Elasticity and Strength of Materials

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Elasticity and Strength of Materials
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The effect of forces on the shape of the body
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When a force is applied to a body, the shape and
size of the body change.
Depending on how the force is applied, the body
may be stretched, compressed bent or twisted.
Elasticity is the property of a body that tends
to return the body to its original shape after
the force is removed. If the applied force is
sufficiently large, however, the body is
distorted beyond its elastic limit, and the
original shape is not restored after removal of
the force. A still larger force will rupture the
body
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Longitudinal Stretch and Compression
Let us consider the effect of a stretching force
F applied to a bar. The applied force is
transmitted to every part of the body, and it
tends to pull the material apart. This force,
however, is resisted by the cohesive force that
holds the material together. The material breaks
when the applied force exceeds the cohesive
force. If the force is reversed, the bar is
compressed, and its length is reduced. A
sufficiently large force will produce permanent
deformation and then breakage.
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Longitudinal Stretch and Compression
Stress S is defined as
The force applied to the bar causes the bar to
elongate by an amount ?l. The fractional change
in length ?l/l is called the longitudinal strain
St that is
In 1676 Robert Hooke observed that while the body
remains elastic, the ratio of stress to strain is
constant (Hookes law) that is,
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A Spring
The force F required to stretch (or compress) the
spring is directly proportional to the amount of
stretch that is
K the spring constant
A stretched (or compressed) spring contains
potential energy.
The energy E stored in the spring is given by
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An elastic body under stress is analogous to a
spring with a spring constant YA/l.
By analogy with the spring, the amount of energy
stored in a stretched or compressed body is
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Bone Fracture Energy Considerations
Knowledge of the maximum energy that parts of the
body can safely absorb allow us to estimate the
possibility of injury under various circumstances.
We shall first calculate the amount of energy
required to break a bone of area A and length l.
Assume that the bone remains elastic until
fracture.
The compression ?l at the breaking point is,
therefore
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Bone Fracture Energy Considerations
Consider the fracture of two leg bones that have
a combined length of about 90 cm and an average
area of about 6 cm2.
The combined energy in the two legs is twice this
value, or 385 J.
This is the amount of energy in the impact of a
70-kg person jumping from a height 0f 56 cm.
By bending the joints of the body we can jump
from a height larger than 56 cm and without any
injury.
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Impulsive Forces
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Impulsive Forces
In a sudden collision, a large force is exerted
for a short period of time on the colliding
object. The force starts at zero, increases to
some maximum value, and then decreases to zero
again in a very short period of time.
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Impulsive Forces
Such a short-duration force is called an
impulsive force.
The average value of the impulsive force Fav can
be calculated.
For example, if the duration of a collision is 6
10-3 sec and the change in momentum is 2 kg
m/sec, the average force that acted during the
collision is
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Fracture Due to a Fall Impulsive Force
Considerations
Calculation of injured effect using the concept
of impulsive force
When a person falls from a height, his/her
velocity on impact with the ground, neglecting
air friction, is
The momentum on impact is
After the impact, the change in momentum is
The average impact force is
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Fracture Due to a Fall Impulsive Force
Considerations
If the impact surface is hard, such as concrete,
and if the person falls with his/her joints
rigidly locked, the collision time is estimated
to be about 10-2 sec.
The breaking stress that may cause a bone
fracture is 109 dyne/cm2.
If the person falls flat on his/her heels, the
area of impact may be about 2cm2.
The force FB that will cause fracture is
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Car Accident
Lamborghini
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Airbags
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Airbags Inflating Collision Protection Devices
An inflatable bag is located in the dashboard of
the car. In a collision, the bag expands,
suddenly and cushions the impact of the passenger.
The forward motion of the passenger must be
stopped in about 30 cm of motion if contact with
the hard surfaces of the car is to be avoided.
where v is the initial velocity of the automobile
and s is the distance over which the deceleration
occurs.
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Airbags Inflating Collision Protection Devices
Such a force would probably injure the passenger.
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Whiplash Injury
Neck bones are rather delicate and can be
fractured by even a moderate force. Fortunately
the neck muscles are relatively strong and are
strong and are capable of absorbing a
considerable amount of energy.
If, however, the impact is sudden, the body is
accelerated in the forward direction by the back
of the seat, and the unsupported neck is then
suddenly yanked back at full speed. Here the
muscles do not respond fast enough and all the
energy is absorbed by the neck bones, causing the
well-known whiplash injury.
Exercise 5-5
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Insect Flight
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Insect Wing Muscles
A number of different wing-muscle arrangements
occur in insects. One of a highly simplified
arrangement is found in the dragonfly.
Upward Movement
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Insect Wing Muscles
Downward Movement
The downward wing movement is produced by the
contraction of muscle B while muscle A is
relaxed. Here the force is applied to the wings
by means of a Class 3 lever.
The physical characteristics of insect flight
muscles are not peculiar to insects. The amount
of force per unit area of the muscle and the rate
of muscle contraction are similar to the values
measured for human muscles. Yet insect wing
muscles are required to flap the wings at a very
high rate.
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Hovering Flight
During the upward movement of the wings, the
gravitational force causes the insect to
drop. The downward wing movement then produces
an upward force that restores the insect to its
original position. The vertical position of the
insect thus oscillates up and down at the
frequency of the wing-beat. The distance the
insect falls between wing-beats depends on how
rapidly its wings are beating. If the insect
flaps its wings at a slow rate, the time interval
during which the lifting force is zero is longer,
and therefore the insect falls farther than its
wings were beating rapidly.
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Hovering Flight
We want to compute the wing-beat frequency
necessary for the insect to maintain a given
stability in its amplitude.
Assuming that the lifting force is at a finite
constant value while the wings are moving down
and that it is zero while the wings are moving up.
During the time interval ?t of the upward
wing-beat , the insect drops a distance h under
the action of gravity.
The downward stroke then restores the insect to
its original position. Typically, it may be
required that the vertical position of the insect
change by no more than 0.1 mm (i.e. h 0.1 mm).
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Since the up movements and the down movements of
the wings are about equal in duration, the period
T for a complete up-and-down wing movement is
twice ?t, that is,
The frequency of wing-beats f, is
Frequency of Bee Wings
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Elasticity of Wings
As the wings are accelerated, they gain kinetic
energy, which is provided by the muscles. When
the wings are decelerated toward the end of the
stroke, this energy must be dissipated. During
the down stroke, the kinetic energy is dissipated
by the muscles and is converted into heat.
Some insects are able to utilize the kinetic
energy in the upward movement of the wings to aid
in their flight and this has to do with a kind of
rubberlike protein called resilin.
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