Bulk modulus

The bulk modulus (

K

{\displaystyle K}

or

B

{\displaystyle B}

or

k

{\displaystyle k}

) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume.

Other moduli describe the material's response (strain) to other kinds of stress: the shear modulus describes the response to shear stress and Young's modulus describes the response to normal (lengthwise stretching) stress. For a fluid, only the bulk modulus is meaningful. For a complex anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law. The reciprocal of the bulk modulus at fixed temperature is called the isothermal compressibility.

Definition

The bulk modulus

K

{\displaystyle K}

(which is usually positive) can be formally defined by the equation

K

=

−

V

d

P

d

V

,

{\displaystyle K=-V{\frac {dP}{dV}},}

where

P

{\displaystyle P}

is pressure,

V

{\displaystyle V}

is the initial volume of the substance, and

d

P

/

d

V

{\displaystyle dP/dV}

denotes the derivative of pressure with respect to volume. Since the volume is inversely proportional to the density, it follows that

K

=

ρ

d

P

d

ρ

,

{\displaystyle K=\rho {\frac {dP}{d\rho }},}

where

ρ

{\displaystyle \rho }

is the initial density and

d

P

/

d

ρ

{\displaystyle dP/d\rho }

denotes the derivative of pressure with respect to density. The inverse of the bulk modulus gives a substance's compressibility. Generally, the bulk modulus is defined at constant temperature as the isothermal bulk modulus, but can also be defined at constant entropy as the adiabatic bulk modulus.