Elasticity:
Elasticity is that property of the object by virtue of which it regain its original configuration
after the removal of the deforming force.
Elastic Limit: Elastic limit is the upper limit of deforming force upto which, if deforming force is removed, the body regains its original form completely and beyond which if deforming force is increased the body loses its property of elasticity and get permanently deformed
Perfectly Elastic Bodies-Those bodies which regain its original configuration immediately and completely after the removal of deforming force are called perfectly elastic bodies. e.g., quartz and phosphor bronze .
Stress
The internal restoring force acting per unit area of a deformed body is called stress.
Stress = Restoring force / Area
Its unit is N/m
2
or Pascal and dimensional formula is [ML
-1
T
-2
].
Stress is a tensor quantity.
Tangential Stress -If deforming force is applied tangentially, then the stress is called
tangential stress.
Strain
The fractional change in configuration is called strain.
Strain = Change in the configuration / Original configuration
It has no unit and it is a dimensionless quantity.
(1) Longitudinal strain= Change in length / Original length
(2) Volumetric strain = Change in volume / Original volume
(3) Shearing strain = Angular displacement of the plane perpendicular to the fixed surface.
Hooke’s Law
Within the limit of elasticity, the stress is proportional to the strain.
1. Young’s Modulus of Elasticity
It is defined as the ratio of normal stress to the longitudinal strain Within the elastic limit.
y = Normal stress / Longitudinal strain
y = FΔl / Al = Mg Δl / πr
2
l
Its unit is N/m
2
or Pascal and its dimensional formula is [ML
-1
T
-2
].
2. Bulk Modulus of Elasticity -It is defined as the ratio of normal stress to the volumetric strain within the elastic limit.
K = Normal stress / Volumetric strain
K = FV / A ΔV
Its unit is N/m
2
or Pascal and its dimensional formula is
3. Modulus of Rigidity (η)
It is defined as the ratio of tangential stress to the shearing strain, within the elastic limit.
η = Tangential stress / Shearing strain
Its urut is N/m
2
or Pascal and its dimensional formula is [ML
-1
T
-2
].
Compressibility-
Compressibility of a material is the reciprocal of its bulk modulus of elasticity.
Compressibility (C) = 1 / k
Its SI unit is N
-1
m
2
and CGS unit is dyne
-1
cm
2
.Limit of Elasticity-
The maximum value of deforming force for which elasticity is present in the body is called its
limit of elasticity.
Breaking Stress-
The minimum value of stress required to break a wire, is called breaking stress.
Breaking stress is fixed for a material but breaking force varies with area of cross-section of the
wire.
Safety factor = Breaking stress / Working stress
Elastomers-
The materials for which strain produced is much larger than the stress applied, with in the limit
of elasticity are called elastomers, e.g., rubber, the elastic tissue of aorta, the large vessel
carrying blood from heart. etc.
Elastomers have no plastic range.
Poisson’s Ratio
When a deforming force is applied at the free end of a suspended wire of length 1 and radius R,
then its length increases by dl but its radius decreases by dR. Now two types of strains are
produced by a single force.
(Write math exp from text)
Relation Between Y, K, η and σ
(i) Y = 3K (1 – 2σ)
(ii) Y = 2 η ( 1 + σ)
(iii) σ = 3K – 2η / 2η + 6K
(iv) 9 / Y = 1 / K + 3 / η or Y = 9K η / η + 3K
Cantilever
A beam clamped at one end and loaded at free end is called a cantilever.
Depression at the free end of a cantilever is given by
δ = wl
3
/ 3YI
G
C
