Elastic constants from ultrasonic velocity

Bulk wave velocities in an isotropic solid are fixed by its elastic constants and density, so measuring both velocities on the same piece gives the constants back. The shear modulus comes straight from the shear velocity, G = ρ·c_S², and Poisson's ratio from the velocity ratio alone, ν = (c_L² − 2c_S²) / [2(c_L² − c_S²)]. Young's modulus and the bulk modulus follow from those two.

This is a genuinely useful non-destructive measurement. It gives the dynamic elastic constants of a component as it actually is — after heat treatment, after service, in a casting whose density differs from the handbook value — without cutting a tensile specimen. It is used to check material substitution, to verify nodularity in ductile iron, to assess porosity in castings and powder-metallurgy parts, and to supply the modulus for a finite element model of the real part rather than a nominal one.

Dynamic constants from ultrasonics run slightly higher than static constants from a tensile test, typically by a few per cent, because the strain is tiny, the loading is essentially adiabatic and there is no plastic relaxation. Quote them as dynamic values and do not substitute them into a design code that calls for static properties.

The result is only as good as the inputs. Both velocities must be measured on the same piece over the same path, with an accurate thickness and a properly measured density, and the material must be reasonably isotropic and fine-grained. Rolled plate, austenitic weld metal and composites are anisotropic — the constants then depend on direction and this isotropic treatment does not apply. Physically valid results also require c_L > √2 · c_S; below that ratio Poisson's ratio comes out negative or undefined.

ν = (c_L² − 2c_S²) / [ 2 (c_L² − c_S²) ]
G = ρ · c_S²
E = 2G(1 + ν) = ρ·c_S² (3c_L² − 4c_S²) / (c_L² − c_S²)
K = ρ ( c_L² − 4/3 · c_S² )
M = ρ · c_L²   (longitudinal / P-wave modulus)