Physics · Ch 9 — Mechanical Properties of Solids

Bulk Modulus

9.5.3

Bulk Modulus

When a solid is subjected to a uniform pressure from all sides — like a cube being squeezed equally on every face — the volume changes but the shape remains the same. This kind of deformation is called volume strain, and the stress that causes it is normal stress (pressure) applied uniformly. The elastic modulus that describes this behaviour is the bulk modulus.

Defining Bulk Modulus

Consider a solid object of volume $V$ immersed in a fluid. The fluid exerts the same pressure $p$ on every point of the object's surface. If we increase this pressure by an amount $\Delta p$ , the volume decreases by $\Delta V$ (a negative change). The bulk modulus $B$ is defined as the ratio of the normal stress (the pressure change) to the volume strain:

B = \frac{\text{normal stress}}{\text{volume strain}} = \frac{-\Delta p}{\Delta V / V}

The negative sign is crucial. Since an increase in pressure ( $+\Delta p$ ) causes a decrease in volume ( $-\Delta V$ ), the ratio $\Delta p / (\Delta V/V)$ would be negative. The negative sign in the definition makes $B$ a positive quantity. So:

B = -\frac{\Delta p}{\Delta V / V}

The SI unit of bulk modulus is the pascal (Pa), the same as pressure. Its dimensions are $[M L^{-1} T^{-2}]$ .

Compressibility

The reciprocal of the bulk modulus is called compressibility, denoted by $k$ :

k = \frac{1}{B} = -\frac{\Delta V / V}{\Delta p}

Compressibility tells you how easily a material's volume changes under pressure. A high compressibility means the material is easy to squeeze; a low compressibility means it resists volume change. For example, gases have very high compressibility, while solids and liquids have very low compressibility.

Watch out

A common mistake is to forget the negative sign in the bulk modulus formula. Always check: if pressure increases, volume decreases, so $\Delta V$ is negative. The negative sign in $B = -\Delta p / (\Delta V/V)$ ensures $B$ comes out positive.

Bulk Modulus for Different States of Matter

The bulk modulus varies enormously across the three states of matter. The table below shows typical values:

Material	Bulk Modulus $B$ ( $10^9 \, \text{N m}^{-2}$ or GPa)
Solids (e.g., steel)	~160
Liquids (e.g., water)	~2.2
Gases (e.g., air)	~ $1.0 \times 10^{-4}$

Solids have the highest bulk modulus because their atoms are tightly packed and resist compression strongly. Liquids have much lower values — they are far more compressible than solids, though still much less compressible than gases. Gases have extremely low bulk modulus; they are very easy to compress.

Important

For an ideal gas undergoing an isothermal process (constant temperature), the bulk modulus equals the pressure: $B = p$ . For an adiabatic process (no heat exchange), $B = \gamma p$ , where $\gamma$ is the ratio of specific heats ( $C_p/C_v$ ). This is a key result connecting thermodynamics to elasticity.

Properties of Bulk Modulus

The textbook lists two important properties of bulk modulus:

›Proof

Property (I): The bulk modulus is always positive.

…

Table 8.3Bulk moduli (B) of some common Materials

Material	B (10⁹ N m⁻² or GPa)
Solids
Aluminium	72
Brass	61
Copper	140
Glass	37
Iron	100
Nickel	260
Steel	160
Liquids
Water	2.2
Ethanol	0.9
Carbon disulphide	1.56
Glycerine	4.76

Table 8.4Stress, strain and various elastic moduli

Type of stress	Stress	Strain	Change in shape	Change in volume	Elastic Modulus	Name of Modulus	State of Matter
Tensile or compressive (σ = F/A)	Two equal and opposite forces perpendicular to opposite faces	Elongation or compression parallel to force direction (ΔL/L) (longitudinal strain)	Yes	No	Y = (FL)/(AΔL)	Young's modulus	Solid
Shearing (σₛ = F/A)	Two equal and opposite forces parallel to opposite surfaces, forces in each case such that total force and total torque on the body vanishes	Pure shear, θ	Yes	No	G = F/(Aθ)	Shear modulus or modulus of rigidity	Solid

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