Elastic Constants Calculator (E, G, ν, K and Lamé λ from any two)

Understanding the elastic constants calculator — concept and principle

This calculator takes any two elastic constants of an isotropic material and returns the rest: Young's modulus E, shear modulus G, Poisson's ratio ν, bulk modulus K, the Lamé constant λ and the P-wave modulus M. Engineers, geophysicists and materials students use it when a data sheet lists only some of these.

For steel-like E = 200 GPa and ν = 0.3, every other isotropic constant follows.

How it is calculated

G = E ÷ (2(1 + ν)); K = E ÷ (3(1 − 2ν))

λ = K − 2G/3; M = K + 4G/3

From E and G: ν = E/(2G) − 1 and K = EG ÷ (3(3G − E)).
From K and G: E = 9KG ÷ (3K + G) and ν = (3K − 2G) ÷ (2(3K + G)).
From E and K: ν = (3K − E) ÷ (6K) and G = 3KE ÷ (9K − E).
Moduli are in GPa; ν has no unit and must lie between −1 and 0.5.

The principle

A linear-elastic isotropic material responds the same way in every direction. Its whole behaviour can then be described by just two independent numbers: one for resistance to change of shape and one for resistance to change of volume. E, G, ν, K and λ are different ways of packaging those two numbers, so any two determine the others. At ν = 0.5 the material is incompressible and K becomes infinite.

Worked example

Default inputs: E = 200 GPa, ν = 0.30 (typical of steel).

G = 200 ÷ (2 × 1.30) = 76.92 GPa.
K = 200 ÷ (3 × (1 − 0.60)) = 200 ÷ 1.2 = 166.7 GPa.
λ = 166.67 − 2 × 76.92 ÷ 3 = 115.4 GPa.
M = 166.67 + 4 × 76.92 ÷ 3 = 269.2 GPa.

Each point is the calculator re-run with only “Young's modulus E (GPa)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Young's modulus E (GPa)	Young's modulus E (GPa)
50	50
100	100
150	150
200	200
250	250
300	300
400	400

How to read the result

Most metals have ν between about 0.25 and 0.35, so G is roughly 37–40% of E. Rubber is close to 0.5, cork close to 0, and a negative ν (an auxetic material) means the material gets fatter when stretched. The P-wave modulus sets the speed of compressional waves in rock or a long bar under lateral constraint, and G sets the shear-wave speed.

Common mistakes

Mixing units, such as E in GPa and G in MPa.
Entering a pair that gives ν outside −1 to 0.5, which no stable isotropic solid can have.
Using these relations for wood, composites or single crystals, which are anisotropic.
Confusing the Lamé constant λ with a wavelength.

Keep in mind: Real material properties depend on alloy, heat treatment, temperature and test method. Use certified data for design, and real designs need applicable codes and a qualified engineer.

Frequently asked questions

Why is the shear modulus smaller than Young's modulus?

For positive ν, G = E ÷ (2(1 + ν)) is always less than half of E.

What is the bulk modulus of steel?

Around 160 GPa, close to the value calculated here.

What is the Lamé constant used for?

It appears with G in the general stress–strain equations and in seismic wave formulas.

Sources: MIT OpenCourseWare 2.001 Mechanics & Materials I (Fall 2006), lecture 13 notes

Checked 9 Nov 2026. MIT OpenCourseWare 2.001 Mechanics & Materials I (Prof. C. Livermore, Fall 2006), lecture 13 notes: ε_lateral = −ν ε_axial, 0 ≤ ν ≤ 0.5 for common materials, G = E/(2(1 + ν)), K = E/(3(1 − 2ν)). The other rearrangements, λ = K − 2G/3 and M = K + 4G/3, are standard isotropic-elasticity identities.

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