Understanding Bulk Modulus in Material Properties

The question asks us to identify the correct definition of Bulk Modulus. Bulk Modulus is a fundamental mechanical property of a substance that describes its resistance to uniform compression.

When a body is subjected to a uniform pressure from all sides, its volume decreases. This change in volume relative to the original volume is known as volumetric strain. The applied pressure is a type of direct stress.

The Bulk Modulus, often denoted by $K$ or $B$, is defined as the ratio of the applied pressure (direct stress) to the resulting volumetric strain.

Mathematically, the formula for Bulk Modulus is:

$\qquad K = \text{Direct Stress} / \text{Volumetric Strain}$

$\qquad K = \frac{\sigma_{direct}}{\epsilon_v}$

Where:

• $\sigma_{direct}$ is the direct stress (pressure) applied uniformly.
• $\epsilon_v$ is the volumetric strain, given by $\Delta V / V_0$, where $\Delta V$ is the change in volume and $V_0$ is the original volume.

Let's examine the given options based on this understanding:

1. Ratio of shear strain is shear stress: This statement seems inverted and relates to shear properties, not uniform compression. It is incorrect.
2. Ratio of volumetric strain to stress: This is the inverse of the correct definition of Bulk Modulus. It represents compressibility, not Bulk Modulus. It is incorrect.
3. Ratio of shear stress to shear strain: This is the definition of Shear Modulus (or Modulus of Rigidity), which describes a substance's resistance to shearing deformation. It is incorrect.
4. Ratio of direct stress to volumetric strain: This definition perfectly matches the standard definition of Bulk Modulus. Direct stress in this context is the uniform pressure applied, and volumetric strain is the relative change in volume. It is correct.

Therefore, the correct definition of Bulk Modulus is the ratio of direct stress to volumetric strain.

Revision Table: Key Elastic Moduli

Additional Information on Bulk Modulus

Bulk Modulus is a measure of how incompressible a substance is. A material with a high Bulk Modulus is difficult to compress, while a material with a low Bulk Modulus is easily compressed.

• Solids generally have a much higher Bulk Modulus than liquids and gases. This is why solids are much harder to compress than liquids or gases.
• The negative sign is often included in the formula for Bulk Modulus ($K = -P / (\Delta V / V_0)$) because an increase in pressure ($+P$) causes a decrease in volume ($\Delta V$ is negative), resulting in a positive value for $K$. Direct stress here refers to the change in pressure relative to the initial pressure.
• The reciprocal of Bulk Modulus is called compressibility ($\beta = 1/K$). Compressibility indicates how much a substance's volume decreases under pressure.