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Elastic Constants Calculator

The elastic constants calculator can help you explore the relationships between the moduli of elasticity for isotropic materials.

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Young's modulus E200 GPa200 GPa — Structural steel
Shear modulus G76.9231 GPa38.46% of E
Bulk modulus K166.6667 GPa0.8333× E — resistance to uniform compression
Poisson's ratio ν0.3typical of a metal
Lamé's first parameter λ115.3846 GPa
Check: 2G(1+ν)200 GPamust equal E — this is the identity that makes only two constants independent
Check: 3K(1−2ν)200 GPamust also equal E
Compressibility6.0e-12 Pa⁻¹the reciprocal of the bulk modulus
Volume change under 100 MPa0.06%uniform pressure — solids are remarkably incompressible
Why ν cannot reach 0.5K would be infiniteat ν = 0.5 the material conserves volume exactly and becomes incompressible — rubber comes close at 0.4999, and water is the limiting case
Why ν cannot be below −1G would be negativenegative-ν materials do exist — auxetics get FATTER when stretched — but none go below −1

The formula

E = 2G(1+ν) = 3K(1−2ν) — only two of the four are independent

Four names, two degrees of freedom

An isotropic solid has only two independent elastic constants. Young's modulus, shear modulus, bulk modulus and Poisson's ratio are four ways of describing those two, linked by E = 2G(1+ν) = 3K(1−2ν). Give any two and the rest follow, which is why this page takes a pair rather than asking for all four — supplying three invites contradiction.

Poisson's ratio is bounded between −1 and 0.5 by those same relations. At 0.5 the material conserves volume exactly and the bulk modulus becomes infinite; rubber sits at about 0.4999, which is why it is treated as incompressible in analysis. Above 0.5 the bulk modulus would be negative, meaning the material expands when squeezed, which nothing does.

Negative Poisson's ratios are real, though. Auxetic materials — certain foams and engineered lattices — get thicker when stretched, and they are genuinely useful for impact absorption. They still cannot go below −1.

For most metals G is close to 0.38E and K close to 0.83E. Those ratios follow directly from ν being near 0.3, so a metal with a wildly different ratio either is not isotropic or has had one of its constants mismeasured.