# Bulk Modulus of Elasticity Formula

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Bulk modulus is one of the interesting concepts we study in the strength of the material topic. To understand what is Bulk modulus, how it’s defined, let us recall the important terms related to it. While studying Hooke’s law we say that stress is directly proportional to strain in the material, the proportionality constant derived from Hooke’s law was the modulus of elasticity.

So, the modulus of elasticity of any material is the ratio of stress applied to the material to strain in the material. Depending upon the type of stress being applied and the resulting strain there are three types of modulus of elasticity, they are:

1. Young's Modulus: It is defined as- the ratio of longitudinal stress to longitudinal strain, is denoted by Y.

2. Bulk Modulus: It is defined as- the ratio of volumetric stress to strain, and is denoted by B.

3. Shear Modulus: It is defined as- the ratio of shear stress to shear strain, and is denoted by 𝜂.

Now our aim is to study bulk modulus, its definition, and formula.

## The Bulk Modulus of Elasticity:

The bulk modulus of elasticity is defined for solids and fluids(gases and liquids). To understand the meaning of modulus, let us consider a ball of volume V and is immersed in the water tub. The ball submerged in the fluid experiences hydraulic pressure, which leads to a decrease in the volume of the body and it will cause a strain known as volume strain or volumetric strain.

The bulk modulus is defined as the ratio of volume stress (the excess pressure applied to the solid) to volume strain. It is denoted by B and mathematically it is expressed as:

⇒ $B=\frac{Volume Stress}{Volume Strain}$

⇒ $B=\frac{-\Delta p}{\frac{\Delta V}{V}}$

Where

ΔP - Change in pressure or the force per unit area

ΔV - Change in volume due to excess pressure

V-The initial volume of the material

The negative sign in the expression explains the decrease in volume of the material due to the excess pressure applied. In other words, if pressure increases it will lead to a decrease in volume. If pressure P is positive, then the change in volume will be negative. The bulk modulus of the material in equilibrium is always positive. SI unit of the bulk modulus is the same as that of the pressure i.e.,N/m2 or Pa (Pascals).

## The Bulk Moduli of a Few Materials are Listed Below:

 State Material B(GPa) Solid AluminumCopper Iron 72140100 Liquid WaterGlycerineMercury 2.24.7625 Gases Air 10-4

### Compressibility:

The reciprocal of the bulk modulus is known as the compressibility of the material and is denoted by K. The compressibility of any material explains how much matter of volume is decreased under the effect of pressure.

⇒ $K=\frac{1}{B}$

From the above table, the bulk modulus of solids is much larger than the liquids and gases. Thus the compressibility of the gases will be larger than than the solids, the relation between compressibility of solid, liquid, and gases can be expressed as:

Ksolid<Kliquid<Kgases

### Examples:

1. The Bulk Modulus For Water is 2.1GPa. Calculate the Change in Volume of 100ml Water that is Subjected to a Pressure of 2MPa.

Sol:

Given,

The bulk modulus of water =2.1GPa

The volume of the water =100ml

The pressure exerted =P=2MPa

Calculate the change in volume.

We know that the bulk modulus of the material is given by:

⇒ $B=\frac{\Delta P}{\frac{\Delta V}{V}}$

On rearranging the above expression for ΔV we get,

⇒ $\Delta V=\frac{P\times V}{B}=\frac{2\times10^{6}\times100}{2.1\times10^{9}}=0.1 ml$

2. What is the Bulk Modulus of a Body that Experiences a Change in Pressure of 4x104N/m2and its Volume Changes From 5cm3 to 4.8cm3.

Ans:

Given,

The volume of the body =5cm3

The pressure exerted =P=4104N/m2

The change in volume of the body = ΔV = 5 - 4.8 = 0.2 cm3

Now, we know that:

⇒ $B=\frac{\Delta P}{\frac{\Delta V}{V}}=\frac{PV}{\Delta V}$

Substituting the corresponding values in the expression,

⇒ $B=\frac{\Delta P}{\frac{\Delta V}{V}}=\frac{4\times10^{4}\times5}{0.2}=10^{6}N/m^{2}$