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• Sooraj S Nair

# Stress, Strain and Hooke’s Law

Updated: May 20

What are these, you ask? Well, lemme tell you!

Table of Contents:

## 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? Unless you are Superman or his cousin (LOL). In this article, we will help you understand why a few objects are more “stretchy” than others.

Ductility is a property of metals that makes them stretchy. You may have seen rubber band stretch, this is by the virtue of ductility or as we call it ‘Elasticity’. Similarly, we can stretch metals like iron and gold by applying forces on them and then they stretch by the virtue of elasticity. Hence they are called “Elastic”.

But “ with power comes responsibility” as Uncle Ben rightly said, when materials “stretch” they experience something called “Stress” and “Strain”. We will discuss these topics in detail today.

"Brain cells create ideas. Stress Kills Brain Cells. Stress is not a Good Idea”- Arthur Frederick Saunders

### What is Stress?

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

σ = F A

where,

σ is the stress applied F is the force applied A is the area of force application

The unit of stress is N/m2

To put it in simple words, Stress is similar to our daily life stress, when you experience trauma or something which is out of your pay grade you experience stress. Stress in materials is very similar. When you stretch a material, they stretch, but they are like Sloth (from Zootopia, if you watch Disney) they are much lazed. Stretching is something they don’t want to do but have to do due to peer pressure (they can’t set a bad example right : P) so they experience stress inside them due to the external forces acting on them.

## Types of Stress:

There are several types of stress in physics but it is mainly categorized into two forms which are: Normal Stress and Tangential or Shearing Stress. Some stress types are discussed in the points below.

### Normal Stress:

As the name suggests, Stress is said to be Normal stress when the direction of the deforming force is perpendicular to the cross-sectional area of the body. It’s like taking a big ball face-on (do not do it, won’t do you good). Normal stress can be further classified into two types based on the dimension of force-

• Longitudinal stress

• Bulk Stress or Volumetric stress

Longitudinal Stress:

Consider a cylinder. When two cross-sectional areas of the cylinder are subjected to equal and opposite forces the stress experienced by the cylinder is called longitudinal stress.

Longitudinal Stress = Deforming Force / Area of cross-section = F/A

As the name suggests, when the body is under longitudinal stress-

• The deforming force will be acting along the length of the body.

• Longitudinal stress results in the change in the length of the body. Hence, thereby it affects a slight change in diameter.

The Longitudinal Stress either stretches the object or compresses the object along its length. Thus, it can be further classified into two types based on the direction of deforming force-

• Tensile stress

• Compressive stress

Tensile Stress

If the deforming force or applied force results in the increase in the object’s length then the resulting stress is termed as tensile stress. For example: When a rod or wire is stretched by pulling it with equal and opposite forces (outwards) at both ends.

Compressive Stress

If the deforming force or applied force results in the decrease in the object’s length then the resulting stress is termed compressive stress. For example: When a rod or wire is compressed/squeezed by pushing it with equal and opposite forces (inwards) at both ends.

Bulk Stress or Volume Stress

When the deforming force or applied force acts from all dimensions resulting in the change of volume of the object then such stress is called volumetric stress or Bulk stress. Like getting showered with arrows in a war field (if you know Amphiaraus from Hercules). In short, when the volume of the body changes due to the deforming force it is termed Volume stress.

### Shearing Stress or Tangential Stress

When the direction of the deforming force or external force is parallel to the cross-sectional area, the stress experienced by the object is called shearing stress or tangential stress. This results in the change in the shape of the body (Not an ideal one though :))

## Summary on Types of Stress Types of stresses

## What is Strain?

Strain is defined as 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 the length of a solid is given below:

ϵ = δl L

where,

ϵ is the strain due to stress applied δl 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.

Long story short, it just messes up the body of the materials. IKR. Basically, the forces are so strong that the shape of the body changes.

## Types of Strain:

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. Elongation in this case.

• Shear strain : It is measured as the displacement of the surface that is in direct contact with the applied shear stress from its original position

• Compressive Strain: It is the change in length (or area) of a body due to the application of compressive strain. Compression in this case.

.

## Hooke’s Law:

When studying springs and elasticity, the 17ᵗʰ century Robert Hooke noticed that within certain limits, the force required to stretch an elastic object such as a metal spring is directly proportional to the extension of the spring. 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

In addition to governing the behavior of springs, Hooke's Law also applies in many other situations where an elastic body is deformed. These can include anything from inflating a balloon and pulling on a rubber band to measuring the amount of wind force is needed to make a tall building bend and sway. So its application is tremendous. Phew!

Above is an illustration of Hooke’s Law, showing the relationship between force and distance when applied to a spring.

Hooke's law is the first classical example of an explanation of elasticity – which is the property of an object or material which causes it to be restored to its original shape after distortion (Lemme help, deformation :P).

This ability to return to a normal shape after experiencing distortion can be referred to as a "restoring force". Understood in terms of Hooke's Law, this restoring force is generally proportional to the amount of "stretch" experienced.

This law had many important practical applications, with one being the creation of a balance wheel, which made possible the creation of the mechanical clock, the portable timepiece, the spring scale, and the manometer (aka. the pressure gauge). Also, because it is a close approximation of all solid bodies (as long as the forces of deformation are small enough), numerous branches of science and engineering as also indebted to Hooke for coming up with this law.

However, like most classical mechanics, Hooke's Law only works within a limited frame of reference. Because no material can be compressed beyond a certain minimum size (or stretched beyond a maximum size) without some permanent deformation or change of state, it only applies so long as a limited amount of force or deformation is involved. In fact, many materials will noticeably deviate from Hooke's law well before those elastic limits are reached.

Still, in its general form, Hooke's Law is compatible with Newton's laws of static equilibrium (Maybe because they were of the same era =D). Together, they make it possible to deduce the relationship between strain and stress for complex objects in terms of the intrinsic materials of the properties it is made of.

## Summary: Summary
• If a body is deformed, internal forces try to restore the body to its original state once the deforming external forces are removed. The sum of total restoring forces per unit area is called stress.

• The deformation produced in the body by the external deforming forces is called strain. It is calculated as the ratio of the change in the body’s dimension to the original value of the same dimension.

• Hooke’s Law states that for small values of strain, stress is directly proportional to strain within Elastic Limits.

• Mathematically, Hooke’s law is commonly expressed as F = –k.x

• All materials do not follow Hooke’s Law.

So that concludes what Stress-Strain and Hooke’s Law is. This is what I had for you today. Hope to see you guys with some more interesting concepts. Until then Ciao!

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