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Relation Between Young's Modulus and Bulk Modulus

What is Young’s 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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FAQs on Relation Between Young's Modulus and Bulk Modulus

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

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?

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 = \[\frac {\Delta P} {(\Delta V/V)}\]


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.

3. What is the modulus?

Modulus, in general, is also known as the absolute value of a function as it derives the exact value of a variable and gives us the exact value of a number. Irrespective of the input we give, the result of this function is always positive. This value is also easy to be represented on the graphs, and a modulus function is written as y = |x| in a standard format. If the input is positive, the result is positive, and if the input is negative, the result is the magnitude of the input. It means that the sign of the input is not considered in this function. This concept is used in many ways, especially in computers, to calculate hours and time. 

4. What are the applications of bulk modulus?

Bulk modulus can be defined as the product of pressure change and 100 required to cause a change in the volume by 1 percent. Bulk modulus is used to measure how incompressible a solid can be. When the value of  “K” increases, the incompressibility of the solid increases. In liquids, the temperature of the liquid and the content of air influence the value of the bulk modulus. It describes the elasticity, i.e., how far can a substance stretch or expand under pressure from all sides. When the pressure is applied to an object, the volume of the object reduces, and it comes back to the normal stage when the pressure is removed. 

5. Why is young’s modulus important?

Young’s modulus is the opposite of bulk’s modulus as it defines how an object can easily stretch under pressure. It is a property of an object that tells us at which point the object will bend or break. Thomas Young in the 18th century defined that when an object is under pressure, it stretches unidirectionally, and this concept is named after him as Young’s modulus. It can be simply derived as the ratio of tensile stress to tensile strain. The object, which is under force returns back to its normal position when the force is removed from it. If the external force increases beyond the elasticity of an object, it starts to deform or completely break. 

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