CHAPTER NINE

MECHANICAL PROPERTIES OF SOLIDS

9.1 INTRODUCTION

In Chapter 7, we studied the rotation of the bodies and then

realised that the motion of a body depends on how mass is

distributed within the body. We restricted ourselves to simpler

situations of rigid bodies. A rigid body generally means a

hard solid object having a definite shape and size. But in

reality, bodies can be stretched, compressed and bent. Even

the appreciably rigid steel bar can be deformed when a

sufficiently large external force is applied on it. This means

that solid bodies are not perfectly rigid.

A solid has definite shape and size. In order to change (or

deform) the shape or size of a body, a force is required. If

you stretch a helical spring by gently pulling its ends, the

length of the spring increases slightly. When you leave the

ends of the spring, it regains its original size and shape. The

property of a body, by virtue of which it tends to regain its

original size and shape when the applied force is removed, is

known as elasticity and the deformation caused is known

as elastic deformation. However, if you apply force to a lump

of putty or mud, they have no gross tendency to regain their

previous shape, and they get permanently deformed. Such

substances are called plastic and this property is called

plasticity. Putty and mud are close to ideal plastics.

The elastic behaviour of materials plays an important role

in engineering design. For example, while designing a

building, knowledge of elastic properties of materials like steel,

concrete etc. is essential. The same is true in the design of

bridges, automobiles, ropeways etc. One could also ask —

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? Answers to such questions begin with the study of how

relatively simple kinds of loads or forces act to deform

different solids bodies. In this chapter, we shall study the

9.1 Introduction

9.2 Elastic behaviour of solids

9.3 Stress and strain

9.4 Hooke’s law

9.5 Stress-strain curve

9.6 Elastic moduli

9.7 Applications of elastic

behaviour of materials

Summary

Points to ponder

Exercises

Additional exercises

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elastic behaviour and mechanical properties of

solids which would answer many such

questions.

9.2 ELASTIC BEHAVIOUR OF SOLIDS

We know that in a solid, each atom or molecule

is surrounded by neighbouring atoms or

molecules. These are bonded together by

interatomic or intermolecular 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 (or intermolecular)

distances. When the deforming force is removed,

the interatomic forces tend to drive them back

to their original positions. Thus the body regains

its original shape and size. The restoring

mechanism can be visualised by taking a model

of spring-ball system shown in the Fig. 9.1. Here

the balls represent atoms and springs represent

interatomic forces.

Fig. 9.1 Spring-ball model for the illustration of elastic

behaviour of solids.

If you try to displace any ball from its

equilibrium position, the spring system tries to

restore the ball back to its original position. Thus

elastic behaviour of solids can be explained in

terms of microscopic nature of the solid. Robert

Hooke, an English physicist (1635 - 1703 A.D)

performed experiments on springs and found

that the elongation (change in the length)

produced in a body is proportional to the applied

force or load. In 1676, he presented his law of

elasticity, now called Hooke’s law. We shall

study about it in Section 9.4. This law, like

Boyle’s law, is one of the earliest quantitative

relationships in science. It is very important to

know the behaviour of the materials under

various kinds of load fr

om the context of

engineering design.

9.3 STRESS AND STRAIN

When forces are applied on a body in such a

manner that the body is still in static equilibrium,

it is deformed to a small or large extent depending

upon the nature of the material of the body and

the magnitude of the deforming force. The

deformation may not be noticeable visually in

many materials but it is there. When a body is

subjected to a deforming force, a restoring force

is developed in the body. This restoring force is

equal in magnitude but opposite in direction to

the applied force. The restoring force per unit area

is known as stress. If F is the force applied normal

to the cross–section and A is the area of cross

section of the body,

Magnitude of the stress = F/A (9.1)

The SI unit of stress is N m

–2

or pascal (Pa)

and its dimensional formula is [ ML

–1

–2

There are three ways in which a solid may

change its dimensions when an external force

acts on it. These are shown in Fig. 9.2. In

Fig.9.2(a), a cylinder is stretched by two equal

forces applied normal to its cross-sectional area.

The restoring force per unit area in this case

is called tensile stress. If the cylinder is

compressed under the action of applied forces,

the restoring force per unit area is known as

compressive stress. Tensile or compressive

stress can also be termed as longitudinal stress.

In both the cases, there is a change in the

length of the cylinder. The change in the length

∆L to the original length L of the body (cylinder

in this case) is known as longitudinal strain.

Longitudinal strain

∆

(9.2)

However, if two equal and opposite deforming

forces are applied parallel to the cross-sectional

area of the cylinder, as shown in Fig. 9.2(b),

there is relative displacement between the

opposite faces of the cylinder. The restoring force

per unit area developed due to the applied

tangential force is known as tangential or

shearing stress.

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MECHANICAL PROPERTIES OF SOLIDS 237

As a result of applied tangential force, there

is a relative displacement ∆x between opposite

faces of the cylinder as shown in the Fig. 9.2(b).

The strain so produced is known as shearing

strain and it is defined as the ratio of relative

displacement of the faces ∆x to the length of

the cylinder L.

Shearing strain

∆

= tan

(9.3)

where

is the angular displacement of the

cylinder from the vertical (original position of

the cylinder). Usually

is very small, tan

is nearly equal to angle

, (if

= 10°, for

example, there is only 1% difference between

and tan

It can also be visualised, when a book is

pressed with the hand and pushed horizontally,

as shown in Fig. 9.2 (c).

Thus, shearing strain = tan

≈

(9.4)

In Fig. 9.2 (d), a solid sphere placed in the

fluid under high pressure is compressed

uniformly on all sides. The force applied by the

fluid acts in perpendicular direction at each

point of the surface and the body is said to be

under hydraulic compression. This leads to

decrease in its volume without any change of

its geometrical shape.

The body develops internal restoring forces

that are equal and opposite to the forces applied

by the fluid (the body restores its original shape

and size when taken out from the fluid). The

internal restoring force per unit area in this case

Robert Hooke

(1635 – 1703 A.D.)

Robert Hooke was born on July 18, 1635 in Freshwater, Isle of Wight. He was

one of the most brilliant and versatile seventeenth century English scientists.

He attended Oxford University but never graduated. Yet he was an extremely

talented inventor, instrument-maker and building designer. He assisted Robert

Boyle in the construction of Boylean air pump. In 1662, he was appointed as

Curator of Experiments to the newly founded Royal Society. In 1665, he became

Professor of Geometry in Gresham College where he carried out his astronomi-

cal observations. He built a Gregorian reflecting telescope; discovered the fifth

star in the trapezium and an asterism in the constellation Orion; suggested that

Jupiter rotates on its axis; plotted detailed sketches of Mars which were later

used in the 19

century to determine the planet’s rate of rotation; stated the

inverse square law to describe planetary motion, which Newton modified later

etc. He was elected Fellow of Royal Society and also served as the Society’s

Secretary from 1667 to 1682. In his series of observations presented in Micrographia, he suggested

wave theory of light and first used the word ‘cell’ in a biological context as a result of his studies of cork.

Robert Hooke is best known to physicists for his discovery of law of elasticity: Ut tensio, sic vis (This

is a Latin expression and it means as the distortion, so the force). This law laid the basis for studies of

stress and strain and for understanding the elastic materials.

(a) (b) (c) (d)

Fig. 9.2 (a) A cylindrical body under tensile stress elongates by

∆

L (b) Shearing stress on a cylinder deforming it by

an angle

every point (hydraulic stress). The volumetric strain is

∆

V/V, but there is no change in shape.

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238 PHYSICS

is known as hydraulic stress and in magnitude

is equal to the hydraulic pressure (applied force

per unit area).

The strain produced by a hydraulic pressure

is called volume strain and is defined as the

ratio of change in volume (∆V) to the original

volume (V

Volume strain

∆

(9.5)

Since the strain is a ratio of change in

dimension to the original dimension, it has no

units or dimensional formula.

9.4 HOOKE’S LAW

Stress and strain take different forms in the

situations depicted in the Fig. (9.2). 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 (9.6)

where k is the proportionality constant and is

known as modulus of elasticity.

Hooke’s law is an empirical law and is found

to be valid for most materials. However, there

are some materials which do not exhibit this

linear relationship.

9.5 STRESS-STRAIN CURVE

The relation between the stress and the strain

for a given material under tensile stress can be

found experimentally. In a standard test of

tensile properties, a test cylinder or a wire is

stretched by an applied force. The fractional

change in length (the strain) and the applied

force needed to cause the strain are recorded.

The applied force is gradually increased in steps

and the change in length is noted. A graph is

plotted between the stress (which is equal in

magnitude to the applied force per unit area)

and the strain produced. A typical graph for a

metal is shown in Fig. 9.3. Analogous graphs

for compression and shear stress may also be

obtained. The stress-strain curves vary from

material to material. These curves help us to

understand how a given material deforms with

increasing loads. From the graph, we can see

that in the region between O to A, the curve is

linear. In this region, Hooke’s law is obeyed.

The body regains its original dimensions when

the applied force is removed. In this region, the

solid behaves as an elastic body.

In the region fr