the theoretical shear strength

Suppose a crystal had no dislocations at all — how strong would it be? To shear a truly perfect crystal you would have to slide one entire plane of atoms bodily over the plane beneath it, breaking every bond across that plane at the same instant and reforming them one atom-spacing along. That takes a colossal stress. The theoretical shear strength is that stress: the shear a flawless, dislocation-free crystal could withstand before slipping. It is the strength metals ideally OUGHT to have, and the gulf between it and their real strength is the founding puzzle of the whole subject.

A simple estimate, due to Frenkel, models the resistance to shear as rising and falling sinusoidally as one plane slides over the next — zero when the atoms sit in registry, zero again a full spacing later, and peaking in between. Working through the geometry gives a maximum of roughly tau_th = G / (2 pi), where G is the shear modulus; more careful models and different bonding push this into the range G/10 to G/30, but the headline is that the ideal strength is a sizeable FRACTION of the shear modulus — thousands of MPa for a typical metal. For copper (G = 48 GPa), tau_th is around 8 GPa.

Now the punchline, which is the honest headline of this entire field. Real annealed metals do not yield anywhere near G/10. They yield at something like G/1000 to G/100000 — copper flows plastically at around 1 MPa, roughly ten thousand times below its ideal strength. Why the vast shortfall? Because real crystals are riddled with dislocations, and a dislocation lets the crystal shear one atomic row at a time (glide) instead of all at once, at a tiny fraction of the stress. That single realisation, made in 1934, explained why metals are soft and launched the whole science of dislocations. It also points the way back up: remove or pin the dislocations — as in a flawless whisker, a heavily worked wire, or a nanostructured metal — and the strength climbs back toward the theoretical value.