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Young’s Modulus

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Young’s Modulus is the ratio of stress and strain. It is named after the famous British physicist Thomas Young. Young’s Modulus provides a relation between stress and strain in any object.  When a certain load is added to a rigid material, it deforms. When the weight is withdrawn from an elastic material, the body returns to its original form, this property is called Elasticity. Elastic bodies have a steady linear Young’s modulus. Young’s modulus of Steel is 2×1011 Nm-2. Young Modulus is also called the Modulus of Elasticity.

In this article, we will learn about Young’s Modulus its formula, unit, and others in this article.

What Is Young’s Modulus?

Young’s Modulus is defined as follows:

Young Modulus is the property of the material which allows it to resist the change in its length according to stress applied to it. Young’s modulus is also called the modulus of elasticity.

It is represented using the letters E or Y.

Young’s Modulus, is the measure of the deformation in the length of the solid such as rods, or wires when the stress is applied along the x-axis. Bulk modulus and Shearing modulus are also used to measure the deformation of the object according to the stress applied.

Before proceeding any further first learn in brief about the stress and strain.

  • Stress is defined as the force applied per unit length of the object.
  • Strain is the change in shape or length of the object with respect to its original length.

Young’s modulus provides a relation between stress and strain. A solid object deforms when a particular load is applied to it. When the force is applied to an object it changes its shape and as soon as the force is removed from the object it regains its original position. This is called the elastic property of the object.

The more elastic the material is more it will resist the change in its shape.

Young’s Modulus of Elasticity

Young’s Modulus is a mathematical constant. It was named after Thomas Young, an 18th-century English physician and scientist. It defines the elastic characteristics of a solid that is subjected to tension or compression only in one direction. For Example, consider a metal rod that returns to its original length after being stretched or squeezed longitudinally.

It is a measurement of a material’s capacity to endure changes in length when subjected to longitudinal tension or compression. It’s also known as the Modulus of Elasticity. It is calculated as the longitudinal stress divided by the strain. In the instance of a tensioned metal bar, both stress and strain may be stated.

Young’s Modulus, also known as Elastic Modulus or Tensile Modulus, is a mechanical property measurement of linear elastic solids such as rods, wires, and so on. Other numbers exist that give us a measure of a material’s elastic characteristics. Bulk modulus and shear modulus are two examples. However, the value of Young’s Modulus is most commonly utilized. This is because it provides information about a material’s tensile elasticity. 

When a material is compressed or stretched, it experiences elastic deformation and returns to its original shape when the load is released. When a flexible material deforms, it deforms more than when a rigid substance deforms. In other words, it can be interpreted as: 

  • A solid with a low Young’s Modulus value is Elastic.
  • A solid with a high Young’s Modulus value is Inelastic or stiff.

Young’s Modulus is described as a material’s mechanical ability to tolerate compression or elongation with respect to its initial length.

Young’s Modulus Formula

Mathematically, Young’s Modulus is defined as the ratio of the stress applied to the material and the strain corresponding to the applied stress in the material as shown below:

Young’s Modulus = Stress / Strain

Y = σ / ϵ 

where 
Y is Young’s Modulus of the material
σ is the stress applied to the material
ϵ is the strain corresponding to the applied stress

Units of Young’s Modulus

SI unit for Young’s modulus is Pascal (Pa)

Dimensional formula for Young’s Modulus is [ML-1T-2].

The values are most often expressed in terms of Megapascal (MPa), Newtons per square millimeter (N/mm2), Gigapascals (GPa), or kilonewtons per square millimeter (kN/mm2). 

Other Form of Young’s Modulus Formula

We know that,

Y = σ / ϵ…(1)

Also,

σ = F/A
ϵ = ΔL/L0

Putting these values in eq(1)

Y = σ / ϵ

   = (F/A)×(L0/ΔL)

Y = FL0 / AΔL

Notations in Young’s Modulus Formula

  • Y is Young’s modulus
  • σ is Stress applied
  • ε is Strain related to the applied stress
  • F is Force exerted by the object
  • A is Actual cross-sectional area
  • ΔL is change in the length
  • L0 is actual length

Young’s Modulus Factors

Young’s Modulus of any material is used to explain the deformation in the length of the material when force is applied to it. As it is clear that the Young Modulus of steel is greater than rubber or plastic it is safe to say that steel is more elastic than both rubber and plastic.

Elasticity is the property of the material which resists the change in its length as soon as the applied stress is removed.

Young’s Modulus of the material explains how a material behaved when stress is applied to it. The lower value of Young’s Modulus in materials tells us that this material is not fit for dealing with large stress and applying large stress will change the shape of the object completely.

How to Calculate Young’s Modulus

Young’s Modulus of any object is calculated using the formula,

Young’s Modulus = Stress / Strain = σ / ϵ

We can also plot a stress-strain curve to find Young’s Modulus of the material.

Stress - Strain Curve

 

The figure discussed above it is the stress-strain curve and the initial slope of the first segment of the curve is Young’s modulus.

If continuously increasing stress is applied to the material it reaches a point when its elasticity gets disappeared and any further stress can create a more significant strain. This point is called the elastic limit of the material.

Further increasing the stress make the material such that it start to deform without even applying stress the point where this started to happen is called the plastic limit.

Young’s Modulus of Some Materials

Young’s Modulus of some common materials are discussed in the table below:

Materials

Young’s Modulus (Y) in Nm-2

Rubber

5 × 108

Bone 

1.4 × 1010

Lead 

1.6 × 1010

Aluminum 

7.0 × 1010

Brass 

9.0 × 1010

Copper

11.0 × 1010

Iron

19.0 × 1010

Mathematical Interpretation of Young’s Modulus

Consider a wire of radius r and length L. Let a force F be applied on the wire along its length i.e., normal to the surface of the wire as shown in the figure. If △L is the change in length of the wire, then Tensile stress (σ = F/A), where A is the area of the cross-section of the wire and the Longitudinal strain (ϵ = △L/L).

Deformation of Bar on applying stress

 

Therefore, Young’s Modulus for this case is given by:

Y = (F/A) / (△L/L) 

   = (F × L) / (A × △L)            

If the extension is produced by the load of mass m, then Force, F is mg, where m is the mass and g is the gravitational acceleration. 

And the area of the cross-section of the wire, A is πr2 where r is the radius of the wire. 

Therefore, the above expression can be written as:

Y = (m × g × L) / (πr2 × △L)

Factors Affecting Young’s Modulus

The Factors on which Young’s Modulus of material depends are,

  • Larger the value of Young’s modulus of the material, the larger the value of the force required to change of length of the material
  • Young’s modulus of an object depends upon the nature of the material of the object
  • Young’s modulus of an object does not depend upon the dimensions (i.e., length, breadth, area, etc) of the object. 
  • Young’s modulus of a substance decreases with an increase in temperature.
  • Young’s modulus of elasticity of a perfectly rigid body is infinite.

Read More,

Solved Examples on Young’s Modulus

Example 1: A cable is cut to half of its length. Why this change has no effect on maximum load cable cab support?

Solution:

The maximum load a cable can support is given by:

F = (YA△L) / L

Here Y and A is constant, there is no change in the value of △L/L.

Hence, no effect on the maximum load.

Example 2: What is Young’s modulus for a perfectly rigid body?

Solution:

The Young’s modulus for a material is,

Y=(F/A) / (△L/L) 

 Here, △L = 0 for rigid body. Hence, Young’s Modulus is infinite.

Example 3: Young’s Modulus of steel is much more than that of rubber. If the longitudinal strain is the same which one will have greater tensile stress?

Solution:

Since the Tensile stress of material is equal to the product of Young’s modulus (Y) and the longitudinal strain. As steel have larger Young’s modulus therefore have more tensile strain.

Example 4: A force of 500 N causes an increase of 0.5% in the length of a wire of an area of cross-section 10-6 m2. Calculate Young’s modulus of the wire.

Solution:

Given that, 

The force acting, F = 1000 N, 

The cross-sectional area of the wire, A = 10-6 m2

Therefore,    

△L/L = 0.5 = 5/1000 = 0.005

 Y = (F/A)/(△L/L)

    = 1012 Nm-2

Example 5: What is the bulk modulus of a perfectly rigid body?

Solution:

Since, the Bulk modulus of a material is defined as,

K= P / (△V/V)

Since, △V = 0 for perfect rigid body.

Hence, the bulk modulus is infinite for perfect rigid body.

FAQs on Young’s Modulus

Q1: What is Young’s Modulus Dimensional Formula?

Answer:

As we know that Young’s modulus is defined as the ratio of stress and strain its dimensional formula is [ML-1T-2].

Q2: What is Young’s Modulus Unit?

Answer:

As we know that Young’s modulus is defined as the ratio of stress and strain its SI unit is Pascal.

Q3: What is the Modulus of Elasticity of Steel?

Answer:

The Modulus of Elasticity of Steel is 2×1011 Nm-2.

Q4: What do you mean by Modulus of Rigidity?

Answer:

Modulus of Rigidity is defined as the ratio of shearing stress (tangential stress) and shearing strain (tangential strain). It is denoted using the letter η.

Q5: What do you mean by Bulk Modulus?

Answer:

Bulk Modulus of any material is defined as the ratio of Pressure (P) applied to the corresponding relative change in the volume or the volumetric strain (∈V) of the material. It is denoted using the letter K.



Last Updated : 04 Feb, 2024
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