MECHANICAL PROPERTIES OF SOLIDS AND ACOUSTICS

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1


MECHANICAL PROPERTIES OF
SOLIDS AND ACOUSTICS



1.1

Elasticity and Plasticity

When the shape or size of a body has been altered by the application of a force or a system
of forces, there is usually some tendency for the body to recover its original shape or size on the
removal of the force. This property of the body by virtue of wh
ich it tends to regain its original
shape or size on the removal of deforming force is called
elasticity.

The property of the body by virtue of which it tends to retain the altered size and shape on
removal of deforming forces is called
plasticity
.

1.2

Stress and Strain

Stress is a quantity that characterizes the strength of the forces causing the deformation, on
a “
force per unit area
” basis. The deforming force per unite area of the body is called
stress
.

The
SI unit of stress is the Pascal (abbreviated Pa, and named for the 17
th

century French scientist and
philosopher Blaise Pascal). One Pascal equals one Newton per square meter.1 Pascal = 1Pa =
1N/m
2
. Strain is a quantity which describes the resulting d
eformation.
Strain

is the fractional
deformation produced in a body when it is subjected to a set of deforming forces. Strain being
ratio has no units.

There are following three types of stress and strain

(i)

Tensile and compressive stress and strain

(ii)

Bulk
stress and strain

(iii)

Shear stress and strain

1.3

Hooke’s Law

This law was proposed by Robert Hooke, the founder of Royal society, in 1676. Hooke’s
law states that within the elastic limit, the stress developed is directly proportional to the strain.
The con
stant of proportionality is the elastic modulus (or modulus of elasticity).

=
elastic modulus
(Hooke’s law)



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Fig. 1.1

Stress
-

Strain diagram

If we plot a graph between stress and strain we
get a curve as shown in Fig. 1.1

and it is
called
stress
-

strain diagram
. It is clear from this graph that Hooke’s law holds good only for the
straight line portion of the curve.

1.4


Elastic Moduli

The coefficient of elasticity or modulus of elasticity indicates how a specimen behaves
when subjected to given stress. This
has the same units as stress that is Nm
-
2

or Pa. There are three
kinds of elastic moduli as given in Table 1.1.

Table 1.1 Three kinds of elastic moduli

Elastic Modulus

Definition

Nature of strain

Young’s modulus (Y)

Tensile stress

Tensile strain

Change
of shape and size

Bulk modulus (B)

Bulk stress

Bulk strain

Change of size but not shape

Shear modulus or
Rigidity modulus (S)

Shear stress

Shear strain

Change of shape but not size



Worked Example 1.1:

A steel rod 2.0m long has a cross sectional
area of 0.30cm
2
. The rod is
now hung by one end from a support structure and a 550kg milling
machine is hung from the rod’s lower end. The Young’s modulus of
steel is 20 ×10
10
Pa. Determine the stress, the strain and the elongation
of the rod.








Elongation =

= (strain) ×
o

= (9.0×10
-
4
) (2.0)





= 0.0018m =
1.8mm.

Strain

Plastic range



0

Stress


Elastic limit

Elastic range

Permanent set

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Disc

Torsionally flexible
elastic wire

Fixed End

1.5

Torsion Pendulum

Definition

A torsion pendulum is an oscillator for which the restoring force is torsion.

Description

The device as shown in Fig.1.2

consisting of a disc or other body of large moment of
inertia mounted on one end of a torsionally flexible elastic rod wire whose oth
er end is held fixed;
if the disc is twisted and released, it will undergo simple harmonic motion, provided the torque in
the rod is proportional to the angle of twist
.

Theory

When the disc is rotated in a horizontal plane so as to twist the wire, the var
ious elements
of the wire undergo shearing strains. Restoring couples, which tend to restore the unstrained
conditions, are called into action. Now when the disc is released, it starts executing torsional
vibrations.

If the angle of twist at the lower end of the wire is θ, then the restoring couple is C θ,
where C is the torsional rigidity of the wire, this couple acting on the disc produces in it an angular
acceleration given by









Fig. 1.
2

Torsion Pendulum

C θ =









(1)

where I is the moment of inertia of the disc about the axis of the wire. The minus sign
indicates that the couple C θ tends to decrease the twist. Equation (1) can be rewritten as










(2)

The above relation shows that the angular acceleration is proportional to the angular
displacement θ and is always directed towards the mean position. Hence the motion of the disc is
simple harmonic motion and the t
ime period of the vibration will be given by


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T = 2π


or T = 2 π


Uses of Torsion Pendulum

(1)

For determining the moment of inertia of an irregular body

For determining the
moment of inertia of an irregular body the torsion pendulum is found
to be very useful. First, the time period of pendulum is determined when it is empty and then the
time period of the pendulum is determined after placing a regular body on the disc and af
ter this
the time period is determined by replacing the regular body by the irregular body whose moment
of inertia is to be determined. It is ensured that the body is placed on the disc such that the axes of
the wire pass through the centre of gravity of
the body placed on the disc.

If
I
,
I
1

and
I
2

are the moments of inertia of the disc, regular body and irregular body and T,
T
1

and T
2

are the time periods in the three cases respectively, then

T = 2 π









(3)


T
1

= 2 π







(4)



T
2
=

2 π









(5)

From relations (3) and (4), we have

T
1
2


T
2

=








(6)

and from relations (3) and (5), we have

T
2
2


T
2

=







(7)





(8)

or








(9)

The moment of inertia of the regular body I
1

is determined with the help of the dimensions
of the body, thus the moment of inertia of the irregular body is calculated
.

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(2)

Determination of Torsional Rigidity

For determining the modulus of rigidity N the time period of the pendulum is found (
i
)
when

the disc is empty, and (
ii
) when a regular body is placed on the disc with axis of wire
passing through the centre of gravity of the body. If T is the time period of the pendulum in first
case and T
1

in the second case, then we have

T = 2 π







(10)


and

T
1
= 2 π








(11)

where I

is the moment of inertia of the disc and I
1

the moment of inertia of the regular body placed
on the disc. From relations (10) and (11), we have

T
1
2


T
2

=






(12)


or










(13)

For a wire of modulus of rigidity N, length
l
and radius
r,
we have









(14)

Equating (13) and (14), we have








(15)

or








(16)

Thus, the value of N can be determined.

Worked Example 1.2
:

A torsion pendulum is made using a steel wire of diameter 0.5mm and
sphere of diameter 3cm. The rigidity modulus of steel is 80 GPa and
density of the material of the sphere is 11300 kg/m
3
. If
the period of
oscillation is 2 second, find the length of the wire.


For sphere, I = 2/5 MR
2

M = volume

density

M = 4/3π (3/2

10
-
2
)
3


11300 = 0.1598 kg


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I = 2/5

0.1598

(3/2
10
-
2

)
2

= 0.14382 × 10
-
4

kgm
2

.

1.
6

Bending of Beams

A beam is a rod or bar of uniform cross
-
section
(circular or rectangular) whose length is
very much greater than it
s thickness as shown in Fig. 1.3
.

The beam is considered to be made up of a large number of thin plane layers called
surfaces placed one above the other. Consider a beam to be bent into a
n arc of a circle by the
application of an e
xternal couple as shown Fig. 1.4
. Taking the longitudinal section ABCD of the
bent beam the layers in the upper half are elongated while those in the lower half are compressed.





Fig. 1.3

A beam

In the middle there is a layer (MN) which is not elongated or compressed due to bending
of the beam. This layer is called the

neutral surface’

and the line (MN) at which the neutral layer
intersects the plane of bending is called the

neutral axis’
.









Fig.1.4

Bending of a beam

It is obvious that the length of the filament increases or decreases in proportion to its
distance away from the ne
utral axis MN.

The layers below MN are compressed and those above MN are elongated and there will be
such pairs of layers one above MN and one below MN experiencing same forces of elongation and
compression due to bending and each pair forms a couple.

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The
resultant of the moments of all these internal couples are called the
internal bending
moment

and in the equilibrium condition, this is equal to the external bending moment.

1.6.1

Bending Moment of a Beam

Consider the section
PBCP


(Fig. 1.5
), the
extended filaments lying above the neutral axis
MN are in state of tension and exert an inward pull on the filament adjacent to them towards the
fixed end of the beam. In the same way the shortened filaments lying below the neutral axis MN

are in a state
of compression and exert an outward push on the filaments adjacent to them towards
the loaded end of the beam. As a result tensile and compressive stresses develop in the upper and
lower halves of the beam respectively and form a couple which opposes to b
ending of the beam.
The moment of this couple is called the
moment of the resistance
. When the beam is in
equilibrium position the bending moment and restoring moment or moment of resistance should be
equal.

To find an expression for the moment of the re
storing couple consider a fiber A

B


at a
distance r from the neutral

axis MN as shown in Fig.1.6
. Let the radius of curvature be
R

of the
part
PB
and ф be the angle subtended by it at the centre of curvature. In unstrained position of the
beam, the
length of the fiber A

B


= MN =
R
ф. In the strained position the length of the fibre

A

B


= (
R + r
) ф.








Fig. 1.5

Calculation of bending moment of a beam










Fig.1.6

Strained position

Strain in the fiber A
1
B
1
, =



M

N


C

B

P


D

A

P

Load

r

N

A


B


R

M

ф


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or







(1)


i.e., strain is proportional to the distance from the neutral axis.

Let the area of the fiber be a
and its neutral axis be at a distance r from neutral axis of the
beam and the strain produced be r/R. We have

Stress =
Y

× Strain =
Y r / R







(2)

where
Y

is the
Y
oung’s modulus of the material

Hence, force on the area
a

F =
Y
(
r/R
) ×
a








(3)

Th
erefore the moment of this force about
MN


=
Y
(
r
/
R
) × a ×
r

=
Y a

r
2

/
R






(4)

As the moment of the forces acting on both the upper and lower halves of the section are
in the same direction, the total moment of the forces acting on the filaments due to
straining








(5)

where
I
g

is the geometrical moment of inertia and is equal to
AK
2
, A being the total area of the
section and
K

being the radius of gyration of the beam

:.
moment of the forces






(6)

In equilibrium bending moment of the beam is equal and opposite to the moment of
bending couple due to the load on one end.

:.
Bending moment of the beam =






(7)

The quantity
YI
g
(=
Y A K
2
) is called the flexural rigidity

of the beam.
Flexural rigidity

is
defined as the bending moment required to produce a unit radius of curvature
.

1
.6.2

Uniform Bending

The beam is loaded uniformly on its both ends, the bent beam forms an arc of a circle.
The elevation in the beam is produced. This bending is called
uniform bending
.

Consider a beam (or bar) AB arranged horizontally on two knife


edges C and D
symmetr
ically so that AC = BD = a as shown in Fig. 1.
7





Fig. 1.7

Uniform Bending

The beam is loaded with equal weights W and W at the ends A and B.

Mechanical Properties of Solids and Acoustics

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D

F

C

o

E

y

R

l
/2

F

The reactions on the knife edges at C and D are equal to W and W acting vertical upwards.

The external bending m
oment on the part AF of the beam is

= W × AF


W × CF = W (AF


CF)

= W × AC = W × a







(1)

Internal bending moment =






(2)



where

Y

-


Young’s modulus of the material of the bar

I
g


-


Geometrical moment of
inertia of the cross
-
section of beam

R

-


Radius of curvature of the bar at F

In the equilibrium position,

external bending moment = internal bending moment









(3)

Since for a given value of W, the value
s of a,
Y
and
I
g

are constants,
R

is constant so that
the beam is bent uniformly into an arc of a circle of radius
R
.

CD =
l
and y

is the elevation of the midpoint E of the beam so that y

= EF

Then from the property of
the circle as shown in Fig. 1.8








Fig. 1.8

Circle Property

EF (2R


EF) = (CE)
2







(4)


y

(2R


y) =







(5)

y

2R =

(s
ince
y
2
is negligible)





(6)


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y

=









(7)


or








(8)

From (3) and (8), Wa =

or








(9)

If the beam is of rectangular cross
-
section,
, where b is the breath and d is the
thickness of beam.

If M is the mass, the corresponding weight W = Mg

Hence








(10)

from which Y the Young’s modulus of the material of the bar is determined.


Worked Example
1.3
:

Uniform

rectangular bar 1 m long 2 cm broad and 0.5 cm thick is
supported on its flat face symmetrically on two knife edges 70 cm apart.
If loads of 200 g are hung from the two ends, find the elevation at the
center of the bar. Young’s modulus of the material o
f the bar is 18
10
10

Pa.

The distance between the nearer knife edge and the point of suspension

a=15×10
-
2

m



Elevation at the centre,






=
4.802 × 10
-
4

m

1.6.3

Non
-
Uniform Bending

If the beam is loaded at its mid
-
point, the depression i produced will not form an arc of a
circle. This type of bending is called non
-
uniform bending.

Consider a uniform beam (or rod or bar) AB of length
l
arranged horizontally on t
wo knife
edges K
1

and K
2

near the en
ds A and B as shown in Fig. 1.9
.


Mechanical Properties of Solids and Acoustics

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Fig. 1.9

Non
-
uniform bending

A weight W is applied at the midpoint E of the beam. The reaction at each knife edge is
equal

to W/2 in the upward direction and ‘y’ is the depression at the midpoint E.

The bent beam is considered to be equivalent to two single inverted cantilevers, fixed at E
each of length

and each loaded at K
1

and K
2

with a weight

In the case of a cantilever of length
l
and load W,

the depression =

Hence, for cantilever of length

and load
, the depression is

y =








(1)

or








(2)

If M is the mass, the corresponding weight W is

W = Mg








(3)

If the beam is a rectangular, I
g

=
, where b is the breadth and d is the thickness of t
he
beam.

Hence








(4)









(5)

or



Nm
-
2






(6)

The value of young’s modulus, Y can be determined by the above equation
.

E

K
2

K
1

B

A

W/2

W/2

W


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1.7

Stress
-
Strain Relation for Different
Engineering Materials

The stress and strain relation can be studied by drawing a graph or curve by taking strain
along the x axis and the corresponding stress along the y axis. This curve is called stress
-

strain
curve. The stress
-
strain relations for
different engineering materials are discussed below.

For ferrous metal

Fig.1.10

shows the stress
-
strain diagram for different types of steel and wrought iron. The
strength of the ferrous metals depends up on carbon content, but at the cost of its ductilit
y, as it is
clearly understood from the figure. The proportion of carbon does not have an appreciable effect
on young’s modulus of elasticity during any hardening process.












Fig. 1.10
. Stress
-

Strain curve for ferrous metals


For non
-
ferrous metal

For hard steels and non
-
ferrous metals stress is specified corresponding to a definite
amount of permanent elongation. This stress is known as proof stress. For aircraft materials the
stress corresponding to 0.1% of strain is the proof stress. The proof st
ress is applied for 15 seconds
and when removed, the specimen should not lengthen permanently beyond 0.1%.










Fig.1.11
. Stress Strain curve for non
-

ferrous metals

Alloy steel or tool steel

Mild steel (Ductile)

High carbon steel

Medium carbon steel

Wrought iron (Most ductile)

Cast iron (Brittle iron)

Strain

Stress

Aluminium bronze



Magnesium
oxide

Brass 70:30

Annealed copper

Strain

Stress

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Fig.1.11

shows stress
-
strain curves for non
-
ferrous materials. The elastic properties of
non
-
ferrous metals vary to a considerable extent, depending upon the method of working and their
compositions in the case of alloys. From the figure it
is clear that the early portion of the stress
-
strain diagram for most of the metals is never quite straight line, but the yield point is well define.

Brittle materials show little or no permanent deformation prior to fracture. Brittle behavior
is exhibited

by some metals and ceramics like magnesium oxide .The small elongation prior to
fracture means that the materials gives no indication of impending fracture and brittle fracture
usually occurs rapidly. It is often accompanied by loud noise.

Saline Features

of stress
-
strain relation



The properties of ductile metals can be explained with the help of stress
-
strain curves.



Higher yield point will represents greater hardness of the metals.



A higher value of maximum stress point will represent a stronger metal.



T
he distance from the ordinates of the load point (or) breaking stress will indicate the
toughness and brittleness of the metal. The shorter the distance then the metal is more
brittle.

1.8

Ductile and Brittle Materials

1.8.1

Ductile materials

A body is said to

have yielded or to have undergone plastic deformation if it does not
regains its original shape when a load is removed. The resulting deformation is called permanent
set. If permanent set is obtainable, the material is said to exhibit ductility. Ductility

measures the
degree of plastic deformation sustained it fracture. One way of specify a material is by the
percentage of elongation (%EL).

Percentage of elongation =


Where L
f

is the length of the specimen at fracture


L
o

is the lengt
h of the specimen without load.

A ductile material is one with a large Percentage of elongation before failure. The original
length of the specimen L
o
is an important value because a significant portion of the plastic
deformation at fracture is confined to

the neck region. Thus, the magnitude of percentage of
elongation will depend on the specimen length.

Table 1.2

Percentage of elongation for ductile materials

Material

Percentage of Elongation

Low
-
Carbon

37%

Medium
-
Carbon

30%

High
-
Carbon

25%


The percentage of elongation of different ductile materials is tabulated above. For ductile
material, the ultimate tensile and compressive strength have approximately the same absolute
value. The steel is ductile material because it far exceeds the 5% elon
gation. High strength alloys,

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such as spring steel, can have 2% of elongation but even this is enough to ensure that the material
yields before it fractures. Hence it is behaved like a ductile material. Gold is relatively ductile at
room temperature. Most
of the material becomes ductile by increasing the temperature.




Properties of ductile materials
:



Easily drawn into wire or hammered thin.



Easily molded or shaped.



Capable of being readily persuaded or influenced tractable.



Easily stretched without breaki
ng in material strength.

Stress


strain behavior of ductile materials

In the case of ductile materials at the beginning of the tensile test, the material extends
elastically. The strain at first increase proportionally to the stress and the specimen retur
ns to its
original length on removal of the stress. The limit of proportionality is the stage up to which the
material obeys Hooke’s law perfectly.

Beyond the elastic limit the applied stress produces plastic deformation so that a
permanent extension remai
ns even after the removal of the applied load. In this stage the resultant
strain begins to increase more quickly than the corresponding stress and continues to increase till
the yield point is reached. At the yield point the material suddenly stretches.

T
he rate of applied load to original cross
-
sectional area is termed the nominal stress. This
continues to increase with elongation, due to strain hardening or work hardening, until the tensile
stress is maximum. This is the value of stress at maximum load a
nd can be calculated by dividing
the maximum load by the original cross
-
sectional area. This stress is called ultimate tensile stress.












Fig 1.12

Stress
-

strain curve for a ductile material.


Fig.1.12

is a stress
-
strain diagram for ductile material (mild steel) showing the limit of
proportionality, elastic limit, yield point, ultimate tensile stress and fracture.

Upper

yield point

Strain

Ultimate stress

Lower yield point

Limit of
proportionality

Elastic limit

Fracture

Stress

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From Fig
.1.12

it is clearly show that at a certain value of load the strain continues at s
low
rate without any further stress. This phenomenon of slow extension increasing with time, at
constant stress is termed creep. At this point a neck begins to develop along the length of the
specimen and further plastic deformation is localized within the

neck. After necking the nominal
stress decreases until the material fractures at the point of minimum cross
-
sectional area.

1.8.2

Brittle Materials

Brittle material is one which is having very low percentage of elongation. Brittle materials
break suddenly under

stress at a point just beyond its elastic limit. A Brittle material exhibits little
or no yielding before failure. Brittle material will have a much lower elongation and area reduction
than ductile ones. The tensile strength of Brittle material is usually

much less than the compressive
strength. The brittle material can be deformed in a ductile only under the conditions of high
pressure.

Ceramic glass and cast iron are having very good brittle nature. Grey cast iron is a best
example for brittle material
whose percentage of elongation is so small. Brittle materials are used
in design of hard ceramic armor, exclusive excavation of rocks, space craft windows, impact of
condensed particle on turbine blades etc.

Determination of Brittle materials



If the percen
tage of elongation is at or below 5%, assume brittle behavior.



If the ultimate compressive strength is greater than the ultimate tensile strength
assume brittle behavior



If no yield strength is occurred suspect brittle behavior

Stress


strain behavior of

brittle materials











Fig. 1.13

Stress


strain curve for a brittle material


Figure 1.13

shows a poorly defined yield point in brittle materials. For the determination
of yield strength in such materials, one has to draw a straight line parallel to the elastic portion of
the stress strain curve at a predetermined strain ordinate value (say 0.
1%). The point at which this
line intersects the stress
-
strain curve is called the yield strength.

Strain

Yield point at off
-
set

Stress

Parallel

Proof stress


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1.9

Some Fundamental Mechanical Properties


The following are the some of the fundamental mechanical properties of metals:

(i) Tensile strength (ii)
Hardness (iii) Impact strength (iv) fatigue and (v) Creep

1.9.1

Tensile Strength

This is the maximum conventional stress that can be sustained by the material. It is the
ultimate strength in tension and corresponds to the maximum load in a tension test.
It is measured
by the highest point on the conventional stress
-
strain curve. In engineering tension tests this
strength provides the basic design information on the materials.

The tensile strength of a material is the maximum amount of tensile stress that
it can be
subjected to before failure. There are three typical definitions of tensile strength.

Yield strength

The stress at which material strain changes from elastic deformation to plastic
deformation, causing it to deform permanently is known as
yield s
trength
.

Ultimate strength


The maximum stress a material can withstand is known as
ultimate strength
.

Breaking strength

The strength co
-
ordinate on the stress
-
strain curve at the point of rupture is known as
breaking strength
.

In ductile materials the loa
d drops after the ultimate load because of necking. This
indicates the beginning of plastic instability. In brittle materials, the ultimate tensile strength is a
logical basis for working stresses. Like yield strength, it is used with a factor of safety.

T
able 1.3

Typical tensile strengths of engineering materials

Material

Tensile Strength kg/mm
2

Alloy steel

60
-
70

Mild Steel

42

Grey CI

19

White CI

47

Aluminum alloy

47


1.9.2

Hardness

Hardness is the resistance of material to permanent deformation of the
surface. However,
the term may also refer to stiffness, temper resistance to scratching and cutting. It is the property of
a metal, which gives it the ability to resist being permanently deformed (bent, broken or shape
change), when a load is applied.

The
hardness of a surface of the material is, of course, a direct result of inter atomic
forces acting on the surface of the material. We must note that hardness is not a fundamental
property of a material, but a combined effect of compressive, elastic and p
lastic properties relative
to the mode of penetration, shape of penetration etc. The main usefulness of hardness is, it has a
constant relationship to the tensile strength of a given material and so can be used as a practical
Mechanical Properties of Solids and Acoustics

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non
-
destructive test for an ap
proximate idea of the value of that property and the state of the metal
near the surface.

Hardness Measurement

Hardness measurement can be in Macro, Micro & nano


scale according to the forces
applied and displacements obtained.

Measurement of the Macro
-
h
ardness of materials is a quick and simple method of
obtaining mechanical property data for the bulk materials from a small sample. It is also widely
used for the quality control of a surface treatments process. The Macro
-
hardness measurement will
be highl
y variable and will not identify individual surface features. It is here that micro
-
hardness
measurements are appropriate.

Micro hardness is the hardness of a material as determined by forcing an indenter into the
surface of the material under load, usua
lly the indentations are so small that they must be measured
with a microscope. Micro hardness measurements are capable of determines the hardness of
different micro constituent with in a structure.

Nano hardness tests measure hardness by using indenter, o
n the order of nano scale. These
tests are based one new technology that allows precise measurement and control of the indenting
forces and precise measurement of the indentation depth.

Hardness Measurement Methods

There are several methods of hardness tes
ting, depending either on the direct thrust of
some form of penetrator into the metal surface, or on the ploughing of the surface as a styles is
drawn across it under a controlled load, or on the measurement of elastic rebound of an impacting
hammer which
possessing known energy. Measurements of hardness are the easiest to make and
are widely used for industrial design and in research. As compared to other mechanical tests,
where the bulk of the material is involved in testing, all hardness tests are made o
n the surface or
close to it.

The following are the most common hardness test methods used in today’s technology.

1.

Rockwell hardness test

2.

Brinell hardness

3.

Vickers

4.

Knoop hardness

5.

S
hore


Brinell, Rockwell and Vickers hardness tests are used to determine hardn
ess of metallic
materials to check quality level of products, for uniformity of sample of metals, for uniformity of
results of heat treatment. The relative micro hardness of a material is determined by the knoop
indentation test. The shore scleroscope meas
ures hardness in terms of the elasticity of the material.

Bri
nell hardness number is the hardness index calculated by pressing a hardened steel ball
(indenter) into test specimen under standard load. The rock well hardness is another index which
widely
used by engineers. This index number is measured by the depth of penetration by a small
indenter. By selecting different loads and shapes of indenter, different Rockwell scales have been
developed. The value of Brinell hardness number is related to tensile

streng
th, which is as shown
in Fig.1.14
.


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.
18












Fig
.1.14

Tensile strength verses Brinell hardness curves


The mechanism of indentation in all indentation tests is that when the indenter is pressed
into the surface under a static load, a large amount
of plastic deformation takes place. The
materials thus deformed flows out in all directions. As a result of plastic flow, sometimes the
material in contact with the indenter produces a ridge around the impression. Large amount of
plastic deformation are ac
companied by large amount of transient creep which vary with the
material and time of testing. Transient creep takes place rapidly at first and more slowly as it
approaches its maximum. For harder materials, the time required for reaching maximum
deformati
on is short (few seconds) and for soft materials the time required to produce the derived
indentation is unreasonably long up to a few minutes.

Hardness of materials is of importance for dies and punches, limit gauges, cutting tools
bearing surfaces etc. S
oftness of a material is opposite extreme of hardness. On heating all
materials become soft.

1.9.3

Impact Strength

Impact strength is the resistance of a material to fracture under dynamic load. Thus, it is a
complex characteristic which takes into account

both the toughness and strength of a material. In
S.I. units the impact strength is expressed in Mega Newton per m
2

(MN/m
2
). It is defined as the
specific work required to fracture a test specimen with a stress concentrator in the mid when
broken by a sin
gle blow of striker in pendulum type impact testing machine.

Impact strength is the ability of the material to absorb energy during plastic deformation.
Obviously brittleness of a material is an inverse function of its impact strength. Course grain
structu
res and precipitation of brittle layers at the grain boundaries do not appreciably change the
mechanical properties in static tension, but substantially reduce the impact strength.

Impact strength is affected by the rate of loading, temperature and
presence of stress
raisers in the materials. It is also affected by variation in heat treatment, alloy content, sulphur and
phosphorus content of the material.

Impact strength is determined by using the notch
-
bar impact tests on a pendulum type
impact test
ing machine. This further helps to study the effect of stress concentration and high
velocity load application.

Mechanical Properties of Solids and Acoustics

1
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19

Factors affecting Impact strength



If the dimensions of the specimen are increased, the impact strength also increases.



When the sharpness of the

notch increase, the impact strength required causing failure
decreases.



The temperature of the specimen under test gives an indication about the type of
fractures like ductile, brittle or ductile to brittle transition.



The angle of the notch also improves

impact
-
strength after certain values.



The velocity of impact also affects impact strength to some extent
.

1.9.4

Fatigue

Fatigue is caused by repeated application of stress to the metal. It is the failure of a
material by fracture when subjected to a
cyclic stress. Fatigue is distinguished by three main
features.


i)

Loss of strength


ii)

Loss of ductility


iii)

Increased uncertainty in strength and service life

Fatigue is an important form of behaviour in all materials including metals, plastics,
rub
ber and concrete. All rotating machine parts are subjected to alternating stresses; aircraft wings
are subjected to repeated loads, oil and gas pipes are often subjected to static loads but the dynamic
effect of temperature variation will cause fatigue.

There are many other situations where fatigue
failure will be very harmful. Because of the difficulty of recognizing fatigue conditions, fatigue
failure comprises a large percentage of the failures occurring in engineering. To avoid stress
concentration
s, rough surfaces and tensile residual stresses, fatigue specimens must be carefully
prepared.

The S
-
N Curve

A very useful way to visual the failure for a specific material is with the S
-
N curve. The
“S
-
N” means stress verse cycles to failure, which when
plotted using the stress amplitude on the
vertical axis and the number of cycle to failure on the horizontal axis. An important characteristic

to this plot as seen in Fig.1.15

is the “
fatigue limit
”.


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.
20














10
4

10
5

10
6

10
7

10
8

10
9


Cycles

Fig.1.15

S
-
N curve for a metal


The point at which the curve flatters out is termed as fatigue limit and is well below the
normal
yield stress. The significance of the fatigue limit is that if the material is loaded below this
stress, then it will not fail, regardless of the number of times it is loaded. Materials such as
aluminium, copper and magnesium do not show a fatigue limit;
therefore they will fail at any
stress and number of cycles. Other important terms are fatigue strength and fatigue life. The
fatigue strength can be defined as the stress that produces failure in a given number of cycles
usually 10
7
. The fatigue life c
an be defined as the number of cycles required for a material to fail
at a certain stress.

1.9.5
Creep


The creep is defined as the property of a material by virtue of which it deforms
continuously under a steady load. Creep is the slow plastic
deformation of materials under the
application of a constant load even for stressed below the yield strength of the material. Usually
creep occurs at high temperatures. Creep is an important property for designing I.C. engines, jet
engines, boilers and t
urbines. Iron, nickel, copper and their alloys exhibited this property at
elevated temperature. But zin, tin, lead and their alloys shows creep at room temperature. In
metals creep is a plastic deformation caused by slip occurring along crystallographic d
irections in
the individual crystals together with some deformation of the grain boundary materials.


Fig.1.16

Creep curve at constant temperature and stress



6

10

14

16

22

18

26

30

34

38

Fatigue strength

Stress

Mechanical Properties of Solids and Acoustics

1
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21

Fig.1.16

shows a typical creep curve. The creep curve usually consists of three points
correspon
ding to particular stages of creep.

(i)

Primary Stage
: In this stage the creep rate decreases with time, the effect of work
hardening is more than that of recovery processes. The primary stage is of great
interest to the designer since it forms an early part
of the total extension reached in a
given time and may affect clearness provided between components of a machine.

(ii)

Secondary Stage
: In this stage, the creep rate is a minimum and is constant with
time. The work hardening and recovery processes are exactly

balanced. It is the
important property of the curve which is used to estimate the service life of the
alloy.

(iii)

Tertiary Stage
: In this stage, the creep rate increases with time until fracture
occurs. Tertiary creep can occur due to necking of the specime
n and other processes
that ultimately result in failure.

The temperature and time dependence of creep deformation indicates that it is a thermally
activated process. Several atomic processes are known to be responsible for creep in crystalline
materials.

T
he yield strength which is determined in short term tests cannot be the criterion of high
temperature strength. Hence it does not consider the behaviour of a material in long
-
term loading.
The actual criteria of high temperature strength are the creep limi
t and long term strength. The
“Creep Limit” is the stress at which a material can be formed by a definite magnitude during a
given time at a given temperature. The calculation of creep limit includes the temperature, the
deformation and the time in which
this deformation appears.

Types of Creep


The creep are classified into three different categories based on the temperature

(i)

Logarithmic Creep

(ii)

Recovery Creep

(iii)

Diffusion Creep

At low temperature the creep rate decreases with time and the logarithmic creep cur
ve is
obtained. At high temperature, the influence of work hardening is weakened and there is a
possibility of mechanical recovery. As a result, the creep rate does not decrease and the recovery
creep curve is obtained. At very high temperature, the cree
p is primarily influenced by diffusion
and load applied has little effect. This creep is termed as diffusion creep or plastic creep.

1.10

Fracture

Fracture is the separation of a specimen into two or more parts by an applied stress.
Fracture is caused by
physical and chemical forces and takes place in two stages: (i) initial
formation of crack and (ii) spreading of crack. Depend upon the type of materials, the applied load,
state of stress and temperature metals have different types of fracture.

There are
four Main types of fracture

i)

Brittle Fracture

ii)

Ductile Fracture


Physics for Technologists

1
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22

iii)

Fatigue Fracture

iv)

Creep Fracture

Fracture is usually undesirable in engineering applications. We may note that flaws such
as surface cracks lower the stress for brittle fracture where as line
defects are responsible for
initiating ductile fractures. Different types of

fracture are shown in Fig
.1.17
.













Fig. 1.17

Different types of fractures



1.10.1

Brittle Fracture

Brittle fracture is the failure of a material with minimum of
plastic deformation. If the
broken pieces of a brittle fracture are fitted together, the original shape & dimensions of the
specimen are restored.

Brittle fracture is defined as fracture which occurs at or below the elastic limit of a
material. The brittle

fracture increases with

(i)

Increasing strain rate

(ii)

Decreasing temperature

(iii)

Stress concentration conditions produced by a notch.

Salient Features of Brittle Fracture

(1)

Brittle fracture occurs when a small crackle in materials grows. Growth continues
until fracture occurs.

(2)

The atoms at the surfaces do not have as many neighbors as those in the interior of a
solid and therefore they form fever bonds. That implies, surface

atoms are at a
higher energy than a plane of interior atom. As a result of Brittle fracture destroying
the inter atomic bonds by normal stresses.

(3)

In metals brittle fracture is characterized by rate of crack propagation with minimum
energy of absorption.

Mechanical Properties of Solids and Acoustics

1
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23

(4)

I
n brittle fracture, adjacent parts of the metal are separated by stresses normal to the
fracture surface.

(5)

Brittle fracture occurs along characteristics crystallographic planes called as
cleavage planes. The fracture is termed as cleavage fracture.

(6)

Brittle
fracture does not produce plastic deformation, so that it requires less energy
than a ductile failure.

Mechanism of Brittle Fracture


The mechanism of Brittle fracture is explained by Griffith theory.
Griffith postulated that
in a brittle material there
are always presence of micro cracks which act to concentrated the stress
at their tips. The crack could come from a number of source, e.g. as a collection of dislocations, as
flow occurred during solidification or a surface scratch.

In order to explain th
e mechanism of ideal brittle fracture, let us consider the stress
distribution in a specimen under constant velocity in the vicinity of crack. When a longitudinal
tensile stress is applied, the crack tends to increase its length causes an increase in surf
ace area of a
crack. As a result, the surface energy of the specimen is also increased. Moreover, there is also
compensation release of energy. This means, an increase in crack length causes the release of
elastic energy “Griffith state that when the el
astic energy released by extending a crack equal to
the surface energy required for crack extension” then the crack will grow.



=









(1)

w
here, e is half of the crack length,


is the true surface energy and E is the Young's mo
dulus.

Equation (1) gives the stress necessary to cause the brittle fracture and the stress is
inversely proportional to the square root of the crack length. Hence the tensile strength of a
completely brittle material is determined by the length of

the largest crack existing before loading.
The relation (1) is known as the Griffith’s equation.

For ductile materials there is always some plastic deformation before fracture. This
involves an additional energy term

p
. Therefore the fracture strength
is given by





=








(2)

Generally

p

>>


for metals.

From the above formula, one can get the size of largest flaw or crack.


1.10.2
Ductile Fracture

Ductile fracture is defined as the fracture which takes place by a slow
propagation of crack
with considerable amount of plastic deformation.

There are three successive events involved in a ductile fracture.



The specimen begins necking and minute cavities form in the necked region. This is
the region in which the plastic defor
mation is concentrated. It indicates that the
formation of cavities is closely linked to plastic deformation.


Physics for Technologists

1
.
24



It has been observed that during the formation of neck small micro cracks are
formed at the centre of the specimen due to the combination of dislo
cations.



Finally these cracks grow out ward to the surface of the specimen in a direction 45°
to the tensile axis resulting in a cup
-
end
-
cone
-
type fracture.











Fig.1.18
V
arious stages in ductile fracture

Fig.1.18
shows the various stages in ductile
fracture. Ductile fracture has been studied
much less extensively than brittle fracture, as it is considered to be a much less serious problem.
An important characteristic of ductile fracture is that it occurs through a slow tearing of the metal
with the e
xpenditure of considerable energy.

The fracture of ductile materials can also explained in terms of work
-
hardening coupled
with crack
-
nucleation and growth. The initial cavities are often observed to form at foreign
inclusions where gliding dislocations ca
n pile up and produce sufficient stress to form a void or
micro
-
crack. Consider a specimen subjected to slow increasing tensile load. When the elastic limit
is exceeded, the material beings to work harden. Increasing the load, increasing the permanent
elon
gation and simultaneously decrease the cross sectional area. The decrease in area leads to the
formation of a neck in the specimen, as illustrated earlier. The neck region has a high dislocation
density and the material is subjected to a complex stress. Th
e dislocations are separated from each
other because of the repulsive inter atomic forces. As the resolved shear stress on the slip plane
increase, the dislocation comes closed together. The crack forms due to high shear stress and the
presence of low angl
e grain boundaries. Once a crack is formed, it can grow or elongated by
means of dislocations which slip. Crack propagation is along the slip plane for this mechanism.
Once crack grows at the expense of others and finally cracks growth results in failure.


Mechanical Properties of Solids and Acoustics

1
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25

Table 1.4

Comparison between Brittle and Ductile fracture

Ductile fracture

Brittle fracture



Material fractures after plastic
deformation and slow propagation
of crack



Material fractures with very little or no
plastic deformation.



Surface obtained at
the fracture is
dull or fibrous in appearance



Surface obtained at the fracture is
shining and crystalling appearance



It occurs when the material is in
plastic condition.



It occurs when the material is in elastic
condition.



It is characterized by the
formation
of cup and cone



It is characterized by separation of
normal to tensile stress.



The tendency of ductile fracture is
increased by dislocations and other
defects in metals.



The tendency brittle fracture is
increased by decreasing temperature,
and i
ncreasing strain rate.



There is reduction in cross


sectional area of the specimen



There is no change in the cross


sectional area.


1.10.3

Fatigue Fracture

Fatigue fracture is defined as the fracture which takes place under repeatedly applied
stresses. It will occur at stresses well before the tensile strength of the materials. The tendency of
fatigue fracture increases with the increase in temperature and higher rate of straining.

The fatigue fracture takes place due to the micro cracks at the

surface of the materials. It
results in, to and fro motion of dislocations near the surface. The micro cracks act as the points of
stress concentration. For every cycle of stress application the excessive stress helps to propagate
the crack. In ductile ma
terials, the crack grows slowly and the fracture takes place rapidly. But in
brittle materials, the crack grows to a critical size and propagates rapidly through the material.

1.10.4

Creep Fracture

Creep fracture is defined as the fracture which takes plac
e due to creeping of materials
under steady loading. It occurs in metals like iron, copper & nickel at high temperatures. The
tendency of creep fracture increases with the increase in temperature and higher rate of straining.

The creep fracture takes place

due to shearing of grain boundary at moderate stresses and
temperatures and movement of dislocation from one slip to another at higher stresses and
temperatures. The movement of whole grains relation of each other causes cracks along the grain
boundaries,

which act as point of high stress concentration. When one crack becomes larger it
spreads slowly across the member until fracture takes place. This type of fracture usually occurs
when small stresses are applied for a longer period. The creep fracture is
affected by grain size,
strain hardening, heat treatment and alloying.



Physics for Technologists

1
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26


Worked Example 1.4:

A Young’s modulus of a certain material is 180
×

10
3

mega Newton/ m
2

and its true surface energy is 1.8 J/m
2
. The crack length is 5 µm.
Calculate the fracture
strength.

The fracture strength is




=





=




=

278 × 10
6

Newton /m
2


1.11


A
coustics of Buildings

Introduction

Acoustics

is the science of sound.
Building acoustics

or
architectural acoustics

deals with
sound in the built environment. From the theaters of ancient Greece to those of the twenty first
century, architectural acoustics has been a key consideration in building design.

1.11.1

Intensity


Intensity I of sound wave at a point is defined

as the amount of sound energy Q flowing
per unit area in unit time when the surface is held normal to the direction of the propagation of
sound wave.


i.e.,



If A = 1m
2

and t = 1 sec, then
I = Q
, where
Q

is sound energy.


The intensity is a physical quantity which depend upon the factors like amplitude
a
,
frequency
f

and velocity
v

of sound together with the density of the medium

.




The intensity I in a medium is given by




I = 2

f
2

a
2


v


The unit of intensity is Wm
-
2
.


The minimum sound intensity which a human ear can sense is called the threshold
intensity. Its value is 10
12

watt/m
2
.
If the intensity is less than this value then our ear

cannot hear
the sound.


This minimum intensity is also known as zero or standard
intensity.
The intensity of a
sound is measured with reference to the standard intensity.

Mechanical Properties of Solids and Acoustics

1
.
27

1.11.2

Intensity

level (relative intensity) I
L


The intensity level or relative intensity of a sound is defined as the ‘logarithmic ratio of
intensity of I of a
sound to the standard intensity I
o
.


i.e.
,



Let I and I
0

represent intensities of two sounds of a particular frequency, and L
t

and L
o

be
their corresponding measures of loudness. Then, according to Weber
-
Fechner law,


L
1

=
K

log
1
0

I



(1)


L
0

=
K

log
10

I
0



(2)

Therefore, the intensity level or relative intensity is


I
L

=


L
1



L
0



=
K

log
10

I



K

log
10

I
0



=
K

(log
10

I


log
10

I
0
)




(1)

If
K

= 1, then
I
L

is expressed

in a unit called
bel
.


From the relation (1), it is seen that, 10 ties increase in intensity i.e., I = 10I
0

corresponds
to 1
bel
. Therefore,
bel

is the

intensity level of a sound whose intensity is
10

times the standard
intensity.


Similarly, 100 times increase in intensity, i
.e., I = 100I
0

corresponds to 2 bel and 1000
times increase in intensity, i.e. I = 1000 I
0

corresponds to 3 bel and so on.



In practice, bel is a large unit. Hence, another unit known as
decibel dB

is more often
used.



i.e.

one

decibel is
th of a bel.

Thus,



The threshold of audibility is 0 dB and the maximum intensity level is 120 dB.
The sound of
intensity level 120 dB produces a feeling of pain in the ear and is therefore
called as the threshold
of feeling
.


Physics for Technologists

1
.
28

1.11.3

Loudness


Loudness is characteristic which is common to all sounds whether classified as musical
sound or noise.


Loudness is a degree of sensation produced on ear. Thus, loudness varies from one
listener to ano
ther. The loudness depend upon intensity and also upon the sensitiveness of the ear.


Loudness and intensity are related to each other by the relation




or


where K is a constant.


From this relation it is s
een that, loudness is directly proportional to the logarithm of
intensity, and is known as Weber
-
Fechner law
.


From the above equation,




where,

is called

as sensitiveness of ear.
Therefore,
sensitiveness decrease with increase of
intensity. Loudness is a physiological quantity.


Worked Example 1.5:
If

the intensity of a source of sound is increased 20 times its value, by how
many decibel does the intensity level increase.











= 10 log
10




I
L

= 13.01 dB
.




Thus
, the sound intensity level is increased by 13 dB when the intensity is
doubled.


Worked Example 1.6
:
The amplitude of a sound wave is doubled; by how many dB

will the
intensity level increase?




We know I


a
2
, therefore when amplitude is doubled, intensity increases
four times.






I = 4I
0

Mechanical Properties of Solids and Acoustics

1
.
29




Hence,






I
L

= 10 log
10

4 = 10 × 0.6020





I
L

= 6.020 dB
.

Thus, the intensity level increase by
6 dB
.



Worked Example 1.7:
What is the resultant sound level when a 70 dB sound is added to a 80 dB
sound?








70 =




7 =






or


I
1

= 10
7

I
o

Similarly,


80 = 10 log
10




1.12

Sound Absorption

When sound is incident on the surface of any medium, it splits into three parts. One part is
reflected from

the surface; another part gets absorbed in the medium, while the remaining part is
transmitted through the medium and emerges on the other side. The property of a surface by which
sound energy is converted into other form of energy is known as absorption.

In the process of
absorption sound energy is converted into heat due to frictional resistance inside the pores of the
material. The fibrous and porous materials absorb sound energy more, than other solid materials.

1.12.1

Sound Absorption Coefficient

Different surfaces absorb sound to different extents. The effectiveness of a surface in
absorbing sound energy is expressed with the help of absorption coefficient. The
coefficient of
absorption

`

’ of a materials is defined as the ratio of sound energy ab
sorbed by its surface to that
of the total sound energy incident on the surface. Thus,


Physics for Technologists

1
.
30





=

In order to compare the relative efficiency of different absorbing surfaces, it is essential to
select a standard in terms of which all

surfaces can be described. A unit area of open window is
selected as the standard. All the sound incident on an open window is fully transmitted and none is
reflected. Therefore, it is considered as an ideal absorber of sound. Thus the unit of absorption
is
the open window unit (O.W.U.), which is named a
“sabin”
after the scientist who established the
unit. A 1m
2

sabin is the amount of sound absorbed by one square metre area of fully open window.
Table 1.5 lists the absorption coefficients of various mater
ials.

Table 1.5 Absorption coefficients of some materials

Material

Absorption coefficient per m
2

at 500 Hz

Open window

Ventilators

Stage curtain

Curtains with heavy folds

Carpet

Audience (One adult in upholstered seat)

Fibrous plaster, Straw board

Perforated compressed fibre board

Concrete

Marble

1.00

0.10 to 0.50

0.20

0.40 to 0.75

0.40

0.46

0.30

0.55

0.17

0.01


The value of `

’ depends on the nature of the material as well as the frequency of sound.
The greater the frequency the larger is the
value of `

’ for the same material. Therefore, the values
of `

’ for a material are determined for a wide range of frequencies. It is a common practice to use
the value of `

’ at 500 Hz in acoustic designs.

If a material has the value of “

” as 0.5, it mea
ns that 50% of the incident sound energy
will be absorbed per unit area. If the material has a surface area of S sq.m., then the absorption
provided by that material is



a

=

. S

If there are different materials in a hall, then the total sound absorpti
on by the different
materials is given by



A = a
1

+ a
2

+ a
3

+ ……



A =

1
S
1

+

2
S
2

+

3
S
3

+ ……

or

A =

where

1
,

2
,

3

………. are absorption coefficients of materials with areas S
1
, S
2
, S
3
, …….

Mechanical Properties of Solids and Acoustics

1
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31

1.12.2

Reverberation

Sound produced in an enclosure does not die out immediately after the source has ceased
to produce it. A sound produced in a hall undergoes multiple reflections from the walls, floor and
ceiling before it becomes inaudible. A person in the hall continues t
o receive successive reflections
of progressively diminishing intensity. This prolongation of sound before it decays to a negligible
intensity is called
reverberation.

Some reverberation is often desirable, especially in a hall used for musical
performance. A
small amount of reverberation improves the original sound. However, too much reverberation
causes boom sound quality in a musical performance, Speeches given in such a hall would be
unintelligible. Reverberation is a familiar phenomenon expe
rienced in halls without furniture.

Note that the reverberation of sound pertains to enclosed spaces only. In open air the
sound spreads out in all directions without repeated reflections.

1.12.3

Reverberation Time

The time taken by the sound in a room to
fall from its average intensity to inaudibility
level is called the reverberation time of the room. Reverberation time is defined as the time during
which the sound energy density falls from its steady state value to its one
-
millionth (10
-
6
) value
after th
e source is shut off. We can also express reverberation time in terms of sound energy level
in dB as follows. If initial sound level is L
i

and the final level is L
f

and reference intensity value is
I ,then we can write



L
i

= 10 log

and L
f

= 10 log





L
i



L
f

= 10 log

As

= 10
-
6
.

L
i



L
f

= 10 log 10
6

= 60 dB

Thus,
the reverberation time is the period of time in seconds, which is
required for
sound energy to diminish by 60 dB after the sound source is stopped
.


1.12.4

Sabine’s Formula for Reverberation Time

Prof.Wallace C.Sabine (1868
-
1919) determined the reverberation times of empty halls and
furnished halls of different sizes and

arrived at the following conclusions.

i)

The reverberation time depends on the reflecting properties of the walls, floor and ceiling
of the hall. If they are good reflectors of sound, then sound would take longer time to die
away and the reverberation time o
f the hall would be long.

ii)

The reverberation time depends directly upon the physical volume V of the hall.

iii)

The reverberation time depends on the absorption coefficient of various surfaces such as
carpets, cushions, curtains etc present in the hall.


Physics for Technologists

1
.
32

iv)

The reve
rberation time depends on the frequency of the sound wave because absorption
coefficient of most of the materials increases with frequency. Hence high frequency would
have shorter reverberation time.

Prof. Sabine summarized his results in the form of the f
ollowing equation.


Reverberation Time, T



or

T =







(1)

where K is a proportionality constant. It is found to have a value of 0.161 when the dimensions are
measured in metric units. Thus,

T =








(2)

Equation (2) is known as
Sabine’s formula

for reverberation time. It may be rewritten as




T =







(3)

or

T =




(4)

1.12.5

Optimum Reverberation Time

Sabine determined the time of reverberation for halls of various sizes and is given in Table
1.6. In these measurements, he used an organ pipe as the source, which was blown at a definite
frequency and under a constant pressure. The instant of cutting off
of the sound and the instant at
which the observer ceased to hear the sound were recorded. And from the results, he deduced the
reverberation time that is likely to be most satisfactory for the purpose for which a hall is built.
Such satisfactory value is
known as the optimum reverberation time.

Table 1.6 Optimum Reverberation Time for Halls

Activity in Hall

Optimum Reverberation Time (s)

Conference halls

Cinema theatre

Assembly halls

Public lecture halls

Music concert halls

Churches

Large halls

1 to
1.5

1.3

1 to 1.5

1.5 to 2

1.5 to 2

1.8 to 3

2 to 3


1.13

F
actors Affecting Acoustics of Buildings

There are several factors that affect the acoustical quality of a hall. We discuss here seven
common acoustical defects and their remedies.

Mechanical Properties of Solids and Acoustics

1
.
33

(1)

Reverberation Time

If a hall is to be acoustically satisfactory, it is essential that it should have the right
reverberation time. The reverberation time should be neither too long nor too short. A very short
reverberation time makes a room
`dead’
. On the
other hand, a long reverberation time renders
speech
unintelligible
. The optimum value for reverberation time depends on the purpose for which
a hall is designed. A reverberation time of 0.6 s is acceptable for speeches and lectures, while a
reverberation
time of 1 to 2 s is satisfactory for concerts. In case of theatres the optimum value
varies with the volume. For small theatres 1.1 to 1.5 s is suitable whereas for large theatres, may
go up to 2.3 s.

Remedies

The reverberation time can be controlled by t
he suitable choice of building materials and
furnishing materials. If the reverberation time of a hall is too long, it can be cut down by increasing
the absorption or reducing volume and if it is too short, it can be increased by changing high
absorption m
aterials to materials of low absorption or increasing volume.

Since open windows allow the sound energy to flow out of the hall, there should be a
limited number of windows. They may be opened or closed to obtain optimum reverberation time.

Carboard sheets
, perforated sheets, felt, heavy curtains, thick carpets etc are used to
increase wall and floor surface absorption. Therefore, the walls are to be provided with absorptive
materials to the required extent and at suitable places. Heavy fold curtains may be

used to increase
the absorption. Covering the floor with carpet also increase the absorption.

Audience also contribute to absorption of sound. The absorption coefficient of an
individual is about 0.45 sabins. In order to compensate for an increase in the
reverberation time
due to an unexpected decrease in audience strength, upholstered seats are to be provided in the
hall. Absorption due to an upholstered chair is equivalent to that of an individual. In the absence of
audience the upholstered chair absorbs

the sound energy and it does not contribute to absorption
when it is occupied.

(2)

Loudness

Sufficient loudness at every point in the hall is an important factor for satisfactory hearing.
Excessive absorption in the hall or lack of reflecting surfaces
near the sound source may lead to
decrease in the loudness of the sound.

Remedies

A hard reflecting surface positioned near the sound source improve the loudness. Polished
wooden reflecting boards kept behind the speaker and sometimes above the speaker wil
l be
helpful.

Low ceilings are also of help in reflecting the sound energy towards the audience.
Adjusting the absorptive material in the hall will improve the situation.

When the hall is large and audience more, loud speakers are to be installed to obtain

the
desired level of loudness.


Physics for Technologists

1
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34

(3)

Focussing

Reflection concave surfaces cause concentration of reflected sound, creating a sound of
larger intensity at the focal point. These spots are known as
sound foci
. Such concentrations of
sound intensity at some p
oints lead to deficiency of reflected sound at other points. The spots of
sound deficiency are known as
dead spots
. The sound intensity will be low at dead spots and
inadequate hearing. Further, if there highly reflecting parallel surfaces in the hall, the

reflected and
direct sound waves may form standing waves which leads to uneven distribution of sound in the
hall.

Remedies

The sound foci and dead spots may be eliminated if curvilinear interiors are avoided. If
such surfaces are present, they should be c
overed highly absorptive materials.

Suitable sound diffusers are to be installed in the hall to cause even distribution of sound
in the hall. A paraboloidal reflecting surface arranged with the speaker at its focus is helpful in
directing a uniform reflect
ed beam of sound in the hall.

(4)

Echoes

When the walls of the hall are parallel, hard and separated by about 34m distance, echoes
are formed. Curved smooth surfaces of walls also produce echoes.

Remedies

This defect is avoided by selecting proper shape
for the auditorium. Use of splayed side
walls instead of parallel walls greatly reduces the problem and enhance the acoustical quality of
the hall.

Echoes may be avoided by covering the opposite walls and high ceiling with absorptive
material.

(5)

Echelon
effect

If a hall has a flight of steps, with equal width, the sound waves reflected from them will
consist of echoes with regular phase difference. These echoes combine to produce a musical note
which will be heard along with the direct sound. This is call
ed
echelon effect
. It makes the original
sound unintelligible or confusing.

Remedies

It may be remedied by having steps of unequal width.

The steps may be covered with proper sound absorbing materials, for example with a
carpet.

(6)

Resonance

Sound waves are capable of setting physical vibration in surrounding objects, such as
window panes, walls, enclosed air etc. The vibrating objects in turn produce sound waves. The
Mechanical Properties of Solids and Acoustics

1
.
35

frequency of the forced vibration may match some frequency of the sound prod
uced and hence
result in
resonance phenomenon
. Due to the resonance, certain tones of the original music may
get reinforced any may result in distortion of the original sound.

In a hall the whole air mass vibrates if sound is continuously produced from a s
ource. The
vibration of air in turn adds to the resonant frequencies of the hall depending on its dimensions. If
lower modes of resonant frequencies are excited by the source, the sound distribution in the hall
will be erratic.

Remedies

The vibrating bodie
s may be suitably damped to eliminate resonance due to them.

In larger halls, the resonant frequencies are quite low. Hence by selecting larger halls
resonance defect can be eliminated.

(7)

Noise

Noise is unwanted sound which masks the satisfactory hearing

of speech and music. There
are mainly three types of noises that are to be minimized. They are (i) air
-
borne noise, (ii)
structure
-
borne noise and (iii) internal noise.

(i)

The noise that comes into building through air from distant sources is called
air
-
bor
ne
noise
. A part of it directly enters the hall through the open windows, doors or other
openings while another part enters by transmission through walls and floors.

Remedies

The building may be located on quite sites away from heavy traffic, market places
, railway
stations, airports etc. They may be shaded from noise by interposing a buffer zone of trees, gardens
etc.

(ii)

The noise which comes from impact sources on the structural extents of the building is
known
-

as the
structure
-
borne noise
. It is directly t
ransmitted to the building by vibrations
in the structure. The common sources of this type of noise are foot
-
steps, moving of
furniture, operating machinery etc.

Remedies

The problem due to machinery and domestic appliances can be overcome by placing
vibr
ation isolators between machines and their supports.

Cavity walls, compound walls may be used to increase the noise transmission loss and
keep the noise in the building at desired level.

(iii)

Internal noise
is the noise produced in the hall or office etc. They
are produced by air
conditioners, movement of people etc.


Physics for Technologists

1
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36

Remedies

The walls, floors and ceilings may be provided with enough sound absorbing materials.
The gadgets or machinery should be placed on sound absorbent material.

Split
-
type air conditioners etc

are to be used.



Worked Example 1.8
:


A classroom has dimensions 20 × 15 × 5 m
3

. The reverberation time is
3.5 sec. Calculate the total absorption of its surfaces and the average
absorption coefficient
.













Worked Example 1.9
:

For an empty assembly hall of size 20 × 15 × 10 m
3

the reverberation
time is 3.5 s. Calculate the average absorption coefficient of the hall.
What area of the wall should be cover
ed by the curtain so as to reduce
the reverberation time to 2.5 s. Given the absorption coefficient of
curtain cloth is 0.5.

Total absorption of the empty hall

A =
owu

Average absorption coefficient


av

=

When the walls are covered with curtain cloth

2.5 =

The area of the wall to be covered with curtain

S =





Mechanical Properties of Solids and Acoustics

1
.
37

1.14

S
ources of Noise

The word noise is derived from the Latin term
nausea
. Noise is
defined as unwanted
sound. Sound, which pleases the listeners, is music and that which causes pain and annoyance is
noise. At times, what is music for some can be noise for others.

Most leading noise sources will fall into the following categories: roads

traffic, aircraft,
railroads, construction, industry, noise in building, and consumer products.

(1)

Road Traffic Noise

In the city, the main sources of traffic noise are the motors and exhaust system of autos,
smaller trucks, buses, and motorcycles. This type of noise can be augmented by narrow streets and
tall buildings, which produce a canyon in which traffic noise rever
berates.

(2)

Air Craft Noise

Now
-
a
-
days, the problem of low flying military aircraft has added a new dimension to
community annoyance, as the nation seeks to improve its nap
-
of
-
the earth aircraft operations over
national parks, wilderness areas, and othe
r areas previously unaffected by aircraft noise has
claimed national attention over recent years.

(3)

Noise from railroads

The noise from locomotive engines, horns and whistles, and switching and shunting
operation in rail yards can impact neighboring
communities and railroad workers

(4)

Construction Noise

The noise from the construction of highways, city streets, and building is a major
contributor to the urban scene. Construction noise sources include pneumatic hammers, air
compressors, bulldozers, l
oaders, and pavement breakers.

(5)

Industrial Noise

Although industrial noise in one of the less prevalent community noise problems,
neighbors of noisy manufacturing plants can be disturbed by sources such as fans, motors, and
compressors mounted on the
outside of buildings. Interior noise can also be transmitted to the
community through open windows and doors, and even through building walls. These interior
noise sources have significant impacts on industrial workers, among whom noise


induced
hearing

loss is unfortunately common.

(6)

Noise in building

Apartment dwellers are often annoyed by noise in their homes, especially when the
building is not well designed and constructed. In this case, internal building noise from plumbing,
boilers, generators,

air conditioners, and fans, can be audible and annoying. Improperly insulated
walls and ceilings can reveal the sound of
-
amplified music, voices, footfalls and noisy activities
from neighboring units. External noise from emergency vehicles, traffic, ref
use collection, and
other city noise can be a problem for urban residents, especially when windows are open or
insufficiently glazed.


Physics for Technologists

1
.
38

(7)

Noise from Consumer products

Certain household equipment, such as vacuum cleaners and some kitchen appliances have
be
en and continue to be noisemakers, although their contribution to the daily noise dose is usually
not very large.

1.15

I
mpacts of Noise

Noise has always been with the human civilization but it was never so obvious, so intense,
so varied & so pervasive as

it is seen in the last of this century. Noise pollution makes men more