MECHANICS OF SOLIDS

Elastic and plastic behavior of solids

Elastic body – A body which returns to its original shape and size on the
removal of the deforming force is called an elastic body. The property of the
material of a body by value of which the body regains its original shape and size
after removing the deforming force is called elasticity.

Contrary to the concept of elasticity in daily life in physics elasticity stands for
opposition to change i.e more rigid a body, more elastic it is .Therefore steel is
more elastic than rubber.

Plastic body – A body which does not regain its original shape and size on the
removal of the deforming force is called a plastic body e.g.: clay, wax

Stress - When a body is deformed restoring force are developed uniformly

inside the body. The restoring force per unit area set up is the body when
deformed by an external force is called stress.

Restoring Force
Tensile stress =
𝐴𝑟𝑒𝑎

If body is compressed under the action of applied force it is compressed as the

restoring force set up in the body is equal and opposite to the external
deforming force, stress can also defined as the external force acting per unit area

External Force applied F

Stress = =
𝐴𝑟𝑒𝑎 𝐴

There are two kinds of stress

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i. Normal stress - The deforming force acting per unit area normal to the
surface of the body is called normal stress. eg. when a wire is pulled by a
𝑚𝑔
force
𝜋𝑟 2

ii. Tangential stress – The deforming force acting per unit area tangential to the
surface is called tangential stress eg: a body being sheared is under of stress
is Nm-2 or Pa in S.I system and dyne cm-2 in cgs system. The dimensional
formula of stress is [ML -1 T -2 ]
Strain
When a deforming force acts on a body, it undergoes change in its
dimensions and the body is said to be deformed or strained. The ratio of change
in dimension of the body to original dimension is called strain. It is not having
a unit or dimensional formula.
Since a body can have three types of deformations ie, in length, in volume and
in shape there are following three types of strain.
i. Longitudinal Strain or linear strain – When the deforming force produces a
change in length, the strain is called longitudinal strain. It is the ratio of
change in length to original length . If ‘l’ is the original length ∆𝑙 is the
change in length then, (fractional change in length) (only in solids)
∆𝑙
Longitudinal strain =
𝑙

ii. Bulk or Volume strain – If the deforming force produces a change in

volume, then the strain is called volumetric strain. It is the ratio of change
in volume to the original volume. If V is the original volume and ∆𝑣 is the
change in volume.
∆𝑣
Volume strain = (fractional compression) or (fractional change in
𝑣

volume)
iii. Shearing Strain – If the deforming force produces a change in shape of the
body then the strain is called shear strain. It is measured as the angle

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 through which the line originally perpendicular to the fixed face gets
turned. This angle is called angle of shear (𝜃)

Hooke’s Law
It states that within elastic limit, stress is directly proportional to strain for
given material. According to Hooke’s Law,
Stress ∝ strain
Stress = constant × strain
or
Stress
= a constant
𝑠𝑡𝑟𝑎𝑖𝑛

This constant of proportionality is called modulus of elasticity or coefficient of

elasticity. Its unit is Pa depending on the three types of strain, there are three
modulus of elasticity namely, Young’s modulus (y), Bulk modulus (B) and
modulus of rigidity (G)
i. Young’s modulus – It is defined as the ratio of normal stress to
longitudinal strain. It is denoted by Y.
𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠
Y=
Longitudinal 𝑠𝑡𝑟𝑎𝑖𝑛

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Consider a wire of length ‘l’ and cross – sectional area A stretched by a force F
let its length increases by ∆𝑙 due to its force
Then, normal stress = F/A
∆𝑙
and Longitudinal 𝑠𝑡𝑟𝑎𝑖𝑛 =
𝑙
𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠
Young’s modulus Y =
Longitudinal 𝑠𝑡𝑟𝑎𝑖𝑛
𝐹/𝐴
Y= ∆𝑙
𝑙

𝐹𝑙
Y=
𝐴 ∆𝑙

For block of mass m hung in air by wire If r is the radius of the wire A
= 𝜋𝑟2
𝐹𝑙 𝑚𝑔𝑙
Y= =
𝜋𝑟 2 ∆𝑙 𝜋𝑟 2 ∆𝑙

The Unit of Y is Pascal or Nm-2 in S.I system and dyne cm-2 in cgs system.
Its dimensional formula is [ML -1 T -2]

ii. Bulk modulus – It is defined as the ratio of normal stress to volumetric

strain. It is denoted by B

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 𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠
∴B=
Volume 𝑠𝑡𝑟𝑎𝑖𝑛

Consider a sphere of volume V and surface area A suppose that a force F

which acts uniformly over the whole surface of the sphere, decreases its
volume by ∆𝑣 .

−∆𝑣
Then, normal stress = F/A and volumetric strain =
𝑣

The negative sign indicates that on increasing the stress volume of the
sphere decreases.
𝐹/𝐴 𝐹𝑣
B= ∆𝑣 =−
𝐴 ∆𝑣
𝑣

But F/A = P, the pressure applied over the surface

𝑃𝑣
∴K=−
∆𝑣

The unit of bulk modulus is Nm-2 or Pa

Compressibility – The reciprocal of bulk modulus of a material is called
compressibility .It is denoted by K.
1 −∆𝑣 1
∴ Compressibility = = K=
𝐵 𝑃𝑣 𝐵

Its unit is N-1m2 or Pa-1

iii. Modulus of rigidity – It is defined as the ratio of tangential stress to shear

strain. It is also called shear modulus. It is denoted by G.
𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠 𝐹/𝐴 𝐹/𝐴 𝐹/𝐴 𝐹/𝐴
G = = ∆𝑥 = =
shear 𝑠𝑡𝑟𝑎𝑖𝑛 𝑇𝑎𝑛 𝜃 𝜃 𝜃
𝑙

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 Consider a rectangular block. Its lower face ABCD is fixed and the upper
face EFGH is subjected to tangential force F. Let A be the area of the each
face and AF = l be the perpendicular distance between them. The tangential
force will shear the rectangular block into a parallelepiped by displacing the
upper face through a distance FF1 = x.
If ∠ FAF1 = 𝜃 Then 𝜃 𝑖𝑠 the angle of shear.
𝐹
Tangential stress = (Shear modulus
𝐴
is
Shear strain = angle of shear = 𝜃 generally less than
𝐹/𝐴
∴G = Youngs modulus
𝜃
𝐹 𝒀
G= , G=
𝐹𝑙 G≈
𝐴𝜃 𝐴𝑥 𝟑
for most materials.)

For solids angle of shear is very small, so that in ∆ 𝐴FF1

𝐹𝐹 1 𝑥
𝜃= = (𝑥 is called lateral displacement). (The shear strain 𝜃
𝐴𝐹 𝑙
is also defined as ratio of lateral
displacement of a layer to its
distance from the fixed layer )

Searl’s apparatus

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Determination of young’s modulus of the material of a wire
The experimental set up consists of two long straight wires of same
length (l) and equal radius (r) suspended side by side from a rigid support. One
is the reference wire used to compensate for any change is length due to
temperature change and the other is the experimental wires are given an initial
small load to keep the wines straight. Now the experimental wire is gradually
loaded with more weights to bring it under a tensile stress. The increase in
length is determined using the vernrier scale arrangement. If ∆𝑙 is the increase
in length, of the wire.

𝑚𝑔 𝑙
Y=
𝜋𝑟 2 ∆𝑙

Poisson’s ratio : Ratio of lateral strain to longitudinal strain is called poisson’s

ratio.

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Let a deforming force be applied to a wire of length ℓ and diameter D, then its
length increases by ∆𝑙 and diameter decreases by ∆𝐷.
∆𝑙 −∆𝐷
Longitudinal strain = , Lateral strain =
𝑙 𝐷
𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛
Poission’s ratio, 𝜎 =
Longitudinal 𝑠𝑡𝑟𝑎𝑖𝑛
−∆𝐷/𝐷 −∆𝐷 𝑙
𝜎= =
∆𝑙/𝑙 𝐷 ∆𝑙

𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 sign shows when length increases, diameter decreases.

Stress – Strain Graph for a metal

A graph is plotted between stress and strain from the graph it is seen that (the
portion OA of the graph is linear showing that stress is proportional to strain.
(ie, Hooke’s law is obeyed) up to A the body regains its original dimension
when the applied force is removed.(The material is said to be elastic). In the
region from A to B stress and strain are not proportional, but the body returns to
its original dimension when the load is removed (The point B is called yield
point or elastic limit and after this point it is permanent deformed . The
corresponding stress that causes permanent deformation is called yield strength.

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 If the load or stress exceeds yield strength the strain increases rapidly for
a small increase in stress. The portion of the curve between B and D shows this.
When the load is removed at any point beyond B, say C, the body does not
regain its original dimension. In this case, even when the stress is zero, strain is
not zero. The material is said to have a permanent set (residual strain)
represented by OE. The deformation is said to be plastic deformation. The strain
persists even after the stress is removed. The strain lags behind the stress. The
lagging behind of strain is called elastic hysteresis.

Further increase of load beyond C produces large increase in strain until

D at which fracture takes place. (The point E is called fracture point or breaking
point). The stress at which the specimen ruptures is called ultimate tensile
strength or the tensile strength if fracture occurs soon after the ultimate strength
the material is said to be brittle (if D and E close). eg. Glass ,ceramics. Materials
which have a good plastic range are called ductile eg: aluminum, copper etc.

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Materials like rubber has large elastic limit and does not obey hook’s law
strictly. Such materials which can be greatly stretched are called elastomers.
Tensile stress
When a rod or a cylinder is stretched by a force applied normal to its
cross-sectional area, the restoring force per unit area is called tensile stress.
Compressive stress
When a rod or a cylinder is compressed under the action of applied force the
restoring force per unit area is called compressive stress.
Applications of Elastic behavior of materials
1. Cranes used for lifting and moving heavy loads have thick metal rope to
which load is attached. Suppose we want to make a crane which has a lifting
capacity of 10 metric tons, it is necessary that the load should not deform the
steel rope permanently. Therefore, the extension should not exceed the
elastic limit of steel, ie. The yield strength Sy = 300 × 106 N/m2. Therefore
the area of cross-section A of the steel rope has to be at least,
𝑚𝑔 10 ×103 ×10
A≥ ≥ ≥ 3.3 × 10-4 m2
𝑠𝑦 300×106

The corresponding radius is about 1 cm for a rope of circular cross-section. But

in practice, for more safety, large numbers of thin wires are braided together, to
have a radius of about 3 cm.
2. A bridge has to be designed in such a way that it does not bend or break
under the load of heavy traffic, force of wind etc. Similarly in the design of
buildings the use of beams is very common. In both the cases, it is important
that the bending of beams should be prevented. The depression 𝛿 at the
centre of a bar, when it is loaded (W) at the centre is given by,
𝑊𝑙 3
𝛿=
4𝑑 3 𝑏𝑦

where , ℓ is the length, b is the breadth and d is the depth of the bar. Y is the
young’s modulus of the material of the bar.

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 To reduce the bending for a given load one should use a material with a large
young’s modulus Y. For a given material, increasing the depth d rather than
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the breadth b is more effective in reducing bending since 𝛿 is proportional 3.
𝑑

3. A deep bar, pillars etc. used in buildings and bridges may bend as shown in
figure. This bending is called buckling.

To check this buckling the cross-section is made I shaped as in fig with a

large load bearing surface.
fig
This shape reduces weight of beam without sacrificing strength and hence
reduces cost.

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4. Maximum height of a mountain on earth can be estimated from the elastic
behavior of earth. The pressure at the base of a mountain is given by
P = h𝞀g
Where h is the height of the mountain 𝞀 is the density of the material of the
mountain (≈ 3 × 103 𝑘𝑔/𝑚3 ) and g is acceleration due to gravity. At the
base of the mountain, the pressure must be less than elastic limit of earth’s
supporting material whose value for a typical rock is 3 × 108 𝑁/𝑚2 .
Therefore,
h 𝞺 g = 3 × 108
3×108 3×108
h= = = 104 m which is more than the height of Mt. Everest.
𝜌𝑔 3×103 ×10

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Energy stored in wire
𝐹
( )
𝐴
Y= ∆𝐿 ,
( )
𝐿

𝐹𝐿 𝑌𝐴𝑥 𝑌𝐴𝑥 𝑑𝑥
Y= F= dw = Fdx =
𝐴𝑥 𝐿 𝐿
ℓ 𝑌𝐴𝑥 𝑑𝑥 𝑌𝐴 ℓ
W= ∫𝑜 𝐿 =
𝐿
∫𝑜 𝑥 𝑑𝑥
ℓ
𝑌𝐴 𝑥2 𝑌𝐴 ℓ2
𝐿
(2) = 𝐿
(2)
𝑜

𝑌𝐴ℓ2 𝐹𝐿 𝐴ℓ2
E= =
2𝐿 𝐴𝑙 2𝐿
1 1
E = Fℓ = 𝑈 Energy = × force × change in length
2 2

𝐸 1 𝐹 ℓ
= ( )( )
𝑉 2 𝐴 𝐿

𝑬 𝟏
= stress × strain
𝑽 𝟐

𝐏𝐨𝐭𝐞𝐧𝐭𝐢𝐚𝐥 𝐞𝐧𝐞𝐫𝐠𝐲 𝐟𝐨𝐫 𝐮𝐧𝐢𝐭 𝐯𝐨𝐥𝐮𝐦𝐞

9 3 1
= +
𝑌 𝐺 𝐾

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