Mechanical Properties of Glass

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29 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Mechanical Properties of Glass



Elastic Modulus and
Microhardness


[Chapter 8


The “Good Book”*]


Strength and Toughness [Chapter 18]


Fracture mechanics tests


Fractography


Stress Corrosion


Fracture Statistics


*A. Varshneya, “Fundamentals of Inorganic Glasses”,

Society of Glass Technology (2006)


jmech@mse.ufl.edu

1

Virtual Course on Glass
-

The Properties of Glass:

Mechanical Properties of Glass
-

Lecture
11

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Virtual Course on Glass
-

The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

2

s


e

Log
v

Log K = Log (Y
s
c
½
)

U

r

K
c

Bond Breaking Leads to Characteristic Features

Elastic Modulus Is Related To The Strength of Nearest
Neighbor Bonds

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Virtual Course on Glass
-

The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

3

U

r

Force = F =
-

dU/dr


Stiffness = S
0

= (dU
2
/dr
2
)
r = r0


Elastic Modulus = E = S / r
0

r
0

F

r

r
0

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Virtual Course on Glass
-

The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

4

Elastic Modulus



Governs Deflection


Strength



Governs Load Bearing Capacity

Toughness



Governs Crack Propagation


S

e


Hardness Measures Surface Properties

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Virtual Course on Glass
-

The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

5

P


P

A = Cross
-
sectional Area =
p

r
2


Stress = P / A


r

P = Load On Sample


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Virtual Course on Glass
-

The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

6

P


P

A = Cross
-
sectional Area =
p

r
2


Strain =
D
L=⼠/


r

L

D
L

L = Length

D
L = Change In Length


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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

7

Infinitesimal cube represents triaxial state of stress.

e
y

= (1 /E)[
s
y

-

n
(
s
x

+
s
z
)]
g
xy

= [2(1+
n
) / E] (
t
xy
)

e
x

= (1 /E)[
s
x

-

n
(
s
y

+
s
z
)]
g
yz

= [2(1+
n
) / E] (
t
yz
)

e
z

= (1 /E)[
s
z

-

n
(
s
y

+
s
x
)]
g
z
x

= [2(1+
n
) / E] (
t
zx
)


Special Cases of Loading Often Occur

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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

8

(a) Tensile stress. (b) Shear stress. (c) Hydrostatic pressure.

In uniaxial loading in the x direction, E (or Y)
relates the stress,
s
x
, to the strain,
e
x
.







s
x

= E
e
x



e
y

=
e
z
=
-

n e
x



s
xy

= G
g


p = K
D
V


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Virtual Course on Glass
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The Properties of Glass: Charge Conduction in Glass
-

Lecture 1

9




e

s

In the case of shear loading, the
shear modulus
is
appropriate

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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

10




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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

11

(a) Tensile stress. (b) Shear stress. (c) Hydrostatic pressure.

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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

12


s


D

V/ V
0

In the case of hydrostatic pressure, the bulk
modulus is appropriate.

There is a relationship between E, G and K
(and of course Poisson’s ratio,
n
)

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

13

G = E / [2 (1+
n
)]


K = E / [3(1
-
2
n
)]



Note:
-
1 ≤
n

≤ 0.5.


(When
n

= 0.5, K ∞ and E 3G. Such
a material is called incompressible.).



There is a relationship between E, G and K
(and of course Poisson’s ratio,
n
)

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

14

G = E / [2 (1+
n
)]


K = E / [3(1
-
2
n
)]

So, when we determine any two parameters,
(for isotropic materials) we can calculate the
others.

There are several techniques used to measure
the elastic modulus:


A. Stress
-
strain directly (load
-
displcament)


1. tension


2. 3
-
pt flexure


3. 4
-
pt flexure


4. Hydrostatic pressure


5. Torque on rod

B. Ultrasonic wave velocity


1. Pulse echo


2. Direct wave

C. Beam Vibration

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The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

15

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

16

P


P

A = Area =
p

r
2


r



Elastic Modulus = Stress / Strain


S or
s

Strain = e or
e


A = Brittle


B = Ductile

S =Stress = P / A



Strain =
D
L=/⁌



To measure E from flexure, need to calculate
the stress and strain.

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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

17


A


A

s

= 3PL / (2 b h
2
)
e = d
/ L


b

h




d

P

Pulse echo technique is often used to measure
modulus

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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

18

C. Kittel, Intro. To Solid State Physics, J. Wiley & Sons

Pulse Echo technique is one of the most
reliable.

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

19

In the simplest case for isotropic materials there
are direct relationships.

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

20

v
L

= [ E /
r
]
1/2


(Longitudinal waves)

v
S

= [ G /
r
]
1/2


(Shear waves)

For the beam vibration technique, we stimulate
the flexural modes.

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

21

Fig 8
-
5


For beam bending:

E = (0.946 L
4

f
2

r

S) / h
2

f = frequency

S = shape factor

H = width and height

L = length

r

= density

In general, E decreases as the size and
concentration of the alkali cations increases

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

22

Fig 8
-
6a


E decreases as the size and concentration of the
alkali cations increase

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

23

E

K

G

n
x

100

Fig 8
-
6b


E decreases as the size and concentration of the
alkali cations increases

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

24

Fig 8
-
6c

E increases with addition of metal oxide (MO)
[except PbO]

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

25

Na
2
O

x MO


5SiO
2

Fig.8
-
7 (Varshneya)

Lithia
-
aluminosilicates have greater E values
than SiO
2

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

26

Fig.8
-
8

In general, bulk moduli of silicate glasses
increase with temperature (except at low
temperatures [0
-

60K])

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Virtual Course on Glass
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The Properties of Glass: Charge Conduction in Glass
-

Lecture 1

27

N.B.
-

the
compressibility,
k,

is
being graphed in the
figure (Fig. 8
-
9).


(The compressibility
is the reciprocal of
the bulk modulus.)

Composition and structure affect the values of
elastic moduli.

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

28

N.B.: at low (< 10mol%)
alkali content, E with
B
2
O
3

addition.

However, with greater
alkali content glasses
addition of B
2
O
3

leads
to a maximum in E.

Complications of silicate glasses makes
predictions difficult

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

29

F

=

[
-
a

/

r
n

]+

b

/

r
m

(Condon
-
Morse)

Force

=

F

=

-

dU/dr

Stiffness

=

S
0

=

(dU
2
/dr
2
)

r

=

r
0

Elastic

Modulus

=

E

=

S

/

r
0





Complications of silicate glasses makes
predictions difficult

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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

30

F

=

[
-
a

/

r
n

]+

b

/

r
m

(Condon
-
Morse)

Force

=

F

=

-

dU/dr

Stiffness

=

S
0

=

(dU
2
/dr
2
)

r

=

r
0

Elastic

Modulus

=

E

=

S

/

r
0


General rules:

1.
E increases as r
0
x

decreases

2.
E increases as valence, i.e.,
q
a

x q
c

3.
E affected by bond type (covalent, ionic,
metallic).

4.
E affected by structure (density, electron
configuration, etc.)




Microhardness is a measure of surface
properties and can be related to elastic
modulus, toughness and surface tension.

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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
-

Lecture 11

31

Hardness = Force / Area

Many hardness tests are available

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

32

The most common microhardness diamond
tips for glasses are Vickers and Knoop

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

33


Hv = 1.854 F / D
2

(Actual area) KHN = 14.23 F / L
2

(Projected area)

Hardness = Force / Area

Fig. 8
-
12

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

34

Note plastic flow in silicate
glass using a Vickers
microhardness indenter.










Plastic flow in Se glass using
a Brinell microhardness
indentation.

Fig. 8
-
13 a & b

Diamond hardness indentations can result in
elastic and plastic deformation.

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

35

Microhardness can be measured dynamically

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

36

H
vL

= 37.84 F / h
2
max


(from loaded depth, h
max
)





H
vf

= 37.84 F / h
2
f


(from unloaded depth, h
f
)






F = a
1
h + a
2

h
2

(equation fit to curve)


H
vL2

(GPa)= 37.84 a
2
{ load independent hardness; a
2

= N/
m
m
2
}

Refs. 34 and 35 in Chapter 8.

Microhardness can be measured dynamically

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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

37

Measure dF/dh on initial
unloading



E
r
= (

p

/ 2



A) [dF/dh]






E
r

=[(1
-
n
2
)/E] + [(1
-
n
i
2
)/ E
i








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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

38

Materials & Methods

o

The energy spent during the nanoindentation process can be
categorized as plastic energy (W
pl
) and elastic energy (W
el
).
The indenter penetrates the sample and reaches the maximum
penetration (h
max
) at P
max
. During the unloading process, the
compressed zone recovers and the final depth of the indent (h
f
)

is often much less than h
max
.


Elastic Moduli and microhardness are two
important mechanical properties.

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Virtual Course on Glass
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The Properties of Glass: Mechanical Properties of Glass
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Lecture 11

39

Elastic modulus is a macroscopic measure of the strength of bonds at the atomic
scale.


Hooke’s law (stress proportional to strain) defines the moduli of linear elastic
solids.


For isotropic glasses only two constants are required


others can be calculated.
Note:
-
1 ≤
n

≤ 0.5. (When
n

= 0.5, K ∞ and E 3G).


Elastic modulus is best measured using the “pulse echo” or similar technique.
For silicate glasses, E 70≈ GPa and
n

≈ 0.22.


Hardness is a measure of the resistance to penetration. Both densification and
material pile
-
up are observed in glasses.


Vickers indentation is the most common diamond indenter for glasses.

For a silicate glass, H v ≈ 5.5 GPa