How this instrument works
Stretch a bar, twist a shaft, squeeze a block from every side: three experiments that look like probes of three separate properties. They are not. In a material with no grain and no preferred direction, all three responses descend from one set of interatomic bonds; elasticity theory pins exactly two of them as independent. Measure Young's modulus with Poisson's ratio in a single tensile test — one axial gauge, one transverse — then shear modulus G plus bulk modulus K follow without further loading. Supplier data sheets exploit that shamelessly, which is why so few of them bother printing four numbers.
Whether it really was two occupied French mathematics for forty years. Navier's 1821 memoir to the Académie built elastic solids from central forces acting between molecules, yielding a single constant; Poisson sharpened that in 1829 into a flat prediction, ν = 1/4 for everything. George Green rederived the field equations from stored energy in 1837, getting two constants for an isotropic body, twenty-one in general. Guillaume Wertheim's measurements around 1848 kept landing nearer 1/3 across metals as well as glasses, while Gabriel Lamé's 1852 lectures handed the surviving pair their enduring symbols, λ and μ. Historians call it the rari-constant versus multi-constant controversy; experiment decided it, and Poisson's tidy quarter lost.
Three conditions hold this conversion up. Isotropy is the demanding one: rolled sheet, drawn wire, timber, fibre laminate and most printed parts answer differently along different axes, and no pair of constants can describe them — orthotropic wood needs nine, a general crystal twenty-one. Even a single iron crystal shears roughly twice as readily across one plane as another, so its true G bears little relation to E ⁄ 2(1 + ν); ordinary steel obeys only because millions of randomly oriented grains average that directionality away. Second, both inputs must come from below yield, where unloading retraces its own path. Third, ν carries hard bounds — stability demands more than −1 and less than 1/2 — and K races toward infinity as that upper limit nears.
- Enter Young's modulus, the stiffness measured along the pull axis, from a tensile test or a certificate. Pa, MPa, psi and ksi all sit on that field's menu.
- Enter Poisson's ratio as a plain decimal. Most structural metals fall between 0.27 and 0.35; 0.30 is a sound default when all you know is that something is steel.
- Read Shear modulus — the constant governing torsion of shafts, deflection of coil springs and shear stress in bolted joints.
- Read Bulk modulus beside it, for hydrostatic loading. Ratios at or above 0.5 are refused, since K diverges there and no finite answer exists.
Worked example — structural steel at 200 GPa
A rolled section arrives with two figures on its certificate: E = 200 GPa and ν = 0.30. Put 200000000000 Pa into Young's modulus and 0.3 into Poisson's ratio. Shear modulus returns 200 ⁄ 2.6 = 76.9 GPa, printed as 76923076923.1 Pa. Bulk modulus returns 200 ⁄ 1.2 = 166.7 GPa, or 166666666667 Pa. Both land where published tables for structural steel put them.
Those two numbers earn their keep at once. Torsion of a driveshaft, wind-up in a coil spring and shear across a bolt group all run on G rather than E, so 76.9 GPa is what a spring designer genuinely needs — and nobody ever measured it. Swap in aluminium at 70 GPa with ν = 0.33 and G falls to 26.3 GPa, barely a third as much: an aluminium spring of identical geometry winds down almost three times as far under equal torque.
Questions
Why do just two constants describe an entire material?
Because isotropy is a severe restriction. A general elastic solid carries twenty-one independent entries in its stiffness matrix; requiring identical behaviour in every direction collapses that count to two. Any pair serves — E and ν, G and K, or Lamé's λ and μ — and every other constant is a rearrangement of whichever two you hold. Engineering settled on E and ν because one tensile coupon wearing an axial and a transverse strain gauge surrenders both at the same moment.
What happens as Poisson's ratio approaches 0.5?
Bulk modulus diverges. Its denominator 3(1 − 2ν) collapses toward zero, sending K to infinity while G stays finite and modest — the fingerprint of an incompressible solid. Rubber at ν ≈ 0.4999 is the standing example: floppy in shear at a few MPa, yet about as hard to squeeze as water at roughly 2 GPa. This instrument declines ν ≥ 0.5 for exactly that reason. Such materials also wreck ordinary finite-element formulations, which is why FE packages ship mixed or hybrid elements for elastomers.
Can Poisson's ratio be negative?
Yes — auxetic materials grow fatter when pulled. Stability permits anything above −1, and Roderic Lakes fabricated a re-entrant polyurethane foam reaching about −0.7 in 1987; α-cristobalite manages it naturally. Such structures now turn up in sports padding and expandable stents. These formulas cope without complaint: at ν = −0.5, G equals E outright and K drops to E ⁄ 6. Ordinary engineering materials never behave this way, so treat a negative entry as deliberate rather than a typing slip.
Which constants from a data sheet should I trust?
Whichever two were actually measured — and confirm that isotropy holds before converting. Cast and wrought metals, most glasses and unfilled thermoplastics behave properly. Rolled sheet, extrusions, timber, fibre composites and printed parts do not, and pushing their axial figures through this conversion can produce a shear modulus wrong by a factor of two. For laminates, ask a supplier for G₁₂ directly rather than deriving it from a longitudinal E.
How do these constants set the speed of sound in a solid?
Directly. A solid carries two bulk wave types and both are fixed here: shear waves travel at √(G ⁄ ρ), pressure waves at √((K + 4G ⁄ 3) ⁄ ρ). Steel at ρ = 7850 kg/m³ therefore gives about 3130 m/s in shear and near 5860 m/s in compression, close to a commonly quoted 5900. Ultrasonic testing runs that logic backwards: time both wave speeds through a coupon and you recover G, K, E and ν without loading anything. Done carefully, it beats a tensile test for accuracy.
Why does my derived shear modulus disagree with a table?
Usually because ν was assumed rather than measured. G forgives that — shifting ν from 0.27 to 0.35 moves G by only about 6% — but K punishes it, climbing some 53% across that same span, since ν sits inside a denominator that is already small. So when bulk modulus feeds your calculation, measure ν properly with a transverse gauge or by ultrasound. When only G matters, a textbook 0.30 for steel will rarely lead you far astray.