Elastic Constants Calculator
Convert between every isotropic elastic constant — Young's modulus E, Poisson's ratio ν, shear modulus G, bulk modulus K, Lamé's first parameter λ and P-wave modulus M. Pick any two you know; the calculator solves for all the rest.
Converter
| Constant | Symbol | Value |
|---|
Worked example
Structural steel, typical values E = 200 GPa and ν = 0.30:
- Shear modulus: G = E / (2(1+ν)) = 200 / 2.6 = 76.9231 GPa
- Bulk modulus: K = E / (3(1−2ν)) = 200 / 1.2 = 166.6667 GPa
- Lamé's first parameter: λ = νE / ((1+ν)(1−2ν)) = 60 / 0.52 = 115.3846 GPa
- P-wave modulus: M = K + 4G/3 = 269.2308 GPa
Check: C11 = M = 269.2308 GPa and C12 = λ = 115.3846 GPa, consistent with the cubic-crystal relations below.
Typical values (isotropic, room temperature)
Reference presets used by this calculator. Real alloys vary — treat these as typical, not exact.
| Material | E (GPa) | ν | G (GPa) | K (GPa) | λ (GPa) |
|---|---|---|---|---|---|
| Steel | 200 | 0.30 | 76.9231 | 166.6667 | 115.3846 |
| Aluminium | 69 | 0.33 | 25.9398 | 67.6471 | 50.3538 |
| Copper | 110 | 0.34 | 41.0448 | 114.5833 | 87.2201 |
| Rubber | 0.01 | 0.49 | 0.003356 | 0.166667 | 0.164430 |
| Auxetic foam (example) | 100 | −0.20 | 62.5000 | 23.8095 | -17.8571 |
Scope note. This tool converts between the isotropic elastic constants (E, ν, G, K, λ, M) — valid for materials whose properties are the same in every direction (most metals, glass, rubber). If you need K straight from a compression test (K = −V·ΔP/ΔV) or the speed of sound from K and density, use the dedicated bulk modulus calculator instead.
Key relations (isotropic, linear elasticity)
- G = E / (2(1+ν)) · K = E / (3(1−2ν))
- λ = νE / ((1+ν)(1−2ν)) = K − 2G/3
- M = K + 4G/3 · E = 9KG / (3K+G) · ν = (3K−2G) / (2(3K+G))
- Only two constants are independent; every other pair determines the rest.
Frequently asked questions
What are elastic constants?
Elastic constants describe how a solid deforms under load while the deformation stays reversible (the linear-elastic regime). For an isotropic material — one with the same properties in every direction, like most metals, glass and rubber — only two independent constants are needed; every other constant follows from those two. The six used most often are Young's modulus (E), Poisson's ratio (ν), shear modulus (G), bulk modulus (K), Lamé's first parameter (λ) and the P-wave modulus (M).
What is Lamé's first parameter (λ)?
λ (lambda) is one of the two Lamé parameters (the other is the shear modulus G, often written μ). It appears directly in Hooke's law for isotropic materials: λ = νE / ((1+ν)(1−2ν)) = K − 2G/3. For steel with E = 200 GPa and ν = 0.30, λ = 115.3846 GPa. In cubic crystals λ equals the stiffness constant C12.
What are C11, C12 and C44? Are they elastic constants too?
C11, C12 and C44 are stiffness-tensor components of (usually cubic) crystals — they describe anisotropic materials, which need more than two constants. This calculator is for isotropic materials only. For a cubic crystal averaged to isotropic behaviour the conversions are C11 = K + 4G/3 (which equals the P-wave modulus M), C12 = K − 2G/3 (which equals λ) and C44 = G. For steel (E = 200 GPa, ν = 0.30): C11 = 269.2308 GPa, C12 = 115.3846 GPa, C44 = 76.9231 GPa.
Why must Poisson's ratio be between −1 and 0.5?
Stability. Positive strain energy requires a positive bulk modulus (K > 0 forces ν < 0.5) and positive shear and Young's moduli (G > 0 and E > 0 force ν > −1). This calculator rejects ν ≥ 0.5 with an explicit message — the classic example is a nearly incompressible material such as rubber, where ν approaches 0.5 from below but never reaches it.
Can Poisson's ratio be negative?
Yes. Auxetic materials (some foams and engineered metamaterials) expand sideways when stretched, giving ν < 0. The calculator accepts any ν in (−1, 0.5). Example: E = 100 GPa, ν = −0.20 gives G = 62.5000 GPa, K = 23.8095 GPa and λ = -17.8571 GPa — note λ can be negative while K and G stay positive.
Why is rubber's bulk modulus so much larger than its Young's modulus?
Because ν ≈ 0.49 is close to the incompressibility limit 0.5, and K = E / (3(1 − 2ν)) divides by a tiny number. For rubber with E = 0.01 GPa and ν = 0.49: K = 0.166667 GPa (about 17× E) while G = 0.003356 GPa. Nearly incompressible does not mean stiff in tension — it means the volume barely changes.
Which two constants should I enter?
Use whichever pair you have: mechanical engineers usually know E and ν; seismologists work with λ and G; geotechnical engineers often quote K and G. Every valid pair yields the same full set of constants. One exception: entering M together with E is ambiguous — two different physical materials (different ν) can share the same M and E — so the calculator asks you to pair M with ν, K, G or λ instead.
How is this different from the bulk modulus calculator?
This tool converts between the isotropic elastic constants (E, ν, G, K, λ, M) starting from any pair of them. The bulk modulus calculator instead computes K directly from a compression test (K = −V·ΔP/ΔV) and also covers the speed of sound from K and density. Use that tool when your starting point is measured pressure–volume data.
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