Elastic Constants Calculator

Enter Young's modulus (E) and Poisson's ratio (ν) for an isotropic material to calculate its shear modulus, bulk modulus, Lamé's first parameter, and P-wave modulus.

Quick Facts

Governing relation
E = 2G(1+ν) = 3K(1-2ν)
Any two independent elastic constants fix the other two for an isotropic material.
Shear modulus
G = E / [2(1+ν)]
Also called the modulus of rigidity; resists shape-changing (shear) deformation.
Bulk modulus
K = E / [3(1-2ν)]
Resists uniform volume change under hydrostatic pressure.
Valid range
-1 < ν < 0.5
Thermodynamic stability bound for isotropic solids; most engineering materials fall between 0.2 and 0.35.

Your Results

Calculated
Shear Modulus (G)
-
G = E / [2(1+ν)]
Bulk Modulus (K)
-
K = E / [3(1-2ν)]
Lamé's First Parameter (λ)
-
λ = Eν / [(1+ν)(1-2ν)]
P-wave Modulus (M)
-
M = λ + 2G

Ready

Enter Young's modulus, its unit, and Poisson's ratio, then press Calculate.

Formula and Method for Elastic Constants of Isotropic Materials

A homogeneous, isotropic elastic material needs only two independent elastic constants to fully describe how it responds to stress — every other elastic constant can be derived from any two of them. This calculator takes Young's modulus (E) and Poisson's ratio (ν), the two constants most often reported on a material data sheet, and derives the shear modulus (G), bulk modulus (K), Lamé's first parameter (λ), and the P-wave (longitudinal) modulus (M).

How the calculation works

The four derived constants follow directly from the theory of linear elasticity (generalized Hooke's law) for isotropic solids: the shear modulus is G = E / [2(1+ν)], the bulk modulus is K = E / [3(1-2ν)], Lamé's first parameter is λ = Eν / [(1+ν)(1-2ν)], and the P-wave modulus — the stiffness that governs compressional (longitudinal) wave speed — is M = λ + 2G = E(1-ν) / [(1+ν)(1-2ν)]. These equations, together with E and ν themselves, give the elastic constants commonly used in solid mechanics, and any pair among them can be used to reconstruct the rest.

Valid range for Poisson's ratio

Thermodynamic stability of an isotropic solid requires -1 < ν < 0.5. As ν approaches 0.5 the bulk modulus grows without bound, describing a nearly incompressible material such as rubber; most metals and ceramics fall between 0.2 and 0.35, and a handful of engineered "auxetic" foams have a negative Poisson's ratio. Entering ν = 0.5 exactly, or a value outside this range, makes the bulk modulus and Lamé's parameter mathematically undefined, so the calculator rejects those inputs.

Common mistakes

  • Mixing modulus units: Young's modulus is reported in GPa for most metals but in MPa or psi for polymers — pick the matching unit before comparing results across materials.
  • Assuming isotropy: these relations only hold for isotropic materials (uniform properties in every direction); wood, fiber composites, and single crystals need additional independent constants and do not fit this two-constant model.
  • Confusing shear modulus with rigidity: "shear modulus" and "modulus of rigidity" are the same quantity (G) — do not add them together or treat them as separate values.

Frequently Asked Questions

What is the relationship between Young's modulus, shear modulus, bulk modulus, and Poisson's ratio?
For an isotropic elastic material, they are linked by E = 2G(1+ν) = 3K(1-2ν). Knowing any two of E, G, K, and ν lets you solve for the other two using algebra on these two equations.
Why do I only need to enter two values to get four results?
An isotropic, linear elastic material has exactly two independent elastic constants. Every other elastic constant — shear modulus, bulk modulus, Lamé's parameters, P-wave modulus — is a fixed combination of whichever two you start with, so entering Young's modulus and Poisson's ratio is enough to derive the rest.
What is a typical value for Poisson's ratio?
Most metals fall between about 0.25 and 0.35 (steel ≈ 0.30, aluminum ≈ 0.33), rubber and other near-incompressible materials approach 0.5, and cork is close to 0. Thermodynamic stability requires -1 < ν < 0.5 for any isotropic solid.
What is Lamé's first parameter used for?
Lamé's first parameter (λ) does not have a simple physical meaning on its own, but together with the shear modulus (G) it appears directly in the Navier-Cauchy equations of elastic wave propagation and stress-strain relations, which is why seismologists and structural engineers use the pair (λ, G) as an alternative to (E, ν).

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