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)
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1
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The Properties of Glass:
Mechanical Properties of Glass

Lecture
11
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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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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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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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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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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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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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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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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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The Properties of Glass: Mechanical Properties of Glass

Lecture 11
10
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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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The Properties of Glass: Mechanical Properties of Glass

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

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

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

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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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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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

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

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

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

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

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

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

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

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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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

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

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

The Properties of Glass: Mechanical Properties of Glass

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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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

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

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

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

Lecture 11
35
Microhardness can be measured dynamically
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The Properties of Glass: Mechanical Properties of Glass

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

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

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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The Properties of Glass: Mechanical Properties of Glass

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
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