Mechanical Properties of Solids and Acoustics

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 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, 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/m2. Strain is a quantity which describes the resulting deformation. 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 constant of proportionality is the elastic modulus (or modulus of elasticity).

Stress = elastic modulus (Hooke‟s law) Strain

1.2

Physics for Technologists

Plastic range

Stress

Elastic limit Elastic range Permanent set Strain 0 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 Tensile stress Tensile strain Bulk stress Bulk strain Shear stress Shear strain

Young‟s modulus (Y) Bulk modulus (B) Shear modulus or Rigidity modulus (S)

Nature of strain Change of shape and size Change of size but not shape Change of shape but not size

Worked Example 1.1: A steel rod 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 milling machine is hung from the rod’s lower end. The Young’s modulus of steel is 20 ×1010Pa. Determine the stress, the strain and the elongation of the rod.

Stress  Strain 

F (550kg )  (9.8m / s 2 )   1.8  108 Pa A 3.0 105 m2  o

Elongation =



Stress 1.8  108 Pa   9.0  104 10 Y 20  10 Pa

 = (strain) ×  o = (9.0×10-4) (2.0) = 0.0018m = 1.8mm.

1.3

Mechanical Properties of Solids and Acoustics

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 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 Fixed End

Torsionally flexible elastic wire

Disc

Fig. 1.2 Torsion Pendulum

C θ = I

d 2 dt 2

(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

d 2 C   dt 2 I

(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 time period of the vibration will be given by

1.4

Physics for Technologists

Displacement Acceleration

T = 2π

or T = 2 π

 2

 C    I  

I/C

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 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. If I, I1 and I2 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=2π

I C

(3)

T1 = 2 π

I  I1 C

(4)

T2 = 2 π

I  I2 C

(5)

From relations (3) and (4), we have T12 – T2 =

42 I1 C

(6)

and from relations (3) and (5), we have T22 – T2 =



or

42 I 2 C

(7)

T12  T 2 4  2 I1 / C I 1   T2 2  T 2 42 I 2 / C I 2 I 2  I1 

T2 2  T 2 T12  T 2

(8)

(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.

1.5

Mechanical Properties of Solids and Acoustics

(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

and

T=2π

I C

(10)

T1= 2 π

I  II C

(11)

where I is 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 T12 – T2 =

or

C

4 2 I 1 C

4 2 I 1

(12)

(13)

T1  T 2 2

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

C

Nr 4

(14)

2l

Equating (13) and (14), we have

4 2 I 1 T1  T 2 2

or N 



Nr 4 2l 8lI1

(T1  T 2 )r 4 2

(15)

(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/m3. If the period of oscillation is 2 second, find the length of the wire.

N

8I T 2r 4

For sphere, I = 2/5 MR2 M = volume  density M = 4/3π (3/2  10-2)3  11300 = 0.1598 kg

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Physics for Technologists

I = 2/5  0.1598  (3/2  10-2 )2 = 0.14382 × 10-4 kgm2 9 2  0.5 3  10  2 4 80  10  2   NT r  2  l   4 8I 8    0.14382  10

1.6

4  5.531m .

Bending of Beams

A 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 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 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.

Mechanical Properties of Solids and Acoustics

1.7

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 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 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 = Rф. In the strained position the length of the fibre AB = (R + r) ф.

P

A

B

M

N

P

D

C Load Fig. 1.5 Calculation of bending moment of a beam

B

A M

N

R

ф

Fig.1.6 Strained position

Strain in the fiber A1B1, =

Change in the length Original length

r

1.8

Physics for Technologists

or Strain 

(R  r)   R  r  R R

(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 Young‟s modulus of the material Hence, force on the area a F = Y(r/R) × a

(3)

Therefore the moment of this force about MN = Y(r/R) × a × r = Y a r2 / 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

 Y

ar2 Y Y  ar 2  I g R R R

(5)

where Ig is the geometrical moment of inertia and is equal to AK2, A being the total area of the section and K being the radius of gyration of the beam :. moment of the forces 

Y Ig R

(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 =

Y Ig R

(7)

The quantity YIg (=Y A K2) 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 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.

1.9

Mechanical Properties of Solids and Acoustics

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 × AF – W × CF = W (AF – CF) = W × AC = W × a

(1) YI g

Internal bending moment =

(2)

R

where Y -

Young‟s 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 Wa 

YI g

(3)

R

Since for a given value of W, the values of a, Y and Ig 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 F F y

C

D E

l/2

o

R

Fig. 1.8 Circle Property

EF (2R – EF) = (CE)2 y (2R – y) =  l 

2

  2

y 2R =

l

(4) (5)

2

(since y2 is negligible) 4

(6)

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Physics for Technologists

y=

l

2

(7)

8R

or

1 8y  R l2

From (3) and (8), Wa =

(8) 8y l

or

2

YI g

W l2 a 8I g y

Y

(9)

3 If the beam is of rectangular cross-section, I g  bd , where b is the breath and d is the 12 thickness of beam.

If M is the mass, the corresponding weight W = Mg Hence Y 

3 Mgl 2 a 2 bd 3 y

(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 of the bar is 18  1010 Pa. The distance between the nearer knife edge and the point of suspension a=15×10-2 m Elevation at the centre, y



3 M g al2 2 Yb d 3

3 20010 3  9.8 1510 2  0.7 2 2 18 1010  2  10 2  (0.5 10 2 ) 3

= 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 two knife edges K1 and K2 near the ends A and B as shown in Fig. 1.9.

1.11

Mechanical Properties of Solids and Acoustics W/2

W/2 E A

B K1

K2

W

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  l  and each loaded at K1 and K2 with a weight W 2

2

In the case of a cantilever of length l and load W, 3 the depression = W l

3I g Y

Hence, for cantilever of length W  l     y =  2  2  3Ig Y

y

or

 l  and load  W  , the depression is     2 2

3

(1)

W l3 48I g Y

(2)

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

(3)

If the beam is a rectangular, Ig =

bd 3 , where b is the breadth and d is the thickness of the 12

beam. 3 Hence y  Mg l3

(4)

bd 48 Y 12

or

y

M gl 3  12 48 bd 3Y

Y

M gl 3 4 bd 3 y

(5) Nm-2

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

(6)

1.12

1.7

Physics for Technologists

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 young‟s modulus of elasticity during any hardening process.

Alloy steel or tool steel High carbon steel

Stress

Medium carbon steel Mild steel (Ductile) Wrought iron (Most ductile) Cast iron (Brittle iron)

Strain

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%. Magnesium oxide

Aluminium bronze

Brass 70:30 Stress

Annealed copper

Strain Fig.1.11. Stress Strain curve for non - ferrous metals

Mechanical 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 stressstrain 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

1.8 1.8.1



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 the toughness and brittleness of the metal. The shorter the distance then the metal is more brittle.

Ductile and Brittle Materials 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).

Lf - Lo Lo

Percentage of elongation =

 100

Where Lf is 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,

1.14

Physics for Technologists

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.

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 returns 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 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. Upper yield point Ultimate stress

Fracture

Stress

Lower yield point Elastic limit Limit of proportionality

Strain 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.

Mechanical Properties of Solids and Acoustics

1.15

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 behavior

Stress – strain behavior of brittle materials

Proof stress

Stress

Yield point at off-set

Parallel

Strain 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.

1.16

1.9

Physics for Technologists

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

1.9.2

Tensile Strength kg/mm2

Alloy steel

60 -70

Mild Steel

42

Grey CI

19

White CI

47

Aluminum alloy

47

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 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

1.17

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 today‟s technology. 1. Rockwell hardness test 2. Brinell hardness 3. Vickers 4. Knoop hardness 5. Shore Brinell, 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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Physics for Technologists

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 tools bearing 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.

Mechanical Properties of Solids and Acoustics

1.19

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.

1.9.4



The angle of the notch also improves impact-strength after certain values.



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

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, 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 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”.

1.20

Physics for Technologists 38 34

Stress

30 26 22 18 16 14

Fatigue strength

10 6 104

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 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 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

Mechanical 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 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 specimen 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. 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 Creep

At 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

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

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Physics for Technologists

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.

1.23

Mechanical Properties of Solids and Acoustics

(4)

In 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 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 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 elastic energy released by extending a crack equal to the surface energy required for crack extension” then the crack will grow.



2 E e

=

(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 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



=

 2E p   e

1

2  

(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 deformation is concentrated. It indicates that the formation of cavities is closely linked to plastic deformation.

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Physics for Technologists



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 dislocations.



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 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 foreign inclusions 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.

Mechanical Properties of Solids and Acoustics

1.25

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 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 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 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 Fracture Creep 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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Physics for Technologists

Worked Example 1.4: A Young‟s 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 µm. Calculate the fracture strength. The fracture strength is

1.11



=

2 E e



=

2  1.8  180 109 3.14  5 / 2  106



=

278 × 106 Newton /m2

Acoustics 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., I 

Q At

 If A = 1m2 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f2 a2 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 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.

Mechanical Properties of Solids and Acoustics

1.27

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.,

 I  I L  K log10    Io 

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 = K log10 I

(1)

L0 = K log10 I0

(2)

Therefore, the intensity level or relative intensity is IL = L1 – L0 = K log10 I – K log10 I0 = K (log10 I – log10 I0)  I  I L  K log10    Io 

(1)

If K = 1, then IL is expressed in a unit called bel. From the relation (1), it is seen that, 10 ties increase in intensity i.e., I = 10I0 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 = 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 dB is more often used. 1dB 

1 bel 10

i.e. one decibel is

Thus,

1 th of a bel. 10

I  I L  K log10   dB  I0 

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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Physics for Technologists

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 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 L  log10 I

or

L  K log10 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

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

where,

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.

 I  I L  10 log10    Io 

 20 I o  = 10 log10    10 log10 20  10  1.3012  Io   IL = 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  a2, therefore when amplitude is doubled, intensity increases four times.  I = 4I0

Mechanical Properties of Solids and Acoustics

Hence,

1.29

 4I  I L  10 log10  o   Io  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?

 I  LL1  10 log10    Io    70 = log10  I1   Io 

I  7 = log10  1   Io  I1  107 Io

or Similarly,

I1 = 107 Io

I  80 = 10 log10  2   Io  

1.12

I2  108 Io

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 absorbed by its surface to that of the total sound energy incident on the surface. Thus,

1.30

Physics for Technologists

=

Sound energy absorbed by the surface Total sound energy incident on the surface

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 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

Absorption coefficient per m2 at 500 Hz 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 different materials is given by A = a1 + a2 + a3 + …… A = 1S1 + 2S2 + 3S3 + …… n

or

A=



n

Sn

1

where 1, 2, 3 ………. are absorption coefficients of materials with areas S1, S2, S3, …….

Mechanical Properties of Solids and Acoustics

1.31

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 to 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 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 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 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 log

Ii I

Li – Lf = 10 log

As

If Ii

and Lf = 10 log

If I

Ii If

= 10-6. Li – Lf = 10 log 106 = 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 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 as carpets, cushions, curtains etc present in the hall.

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iv) The reverberation 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 following equation. Reverberation Time, T  Volume of the Hall  V Absorption  A T= KV

or

(1)

A

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=

0.161V A

(2)

Equation (2) is known as Sabine’s formula for reverberation time. It may be rewritten as T = 0.161 V

(3)

N



n

Sn

1

or

T=

0.161 V  1 S1   2 S 2   3 S 3  .......   n S n

(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

1.13

Optimum Reverberation Time (s)

Conference halls Cinema theatre Assembly halls

1 to 1.5 1.3 1 to 1.5

Public lecture halls Music concert halls

1.5 to 2 1.5 to 2

Churches Large halls

1.8 to 3 2 to 3

Factors 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 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 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 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 the desired level of loudness.

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

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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. T

0.161V  S

S 

0.161(20 15  5)m 3  69 3.5s

average 

69 69   0.07 2(20 15 15 5  20  5) 950

Worked Example 1.9: For an empty assembly hall of size 20 × 15 × 10 m3 the reverberation time is 3.5 s. Calculate the average absorption coefficient of the hall. What area of the wall should be covered 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 = 0.161(201510 138 owu 3.5 Average absorption coefficient av =

138  0.106 2 (20 15  15 10  20 10 )

When the walls are covered with curtain cloth 2.5 =

0.161 (20 15 10 ) 138  0.5 S

The area of the wall to be covered with curtain S = 483  2.5138  110.4m2 2.5 0.5

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1.14

1.37

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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(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 person‟s 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 pollution causes poor quality of crops.

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Mechanical Properties of Solids and Acoustics

(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 level 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 microphone

ii)

the electronic amplifier and calibrated attenuator for gain control

iii)

the frequency weighting or analyzing possibilities

iv)

the data storage facilities

v)

the display

A 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. Pre Amplifier

Weighting network or filters

Amplifier

Rectifier

Averaging System

Microphone

AC Output

Display

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.

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Physics for Technologists

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 that range. (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 at source (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 reduce noise levels. For example, it is a common experience that, many parts of a vehicle will become loose while on a rugged path of journey. If these loose parts are not properly fitted, they will generate noise and cause annoyance to the driver/passenger. Similarly is the case of machines. Proper handling and regular maintenance is essential not only for noise control but also to improve the life of machine.

2. Control in the transmission path The change in the transmission path will increase the length of travel for the wave and get absorbed/refracted/radiated in the surrounding environment. The available techniques are briefly discussed below.

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(i)

Installation of barriers Installation of barriers between noise source and receiver can attenuate the noise levels. For a barrier to be effective, its lateral width should extend beyond the line-ofsight at least as much as the height (See Fig.1.20 The barrier may be either close to the source or receiver, subjected to the condition that, R