Poisson's ratio ν, modulus relations, and 3D deformation — for isotropic linear-elastic materials.
ν = −ε_lateral / ε_axialSign convention: tensile axial strain is positive; the lateral strain of an ordinary material is then negative (it thins). The formula carries the minus sign, so ν comes out positive.
Pick any two elastic constants; the calculator returns ν plus the remaining modulus. Valid only for isotropic linear-elastic solids (−1 < ν < 0.5).
| Material | Typical ν |
|---|---|
| Steel | 0.29–0.30 |
| Aluminium | 0.33 |
| Copper | 0.34 |
| Titanium | 0.34 |
| Concrete | 0.20 |
| Glass | 0.23 |
| Rubber | ≈0.50 (near the incompressible limit) |
| Cork | ≈0.00 (almost no lateral deformation) |
| Auxetic foam | negative (widens when stretched) |
Textbook room-temperature values for isotropic engineering materials. Use a preset in Mode 3 to fill ν.
| Case | Inputs | Result |
|---|---|---|
| ν from strain | ε_axial=0.002, ε_lateral=−0.0006 | ν = 0.3000 |
| ν from moduli | E=200 GPa, G=77 GPa | ν = 0.2987 |
| 3D deformation | ε_axial=0.002, ν=0.3 | ε_lat = -0.0006, ΔV/V = 0.0008 |
Methodology: all computation is 100% client-side in your browser. Definitions and relations assume an isotropic linear-elastic solid under small uniaxial strain, below yielding. They do not apply to anisotropic materials (wood, fibre composites), large deformations, or fluids (G = 0).
Poisson's ratio ν measures how much a material thins when stretched: ν = −ε_lateral / ε_axial, where ε_axial is the strain along the load and ε_lateral the strain perpendicular to it. Example: stretching a bar to ε_axial = 0.002 while it narrows by ε_lateral = −0.0006 gives ν = 0.3000. Enter your measured strains in the "ν from strain" tab.
For an isotropic linear-elastic solid, E = 2G(1+ν) = 3K(1−2ν), so any two of the four constants determine the other two. Example: steel with E = 200 GPa and G = 77 GPa gives ν = E/(2G) − 1 = 0.2987. Use the "Modulus relations" tab to solve any pair.
For an isotropic linear-elastic solid, −1 < ν < 0.5. Most metals sit around 0.28–0.35 (steel 0.29–0.30, aluminium 0.33, copper 0.34). Rubber approaches the 0.5 incompressible limit, cork is near 0, and negative ν means auxetic behaviour. Values outside (−1, 0.5) are impossible for a passive isotropic solid — the calculator flags them.
Under small uniaxial strain the volumetric strain is ΔV/V = ε_axial × (1 − 2ν). Example: ε_axial = 0.002 with ν = 0.3000 gives ΔV/V = 0.0008 — the bar gains about 0.08% volume. Near ν = 0.5 (rubber) the volume barely changes; auxetic materials (ν < 0) gain volume faster. See the "3D deformation" tab.
That tool converts between all 14 pairs of isotropic elastic constants (E, ν, G, K, λ, M) plus cubic-crystal C₁₁/C₁₂/C₄₄ — its headline is constant-to-constant conversion. This calculator is ν-focused: the definition from measured strains, the 3D deformation (lateral strain and volume change), and a material ν table.
That tool is K-focused: computing bulk modulus from compression data (K = −V·ΔP/ΔV) and sound speed in fluids and solids. This calculator is ν-focused: it starts from strains rather than pressure and volume, and shows how ν controls lateral thinning and volume change.
Because ν = 0.5 with ΔV/V = ε(1−2ν) gives zero volume change — the incompressible limit. A passive isotropic solid cannot exceed it: that would mean the material gains more lateral contraction energy than the stretch supplies. Fluids sit exactly at this limit (G = 0).
Yes — materials with negative ν are called auxetics: they get wider when stretched instead of thinner. Engineered foams and some metamaterials show this. For example, an auxetic foam with ε_axial = 0.001 and ε_lateral = +0.0002 has ν = −0.2. The thermodynamic lower bound for isotropic solids is ν > −1.
Room-temperature textbook values: steel 0.29–0.30, aluminium 0.33, copper 0.34, titanium 0.34, concrete 0.20, glass 0.23, rubber ≈ 0.50, cork ≈ 0.00. The calculator's table lists them, and Mode 3 presets fill ν for you.
Only for isotropic linear-elastic solids: same properties in every direction, small deformations, below yielding. They do not apply to anisotropic materials (wood, fibre composites), rubber at large strain, anything past its yield point, or fluids (G = 0, where ν is undefined and the ν → 0.5 limit applies).
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