ELASTICITY
Impact of a force
• Can change location of an object
• Can change orientation of an object
• Can change object’s shape and/or size
• Permanently deformed:
F
F
• Temporarily deformed:
Elasticity: A phenomenon whereby when a deforming
force is removed from a deformed object,
the object retains its original shape and size.
• Elastic force: A reaction force that an object,
subjected to deformation, exerts against
the impact of deformation upon it.
Deforming forces on a surface area along one-dimension
• Compressive force
• Deforming force acting perpendicularly
towards an area (or point) on a body
(can cause contraction of length)
• Tensile force
• Deforming force acting perpendicularly
away from an area (or point) on a body
(can cause extension of length)
• Shearing force
• Deforming force acting parallel to an
area. (can cause slanting an object)
Deformation applied on an area
• An object can undergo deformation when a deforming
force is applied upon part of or its entire surface area
• Magnitude of deforming force per area over which it
acts is known as the stress
Stress, σ =
Magnitude of force
Area
• Extent of deformation that an object undergoes due to
a deforming force is known as strain.
Strain, ε =
Amount of deformation
Original dimension/s
ΔL ΔV
;
Lo Vo
• Hooke’s Law: When an elastic object undergoes
moderate deformation, magnitude of
stress (σ), it’s subjected to, is directly
proportional to the strain (ε) it undergoes.
i.e. σ α ε.
• Stress (along one dimension):
• Tensile and compressive deformations (their
proportionality constant is Young’s modulus, E: σ = Eε
• Shear deformation (proportionality constant:- shear
modulus, G: σ = Gε
Note: we shall also look at volumetric deformation
• Modulus used depends on the type of deformation
that an object is subjected to.
• Moduli magnitudes depends on the molecular/ atomic
make-up of a material, and conditions under which it
was prepared
• Moduli values also depend on the atmospheric
pressure and temperature the object is subjected to.
• Gases and liquids cannot be subjected to one dimensional deformations
Material
E (N.m-2)
G (N.m-2)
Iron
Aluminium
Steel
Brass
Concrete
Wood
Copper
Tungsten
10 × 1010
70 × 109
20 × 1010
10 × 1010
20 × 109
10 × 109
10 × 1010
35 × 1010
70 × 109
25 × 109
80 × 109
35 × 109
10 × 108
40 × 1010
15 × 1010
* At normal pressure and temperature which are constant
e.g. 1
During flooding, a military plane is used to
rescue people, using a container attached to a
steel cable. The total mass of the people being
rescued and the container is 1 900 kg. The
length of the cable is 80 m, and its diameter is
2.0 cm. If the plane moves vertically up at
1.4 m.s-2, by how much does the length of the
cable change, due to the load?
Given: m = 1900 kg, Lo = 80 m,
d = 2 × 10-2 m, a = 1.4 m.s-2
Newton’s: T – Fg = ma
T = m(g + a)
T
Fg
Hooke’s: δ = εEsteel, where Esteel = 2 × 1011 N.m-2,
δ = T/A = m(g + a)/(πd2/4) and ε = (ΔL/Lo)
Therefore, ΔL = Lo(δ /Esteel)
e.g. 2
A hollow copper cylinder, 9.5 cm high, inner
radius of 1.8 cm and outer radius of 30 mm, is
subjected to a tangential force of 5.5 × 103 N.
Find the angle of deflection that the cylinder
shall undergo.
Hooke’s law:
F/A = G(ΔL/Lo),
where A = π(rout2 – rin2)
rin = 1.8 × 10-2 m,
rout = 3.0 × 10-2 m,
h = 9.5 × 10-2 m,
F = 5.5 × 103 N
G = 10 × 1010 N.m-2
But, tanθ = (ΔL/Lo)
Therefore,
θ = tan-1(F/AG)
• Relationship between stress and strain (i.e. Hooke’s Law)
is valid when a moderate force is applied on an elastic
object
• When an elastic object is subjected to ‘too much’
deformation, they can instantly undergo rupture, or
cease from being elastic (but, continue deforming)
• Plasticity – Process whereby an object, that is no longer
elastic , gets further deformed when magnitude of the
deforming force is increased (Such an object is said to be
plastic)
• Materials eventually break-up upon being
subjected to a large magnitude deforming
force.
• Ductile materials – break up after undergoing
significant deformation
• Brittle materials – suddenly break, with
‘little’ deformation
Graph of stress vs strain
E
σ (N.m-2)
C
O
D
E
F
B
ε
• A – Proportionality limit
point
• B – Elastic limit point
• C – Upper yield point
• D – Lower yield point
• E – Ultimate strength
• F – Point of rupture/
fracture
(Typical behaviour of ductile
object, due to deforming
force on it)
• Hooke’s is applicable up to point A (Proportionality limit)
• Elasticity of a material ends at point B (Elastic limit)
• Yield region (beginning of plasticity – interval BD). During
yielding, small change of stress result in sizeable strain of
the material
• Upper yield point (point C) – highest stress within the
yield region
• Lower yield point (D) – lowest point within yield region
• Any stress greater than ultimate strength (E) would make
the material to fracture
• Point of fracture (F) is lower than the ultimate strength
Ductile and Brittle materials graphs*
• Ductile material
• Brittle material
E
O
B
C
E
F
D
A
ε
σ (N.m-2)
σ (N.m-2)
E
E
O
F
B
A
ε
Bulk deformation and modulus
• Deformation along three
dimensions (also
referred to as bulk or
volumetric deformation)
changes the volume of
an object subjected to
such tensile/
compressive stress
‘throughout its surface’
V
ΔV = V – Vo
Arrows represent pressure
• The measure of uniform stress or pressure along all
dimensions per volumetric strain on an object is known
as the bulk modulus, B, of the object: B = -ΔP ÷ (ΔV/Vo)
Bulk Modulus of different materials
Material
Air
Water
Iron
Brass
B (N.m-2)
10 X 104
20 X 109
10 X 1010
80 X 109
Material
Mercury
Aluminium
Steel
Marble
B (N.m-2)
25 X 108
70 X 109
14 X 1010
70 X 1010
e.g. 3
A piece of aluminium is surrounded by air at
a pressure of 1.01 × 105 Pa. The aluminium
is placed in a vacuum chamber where
pressure is reduced to 9.11 × 104 Pa.
Determine the fractional change ΔV/Vo in the
volume of aluminium?
Data: Pi = 1.01 x 105 Pa,
Pf = 9.11 x 104 Pa,
B =70 x 109 N.m-2
Since ΔP = -B(ΔV/Vo),
where ΔP = Pf - Pi
Therefore, ΔV/Vo, = -ΔP/B =
V
Vo
Deforming force on a spring
•Load on a spring, stretches/
shortens it
•Moderate force is proportional
to the amount of extension/
contraction (F α x; Hooke’s law).
Within this range, the spring is
elastic.
•‘Very’ large forces cause
permanent deformation.
Yielding occurs during the
initial stage of plasticity
Note:
• Interval OB – The spring is elastic
• Interval OA – The spring is elastic and length change is
directly proportional to magnitude of deforming force
• Beyond point B (elastic limit) – deforming force result
in spring permanent deformation – plasticity
commences
• Ultimate force (at point E) – Maximum deforming force
magnitude a spring can tolerate
• Point of fracture/rupture, F – Spring breaks up; the
spring cannot be elongated further.
•Within interval OA, Hooke’s law is applicable: Fel α x.
•Force applied by the spring on an object deforming
it, Fel = kx, where k is a spring constant (property of
a spring that informs on how stiff the spring is)
•Spring force is a conservative force.
•An object subjected to a spring force possesses
potential energy (spring potential energy) whose
magnitude is PEel = = 0.5kx2
•Spring force is a conservative force (Homework –
prove)
e.g. 4
A spring, with a spring constant of 15.4 N.m-1,
has a load of 32.9 g hanging from it, determine
the spring potential energy.
Fs
Given: m 32.9 × 10-3 kg,
k = 15.4 Nm-1
Hooke’s law: Fg = kx
=> x = (mg)/k
But, PEel = (0.5)kx2
=
x
m
Fg