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The quantification of the ratio of a material's elasticity is referred to as the Modulus of Elasticity or the Elastic Modulus. When stress is applied to a body, a material resists it to the point of non-permanent deformation. The modulus of elasticity measures the said resistance. The induction of stress is temporary, so is the deformation. The exposure to stress exposes the elastic properties of the materials. The manufacturing and engineering designs are required to consider the region proportional to the elastic limit, with respect to the stress-strain curve. The elastic limit is the point at which every stress inflicted leads to temporary deferment. Once the elastic point is breached the impact is permanent.
Read Also: Relation Between Elastic Constants
Before we dive deeper, let us acquaint ourselves with few terms related to elastic moduli:
Stress and Strain
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When force is applied to a static body, it is deformed to a noticeable extent. The deformation depends upon the magnitude of the force and the material of the body. When subjected to a force, a restoration wave is formed in the body. This wave is equal to the force applied but travels in the opposite direction.
The restoring force is called Stress.
The applied force that results in the relative deformation of the material is called Strain.
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Hooke’s Law of Modulus of Elasticity
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Strain α Stress
σ = E ε
Where, E = Modulus of Elasticity
σ = Stress
ε = Strain
E = \(\frac{?}{?}\) = \(\frac{\frac{F}{A}}{\frac{\bigtriangleup L}{L_0}}\) = \(\frac{FL_0}{A \bigtriangleup L}\)
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Types of Elastic Moduli
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The Modulus of elasticity can be called the quantification of stiffness. Through a range of stress, the Modulus of elasticity remains consistent for most materials.
The various types of Modulus of Elasticity are:
- Young’s Modulus
The Young’s Modulus Experimental Observation shows that the magnitude of strain produced is the same, irrespective of the stress being compressive or tensile.
| Young’s Modulus (Y) | Longitudinal Stress/Longitudinal Strain m-² or pascals. |
- Bulk’s Modulus
The measure of the capacity of a specimen to withstand a shift in volume, when compression is presented on all sides, is called the Bulk Modulus. Bulk modulus is the fraction of the applied pressure and its relative deformation. Bulk modulus is denoted by the symbol K.
| Bulk modulus | Pressure/Strain = p / (Vo − Vn)/Vo |
- Shear Modulus
The ratio of the tangential force impacting per unit area to the deformation in radians per angle is called Shear modulus. Shear modulus, also known as Rigidity modulus, is denoted by the symbol C.
| Shear Modulus (C) | Tangential stress/Shearing strain |
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Graphical Representation Elasticity Moduli
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A Modulus is the slope of a straight-line part of the stress. Focusing on the elastic region, the slope between the two stress-strain points determines the change in stress divided by the change in strain.
- The stress (σ) is divided through force by the cross-sectional region of the sample.
- The strain (ε) is the change in length when divided by the sample’s original length.
Observing that both stress and strain are normalized quantifications, the modulus presents a material property that can be consistently differentiated between the samples of different sizes.
Example:
A large steel sample will have a similar modulus as a small steel sample. But the larger specimen would require a greater degree of force for its surface to be deformed.
Brittle materials such as copper, aluminum, plastic, etc have a steeper slope and higher modulus value than materials such as iron, rubber, or steel. The reason behind the distinction is because the latter are ductile materials.
Unit of Elasticity Moduli
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The Modulus of Elasticity is measured in Pascal (Pa), which is the same measurement for a unit of normal Stress. Because longitudinal strain has no unit and it is concluded by the ratio of the original length and the change in length to it. It can also be measured in
- Megapascals (MPa)
- Gigapascals (GPa)
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Modulus of Elastomers
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Elastomers, as opposed to materials like metal and plastic, do not display a yield point. When exposed to stress, they begin to deform and continue to do so until they break. When it comes to synthetic polymers with elastic properties, the modulus is expressed as the distinguished measure of force at a given elongation.
Here, the graph shows how the Modulus of rubber is greater than that of steel:
Measuring Young’s Modulus of Elasticity
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Young’s Modulus of Elasticity is calculated through the stress-strain curve.
Take a mild-steel rod that is ductile. Do a tension test from the curve. When the stress is unleashed, the elastic region from point O to B is identified. Once the deformation sets in, the points are revealed. The point A denotes the limit of proportionality.
It is determined that the Modulus of Elasticity (E) = Slope of the stress-strain curve all the way up to A.
Read Also: Mechanical Properties of Solid
Sample Questions
Ques: Calculate the fractional compression (ΔV/V) of water at the bottom of the 3000 m deep Indian Ocean. It is given that the water has a bulk modulus equal to 2.2×109 N m–2. (Take g = 10 m s–2)/. (2 marks)
Answer: p = hρ g
= 3000 m×1000 kg m–3×10 ms–2
= 3 × 107 kg m–1 s–2
= 3 × 107 N m–2
Fractional compression ΔV/V, is ΔV/V = stress/B = (3 × 107 N m–2)/(2.2 × 109 N m–2)
= 1.36 × 10-2 or 1.36 %
Ques: A shearing force (on its narrow force) of 9.0 × 104 N is being applied on a square lead slab with the side at 50 cm and a thickness of 10 cm. The lower side is fixed on the floor. How much displaced will the upper edge be? (3 marks)
Answer: When the lead slab is fixed and a force is applied parallel to the applied force:
A = 50 cm × 10 cm
= 0.5 m × 0.1 m
= 0.05 m2
Hence, the stress applied will be
= (9.4 × 104 N/0.05 m2)
= 1.80 × 106 N.m–2
Since the searing strain = (Δx/L)= Stress /G.
Therefore,
Displacement (Δx) = (Stress×L)/G
= (1.8×106 Nm–2 × 0.5m)/(5.6×109 Nm–2)
= 1.6 × 10–4 m = 0.16
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