Lamé Parameters Calculator

Input Panel

Enter any supported elastic input set

Overall page stays single-column. The calculator fields use 3 columns on large screens, 2 on smaller screens, and 1 on mobile.

Calculation mode

Modulus unit

Density unit

Velocity unit

Young’s modulus E

Uses the selected modulus unit.

Poisson’s ratio ν

Dimensionless value.

Density ρ (optional)

Add density to derive wave speeds.

Example Data Table

Typical isotropic material inputs and outputs

Material	E (GPa)	ν	λ (GPa)	μ = G (GPa)	K (GPa)
Steel	210.00	0.30	121.15	80.77	175.00
Aluminum	69.00	0.33	50.88	25.94	68.17
Rubber-like solid	0.010	0.49	0.164	0.0034	0.167

Formula Used

Core equations for isotropic linear elasticity

From E and ν
μ = E / [2(1 + ν)]
λ = Eν / [(1 + ν)(1 - 2ν)]
K = E / [3(1 - 2ν)]
M = λ + 2μ

From K and G
λ = K - 2G/3
E = 9KG / (3K + G)
ν = (3K - 2G) / [2(3K + G)]
M = K + 4G/3

From λ and μ
K = λ + 2μ/3
E = μ(3λ + 2μ) / (λ + μ)
ν = λ / [2(λ + μ)]
M = λ + 2μ

From wave speeds and density
μ = ρVs²
λ = ρ(Vp² - 2Vs²)
K = λ + 2μ/3
M = ρVp²

Stability for isotropic elasticity requires μ > 0 and 3λ + 2μ > 0. These conditions also imply positive shear and bulk response.

How to Use This Calculator

Steps

Select a calculation mode based on the input constants you already know.
Choose modulus, density, and velocity units before entering values.
Enter two or three values, depending on the selected mode.
Click the calculate button to show results above the form.
Review λ, μ, K, E, ν, M, stability, and optional wave speeds.
Use the CSV and PDF buttons to export the results section.

FAQs

Lamé Parameters FAQ

1. What are Lamé parameters?

Lamé parameters are two elastic constants for isotropic linear materials. The first is λ, linked to volumetric response. The second is μ, also called shear modulus, linked to resistance against shape change.

2. Why is μ equal to shear modulus?

In isotropic elasticity, the second Lamé parameter directly measures resistance to shear deformation. Because of that, μ and G describe the same physical property and always have identical numeric values.

3. Can λ be negative?

Yes. A material can have negative λ and still remain physically admissible, provided μ stays positive and the bulk condition 3λ + 2μ > 0 also remains satisfied.

4. What does Poisson’s ratio tell me?

Poisson’s ratio measures lateral strain relative to axial strain. Values near 0.5 suggest nearly incompressible behavior, while lower values indicate stronger volume change under loading.

5. Why add density to the calculation?

Density allows the calculator to convert elastic constants into seismic or ultrasonic wave speeds. With ρ, the page can compute shear-wave velocity Vs and compressional-wave velocity Vp.

6. Which input mode should I choose?

Choose the mode that matches your available lab, textbook, or simulation data. E and ν are common in mechanics, K and G are common in constitutive models, and Vp, Vs, ρ are common in geophysics.

7. What is the P-wave modulus M?

The P-wave modulus is M = λ + 2μ. It controls compressional wave propagation in isotropic solids and is also equal to ρVp² when density and P-wave speed are known.

8. When are Lamé parameters useful?

They are useful in continuum mechanics, finite element modeling, acoustics, seismology, materials science, and structural analysis wherever isotropic elastic stress-strain behavior must be converted between equivalent material constants.