Elastic Limit

We know that elasticity is the ability of a body to regain its original shape after an external force is removed. However, all the bodies have a certain elastic limit up to which they can remain in their original shape. If they are stretched beyond this limit, their orientation changes.

So, we define the elastic limit as the upper limit for deforming force up to which if deforming force is removed from the body, it comes back to its original configuration, and stretching beyond this limit can permanently change the body’s shape.

The maximum value of the force or stress on a material for which it starts showing elastic behaviours is referred to as the elastic limit of that material. It is the highest limit before the deformation of the plastic material takes place. Once the elastic limit is reached by the material, it becomes deformed as more stress or more force is exerted on it. In the case of brittle materials, whenever stress is exerted beyond the elastic limits, it results in a fracture. 

So, if the deforming force is increased, the body loses its elasticity attribute and gets permanently deformed. 

Hooke’s Law for Elastic Limit

According to the experimental study done by Hooke in connection with the extension produced in the wire and load applied, he devised a law known by his name called the Hooke’s law.

Hooke’s Law statement: Within the elastic limit, the extension produced in the wire is directly proportional to the load applied to it.

After some time, this law became applicable to all types of deformations like compression, bending and twisting, etc. In mathematical form, Hooke’s law states that in the elastic limit, stress developed is directly proportional to the strain produced in the body. It is given by:

Stress α Strain

Now, removing the sign of proportionality, we get the equation as:

Stress = E x Strain

Here, E is proportionality constant and is called the coefficient of elasticity or the Modulus of Elasticity of the material of the body. 

Also, E= stress/ strain = a constant 

Here, the stress is the deforming force applied per unit area and strain is the deformation that occurred. Therefore, stress and strain are interlinked.

The unit of stress is Nm² and that of strain is unity (a dimensionless quantity). 

What is Elastic Limit in Physics?

As we got the stress-strain relationship in wire, now let’s understand what elastic limit means in Physics.

Suspend a wire of uniform area vertically from a rigid support and on the other end, attach a hanger on which known weights can be placed. Now, attach a vernier scale V to the wire’s lower end that can slide over the main scale M, as we can see in Fig.1 below:

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Further, keep trying with different weights, place them one-by-one on the hanger, and note the reading. 

After noting down the readings of extensions caused by different known weights on the wire, draw a graph. Going according to the reading, we plot the graph in the following manner:

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Looking at Graph.1, till the portion OA, the Hooke’s law is fully obeyed, which means, the wire could gain its configuration. Therefore, OA is a linear region that represents the elastic limit.

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As the stress doubles, the strain also doubles, as seen in Graph 2, a non-linear relationship is maintained between the stress and strain. Here, point A is considered the proportional limit. But what is the limit of proportionality?

What is the Proportional Limit?

The limit at which the stress is the highest and the stress is directly proportional to the strain which results in a straight line in the stress-strain curve. The gradient becomes equal to the elastic modulus of the material as the proportional limit is reached. In the case of most metals, the elastic limit and proportional limit are equal.

In Graph.1, after point A, point B indicates the limit of elastic behaviour and the beginning of the plastic behaviour, however, till point C, it regains its original shape after the applied force is removed.

At this point, the wire shows an increase in strain without any increase in stress, as the strain increases after point C, the wire begins to flow down and continues till point D. A time comes when it reaches point D, the wire becomes perfectly plastic. 

Further deformation after point D, the weaker sections of the wire break, and point E is the ultimate or fracture point. This is called the yield point. 

In simple words, elastic limit is the point at which the body regains its structure after removing the applied force, while yield point is the point after the permanent deformation and even after unloading, the body (wire) doesn’t gain its original shape.

In order to determine the elastic limit of a given sample, the greatest stress that can be applied to it is measured before any permanent deformation takes place. In the case of metals or other rigid materials, the stress-strain curve is a straight line since the elastic limit and the proportional limit are approximately equal.

On the contrary, materials such as plastic and rubber do not have their stress-strain curve as straight and hence they are said to have an apparent elastic limit.

Difference Between Elastic Limit and Proportional Limit

The difference between elastic limit and the proportional limit is that elastic limit is the greatest pressure that can be applied to a material without causing its deformation. Whereas, the point up to which the stress and strain are directly proportional to each other is referred to as the proportional limit of that material. Another key difference is that in the case of elastic limit the stress and strain possess a linear relationship, while in the case of the proportional limit it does not matter if the relationship between the stress and strain is linear or not.

Difference Between Elastic Limit and Yield Point

The primary difference between the elastic limit and the yield point is that the yield point marks the end of the elasticity, and it is alternatively known as the elastic limit. While the elastic limit within a solid material is the maximum stress which arises before the permanent deformation of the solid body starts to occur. The yield point is introduced by the engineers for engineering convenience in order to define the point of permanent deformation which is marked by the breakage of bonds.

Difference Between Proportional Limit and Yield Point

Proportional limit or the limit of proportionality specifies the direct relation of stress with strain. Till this point, Hooke’s law is fully obeyed.

However, a point at which the stress remains constant, while the strain keeps on elongating the wire, a time comes when it reaches the perfectly plastic stage. This stage occurs at the point called the yield point.

Examples of Elastic Limit

• Rubber is considered one of the most elastic substances.

• Among materials like glass and steel, glass is more elastic. 

• When the shear stress of a hammer blow is exerted on a nail, it gets bent permanently, implying it has reached its elastic limit.

• Quartz and bronze or Phosphorous are considered as nearly plastic bodies.

• Paraffin wax and mud are considered as perfectly plastic bodies.