Elasticity and Strength of Materials

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Nov 29, 2013 (3 years and 9 months ago)

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Elasticity and Strength of Materials

The effect of forces on the shape of the body

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

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.

Longitudinal Stretch and Compression

Stress
S

is defined as

A
F
S

The force applied to the bar causes the bar to elongate by an amount
Δ
l
. The fractional
change in length
Δ
l
/
l

is called
the longitudinal strain

S
t
; that is

In 1676 Robert Hooke observed that while the body remains elastic,
the ratio of stress to strain is constant (
Hooke’s law
); that is,

l
l
S
t



Y
S
S
t

Young’s modulus

Robert Hooke

Y
S
S
t

Young’s modulus

A Spring

The force
F

required to stretch (or compress) the spring
is directly proportional to the amount of stretch; that is

l
K
F


K
: the
spring constant

A stretched (or compressed) spring contains potential energy.

2
)
(
2
1
l
K
E


The energy
E

stored in the spring is given by

2
0
0
2
1

)
(

'
)
'
(
kx
dx
kx
dx
x
F
U
x
x








2
2
1
)
(
Kx
x
U

An elastic body under stress is analogous to a spring with a spring constant
YA
/
l
.

Y
l
l
A
F
S
S
t


/
/

l
l
YA
F


l
YA
K

By analogy with the spring, the amount of energy
stored in a stretched or compressed body is

2
)
(
2
1
l
l
YA
E


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.

l
l
YA
A
S
F
B
B



The corresponding force
F
B
that will fracture the bone is,

The compression
Δ
l

at the breaking point is, therefore

Y
l
S
l
B


Y
AlS
l
l
YA
E
B
2
2
2
1
)
(
2
1



Bone Fracture: Energy Considerations

Y
l
S
l
B


Y
AlS
l
l
YA
E
B
2
2
2
1
)
(
2
1



Data used

1.

A

= 6 cm
2


2.

S
B

= 10
9

dyn/cm
2


3.

Y

= 14
×
10
10

dyn/cm
2

J

192.5
erg

10
25
.
19
10
14
10
90
6
2
1
8
10
18







E
Consider the fracture of two leg bones that
have a combined length of about 90 cm and
an average area of about
6 cm
2
.

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
.

Impulsive Forces

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.

Impulsive Forces

t
t
t



1
2
Duration of the collision

Such a short
-
duration force is
called an
impulsive force
.

The average value of the impulsive force
F
av

can be calculated.

t
mv
mv
F
i
f



av
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

N

10
3
.
3
sec
10
6
m/sec

kg
2
2
3
-
av




F
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

gh
v
2

The momentum on impact is

g
h
W
gh
m
mv
2
2


After the impact, the change in momentum is

g
h
W
mv
mv
i
f
2


The average impact force is

gh
t
m
g
h
t
W
F
2
2




Fracture Due to a Fall: Impulsive Force Considerations

gh
t
m
g
h
t
W
F
2

2





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
10
9

dyne/cm
2
.

If the person falls flat on his/her heels, the area of impact may be about
2cm
2
.

The force
F
B

that will cause fracture is

dyn
10

2
dyn/cm

10

cm

2
9
2
9
2




B
F
gh
t
m
g
h
t
W
F
2

2





2
2
1







m
t
F
g
h

For a man of
70 kg

cm

6
.
41
10
70
10
10
2
980
2
1
2
1
3
2
9
2






















m
t
F
g
h

Car Accident

Lamborghini

Air Bag

Airbags

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.

The average deceleration is given by

s
v
a
2
2

where
v

is the initial velocity of the automobile and
s

is the distance over which
the deceleration occurs.

The average force


dyn

10
17
.
1
30
2
10
70
2
2
3
2
3
2
v
v
s
mv
ma
F








At an impact velocity of
70

km/h,
F

= 4.45
×

10
9

dyn
. If this force is uniformly distributed
over a
1000
-
cm
2

area of the passenger’s body, the stress,
S
, is
4.45
×

10
6

dyn/cm
2
. This is
just below the estimated strength of body tissue.

Airbags: Inflating Collision Protection Devices

s
mv
ma
F
2
2


2
~
v
F
If
v

= 105 km/h

stress

2
7
dyn/cm

10
~
S
Such a force would probably injure the passenger.

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

Insect Flight

Insect Wing Muscles

A number of different wing
-
muscle
arrangements occur in insects.


One of a highly simplified arrangement
is found in the dragonfly.

The wing movement is controlled by
many muscles, which are here
represented by muscles
A

and
B
.

The upward movement of the wings is produced by the
contraction of muscle
A
, which depresses the upper part
of the thorax and causes the attached wings to move up.


While muscle
A

contracts, muscle
B

is relaxed.

Note that the force produced by the muscle
A

is
applied to the wing by means of a
Class 1 lever
.


The fulcrum here is the wing joint marked by the
small circle in the figure.

Wing Joint

Upward
Movement

Insect Wing Muscles

Wing Joint

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
.

Downward
Movement

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
.

This is made possible by
the lever arrangement

of the wings.

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.

The wings of most insects
are designed so that during
the upward stroke the force
on the wings is small.

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
Δ
t

of the
upward wing
-
beat

, the
insect drops a distance

h
under the action of gravity.

2
)
(
2
t
g
h


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
).

sec

10
5
.
4
cm/sec

980
cm

10
2
2
3
2
2
2
/
1















g
h
t

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
Δ
t
, that is,

sec

10
9
sec

)
10
5
.
4
(
2
2
-3
3







t
T

The frequency of wing
-
beats
f
, is

1
-
3
sec

110
10
9
1
1





T
f
This is a typical insect wing
-
beat frequency.

Bibliographic

Entry

Result

(w/surrounding

text)

Standardized

Result

Chapman, R. F.
The Insects: Structure and Functions
.
New York: American Elsevier, 1969.

"In the
Apis

and
Musca

the frequency is about
190/second."

190

Hz

"Invertebrates: Insects."

The World Book Encyclopedia
of Science, The Animal World Edition
. Chicago: World
Book, 1987.

"The number of wing beats varies greatly from 4

20 in
butterflies to 190 beats/second in bees and up to 1000
beats/second in a small fly."

190

Hz

Micucci, Charles.
The Life and Times of the Honey Bee
.
United States: Houghton Mifflin, 1995.

"A honey bee has two pairs of wings that can beat 250
times/second."

250

Hz

Romoser, William J.
The Science of Entomology
. New
York: Macmillan, 1973.

"Insect Wing Beats per sec

Apis
: 190, 108
-
23, 250"

190

Hz

108

123

Hz

250

Hz

Smith, Robert H.
Time Life for Children: Understanding
Science and Nature
. United States: Time, 1993.

"The bee's wings are small for its body, but beat 200
times per second letting the bee fly or hover in one
spot."

200

Hz

Frequency of Bee Wings

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
.

Safe Drive and Safe Ride !!