Elastic modulus measures a material’s resistance to elastic deformation. An external force or load is required to change the shape or size. Due to applying the same force, different materials having the same cross-section and length will have other deformations. 

While designing a structure, knowledge of elastic properties of materials like steel, concrete, etc., is important because the elastic behaviour of materials plays an essential role in design. It is also important in designing bridges, automobiles, ropeways, etc. Let us discuss elastic moduli and their types and resilient modulus.

Elastic Moduli

The elastic modulus is the property of a material that describes its stiffness, and it is one of the most important properties of solid materials. It is the ratio of stress to strain when deformation is elastic. When the deforming force is applied to the material such that the material is in static equilibrium, a resistive force is developed inside the material to oppose the external force. The resistive force per unit area is called stress \(\left({\frac{{{\text{Force}}}}{{{\text{Area}}}}} \right).\) Due to the application of deforming force, there will be a change in the length of the material. The change in length of material per unit length is called strain \(\left({\frac{{\Delta L}}{L}} \right).\) This modulus may be thought of as a material’s resistance to elastic deformation. There are three types of moduli which are discussed below.

Hooke’s Law and Modulus of Elasticity

According to Hooke’s law, “For small deformation, the stress in a body is proportional to the corresponding strain.” i.e.
\({\text{stress}} \propto {\text{strain}}\)
or
\({\text{stress}} = E \times {\text{strain}}\)
Here,
\({\text{stress}} = \frac{{{\text{ Restoring force }}}}{{{\text{ area }}}} = \frac{F}{A}\) and
\({\text{strain}} = \frac{{{\text{ change in length }}}}{{{\text{ original length }}}} = \frac{{\Delta l}}{l}\)
Here \(\left({E = \frac{{{\text{ stress }}}}{{{\text{ strain }}}}} \right)\) is a constant, which is called the modulus of elasticity. Now, depending upon the nature of deforming force applied on the body, stress, strain, the modulus of elasticity are classified into the following three types:

1. Young’s Modulus of Elasticity (Y)

When two equal and opposite forces are applied on a wire in the direction of its length then, the length of the wire is changed. The change in length per unit length \(\left({\frac{{\Delta l}}{l}} \right)\) is called the Longitudinal strain and the restoring force per unit area of cross-section of the wire is called longitudinal stress. For a small change in the length of the wire, the ratio of the longitudinal stress to the corresponding strain is called Young’s modulus of elasticity \(\left( Y \right)\) of the wire. Thus \(Y = \frac{{\frac{F}{A}}}{{\frac{{\Delta l}}{l}}}\) or \(Y = \frac{{Fl}}{{A\Delta l}}\)
The unit of Young’s modulus is the same as that of stress as strain is a dimensionless quantity, i.e. \({\text{N}}\;{{\text{m}}^{ – 2}}\) or Pascal \(\left(\rm{Pa} \right).\)

On plotting the stress and its corresponding strain on the graph, we get a curve, and this curve is called the stress-strain curve or stress-strain diagram. Generally, we take the strain (no units) on the \(x\)-axis and stress (\(\rm{Pa}\) or \(\rm{MPa}\)) on the \(y\)-axis. The slope of the graph under elastic limit gives Young’s Modulus of Elasticity. This stress-strain helps us to understand how a given material behaves with increasing loads. The stress-strain behaviour varies from material to material. The material with a less elastic constant will deform easily.

2. Bulk Modulus of Elasticity (B)

When we apply a uniform pressure all over the surface of a body, then the volume of the body changes and this change in volume per unit volume of the body is called the volume strain and the normal force acting per unit area of the surface is called the normal stress or volume stress. For small strains, the ratio of the volume stress to the volume strain is called the bulk modulus of the material of the body. It is denoted by \(B.\) Then

\(B = \frac{{ – p}}{{\frac{{\Delta V}}{V}}}\) or \(B = \frac{{ – \sigma }}{{\frac{{\Delta V}}{V}}}\)
Here, the negative sign implies that when the pressure increases, volume decreases and vice-versa. SI unit of the bulk modulus is \({\text{N}}\,{{\text{m}}^{ – 2}}\) or \(\rm{Pa}.\)

Compressibility

Compressibility is defined as the reciprocal of the bulk modulus of the material of a body, and it can be written as:
\({\text{compressibility}} = \frac{1}{B}\)

3. Modulus of Rigidity \(\left( \eta \right)\)

When an external force is applied tangentially to a surface of the body by keeping the opposite surface fixed, then it suffers a change in shape, but its volume remains unchanged. The face on which force is applied gets displace in the direction of applied force. Due to this, shear strain occurs in the material. Shearing strain is the ratio of the displacement of a layer in the direction of the tangential force, and the distance of the layer from the fixed surface, and shearing stress is the tangential force acting per unit area of the surface. Modulus of rigidity is defined as the ratio of the shearing stress to the shearing strain of the material of the body. It is denoted by \(\eta .\)

Fig. (b)

Thus,

\(\eta = \frac{{\frac{F}{A}}}{{\frac{{K{K^\prime}}}{{KN}}}}\)

Here,

\(\frac{{K{K^\prime }}}{{KN}} = \tan \theta \approx \theta \)

Therefore,

\(\eta = \frac{{\frac{F}{A}}}{\theta }\) or \(\eta = \frac{F}{{A\theta }}\)

Young’s Moduli, Elastic Limit, and Tensile Strengths of Some Materials

Resilient Modulus \(\left({{M_R}} \right),\)

Resilient modulus, \(\left({{M_R}} \right),\) generally corresponds to the degree to which a material recovers from external shock or disturbance. This property of the material is an estimate of its modulus of elasticity \(\left( E \right).\) In the case of slowly applied load, the slope of the stress-strain curve is linear in the elastic region and yields \(E,\) whereas, for rapidly applied loads, this would yield \(\left({{M_R}} \right).\) It can be expressed as
\({M_R} = \frac{\sigma }{{{ϵ_r}}}\)
where \(\sigma \) is the applied stress and \(ϵ_r\) is the recoverable axial strain.

Summary

Force per unit area is called stress, and elongation or contraction per unit length is called strain. To describe the elastic behaviour of objects as they respond to deforming forces that act on them, we define three elastic moduli, which areYoung’s modulus, shear modulus, and bulk modulus. The ratio between stress and strain is the modulus of elasticity.

Young’s Modulus is the ratio of longitudinal stress to strain. i.e., \(\left({Y = \frac{{Fl}}{{A\Delta l}}} \right).\) Shear Modulus is the ratio of tangential force per unit area to the angular deformation of the body.
i.e.\(\left({\eta = \frac{F}{{A\theta }}} \right),\) and bulk modulus is the ratio of the volume stress to the volume strain.i.e \(\left({B = \frac{{ – \Delta p}}{{\frac{{\Delta V}}{V}}}} \right).\)
The resilient modulus \(\left({{M_R}} \right)\) is the elastic modulus based on the recoverable strain under repeated loads, and it is defined as: \(\left({{M_{\text{R}}} = \frac{\sigma }{{{ϵ_r}}}} \right),\) where \(\sigma \) is the applied stress and \({ϵ_r}\) is the recoverable axial strain.

Elastic Moduli – Sample Problems

Q.1. Find the (a) stress, (b) elongation, and (c) strain on a structural steel rod if the following data are given. It has a radius of \(10\,{\text{mm}},\) and a length of \(1.0\,{\text{m}}\) and a force of \(100\,{\text{kN}}\) stretches it along its length. Young’s modulus, of structural steel is \(2.0 \times {10^{11}}\,{\text{N}}\;{{\text{m}}^{ – 2}}.\)
Solution: Let us consider that the rod is held by a clamp at one end, and at the other end, the force \(F\) is applied parallel to the length of the rod. Then the stress on the rod will be:

\({\text{stress}}\left( \sigma \right) = \frac{F}{A} = \frac{F}{{\pi {r^2}}}\)
\( \Rightarrow \sigma = \frac{{100 \times {{10}^3}}}{{3.14 \times {{\left({10 \times {{10}^{ – 3}}} \right)}^2}}}~{\text{N}}\;{{\text{m}}^{ – 2}}\)
\( \Rightarrow \sigma = 3.18 \times {10^8}~{\text{N}}\;{{\text{m}}^{ – 2}}\)
Then, the elongation will be \(\Delta l = \frac{{\left({\frac{F}{A}} \right)l}}{Y}\)
\( \Rightarrow \Delta l = \frac{{\left({3.18 \times {{10}^8}} \right) \times 1}}{{2.0 \times {{10}^{11}}}}{\text{m}}\)
\( \Rightarrow \Delta l = 1.59 \times {10^{ – 3}}\,{\text{m}}\)
\( \Rightarrow \Delta l = 1.59~{\text{mm}}\)
And, the strain is given by
\({\text{strain}} = \frac{{\Delta l}}{l}\)
\( \Rightarrow {\text{strain}} = \frac{{1.59 \times {{10}^{ – 3}}}}{1}\)
\( \Rightarrow {\text{strain}} = 1.59 \times {10^{ – 3}}\)
\( \Rightarrow{\text{strain}} = 0.16\% \)

FAQs

Q.1. What are the types of elastic moduli?
Ans: Elastic Moduli can be of three types, Young’s modulus, Shear modulus, and Bulk modulus.

Q.2. Is young’s modulus and elastic modulus?
Ans: Young’s modulus, also referred to as elastic modulus, tensile modulus, or modulus of elasticity in tension, is the ratio of stress-to-strain and is equal to the slope of a stress-strain diagram for the material.

Q.3. What do you mean by elasticity?
Ans: The property of a body, under which it tends to regain its original size and shape when the applied force is removed, is known as elasticity.

Q.4. Which is more elastic steel or aluminium?
Ans: The modulus of elasticity for aluminium is \(70\) GPA, while that of steel is in the range of \(190 – 210\) GPA. Steel is more elastic as Young modulus of elasticity more than Aluminium. Hence, Aluminium exhibits more stiffness as compared to steel and it is brittle too.

Q.5. SI unit of modulus of elasticity is?
Ans: Modulus of elasticity is defined as the ratio of longitudinal stress and longitudinal strain. Since the strain does not have any unit, it will have the unit of stress. The SI unit of this modulus is the Pascal \(\left({\rm{Pa}} \right).\)