Short Answer:

Bulk modulus is defined as the ratio of normal stress to the volumetric strain within the elastic limit of a material. It measures a material’s resistance to uniform compression. In other words, it shows how difficult it is to change the volume of a material when it is subjected to equal pressure from all directions.

If a material is more compressible, its bulk modulus is low, and if it is less compressible (more rigid), its bulk modulus is high. It is denoted by the symbol K and measured in N/m² or Pascal (Pa) in the SI system.

Detailed Explanation:

Bulk Modulus

The bulk modulus is one of the fundamental elastic constants of a material that describes its behavior under volumetric loading. When a material is subjected to uniform pressure (like under water or inside a pressure vessel), it experiences a decrease in volume without a change in shape. This property of resisting a change in volume is measured by the bulk modulus (K).

It gives a measure of the incompressibility of a material. A material with a high bulk modulus is very stiff and difficult to compress, while a material with a low bulk modulus is more easily compressible.

Mathematically, the bulk modulus is defined as:

Definition

The bulk modulus (K) of a material is defined as the ratio of the increase in pressure (or stress) to the corresponding decrease in the volume per unit original volume.

Where,

•  = Bulk modulus (N/m² or Pa)
•  = Increase in pressure or external compressive stress (N/m²)
•  = Change in volume (m³)
•  = Original volume (m³)

The negative sign indicates that an increase in pressure causes a decrease in volume. However, in practical calculations, only the magnitude of  is considered, so it is taken as positive.

Explanation of Bulk Modulus

When a body is subjected to equal compressive stresses on all sides, such as a cube placed in a fluid or under uniform pressure, its shape remains the same, but its volume decreases. This type of loading is called hydrostatic loading, and the resistance offered by the body against this uniform pressure is described by its bulk modulus.

The greater the value of the bulk modulus, the smaller the reduction in volume for a given increase in pressure. Therefore, materials with a large bulk modulus are less compressible (stiffer), while materials with a small bulk modulus are more compressible.

For example:

• Steel has a high bulk modulus and is almost incompressible.
• Rubber and gases have low bulk modulus values and are easily compressible.

Derivation of Expression

Consider a cube of material with sides of length L and volume V = L³, subjected to equal compressive stress p on all its faces.

Due to the pressure, the volume of the cube decreases by an amount ΔV. The volumetric strain is given by:

The bulk modulus is then defined as:

or, rearranging,

This equation shows that for a given pressure, the change in volume is inversely proportional to the bulk modulus. A high value of  means very little volume change even under large pressure.

Units and Dimensions

1. SI Unit:
   • Newton per square meter (N/m²) or Pascal (Pa)
   • Often expressed in GPa (Giga Pascal) for solids
2. Dimensional Formula:

1. CGS Unit:
   • dyne/cm²

Typical Values of Bulk Modulus

Material	Bulk Modulus (GPa)
Steel	160 – 170
Copper	140
Aluminum	76
Brass	100
Water	2.2
Air	0.0001

From the table, it can be seen that solids have very high bulk modulus values, liquids have moderate values, and gases have very low values.

Relation Between Bulk Modulus and Other Elastic Constants

Bulk modulus is related to Young’s modulus (E) and Poisson’s ratio (ν) by the following relation:

Where,

•  = Bulk modulus
•  = Young’s modulus
•  = Poisson’s ratio

This relationship shows that the bulk modulus depends on both the material’s stiffness and its tendency to contract laterally when stretched or compressed.

Physical Significance

1. Measure of Incompressibility:
   The bulk modulus measures how resistant a material is to compression. A large value means the material is difficult to compress.
2. Elastic Property:
   It is one of the three fundamental elastic constants (E, G, and K) used to describe material behavior.
3. Pressure Applications:
   Bulk modulus is used in analyzing fluids, pressure vessels, submarines, and hydraulic systems.
4. Sound Velocity in Solids and Fluids:
   The velocity of sound in a material is directly related to its bulk modulus. Higher  results in faster sound propagation.
5. Material Selection:
   In engineering design, bulk modulus helps determine whether a material can withstand pressure without significant volume change.

Example Problem

A copper cube of volume  is subjected to a uniform pressure of . If the bulk modulus of copper is , find the change in volume.

Given,

Using,

 

So, the change in volume is .

Applications of Bulk Modulus

1. Fluid Mechanics:
   Used to determine compressibility of liquids and gases under pressure.
2. Structural Engineering:
   Helps in designing pressure vessels and submarines to withstand high pressures.
3. Material Science:
   Used to study the elasticity and incompressibility of different materials.
4. Acoustics:
   The speed of sound in a fluid depends on its bulk modulus and density.
5. Geophysics:
   Determines how seismic waves travel through different layers of the Earth.

Conclusion

The bulk modulus (K) is the ratio of volumetric stress to volumetric strain and represents a material’s resistance to uniform compression. It is a fundamental elastic property that helps engineers understand how materials respond to hydrostatic pressure. Materials with high bulk modulus values, like metals, are stiff and nearly incompressible, while those with low bulk modulus values, like gases, are easily compressible. Understanding bulk modulus is essential in the design of pressure-based systems and in predicting material behavior under uniform loads.