MECHANICAL PROPERTIES OF SOLIDS
Mohamed Sherif K, HSST Physics, GHSS Athavanad
Deforming Force & Restoring Force
A force which changes the shape and size of a body is called
deforming force
When a deforming force is applied, the body may get deformed.
Then the force developed inside the body, which try to bring the
body back to its original shape and size is called restoring force.
Elasticity
It is the property of a body by virtue of which it tends to regain its
original size and shape after the applied force is removed.
Examples of elastic materials − quartz fibre, phosphor bronze
Plasticity
It is the inability of a body in regaining its original status on the
removal of the deforming forces.
Examples of plastic materials − Wax, mud
9.2 ELASTIC BEHAVIOUR OF SOLIDS
Spring-ball model for the illustration of elastic behaviour of solids
9.3 STRESS AND STRAIN
Stress
The restoring force or deforming force experienced by a unit area
is called stress.
S.I unit = Nm−2
Dimension is [ML-1T-2]
A] Normal Stress
When the elastic restoring force or deforming force acts
perpendicular to the area, the stress is called normal stress.
Normal stress can be sub-divided into the following categories
1) Tensile Stress 2) Compressive Stress 3) Volume Stress
B] Tangential or Shearing Stress
When the elastic restoring force or deforming force acts
parallel to the surface area, the stress is called tangential
stress
9.3 STRESS AND STRAIN
Normal Stress
9.3 STRESS AND STRAIN
Tangential or Shearing Stress
9.3 STRESS AND STRAIN
Strain
•Ratio of change in configuration to the original configuration
•Strain =
•It is a dimensionless quantity
Types of Strain
1. Longitudinal Strain
2. Volumetric Strain
3. Shearing Strain
9.3 STRESS AND STRAIN
Longitudinal Strain
Longitudinal Strain =
9.3 STRESS AND STRAIN
Volumetric Strain
Volumetric Strain =
9.3 STRESS AND STRAIN
Shearing Strain
Shearing Strain
9.4 HOOKE’S LAW
Hooke’s law state that within the elastic limit, stress is directly
proportional to strain
Stress α Strain
stress = k × strain
where k is known as modulus of elasticity
9.5 STRESS-STRAIN CURVE
9.5 STRESS-STRAIN CURVE
Part OA
This part OA obeys Hooke’s law. The point ‘O’ is called elastic limit
or yield point. Corresponding stress is called yield strength. In this
region the material behaves as elastic material.
Part AB
In this region a small increase in stress produce a large change in
strain. At any point between AB, if the deforming force is
removed, the body will never return to the original length. But
results a permanent change in length (Eg: OO’). This is known as
permanent set.
9.5 STRESS-STRAIN CURVE
Part BC :
The stress corresponding to the point ‘C’ is called Ultimate tensile strength
(Su). This is the maximum stress that can be applied to a wire.
Part CD :
Beyond the point C, additional strain is produced even by a reduced stress
and the wire breaks at D
Ductile Solids
Materials have a large CD region is called ductile solids. So these materials
can be drawn into thin wires eg: copper , Aluminum
Brittle Materials
If C and D are very close, the material is said to be brittle. It suddenly breaks
as soon as the ultimate strength (C) is crossed Eg : Glass.
Elastomer
Substances which can stretch to large values of strain are called elastomer.
These materials does not obey Hooke’s law
Eg: Rubber band, Aorta
Elastic Moduli
According to Hooke’s law, within elastic limit,
Stress ∝ Strain
Stress = k × Strain
k=
known as modulus of elasticity
Types of modulus of elasticity
1. Young’s Modulus of Elasticity (Y)
2. Bulk Modulus (B)
3. Rigidity Modulus (G)
Young’s Modulus (Y)
Y=
Y=
∴Y=
Where,
F - Force applied
r - Radius of the wire
l - Original length
Δl - Change in length
Unit → Nm−2 or Pascal (denoted by Pa)
Bulk Modulus (B)
If P is the increase in pressure applied on the spherical body, then
P = F/A
B=
∴B=
V - Original volume
ΔV - Change in volume
Unit → Nm−2 or Pascal
Compressibility (k) −
Reciprocal of bulk modulus of elasticity (B)
i.e., k = 1/B
Rigidity Modulus (G)
G=
G=
Where,
F - Force applied
a - Area
L- Original length
ΔL - Change in length
Units → Nm−2 or Pascal
Stress, Strain & Modulii of Elasticity
Applications of Elastic Behaviour of Materials
Construction of a Beam
Consider a bar of length l, breadth b, and depth (height) ‘d ‘ . Let a load ‘W’
be applied at the mid point of the bar. The mid-point will sag by an
amount ‘δ’ is given by
[Given; the beam has to support a maximum load ‘W’ and Beam length or
span is ‘l’]
Applications of Elastic Behaviour of Materials
From the expression it is clear that, to reduce bending of a Steel Bar;
Use material of large Young’s modulus (Y) . It is better to increase the
depth (δ α 1/d3) rather than the breadth (δ α 1/b).
Applications of Elastic Behaviour of Materials
As depth increases the chance of buckling also increases.
So to avoid buckling and bending I section beams are used.
This shape reduces the weight of the beam without sacrificing the
strength and hence reduces the cost.
Height of a Mountain
At the base of the mountain pressure (hρg) is exerted. Where h-height
of mountain, ρ- density of rocks(3×103). But the elastic limit of typical
rock at the bottom of mountain is 30 × 107 .
Equating; hρg = 30 × 107
h = 10km
From the calculations it is clear that maximum height of mountain must
be less than 10 km.
9.6.5 Poisson’s Ratio
The strain perpendicular to the applied force is called lateral strain
Simon Poisson pointed out that within the elastic limit; lateral stain is
directly proportional to the longitudinal strain.
The ratio of the lateral strain to the longitudinal strain in a stretched
wire is called Poisson’s ratio
Poisson’s ratio = (Δd/d)/(ΔL/L) = (Δd/ΔL) (L/d)
it is a pure number and has no dimensions or units
For steels the value is between 0.28 and 0.30, and for aluminum alloys
it is about 0.33
9.6.6 Elastic Potential Energy in a Stretched Wire
When a wire is put under a tensile stress, work is done against the interatomic forces.
This work is stored in the wire in the form of elastic potential energy
F = YA (l/L).
work done dW = F dl or YAld l /L
W = ½ × Young’s modulus strain2 volume of the wire
= ½ × stress strain volume of the wire
This work is stored in the wire in the form of elastic potential energy (U).
Therefore the elastic potential energy per unit volume of the wire (u) is