One of the amazing concepts we examine in the strength of the material topic is the bulk modulus. Recalling the keywords associated with bulk modulus will help us comprehend what it is and how it is defined. As we examine Hooke's law, we find that stress and strain in the material are exactly proportionate. The modulus of elasticity is the proportionality constant that resulted from Hooke's law.
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It is the basic law of elasticity. It states that the extension produced in a wire is directly proportional to the load attached to it.
Thus, according to Hooke's law, extension
However, this proportionality holds good up to a certain limit, called the elastic limit.
According to the Hooke's law,
or stress
or
or
This constant of proportionality is called the modulus of elasticity or the coefficient of elasticity of the material. Its value depends upon the nature of the material of the body and the manner in which the body is deformed. There are three moduli of elasticity namely Young's modulus
What is Stress
Stress is defined as the restoring force per unit area set up in the body when deformed by the external force. Thus,
When a deforming force acts on a body, it undergoes a change in its dimensions and the body is said to be deformed or strained.
The ratio of change in the dimension of the body to its original dimension is called strain.
Since a body can have three types of deformations i.e. in length, volume or shape, likewise there are the following three types of strains:
It is defined as the increase in length per unit of original length when deformed by external force. It is also called linear strain or tensile strain.
Thus, longitudinal strain
Thus, volumetric strain
where V is the original volume and
It is defined as the angle
Bulk Modulus is defined as the ratio of the normal stress to the volumetric strain. It is denoted by K.
Thus, in accordance with Hooke's law, we have
Consider a sphere of volume V and surface area
and volumetric strain
The negative sign indicates that on increasing the stress, the volume of the sphere decreases. Therefore,
or
Now,
Therefore,
The units of bulk modulus are Pa or
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We will discuss some important factors
Compressibility. The reciprocal of the bulk modulus of a material is called its compressibility. Therefore,
The units of compressibility are reciprocal of those of the bulk modulus i.e. the units of compressibility are
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Generally, the bulk modulus is used for liquids rather than solids. Because bulk modulus resembles incompressibility, and we know the liquids are incompressible.
The ratio of the normal stress to the volume stain is called the bulk modulus of elasticity.
The formula of the bulk modulus is
B = pV/ΔV
The dimension of the bulk modulus or volume elasticity is [ML-²T-²].
The bulk modulus of steel is 160 Gpa and the bulk modulus of water is 2.2 GPa.
The SI unit of bulk modulus of elasticity is ‘newton/ metre^ (Nm-²) or ‘pascal’ (Pa).
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