Relation Between E, G and ν

An isotropic material — one with no grain direction, which covers most metals, glass and concrete well enough — has only two independent elastic constants. Pick any two of E, G, ν and K, and the rest follow. This particular identity falls out of resolving a pure shear into equal tension and compression on planes at 45°, and it is the fastest sanity check in the business: steel's E = 200 GPa with ν = 0.30 demands G = 200 ÷ (2 × 1.3) = 76.9 GPa, which is what handbooks list (77 GPa).

Whether elasticity needed two constants or one was a genuine nineteenth-century brawl. Navier, Poisson and Cauchy's early "rari-constant" molecular theory insisted a single constant sufficed and forced ν = 1/4 for everything; Green and Stokes argued for two on energy grounds. Careful measurement settled it — real materials refuse to sit at ν = 0.25. The trap: this relation applies only to isotropic materials. Wood, fibre composites and rolled sheet with strong texture have direction-dependent moduli, and plugging one axis's E into this formula will give you a shear modulus that is off by a factor of ten.