Relation Between Young's Modulus and Bulk Modulus

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What is Young’s Modulus?

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The mechanical property of a material to withstand the compression or the elongation concerning its length is called Young’s Modulus which is also referred to as the Elastic Modulus or Tensile Modulus is denoted as E or Y.

Young’s Modulus is a measure of mechanical properties of linear elastic solids such as the rods, wires. Other numbers give us a measure of elastic properties of a material, such as the Bulk modulus and shear modulus, but the value of Young’s Modulus is most commonly used in the world. Young’s Modulus is used very generally because it gives us information about the tensile elasticity of a material which is the ability to deform along an axis.

Young’s modulus describes the relationship between stress i.e. force per unit area and strain i.e. proportional deformation in an object. The Young’s modulus is named after Thomas Young who was a British scientist. Any solid object will deform when a particular load is applied to it. But if the object is elastic, then the body regains into its original shape when the pressure is removed from the object. Many materials are not linear elastic beyond a small amount of deformation and Young's modulus applies only to linear elastic substances.

Young’s Modulus Formula is E = σ/ϵ

Young’s Modulus Formula From Other Quantities: 

E = FL0  / AΔL 

Notations that are Used in the Young’s Modulus Formula are as follows:

  • E is Young’s modulus in Pa

  • σ is uniaxial stress in Pa
  • ε is a strain or proportional deformation

  • F is the force exerted by the object under tension

  • A is the actual cross-sectional area

  • ΔL is a change in length

  • L0 is the actual length

Units and Dimension of Young’s Modulus Formula

  • SI unit- Pa

  • Imperial Unit- PSI

  • Dimension- ML-1T -2

With the value of Young’s modulus for a material, we can find the rigidity of the body. This is only because it tells us about the ability of the body to be able to resist deformation on the application of force.

The Young’s Modulus values ( x 109 N/m2)

 of different material are given:

  • Steel– 200

  • Glass– 65

  • Wood– 13

  • Plastic (Polystyrene)– 3

What is a Bulk Modulus?

The bulk modulus is defined as the proportion of the volumetric stress related to the volumetric strain of specified material, while the material deformation is within the elastic limit. In more simple words we can say that the bulk modulus is nothing but a numerical constant that is used to measure and describe the elastic properties of a solid or of fluid when a particular pressure is applied on all the surfaces.

The bulk modulus of elasticity is one of the measures of the mechanical properties of solids and whereas the other elastic modules include Young’s modulus and the Shear modulus. The bulk elastic properties of a material are always used to determine how much the material will compress under a given amount of external pressure. Here it is very crucial to find and also to note the ratio of the change in pressure to the fractional volume of compression.

The value is denoted with a symbol ‘K’ and it has the dimension of force per unit area. It is expressed in the units per square inch i.e. psi in the English system and newtons per square meter (N/m2) in the metric system.

Relation Between Elastic Constants 

The Young’s modulus, the bulk modulus as well as the Rigidity modulus of an elastic solid are together called the Elastic constants. When a deforming force is acting on a solid, it will result in a change in its original dimension. In such cases, we can use the relation between the elastic constants to understand the magnitude of the deformation.

Elastic Constant Formula

Where K is the Bulk modulus, G is shear modulus or modulus of rigidity and E is Young’s modulus or modulus of Elasticity.

Individually Young’s modulus and bulk modulus, as well as the modulus of rigidity, are related as-

  • The formula for the relation between modulus of elasticity and modulus of rigidity is  E = 2G(1 + μ) and the SI unit is N/m2 or pascal(Pa)

  • The formula for the relation between Young’s modulus and bulk modulus is E = 3K(1 − 2μ) and the SI unit is N/m2 or pascal(Pa)

Relation Between Bulk Modulus and Young’s Modulus

The Young’s Modulus is the ability of any material to resist the change along its length whereas the Bulk Modulus is the ability of any material to resist the change in its volume. The bulk modulus and young’s modulus relation can be mathematically expressed as;

Young’s Modulus And Bulk Modulus Relation

K= Y/ 3 [1−(2/μ ) ]

Where K is the Bulk modulus, Y is Young’s modulus, and μ is the Poisson’s ratio.

FAQ (Frequently Asked Questions)

1. What are the Factors of Young’s Modulus?

Ans: By understanding the modulus of elasticity of steel, we can easily say that steel is more rigid in nature than wood or even polystyrene because its tendency to experience deformation under an applied load is very less. Young’s modulus is also used to determine how much a material will deform when a certain load is applied.

Another thing one must always keep in mind is that the lower the value of Young’s Modulus in the materials, the more is the deformation experienced by the body. This type of deformation in the case of objects such as clay and also wood can vary in the one sample itself. One part of the clay sample might deform more than the other sample whereas a steel bar will experience an equal deformation throughout every sample.

2. What is Bulk Modulus of Elasticity Formula?

Ans: Bulk Modulus of Elasticity Formula:

The bulk modulus of the elasticity formula is given by the ratio of pressure applied to the corresponding relative decrease in the volume of the material.

Mathematically, it is represented as

B = ΔP /(ΔV/V)

Where B is the Bulk modulus, ΔP is the change of the pressure or force applied per unit area on the material, ΔV is the change of the volume of the material due to the compression, V is the Initial volume of the material in the units of in the English system and N/m2 in the metric system.