Young’s Modulus

Understanding the Stress-Strain Curve

The relation between the stress and the strain can be found experimentally for a given material under tensile stress. In a standard test or experiment of tensile properties, a wire or test cylinder is stretched by an external force. The fractional change in length or what is referred to as strain and the external force required to cause the strain are noted. The applied external force is gradually increased step by step and the change in length is again noted. Then, a graph is plotted between the stress (equal in magnitude to the external force per unit area) and the strain. A graph for metal is shown in the figure below: 

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It is also possible to obtain analogous graphs for compression and shear stress. The stress-strain curves usually vary from one material to another. Such curves help us to know and understand how a given material deforms with the increase in the load.

From the graph in the figure above, we can see that in the region between points O to A, the curve is linear in nature. In this region, Hooke's law is completely obeyed. The body regains its original shape and size when the applied external force is removed. In this particular region, the solid body behaves and exhibits the characteristics of an elastic body. In the region from A to B - stress and strain are not proportional to each other. Nevertheless, the body still returns to its original size and shape when the corresponding load is removed. The point B in the curve is known as yield point, also known as the elastic limit, and the stress, in this case, is known as the yield strength of the material. If the load increases further, the stress also exceeds the yield strength, and strain increases, even for a very small change in the stress. The portion of the curve between points B and D explains the same. When the load is removed, say at some point C between B and D, the body does not regain its shape and size. In this specific case, even when the value of stress is zero, the value of strain is not zero. The material is said to then have a permanent set. The deformation is known as plastic deformation. The point D on the graph is known as the ultimate tensile strength of the material. Beyond point D, the additional strain is produced even by a reduced applied external force, and fracture occurs at point E. If the ultimate strength and fracture points D and E are close enough, the material is called brittle. If they are far apart, the material is called ductile. The stress-strain behaviour varies from one material to the other material. For example, rubber can be pulled off its original length, but it shall still return to its original shape. The substances, which can be stretched to cause large strains, are known as elastomers.

 

Elastic Moduli

The region of proportionality within the elastic limit of the stress-strain curve, which is the region OA in the above figure, holds great importance for not only structural but also manufacturing engineering designs. The ratio of stress and strain or modulus of elasticity is found to be a feature, property, or characteristic of the material.

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Young’s Modulus

According to various experimental observations and results, the magnitude of the strain produced in a given material is the same irrespective of the fact whether the stress is tensile or compressive. Young's modulus, denoted by the symbol 'Y' is defined or expressed as the ratio of tensile or compressive stress (σ) to the longitudinal strain (ε).

 Y = σ ε

We have Y = (F/A)/(∆L/L) = (F × L) /(A × ∆L)

As strain is a dimensionless quantity, the unit of Young’s modulus is the same as that of stress, that is N/m² or Pascal (Pa). The table below has specified the values of Young’s moduli and yield strengths of some of the material.

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From the data specified in the table above, it can be seen that for metals, the value of Young's moduli is comparatively large. Hence, these materials require a relatively large external force to produce little changes in length. For increasing the length of a thin steel wire of 0.1 cm² and cross-sectional area by 0.1%, a force of 2000 N is needed. The applied force required to produce the same strain in aluminium, brass, and copper wires with the same cross-sectional area is 690 N, 900 N, and 1100 N, respectively. It implies that steel is more elastic than copper, brass, and aluminium. The same is the reason why steel is preferred in heavy-duty machines and structural designs. Wood, bone, concrete, and glass have a small Young's moduli.

 

How to Determine Young’s Modulus of the Material of a Wire?

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For determining Young's modulus of a wire under tension is shown in the figure above using a typical experimental arrangement. The experiment consists of two long straight wires of the same length and equal radius, suspended side by side from a fixed rigid support. The wire, A called the reference wire, carries a millimetre main scale M and a pan to place weight. The wire B, called the experimental wire, of a uniform area of cross-section, also carries a pan, in which the known weights can be placed. A Vernier scale, V, is attached at the bottom of the experimental wire B's pointer, and also, the main scale M is fixed to the reference wire A. The weights placed in the pan exert a downward force and stretch the experimental wire under tensile stress. The elongation of the wire or the increase in length is measured by the Vernier arrangement. The reference wire, in this case,  is used to compensate for any change in length that may occur due to change in room temperature as it is a matter of fact that yes - any change in length of the reference wire because of temperature change will be accompanied by an equal chance in the experimental wire. Both the experimental and reference wires are initially given a small load to keep the wires straight, and the Vernier reading is recorded. Now, the experimental wire is gradually loaded with more weights to bring it under tensile stress, and the Vernier reading is recorded once again. The difference between the two vernier readings gives the elongation or increase produced in the wire. Let 'r' and 'L' denote the initial radius and length of the experimental wire, respectively. So, the area of cross-section of the wire would be πr². Let 'M' denote the mass that produced an elongation or change in length ∆L in the wire. Therefore, the applied force is equal to Mg, where g is known as the acceleration due to gravity. The Young’s modulus of the material of the experimental wire is given by the formula specified below:

 Y =σ ε =Mg.l/πr² (change in l).