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Mechanical
Bulk modulus from Young’s modulus and Poisson’s ratio
Resistance of an isotropic elastic material to uniform compression.
- Ref
- MEC-84
- Rev
- v1
- Field
- Mechanical
- Result
- K [GPa]
The bulk modulus K measures how much a material resists volume change under all-round pressure. For an isotropic elastic solid, K = E/[3(1 − 2ν)]. It grows without limit as ν approaches 0.5, which is the incompressible limit (rubber, for example). Together with G it gives the other elastic constants. The volume strain under a pressure p is p/K.
Resistance of an isotropic elastic material to uniform compression. It grows without limit as Poisson’s ratio approaches 0.5, the value for an incompressible material.
Calculator
- Shear modulus G from the same E and ν76.92 GPa
- Ratio K/E0.8333
- Volume change under 100 MPa all-round pressure0.06 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With E = 200 GPa and ν = 0.3, the bulk modulus is 166.7 GPa. A pressure of 100 MPa on all faces would shrink the volume by only 0.06 %.
- For ν = 0.3, K is 0.83 E; for ν = 0.4 it is 1.67 E, and as ν tends to 0.5 K grows without limit, which is why rubber is easy to deform but almost impossible to compress.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (GPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ν Poisson’s ratio (below 0.5) | 0 to 0.49 | 66.67 | 3,333 | 3,267 | +10% | +17.6 % |
| E Young’s modulus | 1 to 500 GPa | 0.8333 | 416.7 | 415.8 | +10% | +10 % |
Sweep of Poisson’s ratio (below 0.5)
| Poisson’s ratio (below 0.5) | Bulk modulus (GPa) |
|---|---|
| 0 | 66.67 |
| 0.06125 | 75.97 |
| 0.1225 | 88.3 |
| 0.1837 | 105.4 |
| 0.245 | 130.7 |
| 0.3063 | 172 |
| 0.3675 | 251.6 |
| 0.4287 | 467.8 |
| 0.49 | 3,333 |
Sweep one variable
Variables and how each one contributes
Bulk modulus
What the equation gives you. Everything else feeds it.
Young’s modulus
Young’s modulus, in GPa. Raises K in direct proportion. Raising it by 10 GPa (to 210 GPa) moves the result by +5 %, from 166.7 to 175 GPa.
Poisson’s ratio (below 0.5)
Poisson’s ratio, below 0.5. Raises K, steeply as ν approaches 0.5, where the material becomes incompressible. Raising it by 0.03 (to 0.33) moves the result by +17.6 %, from 166.7 to 196.1 GPa.
Worked example with the values above
- Denominator: 3(1 − 2ν) = 3 × (1 − 2 × 0.3) = 1.2.
- Divide the modulus: 200 / 1.2.
- K = 166.7 GPa.
Assumptions and limits
- An isotropic, linear elastic material, with ν less than 0.5.
- Small volume strains.
Common mistakes
- Using ν = 0.5 for rubber, which gives an infinite K.
- Applying it to an anisotropic material.
Statistics and uncertainty
Close to ν = 0.5 the formula is extremely sensitive: a 0.01 error in ν at ν = 0.45 changes K by about 20 %. For nearly incompressible materials, measure K directly.
Where it is used
- Compressibility of solids and the pressure rise in a confined, heated solid.
- Hydrostatic response in pressure-vessel and underwater applications.
Maths you need
- Stress and strain: force per area and relative change in length
- Elastic modulus: stress divided by strain in the elastic range