Lamé Parameters Calculator

For a linear, isotropic solid, two independent elastic constants define the rest. This tool computes Lamé parameters:

• μ is the shear modulus, often written as G.
• λ is Lamé’s first parameter.

Common conversions used:

• From E and ν: μ = E / (2(1+ν)), λ = Eν / ((1+ν)(1−2ν))
• From K and μ: λ = K − 2μ/3
• Derived values: K = λ + 2μ/3, ν = λ / (2(λ+μ)), E = μ(3λ+2μ)/(λ+μ), M = λ + 2μ

1. Select an input method that matches your available constants.
2. Choose the input unit for any modulus values you enter.
3. Set your preferred output unit, decimals, and notation.
4. Click Calculate to display results above the form.
5. Use Download CSV or Download PDF for reporting.

Values are typical examples and vary by alloy, heat treatment, and temperature.

1) Lamé parameters in isotropic elasticity

Lamé parameters λ and μ define the stress–strain law for a homogeneous, isotropic, linear elastic solid. They are widely used in continuum mechanics because many governing equations, boundary‑value problems, and numerical solvers become compact when written in Lamé form.

2) Physical meaning of λ and μ

The shear modulus μ (also written as G) controls resistance to shape change at constant volume. The parameter λ couples volumetric strain to normal stress and strongly influences compressibility. Together they determine bulk modulus, Young’s modulus, and Poisson’s ratio.

3) What this calculator computes

You can start from common input pairs such as E–ν, K–G, or direct λ–μ. The tool converts your inputs into λ and μ, then derives E, K, ν, and the P‑wave modulus M = λ + 2μ for cross‑checks.

4) Typical engineering data for scale

Room‑temperature examples show realistic magnitudes. Steel often uses E ≈ 200 GPa and ν ≈ 0.30, giving μ ≈ 76.9 GPa and λ ≈ 115.4 GPa. Aluminum can use E ≈ 69 GPa, ν ≈ 0.33, giving μ ≈ 25.9 GPa and λ ≈ 50.5 GPa.

5) Stability limits and near‑incompressibility

For stable isotropic solids, a practical guideline is μ > 0 and K > 0, which typically corresponds to -1 < ν < 0.5. As ν approaches 0.5, the material becomes nearly incompressible and λ can become much larger than μ, which may cause stiff numerical behavior.

6) Linking to wave and dynamic properties

Lamé parameters connect directly to wave response through M = λ + 2μ. In materials testing, shear modulus relates to shear wave behavior, while M reflects longitudinal response. When paired with density and measured wave speeds, these moduli help validate elastic constants from ultrasonic or vibration data.

7) Value in simulation and material models

Some finite element formulations and constitutive implementations prefer λ and μ, especially for mixed methods or penalty approaches. Converting from datasheet E–ν into Lamé form improves consistency when multiple material models share the same elastic baseline and when comparing compressibility across candidates.

8) Units, reporting, and traceability

All moduli carry stress units, so consistent unit handling is essential. This calculator supports Pa, kPa, MPa, GPa, and psi for input and output. Results appear above the form for quick review, and CSV/PDF exports provide a compact record for design reviews and documentation.

1) What are Lamé parameters used for?

They parameterize isotropic linear elasticity. Many constitutive laws and numerical solvers use λ and μ directly to compute stress from strain in three dimensions.

2) Is μ the same as shear modulus G?

Yes. In isotropic elasticity, μ equals the shear modulus G. The calculator labels this clearly and converts between input pairs that include G.

3) Why does λ grow very large near ν = 0.5?

As ν approaches 0.5, the material becomes nearly incompressible. The term (1−2ν) in the λ formula shrinks, making λ increase sharply relative to μ.

4) Which input method should I choose?

Use the pair you trust most from your data sheet or test results: E and ν are common; K and G are convenient; λ and μ work for direct model inputs.

5) What unit should I use for moduli?

Any consistent stress unit works. Choose an input unit matching your data, then pick an output unit for reporting. Poisson’s ratio stays dimensionless.

6) What does the P-wave modulus M represent?

M equals λ + 2μ and relates to longitudinal wave response in an isotropic solid. It is useful when linking elastic properties to ultrasonic or seismic measurements.

7) Why do I see stability warnings?

Warnings appear when derived parameters fall outside typical stability limits, such as ν not in -1 < ν < 0.5 or nonpositive K or μ. Recheck inputs and units.