Elastic Modulus>Bulk Modulus

Bulk modulus is a specific type of elasticity that deals with how the volume of an object changes when a deforming force is applied. Because it relates to volume changes, it is often referred to as the "elasticity of volume." It is a property found in solids, liquids, and gases.

"Bulk modulus is defined as the ratio of volume stress to volume strain."

The mathematical representation of Bulk Modulus (K) is:

K = \[\frac{\text{Volume Stress}}{\text{Volume Strain}}\]

K = \[\frac{dP}{\left(\frac{dV}{V}\right)}\] = V \[\frac {dP}{dV}\]

Where:

• K: Bulk Modulus
• dP: Change in pressure (Volume Stress)
• dV: Change in volume
• V: Original volume

• Symbol: Denoted by the letter K.
• Alternate Name: Elasticity of volume.
• Applicability: Applicable to solids, liquids, and gases.
• Function: Measures the resistance offered by a material when an attempt is made to change its volume.
• SI Unit: N/m² (Newtons per square meter).
• Dimensions: [L^-1 M¹ T^-2].
• Relation to Compressibility: It is the reciprocal of compressibility.

Imagine a sphere made of rubber immersed completely in a liquid. The liquid compresses it uniformly from all sides.

• Force applied: Compressive force (F).
• Pressure change: The change in pressure on the sphere is dP.
• Volume change: The volume changes by dV from its original volume V.

Calculation of Volume Strain:
Volume strain is the volume change divided by the original volume.
Volume Strain = −\[\frac{dV}{V}\]

Note: The negative sign indicates that there is a decrease in volume under compression. However, the magnitude is considered just \[\frac{dV}{V}\].

Concept of Compressibility
Compressibility is the opposite of Bulk Modulus.

• Definition: It is the reciprocal of the bulk modulus of elasticity.
• Formula: \[\text{Compressibility} = \frac{1}{\text{Bulk Modulus}}\]
• Meaning: It represents the fractional decrease in volume per unit increase in pressure.
• Comparison: Materials with a small Bulk Modulus have large compressibility (they are easier to compress).
• SI Unit: m²/N or Pa^-1.

Material Data (Comparison)
Different materials have different Bulk Moduli (K × 10¹⁰ N/m²):

• Lead: 4.1 (Lower resistance to compression)
• Steel: 16.0
• Gold: 18.0 (Higher resistance to compression)

Water: Bulk modulus = 2.18 × 10⁸ Pa; Compressibility = 45.8 × 10^-10 Pa^-1.

Problem:
A metal cube with a side length of 1m is subjected to a force acting normally on its whole surface. The volume changes by 1.5 × 10^-5 m³. The Bulk Modulus of the metal is 6.6 × 10¹⁰ N/m². Calculate the change in pressure.

Solution:

1. Identify Given Values

• Side of cube (l) = 1m
• Original Volume (V) = l³ = 1³ = 1m³
• Change in volume (dV) = 1.5 × 10^-5 m³
• Bulk Modulus (K) = 6.6 × 10¹⁰ N/m²

2. Identify Formula
K = \[V\frac{dP}{dV}\]

3. Rearrange to find Pressure ($dP$)
dP = \[\mathrm{K}\frac{dV}{V}\]

4. Substitute and Calculate
dP = \[\frac{6.6\times10^{10}\times1.5\times10^{-5}}{1}\]
dP = 9.9 × 10⁵ N/m²

Final Answer:
The change in pressure is 9.9 × 10⁵ N/m².