A material can resist not only changes of length or shape, but also changes of volume. The elastic property that measures resistance to uniform compression is the bulk modulus.

Suppose an approximately uniform pressure change $\Delta p$ acts on all sides of a material sample and changes its volume from $V$ to $V+\Delta V$. For a small elastic change,

$$\boxed{K=-\frac{\Delta p}{\Delta V/V}},$$

where $K$ is the bulk modulus.

The fractional volume change

$$\varepsilon_V=\frac{\Delta V}{V}$$

is the volumetric strain. Compression gives $\Delta V<0$ when the pressure increase is positive, so the minus sign makes $K$ positive for ordinary stable materials.

Equivalently,

$$\boxed{\frac{\Delta V}{V}=-\frac{\Delta p}{K}}.$$

A large bulk modulus means that a large pressure change is required to produce a given fractional volume change.

Differential form

When the response varies with pressure, the local bulk modulus is defined by

$$\boxed{K=-V\frac{dp}{dV}}.$$

The small-change formula follows when $K$ is approximately constant over the pressure interval.

Compressibility

The reciprocal quantity

$$\boxed{\kappa=\frac1K}$$

is the compressibility. In differential form,

$$\kappa=-\frac1V\frac{dV}{dp}.$$

Large $K$ means small compressibility; small $K$ means the material's volume changes more readily under pressure.

Liquids are often treated as incompressible in introductory fluid mechanics because their fractional volume changes can be very small under ordinary pressure variations. That approximation does not mean their bulk modulus is infinite in reality.

Worked example

A liquid has bulk modulus

$$K=2.2\times10^9,\mathrm{Pa}.$$

A sample initially has volume

$$V=1.0\times10^{-3},\mathrm{m^3}$$

and experiences a pressure increase

$$\Delta p=5.0\times10^6,\mathrm{Pa}.$$

Assuming the bulk modulus is constant,

$$\frac{\Delta V}{V} =-\frac{5.0\times10^6}{2.2\times10^9} \approx-2.27\times10^{-3}.$$

Therefore

$$\Delta V \approx(-2.27\times10^{-3})(1.0\times10^{-3}) =-2.27\times10^{-6},\mathrm{m^3}.$$

The volume decreases by about

$$\boxed{2.3,\mathrm{cm^3}}.$$

Even a pressure increase of several megapascals changes the volume by only a small fraction in this example.

Relation to other elastic moduli

Young's modulus describes resistance to uniaxial extension or compression. Shear modulus describes resistance to shape distortion at approximately unchanged volume. Bulk modulus instead describes resistance to an approximately uniform change of volume.

These are different mechanical responses. For an isotropic linear elastic solid they are related through the material's other elastic constants, but the bulk modulus is also meaningful for fluids, which resist compression even though a fluid at rest cannot sustain a static shear stress.

Bulk modulus is especially important whenever pressure disturbances compress and expand a medium, including acoustic waves and high-pressure fluid systems.