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Elasticity
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Deformation
When balanced forces are applied on a material body,
dimension/s of the body (i.e. size volume, shape or all
three)
may
change.
Such
changes
are
called
deformation.
Deforming Forces
The forces which cause deformation of the body are
called deforming forces. e.g. When rubber band is
stretched from both the sides, its length increases. This
is called deformation and the forces are called
deforming forces.
Elasticity
It is the property possessed by a material body by
virtue of which the body opposes any change in its
dimensions, within elastic limit & regain them when
deforming forces are removed.
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Elastic body
The body which possesses the property of elasticity is
called elastic body. OR It is the body which opposes
the deformation, within the elastic limit and regain its
original dimensions after removal of deforming forces.
e.g. rubber, copper, steel, gold, silver etc.
Plastic body
The body which do not possess the property of
elasticity OR the body which do not oppose the
deformation and can not regain its original dimensions
after removal of deforming forces is called a plastic
body. OR A body which can be deformed when very
small deforming force is applied and which doesn’t
regain its original dimensions when the forces are
removed, is called plastic body. e.g. clay, chalk,
Plasticine etc.
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Restoring Forces
When deforming forces are applied on a material body,
some internal forces are developed in the body, which
try to oppose the changes in the dimension/s. These
forces are called restoring forces.
If the magnitude of deforming forces is greater than
that of restoring forces, the deformation takes place.
As the restoring forces are directly proportional to the
deformation, this increases the magnitude of deforming
forces. If it is still less then that of the deforming forces,
deformation continues. The process continues till the
equilibrium is reached, when the restoring forces
balance the deforming forces and the deformation
stops.
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Stress
stress can be defined as applied force per unit area.
Stress
internal restoring force
area
Stress
applied deforming force
area
Depending upon changes in size, volume and
shape, there are three kinds of stresses.
Longitudinal
or
Tensile
1.
Tensile stress
Force
Area
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Mg
r2
stress
:-

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2. Volume stress :-
Volume stress
Applied Force
Area
Change in pressure
3. Shearing Stress or Shear :-
Shearing Stress
Applied force
Area
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F
A
dP

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Explanation of elasticity on the basis of molecular
model
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Strain
Strain
Change in original dimensions
Original dimensions
There are three types of strains.
1. Longitudinal or Tensile strain
Tensile Strain
Change in the original length
Original length
Tensile Strain
L

2. Volume Strain
Volume Strain
Change in the volume
Original Volume
Volume Strain
dV
V
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3. Shearing Strain
Shearing Strain
Lateral displacement of any layer
its distance from a fixed layer
EE'
h
tan
Usually θ is very small. For small values of θ,
measured in radian, tan θ = θ
∴ Shearing strain = θ
As the strain is a ratio of two similar quantities, its
value is purely numerical or it doesn’t have any unit
and hence, the dimensions. Or Its dimensions can
0 0 0
be written as [M L T ].
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Q.1
A body remains perfectly elastic if
(a.1) The compression is large
(b.1) The extension is large
(c.1) The compression or extension is small
(d.1) It does not undergo a deformation
Q.2
The dimensional formula for stress is the same
as that for
(a.2) Work
(b.2) Power
(c.2) Pressure
(d.2) Force
Q.3 Steel is more elastic than rubber because for a
given load the stain produced in steel, as compared
to that produced in rubber is
(a.3) More (b.3) Less
(c.3) Equal (d.3) Very large
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Hooke’s law
Within elastic limit, the ratio of stress to strain is
constant for the given material. This constant is the
property of the material. It is called modulus of
elasticity.
Stress
Strain
Modulus of Electricity
Depending upon different stresses and strains, there
are three elastic modulii or elastic constants.
1. Young’s Modulus (Y)
It the ratio of tensile stress to tensile strain.
Y
tensile stress
tensile strain
When a metal rod of length L and radius r is
elongated through l by applying force Mg,
Tensile Stress
Mg
, Tensile Strain
2
r
12

L

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MgL
r 2
Y
2. Bulk Modulus (k)
It is the ratio of volume stress to volume stain.
k
Volume Stress
Volume Strain
If a balloon of volume V is compressed by changing
pressure on it by dP, its volume changes by dV
∴ Volume Stress = dP and
VolumeStrain
k
dV
V
dp
V
dV
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3. Modulus of Rigidity (n)
It is the ratio of shearing stress to shearing strain.
Shearing Stress
Shearing Strain
If a force F acting on area A of a body moves the
layers of the body through angle θ
Shearing stress = F/A, Shearing strain = tanθ
F
A tan
F
A
As strain is a unitless quantity, modulus of elasticity
has the units of stress, that are, N/m2 in S.I. or
1 -1 -2
dyne/cm2 in C.G.S. Its dimensions are [M L T ]
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Poisson’s Ratio (σ).
“Within the elastic limit, the ratio of lateral strain to the
tensile strain is constant, which is known as Poisson’s
ratio”.
∴ Lateral strain = d/D and Tensile strain = ℓ / L
As
Lateral Strain
Tensile Strain
dL
D
For homogeneous isotropic material,
1 σ 0.5
In actual practice σ is always positive.
0.2 σ 0.4
σ for cork → 0, metal → 0.3, rubber → 0.5
Poisson's ratio is unitless and dimensionless quantity.
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