into the half of the range the material is good at; tighten a bolt hard and the load it carries fluctuates by a fifth of what is applied to it. Both put the structure somewhere no arrangement of external loads could.

mechanics · Free-body The pressure at which an oil becomes a glass. Viscosity against pressure for 4 liquids on a logarithmic axis, from Barus's rule with the pressure coefficient each one actually has. The rule is an exponential, so a gigapascal multiplies an ordinary oil's viscosity by a hundred million or more, and the curves cross the line conventionally taken to mark a glass — a million million pascal-seconds — at 1.37 GPa for a mineral oil, 0.98 GPa for a traction fluid. The vertical line is the peak pressure inside the loaded contact this figure is about, 1.32 gigapascals, computed from the Hertz solution for that geometry and load. Water is drawn for contrast: its pressure coefficient is thirty times smaller, it never approaches a glass over this range, and that is why it is useless as a lubricant in a rolling contact however clean it is. Each curve is drawn solid up to the glass line and dashed above it, because past that point the material is not a liquid and its viscosity is not what decides how it shears — the exponential continues, and the substance it was written for does not.

The oil that is a glass for a quarter of a millisecond

Every other account of viscosity here varies the temperature. Pressure does something larger and in the same direction for every liquid: viscosity rises exponentially, by a factor of ten to the eight or more at the gigapascal inside a loaded gear tooth. That is not a curiosity — it is the only reason there is a film there at all. Remove the pressure dependence from the calculation and the predicted film is five nanometres, under the roughness, and the surfaces touch.

fluids · Viscosity

Named alongside it

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

Constraint countingStatically indeterminateStiffnessFrictionPreloadReaction forceStatic equilibriumContactContact areaContact mechanicsDissipationHysteresis

All concepts