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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 2

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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.

Elastic body

The body which possesses the property of elasticity is

called elastic body. OR It is the body which opposes 3

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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.

Restoring Forces

When deforming forces are applied on a material body,

some internal forces are developed in the body, which 4

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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.

Stress

stress can be defined as applied force per unit area.

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Depending upon changes in size, volume and

shape, there are three kinds of stresses.

1. Longitudinal or Tensile stress :-

2. Volume stress :-

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3. Shearing Stress or Shear :-

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Explanation of elasticity on the basis of molecular

model

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Strain

There are three types of strains.

1. Longitudinal or Tensile strain

2. Volume Strain

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3. Shearing Strain

Usually θ is very small. For small values of θ,

measured in radian, tan θ = θ

∴ Shearing strain = θ

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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

be written as [M0L0T0].

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) ForceQ.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.

Depending upon different stresses and strains, there

are three elastic modulii or elastic constants.

1. Young’s Modulus (Y)

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It the ratio of tensile stress to tensile strain.

When a metal rod of length L and radius r is

elongated through l by applying force Mg,

2. Bulk Modulus (k)

It is the ratio of volume stress to volume stain.

If a balloon of volume V is compressed by changing

pressure on it by dP, its volume changes by dV

∴ Volume Stress = dP and

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3. Modulus of Rigidity (n)

It is the ratio of shearing stress to 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

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As strain is a unitless quantity, modulus of elasticity

has the units of stress, that are, N/m2 in S.I. or

dyne/cm2 in C.G.S. Its dimensions are [M1L-1T-2]

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

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For homogeneous isotropic material,

In actual practice σ is always positive.

σ for cork → 0, metal → 0.3, rubber → 0.5

Poisson's ratio is unitless and dimensionless quantity.

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