Elasticity and Continuum Mechanics

The field's other foundational idea is not about materials at all, and Euler had it before anyone could measure a stress. A slender column under compression does not fail by being crushed; it bows sideways, at a load

$$P_{\text{cr}} = \frac{\pi^{2}EI}{L^{2}},$$

where $EI$ is the bending stiffness. Nothing in the column has exceeded any limit of the material. The straight configuration has simply stopped being stable. Structures fail this way more often than they fail by fracture, and the inverse-square dependence on length is unforgiving: double the column and it carries a quarter.

Buckling is a loss of stability, and stability problems behave differently from strength problems in a way that has killed people. A strength calculation is forgiving: exceed the limit by 10% and a ductile material yields locally and redistributes the load. A buckling calculation is not, because beyond the critical load there is no neighbouring equilibrium to fall back into — the column leaves its straight configuration and keeps going. The collapse of the Quebec Bridge in 1907, which killed 75 workers, followed from compression chords whose capacity had been estimated from tests on short specimens and applied to members several times longer, where the inverse-square law governs.

Worse, real structures are imperfect, and buckling is unusually sensitive to imperfection. A perfect cylinder under axial compression has a critical load that can be calculated exactly; a real one, out of round by a fraction of its wall thickness, may carry only a third of it. This imperfection sensitivity is why thin-shell structures — rocket bodies, submarine hulls, silo walls — are designed with large margins against a theoretical figure nobody expects to reach, and why the relevant codes are built on test data rather than on the elegant formula.

Estimate the strength of glass from its bonds. The theoretical cohesive strength of a brittle solid is roughly

$$\sigma_{\text{th}} \approx \frac{E}{10},$$

and for silica glass $E \approx 70$ GPa, giving about 7 GPa. Measured strength of an ordinary glass rod: around 50 MPa. The discrepancy is a factor of 140, which is not the kind of error that gets fixed by better measurement.

Alan Griffith found the clue in a size effect. Thin glass fibres, freshly drawn, were far stronger than thick rods of the same glass, and the thinner the fibre the stronger it got, approaching the theoretical value as the diameter fell. Strength therefore depends on the specimen, not only on the substance — which means the controlling factor is a flaw, and a thinner fibre has less room for a large one.

His criterion is an energy balance. Extending a crack of length $a$ in a stressed plate releases elastic energy, because the material near the crack faces unloads, and the energy released grows as $a^{2}$. It costs surface energy, which grows as $a$. For small cracks the cost wins and the crack is stable; past a critical length the release wins and the crack runs away. Equating the two gives

$$\sigma_{f} = \sqrt{\frac{2E\gamma}{\pi a}},$$

with $\gamma$ the surface energy per unit area. Put in glass values: $E = 70$ GPa, $\gamma \approx 0.3$ J/m², and a flaw $a = 1$ µm:

$$\sigma_{f} = \sqrt{\frac{2(70\times10^{9})(0.3)}{\pi(10^{-6})}} = \sqrt{1.34\times10^{16}} = 1.2\times10^{8}\ \mathrm{Pa} = 116\ \mathrm{MPa}.$$

A micron-sized flaw accounts for the observed strength within a factor of two. Run the formula backwards for the measured 50 MPa:

$$a = \frac{2E\gamma}{\pi\sigma_{f}^{2}} = \frac{4.2\times10^{10}}{\pi(2.5\times10^{15})} = 5.3\times10^{-6}\ \mathrm{m}.$$

A scratch five microns deep — invisible, routinely present on any handled surface — is the difference between 7 GPa and 50 MPa.

Three consequences follow, and all three are engineering practice. Strength depends on $1/\sqrt{a}$, so it is governed by the single largest flaw, which makes it a statistical quantity: large specimens are weaker than small ones because they contain more chances of a big flaw, and brittle materials are specified with a failure probability rather than a strength. Surface condition matters more than bulk composition, which is why glass is tempered or chemically strengthened to put the surface in compression, closing cracks instead of opening them. And toughness, the energy consumed in fracture, is a separate property from strength — a material can be strong and shatter, or weak and absorb enormous energy before failing.

Metals present the mirror-image puzzle. Their shear strength should be about $G/10$, roughly 8 GPa for steel, and they yield at a few hundred megapascals — a thousand times too weak. The resolution, found independently by G. I. Taylor, Egon Orowan and Michael Polanyi in 1934, is the dislocation. Rather than sliding one whole plane of atoms over another, breaking every bond at once, the crystal moves a line defect through itself, breaking one row of bonds at a time — the difference between dragging a carpet and pushing a ruck along it. The defects were not seen until 1956, and in the meantime an entire metallurgy was built on them: alloying elements, precipitates and grain boundaries all work by impeding dislocation motion, and work hardening is dislocations getting in each other's way.

A continuum with elasticity supports waves, and a solid supports two kinds that a fluid does not: shear waves, in which the motion is transverse, and surface waves. Lord Rayleigh found the latter in 1885 — a disturbance that decays within about a wavelength of depth and travels just below the shear speed.

This is what made the Earth's interior accessible. An earthquake generates compression waves, shear waves and surface waves; each travels at a speed set by the elastic moduli and density of the rock it is in; each arrives at a seismograph at a different time, refracted and reflected along the way. In 1926 Harold Jeffreys concluded from the absence of shear waves through the centre that the outer core must be liquid, since a fluid has no shear modulus and cannot carry them. The same inversion, with thousands of stations and modern computation, now images subducted slabs and mantle plumes in three dimensions.

Making those computations possible required something else that came out of this field. Stresses in a complicated structure have no closed-form solution, and the method devised for aircraft and dams in the 1940s and 1950s — divide the object into elements, write each one's energy in terms of its corner displacements, minimise the total — turned out to be a general way to solve partial differential equations. The finite element method is now the largest single use of engineering computation, and it began as a way of finding out whether a wing would hold.

What happens when a material is neither a simple solid nor a simple fluid — when it flows if pushed slowly and resists if pushed quickly, or when its elasticity comes from entropy rather than from bonds — is soft matter.