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1
MECHANICAL PROPERTIES OF
SOLIDS AND ACOUSTICS
1.1 Elasticity and PlasticityWhen 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 which 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, ona 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 17th
century French scientist and
philosopher Blaise Pascal). One Pascal equals one Newton per square meter.1 Pascal = 1Pa =
1N/m2. Strain is a quantity which describes the resulting deformation. Strainis 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 Hookes Law
This law was proposed by Robert Hooke, the founder of Royal society, in 1676. Hookes
law states that within the elastic limit, the stress developed is directly proportional to the strain.
The constant of proportionality is the elastic modulus (or modulus of elasticity).
Stress
Strain= elastic modulus (Hookes law)
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Physics for Technologists1.
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 Hookes 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
Youngs modulus (Y)Tensile stress
Tensile strain
Change of shape and size
Bulk modulus (B)Bulk stress
Bulk strainChange of size but not shape
Shear modulus orRigidity modulus (S)
Shear stress
Shear strainChange of shape but not size
Worked Example 1.1: A steel r od 2.0m long has a cross sectional area of 0.30cm2. The rod is
now hung by one end from a support structure and a 550kg mil li ng
machine is hung from the rods lower end. The Youngs modulus ofsteel i s 20 10
10Pa. Determine the stress, the strain and the elongation
of the rod.
28
5 2
(550 ) (9.8 / )1.8 10
3.0 10
F kg m sStress Pa
A m
84
10
1.8 109.0 10
20 10o
Stress PaStrain
Y Pa
Elongation = = (strain) o = (9.010-4) (2.0)
= 0.0018m = 1.8mm.
Strain
Plastic range
0
Stress
Elastic limit
Elastic range
Permanent set
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Mechani cal Properties of Solids and Acoustics 1.3
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 other 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 various 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 torsionalvibrations.
If the angle of twist at the lower end of the wire is , then the restoring couple is C ,
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 =2
2
dI
dt
(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 tends to decrease the twist. Equation (1) can be rewritten as
2
2
d C
dt I
(2)
The above relation shows that the angular acceleration is proportional to the angular
displacement and is always directed towards the mean position. Hence the motion of the disc is
simple harmonic motion and the time period of the vibration will be given by
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Physics for Technologists1.4
T = 2Displacement
Acceleration
2C
I
or T = 2 C/I
Uses of Torsion Pendulum
(1) For determi ning the moment of inerti a of an ir regular 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 after 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.
IfI,I1 andI2 are the moments of inertia of the disc, regular body and irregular body and T,
T1 and T2 are the time periods in the three cases respectively, then
T = 2 I
C(3)
T1= 2 1
I I
C
(4)
T2 =2 2I I
C (5)
From relations (3) and (4), we have
T12T
2=
2
14 I
C
(6)
and from relations (3) and (5), we have
T22T2 =2
24 IC
(7)
2 2 2
1 1 1
2 2 2
2 2 2
4 /
4 /
T T I C I
T T I C I
(8)
or2 2
22 1 2 2
1
T TI I
T T
(9)
The moment of inertia of the regular body I1 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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Mechani cal Properties of Solids and Acoustics 1.5
(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 T1 in the second case, then we have
T = 2 I
C(10)
and T1= 2 C
I
II(11)
where Iis the moment of inertia of the disc and I1 the moment of inertia of the regular body placed
on the disc. From relations (10) and (11), we have
T12T2 =
C
I124 (12)
or22
1
1
24
TT
IC
(13)
For a wire of modulus of rigidity N, length land radius r, we have
l
NrC
2
4 (14)
Equating (13) and (14), we have
l
Nr
TT
I
2
4 4
22
1
1
2
(15)
or422
1
1
)(
8
rTT
lIN
(16)
Thus, the value of N can be determined.
Worked Example 1.2: A torsion pendulum is made using a steel wi re of diameter 0.5mm and
sphere of diameter 3cm. The r igi dity modulus of steel is 80 GPa and
density of the mater ial of the sphere is 11300 kg/m3. I f the per iod of
oscil lation is 2 second, find the length of the wire.
42
8
rT
IN
For sphere, I = 2/5 MR2
M = volume density
M = 4/3 (3/2 10-2)3 11300 = 0.1598 kg
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Physics for Technologists1.
I = 2/5 0.1598 (3/2 10-2 )2 = 0.14382 10-4 kgm2
40.59 2 3
80 10 2 102 4NT r 2
48 I8 0.14382 10
5.531ml .
1.6 Bending of BeamsA beam is a rod or bar of uniform cross-section (circular or rectangular) whose length is
very much greater than its 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 an arc of a circle by the
application of an external 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 surfaceand 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 neutral 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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Mechani cal Properties of Solids and Acoustics 1.7
The resultant of the moments of all these internal couples are called the in ternal bending
momentand in the equilibrium condition, this is equal to the external bending moment.
1.6.1 Bending Moment of a Beam
Consider the sectionPBCP (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 bending 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 restoring couple consider a fiber AB at a
distance r from the neutral axis MN as shown in Fig.1.6. Let the radius of curvature be R of thepartPB and be the angle subtended by it at the centre of curvature. In unstrained position of the
beam, the length of the fiber AB = MN = R. In the strained position the length of the fibre
AB= (R + r) .
Fig. 1.5 Calculation of bending moment of a beam
Fig.1.6 Strained position
Strain in the fiber A1B1, =lengthOriginal
lengththeinChange
M
N
C
B
PD
AP
Load
rN
A B
R
M
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Physics for Technologists1.
orR
r
R
RrRStrain
)((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 thebeam and the strain produced be r/R. We have
Stress = Y Strain = Y r / R (2)
where Yis the Youngs modulus of the material
Hence, force on the area a
F = Y(r/R) a (3)
Therefore the moment of this force aboutMN
= Y(r/R) a r= Y ar2
/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
gIR
Yar
R
Y
R
raY 2
2
(5)
whereIg is the geometrical moment of inertia and is equal to AK2, A being the total area of the
section andKbeing the radius of gyration of the beam
:. moment of the forces gIR
Y (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 = gIR
Y(7)
The quantity YIg (=Y A K2) is called the flexural rigidityof the beam. F lexural ri gidityis
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 uni form bending.
Consider a beam (or bar) AB arranged horizontally on two knife edges C and D
symmetrically 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.
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Mechani cal Properties of Solids and Acoustics 1.9
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 moment on the part AF of the beam is
= W AFW CF = W (AFCF)
= W AC = W a (1)
Internal bending moment =R
YIg (2)
where
Y - Youngs modulus of the material of the bar
Ig - 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
R
YIWa
g (3)
Since for a given value of W, the values of a, YandIg are constants,R is constant so that
the beam is bent uniformly into an arc of a circle of radiusR.
CD = land yis 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 (2REF) = (CE)2
(4)
y(2Ry) =2
2
l (5)
y2R =
2
4
l(since y2 is negligible) (6)
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Physics for Technologists1.10
y =
2
8R
l(7)
or2
8
R
1
l
y (8)
From (3) and (8), Wa =8
YIg2
y
l
oryI
alWY
g8
2
(9)
If the beam is of rectangular cross-section,12
3bdIg , where b is the breath and d is the
thickness of beam.
If M is the mass, the corresponding weight W = Mg
Henceybd
aMglY
3
2
2
3 (10)
from which Y the Youngs modulus of the material of the bar is determined.
Worked Example 1.3: Unif orm rectangular bar 1 m long 2 cm broad and 0.5 cm thick is
supported on i ts flat face symmetri call y on two kn if e edges 70 cm apart .
I f loads of 200 g are hung f rom the two ends, fi nd the elevation at the
center of the bar. Youngs modulus of the mater ial of the bar is 1810
10Pa.
The distance between the nearer knife edge and the point of suspension
a=1510-2 m
Elevation at the centre,
3
2
2
3
dYb
lagMy
32210
223
)105.0(102101827.010158.9102003
= 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 larranged horizontally on two knife
edges K1 and K2 near the ends A and B as shown in Fig. 1.9.
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Mechani cal Properties of Solids and Acoustics 1.11
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
2
l and each loaded at K1 and K2 with a weight2
W
In the case of a cantilever of length land load W,
the depression =YI
lW
g3
3
Hence, for cantilever of length
2
land load
2
W , the depression is
y = YI
lW
g3
22
3
(1)
orYI
lWy
g48
3
(2)
If M is the mass, the corresponding weight W is
W = Mg (3)
If the beam is a rectangular, Ig =12
3bd, where b is the breadth and d is the thickness of the
beam.
Hence3
3
4812
Mg ly
bdY
(4)
Ybd
glMy
3
3
48
12 (5)
orybd
glMY
3
3
4 Nm
-2(6)
The value of youngs modulus, Y can be determined by the above equation.
E
K2K1
BA
W/2W/2
W
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Physics for Technologists1.12
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 ductility, as it is
clearly understood from the figure. The proportion of carbon does not have an appreciable effect
on youngs 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 stress 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 bronzeMagnesium
oxide
Brass 70:30
Annealed copper
Strain
Stress
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Mechani cal Properties of Solids and Acoustics 1.13
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.
The distance from the ordinates of the load point (or) breaking stress will indicate thetoughness 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 = 100L
L-L
o
of
Where Lfis the length of the specimen at fracture
Lo is the length of the specimen without load.
A ductile material is one with a large Percentage of elongation before failure. The original
length of the specimen Lo 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% elongation. High strength alloys,
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Physics for Technologists1.14
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 breaking in material strength.
Stressstrain 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 returns to its
original length on removal of the stress. The limit of proportionality is the stage up to which thematerial obeys Hookes law perfectly.
Beyond the elastic limit the applied stress produces plastic deformation so that a
permanent extension remains 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.
The 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 and 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
S
tress
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Mechani cal Properties of Solids and Acoustics 1.1
From Fig.1.12 it is clearly show that at a certain value of load the strain continues at slow
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 percentage 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 behaviorStressstrain behavior of brittle materials
Fig. 1.13 Stressstrain 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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Physics for Technologists1.16
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 strength.
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 load 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.Table 1.3 Typical tensile strengths of engineering materials
Material Tensile Strength kg/mm2
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 plastic properties relative
to the mode of penetration, shape of penetration etc. The main usefulness of hardness is, it has aconstant relationship to the tensile strength of a given material and so can be used as a practical
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Mechani cal Properties of Solids and Acoustics 1.1
non-destructive test for an approximate 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-hardness 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 highly 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, usually 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, on 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 testing, 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 on the surface or
close to it.
The following are the most common hardness test methods used in todays technology.
1. Rockwell hardness test2. Brinell hardness3. Vickers4. Knoop hardness5. ShoreBrinell, Rockwell and Vickers hardness tests are used to determine hardness 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 measures hardness in terms of the elasticity of the material.
Brinell 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 strength, which is as shown
in Fig.1.14.
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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 accompanied 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
deformation 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 toolsbearing surfaces etc. Softness 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 m2
(MN/m2). It is defined as the
specific work required to fracture a test specimen with a stress concentrator in the mid when
broken by a single 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
structures 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 testing machine. This further helps to study the effect of stress concentration and high
velocity load application.
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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 offractures 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 FatigueFatigue 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 mainfeatures.
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,
rubber 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
concentrations, 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 whenplotted 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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104
105
106
107
108
109
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 107. The fatigue life can be defined as the number of cycles required for a material to fail
at a certain stress.
1.9.5 CreepThe 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 turbines. 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 directions 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
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Mechani cal Properties of Solids and Acoustics 1.21
Fig.1.16 shows a typical creep curve. The creep curve usually consists of three points
corresponding to particular stages of creep.
(i) Primary Stage: In this stage the creep rate decreases with time, the effect of workhardening 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 withtime. 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 fractureoccurs. Tertiary creep can occur due to necking of the specimen and other processes
that ultimately result in failure.
The temperature and time dependence of creep deformation indicates that it is a thermallyactivated process. Several atomic processes are known to be responsible for creep in crystalline
materials.
The 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 limit 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 CreepAt low temperature the creep rate decreases with time and the logarithmic creep curve 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 creep is primarily influenced by diffusion
and load applied has little effect. This creep is termed as diffusion creep or plastic creep.
1.10 FractureFracture 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 Fractureii) Ductile Fracture
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iii) Fatigue Fractureiv) Creep FractureFracture 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 continuesuntil fracture occurs.
(2) The atoms at the surfaces do not have as many neighbors as those in the interior of asolid 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 minimumenergy of absorption.
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Mechani cal Properties of Solids and Acoustics 1.23
(4) In brittle fracture, adjacent parts of the metal are separated by stresses normal to thefracture surface.
(5) Brittle fracture occurs along characteristics crystallographic planes called ascleavage planes. The fracture is termed as cleavage fracture.
(6) Brittle fracture does not produce plastic deformation, so that it requires less energythan 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 the 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 surface area of acrack. 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 elastic energy released by extending a crack equal to
the surface energy required for crack extension then the crack will grow.
=e
E2
(1)
where, e is half of the crack length, is the true surface energy and E is the Young's modulus.
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 Griffiths 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
=
1
2p
2E
e
(2)
Generally p >> for metals.
From the above formula, one can get the size of largest flaw or crack.
1.10.2 Ductile FractureDuctile 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 isthe region in which the plastic deformation is concentrated. It indicates that the
formation of cavities is closely linked to plastic deformation.
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It has been observed that during the formation of neck small micro cracks areformed at the centre of the specimen due to the combination of dislocations.
Finally these cracks grow out ward to the surface of the specimen in a direction 45to the tensile axis resulting in a cup-end-cone-type fracture.
Fig.1.18 Various 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 expenditure 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 foreigninclusions where gliding dislocations can 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
elongation 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. The 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 angle 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.
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Table 1.4 Comparison between Brittle and Ductile fracture
Ductile fracture Brittle fracture
Material fractures after plasticdeformation and slow propagation
of crack
Material fractures with very little or noplastic deformation.
Surface obtained at the fracture isdull or fibrous in appearance
Surface obtained at the fracture isshining and crystalling appearance
It occurs when the material is inplastic condition.
It occurs when the material is in elasticcondition.
It is characterized by the formationof cup and cone
It is characterized by separation ofnormal to tensile stress.
The tendency of ductile fracture isincreased by dislocations and other
defects in metals.
The tendency brittle fracture isincreased by decreasing temperature,
and increasing 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 FractureFatigue 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 offatigue 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 materials, 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 FractureCreep fracture is defined as the fracture which takes place 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.
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Worked Example 1.4: A Youngs modulus of a certain material is 180 103
mega Newton/ m2
and its true surface energy is 1.8 J/m2. The crack length is 5 m.
Calculate the fracture strength.
The fracture strength is
=e
E2
=9
6
2 1 8 180 10
3 14 5 2 10
.
. /
= 278 106
Newton /m2
1.11 Acoustics of Buildings
Introduction
Acousticsis the science of sound. Buil ding acousticsorarchi tectural acousticsdeals 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.,Q
IAt
If A = 1m2 and t = 1 sec, thenI = Q, where Q is sound energy.
The intensity is a physical quantity which depend upon the factors like amplitude a,
frequencyfand velocity v of sound together with the density of the medium .
The intensity I in a medium is given by
I = 2f2 a2v
The uni t of in tensity is Wm-2.
The minimum sound intensity which a human ear can sense is called the threshold
intensity. Its value is 1012
watt/m2. 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.
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1.11.2 Intensity level (relative intensity) IL
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 Io.
i.e., 10logLo
II KI
Let I and I0 represent intensities of two sounds of a particular frequency, and Lt and Lo be
their corresponding measures of loudness. Then, according to Weber-Fechner law,
L1 =Klog10I (1)
L0 =Klog10 I0 (2)
Therefore, the intensity level or relative intensity is
IL = L1L0
= Klog10IKlog10I0
= K(log10 Ilog10 I0)
10logL
o
II K
I
(1)
IfK= 1, thenILis expressed in a unit called bel.
From the relation (1), it is seen that, 10 ties increase in intensity i.e., I = 10I0 correspondsto 1 bel. Therefore, belis the intensity level of a sound whose intensity is 10 times the standard
intensity.
Similarly, 100 times increase in intensity, i.e., I = 100I0 corresponds to 2 bel and 1000
times increase in intensity, i.e. I = 1000 I0 corresponds to 3 bel and so on.
In practice, bel is a large unit. Hence, another unit known as decibel dBis more often
used.
11
10
dB bel
i.e. one decibel is1
10th of a bel.
Thus,10
0
logLI
I K dBI
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.
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1.11.3 LoudnessLoudness 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 onelistener to another. The loudness depend upon intensity and also upon the sensitiveness of the ear.
Loudness and intensity are related to each other by the relation
IL 10log
or 10logL K I where K is a constant.
From this relation it is seen that, loudness is directly proportional to the logarithm of
intensity, and is known as Weber-Fechner law.
From the above equation,
dL
dI I
where,dL
dIis 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.
1010 logLo
II
I
= 10 log10 1020
10 log 20 10 1.3012o
o
I
I
IL = 13.01 dB.Thus, the sound intensity level is increased by 13 dB when the intensity isdoubled.
Worked Example 1.6: The amplitude of a sound wave is doubled; by how many dB will the
intensity level increase?
We know I a2, therefore when amplitude is doubled, intensity increases
four times.
I = 4I0
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Hence,10
410 log oL
o
II
I
IL = 10 log10 4 = 10 0.6020
IL = 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?
1 1010 logL
o
IL
I
70 = 110
logo
I
I
7 = 110
logo
I
I
71 10o
I
I
or I1 = 107
Io
Similarly, 80 = 10 log10 2
o
I
I
82 10o
I
I
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 istransmitted 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 CoefficientDifferent surfaces absorb sound to different extents. The effectiveness of a surface in
absorbing sound energy is expressed with the help of absorption coefficient. The coeff icient of
absorption of a materials is defined as the ratio of sound energy absorbed by its surface to that
of the total sound energy incident on the surface. Thus,
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=surfacetheonincidentenergysoundTotal
surfacethebyabsorbedenergySound
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 1m2
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 materials.
Table 1.5 Absorption coefficients of some materials
Material Absorption coefficient per m2
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 means 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 absorption by the differentmaterials is given by
A = a1 + a2 + a3+
A = 1S1 + 2S2 + 3S3+
or A = n
nn S1
where 1, 2, 3. are absorption coefficients of materials with areas S1, S2, S3, .
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Mechani cal Properties of Solids and Acoustics 1.31
1.12.2 ReverberationSound 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 to receive successive reflections
of progressively diminishing intensity. This prolongation of sound before it decays to a negligibleintensity 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 experienced 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 TimeThe 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 the 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 Li and the final level is Lf and reference intensity value is
I ,then we can write
Li = 10 logI
Ii and Lf= 10 logI
If
LiLf= 10 logf
i
I
I
As
i
f
I
I= 10
-6. LiLf= 10 log 10
6= 60 dB
Thus, the reverberation t ime is the period of t ime in seconds, which is requir ed for
sound energy to dimin ish by 60 dB af ter the sound source is stopped.
1.12.4 Sabines Formula for Reverberation TimeProf.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 ceilingof the hall. If they are good reflectors of sound, then sound would take longer time to die
away and the reverberation time of 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 ascarpets, cushions, curtains etc present in the hall.
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iv) The reverberation time depends on the frequency of the sound wave because absorptioncoefficient 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 following equation.
Reverberation Time, T Volumeof theHall V
Absorption A
or T =A
VK (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 =A
V161.0(2)
Equation (2) is known asSabines formula for reverberation time. It may be rewritten as
T =
N
nn S
V
1
161.0
(3)
or T =nn
SSSS
V
.......161.0
332211
(4)
1.12.5 Optimum Reverberation TimeSabine 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 hallsCinema theatre
Assembly halls
Public lecture halls
Music concert halls
Churches
Large halls
1 to 1.51.3
1 to 1.5
1.5 to 2
1.5 to 2
1.8 to 3
2 to 3
1.13 Factors Affecting Acoustics of Buildings
There are several factors that affect the acoustical quality of a hall. We discuss here sevencommon acoustical defects and their remedies.
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(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 whicha 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 the 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 materials 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 timedue 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 will 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 thedesired level of loudness.
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Physics for Technologists1.34
(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 points 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 covered 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 reflected 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 called echelon ef fect. 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 aswindow panes, walls, enclosed air etc. The vibrating objects in turn produce sound waves. The
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Mechani cal Properties of Solids and Acoustics 1.3
frequency of the forced vibration may match some frequency of the sound produced 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 source. 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 bodies 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-bornenoise. 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 isknown- as the structure-borne noise. It is directly transmitted 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
vibration 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) I nternal noiseis the noise produced in the hall or office etc. They are produced by airconditioners, movement of people etc.
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Physics for Technologists1.36
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 m3. The reverberati on time is
3.5 sec. Calculate the total absorption of i ts sur faces and the average
absorption coeff icient.
S
VT
161.0
695.3
)51520(161.0 3 smS
average
69 69
2(20 15 15 5 20 5) 950
0.07
Worked Example 1.9: For an empty assembly hal l of size 20 15 10 m3the reverberation
time is 3.5 s. Calcul ate the average absorption coeff ici ent of the hal l .
What area of the wall should be covered by the cur tain so as to reduce
the reverberation time to 2.5 s. Given the absorption coefficient of
cur tain cloth i s 0.5.
Total absorption of the empty hall
A = 1385.3
101520(161.0
owu
Average absorption coefficient
av = 106.0)102010151520(2
138
When the walls are covered with curtain cloth
2.5 =S5.0138
)101520(161.0
The area of the wall to be covered with curtain
S = 483 2.5 138
2.5 0.5
2
110.4m
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Mechani cal Properties of Solids and Acoustics 1.37
1.14 Sources 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 reverberates.
(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 other 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, loaders, 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, refuse collection, and
other city noise can be a problem for urban residents, especially when windows are open or
insufficiently glazed.
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Physics for Technologists1.38
(7) Noise from Consumer products
Certain household equipment, such as vacuum cleaners and some kitchen appliances have
been and continue to be noisemakers, although their contribution to the daily noise dose is usually
not very large.
1.15 Impacts 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
irritable. The effect of noise pollution is multifaceted & inter related. The impacts of noise on
human being, animal and property are as follows.
(1) It decreases the efficiency of a man
Regarding the impact of noise on human efficiency, there are number of experiments
which point out the fact that human efficiency increases with noise reduction. Thus human
efficiency is related with noise.
(2) Lack of Concentration
For better quality of work there should be concentration. Noise causes lack of
concentration. In big cities, mostly all the offices are on main road, the noise of traffic or the loud
speakers of different types of horns divert the attention of the people working in offices.
(3) Fatigue
Because of noise pollution, people cannot concentrate on their work. Thus they have to
give their more time for completing the work and they feel tiring
(4) Abortion is caused
There should be cool and calm atmosphere during the pregnancy. Unpleasant sounds
make a lady of irritative nature. Sudden noise causes abortion in females.
(5) It causes Blood Pressure
Noise pollution causes certain diseases in human. It attacks on the persons peace of mind.
The noises are recognized as major contributing factors in accelerating the already existing
tensions of modern living. The tensions result in certain disease like blood pressure or mental
illness etc.
(6) Temporary or Permanent Deafness
The effect of noise on audition is well recognized, in Mechanics, locomotive drivers,
telephone operators etc. All have their hearing. impairment as a result of noise at the place of
work. Physicist, physicians & psychologists are of the view that continued exposure to noise level
above 80 to 100 dB is unsafe. Loud noise causes temporary or permanent deafness.
(7) Effect on Vegetation
It is well known to all that plants are similar to human being. They are also as sensitive as
man. There should be cool & peaceful environment for their better growth. Noise pollutioncauses poor quality of crops.
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Mechani cal Properties of Solids and Acoustics 1.39
(8) Effect on Animals
Noise pollution damage the nervous system of animals. Animal looses the control of its
mind. They become dangerous.
(9) Effect on Property
Loud noise is very dangerous to building, bridges and monuments. It creates waves which
struck the walls and put the building in danger condition. It weakens the edifice of buildings.
1.16 Sound Level Meter
Definition
The instrumentation to determine sound level or noise level is referred as a sound l evel
meter.
Principle
The pressure of the sound waves under study actuates the microphone thus converting the
acoustical energy into electrical current which in turn serve to operate the display device.
Design
The various elements in a sound level meter are
i) the transducer; that is, the microphoneii) the electronic amplifier and calibrated attenuator for gain controliii) the frequency weighting or analyzing possibilitiesiv) the data storage facilitiesv) the displayA block diagram of a simple sound level meter is shown in Fig.1.19.The most important
element of sound level meter is the microphone.
Fig.1.19 Block diagram of a sound level meter
Microphones
The microphone is the interface between the acoustic field and the measuring system. It
responds to sound pressure and transforms it into an electric signal which can be interpreted by the
measuring instrument.
Microphone
Pre
Amplifier
Weightingnetwork or
filtersAmplifier Rectifier
Averaging
System
DisplayAC
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The microphone can be of the following types : piezoelectric, condenser, electret or
dynamic. In a piezoelectric microphone, the membrane is attached to a piezoelectric crystal which
generates an electric current when submitted to mechanical tension. The vibrations in the air,
resulting from the sound waves, are picked up by the microphone membrane and the resulting
pressure on the piezoelectric crystal transforms the vibration into an electric signal. These
microphones are stable, mechanically robust and not appreciably influenced by ambient climatic
conditions. They are often used in sound survey meters.
In a condenser microphone, the microphone membrane is built parallel to a fixed plate and
forms with it a condenser. A potential differential is applied between the two plates using a d.c.
voltage supply (the polarization voltage). The movements, which the sound waves provoke in the
membrane, given origin to variations in the electrical capacitance and therefore in a small electric
current. These microphones are more accurate than the other types and are mostly used in
precision sound level meters. However, they are more prone to begin affected by dirt and
moisture.
A variation on the condenser microphone which is currently very popular is the electret.In this case the potential difference is provided by a permanent electrostatic charge on the
condenser plates and no external polarizing voltage. This type of microphone is less sensitive to
dirt and moisture than the condenser microphone.
In dynamic microphone, where the membrane, is connected to a coil, centred in a magnetic
field, and whose movements, triggered by the mechanical fluctuations of the membrane, give
origin to a potential differential in the poles of the coil. The dynamic microphone is more
mechanically resistant but its poor frequency response severely limits its use in the field of
acoustics
Working
The electrical signal from the transducer is fed to the pre-amplifier of the sound level
mater and a weighted filter over a specified range of frequencies. Further amplification prepares
the signal either for output to other instruments such as a tape recorder or for rectification and
direct reading on the meter. The scale on the indicating device is such that the linear signal may be
read in dB. The two main characteristic are:
(1) The frequency response
That is, the deviation between the measured value and true value as a function of the
frequency. As the ear is capable of hearing sounds between 20Hz and 20KHz, the frequency
response of the sound level meter should be good, with variations smaller than 1dB, over thatrange.
(2) The dynamic range
That is, the range in dB over which the measured value is proportional to the true value, at
a given frequency (usually 1000Hz). This range is limited at low levels by the electrical
background noise of the instrument and at high levels by the signal distortion caused by
overloading the microphone or amplifiers.
1.17 Control of Noise Pollution
The techniques employed for noise control can be broadly classified as (1) control atsource (2) control in the transmission path and (3) using protective equipment.
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1. Noise Control at Source
The noise pollution can be controlled at the source of generation itself by employing
following techniques.
(i) Reducing the noise levels from domestic sectors
The domestic noise coming from radio, tape recorders, television sets, mixers,
washing machines, cooking operations can be minimized by their selective and
judicious operation. By usage of carpets or any absorbing material, the noise generated
from felling of items in house can be minimized.
(ii) Maintenance of automobiles
Regular servicing and tuning of vehicles will reduce the noise levels. Fixing of
silencers to automobiles, tow wheelers etc., will reduce the noise levels.
(iii) Control over vibrations
The vibrations of materials may be controlled using proper foundations, rubber
padding etc., to reduce the noise levels caused by vibrations.
(iv) Low voice speaking
Speaking at low voices enough for communication reduces the excess noise levels.
(v) Prohibition on usage of loud speakers
By not permitting the usage of loudspeakers in the habitant zones except for important
meetings / functions. Now-a-days, the urban administration of the metro cities in
India, is becoming stringent on usage of loudspeakers.
(vi) Selection of machinery
Optimum selection of machinery tools or equipment reduces excess noise levels. For
example selection of chairs, or selection of certain machinery / equipment which
generate less noise (sound) due to its superior technology etc. is also an important
factor in noise minimization strategy.
(vii) Maintenance of machines
Proper lubrication and maintenance of machines, vehicles etc., will r
Recommended