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Elasticity is the ability of a
material to return to its original
shape and size after being
stretched, compressed, twisted or
bent. Elastic deformation (change
of shape or size) lasts only as long
as a deforming force is applied to
the object, and disappears once
the force is removed. Greater
forces may cause permanent
changes of shape or size, called
plastic deformation.
In the modern statement of
Hooke's law, the terms "stress"
and "strain" have precise
mathematical definitions. Stress is
the applied force divided by the
area the force acts on. Strain is
the added length divided by the
original length.
Stress and Strain:
Stress: is a quantity that is proportional to
the force causing a deformation. Stress is
the external force acting on an object per
unit cross sectional area.
Strain: is a measure of the degree of
deformation. It is found that for
sufficiently small stresses strain is
proportional to stress.
The constant of the proportionality
depends on the material being deformed
and on the nature of deformation
We call this proportionality constant the
elastic modulus.
The elastic modulus is therefore the ratio
of stress to the resulting strain.
Elastic Modulus=Stress/Strain
In a very real sense it is a comparison of
what is done to a solid object (a force is
applied) and how that object responds (it
deforms to some extent)
We consider three types of deformation
and define an elastic modulus for each:
Young’s Modulus: which measures the
resistance of a solid to a change in its
2. Shear Modulus: which measures the
resistance to motion of the planes of a
solid sliding past each other
3. Bulk Modulus: which measures the
resistance of solids or liquids to changes
in their volume
Young’s Modulus:
Consider a long bar of cross sectional
area A and initial length Li that is
clamped at one end. When an external
force is applied perpendicular to the
cross section internal forces in the bar
resist distortion “stretching” but the bar
attains an equilibrium in which its
length Lf is greater than Li and in which
the external force is exactly balanced
by internal forces.
In such a situation the bar is said to be
stressed. We define the tensile stress as
the ratio of the magnitude of the external
force F to the cross sectional area A. the
tensile strain in this case is defines as the
ratio of the change in length ΔL to the
original length Li.
Y=tensile stress/ tensile strain
The Elastic Limit:
The elastic limit of a substance is defined as the
maximum stress that can be applied to the substance
before it becomes permanently deformed. It is possible to
exceed the elastic limit of a substance by applying
sufficiently large stress, as seen in in the figure
Initially a stress strain curve is a straight
line. As the stress increases, however the
curve is no longer a straight line.
When the stress exceeds the elastic limit
the object is permanently distorted and it
does not return to its original shape after
the stress is removed.
What is Young’s modulus for the elastic solid
whose stress strain curve is depicted in the
figure ??
Young’s modulus is given by the ratio of stress to
strain which is the slope of the elastic behavior
section of the graph in slide 9 reading from the
graph we note that a stress of approximately
3x10⁸N/m² results in a strain of 0.003. The
slope, and hence Young’s modulus are therefore
Shear Modulus:
Another type of deformation occurs when
an object is subjected to a force
tangential to one of its faces while the
opposite face is held fixed by another
force. The stress in this case is called a
shear stress.
If the object is originally a rectangular
block a shear stress results in a shape
whose cross section is a parallelogram. To
a first approximation (for small
distortions) no change in volume occurs
with this deformation.
We define the shear stress as F/A, the
ratio of the tangential to the area of A of
the force being sheared.
The shear strain is defined as the ratio
ΔX/H where ΔX is the horizontal distance
that the sheared force moves and H is the
height of the object.
In terms of these quantities the shear
modulus is
S= shear stress/ shear strain
S= (F/A)/ (ΔX/H)
Bulk Modulus:
Bulk modulus characterizes the response
of a substance to uniform squeezing or to
a reduction in pressure when the object is
placed in a partial vacuum. Suppose that
the external forces acting on an object are
at right angles to all its faces, and that
they are distributed uniformly over all the
A uniform distribution of forces occur
when an object is immersed in a fluid. An
object subject to this type of deformation
undergoes a change in volume but no
change in shape. The volume stress is
defined as the ratio of the magnitude of
the normal force F to the area A.
 The quantity P=F/A is called the pressure.
If the pressure on an object changes by
an amount ΔP= ΔF/A the object will
experience a volume change ΔV.
The volume strain is equal to the change
in volume ΔV divided by the initial volume
B= volume stress/volume strain
B=-(ΔF/A)/(Δ V/Vi)
B=- Δ P/(ΔV/Vi)
When a solid is under uniform pressure it undergoes a change in
volume but no change in shape. This cube is compressed on all
sides by forces normal to its 6 faces.
The term viscosity is commonly used in
the description of fluid flow to characterize
the degree of internal friction, or viscous
force is associated with the resistance
that two adjacent layers of fluid have to
moving relative to each other. Viscosity
causes part of the kinetic energy of a fluid
to be converted to internal energy.
Units of Measure
Dynamic viscosity and absolute viscosity are synonymous. The
IUPAC symbol for viscosity is the Greek symbol eta (η), and
dynamic viscosity is also commonly referred to using the Greek
symbol mu (μ). The SI physical unit of dynamic viscosity is the
Pascal-second (Pa·s), which is identical to 1 kg·m−1·s−1. If a
fluid with a viscosity of one Pa·s is placed between two plates,
and one plate is pushed sideways with a shear stress of one
Pascal, it moves a distance equal to the thickness of the layer
between the plates in one second.
The name Poiseuille (Pl) was proposed for this unit (after
Jean Louis Marie Poiseuille who formulated Poiseuille's law
of viscous flow), but not accepted internationally. Care
must be taken in not confusing the Poiseuille with the poise
named after the same person.
The cgs physical unit for dynamic viscosity is the poise (P), named
after Jean Louis Marie Poiseuille. It is more commonly expressed,
particularly in ASTM standards, as centipoise (cP). The centipoise
is commonly used because water has a viscosity of 1.0020 cP (at
20 °C; the closeness to one is a convenient coincidence).
1 P = 1 g·cm−1·s−1
The relation between poise and Pascal-seconds is:
10 P = 1 kg·m−1·s−1 = 1 Pa·s
1 cP = 0.001 Pa·s = 1 mPa·s