Last updated: 2026-07-13

Bulk Modulus

Bulk modulus is a single number that captures how strongly a material resists being squeezed from every side at once. Press on a block of steel and it barely shrinks; press on a pocket of air and it collapses easily. That difference is exactly what the bulk modulus measures. This calculator works it out from a simple compression test: a known pressure increase applied to a sample of known volume, and the small drop in volume that follows.

Where the formula comes from

Raising the pressure on a sample by $\Delta P$ makes its volume fall by $\Delta V$. The volumetric strain is that loss expressed as a fraction of the original volume, $\varepsilon_V = \Delta V / V_0$, a pure ratio. Over the elastic range the fractional shrink is proportional to the applied pressure, and the constant linking them is the bulk modulus:

The minus sign is there because the volume falls as the pressure rises, so $\Delta V$ is genuinely negative and the ratio would otherwise come out negative. Working with the size of the volume reduction, the calculator writes the same relation as $K = \Delta P / (\Delta V / V_0)$, which is always positive.

Formula

Because the volumetric strain has no units, $K$ carries the units of pressure — pascals — and is most often quoted in gigapascals (GPa).

Worked example

Bulk modulus and compressibility

The compressibility $\beta = 1/K$ is just the bulk modulus turned upside down. A stiff material has a large $K$ and therefore a small $\beta$. Because $K$ is measured in pascals, $\beta$ comes out in inverse pascals ($\text{Pa}^{-1}$), which is why the calculator shows it in scientific notation — for water it is about $4.6\times10^{-10}\ \text{Pa}^{-1}$. Some fields prefer to state compressibility per gigapascal instead, where water sits near $0.46\ \text{GPa}^{-1}$.

Typical values

The range across states of matter is enormous. Air at constant temperature has a bulk modulus of only about 0.1 MPa, so it compresses readily. Water is roughly 2.2 GPa, mercury near 28 GPa, ordinary glass 35–55 GPa, structural steel around 160 GPa, and diamond above 400 GPa. This is why hydraulic machines are filled with oil rather than air: a nearly incompressible fluid transmits force with almost no loss of motion, while a gas would simply spring back.

Fluids versus solids

For a fluid, the bulk modulus is the only elastic constant that matters — a liquid has no fixed shape, so it resists a change in volume but not a change in form. A solid resists both, and its bulk modulus sits alongside Young’s modulus and the shear modulus in a fuller description of how it deforms. For an isotropic solid the three are linked through Poisson’s ratio, so knowing any two fixes the third.

Limitations

The relationship is linear only while the material stays elastic and springs back once the pressure is released; at extreme pressures the response curves and the bulk modulus itself changes with pressure. For gases the result also depends on how the compression happens — the isothermal bulk modulus (slow, constant temperature) differs from the adiabatic one (fast, no heat exchange). The calculation assumes a uniform pressure acting over the whole surface and a sample that is free to change volume without leaking or changing phase.