[Physics Class Notes] on Young’s Modulus Pdf for Exam

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:

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 the 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 behavior 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.

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 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 materials.

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 aluminum, 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 aluminum. The same is the reason why steel is preferred in heavy-duty machines and structural designs. Wood, bone, concrete, and glass have small Young’s moduli.

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

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 millimeter 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).

Characteristics of Young’s Modulus

Knowing when an item or material will bend or break is one of the most critical tests in engineering, and the characteristic that informs us this is Young’s modulus. 

Young’s modulus (E) is a material parameter that indicates how readily it can stretch and flex and is defined as the ratio of tensile stress (σ) to tensile strain (ε). 

The stress-strain charts for various materials might seem extremely different. Brittle materials are particularly strong because they can endure a lot of stress, don’t stretch much, and shatter quickly. Plastic materials are not very robust, but they can sustain a great deal of strain. The gradient of the line in a stress-strain dia
gram represents Young’s modulus.

The study of material mechanical characteristics is vital because it helps us understand how materials behave and allows us to build new goods and enhance old ones.

Many items on a tiny scale contain both biological (e.g., pharmaceutical medications, reproductive therapies, tissue engineering) and non-biological microparticles (e.g. chemicals, agriculture, household care). Understanding their mechanical qualities allows us to forecast their behavior in manufacturing and processing, allowing us to maximize their performance potential.

The Young’s modulus of a material is an important attribute to understand in order to forecast how the material will behave when applied to a force. This is critical for practically everything around us, including buildings, bridges, automobiles, and more.

The Young’s Modulus of a substance is an unchangeable basic feature of all materials. It is, however, temperature and pressure dependent.

The Young’s Modulus (or Elastic Modulus) of a material is essentially its stiffness. In other words, how readily it can be bent or stretched.

How to Calculate Young’s Modulus

To be more specific, the Physics and numerical values are calculated as follows:

Stress / Strain = Young’s Modulus

The slope of the first segment of the curve (i.e. m in y = mx + b) is Young’s modulus.

When a material reaches a particular stress, it begins to deform. It has progressed to the point where the material structure is stretching rather than deforming. However, if the material is stressed above this point, the molecules or atoms inside will begin to deform and irreversibly modify the substance.

A suitable analogue would be a rubber band: when you stretch a rubber band, you are extending it rather than deforming it. However, if you pull it too hard, the rubber band will disintegrate and bend. When this happens, it usually doesn’t take long before it breaks.

Doctors and scientists value Young’s Modulus because it predicts when a structural implant may deform. This will teach them how to mechanically create a component for usage in a body.