PH8151 Engineering Physics Chapter 1

Room-Temperature Young’s, Shear and Bulk 1.5. FACTORS AFFECTING ELASTICITY

Moduli and Poisson’s Ratio for Various materials

(Figure 6.5), application of the load corresponds to moving from the origin up and

along the straight line. Upon release of the load, the line is traversed in the oppo-

site direction, back to the origin.

There are some materials (e.g., gray cast iron, concrete, and many polymers)

for which this elastic portion of the stress–strain curve is not linear (Figure 6.6);

hence, it is not possible to determine a modulus of elasticity as described previously.

For this nonlinear behavior, either tangentor secant modulusis normally used. Tan-

gent modulus is taken as the slope of the stress–strain curve at some specified level

of stress, whereas secant modulus represents the slope of a secant drawn from the

origin to some given point of the curve. The determination of these moduli is

illustrated in Figure 6.6.

On an atomic scale, macroscopic elastic strain is manifested as small changes

in the interatomic spacing and the stretching of interatomic bonds. As a conse-

quence, the magnitude of the modulus of elasticity is a measure of the resistance

to separation of adjacent atoms, that is, the interatomic bonding forces. Further-

more, this modulus is proportional to the slope of the interatomic force–separation

curve (Figure 2.8a) at the equilibrium spacing:

(6.6)

Figure 6.7 shows the force–separation curves for materials having both strong and

weak interatomic bonds; the slope at r 0 is indicated for each.

Er a

dF

dr

b

r 0

sñ!

6.3 Stress–Strain Behavior • 157

Strain

Stress

0

0

Slope = modulus

of elasticity

Unload

Load

Table 6.1 Room-Temperature Elastic and Shear Moduli and Poisson’s Ratio

for Various Metal Alloys

Modulus of

Elasticity Shear Modulus Poisson’s

Metal Alloy GPa 106 psi GPa 106 psi Ratio

Aluminum 69 10 25 3.6 0.

Brass 97 14 37 5.4 0.

Copper 110 16 46 6.7 0.

Magnesium 45 6.5 17 2.5 0.

Nickel 207 30 76 11.0 0.

Steel 207 30 83 12.0 0.

Titanium 107 15.5 45 6.5 0.

Tungsten 407 59 160 23.2 0.

Figure 6.5 Schematic stress–strain diagram showing linear

elastic deformation for loading and unloading cycles.

JWCL187_ch06_150-196.qxd 11/5/09 9:36 AM Page 157

Material

(Figure 6.5), application of the load corresponds to moving from the origin up and

along the straight line. Upon release of the load, the line is traversed in the oppo-

site direction, back to the origin.

There are some materials (e.g., gray cast iron, concrete, and many polymers)

for which this elastic portion of the stress–strain curve is not linear (Figure 6.6);

hence, it is not possible to determine a modulus of elasticity as described previously.

For this nonlinear behavior, either tangentor secant modulusis normally used. Tan-

gent modulus is taken as the slope of the stress–strain curve at some specified level

of stress, whereas secant modulus represents the slope of a secant drawn from the

origin to some given point of the curve. The determination of these moduli is

illustrated in Figure 6.6.

On an atomic scale, macroscopic elastic strain is manifested as small changes

in the interatomic spacing and the stretching of interatomic bonds. As a conse-

quence, the magnitude of the modulus of elasticity is a measure of the resistance

to separation of adjacent atoms, that is, the interatomic bonding forces. Further-

more, this modulus is proportional to the slope of the interatomic force–separation

curve (Figure 2.8a) at the equilibrium spacing:

(6.6)

Figure 6.7 shows the force–separation curves for materials having both strong and

weak interatomic bonds; the slope at r 0 is indicated for each.

Er a

dF

dr

b

r 0

sñ!

6.3 Stress–Strain Behavior • 157

Strain

Stress

0

0

Slope = modulus

of elasticity

Unload

Load

Table 6.1 Room-Temperature Elastic and Shear Moduli and Poisson’s Ratio

for Various Metal Alloys

Modulus of

Elasticity Shear Modulus Poisson’s

Metal Alloy GPa 106 psi GPa 106 psi Ratio

Aluminum 69 10 25 3.6 0.

Brass 97 14 37 5.4 0.

Copper 110 16 46 6.7 0.

Magnesium 45 6.5 17 2.5 0.

Nickel 207 30 76 11.0 0.

Steel 207 30 83 12.0 0.

Titanium 107 15.5 45 6.5 0.

Tungsten 407 59 160 23.2 0.

Figure 6.5 Schematic stress–strain diagram showing linear

elastic deformation for loading and unloading cycles.

JWCL187_ch06_150-196.qxd 11/5/09 9:36 AM Page 157

Young’s

Modulus

(GPa)

(Figure 6.5), application of the load corresponds to moving from the origin up and

along the straight line. Upon release of the load, the line is traversed in the oppo-

site direction, back to the origin.

There are some materials (e.g., gray cast iron, concrete, and many polymers)

for which this elastic portion of the stress–strain curve is not linear (Figure 6.6);

hence, it is not possible to determine a modulus of elasticity as described previously.

For this nonlinear behavior, either tangentor secant modulusis normally used. Tan-

gent modulus is taken as the slope of the stress–strain curve at some specified level

of stress, whereas secant modulus represents the slope of a secant drawn from the

origin to some given point of the curve. The determination of these moduli is

illustrated in Figure 6.6.

On an atomic scale, macroscopic elastic strain is manifested as small changes

in the interatomic spacing and the stretching of interatomic bonds. As a conse-

quence, the magnitude of the modulus of elasticity is a measure of the resistance

to separation of adjacent atoms, that is, the interatomic bonding forces. Further-

more, this modulus is proportional to the slope of the interatomic force–separation

curve (Figure 2.8a) at the equilibrium spacing:

(6.6)

Figure 6.7 shows the force–separation curves for materials having both strong and

weak interatomic bonds; the slope at r 0 is indicated for each.

E r a

dF

dr

b

r 0

sñ!

6.3 Stress–Strain Behavior • 157

Strain

Stress

0

0

Slope = modulus

of elasticity

Unload

Load

Table 6.1 Room-Temperature Elastic and Shear Moduli and Poisson’s Ratio

for Various Metal Alloys

Modulus of

Elasticity Shear Modulus

Poisson’s

Metal Alloy GPa 106 psi GPa 106 psi Ratio

Aluminum 69 10 25 3.6 0.

Brass 97 14 37 5.4 0.

Copper 110 16 46 6.7 0.

Magnesium 45 6.5 17 2.5 0.

Nickel 207 30 76 11.0 0.

Steel 207 30 83 12.0 0.

Titanium 107 15.5 45 6.5 0.

Tungsten 407 59 160 23.2 0.

Figure 6.5 Schematic stress–strain diagram showing linear

elastic deformation for loading and unloading cycles.

JWCL187_ch06_150-196.qxd 11/5/09 9:36 AM Page 157

Shear

Modulus

(GPa)

(Figure 6.5), application of the load corresponds to moving from the origin up and

along the straight line. Upon release of the load, the line is traversed in the oppo-

site direction, back to the origin.

There are some materials (e.g., gray cast iron, concrete, and many polymers)

for which this elastic portion of the stress–strain curve is not linear (Figure 6.6);

hence, it is not possible to determine a modulus of elasticity as described previously.

For this nonlinear behavior, either tangentor secant modulusis normally used. Tan-

gent modulus is taken as the slope of the stress–strain curve at some specified level

of stress, whereas secant modulus represents the slope of a secant drawn from the

origin to some given point of the curve. The determination of these moduli is

illustrated in Figure 6.6.

On an atomic scale, macroscopic elastic strain is manifested as small changes

in the interatomic spacing and the stretching of interatomic bonds. As a conse-

quence, the magnitude of the modulus of elasticity is a measure of the resistance

to separation of adjacent atoms, that is, the interatomic bonding forces. Further-

more, this modulus is proportional to the slope of the interatomic force–separation

curve (Figure 2.8a) at the equilibrium spacing:

(6.6)

Figure 6.7 shows the force–separation curves for materials having both strong and

weak interatomic bonds; the slope at r 0 is indicated for each.

Er a

dF

dr

b

r 0

sñ!

6.3 Stress–Strain Behavior • 157

Strain

Stress

0

0

Slope = modulus

of elasticity

Unload

Load

Table 6.1 Room-Temperature Elastic and Shear Moduli and Poisson’s Ratio

for Various Metal Alloys

Modulus of

Elasticity Shear Modulus Poisson’s

Metal Alloy GPa 106 psi GPa 106 psi Ratio

Aluminum 69 10 25 3.6 0.

Brass 97 14 37 5.4 0.

Copper 110 16 46 6.7 0.

Magnesium 45 6.5 17 2.5 0.

Nickel 207 30 76 11.0 0.

Steel 207 30 83 12.0 0.

Titanium 107 15.5 45 6.5 0.

Tungsten 407 59 160 23.2 0.

Figure 6.5 Schematic stress–strain diagram showing linear

elastic deformation for loading and unloading cycles.

JWCL187_ch06_150-196.qxd 11/5/09 9:36 AM Page 157

Poisson’s

Ratio

Bulk

Modulus

(GPa)

70

80

123

45

180

140

110

310

Solids

Liquids

Water

Ethyl Alcohol

Mercury

2.

1.

2.

Gases

Air, H 2 , He, CO 2 1.01 x 10 -

Table 1.1: Room-Temperature Young’s, Shear and Bulk Moduli and Poisson’s Ratio for

Various materials

For isotropic materials, shear and elastic moduli are related to each other and to

Poisson’s ratio according to

Y =2G(1 +‹)

In most metals G is about 0.4Y; thus, if the value of one modulus is known, the other

may be approximated.

Learning Resource : Poisson’s Ratio

A well prepared introduction to Poisson’s ratio with a lucid

explanation of its usefulness in selecting materials that suit a

given application - for example, the reason why cork is used as

wine bottle stopper.The Ecient EngineerYoutube channel.

URL:https://youtu.be/tuOlM3P7ygA

1.5 Factors Aecting elasticity

Solids are made up of a large number of atoms or molecules arranged in a tightly packed

manner. In the case of crystalline solids, this arrangement also will be in regular periodic

fashion such that the neighbourhood of any atom is identical to that of any other atom

in the solid. Therefore, materials properties are the net result of the collective physical

characteristics of the constituent atoms, how these atoms interact with one another and

geometrical structure in which they are arranged in the bulk material. Each atom in

the solid experiences forces due to the surrounding atoms. The size and shape of a solid

PH8151 17 LICET