Relation Between Young's Modulus and Bulk Modulus

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 measures the mechanical properties of linear elastic solids such as rods and 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 = \[\frac {\sigma} {\varepsilon}\]

Young’s Modulus Formula From Other Quantities: 

E = \[\frac {FL_0} {A\Delta L}\]

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

Tensile Stress and Tensile Strength in Young’s Modulus:

Tensile stress is the force that causes an object to stretch. Ductile materials can bear higher tensile stress, and brittle materials can not withstand higher tensile stress as they break away easily. Elastic modulus is a tensile stress property, and it is the ratio of stress and strain when the change in the object is completely elastic. Fracture stress is another property of tensile stress. On the other hand, tensile strength is the maximum force an object can withstand before breaking or tearing down. When the stress is less than the tensile strength of an object, it expands initially and returns back to its normal shape and size once the force is removed but, if the stress exceeds the tensile strength of an object, it starts tearing down. 

Young’s modulus is expressed as the ratio of tensile stress and tensile strain. Here, tensile strain is the damage caused by a force when it tries to expand an object. Young’s modulus is very important to judge the strength of an object, and the highest young modulus can be seen in diamond. Objects that are flexible generally have low Young’s modulus as they can easily change their volume when they are subjected to external force or pressure. 

So, we can conclude that objects with high Young’s modulus are very inelastic and could not be stretched but the objects with less Young's modulus value are very elastic and easily alter when they are subjected to external force or pressure. This principle is very useful in deciding the construction material to be used. For example, the builders use concrete, which has a high modulus value, to build bridges and roads as they are subjected to heavy weights every day. Similarly, steel is chosen to build railways as the steel has a high modulus value to withstand the heavyweight of the train.

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 used to measure and describe the elastic properties of a solid or 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 the 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 the 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 follows-

• 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= \[\frac {Y}{3}\]

1−(2/μ)

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

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