# Elastic Behavior of Materials

### Introduction

When we stretch a slingshot, it has been deformed due to the applied force, and again its original shape has been regained, when we stop applying the force, which is called elasticity, that means when stress is being applied the body resists any permanent change. The body regains its original shape, and size with the removal of applied stress. Let us say that if a thin steel rod has been taken for makes it bend. The application of force should be stopped when it bends a little. The rod does not regain its original shape. Based on the elastic and plastic nature of the materials, different types of behavior of the material can be seen, which can be explained using Hooke’s law.

The ability of a body to resist any permanent change to it when stress is applied is known as Elasticity. Different materials show different elastic behavior. It is very important to study the elastic behavior of a material. Most of the engineering design requires knowledge of the elastic behavior of materials in the construction of various structures like bridges, columns, pillars, beams, etc.

### Example

Let us consider a beam resting at both ends subjected to a load W at its midpoint. The beam has a length, l, width, b, and thickness, a. When a load is exerted at its midpoint, it bends as shown. In the process, the upper surface is compressed whereas the lower surface is extended. The beam will sag or deflect due to the load.

The beam bends less for a given load if the width b is greater and the length is smaller. This is due to the fact that the deflection of the beam due to the load is inversely proportional to the cube of the width and directly proportional to the cube of the length of the beam. But on increasing the width, b, unless the load is placed at the right place, there is every chance that the beam will bend. Such bending is called 'buckling'. Hence the beam can buckle under asymmetric loading, which is the case in bridges that carry differently distributed traffic at different times. Hence to avoid this, the cross-section of the beam is chosen to be an I-shape. A large load-bearing surface and enough depth to prevent bending are being provided by this shape.

In this case, it is given as;

$\delta = \frac{Wl^{3}}{4bd^{3}Y}$

Where,

$\delta$ is the sag.

Y is Young’s modulus of elasticity

Using the above equation we can easily say that to reduce the amount of bending for a certain load, Young’s modulus of elasticity of the material used must be large. Since sag is inversely proportional to the cube of depth, the depth d must be considered. But the problem faced on increasing the depth is that bending increases and this is known as buckling. Therefore, a compromise is made between the different cross-sectional shapes.

### Application of Elastic Behavior of Materials

1. The theory of elasticity is used to design safe and stable man-made structures such as skyscrapers and overbridges to make life convenient. Cranes used to lift loads use ropes that are designed so that the stress due to the maximum load does not exceed the breaking stress. It is also found that a collection of thinner wire strands when compacted together make the rope stronger than a solid rope of the same cross-section. That is the reason, crane ropes are made of several strands instead of one.

2. Structures such as bridges and tall buildings that have to support static or dynamic loads are generally constructed using pillars and beams to support them. The beams used in buildings and bridges should have to be carefully designed so that they do not bend excessively and break under the stress of the load on them. Beams and pillars are designed to remain stable and safe within the range of the maximum load they are designed to carry.

### Fun Facts

1. If you can twist, bend, stretch or squeeze it, and when you let go it returns to its original shape, it's an elastic object.

2. To a greater or lesser extent, most solid materials exhibit elastic behaviour, but there is a limit to the magnitude of the force and the accompanying deformation within which elastic recovery is possible for any given material.

3. Gases and liquids also possess elastic properties since their volume changes under the action of pressure.

4. Elasticity is the ability of a material to regain its own original shape after being stretched according to which rubber is the most elastic substance and glass will have the least elasticity.

5. When all three balls are dropped from the same height, the rubber ball will bounce the highest because it has the greatest elasticity. It gets compressed or squashed when the rubber ball hits the ground because it is very elastic, it quickly returns to its original shape.