Bulk modulus (K)

Bulk modulus is the modulus of elasticity associated with a change in the volume of an object under an applied deforming force — hence it is also called the elasticity of volume. Unlike Young's modulus and the modulus of rigidity (both of which apply only to solids), bulk modulus is a property that solids, liquids and gases all possess, since all three states of matter have a definite volume that can be compressed.

Picture a rubber sphere fully immersed in a liquid, so that it is compressed uniformly from every side (exactly the volume-stress geometry of Fig. 6.2). Suppose the applied compressive force produces a small change in pressure $dP$ on the sphere, and a corresponding small change in its volume $dV$, starting from an original volume $V$. The volume strain produced is defined as

The negative sign here is not arbitrary — it explicitly records that an increase in pressure ($dP$ positive) produces a decrease in volume ($dV$ negative), so the ratio $dV/V$ on its own would be negative; the minus sign in front makes the volume strain itself come out as a positive quantity, equal in magnitude to $|dV|/V$.

Bulk modulus $K$ is then defined as the ratio of volume stress to volume strain:

Its SI unit is $\text{N/m}^2$, and its dimensional formula is $[L^{-1}M^1T^{-2}]$, the same as for stress. Table 6.2 lists bulk moduli for common materials — a large bulk modulus (like steel or gold) means a material strongly resists having its volume squeezed down, while a small bulk modulus means the opposite.

The reciprocal of the bulk modulus is called the compressibility of a material:

$\text{Compressibility} = \dfrac{1}{K}$ …