# Stress And Strain

## Introduction to Stress and Strain

You may have noticed that certain objects can stretch easily, but stretching objects like an iron rod sounds impossible, right? In this article, we will help you understand why few objects are more malleable than others. Mainly, we will be discussing Stress-strain curves because they are useful in understanding the tensile strength of a given material. We will also learn how does a force applied on a body generates stress and know the stress-strain relationship with the help of a stress-strain curve.

## What is Stress?

In mechanics, stress is defined as a force applied per unit area. It is given by the formula

$\sigma = \frac{F}{A}$

where,

$\sigma$ is the stress applied
F is the force applied
A is the area of force application

The unit of stress is $N/m^{2}$

Stress applied to a material can be of two types. They are:

Tensile Stress: It is the force applied per unit area which results in the increase in length (or area) of a body. Objects under tensile stress become thinner and longer.
Compressive Stress: It is the force applied per unit area which results in the decrease in length (or area) of a body. The object under compressive stress becomes thicker and shorter.

### What is Strain?

The strain is the amount of deformation experienced by the body in the direction of force applied, divided by initial dimensions of the body. The relation for deformation in terms of length of a solid is given below.

$\epsilon = \frac{\delta l}{L}$

where,

$\epsilon$ is the strain due to stress applied
$\delta l$ is the change in length
L is the original length of the material.

The strain is a dimensionless quantity as it just defines the relative change in shape.

Depending on stress application, strain experienced in a body can be of two types. They are:

Tensile Strain: It is the change in length (or area) of a body due to the application of tensile stress.
Compressive Strain: It is the change in length (or area) of a body due to the application of compressive strain

When we study solids and their mechanical properties, information regarding their elastic properties is most important. These can be obtained by studying the stress-strain relationships, under different loads, in these materials.

## Stress-Strain Curve

The stress-strain relationship for materials is given by the material’s stress-strain curve. Under different loads, the stress and corresponding strain values are plotted. An example of a stress-strain curve is given below.

Stress-Strain Curve

### Explaning Stress-Strain Graph

The stress-strain graph has different points or regions as follows:

• Proportional limit
• Elastic limit
• Yield point
• Ultimate stress point
• Fracture or breaking point

#### (i) Proportional Limit

It is the region in the stress-strain curve that obeys the Hooke’s Law. In this limit, the ratio of stress with strain gives us proportionality constant known as young’s modulus. The point OA in the graph is called the proportional limit.

#### (ii) Elastic Limit

It is the point in the graph up to which the material returns to its original position when the load acting on it is completely removed. Beyond this limit, the material doesn’t return to its original position and a plastic deformation starts to appear in it.

#### (iii) Yield Point

The yield point is defined as the point at which the material starts to deform plastically. After the yield point is passed, permanent plastic deformation occurs. There are two yield points (i) upper yield point (ii) lower yield point.

#### (iv) Ultimate Stress Point

It is a point that represents the maximum stress that a material can endure before failure. Beyond this point, failure occurs.

#### (v) Fracture or Breaking Point

It is the point in the stress-strain curve at which the failure of the material takes place.

## Hooke’s Law

In 19th-century, while studying springs and elasticity, English scientist Robert Hooke noticed that many materials exhibited a similar property when the stress-strain relationship was studied. There was a linear region where the force required to stretch the material was proportional to the extension of the material. This is known as Hooke’s Law.

Hooke’s Law states that the strain of the material is proportional to the applied stress within the elastic limit of that material.

Mathematically, Hooke’s law is commonly expressed as:

F = –k.x

Where,

• F is the force
• x is the extension length
• k is the constant of proportionality known as spring constant in N/m