Why are only two elastic constants independent?

For isotropic (direction-independent) linear elastic materials, the stress-strain relationship is fully described by two independent parameters. All other elastic constants can be derived from any pair.

What is a typical Poisson's ratio?

Most metals are 0.25–0.35. Rubber is nearly 0.5 (incompressible). Cork is near 0. Auxetic materials have negative Poisson's ratio (they expand laterally when stretched).

What is Young's modulus used for?

Young's modulus (E) is the most commonly used elastic constant in structural engineering. It determines deflection under load, natural frequencies, and critical buckling loads.

When should I use bulk modulus?

Bulk modulus is important for hydrostatic loading (uniform pressure from all directions), such as deep-sea or geological applications. It equals the inverse of compressibility.

Do these formulas work for all materials?

Only for isotropic, linearly elastic materials. Anisotropic materials (wood, composites, crystals) require up to 21 independent elastic constants.

What are Lamé parameters used for?

Lamé's first parameter (λ) and second parameter (μ = G) appear in the generalized Hooke's law tensor form used in continuum mechanics and finite element analysis. They are useful when you need a compact material model rather than just E, G, K, and ν.

Elastic Constants Calculator

Relate Young's modulus, shear modulus, bulk modulus, and Poisson's ratio. Find any elastic constant from the others with material presets and conversion formulas.

Solve For

Shear Modulus (G)

GPa

Bulk Modulus (K)

GPa

Poisson's Ratio (ν)Typically 0.1–0.45 for real materials

Young's Modulus (E)

206.180 GPa

Ratio of tensile stress to tensile strain — stiffness in tension/compression

Shear Modulus (G)

79.300 GPa

Ratio of shear stress to shear strain — rigidity under twisting

Bulk Modulus (K)

160.000 GPa

Resistance to uniform compression — inverse of compressibility

Poisson's Ratio (ν)

0.3000

Ratio of lateral strain to axial strain; 0.5 = incompressible, 0 = no lateral deformation

Lamé First Parameter (λ)

107.133 GPa

λ = K − 2G/3 — used in continuum mechanics stress-strain relations

P-Wave Modulus (M)

265.733 GPa

M = K + 4G/3 — governs compression wave speed in the material

Poisson's Ratio Visual

0.3000

-1 (auxetic)0 (cork)0.5 (incompressible)

Material Elastic Properties

Material	E (GPa)	G (GPa)	K (GPa)	ν
Steel	200	79.3	160	0.3
Aluminum	69	25.5	76	0.35
Copper	120	44	140	0.34
Glass	70	29	41	0.22
Rubber	0.01	0.003	2	0.499
Titanium	116	44	110	0.32
Concrete	30	12.5	18	0.2
Bone	18	3.3	12	0.3

Conversion Formulas

From	To E	To G	To K	To ν
E, ν	—	E/(2(1+ν))	E/(3(1−2ν))	—
E, G	—	—	EG/(3(3G−E))	E/(2G)−1
K, G	9KG/(3K+G)	—	—	(3K−2G)/(2(3K+G))
K, ν	3K(1−2ν)	3K(1−2ν)/(2(1+ν))	—	—

Planning notes, formulas, and examples

About the Elastic Constants Calculator

For any isotropic linear elastic material, all elastic behavior can be described by just two independent constants. The four commonly used elastic constants — Young's modulus (E), shear modulus (G), bulk modulus (K), and Poisson's ratio (ν) — are all interrelated, so knowing any two allows you to calculate the other two.

Young's modulus measures stiffness in tension or compression. Shear modulus (also called the modulus of rigidity) measures resistance to shearing deformation. Bulk modulus measures resistance to uniform compression. Poisson's ratio describes how much a material contracts laterally when stretched axially — most materials have values between 0.2 and 0.45, with rubber approaching 0.5 (incompressible) and cork near 0 (no lateral expansion).

This calculator lets you solve for any one of the four elastic constants given two others. It includes material presets for common engineering materials, a visual Poisson's ratio indicator, and complete conversion formula references for all constant pairs.

When This Page Helps

Converting between elastic constants means keeping several coupled formulas straight, and that is where algebraic slips happen. This calculator handles all six two-variable combinations, includes the Lamé parameters and P-wave modulus used in continuum mechanics and seismology, and gives you material presets so you can verify a result against a known material instead of trusting memory alone.

How to Use the Inputs

Select which elastic constant to solve for from the dropdown.
Enter the two known elastic constants in the appropriate fields.
Alternatively, click a material preset to auto-fill all four constants.
Read the computed values for all four elastic constants plus Lamé's first parameter and P-wave modulus.
Check the Poisson's ratio visual indicator — values near 0.5 indicate incompressible material.
Reference the conversion formula table for the relationships between constant pairs.

Formula used

Relationships (isotropic materials):
  E = 2G(1 + ν) = 3K(1 − 2ν) = 9KG/(3K + G)
  G = E/(2(1 + ν)) = 3K(1 − 2ν)/(2(1 + ν))
  K = E/(3(1 − 2ν)) = EG/(3(3G − E))
  ν = E/(2G) − 1 = (3K − 2G)/(2(3K + G))

Lamé Parameters:
  λ = K − 2G/3
  μ = G (second Lamé parameter)

P-Wave Modulus:
  M = K + 4G/3

Example Calculation

Result: 206.2 GPa

With G = 79.3 GPa and ν = 0.3 (typical steel), Young's modulus is E = 2 × 79.3 × (1 + 0.3) = 206.2 GPa, and bulk modulus is K = 206.2 / (3 × (1 − 0.6)) = 171.8 GPa.

Tips & Best Practices

Poisson's ratio must be between -1 and 0.5 for thermodynamic stability — the calculator enforces this constraint.
For most engineering metals, ν ≈ 0.3 is a reasonable approximation when exact data is unavailable.
The P-wave modulus determines the speed of compression waves in the material: v_p = √(M/ρ).
Temperature significantly affects elastic constants — most materials get softer (lower E, G, K) at higher temperatures.
Composites and many biological materials are anisotropic — these formulas do not apply to their directional properties.
Rubber's Poisson's ratio near 0.5 means it deforms in shape but barely changes volume, which is why it is used for seals.

Understanding the Four Elastic Constants

Young's modulus (E) measures tensile stiffness, shear modulus (G) measures resistance to shape change, bulk modulus (K) measures resistance to uniform compression, and Poisson's ratio (ν) captures how much a material narrows when stretched. For isotropic linear materials, any two of those are enough to recover the others.

How Engineers Use Them

Structural engineers use E for deflection and buckling checks. Mechanical engineers use G for torsion and shaft design. Geophysicists use K and G to interpret seismic wave speeds. If the constants you measure do not agree with one another, that is usually a sign to recheck the test method or the specimen assumptions.

Limits of the Model

These relationships only apply cleanly to isotropic, linearly elastic materials. Wood, composites, and many crystals need a fuller directional stiffness model, so this calculator is best used as a quick isotropic check rather than a universal material law.

Sources & Methodology

Last updated: March 7, 2026

Frequently Asked Questions

For isotropic (direction-independent) linear elastic materials, the stress-strain relationship is fully described by two independent parameters. All other elastic constants can be derived from any pair.