Important notes on Derivation of Bulk modulus

Introduction

Bulk modulus is an important property that measures the compressibility of a material. The bulk modulus and compressibility are inversely proportional to each other. Young’s modulus is the ratio of tensile stress to strain. The bulk modulus is the extended property of Young’s modulus. All the 3 dimensions are extended in bulk modulus. In this article, we will derive the bulk modulus using Hooke’s law. And we will look into the detailed applications of bulk modulus property.

Bulk modulus

The bulk modulus is defined as the ratio of volumetric stress and strain. In material, it is a measurement of a resistant property to compression. It can be shown as the ratio of infinitesimal pressure change to volume change. The change in volume can be determined by applying uniform stress on the material surface. It can be studied in all states of matter but is mostly found in a fluid state. It is very difficult to study the bulk modulus in solids because the solid state has various properties in different axes.

Unit of Bulk modulus

The bulk modulus can be expressed in the unit of pascals or N/m²

It is commonly denoted as B.

The dimensions can be written as [ L^-1 M¹ T^-2 ]

Determination of Bulk modulus

The bulk modulus is calculated by many techniques like diffraction. Diffraction is the method in which the electrons are focussed on a micro-crystalline sample. This micro-crystalline sample is well grounded. Other methods like modeling can be done to calculate the bulk modulus. In this technique, the known tested samples are used.

The modeling technique includes apparatus like an interferometer, coupler, ripple, and tester. The interferometer is of three kinds. They are Electronic Speckle Pattern Interferometer, Holographic interferometer, and Doppler interferometer. The tester is used to determine the vibration effect. The ripple measures the flow of vibration and normal impedance.

Bulk modulus Formula

Bulk modulus can be given by the below formula,

B = ∆p * ∆V / V   —(1)

where B represents the Bulk modulus

V represents volume

∆V represents a change in volume

∆p represents a change in pressure

The above formula can be used to calculate the compressibility of various materials such as glass, diamond, and steel.

Derivation of Bulk modulus

We can derive the formula of bulk modulus by Hooke’s law. Hooke’s law states that the ratio of strain and stress is constant.

Stress/strain = constant

Stress = B * strain

Here  B is the proportionality constant.

∆p= B * ( ∆V/V )

B = ∆p /( ∆V/ V)  —(2)

Let us consider the material with volume V and the volume change is represented as V. The pressure P is applied and the temperature reduces gradually with the applied pressure.

Normal stress = p —(3)

Volume strain = – ∆V/V  —–(4)

B = Stress/ Strain —-(5)

Substitute (3) (4) in (5),

B = ∆p x 1/ (∆V/V)

B = -∆p * V/∆ V —(6)

Cases (i)

When p is +ve, V will become a negative value.

Case (ii)

When V is -ve, Bulk modulus(B) will become positive.

To acquire high strain, high pressure is applied. This in turn increases the bulk modulus and strain. A higher strain can be acquired in the gas state than in solids and fluids. The expression (6) is used to calculate the elasticity of a material.

Compressibility of material

The compressibility of a material can be defined as the reciprocal of Bulk modulus. The bulk modulus and Compressibility are inversely proportional to each other. The compressibility property is higher in gases and fluids. The solids are difficult to compress.

Compressibility can be expressed as below,

K = 1/B

K= – ∆V/∆p.V

The unit is m² / N

The dimension is represented as [ M^-1 L¹T² ].

Applications of bulk Modulus

• The bulk modulus property is widely used in hydraulic applications. The fluids are affected by trapped gas and high temperature. It affects the bulk modulus property of fluids. Hence, Bulk modulus can be easily studied in fluids. It can be used in the production of hydraulic spares like motors and valves. The threshold for temperatures can be set on the design during the manufacturing processes.
• Solids have three dimensions that are rarely affected by volumetric stress. So, it is easier to find Young’s modulus and shear modulus experiments.
• Finally, it is also helpful in the production of echo-location transducers and sonar transducers

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