Modulus of Elasticity

Modulus of Elasticity or Elastic Modulus is the measurement of resistance offered by a material against the deformation force acting on it. Modulus of Elasticity is also called Young’s Modulus. It is given as the ratio of Stress to Strain. The unit of elastic modulus is megapascal or gigapascal

Modulus of Elasticity is one of the most important concepts of material science and engineering. It is the property of material that measures stiffness to elastic deformation under stress. In this article, we will discuss about Modulus of Elasticity, including its definition, types, and fundamental concepts along with its various applications in day-to-day life.

What is Modulus of Elasticity

Modulus of Elasticity, or Young Modulus of Elasticity or Elastic Modulus, is a fundamental mechanical property of materials that measures their stiffness or resistance to elastic deformation under stress. It is also defined as the ratio of tensile stress (σ) to tensile strain (ε), where

Stress is the amount of force applied per unit area (σ = F/A) and strain is extension per unit length (ε = dl/l).

• Modulus of Elasticity was introduced by a famous British polymath and physician Sir Thomas Young who was born on 13th June, 1773. He was well known for the discovery of many theories and experiments such as – Young’s modulus, Young’s interference experiment, Astigmatism, Young-Helmholtz theory and Young Temperament.
• This property is essential in engineering and materials science, as it determine a material’s ability to support loads and maintain its shape.
• One of the most important tests in engineering is knowing when an object or material will bend or break, and the property that tells us this is the Young’s modulus. It is the measure of how easily a material stretches and deforms.
• Modulus of Elasticity is also known as Young Modulus, tensile modulus and elastic modulus.
• Whether it is a small piece of rubber or large piece, its modulus of elasticity will always be same.

Modulus of Elasticity Definition

Modulus of Elasticity is the material’s resistance to the deformation force. It is the ratio of stress acting on the body to the strain produced.

Modulus of Elasticity Example

Steel is one of the most common example of elastic modulus which is often used in construction because it has high strength and durability It is able to resist deformation and maintain its shape under stress. This allows steel to bear heavy loads without even bending or breaking. It is a brittle material and is instructed to use with safety and precaution. Diamond has one of the highest elastic modulus around 1220 GPa.

• Elastic Modulus of Steel is 210GPa
• Elastic Modulus of Concrete is 30 to 50 GPa
• Elastic Modulus of Aluminum is 70 GPa
• Elastic Modulus of Wood is 5.5 GPa to 17 GPa

How to Calculate Modulus of Elasticity

Modulus of Elasticity, in terms of the stress-strain curve, is the slope of the stress-strain curve in the region of elastic behavior, where stress is linearly proportional to strain.

From the Graph we can measure the Modulus of Elasticity in the following Manner

• Modulus of Elasticity is the slope of the stress-strain curve.
• The region starting from 0 to the Yield strength is known as the elastic region. After that region, the plastic deformation region occurs.
• Yield strength is the stress required to produce a small amount of plastic deformation.
• Ultimate strength is the maximum stress a material can hold while being stress or pulled.
• After the ultimate stress point failure occurs as the material cannot hold the stress and hence breaks.

Determining of Modulus of Elasticity of Wire

Determining Young’s modulus (also known as modulus of elasticity) of a material typically involves performing a tensile test on a wire or specimen.

• Cut a length of the wire to be tested. The wire should be uniform in diameter along its length and free from defects.

• Fix one end of the wire to a fixed clamp or holder. The other end should be attached to a load cell or device capable of applying a controlled tensile force.

• Set up a measurement system to accurately measure the applied force (tensile load) and the corresponding elongation (strain) of the wire.

• Apply a gradually increasing tensile force to the wire while simultaneously measuring the elongation. Ensure that the force is applied uniformly along the length of the wire and that the loading rate is controlled.

• Record the applied force (load) and the corresponding elongation (strain) at regular intervals throughout the test until the wire fractures. Plot a stress-strain curve using the recorded data.

• From the stress-strain curve, determine the elastic (linear) region where the stress is directly proportional to the strain. Calculate the slope of the linear portion of the curve, which represents the Young’s modulus of the material according to Hooke’s law

• Young’s modulus (E) is calculated using the formula: E = Stress​/Strain

Learn,

• Stress and Strain
• Hooke’s Law

Modulus of Elasticity Formula

Formula of Elastic Modulus is given as the ratio of stress and strain

E = σ / ε

where

• E is the Elastic Modulus,
• ε is the resulting strain of the material
• σ is the stress applied to the material.

We can also define the modulus of elasticity by:

E = (F × L) / (A × δL)

where

• F = force exerted by the object under tension
• L = length
• A = cross-sectional area
• δL = change in length
• E = Young modulus in Pa

Modulus of Elasticity Dimensional formula

The Dimensional Formula for Modulus of Elasticity can be written as: [M¹L^-1T^-2]

Where,

• M = mass
• L = length
• T = time

Derivation of Dimensional Formula of Modulus of Elasticity

We can derive the dimensional formula of Modulus of Elasticity as follows:

Modulus of Elasticity = Stress / Strain

We know, Stress = Force / Area.

Force = mass × acceleration = [M¹L¹T^-2]

Area = length × breadth [M⁰L²T⁰]

Now, Stress = [M¹L^-1T^-2] {Force / Area}

And, the dimensional formula of linear strain = [M⁰L⁰T⁰]

Therefore, the modulus of elasticity is dimensionally represented as [M¹ L^-1 T^-2].

Unit of Elastic Modulus

SI Unit of Modulus of Elasticity is Nm^-2 or Pascal

Elastic Modulus is expressed as a force per unit area. Its unit of measurement is pascals (Pa) or pounds per square inch (psi). If elastic modulus is of larger magnitudes, they are expressed by MPa and GPa. It is a measure of a material’s stiffness or resistance to elastic deformation under stress.

Modulus of Elasticity of Different Materials

The values of some common elastic modulus materials are mentioned below:

Elastic Modulus of Different Materials
Materials	Modulus of Elasticity (GPa)
ABS (Plastic)	2.3
Acrylic	3.2
Aluminum	69
Aluminum 6061	68.9
Copper	117
Diamond	1220
Glass	50-90
Magnesium	45
Nylon	2-4
Polypropylene	1.5-2
Rubber	0.01-0.1
Steel	200-210
Titanium and its alloys	105-120
Wood	10-20

Modulus of Elasticity vs Modulus of Rigidity

Modulus of Elasticity and Modulus Rigidity are two component that acts when the deforming force acts parallel and tangential respectively.

Difference Between Modulus of Elasticity and Modulus of Rigidity
Modulus of Elasticity	Modulus of Rigidity
Calculate the deformations of an object when the deforming force is applied parallel to the surface.	Calculate the deformation of an object when the deforming force is tangential to the surface.
Denoted by E	denoted by G
E = Stress/Strain	G = Shear Stress/Shear Strain
stretching or compressing type of deformation	sliding or twisting type of deformation
E = 3K(1 – 2v)	G = E/(2 + 2v)

Elastic Constants

Elastic Constants are used to determine the deformation produced by stress. They are used to obtain relationship between stress and strain. There are basically 4 types of elastic constants which we use, namely:

• Bulk modulus (K)
• Young’s modulus (E)
• Shear modulus (G)
• Poisson’s ratio (µ)

Bulk Modulus of Elasticity is explained as having the ability of a material to resist deformation in terms of change in volume at the time of subject compression under pressure.

Young’s Modulus of Elasticity is based on the elastic constant which is defined as the proportionality constant between stress and strain.

Modulus of rigidity that is also known as Shear Modulus is defined as the measure of elastic shear stiffness of a material. This property depends on the material of the member which means the more elastic the member, the higher the modulus of rigidity.

Poisson’s ratio is defined as the ratio of the change in width per unit width of a material in order to the change in its length per unit length which will be given as a result of strain.

Relation Between Elastic Constants

The relationship between these constants can be known by the given expression:

1/K − 3/G = 9/E

Where,

• K is the Bulk Modulus
• G is the shear modulus
• E is the Young’s modulus.

Elastic Modulus Applications

Modulus of Elasticity is used in day to day life in many ways. It is used in engineering as well as medical science.

• We can use the elastic modulus to calculate how much a material will stretch and also how much potential energy will be stored.
• The elastic modulus allows us to determine how a given material will respond to stress.
• Elastic modulus is used to characterize biological materials like cartilage and bone as well.

Elastic Modulus Examples

Example 1. A wire is 2m long and has cross-sectional area of 10^-6m² A load of 980 N is suspended. Calculate the stress, the strain and, the energy stored in the wire

Given: Y = 12 × 10¹⁰ N m^-2

Solution:

Stress = F / A = 980 / 10^-6 = 98 × 10⁷ N m^-2

Strain = Stress / Y = 98 × 10⁷ / 12 × 10¹⁰ = 8.17 × 10^-3

Energy = 1/2 × (stress × strain) × volume = 1/2 × (98 × 10⁷ × 8.17 × 10^-3) × 2 × 10^-6 = 8 Joules.

Example 2. A Metallic Cube with side 0.20 m is undergoes a shearing force of 1000 N. The top surface is displaced through 0.40 cm with respect to the bottom. Find the shear modulus of elasticity of the metal.

Solution:

Here, L = 0.20 m, F = 1000 N, x = 0.40 cm = 0.004 m and

Area A = L²= 0.04 m²

Therefore, Shear modulus = (1000/ 0.04) × (0.20 / 0.004) = 1.25 × 10⁶ N m^-2.

Example 3. What must be the elongation of a wire 5 m long so that the strain is 1% of 0.1? If the wire has cross-selection of 1 mm² and is stretched by 10 kg, what is the stress?

Solution:

L = 5m, Strain = 1% of 0.1 = 1 × 10^-2 × 0.1 = 1 × 10^-3, Area of cross-section = 1 mm² = 1 × 10^-6 m²

F = 10 kg= 10 × 9.8 N

Extension = Stress × L = 5mm

Stress = Force / Area = (10 × 9.8) / (1 × 10^-6) = 9.8 × 10⁷ N/m².

Modulus of Elasticity Numericals

Q1: A rod 10 m long has a cross-sectional area 1.5 × 10^-4 m². It is subjected to a load of 10 kg. If Young’s modulus of the material is 4 × 10¹⁰ N/m², calculate the elongation produced in the wire.

Q2: A metallic cube of side 100 cm is subjected to a uniform force acting normal to the whole surface of the cube. The pressure is 106 pascal. If the volume changes by 1.25 x 10^-3 m³, calculate the bulk modulus of the material.

Q3: A metal cube of side 0.45 m is subjected to a shearing force of 8000 N. The top surface is displaced through 0.30 cm with respect to the bottom. Calculate the shear modulus of metal.

Q4: A mild steel wire of radius 0.5 mm and length 3 m is stretched by a force of 49 N. calculate a) longitudinal stress, b) longitudinal strain c) elongation produced in the body if Y for steel is 2.1 × 10¹¹ N/m².

Modulus of Elasticity FAQs

Is young modulus and modulus of elasticity different from each other?

No, both are the same. Modulus of Elasticity is also known as Young Modulus because it was proposed by a scientist name Sir Thomas Young.

What is strain?

Strain is the ratio of the change in the dimensions like the length, volume or size of the body to the actual dimension of the body.

What is stress?

Stress is the restoring force or deforming force per unit area of the body.

State Hooke’s law.

Hooke’s law states that stress is directly proportional to strain. Stress ∝ Strain.

What is Young’s modulus?

Modulus of Elasticity, also known as Young Modulus, is a fundamental mechanical property of materials that measures their stiffness or resistance to elastic deformation under stress.

How to calculate elastic modulus?

To calculate Modulus of Elasticity, a substance is subjected under stress corresponding to which strain is observed. The ratio of stress and strain up to which stress and strain shows linear relation gives the Modulus of Elasticity

Which material has the highest modulus of elasticity?

Diamond has the highest modulus of elasticity till date.

What is the relation between bulk modulus, shear modulus and young modulus?

The relation between Elastic Constants is given as 1/K − 3/G = 9/E , where K is the Bulk Modulus, G is the shear modulus and E is the Young’s modulus.

What is unit of elastic modulus?

The unit of Elastic Modulus is Nm^-2 or Pascal

What is formula for elastic modulus?

The formula for Elastic Modulus is Stress/Strain