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
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Lecture
11
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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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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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Lecture 11
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Elastic Modulus
–
Governs Deflection
Strength
–
Governs Load Bearing Capacity
Toughness
–
Governs Crack Propagation
S
e
Hardness Measures Surface Properties
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P
P
A = Cross
-
sectional Area =
p
r
2
Stress = P / A
r
P = Load On Sample
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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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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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(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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e
s
In the case of shear loading, the
shear modulus
is
appropriate
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Lecture 11
11
(a) Tensile stress. (b) Shear stress. (c) Hydrostatic pressure.
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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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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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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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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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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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C. Kittel, Intro. To Solid State Physics, J. Wiley & Sons
Pulse Echo technique is one of the most
reliable.
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In the simplest case for isotropic materials there
are direct relationships.
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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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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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Fig 8
-
6a
E decreases as the size and concentration of the
alkali cations increase
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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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Fig 8
-
6c
E increases with addition of metal oxide (MO)
[except PbO]
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Na
2
O
x MO
5SiO
2
Fig.8
-
7 (Varshneya)
Lithia
-
aluminosilicates have greater E values
than SiO
2
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Fig.8
-
8
In general, bulk moduli of silicate glasses
increase with temperature (except at low
temperatures [0
-
60K])
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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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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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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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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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Hardness = Force / Area
Many hardness tests are available
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The most common microhardness diamond
tips for glasses are Vickers and Knoop
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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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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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Microhardness can be measured dynamically
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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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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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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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Lecture 11
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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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