Young's, Bulk and Shear Moduli

The three elastic moduli measure a solid's resistance to longitudinal stretching, uniform compression and shape distortion. NEET routinely tests their definitions, units and the relations among $Y$, $K$ and $\eta$ through Poisson's ratio $\sigma$.

Concept

For each kind of deformation, the modulus is the ratio of stress to strain within the elastic limit.

• Young's modulus $Y$: longitudinal stress per unit longitudinal strain.
• Bulk modulus $K$: pressure per unit fractional volume change.
• Shear (rigidity) modulus $\eta$: tangential stress per unit shear angle.

Liquids and gases possess only a bulk modulus; they cannot sustain shear or longitudinal stress in static equilibrium. Compressibility is the reciprocal of $K$.

Formula Derivation

The defining equations are

$Y = \frac{F L}{A\,\Delta L}, \quad K = -\frac{\Delta P}{\Delta V/V}, \quad \eta = \frac{F/A}{\phi}.$

Poisson's ratio is

$\sigma = -\frac{\text{lateral strain}}{\text{longitudinal strain}}, \quad 0 \le \sigma \le 0.5.$

The three moduli are related by

$Y = 2\eta(1 + \sigma) = 3K(1 - 2\sigma), \quad \frac{9}{Y} = \frac{1}{K} + \frac{3}{\eta}.$

For an isothermal ideal gas $K_{\text{iso}} = P$, while for an adiabatic process $K_{\text{adi}} = \gamma P$.

NEET-style Worked Example

A solid copper cube of side $10\ \text{cm}$ is subjected to a uniform pressure of $10^{7}\ \text{Pa}$. If $K = 1.4 \times 10^{11}\ \text{Pa}$, find the fractional change in volume.

$\frac{\Delta V}{V} = -\frac{\Delta P}{K} = -\frac{10^{7}}{1.4 \times 10^{11}} \approx -7.14 \times 10^{-5}.$

So $\Delta V \approx -7.14 \times 10^{-5} \times 10^{-3}\ \text{m}^{3} = -7.14 \times 10^{-8}\ \text{m}^{3}$.

Common Confusions

• $Y$, $K$ and $\eta$ all have units of pressure; only their physical meaning differs.
• The negative sign in $K = -\Delta P/(\Delta V/V)$ makes $K$ positive: increasing pressure shrinks volume.
• Gases have $K_{\text{adi}} = \gamma K_{\text{iso}}$, but the formula $K = P$ applies only to ideal gases at fixed temperature.
• Liquids resist compression strongly: $K_{\text{water}} \approx 2.2 \times 10^{9}\ \text{Pa}$ is far greater than $K$ for air.

Key Takeaways

• $Y$ for stretching, $K$ for compression, $\eta$ for shape.
• Liquids: only $K$. Gases: $K$ depends on the process.
• $Y = 3K(1 - 2\sigma) = 2\eta(1 + \sigma)$ link the moduli through $\sigma$.
• Compressibility $= 1/K$.
• $K_{\text{adi}} = \gamma P$ for an ideal gas in adiabatic compression.