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."

OR

The ratio of hydraulic stress to the corresponding hydraulic strain (change in volume) is called the Bulk Modulus, denoted by B.

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.