Page 1
CHAPTER EIGHT
MECHANICAL PROPERTIES OF SOLIDS
8.1 INTRODUCTION
In Chapter 6, 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
8.1 Introduction
8.2 Stress and strain
8.3 Hooke’s law
8.4 Stress-strain curve
8.5 Elastic moduli
8.6 Applications of elastic
behaviour of materials
Summary
Points to ponder
Exercises
Rationalised-2023-24
Page 2
CHAPTER EIGHT
MECHANICAL PROPERTIES OF SOLIDS
8.1 INTRODUCTION
In Chapter 6, 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
8.1 Introduction
8.2 Stress and strain
8.3 Hooke’s law
8.4 Stress-strain curve
8.5 Elastic moduli
8.6 Applications of elastic
behaviour of materials
Summary
Points to ponder
Exercises
Rationalised-2023-24
168 PHYSICS
8.2 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 (8.1)
The SI unit of stress is N m
–2
or pascal (Pa)
and its dimensional formula is [ ML
–1
T
–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. 8.1. In
Fig.8.1(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
?
=
L
L
(8.2)
However, if two equal and opposite deforming
forces are applied parallel to the cross-sectional
area of the cylinder, as shown in Fig. 8.1(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.
As a result of applied tangential force, there
is a relative displacement ?x between opposite
faces of the cylinder as shown in the Fig. 8.1(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
?
=
x
L
= tan ? (8.3)
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. 8.2 (c).
Thus, shearing strain = tan ? ˜ ? (8.4)
In Fig. 8.1 (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
(a) (b) (c) (d)
Fig. 8.1 (a) A cylindrical body under tensile stress elongates by ?L (b) Shearing stress on a cylinder deforming it by
an angle ? (c) A body subjected to shearing stress (d) A solid body under a stress normal to the surface at
every point (hydraulic stress). The volumetric strain is ?V/V, but there is no change in shape.
elastic behaviour and mechanical properties of
solids which would answer many such
questions.
Rationalised-2023-24
Page 3
CHAPTER EIGHT
MECHANICAL PROPERTIES OF SOLIDS
8.1 INTRODUCTION
In Chapter 6, 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
8.1 Introduction
8.2 Stress and strain
8.3 Hooke’s law
8.4 Stress-strain curve
8.5 Elastic moduli
8.6 Applications of elastic
behaviour of materials
Summary
Points to ponder
Exercises
Rationalised-2023-24
168 PHYSICS
8.2 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 (8.1)
The SI unit of stress is N m
–2
or pascal (Pa)
and its dimensional formula is [ ML
–1
T
–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. 8.1. In
Fig.8.1(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
?
=
L
L
(8.2)
However, if two equal and opposite deforming
forces are applied parallel to the cross-sectional
area of the cylinder, as shown in Fig. 8.1(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.
As a result of applied tangential force, there
is a relative displacement ?x between opposite
faces of the cylinder as shown in the Fig. 8.1(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
?
=
x
L
= tan ? (8.3)
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. 8.2 (c).
Thus, shearing strain = tan ? ˜ ? (8.4)
In Fig. 8.1 (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
(a) (b) (c) (d)
Fig. 8.1 (a) A cylindrical body under tensile stress elongates by ?L (b) Shearing stress on a cylinder deforming it by
an angle ? (c) A body subjected to shearing stress (d) A solid body under a stress normal to the surface at
every point (hydraulic stress). The volumetric strain is ?V/V, but there is no change in shape.
elastic behaviour and mechanical properties of
solids which would answer many such
questions.
Rationalised-2023-24
MECHANICAL PROPERTIES OF SOLIDS 169
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
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
?
=
V
V
(8.5)
Since the strain is a ratio of change in
dimension to the original dimension, it has no
units or dimensional formula.
8.3 HOOKE’S LAW
Stress and strain take different forms in the
situations depicted in the Fig. (8.1). 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 (8.6)
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.
8.4 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. 8.2. 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 from A to B, stress and strain
are not proportional. Nevertheless, the body still
returns to its original dimension when the load
is removed. The point B in the curve is known
as yield point (also known as elastic limit) and
the corresponding stress is known as yield
strength (s
y
) of the material.
If the load is increased further, the stress
developed exceeds the yield strength and strain
increases rapidly even for a small change in the
stress. The portion of the curve between B and
D shows this. When the load is removed, say at
some point C between B and D, the body does
not regain its original dimension. In this case,
even when the stress is zero, the strain is not
zero. The material is said to have a permanent
set. The deformation is said to be plastic
deformation. The point D on the graph is the
ultimate tensile strength (s
u
) of the material.
Beyond this point, additional strain is produced
even by a reduced applied force and fracture
occurs at point E. If the ultimate strength and
fracture points D and E are close, the material
is said to be brittle. If they are far apart, the
material is said to be ductile.
Fig. 8.2 A typical stress-strain curve for a metal.
Rationalised-2023-24
Page 4
CHAPTER EIGHT
MECHANICAL PROPERTIES OF SOLIDS
8.1 INTRODUCTION
In Chapter 6, 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
8.1 Introduction
8.2 Stress and strain
8.3 Hooke’s law
8.4 Stress-strain curve
8.5 Elastic moduli
8.6 Applications of elastic
behaviour of materials
Summary
Points to ponder
Exercises
Rationalised-2023-24
168 PHYSICS
8.2 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 (8.1)
The SI unit of stress is N m
–2
or pascal (Pa)
and its dimensional formula is [ ML
–1
T
–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. 8.1. In
Fig.8.1(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
?
=
L
L
(8.2)
However, if two equal and opposite deforming
forces are applied parallel to the cross-sectional
area of the cylinder, as shown in Fig. 8.1(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.
As a result of applied tangential force, there
is a relative displacement ?x between opposite
faces of the cylinder as shown in the Fig. 8.1(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
?
=
x
L
= tan ? (8.3)
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. 8.2 (c).
Thus, shearing strain = tan ? ˜ ? (8.4)
In Fig. 8.1 (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
(a) (b) (c) (d)
Fig. 8.1 (a) A cylindrical body under tensile stress elongates by ?L (b) Shearing stress on a cylinder deforming it by
an angle ? (c) A body subjected to shearing stress (d) A solid body under a stress normal to the surface at
every point (hydraulic stress). The volumetric strain is ?V/V, but there is no change in shape.
elastic behaviour and mechanical properties of
solids which would answer many such
questions.
Rationalised-2023-24
MECHANICAL PROPERTIES OF SOLIDS 169
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
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
?
=
V
V
(8.5)
Since the strain is a ratio of change in
dimension to the original dimension, it has no
units or dimensional formula.
8.3 HOOKE’S LAW
Stress and strain take different forms in the
situations depicted in the Fig. (8.1). 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 (8.6)
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.
8.4 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. 8.2. 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 from A to B, stress and strain
are not proportional. Nevertheless, the body still
returns to its original dimension when the load
is removed. The point B in the curve is known
as yield point (also known as elastic limit) and
the corresponding stress is known as yield
strength (s
y
) of the material.
If the load is increased further, the stress
developed exceeds the yield strength and strain
increases rapidly even for a small change in the
stress. The portion of the curve between B and
D shows this. When the load is removed, say at
some point C between B and D, the body does
not regain its original dimension. In this case,
even when the stress is zero, the strain is not
zero. The material is said to have a permanent
set. The deformation is said to be plastic
deformation. The point D on the graph is the
ultimate tensile strength (s
u
) of the material.
Beyond this point, additional strain is produced
even by a reduced applied force and fracture
occurs at point E. If the ultimate strength and
fracture points D and E are close, the material
is said to be brittle. If they are far apart, the
material is said to be ductile.
Fig. 8.2 A typical stress-strain curve for a metal.
Rationalised-2023-24
170 PHYSICS
As stated earlier, the stress-strain behaviour
varies from material to material. For example,
rubber can be pulled to several times its original
length and still returns to its original shape. Fig.
8.3 shows stress-strain curve for the elastic
tissue of aorta, present in the heart. Note that
although elastic region is very large, the material
does not obey Hooke’s law over most of the region.
Secondly, there is no well defined plastic region.
Substances like tissue of aorta, rubber etc.
which can be stretched to cause large strains
are called elastomers.
8.5 ELASTIC MODULI
The proportional region within the elastic limit
of the stress-strain curve (region OA in Fig. 8.2)
is of great importance for structural and
manufacturing engineering designs. The ratio
of stress and strain, called modulus of elasticity,
is found to be a characteristic of the material.
8.5.1 Young’s Modulus
Experimental observation show that for a given
material, the magnitude of the strain produced
is same whether the stress is tensile or
compressive. The ratio of tensile (or compressive)
stress (s) to the longitudinal strain (e) is defined as
Young’s modulus and is denoted by the symbol Y.
Y =
s
e
(8.7)
From Eqs. (8.1) and (8.2), we have
Y = (F/A)/(?L/L)
= (F × L) /(A × ?L) (8.8)
Since strain is a dimensionless quantity, the
unit of Young’s modulus is the same as that of
stress i.e., N m
–2
or Pascal (Pa). Table 8.1 gives
the values of Young’s moduli and yield strengths
of some material.
From the data given in Table 8.1, it is noticed
that for metals Young’s moduli are large.
Fig. 8.3 Stress-strain curve for the elastic tissue of
Aorta, the large tube (vessel) carrying blood
from the heart.
Table 8.1 Young’s moduli and yield strenghs of some material
# Substance tested under compression
Rationalised-2023-24
Page 5
CHAPTER EIGHT
MECHANICAL PROPERTIES OF SOLIDS
8.1 INTRODUCTION
In Chapter 6, 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
8.1 Introduction
8.2 Stress and strain
8.3 Hooke’s law
8.4 Stress-strain curve
8.5 Elastic moduli
8.6 Applications of elastic
behaviour of materials
Summary
Points to ponder
Exercises
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168 PHYSICS
8.2 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 (8.1)
The SI unit of stress is N m
–2
or pascal (Pa)
and its dimensional formula is [ ML
–1
T
–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. 8.1. In
Fig.8.1(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
?
=
L
L
(8.2)
However, if two equal and opposite deforming
forces are applied parallel to the cross-sectional
area of the cylinder, as shown in Fig. 8.1(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.
As a result of applied tangential force, there
is a relative displacement ?x between opposite
faces of the cylinder as shown in the Fig. 8.1(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
?
=
x
L
= tan ? (8.3)
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. 8.2 (c).
Thus, shearing strain = tan ? ˜ ? (8.4)
In Fig. 8.1 (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
(a) (b) (c) (d)
Fig. 8.1 (a) A cylindrical body under tensile stress elongates by ?L (b) Shearing stress on a cylinder deforming it by
an angle ? (c) A body subjected to shearing stress (d) A solid body under a stress normal to the surface at
every point (hydraulic stress). The volumetric strain is ?V/V, but there is no change in shape.
elastic behaviour and mechanical properties of
solids which would answer many such
questions.
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MECHANICAL PROPERTIES OF SOLIDS 169
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
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
?
=
V
V
(8.5)
Since the strain is a ratio of change in
dimension to the original dimension, it has no
units or dimensional formula.
8.3 HOOKE’S LAW
Stress and strain take different forms in the
situations depicted in the Fig. (8.1). 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 (8.6)
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.
8.4 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. 8.2. 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 from A to B, stress and strain
are not proportional. Nevertheless, the body still
returns to its original dimension when the load
is removed. The point B in the curve is known
as yield point (also known as elastic limit) and
the corresponding stress is known as yield
strength (s
y
) of the material.
If the load is increased further, the stress
developed exceeds the yield strength and strain
increases rapidly even for a small change in the
stress. The portion of the curve between B and
D shows this. When the load is removed, say at
some point C between B and D, the body does
not regain its original dimension. In this case,
even when the stress is zero, the strain is not
zero. The material is said to have a permanent
set. The deformation is said to be plastic
deformation. The point D on the graph is the
ultimate tensile strength (s
u
) of the material.
Beyond this point, additional strain is produced
even by a reduced applied force and fracture
occurs at point E. If the ultimate strength and
fracture points D and E are close, the material
is said to be brittle. If they are far apart, the
material is said to be ductile.
Fig. 8.2 A typical stress-strain curve for a metal.
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170 PHYSICS
As stated earlier, the stress-strain behaviour
varies from material to material. For example,
rubber can be pulled to several times its original
length and still returns to its original shape. Fig.
8.3 shows stress-strain curve for the elastic
tissue of aorta, present in the heart. Note that
although elastic region is very large, the material
does not obey Hooke’s law over most of the region.
Secondly, there is no well defined plastic region.
Substances like tissue of aorta, rubber etc.
which can be stretched to cause large strains
are called elastomers.
8.5 ELASTIC MODULI
The proportional region within the elastic limit
of the stress-strain curve (region OA in Fig. 8.2)
is of great importance for structural and
manufacturing engineering designs. The ratio
of stress and strain, called modulus of elasticity,
is found to be a characteristic of the material.
8.5.1 Young’s Modulus
Experimental observation show that for a given
material, the magnitude of the strain produced
is same whether the stress is tensile or
compressive. The ratio of tensile (or compressive)
stress (s) to the longitudinal strain (e) is defined as
Young’s modulus and is denoted by the symbol Y.
Y =
s
e
(8.7)
From Eqs. (8.1) and (8.2), we have
Y = (F/A)/(?L/L)
= (F × L) /(A × ?L) (8.8)
Since strain is a dimensionless quantity, the
unit of Young’s modulus is the same as that of
stress i.e., N m
–2
or Pascal (Pa). Table 8.1 gives
the values of Young’s moduli and yield strengths
of some material.
From the data given in Table 8.1, it is noticed
that for metals Young’s moduli are large.
Fig. 8.3 Stress-strain curve for the elastic tissue of
Aorta, the large tube (vessel) carrying blood
from the heart.
Table 8.1 Young’s moduli and yield strenghs of some material
# Substance tested under compression
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MECHANICAL PROPERTIES OF SOLIDS 171
u
u
Therefore, these materials require a large force
to produce small change in length. To increase
the length of a thin steel wire of 0.1 cm
2
cross-
sectional area by 0.1%, a force of 2000 N is
required. The force required to produce the same
strain in aluminium, brass and copper wires
having the same cross-sectional area are 690 N,
900 N and 1100 N respectively. It means that
steel is more elastic than copper, brass and
aluminium. It is for this reason that steel is
preferred in heavy-duty machines and in
structural designs. Wood, bone, concrete and
glass have rather small Young’s moduli.
Example 8.1 A structural steel rod has a
radius of 10 mm and a length of 1.0 m. A
100 kN force stretches it along its length.
Calculate (a) stress, (b) elongation, and (c)
strain on the rod. Young’s modulus, of
structural steel is 2.0 × 10
11
N m
-2
.
Answer We assume that the rod is held by a
clamp at one end, and the force F is applied at
the other end, parallel to the length of the rod.
Then the stress on the rod is given by
Stress
F
A
=
=
F
r p
2
=
×
( )
×
- 100 10 N
3.14 10 m
3
2
2
= 3.18 × 10
8
N m
–2
The elongation,
( ) F/A L
L
Y
? =
=
( )
( ) ×
×
8 –2
11 –2
1m 3.18 10 N m
2 10 N m
= 1.59 × 10
–3
m
= 1.59 mm
The strain is given by
Strain = ?L/L
= (1.59 × 10
–3
m)/(1m)
= 1.59 × 10
–3
= 0.16 % ?
Example 8.2 A copper wire of length 2.2 m
and a steel wire of length 1.6 m, both of
diameter 3.0 mm, are connected end to end.
When stretched by a load, the net
elongation is found to be 0.70 mm. Obtain
the load applied.
Answer The copper and steel wires are under
a tensile stress because they have the same
tension (equal to the load W) and the same area
of cross-section A. From Eq. (8.7) we have stress
= strain × Young’s modulus. Therefore
W/A = Y
c
× (?L
c
/L
c
) = Y
s
× (?L
s
/L
s
)
where the subscripts c and s refer to copper
and stainless steel respectively. Or,
?L
c
/?L
s
= (Y
s
/Y
c
) × (L
c
/L
s
)
Given L
c
= 2.2 m, L
s
= 1.6 m,
From Table 9.1 Y
c
= 1.1 × 10
11
N.m
–2
, and
Y
s
= 2.0 × 10
11
N.m
–2
.
?L
c
/?L
s
= (2.0 × 10
11
/1.1 × 10
11
) × (2.2/1.6) = 2.5.
The total elongation is given to be
?L
c
+ ?L
s
= 7.0 × 10
-4
m
Solving the above equations,
?L
c
= 5.0 × 10
-4
m, and ?L
s
= 2.0 × 10
-4
m.
Therefore
W = (A × Y
c
× ?L
c
)/L
c
= p (1.5 × 10
-3
)
2
× [(5.0 × 10
-4
× 1.1 × 10
11
)/2.2]
= 1.8 × 10
2
N ?
u
Example 8.3 In a human pyramid in a
circus, the entire weight of the balanced
group is supported by the legs of a performer
who is lying on his back (as shown in Fig.
8.4). The combined mass of all the persons
performing the act, and the tables, plaques
etc. involved is 280 kg. The mass of the
performer lying on his back at the bottom of
the pyramid is 60 kg. Each thighbone (femur)
of this performer has a length of 50 cm and
an effective radius of 2.0 cm. Determine the
amount by which each thighbone gets
compressed under the extra load.
Fig. 8.4 Human pyramid in a circus.
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