· Elastic Limit:
The maximum stress from which an elastic body will recover its original size
or shape after the removal of the deforming force is called elastic limit.
· Elastomers: Materials which undergo
large elongation under the effect of load at room temperature and regain
their original shape or size when the load is removed are called elastomers.
For example, rubber is an elastomer. Rubber comes to its original length,
even when its length is increased several times its original length. Thus, it
has a large elastic limit but at the same time it doesn't obey Hooke's law.
Further, rubber doesn't have plastic range, i.e. it just breaks when stretched
beyond certain limit.
· Young's Modulus (Y):
(i) Young's modulus is possessed by solids only.
· Bulk Modulus (K):
(i) Compressibility: Compressibility is the measure of how easily a
material can be compressed. In other words, compressibility is just the
reciprocal of the bulk modulus of the material.
Compressibility, k =
(ii) Bulk modulus is possessed by solids, liquid and gases. Bulk modulus for
solids and liquids is large because large forces are required to produce even
a very small change in their volume. Since solids and liquids are relatively
incompressible, they have very low compressibility and hence large bulk
modulus. Bulk modulus of solids and liquids is almost independent of the
changes in temperature and pressure. On the other hand, gases can easily be
compressed and so have large compressibility and correspondingly small bulk
modulus. Bulk modulus of gases depends on temperature and pressure.
· Shear Modulus (h):
(i) Shear modulus is possessed by solids only. Fluids (liquids or gases)
cannot sustain shear stress as they flow under the influence of a shear
stress (tangential stress).
(ii) The shear modulus of a solid is nearly one-third of the value of its
· Relation Between Y, K, h and s:
(i) Y = h (1 + s)
(ii) Y = 3K(1 - 2s)
where Y = Young's modulus
K = Bulk modulus
h = Shear modulus
s = Poisson's ratio
· Range of Poisson's Ratio:
(i) Y = h (1 + s)
On the L.H.S., Y is always positive. On the R.H.S., h is always positive.
For Y to be positive, (1 + s) > 0
i.e. s > -1 …(1)
(ii) Y = 3K(1 - 2s)
On the L.H.S., Y is always positive. On the R.H.S., K is always positive. For
Y to be positive, (1 - 2s) > 0
i.e. 2s < 1
\ s <
From (1) and (2), we get,
-1 < s <
· Elastic After-Effect: Within the
elastic limit, certain material bodies such as phosphor-bronze, quartz,
silver etc recover their original state almost immediately after the
deforming force is removed. However, most of the material bodies, in general,
take appreciably long time to recover their original state.
This delay in the recovery of the original state of a body after the
deforming force ceases to act on the body is called elastic after-effect.
For example, a glass fibre will take hours to return to its original state
when the torsional twist ceases to act on it. On the other hand, a quartz or
phosphor-bronze fibre will immediately regain its original state under
similar conditions. For this reason, the suspension fibre in a moving coil
galvanometer is made of quartz or phosphor-bronze.
· Elastic Fatigue: It is defined as
the property due to which an elastic body becomes less elastic under the
action of repeated alternating deforming forces.
For example, a wire performing torsional vibrations is subjected to repeated
alternating deforming forces (restoring torque). If the wire performs the
same set of motion again and again for long interval of time, then its
vibrations die out very quickly as the wire is said to be fatigued (or
tired). If an elastic body is subjected to repeated strains beyond its
elastic limit, it ultimately breaks. This is the reason why bridges are
declared unsafe after long use.
· Elastic Hysteresis: Those materials
which exhibit elastic after-effect take appreciably long time to recover
their original state after the deforming force is removed. In such materials,
the strain persists even when the stress is removed. This lagging behind of
the strain is called elastic hysteresis. Materials with practically no
elastic after-effect show no elastic hysteresis. The large elastic hysteresis
of some kinds of rubber makes these materials very valuable as vibration
Tensors: A tensor is a geometric object that requires for its
full description more than just one number, as scalar, and even more than
three numbers, as a vector.
Examples of tensors include: Stress tensor, strain tensor, inertia tensor,
energy-momentum tensor, tensor of the electromagnetic field, metric tensor,
curvature tensor, etc.
Stress is a tensor because in addition to the vector which defines the force,
stress also depends on a second vector which represents the surface upon
which the stress force is acting upon.
Stress can be resolved into two components:
(i) Normal stress - The stress component which is normal to the surface.
(ii) Shear stress - The stress component which acts in the plane of the surface.
A normal stress which acts in the direction away from a surface is called a
tensile stress, and it has a positive value.
A normal stress which acts in the direction toward a surface is called a
compressive stress, and it has a negative value.