AL Solids

P.23

P.23Types of solids

Crystalline (Long range order) e.g. metals, sugar, salt

P.23Types of solids

Amorphous (Disorder)

e.g. glass

P.23Crystal structures

Kinds of bonds formed

- Hydrogen bond

- Ionic bond

- Covalent bond

- Metallic bond

Size and shape

- fcc

- bcc

P.24Size of a molecule

Monolayer experiment

P.24Size of a molecule

Using Avogadro constant

nm14.0r

cm109x6x

64r

3

4

cm109x6x

64=atomeach of volume

106x=atomsCu of number

cm9

64=Cu of volumemolar

gcm9=Cu ofdensity

64=Cu of mass atomic

323

3

323

23

3

3-

P.25Elasticity of metals

P.25Hooke’s Law

The extension is proportional to the force in a wire if the proportional limit is NOT exceeded.

P.25Stress and Strain

Strength

The Greatest force a material can withstand before breaking

Stiffness (force constant)

The opposition a material sets up to being distorted by having its change of size and/or shape.

Stress

The force acting on unit cross-sectional area

Stress = F / A

Strain

The extension of unit length

Strain = e / l

P.26

Stress S=(Weight W) / (Area A)

New volume V’ = 23 V = 8V

New weight W’ = V’ g = 8 V g = 8 W

New area A’ = 22 A = 4 A

New Stress S’ = (8 W) / (4 A) = 2 S

P.27Stress and strain behaviour of ductile material

P.27Stress and strain behaviour of ductile material

P.27Ductile materials

Lengthen and undergo plastic deformation until they break

P.27Brittle materials

Lengthen and break just after elastic limit L is reached.

P.28Fatigue

Under varying stress, fracture may occur even the maximum stress has not reached.

P.28Creep

Under high temperature, the metal continues to deform even under constant stress.

P.28Young’s modulus E

Under elastic limit, stress α strain

Estress

strain

F

Ae

( )

( )

E

F

Ae

F

eA

FEA

e ke

( )

( )

( )

EA

Stiffness

P.29Experiment

EF

eA

FEA

e

slope m

Em

A

( )

=EA

P.26

mg = ke

E = (F l) / (A e)

F = [(E A) / (l)] e

k = [(E A) / (l)]

For same material, Young modulus will be the same

New l’ => new k’=2k

m’g = k’e’ + k’e’

2m g = 2ke’ + 2ke’

e’ = (0.5) e

P.30

Q: larger strain before break

P: E=stress/strain = slope = stiffnessstrength = greater breaking stress

P.31

=/E. Strain is independent of the unstretched length

NNY

=/E =(F/(EA)). When A increases, decreases.

=/E =(F/(EA)). Ebrass < Esteel. When E decreases, increases.

P.31

%E = %m + %g + %l + 2%d + %e

%l = 0.51%

%d = 1.64%

%m = 0.1%

%e = 2.56%

%g = 1.02%

ed

mgl

er

lmg

Ae

FLE 22

2

)(

)(

P.31Energy stored in a wire

1

21

21

2

2

Fe

EAee

EAe

( ) )

( E =F / A

e /

W FeEAe

e EAe

1

2

1

2

1

2

2

( )

P.32Energy per unit volume of a wire

1

2

1

21

21

2

Fe

VFe

AF

A

e

x stress x strain

( )( )

( ) ( )

P.32

Energy on the wire

FeW2

1

Ae

FlE

)(1033.81003.01022

)1(100

22

1 3411

22

JEA

lF

EA

FlFW

P.32

Energy on the wire

FeW2

1

Ae

FlE

)(1

)1.0(2

)02.0(101105

22

1 2682

Jl

EAee

l

EAeW

2

2

1..1 mvEKW

)/(20 smv

P.32

)(20.1

1001092

101.01017 6

2310

NF

F

l

lEA

l

EAeF

P.32

7542.28

1

00019998.0)21022.1(103.12 2310

l

EAeF

00019998.010004.1'

0004.102.01' 22

lle

l l’

o854.88

5002.0

1tan

)(115.010/)854.88cos(7542.282/cos2 kggFm

mg

P.32

Their extensions should be the same.

2brassE

E

F

Fconst

E

F

l

EAeF steel

brass

steel

x

Taking moment at point O

O

)(600

9005.0

)900(

mmx

xx

xFxF steelbrass

P.33

These two wires are connected in series. They have same force.

2122

21

1

1 eekel

AEe

l

AEF

l

EAeF combined

1

11 /

l

AEFe

2

22 /

l

AEFe combinedkFee /21

combinedkFl

AEF

l

AEFee ///

2

2

1

121

1221

21

2

2

1

1

2

2

1

1

lElE

AEE

lAE

lAE

lAE

lAE

kcombined

P.33

mg = k x

YYN

m x

x = mg / k x depends on weight of the ball and force constant k

The ball eventually stopped, energy is lost due to air resistance.

P.33

P.33

P.33

P.33

P.34

P.34

P.34

P.35

P.35

P.35

P.35

P.36

P.36

P.36

P.36

P.36

P.38Stress – strain curves for glass, copper and rubber

P.38

Stiffer material needs a greater stress to produce same strain. Young modulus is larger which has steeper slope.

Y

YY

Stronger material can withstand a greater stress before breaking

Z breaks at strain=1.6, hence e = 1.6l . The length of Z at breaking point = 2.6 l

P.38

Ductility is independent of its melting point.

Young modulus is a measure of stiffness. It is NOT a direct measurement of the ductility of material.

Copper has performed plastic deformation between the elastic limit and the facture point. Glass has narrow gap only.

Extension under same stress depends on the proportional part. It is independent of plastic deformation.

P.39Intermolecular forces

Va

r

b

rp q

FdV

dr

pa

r

qb

rp q 1 1

P.40Equilibrium spacing of molecules

Stable equilibrium occurs when P.E. is minimum.

)(

1

11

11

0

qbpa

r

rr

rr

qp

o

qp

qp

o

qb

o

pa

F

qbpaF

P.40Elasticity and Hooke’s Law

Near ro, F-r curve is nearly a straight line

kdF

dr

EF A

e

/

/

Er r r

F E r r ro o

o o

force between two molecules F / r 2o

( ) /

( )

P.40Thermal expansion

At higher temperature, molecules have some energy and oscillate between X and Y.

Due to asymmetry, mean position will be G.

Metal is expanded.

At very high temperature, the molecule separation will be very large. The corresponding energy needed is called latent heat or binding energy.

P.41

kdF

dr

EF A

e

/

/

Er r r

F E r r ro o

o o

force between two molecules F / r 2o

( ) /

( )

1183 105.1106.4/109.6 EESlope ro

P.41

321

r

C

r

CV

42

21 3

r

C

r

C

dr

dVF

0342

20

1

0

r

C

r

C

dr

dVF

22

10

3r

CC

1

20

3

C

Cr

P.41

126 r

b

r

aV

0126

130

70

r

b

r

a

dr

dV

6

1

0

2

a

br 6

06

0

22

2min 22422 r

a

ar

a

b

a

ab

b

aba

V

P.41

137

126

r

b

r

a

dr

dVF

015642

1482

2

r

b

r

a

dr

Vd

dr

dF

0

6

16

16

06

1

7

13

7

13

7

26r

a

ar

a

br

137

126

r

b

r

aF

130

6

13

60

70

6

7max

713

6

713

6

r

ar

r

aF

13

71

13

76

70

6

7

6

7

max

r

aF

6

7

70

max 13

7

13

36

r

aF

P.41

(1) : Spacing of molecules at equilibrium

(2) : Max. distance before breaking

(3) : force constant => stiffness

(4) : max. tension that the material can tolerate before breaking, it is called tensile strength.

P.42

U=0

P.42

36 r

B

r

A

dr

dVF

25 r

b

r

aV

P.42

A1 is +ve energy that the system repel back to eq. pt.

A2 is -ve energy that the system attract back to eq. pt. with max. attractive force.

A3 is -ve energy that the material breaks and go to infinity

A2 + A3 is energy needed to move the molecules from eq. pt to infinity

P.43

P.43

P.43