Calculate the shear modulus G from Young's modulus E and Poisson's ratio ν using G = E/(2·(1+ν)). Isotropic elastic materials have only two independent material constants – if one of the three quantities E, G and ν is unknown, it follows exactly from the other two.
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Calculate the shear modulus G from Young's modulus E and Poisson's ratio ν using G = E/(2·(1+ν)). Isotropic elastic materials have only two independent material constants – if one of the three quantities E, G and ν is unknown, it follows exactly from the other two.
Structural steel with E = 210 GPa and ν = 0.3 gives G = 210 GPa/(2·1.3) = 80.77 GPa. This matches the tabulated steel value of about 81 GPa and confirms the relation independently. Using ν = 1/3 instead gives G = 78.75 GPa – the choice of ν therefore influences G only weakly, because ν appears in the denominator only as (1+ν).
Valid only for homogeneous, isotropic, linear-elastic materials – only there do exactly two independent elastic constants exist, so that the third follows exactly. The relation does not apply to anisotropic materials such as fibre composites, wood, single crystals or directionally solidified structures; there E, G and ν are direction-dependent and must be measured independently. Nonlinear and viscoelastic behaviour (plastics and elastomers under time and temperature), yielding above the elastic limit and the temperature dependence of the constants are likewise not covered.
Material tables often list only Young's modulus, although torsion, shear and spring calculations need the shear modulus. This calculator determines G from E and Poisson's ratio ν – and conversely E or ν from the other two.
Young's modulus E describes stiffness against length change (σ = E·ε), the shear modulus G stiffness against angular distortion (τ = G·γ), and Poisson's ratio ν states how strongly a stretched bar contracts transversely. For a homogeneous, isotropic, linear-elastic material, however, only two independent material constants exist – the third is fixed by G = E/(2·(1+ν)). This is not an approximation but follows necessarily from the elasticity law.
A sponge stretched lengthwise becomes thinner. How much is what ν states. And how easily the same sponge can be sheared sideways is directly related to how easily it stretches and contracts. That is why E and ν already contain all the information about G.
G = E / (2 · (1 + ν))
Shear modulus: G = E/(2·(1+ν))Young's modulus: E = 2·G·(1+ν)Poisson's ratio: ν = E/(2·G) − 1Hooke's law for shear: τ = G·γBounds of the isotropic range: E/3 ≤ G ≤ E/2| Symbol / input | Meaning |
|---|---|
| Shear modulus G | The shear modulus (modulus of rigidity) is the proportionality factor in Hooke's law for shear, τ = G·γ: it links shear stress τ to shear angle γ. G has the same dimension as E but is only about 38 % of it for metals. G is used wherever shear or torsion occurs – in the torsional rigidity of a shaft or the rate of a helical spring. Typical values: steel ≈ 81 GPa, aluminium ≈ 26 GPa. |
| Young's modulus E | Young's modulus describes stiffness in tension and compression, σ = E·ε. Source: material tables – structural steel ≈ 210 GPa, aluminium alloys ≈ 70 GPa, copper ≈ 120 GPa, grey cast iron ≈ 100 GPa. For unknown materials, take it from a tensile test as the slope of the stress-strain curve in the linear range. |
| Poisson's ratio ν | Poisson's ratio is the ratio of transverse to longitudinal strain in a tensile test, entered without a sign: a bar that stretches by 0.1 % lengthwise becomes 0.03 % thinner at ν = 0.3. Source: material tables – most metals have ν ≈ 0.3, rubber approaches 0.5, cork is near 0, concrete ≈ 0.2. Enter it as a decimal, not a percentage. The thermodynamically possible range for isotropic materials ends at ν = 0.5 (constant volume, incompressible). |
E is Young's modulus from a material table (structural steel ≈ 210 GPa, aluminium ≈ 70 GPa, copper ≈ 120 GPa). ν is Poisson's ratio, entered without a sign and as a decimal: ≈ 0.3 for almost all metals, ≈ 0.2 for concrete, close to 0.5 for rubber, near 0 for cork. Both values must refer to the same material at the same temperature.
Choose the target and enter the two known constants. In practice this is almost always E and ν from a material table, with G the unknown. If instead E and G are available from tests, ν can be determined and the isotropy assumption checked: a back-calculated ν outside roughly 0 to 0.5 is a strong indication that the material is not isotropic or that the measurements are faulty.
Structural steel with E = 210 GPa and ν = 0.3: G = 210 GPa/(2·(1+0.3)) = 210/2.6 GPa = 80.77 GPa. The tabulated steel value is about 81 GPa, which confirms the relation independently. Aluminium with E = 70 GPa and ν = 0.33 gives G = 26.3 GPa, again in good agreement with the tabulated ~26 GPa. Using ν = 1/3 instead of 0.3 for steel gives G = 78.75 GPa: uncertainty in ν has only a weak effect, because ν enters merely as (1+ν).
For metals with ν ≈ 0.3, G is always about 38 % of E – a useful mental check. The bounds of the isotropic range show how little room there is: at ν = 0, G would be E/2; at ν = 0.5 (constant volume, incompressible like rubber), G = E/3. G can therefore never be less than a third nor more than half of E. A G value outside that band means either an anisotropic material or an error in the data.
The calculation runs in pascal internally. E and G may be entered independently in GPa, MPa, kPa, psi or ksi. ν is dimensionless and entered as a decimal – 0.3, not 30 %. Since G and E always keep the same ratio, the result is independent of the unit system chosen.
Torsion calculations on shafts (angle of twist, torsional rigidity G·It), rates of helical compression and torsion springs, shear deformation of elastomer mounts, shear loading of bonded and riveted joints, material input for FE models, and filling gaps in incomplete material datasheets.
Valid only for homogeneous, isotropic, linear-elastic materials; only there do exactly two independent elastic constants exist. The relation does not apply to fibre composites, wood, single crystals, strongly textured rolled sheet or directionally solidified structures – there E, G and ν are direction-dependent and must be measured separately. Nonlinear and viscoelastic behaviour of plastics and elastomers, yielding above the elastic limit and the temperature dependence of all three constants also lie outside the model.
Common mistake: ν is frequently entered as a percentage (30 instead of 0.3), which gives a completely wrong G. Confusing E and G in torsion formulas is equally common – an angle of twist computed with E instead of G comes out roughly 2.6 times too small. Another error is applying the relation to fibre composites or wood: there it is simply wrong, and a G derived from E and ν can substantially overestimate the shear stiffness.
Torsion calculations on shafts (angle of twist, torsional rigidity G·It), rates of helical compression and torsion springs, shear deformation of elastomer mounts, shear loading of bonded and riveted joints, material input for FE models, and filling gaps in incomplete material datasheets.
E is Young's modulus from a material table (structural steel ≈ 210 GPa, aluminium ≈ 70 GPa, copper ≈ 120 GPa). ν is Poisson's ratio, entered without a sign and as a decimal: ≈ 0.3 for almost all metals, ≈ 0.2 for concrete, close to 0.5 for rubber, near 0 for cork. Both values must refer to the same material at the same temperature.
Valid only for homogeneous, isotropic, linear-elastic materials; only there do exactly two independent elastic constants exist. The relation does not apply to fibre composites, wood, single crystals, strongly textured rolled sheet or directionally solidified structures – there E, G and ν are direction-dependent and must be measured separately. Nonlinear and viscoelastic behaviour of plastics and elastomers, yielding above the elastic limit and the temperature dependence of all three constants also lie outside the model.