## Introduction

Can we design an **aeroplane** which is very light but sufficiently strong? Can we design an **artificial limb** which is lighter but stronger? Why does a **railway track** have a particular shape like **I**? Why is **glass brittle** while **brass** is not?

In this chapter, we will introduce the concepts of stress, strain and elastic modulus and a simple principle called **Hooke’s law** that help us predict what deformation will occur when forces are applied to a real kind of body.

## Elastic Behaviour of Solids

In a solid, each atom or molecule is surrounded by neighbouring **atoms** or **molecules**. These are bonded together by interatomic or intramolecular forces and stay in a stable equilibrium position. When a solid is deformed, the atoms or molecules are displaced from their equilibrium positions causing a change in the interatomic distance.

When the **deforming force** is removed, the interatomic forces tend to drive them back to their original position. Thus the body regains its original shape and size.

### (i) Deforming Force

If a force applied on a body produces a change in the normal positions of the molecules of the body, it is called **deforming force**.

### (ii) Elasticity

The property of the body due to which, it tries to regain its original configuration when the deforming forces are removed is called **elasticity**.

### (iii) Perfectly Elastic body

A body which completely regains its original configuration after the removal of deforming force, is called **perfectly elastic body**. Quartz and phosphor bronze are closest to perfectly elastic body known.

**Read also:** Mechanical Properties of Fluids Class 11 Physics Notes Chapter 10

### (iv) Perfectly Plastic Body

A body which does not regain its original configuration at all on the removal of deforming force, how so ever small the deforming force may be is called **perfectly plastic body**. For example, clay behaves like a perfectly
plastic body.

### (v) Restoring Force

When a deforming force is applied to a body to change its shape, the body develops an opposing force due to its elasticity. This opposing force tries to restore the original shape of the body, it is called **restoring force**.

## Stress and Strain

### (A) Strain

The **strain** is the relative change in dimensions of a body resulting from the external forces.

Strain = change in length / original length

It is a **fractional quantity** so, it has **no unit**.

**Read also:** Hydrogen Class 11 Notes Chemistry Chapter 9

#### (i) Tensile Strain

The **tensile strain** of the object is equal to the fractional change in length, which is the ratio of the elongation Δl to the original length l.

Tensile Strain`=\frac{Δl}{L}`

#### (ii) Shear Strain

We define **shear strain** as the ratio of the displacement x to the transverse dimension L.

Shear Strain`=\frac{x}{L}`

#### (iii) Bulk Strain

The fractional change in volume that is, the ratio of the volume change ΔV to the original volume V is called **Bulk Strain**.

Bulk Strain`=\frac{ΔV}{V}`

### (B) Stress

The restoring force developed per unit area in a body is called **stress**.

Stress = Restoring Force / area

In SI system, stress is measured in N / m² (pascal) and in CGS system in dyne/cm². The dimensional formula for stress is [M L^{–1}T^{–2}]

**Read also:** Conceptual Questions for Class 11 Physics Chapter 9 Mechanical Properties of Solids

#### (i) Tensile Stress

We define the **tensile stress** at the cross-section as the ratio of the force F_{⊥} to the cross-sectional area A.

Tensile Stress`=\frac{F_{⊥}}{A}`

#### (ii) Shear Stress

We define the **shear stress** as the force F_{||} acting tangent to the surface, divided by the area A on which it acts.

Shear Stress`=\frac{F_{||}}{A}`

#### (iii) Bulk Stress

If an object is immersed in a fluid (liquid or gas) at rest, the fluid exerts a force on any part of the surface of the object. This force is perpendicular to the surface. The force F_{⊥} per unit area that the fluid exerts on the surface of an immersed object is called the pressure p in the fluid (Bulk Stress).

## Hooke’s Law

For small deformations the stress and strain are proportional to each other. This is known as **Hooke’s law**.

Thus,

stress ∝ strain

stress = k × strain

where k is the proportionality constant and is known as **modulus of elasticity**.

## Elastic Moduli

The ratio of stress and strain, called modulus of elasticity, is found to be a characteristic of the material.

### (i) Young’s Modulus

For a sufficiently small tensile stress, stress and strain are proportional. The corresponding elastic modulus is called Young’s modulus, denoted by Y.

Y = Tensile-stress / Tensile-strain`=\frac{F/A}{{Δl}/L}`

`Y=\frac{FL}{AΔl}`

### (ii) Shear Modulus

If the forces are small enough that Hooke’s law is obeyed, the shear strain is proportional to the shear stress. The corresponding elastic modulus is called the shear modulus, denoted by G. It is also called the modulus of rigidity.

G = Shear-stress / Shear-strain`=\frac{F/A}{{Δx}/L}`

`G=\frac{F\times L}{A\times Δx}`

`G=\frac{F}{Aθ}`

SI unit of shear modulus is Nm^{–2} or Pa.

### (iii) Bulk Modulus

When **Hooke’s law** is obeyed, an increase in **Bulk stress** produces a proportional **Bulk strain**. The corresponding elastic modulus (ratio of stress to strain) is called the **Bulk modulus**, denoted by B.

When the pressure on a body changes by a small amount Δp, from p to (p+Δp), and the resulting Bulk strain is ΔV/V, Hooke’s law takes the form

B = Normal-stress / Volume-strain`=-\frac{Δp}{{ΔV}/V}`

We include a minus sign in this equation because an increase of pressure always causes a decrease in volume. The Bulk modulus B itself is a positive quantity.

SI unit of bulk modulus is Nm^{–2} or Pa.

#### Compressibility

The reciprocal of the Bulk modulus is called the **compressibility** and is denoted by K. From equation

`K=\frac{1}{B}`

The units of compressibility are those of reciprocal pressure, Pa^{–1} or atm^{–1}.

## Elastic Potential Energy

The excess of the energy of interaction between all atoms/molecules of a deformed object is **elastic energy**. When we remove the external force the body becomes undeformed and the elastic energy, will be retrieved back and converted into vibrational energy followed by heat, light, sound etc.

The elastic potential energy

ΔU`=\frac{1}{2}`stress × strain × volume

Also, the elastic potential energy per unit volume, i.e.,

`\frac{ΔU}{volume}=\frac{1}{2}`stress × strain

## Poisson’s Ratio

When a body is linearly extended, it contracts in the direction at right angles. **Poisson’s ratio**, σ is the ratio of the lateral strain to the longitudinal strain.

Longitudinal strain = Δl/L

Lateral strain = – ΔR/R

The Poisson’s ratio is given as,

σ = lateral strain / longitudinal strain

`σ=\frac{–{ΔR}/R}{{Δl}/L}`

–ve sign shows that if the length increases, then the radius of wire decreases. Poisson’s ratio is a unit less and dimensionless quantity.

### Relation between Y, K, η and σ

Y = 3K (1 – 2σ)

Y = 2η (1 + σ)

σ = (3K – 2η) /(2η + 6K)

9/Y = 1/K + 3/η

## Applications of Elastic Behaviour of Materials

In our **daily life**, most of the materials which we use, undergo some kind of stress. That is why, while designing a structure of the material we give due consideration to the possible stresses, the material might suffer at one stage or the other. The following examples illustrate this concept.

The metallic parts of the machinery are never subjected to a stress beyond elastic limit, otherwise they will get permanently deformed.

The crane which is used to lift and move the heavy load is provided with thick and strong metallic ropes to which the load to be lifted is attached. The rope is pulled by using pulleys and motor.

The bridges are designed in such a way that they do not bend much or break under the load of heavy traffic, force of strongly blowing wind and its own weight.

Maximum height of a mountain on earth can be estimated from the elastic behaviour of earth. At the base of mountain, the pressure is given by p = ρgh, where h is the height of mountain, ρ is the density of material of mountain and g is the acceleration due to gravity.

## Summary

**Elasticity :**Elasticity is that property of the material of a body due to which the body opposes any change in its shape and size when deforming forces are applied on it and recovers its original configuration partially or wholly as soon as the deforming forces are removed.**Stress :**It is defined as the internal restoring force per unit area of cross-section of object.**Strain :**The change in dimensions of an object per unit original dimensions is called strain.**Hooke’s law :**For small deformation, the stress is proportional to strain.**Young’s modulus :**The ratio of tensile (or compressive) stress to the corresponding longitudinal strain is called Young’s modulus.**Bulk modulus :**The ratio of volumetric stress to volumetric strain is called Bulk modulus.**Shear modulus :**It is the ratio of shear stress to shearing strain.**Poisson’s ratio :**The lateral strain is proportional to longitudinal strain within the elastic limit and the ratio of two strains is called Poisson’s ratio.**Elastic after effect :**The slow process of recovering the original state by an object after the removal of the deforming force is called elastic after effect.