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 asA

FS

The force applied to the bar causes the bar to elongate by an amount Δl. The fractional change in length Δl/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 (Hooke’s law); that is,

l

lSt

YS

S

t

Young’s modulus

Robert Hooke

YS

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

lKF K: the spring constant

A stretched (or compressed) spring contains potential energy.

2)(2

1lKE

The energy E stored in the spring is given by

2

0

0

2

1

)(

')'(

kx

dxkx

dxxFU

x

x

2

2

1)( KxxU

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

Yll

AF

S

S

t

/

/

ll

YAF

l

YAK

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

2)(2

1l

l

YAE

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.

ll

YAASF BB The corresponding force FB

that will fracture the bone is,

The compression Δl at the breaking point is, therefore

Y

lSl B

Y

AlSl

l

YAE B

22

2

1)(

2

1

Bone Fracture: Energy Considerations

Y

lSl B

Y

AlSl

l

YAE B

22

2

1)(

2

1

Data used

1. A = 6 cm2

2. SB = 109 dyn/cm2

3. Y = 14 ×1010 dyn/cm2

J 192.5erg 1025.191014

10906

2

1 810

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 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.

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

ttt 12 Duration of the collision

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

The average value of the impulsive force Fav can be calculated.

t

mvmvF if

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 103.3sec106

m/sec kg2 23-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

ghv 2

The momentum on impact isg

hWghmmv

22

After the impact, the change in momentum is g

hWmvmv if

2

The average impact force is ght

m

g

h

t

WF 2

2

Fracture Due to a Fall: Impulsive Force Considerations

ght

m

g

h

t

WF 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 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 dyn10 2dyn/cm 10 cm 2 9292 BF

ght

m

g

h

t

WF 2

2

2

2

1

m

tF

gh

For a man of 70 kg cm 6.411070

10102

9802

1

2

13

292

m

tF

gh

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 bys

va

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 1017.1302

1070

223

232

vv

s

mvmaF

At an impact velocity of 70 km/h, F = 4.45 × 109 dyn. If this force is uniformly distributed over a 1000-cm2 area of the passenger’s body, the stress, S, is 4.45 × 106 dyn/cm2. This is just below the estimated strength of body tissue.

Airbags: Inflating Collision Protection Devices

s

mvmaF

2

2

2~ vF

If v = 105 km/h stress 27 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 Δt of the upward wing-beat , the insect drops a distance h under the action of gravity.

2

)( 2tgh

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 105.4cm/sec 980

cm 1022 32

22/1

g

ht

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 Δt, that is,

sec 109sec )105.4(22 -33 tT

The frequency of wing-beats f, is 1-

3sec 110

109

11

T

f

This is a typical insect wing-beat frequency.

Bibliographic EntryResult(w/surrounding text)

StandardizedResult

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 secApis: 190, 108-23, 250"

190 Hz108–123 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 !!