THE ORIGINS OF INTERNAL STRESSES
IN POLYCRYSTALLINE AL 203
AND THEIR EFFECTS ON MECHANICAL PROPERTIES
by
JOHN EDWARD BLENDELL
B.S., Alfred University
1974
B.A., Alfred University
1974
S.M., Massachusetts Institute of Technology
1976
Submitted in partial fulfillment of the requirements
for the degree of
DOCTOR OF SCIENCE
at the
Massachusetts Institute of Technology
June, 1979
Signature of Author.
Department of Materials Science and, Engineering
May 4, 1979
/
Certified by......
.........
..
.............-
....
.
.. .
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Thesis Supervisor
Accepted by.............
..........
Chairman, Department4 al Committee(on Graduate Students
Archives
iJ
Jt.i
ABSTRACT
The Origins of Internal Stresses in Polycrystalline
Al203
and Their Effects on Mechanical Properties
by
John Edward Blendell
Submitted to the Department of Materials Science and
Engineering on May 4, 1979 in partial fulfillment of
the requirements for the degree of Doctor of Science.
A model for the stresses arising from thermal expansion anisotropy
during cooling polycrystalline materials has been developed. The model
incorporates the relaxation of stresses at high temperatures by diffusional creep with the generation of stresses due to thermal expansion
differences across a grain boundary. The room temperature stresses are
a function of grain size and cooling rate, which was assumed to be
linear. It was found that the calculated room temperature stresses
significantly increase with grain size and cooling rate. From the results of the calculation, an effective lower limit for stress relaxation could be defined. Using this effective lower limit and a linear
dependence of stress on temperature below the limit yields a simple
method for estimating the room temperature stress.
The stresses were measured directly by a spectroscopic technique
and inferred from measurement of the toughness. The samples were commercial A12 03 which were annealed at 2150K for various times and cooled
at different linear rates.
While no grain size dependence of the stresses was observed, the
cooling rate dependence was clearly demonstrated. The magnitude of the
stresses measured by the spectroscopic technique were in good agreement
with the value predicted from the model. There exists at present no
agreement in the literature on the effect of thermal expansion stresses
on the toughness. Thus, the observed increase in the toughness with
increasing stress could not be used to determine the magnitude of the
stresses.
The room temperature stresses can be minimized by slow cooling in
the range of temperature where stress relaxation has been shown to be
important.
Thesis Supervisor:
Title:
Robert L. Coble
Professor of Ceramics
Acknowledgement
I would like to thank Bob Coble for his guidance and friendship
during my time at M.I.T.
I also wish to thank
It has been very enjoyable working with him.
Dick Charles for his help in developing the stress
relaxation model, Rowland Cannon for many illuminating discussions and
helpful suggestions and Lou Grabner for the use of his equipment and
assistance in making the fluorescence measurements at N.B.S.
I would also like to extend my deepest appreciation to Carol Handwerker, for without her assistance and support this thesis would not now
be finished.
I also thank Pat Kearney, Al Freker and John Centerino for their
assistance with my experiments and to Diane Saunders for typing this manuscript.
Special thanks go to my many friends at M.I.T., especially Todd
Gattuso, John Blum, Pat Foley, Joe Dynys, Paul Lemaire, Jim Hodge and the
Rock for helping me maintain my sanity during my stay at M.I.T.
The support of the U.S. Department of Energy under Contract No.
EY-76-S-02-2390.AO01
is gratefully acknowledged.
4
Table of Contents
Page
Title ..................................................................
Abstract ..............................................................
2
Acknowledgement.......................................................
3
Table of Contents..................... ................................
4
List of Figures........................................................
6
List of Tables ........................................................
8
Introduction......................................................
9
I
II Stress Due to Thermal Expansion Anisotropy........................ 16
1. Stress Relaxation.............................................
III
18
1.1
Relaxation of TEA Stresses ................................ 18
1.2
Effective Diffusion Coeffficient.......................... 24
1.3
Calculation of TEA Stresses............................... 26
1.4
Relaxation of Macrostress ................................. 29
2. Summary ............................................. .........
31
Preparation of Samples...........................................
36
IV Spectroscopy............................................ ........ . 41
1. Piezospectroscopic Effect ..................................... 41
1.1
Single Crystal............................................
41
1.2
Polycrystal Effects......................................
44
2. Experimental.................................................. 49
3. Discussion.................................................... 49
4. Summary................................................ ........ 53
V
Indentation Fracture.............................................
62
I. Crack Formation................
1IU 1
Pl
hn
1.2
Sharp Indenters.
II ",-.L+t,
- I
.a
e
..II.CIIL
*
. . . .
. . *. . .
. . . 0
.....
· · ·
.......................
· ·
.....
2. Fracture Mechanics .........................
..
3. Experimental..
.e....
...........................
o...
· · ·
4. Discussion.................................
.....
· ·
5. Summary....................................
.....
· · ·
VI Fracture and Toughness.
1. Fracture Strength.
........................
.....
..........................
...
e..
2. Fracture Toughness .........................
..
oo..
3.
..
o..
Experimental...............................
4. Discussion.................................
5. Summary....................................
VII Summary......................................
VI II Suggestions for Future Work....................
Appendix A - Development of Relaxation Model..
· · ·
· · ·
· · ·
· · ·
.....
.....
0.....
...
· · ·
e..
.....
· · ·
Appendix B - Analytical and Numerical Soltuion to
Relaxation Equation...............
101
Appendix C - Calculation of Av for Quenched Pl te...
105
Appendix D - Spectroscopic Data................
107
..... o
Appendix E - Fracture Toughness Data...........
.....
113
References...................................
.....
120
Biographical Note. .............................
.....
121
List of Figures
I-1
Strength as a function of quenching temperature for A1203.
1-2
Effect of grain size on the toughness of anisotropic materials.
II-1
Calculated stress due to TEA as a function of temperature for
ý = -1.0 K/min.
11-2
Calculated stress due to TEA as a function of temperature for
G = 50 m.
11-3
Critical temperatures (TC , TR, TE) as a function of cooling rate.
III-1
Geometry of samples.
111-2
Typical microstructure of samples.
IV-l
Typical fluorescence spectrum of ruby.
IV-2
Line width and lineshifts for ruby as a function of temperature.
IV-3
Line shift for ruby as a function of applied pressure.
IV-4
Orientation of X1 , X2,, X3 axes.
IV-5
Experimental arrangement used to measure fluorescence spectrum.
IV-6
Adsorption as a function of wavelength of the filters.
IV-7
Stress measured from linewidth as a function of cooling rate.
IV-8
Stress measured from linewidth as a function of cooling rate.
V-l
Geometry of blunt and shart indenters.
V-2
Typical indentation impressions.
V-3
Toughness of A12 0 3 determined from indentation as a function
of cooling rate.
VI-l
Experimental arrangement used to apply internal pressure.
VI-2
Fracture stress as a function of G
VI-3
Typical strength histograms.
2
VI-4
Typical notch used for toughness measurements.
VI-5
Typical fracture surface of notched samples.
VI-6
Toughness measured from notched samples as a function of
cooling rate and anneal time.
D-1 Width of R, and R2 lines as a function of cooling rate.
D-2 Width of R l and R2 lines as a function of anneal time.
D-3 Shift of R1 and R 2 lines as a function of cooling rate.
D-4 Shift of R1 and R 2 lines as a function of anneal time.
E-l
Indent impression and crack size as a function of cooling rate.
E-2
Indent impression and crack size as a function of anneal time.
8
List of Tables
Table III-1
Grain size as a function of anneal time
Table
IV-l
Values of Piezospectroscopic tensor (i..)
Table
D-l
Line shift and line width as a function of anneal
time and cooling rate.
Table
E-l
KIC as determined from indentation test.
Table
E-2
KIC as determined from notched ring test.
I. Introduction
In a two-phase polycrystalline solid in which the phases have different expansion coefficients or in a polycrystalline material, with random
orientation, which has anisotropic thermal expansion, heating or cooling
leads to changes in localized stresses across the grain boundaries (Boas &
Honeycomb, 1946).
The magnitude of the local stress depends on the dif-
ference in thermal expansion across the boundary, the elastic moduli, and
the difference between the temperature at which the body was stress-free and
the current ambient temperature.
In A1203 the anisotropy in thermal expan-
sion will cause localized stresses to be developed in a polycrystalline body
as it is cooled from the sintering temperature.
It is observed that in A 20,, and other anisotropic oxides, boundary
cracking is much more prevalent in coarse than in fine grained samples.
It
is assumed that the localized stresses due to the thermal expansion anisotropy (TEA) cause the boundary cracking.
However, since none of the vari-
ables used in a simple model to predict the TEA stress are grain-size
dependent, the observed cracking behavior is not consistent with a critical
stress model alone.
This can be reconciled by use of an energy criterion
for fracture (Davage & Green, 1968).
The stored elastic strain energy due
to the TEA stress is a function of the volume of material under stress,
which scales with the third power of the grain-size.
The energy needed to
form new surface during fracture is a function of the area fractured, which
scales with the square of the grain-size.
Hence, at some grain-size, there
will be enough stored elastic strain energy to supply the energy for new
surface formation and spontaneous fracture of the grain boundary will occur.
In a polycrystalline array, with random orientations of the grains,
only some of the boundaries will have the highest possible TEA stress and
therefore only some of the boundaries will crack.
Also there is usually a
range of grain-sizes present in a sample, so it is not expected that at
some critical grain-size complete fracture of the sample will occur.
An implicit assumption in this simple model is that there exists an
effective freezing temperature, or a temperature above which all stress
are relaxed and below which all stress cannot relax, and that the effective freezing temperature is grain-size independent.
The actual tempera-
ture and grain-size dependence of stress relaxation can be directly determined from the creep data or from creep models using available diffusivity data.
The diffusional creep models would predict that at small
grain sizes, relaxation of stress can occur at lower temperatures than
for large grain sizes.
Also, the relaxation temperature is a function of
the time scale of the experiment.
The isothermal-hold time to relax all
stress increases rapidly as the temperature decreases because of the
temperature dependence of the diffusivity that controls the relaxation.
An example of the relaxation of stresses at high temperatures by creep is
observed in the quench strenghening of A12 0 3 .
It has been shown
(by Marshall, et. al., 1978) that the strengthening of quenched
A120 3 is due to a compressive surface layer.
In order for a com-
pressive surface layer to be developed during quenching, the material in
the center of the sample must flow and relax the stress due to the temperature distribution.
The sample must be at a sufficiently high temperature
so that flow can occur, as seen in Figure 1-1 which presents the data of
Stolz and Varner (1977).
If the starting temperature is too low, the sam-
ple can be weakened by thermal shock.
Thus, it is expected that the magnitude of the TEA stresses are a
function of the grain-size and the cooling rate, which defines the time
scale for relaxation.
This dependence of TEA stresses on grain-size and
cooling rate must be considered when using the energy balance to calculate
the critical grain-size for microcracking.
In this study, a model for the
stresses due to TEA which includes stress relaxation by creep has been developed.
The model assumes that the stress relaxation is given by the
stress relaxation of a rheologically simple material, or that the behavior
is linear visco-elastic.
The stress due to TEA is predicted to increase
with increasing grain-size and cooling rate.
divided into three regions.
The stress function can be
Above some temperature, TC, stress relaxation
is very fast and all stresses will relax.
Below a different temperature,
TR < TC, no stress relaxation will occur and the stress wil be a linear
function of temperature as in the simple model.
but not all of the stress will relax.
Between TC and TR, some
The values of TC and TR will vary
with grain-size and cooling rate.
While it is expected that TEA stresses can effect the mechanical properties, the exact effect is not known.
the fracture stress of BaTiO
3
Pohanka, et. al.
(1976) measured
above and below the Curie temperature (120 0 C).
They found that below the Curie temperature, where BaTiO
3
is tetragonal,
the strength was lower than above the Curie temperature, where BaTiO
cubic.
3
is
Since all other factors are the same (i.e., grain-size, porosity,
surface finish, test technique) and the temperature change is small
(150 0 C to 250 C), the reduction in strength must be due to the TEA stress
that occurs in the tetragonal phase upon cooling below the Curie temperature.
Thus it is expected that for A12 03 , the strength would also be re-
duced by the TEA stress.
The effect of TEA stresses on the fracture toughness is not clear.
model by Evans, et. al.
A
(1977) predicts a decrease in toughness with in-
creasing grain-size, but a model by Rice, et. al. (1978)
predicts an increase
in the toughness with increasing grain-size up to a certain grain-size and
then a decrease in toughness (Figure 1-2).
increase in the TEA stress with grain-size.
based on an energy balance approach.
Neither model considers the
Both of the above models were
Different measurements of the tough-
ness as a function of grain-size have given inconsistent results.
In A1203
the general trend is for the toughness to increase with grain-size, but
this may be due to the test technique or the mode of fracture.
It is ex-
pected that the toughness will increase with increasing TEA stress based
on the best available data and the model by Rice.
Thus it is expected that the effects of TEA stress would be to lower
the strength and increase the toughness of a material as the TEA stress
increases.
vestigated.
In this study, the possible influence of TEA stresses was inThe fracture stress and toughness of A12 03 samples with
various grain-sizes and cooling rates were measured.
The toughness was
measured from microhardness indentations and by a notched ring technique.
The fracture stress was measured by an expanded ring test.
A direct
measurement of the TEA stresses was performed using a spectroscopic tech-
13
nique.
It has been well established that stresses affect the shape and
position of the chromium fluorescence lines; by measuring the positions
of these lines, the TEA stress can be calculated.
The model developed to predict TEA stresses will be presented in the
next section.
The experiments performed to measure the effects of the TEA
stresses will be presented in the following sections.
Ojuu
11111I
- 600
E
Z
b
4 00
200
O
-I
T
-I
-
I
I
I
I
I
I
I
I
1000
1200
,I
1400
II
I_
1600
Temperature (oC)
I-1
Strength as a function of quenching temperature for Al 203.
Data from Stolz and Varner (1977) for quench of 5mm by
5mm square bars, G = 10p into silicon oil (Blp > -1000K/min).
C2
n
0
0
0
0
40
'a
U)
20
c 0~414
C\J
>0
x•
I0
o 0.2 0.4 0.6
G/Gcr,,it
(a)
1-2
0.8
1.0
1
100
10
6(11m)
(b)
Effect of grain size on the toughness of anisotropic materials.
(a) from Rice, et. al. (1978) for all A1 2 0 3 (Yf O KIC)
(b) from Evans, et. al. (1977) G
is the grain size at which
crit
spontaneous microcracking occurs.
1000
II. Stresses Due to Thermal Expansion Anisotropy
That thermal expansion anisotropy (TEA) affects the mechanical properties of materials was proposed by Howe (1910) for the weathering of rock
and by Lord Rayleigh (1934) for the loss of strength of marble upon heating.
Boas and Honeycombe (1946) examined the plastic deformation of non-cubic
metals upon heating and cooling.
They concluded that the deformation was
due to stresses introduced by the thermal expansion anisotropy and calculated the magnitude of the stress on a boundary of two crystals as
=
(
-
E *E
) E +E
AT
-1
where a is the stress, aI, El and a2, E2 are the expansion coefficients and
elastic moduli for the two grains and
change.
T = (TI - TF) is the temperature
Lazzo (1943) calculated the stress due to TEA for iron and steel
and also the stresses due to inhomogeneities and phase transformations.
Armstrong and Borch (1971) extended the analysis to include the single
crystal elastic constants for a hexagonal crystal structure.
The stress is
given by:
a
=
(oi -
2 )AT
where S is the effective compliance.
11-2
The effective compliance is a complex
function of the elastic constants, crystal symmetry and relative orientation of the grains across the grain boundary.
The TEA stresses (aTEA) for
several metals and BeO were calculated using the melting temperature as the
initial temperature and it was found that, for Be, the magnitude of aTEA
was comparable to the yield stress.
The TEA stresses could cause plastic
deformation or, at least, influence the fracture behavior in Be.
A similar
analysis is given by Likhaehev (1961).
Beussen (1961) calculated the local stress distribution and found the
normal and shear stresses at a grain boundary to be:
SN
=
Os
=
-
P -
EAT
II-3a
11-3b
I EAT
where aN and x are the expansion coefficients normal and parallel to the
grain boundary and a is the average macroscopic expansion coefficient.
Any
difference in elastic modulus was assumed not to have a large effect.
The stresses due to an inclusion or inhomogeneity in an otherwise uniform matrix have been considered by many authors.
The model usually con-
sidered is for a solid of revolution having homogeneous elastic properties.
The stress due to the difference in properties can be calculated at any location in the matrix or inclusion.
The actual stress state in a polycrys-
talline array will vary with location, but the stress in an individual grain
can be estimated using the above model.
The TEA stress can be completely relaxed if the boundary between the
grain fractures.
If the coherency of the boundary is maintained, the TEA
stress can be relaxed by plastic deformation or diffusional creep.
A model
for the relaxation of TEA stress in A120 3 , assuming that no grain boundary
fracture occurs, will be presented in the next section.
This model will
also be applied to the relaxation of stresses due to non-linear temperature
distributions.
II-1.
Stress Relaxation
When a polycrystalline anisotropic material is cooled from high tem-
perature, stresses develop according to the relationships discussed above.
An
At high temperatures the stresses can relax by both slip and diffusion.
analysis of the deformation data for Al
203
by Heuer, et. al.
cates that at 1800K, for grain sizes less than Imm
(1979),
indi-
(1000m) and stress
levels below 10MPa (1.4 ski), the deformation is controlled by diffusion.
Even at 2200K the contribution of slip to the deformation is negligible.
Therefore, the stress relaxation of intermediate
grain-sized Al 203 can be
considered simply as a diffusional process, provided microcracking does not
occur.
11-1.1
Relaxation of TEA Stresses
For macroscopic stress on a polycrystalline Al
20
sample, the steady-
state deformation behavior can be described by the Nabarro-Herring creep
equation:
(Nabarro (1984), Herring (1950))
S
where
c =
dE:
=
11-4
CNH o
is the strain rate and a is the applied stress.
The effec-
tive diffusional viscosity, CNH is given for an elemental solid as:
CNH
where
G
1
qeff
14QD
14eff
G2 kT
11-5
is the average, grain-size, k is Boltzmann's constant, n is the
atomic volume and Deff, is the effective diffusion coefficient, which takes
into account the different possible paths for transport.
After a constant
strain is applied to a sample obeying the Nabarro-Herring creep equation
(11-4), the stress relaxation will be governed by
_
-d
do
dt
I -6a
Seff
and the stress as a function of time is
o(t)
=
o(t=0) exp
l-6b
t/TR]
where S is the effective compliance and TR is
eff
R
II-7a
G
for a and c being the shear stress and strain, and G is the shear modulus;
or
3
R
Ieff
I-7b
E
for a and e being the uniaxial stress and strain, and E is Young's modulus.
During heating or cooling, the strain due to TEA will be constant for
a step change in temperature.
Assuming that the expansion coefficient is
constant in the range of temperatures used, for a constant rate of temperature change the strain rate will be constant.
be governed by an equation of the form:
d
dt
The stress relaxation will
(Lee (1960))
+
S
11-8
Sef f
where e is related to the cooling rate and expansion coefficient, and E/S
is related to the generation of stress as the temperature is changed.
S-- all stress can relax.
STeff
S
When
Sneff
<S
no stress can relax.
When
For a spherical inclusion
of radius a with elastic properties oa,
v, imbedded in a matrix having elastic properties c 2 , E2,
El,
v 1, which upder-
goes a change in temperature of AT, the stress in the inclusion is hydrostatic and is given as:
(Sesing (1961))
OH
H
1(+2 =
11-9
OL)AT
1+v1
1-2v 2
2E 1
E
-
The radial and tangential stress in the matrix are:
CRR
aO = -
11-10a
()
- -H
1
a3
H(r
where r is the distance from the center of the inclusion.
The stresses in a uniform matrix due to the presence of a semi-infinite
cylinder are given by Myklestad (1942).
The model assumes that the cylinder
is identical to the matrix except that the cylinder is hotter than the
matrix by AT.
The stress state is not hydrostatic.
The shear stress is
infinite at the boundary of the cylinder and decays to zero in the matrix.
The normal stresses are approximately constant through the cylinder and
decay into the matrix.
The TEA stresses in two grains will be similar, but
will be modified by the finite dimensions of the grains.
It will be
assumed here that the stress is constant across the grain and are given by
equation 11-2 before any relaxation occurs.
If the two grains were isolated, the shear stress could relax by grain
boundary sliding.
In a polycrystalline array the grains are constrained by
the rest of the matrix, provided cracking of the boundaries or grains does
not occur, and boundary sliding will not occur.
The relaxation of the
shear stress requires diffusion to maintain the coherency of the sample.
Some relaxation of stress may occur in grains on the surface of the sample
which are not constrained.
Only a small number of grains can relax, and
this effect will not change the average stress values in the bulk.
Since
the scale of the stress is on the order of the grain size, the deformation
of an individual grain will be taken as the deformation of a sample for an
equivalent applied macrostress.
The stress relaxation, at steady state for
a constant cooling rate will be governed by Eq.
11-8.
cooling, there are no stresses present in the sample.
the sample has
to relax.
At the start of
It is assumed that
been held at high temperature long enough for all stresses
As the temperature is lowered, stresses will be developed in the
grains in accordance with Eq. 11-8.
The concentration of vacancies at the
boundary will change due to the normal stress and is given by
C
=
C
+
11-11
kT
where C
is the stress free vacancy concentration.
There will be a tran-
sient during which the concentration gradient drops from the initial step
function to the steady state value (V2 C = 0).
The stress state along the
boundary can not remain uniform if the boundary maintains coherency and
shape (assumed straight).
The stress will redistribute, dropping to zero
at the grain corners and increasing at the center of the boundary making
the flux constant along the length of the boundary.
This transient has
been analyzed by Lifshitz and Shikin (1965) for lattice diffusion and by
Cannon (1974) for boundary diffusion.
The transient is dependent on two parameters:
the relaxation of the
vacancy concentration and the relaxation of the stress to the steady state
distributions.
The change in the vacancy concentration will cause a strain
which will relax some of the stress.
If the strain due to vacancies (Evac )
is larger than the elastic strain due to the stress (Eelastic), the transient will be controlled by vacancy diffusion and if Eelastic >
elastic
relaxation will be controlled by atom diffusion.
E
vac
the
The increase in the
number of vacancies for an applied stress (a) is
AN
AC * V
=
where V is the volume under stress.
11-2
The change in the length of the volume
due to the change in the vacancy concentration is
AL
AN *-2
-
AN11-3
A
where A is the area over which the stress is applied.
11-11,
Combining equations
11-12, and 11-13 gives the vacancy strain as
E
vac
-
C0 2 o
Co Q2
kT
11-14
The elastic strain is
=
elastic
G/EtE
/E
11-15
If X is defined as the ratio of the vacancy strain to the elastic strain,
then
X
Co0 2 E
kT
11-16
For
lXJ
>> 1 the elastic strains are not important.
Lifshitz and Shikin
(1965) give the relaxation time, at constant stress, for a square grain as:
d2
TR
R
11-17
2DD
V
where d is the width of the grain and Dv is the vacancy diffusion coefficient in the lattice.
For Al 0,
(and most materials) the value of X is
always less than I. In this case, the elastic strains are important and
their relaxation occu rs with the vacancy concentration at quasi-steady
state in the grain
V2C - 0
11-18
The relaxation time, for a constant stress is
TR
=
d2kT
2II-19a
r2D ~2E
L2
LQE
if lattice diffusion is controlling (Lifshitz and Shikin (1965)), and
S- _
R
d2kT
2(2.867) 2Db Q
11-19gb
if boundary diffusion is controlling (Cannon (1974)) where DL is the lattice
diffusion coefficient and 6Db is the boundary diffusion coefficient.
For constant strain rate Cannon (1974) gives the relaxation time for
boundary diffusion control as
d2kT
T
R
-=
dk
21T2 6D bQE
11-20a
It will be assumed that the relaxation time for lattice diffusion control at
constant strain rate will be similar to Equation II-19a,
R
c
d2
--
I1-20b
where the constant will be less than r2.
The relaxation time for stress re-
distribution at constant strain rate is smaller than the relaxation time for
an applied stress, for boundary diffusion control as can be seen from Eqs.
11-20a, 11-5 and 11-7, with G = 1.2d.
The ratio of the relaxation time is
ss
2
672(1.2)3
(.2)
TR
R
>
1
I -21a
TR
where for boundary diffusion
D
eff
=
11-21b
For lattice diffusion control, the ratio of the relaxation times is aproximately
one, assuming that the relaxation time for constant strain rate is
T2 times the relaxation time for constant strain, Equation II-19a.
During the transient, the stress relaxes faster than at steady state.
After the transient has relaxed to the steady state, the stress increase due
to the constant strain
bution.
rate will not change the shape of the stress distri-
The transient will affect the stress relaxation at short times;
ignoring it would cause a slight under-estimation of the amount of stress
relaxation.
11-1.2
Effective Diffusion Coefficient
For an elemental solid the effective diffusion coefficient is the sum
of the diffusion coefficients for the possible paths
.v6D'
Deff
=
DL
+
11-22
where -
accounts for the difference in path length as a function of grain
size.
In a compound each species would have an effective diffusion coefficient (D ff).
At steady state, the fluxes of all species must be in the
stoichiometric ratio.
Since the diffusion coefficient of each species is
different, there will be a partial separation of the species under their
common driving force.
separation will
For the oppositely charged anions and cations, the
introduce an electric field which couples the fluxes.
The
electric field coupling causes the fluxes to remain in the stoichiometric
ratio in spite of the difference in values for their (tracer) diffusivities.
For a material with composition A B
,
with the diffusion coefficient of
each species given by Equation 11-22, Gordon (1972) gives the diffusion
coefficient as
(a+
D
eff
=
A
eff
B
)DA D
eff eff
+
11-23
BtD
eff
The atomic volume used must account for the molecular volume:
QA B
11-24
=
With these modifications, Equation 11-5 is applicable to binary compounds
as well as to elemental solids.
For A1,0 3 the lattice diffusion coefficient has been measured by
Paladino and Kingery (1962) and by Oishi and Kingery (1960).
It was found
that
0
D
<<
9<
Al
D
P
I -25a
From the analysis of Paladino and Coble (1963) on the creep of polycrystalline A1 20 3 it can be inferred that
(D
0
b
>
SD
Al
b
IAI-25b
Substituting in Equation 11-23 and assuming that oxygen lattice diffusion
(D0) does not contribute significantly to the effective diffusion at intermediate or fine grain sizes yields
5 DA +Db
Al 203
Dff
TDb
G
=
D9+
3
.
A11-26
A
T6D _
6D b
b
+ 2
G -1 1 G
The diffusion coefficients were calculated from measured creep data on A1 03
(see Cannon and Coble (1975))
Al
D
6DbAl
= x8.610RT
0
3.3 x 10- exp
6Db
11-1.3
II-27a
0
cm2/sec
I -27b
I
RT
cm3/sec
RT
I I-27c
Calculation of TEA Stresses
The maximum TEA stress will occur when the c-axis of one grain is
oriented perpendicular to the c-axis of the adjoining grain.
Taking the
initial boundary length as X0 , for a temperature change of AT = To - T, the
stress in a grain as a function of time will be (Appendix A)
EX 1- X2
--t)
[
2
A-AT
11-28
Xo
where As is the difference in the expansion coefficient of the grains and
X 1 and X 2 are the lengths of the grains as a function of time if the grains
were able to creep in response to the applied stress.
It is assumed that
the elastic properties of the two grains are the same.
Al 0, is elastical-
ly anisotropic, but the variation in properties with crystallographic oriIt is also assumed that E and al and a2 are constant
entation is small.
over the temperature range of interest.
The stress relaxation is
.V
-
V
+ A
dt
For
dT
d-. = 0
dt
-J
dt
.
11-29
which corresponds to a constant strain, the relaxation is of
the form of Equation 11-6a.
Converting equation 11-20 from a time depen-
dence to a temperature dependence using
=
dT
I -30a
3dt
where 1 is a constant, the cooling rate, and converting the strain rate to
a stress using Equation 11-4
d'X
- X]
[-X
id
II-30b
neff
dt
yields the governing equation for stress relaxation
do
dT
Eo
=
2n eff
AGE
+
2
11-31
Equation 11-31 is linear in stress and can be solved by use of an integrating factor.
When this is done the resulting function for the stress can
not be evaluated analytically (Appendix B).
The stresses were evaluated numerically by assuming a step function of
temperature with time and letting the width of the step decrease until
further reductions have no effect on the stress calculated.
The stress as
a function of temperature is shown in Figures 11-1 and 11-2 for the grain
sizes and cooling rates used.
regions.
The solution can be divided into three
Above some temperature, TC the first term of Equation 11-31 will
dominate and all stress will relax as the sample is cooled.
Below a dif-
ferent temperature, TR < TC, the second term of Equation 11-31 will dominate
and no stress will relax.
Since the second term of Equation 11-31 is not
a function of cooling rate, grain size, or temperature, changes in the first
term of Equation 11-31 will determine the values of TC and TR.
As B in-
creases,
2decreases
E
and TC and TR move to higher temperatures. The
2 eff
grain size enters through the value offleff* From an approximation to the
analytic solution (Appendix B) TC and TR are given as a function of cooling
rate and grain size in Figure 11-3.
The linear region of Equation 11-31, that is below TR, was extrapolated
to zero stress.
The extrapolated temperature, TE, was found to be approx-
imately equal to (TC + 2TR)/3.
The value of TE from the numerical solution
to Equation 11-31 is shown in Figure 11-3.
By using TE and Equation 11-1
EIE 2
a
=
(a-
)
E + E AT
11-1
with AT = TE - Troom, the room temperature stresses due to TEA can be pre-
dicted as a function of grain size and cooling rate.
11-1.4
Relaxation of Macrostresses
It has been demonstrated that the flexural strength and impact resistance of polycrystalline Al203 can be improved by rapid cooling of the
sample from high temperature (Kirchner, et. al.
Varner (1977)).
(1971),
(1973), Stolz and
The improvement in the mechanical properties is due to a
residual surface compressive layer which inhibits the growth of surface
flaws, by analogy with the thermal tempering of glass.
For thermal tempering to occur, the material must be able to flow (or
creep) in response to a macrostress.
Flow will only occur if the tempera-
ture of the sample is above TR as discussed in Section 11-1.3; thus at the
start of the quench, the sample must be at a temperature (TQ) above TR.
If
TQ < TR the sample can be weakened by thermal shock, as the data of Stolz
and Varner shows (Figure 1-1).
TQ for tempering is taken as TR, which is a
function of grain size and quench rate.
The residual stresses in quenched glasses have been analyzed by Narayamaswamy (1978), Aggarwala and Siebul (1961), and Lee, et. al.
(1965).
Buessem and Gruver (1972) calculated the residual stresses in quenched
A120, rods.
In this study the cooling rates used were much lower than the cooling
rate used in quenching studies.
The residual stresses due to thermal tem-
pering can be calculated from a simple analysis.
Due to the low cooling
rates, the rate of change of the temperature at all points in the sample
will be the same, after an initial transient, and is assumed to be equal
to the constant cooling rate.
dT(X,t)
=
dTX
dt
11-32
.
The temperature distribution will be
KV2 T
=
dT(Xt)
S
=
11-33
dt
where K is the thermal diffusivity.
The ring sample can be modeled as a
flat plate cooled from both sides with the radial temperature distribution
assumed to be equivalent to that through the thickness of the plate.
the center of the plate be at X
=
0 and the edges at
+
L
22.
Let
The residual
stress will be
0000
=
for this geometry.
0 a0 0
The temperature at the edge at any time is
T(±L , t)
where
T(X,O)
11-34
=
T(X,O)
+
St
11-35
is the initial uniform temperature.
A solution of Equation
11-33 is
T(X,t)
=
T(0,t)
-
ýX
.
11-36
Substituting into Equation 11-35 yields
=
S
2K
11-37
For a flat plate with a temperature variation through the thickness only,
the stress is (Boley and Weiner (1960))
L
L
2
SE
-T(X) +
2
T(X)dX
+
12x
-L
11-38
T(X) xdx
-L
2
2
For the temperature distribution of Equation 11-36
Y= aE
1-v 2K
X2
11-39
L
12
It is assumed that all stresses are relaxed as the sample is cooled through
TE and that no relaxation occurs below TE.
The room temperature stress will
Taking E = 4.14 x 1011Pa, v = .239 (Simmons and
be given by Equation 11-39.
Wang (1971)), K = 0.017 cm2/sec (Malta and Hasselman (1975)),
ai
L =
(Kingery, et. al.
8.8 x 106/K
=
o(X = 0)
0.13 MPa
(1976)) and 1
=
100 K/min gives
(tension)
I -40a
c(X =
22 =
+L)
0.26 MPa
(compression)
The tempering stresses are small and should have very little effect on the
mechanical properties of samples cooled at 100 K/min or less.
11-2.
Summary
A model for the relaxation of stresses due to thermal expansion aniso-
tropy by diffusional creep was developed.
The generation of stress is
goverened by
-
2
EAoAT
E
11-41
where
T is the temperature change.
The resulting equation for the stress
relaxation during cooling at a constant rate was integrated to determine
the stress as a function of temperature, cooling rate and grain size.
The
stress could not be evaluated analytically, so a numerical technique was
used.
TE
The room temperature stress can be estimated by assuming that at
LC
3
the stresses are all relaxed and develop according to
Equation 11-41 below TR where TR is a function of the grain size and cooling rate.
This technique underestimates the stress at room temperature,
since TE is slightly lower than the actual extrapolated temperature.
The relaxation of stress by creep was extended to consider the thermal
tempering of polycrystalline Al
203.
A minimum temperature for tempering by
quenching was found, TQ, which is equal to TR.
This agrees with data on
the strength of Al203 as a function of quenching temperature.
The magnitude of the residual stress due to thermal tempering in the
samples used in this study was calculated for the cooling rate used and
found to be
very small.
0
~hh
IUU
90
80
70
S60
S50
b40
30
20
I0
n
II-1
0
500
1500
1000
Temperature (K)
2000
Calculated stress due to TEA as a function of temperature in A12 03
for ý= -1.OK/min. Above T all stress can relax and below TR no
stress can relax and the stress is a linear function of temperature.
T E = TC+2T R and is approximately the extrapolation of the linear reigon.
0
0
O
11-2
500
1000
150U
eUUU
Temperature (K)
Calculated stress due to TEA as a function of temperature in A1 2 03
G = 50 m. For
3 = -100K/min, TC > Tanneal'
See Figure II-1 for explanation of
T
TE, and T
C ,
R.
for
/3 (K/min)
0.4
1000
100
10
1.0
0.1
TM
TM
Tanneal
Tanneal
2000
0.5
F-
1700 a
0.6
rr)
E
0.7
1400
0.8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
1250
Lnp (K/sec)
11-3
Critical temperatures (TC, TR,
TE) as a function of cooling rate.
The change in slope shows the shift in the controlling diffusion
mechanism as the grain size increases.
III
Preparation of Samples
Samples of polycrystalline Al20, with various grain sizes were cooled
at different constant rates in order to measure the magnitude
of the re-
sidual stresses, due to thermal expansion anisotropy, at room temperature.
All samples were purchased from the General Electric Co., Cleveland, Ohio.
Standard Lucalox 0
lamp tubing with a nominal 9mm O.D. and 7mm I.D. was
cut into rings 2.5 mm high (Figure 111-1).
The rings were annealed at 2150K
(1875 0 C) for different lengths of time and examined optically to measure
the range of grain sizes.
The samples used to measure the stress were
annealed in vacuum at 2150K and cooled at a constant rate of 0.1,
or 100K/min.
The grain size was estimated from the time at 2150K, using
G3
where
G
1.0, 10
-
0
=
kT
III-1
is the average grain size (and is taken as 1.5 L where L is the
average intercept length) and a calculation of the amount of grain growth
that occurs during cooling.
Grain growth is controlled by diffusion, so the temperature dependence
of the rate of grain growth is of the form
R
=
Aexp (-Q/RT)
111-2
.
The activation energy (Q) was found by Bruch (1962) to be 150,000 cal/mole.
When a sample is cooled for the annealing temperature T
anneal to room
temperature, Troom, some grain growth will occur.
The amount will be
given by integrating the rate of grain growth over the temperature interval.
The same amount of grain growth at Tanneal would occur in a time,
teff and is given as
Troom
T
tef f
eff
D(T)dt
anneal
D(T
111-3
)
anneal
It is assumed that A and Q are not functions of temperature.
Converting to
the temperature derivative
=
dT
111-4
Bdt
where ý is a constant, yields
T
room
exp(-Q/RT)dT
T
eff
1
anneal
B
exp(-Q/R
anneal )
The value of teff calculated from Equation 111-5 did not vary much for
Q = 140 kcal/mole
to
Q = 160 kcal/mole.
Since teff scales with 1/,
teff was only calculated for a value of B = 1 K/min, and was found to be
approximately equal to one hour.
The grain sizes were measured on samples that had been thermally
etched at 1850K for 15 minutes or etched in boiling H2PO4 . The results for
the different etching techniques were identical.
The grain sizes for the
different anneal times are given in Table III-1, and a typical microstructure is shown in Figure 111-2.
(It is observed that the grain size in-
creases somewhat towards the edges of the sample.
This is presumably due
to evaporation of MgO from the surface and a corresponding increase in the
grain growth rate.
(Harows and Budworth (1970) and Bruch (1962)).
The
value of G calculated using teff equal to one hour for B equal I K/min
agreed with the measured G on slow cooled samples.
The inner and outer radii
(ri and r ), and the thickness (L) of the
I
O
ringe samples were measured using a filar eye piece.
measured using a micrometer.
The height was
The measurements are shown on Figure II1-1.
TABLE III-1
Grain size as a function of anneal time as 2150K
Anneal (hrs)
G(•)
0
50
1
67
2
72
78
83
93
132
--
F
h=10cm
+.4 mm
A
r =4.035mm
I~
L= 0.78mm
=3.65 mm
t = 25mm
II1-1 Geometry of samples.
h
10cm,
Rings were cut from as received tubes.
r. I = 3.65mm,
r
= 4.42mm,
L = 0.78 mm , t = 2.5mm.
-
I
I
L
I
II
I
II
~
II
· ·
I
I
111-2 Typical Microstructure of Samples. Width of sampleshown is
0.68 mm. Grain size is seen to increase towards the edges
(left and right).
I
1.
IV. Spectroscopy
The optical fluorescence of ruby (Al 03 doped with Cr) as a function of
wave length is well documented and shows sharp lines due to the Cr
(Schawlow (1961), Deutschbein (1934)).
In a cubic field a Cr ion would
cause a single line in the spectrum corresponding to the
tion (Sugano and Tanabe (1958)).
In Al
ion
203
2E
--
4
A 2 transi-
the Cr3+ ion is in a distorted
field and as a result the twofold orbital degeneracy of the cubic field is
lifted and the line splits into two lines, called the R 1 and R2 lines.
Schawlow (1961) has examined the shift in the wavelength of the R, and R2
lines with uniaxial pressure.
Piermarini, et. al.
(1975) used the shift of
the R1 line to calibrate a high pressure load cell.
In this section a method for using the pressure shift of the Ri and
R2 lines to determine the residual stress in A12 03 is described.
Data on
A12 0, rings with the thermal history given in Section III will be presented.
IV-l.
Piezospectroscopic Effect
The effect of pressure of the fluorescence spectrum of ruby will be
different for a single crystal and polycrystalline sample.
First will be
considered the effect on a single crystal of ruby and next the modifications for a polycrystalline sample are discussed.
IV-l.l
Single Crystal
In ruby the R1 and R2 lines are split by 29-1cm(14 A)' below 80K.
*Wavenumber
(cm -
1)
x wavelength
O
(A)
=
lx
10 8
The R, line occurs at 14,422 cm-1(6934 A) and the R 2 line at 14,451 cm-1
(6920 A) below 80K (Sugano and Tsujikawa (1958)).
The line width is very
small for strain free samples, being approximately 0.3 cm- ' at 80K
(Schawlow (1961)).
A typical spectrum is shown in Figure IV-l.
The tem-
perature dependence of the line width and line shift was measured by
McCumber and Sturge (1963), Deutschbein (1934), and Gibson (1916) . The
data from McCumber and Sturge is shown in Figure IV-2.
There is little
change in the linewidth below 80K.
The pressure dependence of the R i and R 2 lines has been measured by
Schawlow (1961)
and by Noack and Holzapfel (1977).
is shown in Figure IV-3.
Piermarini, et. al.
The data from Schawlow
(1975) measured the shift of
the R 1 line at high pressure (20 GPa) using the lattice parameter variation
of NaCI to measure the pressure.
In addition, Piermarini, et. al.
(1973)
measured the hydrostatic limits of liquids by noting theonset of line broadening of the R 1 line.
Munro (1977) has developed a theory that predicts
the line shift for the R1 with hydrostatic pressure.
The change in posi-
tion with pressure was found to be
dv
0.0076 cm-'/MPa
IV-1
dP
for the
2E
-+
4A
2
transition; with which Noack and Holzapfel's (1977)
measured value of dv/dP agrees.
Piermarini, et. al.
(1975) measured the
line shift for R 1 and found
P(MPa)
which yields
=
-132.8 Av
+
.03Av 2
IV-2a
43
-.0075 cm 1 /MPa
dv
dP
.
IV-2b
Grabner (1978) developed a model that combines the line shift and line
width change to calculate the stress state.
The model is for a uniform
stress, but if large gradients in the stress are not present, the model
will still be applicable.
The change in the spectrum with stress is given as
Av
=
i..o..
IJ IJ
IV-3
where a.. is the stress tensor, Av is the change in the spectrum and is
IJ
composed of a line shift and a line broadening term and f..
is the Piezo-
IJ
spectroscopic tensor.
From symmetry (Nye (1957))
-T
0
0
0
'"11
0
0
0
r33
=
the form of v.. is
IJ
Tr..
aj
IV-4
for A1 20 3 , where X1 is along the al-axis and X 3 is along the c-axis.
Since the hydrostatic and shear components of the stress contribute
only to the line shift and line broadening, respectively, the stress tensor
is split into a hydrostatic tensor and a shear tensor (Dieter (1976))
F..
13
0
ij kk
3 6.j
1i kk
+
I'
where a.. is the Kronecker tensor and aij is
IJ
IJ
IV-5
2a
-G22 -CG33
3
a 12
2L2-i1 -122
a2 1
ar.
Ij
G13
3
iv-6
a23
2a23 a-11 a31
a 32
22
3
Substituting in Equation IV-3 yields
Av=AvH
S= (2
(11 +(22 +(33
Let
3
and
3
( 1 1 +a22+ 03 3 )
+ r 33 )
G3 3 - G11 -
3
22
(2
+ (T+r
+3
3 3+
1)
33
- 1 - a2 2 )
3
•
IV-7
be called aH and aS respectively, to
3H
S
refer to the hydrostatic and shear components of the stress.
Measured and calculated values for the Piezospectroscopic coefficients
are given in Table IV-l.
A value for 27
of -0.0075 cm-'/MPa will be assumed.
1 1+T33
For RI,
for both the RI and R. lines
3, 3
- r11 = 0.0016 cm-'/MPa
and for R 2 , r 33 - T 11 = 0.0008 cm-'/MPa will be assumed.
Substituting into
equation IV-7 yields
(Av
+ Av )
H+
(AvH + AVS)R
=
-0.0075a
=
-0.0075a
+ 0.00160
S
IV-8a
+ 0.00080
S
IV-8b
H8b
HSS)R
H
IV-1.2 Polycrystal Effects
The stresses in the polycrystalline samples used in this study are
both macroscopic and microscopic.
The macroscopic stresses are due to
to nonlinear temperature distributions during cooling.
stresses are due to the thermal expansion anisotropy.
The microscopic
The origins and
theoretical values of these microstresses were discussed in Section II.
The fluorescence spectrum is from a small region of the sample (approximately half the thickness).
In this region the microstress will inte-
grate to zero, but the macrostress does not.
very small in these samples.
It was calculated to be
The hydrostatic component of microstress
will contribute to line shift; but the shift will be in both directions
due to both tensile and compressive microstresses.
The effect of a hydro-
static microstress will therefore appear as line broadening.
Any line
shift is due to the hydrostatic component of the macrostress and the line
broadening is caused by macroscopic shear stresses and microstresses.
TABLE IV-1
VALUES OF THE PIEZOSPECTROSCOPIC TENSOR
T
Line
T 11
" 33
..
,j(cm- I/MPa)
27 1 1 +Tr
+ 33
iT33 -
11
Note
Reference
-.0031 -.0018
-.0080
+0.0013 T= 77K
Schawlow (1961
(14 422cm- ) -.0031 -.0014
-.0076
+0.0017 T= 77K
Kaplynaskii &
Przhevushii (1962)
-. 0075
T= 300K
Piermarini, et. al.
(1975)
-.0077
T= 300K
Forman, et. al.
(1972)
-.0075
T= 300K
Barnett, et. al.
(1972)
-.0090
T= 300K
Langer & Euwema
(1967)
-.0094
T= 300K
Paetzold (1951)
-.0075
T = 80K
Paetzold (1951)
-.0076
T=4 to Noack & Holzapfel
300K (1977)
-.0076
Calculated
-.0028 -.0023
-.0079
+ .0005 T = 77K
Schawlow (1961)
(14451cm - 1) -.0027 -.0019
-.0073
+ .0008 T = 77K
Kaplynaskii &
Przhevushii (1962)
Munro (1977)
As discussed in Section 11-1.4 the stress in a flat plate cooled symmetrically through the thickness is given as
0
L.
a..
= 0
IJ
0
0-
a33
0
0
o 3
IV-9
The orientation of the X 1, X 2, X 3 axis for the ring sample is shown in
Figure IV-4.
To determine the change in the spectrum, the stress must be
referred to the crystallographic axis of the individual crystallites al,
As in the previous section
a 2, C (or X', X2, X).
1
..j
IJ
3
2
Ij
=
2
o
0 0
IV-10a
a..
ij kk
IJ
V.
-2
3a333 0
2
0
30
S
kk+
0
Ta33
a..
6
I
2
0
"C'3~ 3
0
IV-10b
1
3-3
Using the tensor transformation law
a..
IJ
where a..
IJ
=
IV-ll
aijaj
ij jllkl kl
is the cosine of the angle between the x. and x. axis, gives
I
J
for a..
IJ
a..
IJ
=
2
3
i
J
s
-
3:.
3 C.
ij
IV-12
C.. is the matrix of the a..
iJ
IJ
C..
-
ij
a 11 -1/3
all a12
a1 2 a 13
a 2 1 all
a2
a 2 1 a3 1
La31all
- 1/3
a,31
IV-13
a 2 - 1/3
21
Substituting in Equation IV-3 gives the change in the spectrum (Appendix
C)
A•
=
(2T 1 13+7
2
) -
33
(Tr
11 )C
33 -
IV-14
Since the crystallites will occur with random orientations of the crystallographic axis, the effect of the nonhydrostatic stress term should be
integrated over all orientations to determine the effect of tempering
stresses on line broadening.
Let a be defined as
cosine -1cosinea
a31
=
c
IV-15a
and assume that all orientations are equally probably so the range of a
is
< --rr
0 <
IV-15b
-2
Integrating C33 over the variation in a yields
A
/2 C d
0
IV-16
= -4
The change in the spectrum due to thermal tempering is then
Av
=
0
A•= ý CF33
(2r 1z 1 +
s )3 +T
r33 )
(
"
-
o'33 (7r33 - Trl)
)
IV-17
IV-2
Experimental
The fluorescence spectra was determined for samples prepared as des-
cribed in Section III and for two samples of unknown thermal history.
Because the line width increases rapidly with temperature above 80K, all
results were obtained with the sample immersed in liquied N,.
The
temperature of the sample may rise slightly due to the heating of the
beam, but the temperature was not above 100K.
The experimental arrange-
ment is shown in Figure IV-5, illustrating equipment used at the National Bureau of Standards in Washington, D. C. The
lamp.
light source was a Hg
The incident light was passed through a CuSO
4
solution and the
fluoresced light was passed through a red glass filter.
spectra for the filters is shown in Figure IV-6.
The adsorption
A Spex Monochrometer
was used with 360pm wide slits, and the second order diffraction was
measured.
IV-3
The line shift and line width are listed in Appendix D.
Discussion
The depth of the sample that contributes to the fluorescence spec-
tra depends on the wave length used and the size and density of scattering centers (usually pores).
The samples used in this study were ap-
proximately pore-free so scattering is not important.
reports that for a 0.5mm sample of Al
wave length of 500mm (5000 A).
T
where t is the thickness.
203
Peelen (1977)
the transmission is 25% at a
The transmission is given as
e
e~
From Peelen's data 6 = 2.77.
IV-18
Thus for a
sample 0.8 mm thick the transmission is 10%.
0.8 mm corresponds to a
depth of penetration of 0.4mm for the incident light and 0.4 mm for the
fluoresced light.
The spectra will be representative of the stresses
over half of the sample thickness.
The data (see Appendix D) show an increase in the width of the lines
with increasing cooling rate but no dependence on G.
No dependence of
the line shift on either cooling rate of G
The magnitude of
is seen.
the macrostress due to thermal tempering was calculated in Section 11-1.4
and was found to be less than 0.3 MPa for the highest cooling rates used
in this study.
of 0.2 x 10
3
cm
This stress would cause a peak shift (from Equation IV-8)
which is too small to be observed.
Thus the observed
shift is not due to macrostress and must be due either to microstress or
experimental error.
The source of experimental error is in the measure-
ment of the absolute peak location.
This would tend to increase the
scatter of the data but would not cause a net shift in one direction, as
observed in the data.
Since the magnitude of the macrostress is small, peak broadening
is due entirely to the microstress but observed values are the sum of
the intrinsic (single crystal)
line widths and the broadening due to hy-
drostatic and shear microstress.
Since the hydrostatic stress is both
tensile and compressive, the hydrostatic stress will shift the peaks in
both directions, so the observed broadening should be twice the value of
the shift.
It was assumed by Grabner (1978) that the orientation of the stress
was random.
For microstresses due to TEA
this will not be the case.
Since the expansion coefficient parallel to the c-axis is larger than
the coefficient perpendicular to the c-axis, upon cooling there will be
tensile stresses developed parallel to the c-axis.
Since the line shift
is larger for stress perpendicular to the c-axis than parallel, there
will be a net line shift due to the non-random orientation of the microstress and will be larger for the R1 line than for theR2 line.
The line
shift is
=
AV
(
RI = -0.0013
cm-1/MPa
IV-19a
- 33 )R = -0.0005 cm '/MPa
IV-19b
11 -S
AvR
and the stress will be given as
aH
=
-1250(AvRi - AR )MPa/cm- 1
IV-19c
The cH is shown in Figure IV-7 along with the stress calculated in Section
11-1.3.
The scatter is too large to allow any comparison between the
measured and calculated values of the stress.
calculated from the broadening.
The stress can also be
The broadening is
Av=Av single crystal + Avshear
+ 2Av hydrostatic .
IV-20a
Substituting in Equation IV-8 gives
(Av - Av single crystal)R
1, .05
.015c
H
H
+ .0016y
(Av - Avsinglecrystal)R2 = .015 H + .0008a S
IV-20b
IV-20ba
IV-20c
and rearrangement gives the hydrostatic stress
H
=o
2(Av -Av single crystal)R - (Av - Av single crystal)
0.01-55R
IV-20d
McCumber and Sturge (1963) and Schawlow (1961) have measured the line
width of the R
lines as a function of temperature (see Figure IV-2).
Below 80K, the line widths of R i and R 2 both equal 0.15cm- 1.
The thermal
history of the sample is not known (it was supplied by Grabner at N.B.S.)
but it would have to be highly strained (quenched) to account for the
line width increase.
From the difference in width of the Ri and R2 lines,
and the peak locations it can be inferred that the increase in width is
not due to temperature error (i.e.,
the sample being above 1OOK).
data from McCumber and Sturge at 100K will be used.
The
The hydrostatic
components of the microstress, calculated from Equation IV-20d are shown
in Figure IV-8.
Also shown are the stresses calculated in Section 11-1.3.
The slopes of the two curves show good agreement.
The magnitude of the
measured stress is higher than the calculated stresses, but the value of
the measured stress is dependent on assumed values for the single crystal
line widths.
The shear stress calculated using Equation IV-20 shows a high degree
of scatter.
The small broadening coefficient for shear stresses causes
the effect to be obscured by scatter in the measurements.
From the data
in Appendix D, it is seen that the slope for the width of the R1 line as
a function of cooling rate is higher than the slope of the R2 line.
This is as expected since (w,3
- I11)R1 is twice (r33 - Tl)R2-
IV-4
Summary
Using a model developed by Grabner, the stress in polycrystalline
A120 3 was measured.
The shift and broadening of the fluorescence lines,
due to the presence of Cr
ions were measured.
The stress due to TEA
was calculated both from the line shift and from the line broadening and
compared to the predictions of Section II. The grain size variation of
stress is not observed.
This is due to the small dependence of the stress
on grain size over the range of grain sizes used and due to variations of
the grain size in a sample (see Figure 111-2).
Due to errors in the
measurements of the absolute peak position, there was a large scatter in
the stress calculated from the line shift, although the magnitude was in
good agreement with the predicted stress variation with cooling rate.
The stress calculated from the measured line broadening showed good
agreement with the predicted stress as a function of cooling rate, although the predicted magnitude of the stress was lower.
The spectroscopic technique reported here is a good method to
rapidly and non-destructively determine the microstresses in polycrystalline A1,0 3 . These stresses are seen to increase with increasing cooling rate, which is predicted for stress relaxation by creep, and agree
with the predicted magnitude.
R2
4-
I
I
I
Ia
U)
c--
I
I
WI
300K
77 K
,,
II
6900
IV-1
-L
w
6920
6940
Wavelength (A
6960
Typical fluorescence spectrum of ruby. The peaks narrow and shift
to shorter wavelengths as the temperature is dropped.
w
IA~
0
0
48.0
IVU
100
48.0
I0
5
5.00<
5.0
10
C)
o<q:
.
0)
C
4-
I-
0.5
I05
0.5
U)
0.I
0.05
IO0
IV-2
100
1000
4-
4-
1.0
0.5 Cn
0.1
0.05
IO0
100
T(K)
T(K)
(a)
(b)
1000
Linewidth and lineshift for ruby as a function of temperature
from McCumber and sturge (1963).
Below = 50 K the linewidth at
half height is independent of temperature. Shift is to shorter wavelengths.
A
U
0
-0.2
0.1
_ -0.4
0.2
o -0.6
0.6
4-
oO
i
0.1
-
) -0.2
-0.4
0.2
-n
n-
;
0
100
200 0
100
200
r (MPa)
(a)
IV-3
o<
(b)
Lineshift for ruby as a function of applied pressure. From
Shalow (1961). T = 77K. Shift is to longer wavelengths.
a) pressure applied parallel to c-axis.
b) pressure applied perpendicular to c-axis.
Srr
'A
z
X1
f
I
IV-4
Orientation of Xl, X 2, X 3 axes. It is assumed that there are no
macrostresses in the X1 direction.
FilterLens
N.
LASER ------- ----
le
Al
---' Ln
Lens.s
Slit
Fitirrorer
IV-5
Slit
Detector
Monochrometer
Experimental arrangement used to measure fluorescence spectrum.
A 200 watt Hg vapor lamp and a spex monochrometer were used, at
N.B.S.
I
2
5
10
30
50
O
U 80
4
90
95
98
99
4000
IV-6
5000
6000
Wavelength (A)
7000
Adsorption as a function of wavelength of the filters. CuS04
filter was used on the incident light and glass filter was
used on fluorescent light.
Anneal
(hrs)
150
b
100 -U
,-
I
0
3
4
6
12
20
.
x
A
x
0L
o
x
N
6,,,...
100'm (calc)
50
0
IV-7
0.I
I.0O
10
100
9 (K/min)
Stress measured from lineshift_as a function of cooling rate.
line is calculated stress for G = 100p from Section II.
Solid
· __
•
JI
IO3U
0
x
o
Anneal
(hrs)
I
031
0
1
S4
x6
12
A 12
120
*20
A
x
110
b
100
(caic)
90
0.1
1.0
10
100
/9 (K/min)
IV-8
Stress measured from linewidth as a function of cooling rate.
Solid line is calculated stress for G = 100pt, from Section II.
V
Indentation Fracture
The indentation of a brittle solid that occurs during contact load-
ing can be generally divided into two classes, blunt or sharp, depending
upon the shape of the contacting particle and the nature of the stress
field beneath the contact.
tic contact.
surface.
sharp.
Blunt indentation refers to completely elas-
This is usually the case for a spherical indenter on a flat
If the contact is not fully elastic the indenter is called
Although most particle contacts are sharp, the effect of blunt
indenters, first considered by Hertz (1881),
model than the case of sharp indenters.
is a simpler situation to
Hertz considered the completely
elastic contact between two curved bodies and described the resulting
cone crack.
The stresses arising from a sharp indenter were first given by
Boussinesq (1885), who considered a load with zero contact area.
As a
result the stress field contains a singularity at the contact point.
is assumed that
It
inelastic deformation about the singularity will distri-
bute the load about a finite contact area.
The details of the stress fields beneath blunt and sharp indenters
will not be presented here, having been well covered in reviews of indentation fracture by Lawn and Wilshaw (1975) and Evans and Wilshaw (1976).
The use of indentation to determine the fracture parameters of a
sample has been studied by many authors.
In the following section the
basic principles of microindentation fracture will be discussed, and the
measurements on Al 03 and the calculated fracture properties will be
presented.
V-i
Crack Formation
This section describes the formation of cracks for blunt and sharp
indenters and the geometry of the resulting impression and cracks.
V-1.1
Blunt Indenters
The classic blunt indenter is a sphere on a flat plate (Figure V-la).
Since the contact is fully elastic, cracks cannot be nucleated; pre-existing flaws on the contact surface will be the fracture initiation site.
The tensile stress on the surface is highest just outside the contact
radius.
At a critical load the tensile stress on a suitably oriented
pre-existing flaw will be large enough to propagate the flow.
The flaw
will form a ring crack, which is characteristic of blunt indentation.
Upon increasing the load, the ring crack grows in a stable manner into
the sample following the stress trajectories to develop a cone crack.
high loads
At
inelastic deformation occurs and crack patterns typical of
sharp indenters are formed.
The transition from elastic to inelastic
behavior is a function of the sphere radius, the hardness and fracture
toughness of the sample and possibly the pre-existing flaws on the surface if their size is small or their density is low (Evans and Wilshaw
(1976)).
V-1.2
Sharp Indenters
A Vickers pyramid will be used as an example of a sharp indenter
(Figure V-lb).
Pre-existing flaws are not necessary because the singul-
arity in the stress field gives rise to plastic deformation and crack
initiation at very low loads (Langford (1978), Seaton (1971)).
The cracks formed are penny-shaped sub-surface cracks in the plane
defined by the load axis and the indenter diagonals.
load the crack will grow and intersect the surface.
With increasing
Upon removal of the
load, the stresses due to the plastic deformation cause the formation of
a second set of cracks called lateral cracks (Lawn & Swain (1975)).
Evans and Wilshaw (1976) have observed the formation of cracks for a
spherical inelastic contact (small radius spheres) for a variety of
materials.
Hockey and Lawn (1975) have observed the cracking about
Vickers indentations in A1 2 0 3 and SiC.
Lawn, et. al.
(1975) and Lawn and
Fuller (1975) also have observed the crack patterns about indents.
Above a critical load cracks about both blunt and sharp
indenters
are similar and can be modeled as penny-shaped with the surface trace approximately equal to the depth of the crack below the surface (Lawn and
Fuller (1975)).
V-2
Fracture Mechanics
The fracture mechanics reviewed here will be that which is applicable
for the median crack typically formed by a Vickers indenter.
These re-
sults will apply to all types of indenters at high loads.
The pressure beneath a Vickers pyramid is
P0
=
-
P
sin 0
V-1
2a 2
where P is the applied load, a is the impression half diagonal (see Figure
V-lb) and
0 = 68
0
is the angle between the indenter face and the load axis.
for a Vickers pyramid (Dieter 1976).
The stress that causes the median crack to propagate was determined
by Boussinesq (1885) and is given as
SWP
S(Z)
1•-
2 v-
V-2
V-2
where z is the depth below surface and v is Poisson's ratio.
K I for a
penny-shaped crack in the inhomogeneous stress distribution given in Equation V-2 is needed.
Lawn and Swain (1975) used K I for a straight-edged
internal crack in the inhomogeneous stress state.
edged crack was multiplied by vr2/7
The K I for the straight
to derive the KI for the penny-shaped
crack, in analogy with the homogeneous case.
Mendiratta and Petrovic
(1976) suggest that the multiplicative factor for converting to a pennyshaped crack is
]V'/Q
Q =
where
Jr/ 2in
-2
2 + fc-)
1/2
cos2
dO
V-3a
where c is the surface trace and c' is the depth of the crack below the
surface.
Lawn & Fuller (1975) and Marshal & Lawn (1977) showed c = c'
and therefore
r=
•
The two solutions differ by a factor of vI2-.
V-3b
For the straight edge crack
KI is given by Lawn and Wilshaw (1975) and KI for the penny-shaped crack
is
KI
c GO(z)
c
K V
V2)112
2
0 (c 2
- z2)
V
V-4
dz
.
V
The integration cannot be performed due to the singularity in the stress
at z = 0.
depth, zo.
However, the stress will relax by plastic deformation to a
Above zo all tensile stresses will relax to zero.
K I will be
given as
_2
K
22.)T
c
P
z0
wz
(1-2v•)
1
(c 2 -
z )
½ dz
.
The value of z0 for a given indentation depth is not known.
et. al.
/V-5a
Petrovic,
(1975) found that KIC measured in bending for Si3 N,4' using a
Knoop indentation as a strength controlling flaw was lower than
the KIC
measured by other techniques.
Upon removal of a surface layer to a depth
of four to eight times the indent depth, their measured KIC values increased, and was in good agreement with other techniques.
It is assumed
that the plastic deformation about the indent caused the reduction in
KIC.
It can be inferred that the deformation zone is not greater than
eight times the indent depth.
Assuming that the same will be true for
Vickers indent in A12 0 3 , then the depth of the zone of plastic deformation, zo, will be given as
z
6a
= 6X = tan
where
Xis
the
indent
depth. Substanituting
68
V-6a
for a Vickers, yields
where X isthe indent depth. Substituting 0 = 68 for a Vickers, yields
zo
=
0.4a
6a
V-6b
where 6 is between four and eight.
zo
=
Lawn and Swain (1975) assume that
V-6c
2a
Equation V-6c will be used, as the actual value of 6 is not known, so the
data can be compared to the literature data.
Equation V-5b is then
given by
(c 2 - 4a 2 )
c 3 / 2 2a
1-2
I
5/2
V-7
From measurements of the indent size, crack length and load, K I can be
calculated.
A different approach was used by Evans and Wilshaw (1976).
Instead
of a direct calculation of K I , the general dimensional form was developed
K
HV FF
v-8
FF () F (v)
where H = Po is the hardness, p is the friction coefficient and j is the
constant (=3).
By empirically fitting indentation data to V-8 the func-
tions F1 , F2 , F3 , F4 could be determined.
Evans and Charles (1976) found
little effect for changes in v and p and that F3 (v)
considered constant over a wide range of materials.
and F,()
could be
Curve fitting the
data gave
F
and then Fr(
=
is given as
I
V-9a
F
0.055 log
=
(8.4 .
.
V-9b
Substituting in V-8 yields
0.055 log (8.4 a H
K =
V-9c
The data of Evans and Charles shows a good fit to Equation V-9c.
V-2.1
Residual Surface Compressive Stress Effects
The fact that indentation techniques for determining K I test only
the surface region of a sample, indicates that surface stresses will have
an effect on the measured K I.
A residual surface compressive stress will
increase the measured K I value over the bulk value.
(1977), Swain, et. al.
(1977)).
(Marshall and Lawn
A detailed analysis of K I measurements
for tempered materials is given by Lawn and Marshall (1977).
The increase in K
is due to the stress field from the surface com-
pression opposing the stress field due to indentation.
For a material
with a residual stress, the K I for an indentation crack is,
a =O
KI =K
I
where
oR=O
KI
a
+
I
K
V-10
I
is the stress intensity factor in a sample with no residual
oR
stress, and K
is the effect due to the residual stress field.
From Equation V-7
SR= 0
-
KI R3/2
and for a crack of length 2c in a stress field of
V-l la
R
R
1/2
O OR(C
KI
V-llb
)
Combining Equations V-11a and V-llb and assuming that Equation V-7 will
be valid in a residual stress field (Lawn & Marshall (1977), yields
P aR
(3/2
PP ' CR +
+
CR(c)
.
V-12
Measurements of the crack size as a function of load on stress free and
residual stressed samples allows oR to be determined.
V-3
Experimental
A1 20 3 rings, prepared as discussed in Section III were polished with
1lp size diamond paste.
grain boundaries.
Some samples were etched in hot HPO4 to show the
The samples were then indented using a Vickers diamond
pyramid on a Leitz microhardness tester.
were measured from optical micrographs.
The crack size and indent size
It was more difficult to deter-
mine the crack length on etched samples as the cracks were often obscured
by the grain boundaries.
indent size impossible.
At high loads chipping made measurement of the
Typical photographs are shown in Figure V-2.
Five indents were made on each sample and the measurements were averaged.
The results are given in Appendix E.
Indents at loads above 20N gave results with a high degree of
scatter.
Variable alignment of the indenter and vibration of the load
caused chipping and uneven indents, which made measurement of the crack
size impossible.
V-4
Discussion
The value of K I calculated from Equation V-7 and V-9c are given in
Appendix E and shown as a function of cooling rate in Figure V-3.
The two
analyses give the K I value from Equation V-7 approximately one-half of the
K I value from Equation V-9c.
In this discussion, the K I value given by
Equation V-9c will be assumed to be correct.
Within the scatter of the
data, no trends in KIC as a function of grain size or cooling rate can be
observed.
As discussed in Section 11-1.4, the magnitude of the residual
surface compression is small.
The effect on KIC of the residual stress
is too small to be observed.
A residual surface stress of 0.3 MPa would
increase KIC by less than 0.004 MN/m
3/2
2 . The
increase in KIC with load
shown in Figure V-3 is due to the crack size being large relative to the
indent size and therefore not affected by the residual stress due to
plastic deformation around the indent.
As the crack size increases, the
crack front intersects a larger number of grains which gives a better
measurement of the bulk properties (Rice, et. al.
(1978)).
The single crystal value of KIC for A120 3 was reported by Weiderhorn
(1969) and Evans and Wilshaw (1976) as 2.1 MN/m 3 / 2 . At a 20N
load,
the KIC value measured in this study for a well annealed and slow cooled
single crystal of A1 203 from the A. D. Little Company was 3.4 MN/m 3 / 2
The increase is due to KIC being measured in a different orientation than
that reported by Weiderhorn (Becher (1976)).
The KIC values measured on polycrystalline samples ranged from
3/2
4.6o MN/m 3/2
3.95 MN
MN/m
to 4.60
Because no clear trends were observed in
the data as a function of grain size and cooling rate, a different method
was used to also measure KIC values.
This technique and the microstruc-
tural dependence of KIC will be discussed in Section VI.
V-5
Summary
The use of microhardness indentation to measure KIC was reviewed.
Using the relationships developed, KIC was measured as a function of
cooling rate and grain size for A12 0 3 polycrystals.
Scatter in the data,
due to errors in measurement of the indent and crack size and measurement
of a surface effect
not a bulk property obscures any trends.
Indentation can be used to measure large changes in the value of
KIC, but the effects due to TEA are too small to be observed.
0
0
P
4
Co
J•
(a)
V-I
(b)
Geometry of blunt and sharp indenters. a) Hertzian or blunt indenter,
C is the depth of pre-existing flaws on surface, r is the radius of
o the indenter, c is the depth of the crack. b) Vicker's or sharp
indenter, 2a is the impression diagonal, 2c is the crack length and
Zo is the depth of the deformation zone.
-r-.
,-;----_.
---------·
V.
. .
- t. ,.·
•
,
•,
r~,-----
I~
.,1
!
"*
*
..
•
•
,
·•
V-2
V-2
(a)
V-2
Typical Vicker's Indentation Impressions on A1,0 3
b) Load = 20N
c) Load - 59N
a) Load = 3n
V-2
(c)
120N
4.5
3.0
OU
E
z
r'
0
4.0
A
1zE
3N
Anneal(hrs)
04
-- x 6
A 12
020
0
2.5
(-
2
30
3.5
3.0
V-3
- ADL
* single
crystal
I
I
0.1
ADL
2.0
1.0
I
i
10
100
single
Acrystaol
I
I
I
I
0.1
I.0
10
100
8 (K/min)
,3 (K/min)
(a)
(b)
Toughness of A12 03 determined from indentation as a function of cooling rate.
Single crystal was cooled at 0.5k/min.
a) Load = 20N
b) Load = 3N
Filled squares are from samples that were cut in half.
VI
Fracture and Toughness
The fundamentals of fracture were derived by Inglis (1913), Grif-
fith (1920), and Irwin (1958).
The fracture stress was related to
material properties and the size and shape of small flaws which act as
stress concentrators.
For plane strain
2Ey
af
(lV2)rcrl
_
=
- 1/2
_Vl-1
where acf is the fracture stress, E is Young's modulus, v is Poisson's
ratio, yf is the fracture surface energy and c is the crack length.
the derivation, the crack tip is assumed to be sharp.
In
The stress in the
plane of the crack at a distance r ahead of the crack tip is
=
a
VI-2
(2Trr)
where K I is the stress intensity factor.
At fracture, K I assumes a
critical value which is given as
[2Ey
- f
K
IC
1f/2
VI-3a
1-v
for plane strain and for plane stress is given as
KIC
=
VI-3b
2Ef/2
The critical stress intensity factor, KIC, is also called the toughness
and is assumed to be a material property.
However, KIC is found to vary
with sample size, microstructure and test technique.
The variation of af and KIC with thermal expansion anisotropy stress
and microstructure will be discussed.
The fracture stress of unnotched
A12 03 rings and the toughness of notched A1 2 0 3 rings were measured and
will be presented.
VI-l
Fracture Strength
A sample will fracture when the stress on a suitably oriented flaw
(or crack) reaches the theoretical strength of the material.
The strength
controlling flaw can be related to the microstructure for large grain
sized materials and to extrinsic influences in fine grained sized materials.
Sources of extrinsix flaws include surface damage during handling
and polishing (Cranmer, et. al.
(1977)), porosity and inclusions.
It is observed in anisotropic materials that fine grain sized samples
do not spontaneously microcrack along grain boundaries, but large grain
sized samples do microcrack.
strength controlling flaws.
These microcracks, due to TEA can be
Davidge and Green (1968) developed a model
to predict the spontaneous fracture due to TEA stresses on the basis of
an energy criteria.
Kuszyk and Bradt (1973) and Cleveland and Bradt
(1978) extended the model to quantitatively predict the critical grain
size, G
crit
,crit
for microcracking in the pseudobrookite oxides.
The model
predicts spontaneous fracture when the stored elastic strain energy, Us ,
equals the surface energy needed to form a crack of some length that is
related to a microstructural feature, generally assumed to be the grain
size.
Us is a function of the volume under stress, which is related to
the grain size, and the surface energy needed is a function of the area
of new surface formed.
Us is usually calculated from the stress given by Equation 11-41.
The use of T E (which is a function of G and cooling rate) would give an
increase in Us with grain size.
The critical grain size, assuming dode-
caedron shaped grains is given by Cleveland
=
crit
T = T E - T room,
where
and Bradt (1978) as:
14.4yf
VI-4
EA2AT
where TE is given in Figure 11-3.
The spontaneous
fracture of grain boundaries reduces the effective cross-sectional area
supporting the applied load.
stress.
This would lower the measured fracture
Even if the boundaries are not fractured, the fracture stress
will still be reduced as shown by Pohanka, et. al.
(1976).
They measured
the fracture strength of BaTiO, at 1500C and at 250 C. Above 120 0 C,
BaTiO
3
is tetragonal and TEA stresses will develop as the sample is
cooled below 120 0C.
The measured fracture stress was lower at 250 C than
at 150 0 C for all grain sizes.
The difference in strength was independent
of grain size, as would be expected since the temperature is too low for
stress relaxation.
The fracture strength of ceramics as a function of grain size is
usually of the form
a
+
C
f
at small grain sizes and for large grain sizes as
VI-5a
af
M G-
½
VI-5b
It is expected that the fracture stress will be reduced due to the presence of TEA stresses, but since there are no measurements of the strength
of Al
20
with no TEA stress (infinitely slow cooled), no comparison can
be performed.
VI-2
Fracture Toughness
There is disagreement in the literature about the effect of grain
size and TEA stresses on the fracture toughness.
In this discussion it
will be assumed that Equation VI-3 is valid, so that the
toughness, KIC,
and the fracture surface energy, Yf, are directly related, and will be
used interchangeably.
For cubic materials, a survey by Rice, et. al.
(1978) concluded that there was no dependence of Yf on grain size.
Evans
and Langdon (1976) also say that if extensive microcracking does not
occur, there should be no variation in KIC with grain size.
However,
Monroe and Smyth (1978) found a decrease in KIC with increasing G in
Y,
2 0 which is cubic.
Simpson (1973) measured the KIC of Al 203 as a
function of grain size and found a decrease with increasing G, however,
Simpson, et. al.
(1975) and Simpson (1973) conclude that the observed
decreased in KIC with grain size was due to the test technique used.
They conclude that the single edge notched beam, (SENB), technique for
measuring KIC was subject to increasing errors as the grain size increased,
which tended to decrease KIC.
This may be the causeof the decrease
observed by Monroe and Smyth (1978) in Y2 0 3.
Pratt (1977) found a slight
decrease in KIC with increasing G in A120 3 , and also found no difference
among the different test techniques.
The grain size used by Pratt was
smaller than the grain size at which Simpson, et. al.
errors in the SENB technique.
(1975) observed
It may be that Pratt would have observed
differences among the different techniques at larger grain sizes.
Gutshall and Gross (1969) observed a two-fold increase in KIC for
AL2 0 3 as G increased from 10pm to 50Pm.
They also observed that the
fracture mode changed from intergranular to transgranular, and they assumed that this change in mode was the cause of the increased KIC.
Rof
(1979) observed a change in fracture mode in Zn0 2, from intergranular to
transgranular as the grain size increased from 2pm to 30m.
However, he
found that the KIC decreased, in direct opposition to the data of Gutshall
and Gross.
Rof assumes that the decrease in KIC as G increases in his
samples is due to a decrease in microcracking as the grain size increases.
The effect of TEA stresses was not considered except by Rice, et. al.
(1978).
It has been proposed that microcracking may increase the KIC of a
material.
Claussen, et. al.
(1977) and Porter and Heuer (1977) have ob-
served an increase in KIC in two-phase materials, with a residual internal
stress.
Rice, et. al.
(1978), Evans, et. al.
(1977), and Pompe, et. al.
(1978) have developed models to predict the effect of microcracking on
K IC.
Pompe, et. al. considered microcracking in a process zone ahead of
the crack in a two phase material.
They found that the KIC could be in-
creased or decreased depending on the density and shape of the microcracks.
Rof (1979) assumed that the process zone (region in which the
stress can cause microcracking) was constant with grain size.
At small
grain sizes, the process zone is larger than the grain size and microcracking occurs.
Energy is dissipated in forming cracks.
In large
grained samples, the process zone is smaller than the grain size and no
microcracking occurs.
ing could increase KIC.
Rice, et. al.
(1978) also assumed that microcrack-
In anisotropic materials, TEA stresses will
assist microcracking, and since the additional energy needed to cause
microcracking decreases as Gcrit is approached, KIC would increase with
G. Above Gcrit, spontaneous microcracking occurs and KIC decreases as
the effective cross-sectional area is reduced (see Figure 1-2).
et. al.
Evans,
(1977) proposed a model that predicts a decrease in KIC with
based on energy considerations.
The literature data on the effect of grain size and TEA stresses on
KIC is conflicting, and a comprehensive theory has not been developed.
VI-3
Experimental
The fracture strength and toughness were measured on A12 0 3 rings,
prepared as described in Section III,
fractured by internal pressure.
This technique increases the volume of the sample under the maximum stress
relative to bend tests.
Internal pressure was applied using a silicon
rubber plug to transmit the load as shown in Figure VI-1.
The load was
applied by a standard testing machine at a crosshead speed of 0.13mm/min
(.005"/m in).
The tensile stress in the ring is (Sedlacek and Halden (1962))
S0(r) =
r
Pr
I+
2
r- r
iv-6
r
where P is the pressure, ro and r. are the outer and inner radii and r is
distance along a radius.
For internal pressure, the tensile stress is
highest at the inner edge of the ring.
The rubber plug was assumed to
behave hydrostatically, so the pressure, P, was directly related to the
applied load.
Rings were tested with unpolished and polished surfaces, but no
difference in the fracture stress with surface finish was observed.
Figures VI-2 and VI-3 show the fracture stress as a function of grain
size and typical strength histograms.
The fracture toughness, KIC, was measured on notched rings.
notch was made using a 0.15mm wide diamond saw blade.
The
A typical notch
is shown in Figure VI-4 and shown in Figure VI-5 is a fracture surface.
KIC is given as (Rowecliff et. al.
K
IC
(1977))
w(f )
VI-7
where a is the notch depth, w is the sample width, af is given by Equation VI-6 and f(2-) depends on the ring geometry r-. Rowecliff, et. al.
w
r.
determined f(.)
for several values of - . For the rings used in this
w
r.
r0
ro
I
study
-- = 1.21.
r.
For
-
r
= 1.25, f( ·) is given as
f(a) = .265 + 4.15(--)
w
w
w
- 4.
5
(a)
w
2
+ 5.42(a)3
w
.
vi-8
The notch depth was measured on each sample and samples in which cracking
at the bottom of the notch was observed were discarded.
KIC was calcul-
ated as the average of at least 15 samples for each anneal time and cooling rate, and is shown in Figure VI-6 and given in Appendix E.
VI-4
Discussion
The fracture strength shows the expected G-
effect of cooling rate is observed.
dependence, but no
This is due in part to the large
scatter of the data, which are plotted as the average strength, as shown
by the strength histograms.
Using Equation VI-I and yf = 20N/m(20J/m 2 )
the critical flaw size was calculated and found to be >250 m for the
strongest samples.
grain sizes.
This calculated flaw size is much larger than all the
The strength controlling flaw is probably due to a very
large grain or an inclusion.
large scatter.
The random nature of such defects causes the
Since measurements could not be made on equivalent
samples without TEA stresses (as could be done for BaTiO ), the effect of
TEA stress could not be observed.
While the literature shows no clear trends of the effect of grain
size on toughness, it is expected that residual stress will increase the
toughness as observed by Claussen (1976), Claussen, et. al.
(1977) and
Porter and Heuer (1977). Residual stresses due to the presence of a
second phase can increase the toughness of a material.
The increase in
toughness arises from the formation of microcracks that do not link up to
the main crack.
The amount of surface formed is larger and hence more
energy must be supplied to the crack to propagate it through the sample.
It is also possible that non-reversible interactions of the stress field
due to the second phase and the applied stress field ahead of the crack
could cause toughening.
In an isotropic material, the applied stress could add to the TEA
stress, causing microcracking on suitably oriented grain boundaries, as
proposed by Rice, et. al.
(1978).
As the grain size increases, the stored
elastic energy approaches the surface energy needed for spontaneous microcracking and the amount of energy absorbed decreases.
Also, since the TEA
stress is grain size dependent, the stored elastic strain energy increases faster than predicted by Cleveland and Bradt (1978) and Rice, et.
al.
(1978).
Hence, KIC increases with grain size at small grain sizes,
and reaches a broad maximum near G
crit
, as the amount of energy absorbed
per microcrack (and therefore the increase in toughness) decreases.
The
variation in G within a sample will also contribute to the broadening of
the curve.
The data of this study do not show a grain size dependence
of KIC*
Although the scatter inthe data is large, an increase in KIC is observed
as a function of increasing cooling rate, or increasing TEA stress.
data for 1 = 100K/min show the highest scatter
values corresponding to the largest grain sizes.
The
with the lowest KIC
This decrease in KIC
may be due to cracks formed during notching of the samples, or slow
crack growth during testing.
Both factors would lower KIC.
size dependence is not observed at other cooling rates.
This grain
The magnitude of
KIC measured by the notched ring technique is lower than the value
of KIC
measured by the indentation technique and reported literature values.
This
decrease is probably due to the notch not being aligned along a radius.
Variations in the eccentricity of the rings made such alignment difficult,
and is a factor in the scatter of the data.
The fact that at large grain
sizes (> 100-m) the thickness of the sample below the notch (=400om) was
close to the grain size
will also cause a high degree of scatter.
In
some cases, a large grain may occur at the root of the notch and the
measured KIC, in this case will be more representative of the single crystalline KIC than the polycrystalline KIC.
This behavior was not observed
on the sample examined (see Figure VI-5).
VI-5
Summary
The fracture strength and fracture toughness of A1 20 3 ring samples
was measured.
The strength was found to be a function of G
2,
as is
typical for ceramics, but no dependence of strength on cooling rate was
observed.
The scatter of the data was very large.
The toughness in-
creased with increasing cooling rate but no dependence on grain size was
observed.
The literature data on KIC as a function of grain size for A1 0
2 3
and other materials
are contradictory.
This is due to microstructural
variation in samples (grain shape, pore size and shape, impurities) and
test technique.
KIC is reported by Simpson, et. al.
same samples as of function of the test technique.
(1975) to vary on the
Theoretical predictions
also contradictory, but it is assumed that an increase in KIC with grain
86
size, for small grain sizes, in an anisotropic material is expected.
large grain sizes, KIC is expected to decrease.
At
The effect of other
variables, such as the grain shape, porosity and the effect of grain size
in the absence of TEA stresses make elucidation of the TEA effects difficult.
Fracture toughness and strength can be used to measure the effect of
TEA stresses, but many factors can obscure the results.
Load
Silicon
rubber
Li-, VJ/
U
I1I
Pd* d
L
"-Sample
i.
!?:.'.
I
I
N
~~~I~ 1
VI-I Experimental arrangement used to apply internal pressure.
The load was applied with an Instron Testing Machine at a
constant crosshead speed. The silicon rubber was wrapped
in 0.03mm Ni foil to prevent it being cut by the sample.
The top clearance was at least 0.03 mm.
S
0
0
400
150
0
G(/0m)
100
40
10
15
100
0L
50
0
0
VI-2
5
G-1/2 (cm)-1/2
Data points represent
Fracture stress as a function of G 2.
average of at least 20 samples broken by internal pressure at
a constant crosshead speed. Solid line is least square fit to
the data.
5
0
I0
E
0
I0
5
0
5
15
10
20
25
c-f (ksi)
VI-3
Typical stren gth histograms.
vertical bar.
Average is indicated by the
30
-
-·-~--.~-~-~11--;-
-III---~-~....
VI-5
IC
IIIIIIII~
~~1,-.
__ ·
__.
I
I
Typical Fracture Surface of Notched Samples. a) and b) are
opposite ends of the same sample. The root of the notch can
be seen at the right of the photo.
II
_~I~
_
_
T
-----------
-----
·--
·--------
~-·--
I,-_
__
.~ss~--
-I
I
A
VI-4
Typical notch used for toughness measurements.
cut notch was 0.15mm wide.
Saw blade used to
II
0
*
S
0
S(K /min)
U
A 0o
, 100
U
CJ
E
2
x
Xx A
m
A
r')
E
z
n
e 4
x6
A12
"=
0
0
X
C)
Anneal hrs.
D0 1
03
@ 0.1
x 1.0
AA
()e
x
o
A
0a
I
1
I
I
10
15
20
Anneal (hrs)
(a)
VI-6
Il
I
0.1 1.0 I0 100
18( K/min)
(b)
Toughness measured from notched samples as a function of cooling rate
and anneal time. Each data point is the average of at least 15 samples
broken by internal pressure at a constant crosshead speed.
VII Summary
Al 203 is a material with anisotropic thermal expansion.
This ani-
sotropy causes stresses in polycrystalline samples when the temperature is
changed.
A model has been developed to predict the stresses due to ther-
mal expansion anisotropy (TEA) in a polycrystalline sample as it is
cooled from high temperature.
The model combines stress generation due
to TEA with stress relaxation by diffusional creep.
At high temperature
(above TC) the stress relaxation is rapid and all stresses can relax and
at low temperature the relaxation is very slow, so no stresses relax.
Below a critical temperature, TR, where no stress relaxation occurs, the
stress increases linearly with temperature.
The value of TR and TC are
functions of both the grain size (which defines the diffusion distance)
and the cooling rate (assumed constant).
The linear region of the stress
temperature function was extrapolated to zero stress to define TE, which
was found to be approximately
T
T
E
=
+ 2T
3
R
VII-1
Using TE and Equation 11-41
OTEA
where
T = TE - T
room
=
1/2 EAaAT
and Aa is the expansion mismatch.
11-41
The room tempera-
ture stress due to TEA could be estimated.
Stress relaxation by creep was also used to analyze the residual
stresses in a polycrystalline A1 2 0 3 sample due to quenching.
It was found
that the temperature of the sample before quenching must be above a
critical temperature, TQ, or the sample will not be strengthened due to
thermal tempering.
This agrees with literature data on the thermal
tempering of A12 03 .
Polycrystalline Al 203 samples with various grain sizes were cooled
at different constant cooling rates.
The stresses due to TEA were
measured directly bya spectroscopic technique and were inferred from
measurements of the toughness.
The spectroscopic measurement of TEA stresses used the change in the
fluorescence spectrum with pressure.
fluorescence lines due to Cr
3+
The shift and broadening of the
impurities in Al 0, as a function of pres-
sure has been well documented in the literature.
The calculation of the
stress from the line shift and line broadening measurements is based on a
model developed by Grabner (1978).
The measured stresses did not show the
expected grain size dependence, but this is felt to be due to the distribution of grain sizes in a sample.
The calculated stresses were based on
a sample having one grain size, nota wide distribution of grain sizes.
The measured stresses were found to vary with the cooling rate, and the
magnitude was in good agreement with the calculated values.
This tech-
nique isa very good technique for measuring the residual stresses in Al 0,.
The effect of TEA stresses on the toughness of Al 203 is not clear.
While it has been demonstrated by Pohanka, et. al.
(1976) that TEA stres-
ses can lower the fracture strength of a material, high scatter in measured
toughness or predictions.
In this study the toughness was measured by an
indentation technique and a notched-ring technique.
The crack and indent size from a Vickers indentation were measured
and the toughness was calculated from an empirical fit to measured data
given by Evans and Charles (1976) as
KC
=0.1553
iCa
0.055 log (8.L
)
1 (3.73x 10-
a
-
VII-2
where a and c are the indent size and crack size and P is the load.
No
clear dependence of KIC on either grain size or cooling rate was observed.
The toughness was also measured using a notched ring which was
stressed to fracture by internal pressure.
The toughness was calculated
from an expression given by Rowcliffe, et. al.
KIC
7TaGf[0.265 + 4.
15
(1978) as
( ) - 4.5(,)2 + 6.2()3
VII-3
where w and a are the sample width and notch depth and of is the fracture
strength.
The toughness was found to increase with increasing cooling
rate (and hence increasing TEA stress), but again no dependence on the
grain size was observed.
Since the effects of TEA stresses on toughness
are unclear, no direct comparisons between theory and experimental data
can be made.
However, it is felt that the increase in toughness with
increasing cooling rate is due to TEA stress.
The lack of a dependence of the toughness or the spectroscopically
measured stresses on grain size is due to two reasons.
The first is the
wide grain size distribution in the samples which is not considered in
the model.
The second is the relatively small change in the predicted
magnitude of the room temperature TEA stresses with the grain sizes used.
From Figure II-1 it can be seen that the stresses vary by less than 10%
over the range of grain
sizes used.
The variation of TEA stress with
the cooling rates used is much larger (20%) and a dependence of the
measured stresses on cooling rate is observed.
VIII Suggestions for Future Work
This study suggests many other areas of study related to thermal
expansion anisotropy.
Since the fluorescence technique is at present ap-
plicable only to A12 03 it would be of great value to measure the pressure
coefficients for other materials.
A narrow grain size distribution and a
wider range of grain sizes (from <1 p to as large as can be made) may
show the predicted grain size dependence of the TEA stress.
grain-size distribution should be measured.
Also, the
The critical grain size for
spontaneous microcracking could be determined.
Using an ultrasonic de-
termination of the elastic modulus, the grain size at which microcracking
occurs could be determined as a function of cooling rate and compared to
the predicted values of the TEA stress.
A systematic study of the effect of grain size on the fracture
toughness, in both cubic and non-cubic material, would be very valuable.
Using different cooling rates, the TEA stresses in anisotropic materials
could be held constant over a range of grain sizes.
Also, measurements
of the acoustic emission during slow crack growth, as a function of TEA
stress may show the extent of microcracking occuring due to TEA.
The study of TEA stresses in other materials may prove illuminating.
As shown by Pohanka, et. al.
(1976) BaTiO
show the effect of TEA stress.
3
can be used to definitively
Other material with a very high thermal
anisotropy (such as pseudobrookite oxides or PbTiO 3 ) may be easier to
study.
The grain size dependence of the minimum temperature necessary for
strengthening by quenching could also provide confirmation of the validity of the creep relaxation model developed here.
A possibility for extension of the model would be to consider other
cooling rates, in particular to investigate the effect of oscillating
the temperature near TR on the reduction of the room temperature stress.
Appendix A
Consider two grains, with a common boundary of length X 0 at temperature T o . As the temperature is changed stresses will be developed in
the two grains if the expansion coefficients are different.
In this dis-
cussion it will be assumed that the temperature is decreased and that
az > a .
If the two grain were not joined, their lengths at any tempera-
ture would be
X,
=
X2
=
_0 1 + oI(T 0 -T T
A-la
0 I + a 2 (T 0 -T]
A-lb
If the grains are assumed to be joined by a visco-elastic medium, their
lengths as a function of time for an instantaneous change in temperature
of T0 -T are
X1
X
=
2 =
X0[ + oi(T - T)
X0
L
+ a
+
(T - T)
t)]
A-2a
A-2b
E j
At t = 0, no relaxation can occur in the viscoelastic medium and X1 = X
and the stress is
(a
o(t=O)
=
-u2)(To -T)E
2
-
T)
A-3
This is the form of the stress function that is usually assumed (i.e.,
no relaxation of stress).
When the grains are allowed to creep under the
stress, the stress will exponentially decay to zero.
Rearranging
Equation A-2 gives
=
where
X1
and
(- - a 2 )(To - T)
. S(t) X
are function of time.
X2
da
E
d
2dt
dt
Differentiating yields
(")dTl
X
A-4
I
A-5
dt
The strain due to the thermal expansion anisotropy is
X, TEA
where
XI
X2
X
is length of the grains at t = 0 and is
=
X,
X
I +
2
(T - T)
.
A-7
Neglecting higher order terms
Xz - X2
X1 - X2
X
X0
A-8
and Equation A-5 becomes
do
E
+
d2
_ o-)d-
A-9
2dt
It is assumed that the strain rate is related to the stress by the
Nabarro-Herring creep equation
E
=
a/neff
A-10
100
and that the temperature is a linear function of time
dT
A-li
dt
Substituting Equations A-10 and A-l1
into A-9 yields
GE
2nef
+
f
al - a 2
2
E .
A-12
101
Appendix B
The stress in a grain due to thermal expansion anisotropy for a
material being cooled at a constant rate, 1, is the solution of
do
dt
_
l
21nef
+
f
A_+
2
B- 1
where p is the effective modulus, neff is the effective diffusional viscosity and As is the magnitude of the expansion difference.
Equation
B-1 is linear in stress and can be solved by use of an integrating
factor.
Equation B-I can be rewritten as
do
-dc
dt
where
P =
-F
23rlf
2 eff
+
=
Pa
Q
B-2
Q = a
2
2
and
e PdT Q dt
=
ePdT
Then the stress is
+
C
B-3
where C is determined from the boundary condition
a(T = Tana )
=
0 .
The difficulty arises in evaluating
JPdT
neff
= -
neff
eff
G2 kT
=
14
JPdT
fl/nff dT
is given from Equation 11-5 or
EB-6
Deff
eff
B-4
B-5
102
so Equation B-5 becomes
2G2 k f
expanding on Def f as
PdT
Let
A =
141
D o
2G 2k
Qff
RT ) yields
D exp
=
-
dT
b = Q/R
;
I PdT
and
= Af
PdT
Then
exp(-b0)d0
B-9
b >> 1 as
which can be approximated for
f
1
0 =
B-8
= A L-exp (-be)
B-10
-b
substituting back into Equation B-3 gives
A
c = exp
ex(-b
expb
)I
-
exp A
e x pb ( -Ob
(be
QdT + C'
B-11
bexp
where
C' = exp
exp
be
C
The expansion of the exponentials and integration of Equation B-ll
to a series which does not rapidly converge.
leads
Hence an analytic expression
for a as a function of temperature cannot be given.
The stress as a
function of temperature was evaluated numerically at M.I.T.'s Information
103
Processing Center.
The critical temperatures, as discussed in Section II, can be
approximated by assuming that the critical relaxation time (time for the
stress to relax to l/e of its original value) is proportional to the inverse
of the cooling rate, and solving the stress relaxation at constant
strain.
This is equivalent to making the temperature a step function of
time with the step lengths proportional to the cooling rate.
This aprox-
imation agreed well with the result of the numerical solution, where the
step width was decreased until further reduction had no effect on the
results.
TC was taken as the temperature of which TR equaled 1/W and T R was
taken as the temperature at which
TR
equaled
relaxes to 0.999 of its original value.
1000
--
or thus the stress
This was chosen from the nume-
rical solution.
The stress as a constant strain is given as
d-a
-
2ref
B-12a
2rn eff
dT
and
In-
1f
c•
I
=
Substituting TR for AT/M
2G2 k
412G TC
-
IAT
2ýrl
eff
B-12b
yields the critical temperatures
=
D exp(-Q/RTc)
B-13a
104
2G2k TR
14
1000
D exp(-Q/RTR)
0
R
TC and TR are shown in Figure 11-3.
vs. In
B-13b
It is seen that the slope of I/T
increases with grain size showing the change in the effective
activation energy for diffusion as the grain size increases.
105
Appendix C
For a flat plate, with X 2 and X 3 oriented in the plane of the plate
and X, normal to the plate, the stress for a temperature variation in
the X1 direction only is
S=
0
0
0
0
a 33
0
0
0
a3
.
C-1
This stress can be separated into a hydrostatic component and a shear
component.
2-
0
a..
=
Ii
2
0
-j-
0
0
2
0
3u33
33
33
-j-03
+
0
0
3jc s
33
0
0
30a33
0
0
C-2
-3-a 3 3
To calculate the line shift due to the stress in a given grain, the
stress in the X1, X2 , X3 coordinate system must be transformed to the
crystllographic coordinate system of the grain al, a2 , c
X".
or
X , X,
The tensor transformation law is
ij
=
aikajlakl
C-3
where a.. is the cosine of the angle between X. and X., and repeated
IJ
J
subscripts imply summation over that index.
is invariant under transformation.
yields
Thy hydrostatic component
Substituting Equation C-2 into C-3
106
a11
3
3 ijii a 33
ij
a11
33
-0
is the Kronecker delta.
c-4
1
a21 -"
31 11
IJ
a11 a13
22
a 21 all
where 6..
12
321
a21 a 3
a31
3
The effect of stress on the fluores-
cence spectrum is given as
Av
where
..
Ij
=
C-5
T..a..
IJ ij
for Al203 is
0
0
0
r11
0
0
0
T3 3
TFT
11
Ij
c-6
Substituting C-4 into C-5 gives
Av
3 a=
a 33 (27
+1 33
11 +F
) -a 3
11 (a211
+a 221 -
32+
T3
(aa
-)
C-7
Using the relationship between the direction cosines
a
11
+ a2
22
+ a
33
=
C-8
1
gives for the shift in the spectrum as
Av
2
2
33
(2711 + T3)
33
-33
(33 - 711 )(a 311
3
C33 (T33
- 3)1
3
C-9
107
Appendix D
The fluorescence spectrum was recorded as a function of wave length
using a chart recorder.
The widths of the peaks were measured from the
chart and the peak locations were measured from marks put on the chart
by the Spex monochrometer.
The peak locations and widths are given in
Table D-l and shown as a function of cooling rate and grain size in
Figures D-l through D-4.
108
Table D - 1
Sample
Thermal History
Anneal (hrs.)
1
3
4
6
12
20
1
3
4
6
12
20
0
2
3
5
11
19
2
8
as received
Single crystal ruby
B (K/min)
100
100
100
100
100
100
10
10
10
10
10
10
1.0
1.0
1.0
1.0
1.0
1.0
0.1
0.1
Width (cm- )
R
2.335
2.256
2.260
2.246
2.232
2.273
2.129
2.263
2.132
2.113
2.123
2.165
2.207
2.024
2.058
2.134
2.060
2.094
1.937
2.031
2.239
0.744
R2
2.170
2.160
2.170
2.181
2.137
2.166
2.059
2.122
2.051
2.036
2.016
2.056
2.035
1.866
1.943
2.106
1.952
2.018
1.909
1.962
2.062
0.368
Shift (cm-
)
R1
R2
-0.096
-0.200
-0.221
-0.200
-0.263
-0.304
-0.387
-0.387
-0.575
-0.159
-0.304
-0.325
-0.491
-0.263
-0.304
-0.429
-0.200
-0.429
-0.304
-0.200
-0.429
-
+0.102
-0.002
-0.860
-0.002
-0.023
-0.106
-0.211
-0.378
-0.294
-0.002
-0.044
-0.148
-0.336
-0.169
-0.357
-0.253
-0.044
-0.148
-0.232
-0.065
-0.399
-
Anneal
(hrs)
2.3
2
S01
H-
1I
0
2.1
4-+
2.0
1.9 A
0.1
gA
0
I
0<
C\
0
A 12
2.0
_~x
-D
i
I
i1
0 3
e 4
x6
A 12
i 20
9
A
1.0 10 100
P( K/min)
--
o
w
v
x
A
-I
0 20
A
A
1.8
I
0.1
(a)
D-1
Anneal
(hrs)
01 1
0 3
. 4
x6
2.2
Width of R1 and R 2 lines as a function of cooling rate.
0
I
1
1
1.0
10
100
(K/min)
(b)
2.3
U
2.2
x
2.1
C\j
Am
x
x
Ix
I•
Wk
2.0
•
AA
A
x
1.9
1.9
x
x
1.8
I
I
2.1
x
-c
2.C
, (K/min)
* 0.1
x 1.0
A 10
a100
2.2
0<
0<
4-,
2.3
, (K/min)
* 0.1
x 1.0
A 10
S100
O
1
I
I
10
15
20
Anneal( hours)
(a)
D-2
Width of RI and R 2 lines as a function of anneal time
I
I
I
I
I0
15
20
Anneal ( hours)
(b)
0
0)
0.6
0.5
0.5
Anneal
(hrs)
0 1
0
0.4
03
* 4
x 6
A 12
S20
'0.4
0
4
*
6
*
A 12
20
x
0
A
0.2
0
0
V-
4-
Go
o0 3
o
0.3
' 0.3
Anneal
(hrs)
0 1
.0.2
Ax
0
U 0.I
*
A
A
I
I
I
0
0.1
I
0
0.1
I
I
1.0 I0
100
I
-0.1
0.1 1.0 I0
P( K/min
(a)
D-3
Shift of R1 and R 2 lines as a function of cooling rate.
'3
r
100
K/min )
(b)
0
. 6n
0.5
/(K/min)
S0.1
x 1.0
A
0.5
0.3
x
U
EU.
x
*0
IC)
to
x
x
x·
0.1
0
0.I--m
0
-
x 1.0
A 10
S100
N0.2
4-
C, 0.2 _
A
* 100
0.4
0.3
~(K /min)
. 0.1
0.4
x
E
0
I
II
I
I0
15
Anneal (hours)
(a)
D-4
20
-0I C
)
.
I
SI
15
Anneal (hours)
(b)
Shift of RI and R 2 lines as a function of anneal time.
I0
I
20
113
Appendix E
Samples were indented using a Leitz microhardness testor with a
Vicker's diamond pyramid using loads of 3N, 20N and 59N.
The indents
were on polished surface and measured from optical photographs.
indents were done on each sample and the sizes were averaged.
Five
Figures
E-l and E-2 give the average crack length and indent size for the
samples;
size.
at loads of 3N and 20N, as a function of cooling rate and grain
KIC was determined from
K
0.155
3/2
P
.055 log (8.4 a )(3.73x 10-13 P0.4
E-
where P is the load in Newtons, a is the indent half diagonal in meters
and c is the crack length in meters (see Figure V-l).
KIC from the in-
dentation measurements is given in Table E-l.
KIC was also measured by a notched ring technique.
Notched rings
were fractured by internal pressure and KIC was given as
KIC =
0.265 + 4.5(-)
2
4.5(()2+
2
4 1 a·
5.42(
a
1i
a
where w is the width of the ring, a is the notch depth and of is the
fracture stress (see Figure VI-4).
KIC is given in Table E-2.
E-2
114
Table E - 1
Sample
Thermal History
Anneal (hrs.)
3 (K/min)
4
6
12
20
20
4
6
12
20
20
3
5
11
19
19
2
8
8
ADL Single crystal
*Sample cut in half
100
100
100
100
100*
10
10
10
10
10*
1.0
1.0
1.0
1.0
1.0"*
0.1
0.1
0.1*
0.5
KIC
(MN/m
3N
20N
1.96
2.47
2.14
2.51
2.24
2.55
2.36
2.23
2.55
2.82
2.64
2.42
1.94
2.15
2.53
1.75
2.12
2.38
2.12
4.51
4.55
4.15
4.48
4.40
4.51
4.35
4.00
4.45
4.06
4.47
3.94
4.29
4.41
4.02
4.52
4.40
3.40
115
Table E -
Sample
Thermal History
2
MN/m
Number
of
Samples
2.69
1.97
2.24
1.81
1.36
1.81
1.93
1.55
1.47
2.08
1.93
2.06
1.36
2.08
2.03
1.71
2.00
2.25
1.61
1.20
2.00
20
20
16
20
19
20
19
20
19
18
18
16
19
16
17
19
20
20
20
19
10
KIC
34
Anneal (hrs.)
1
3
4
6
12
20
1
3
4
6
12
20
0
2
3
5
11
19
2
8
as received
(3 (K/m)
100
100
100
100
100
100
10
10
10
10
10
10
1.0
1.0
1.0
1.0
1.0
1.0
0.1
0.1
13N I
16
--
E
Anneal
.
x
A
ADL
13
a)
N
0
-
13N
Anneal
(hrs)
. 4
4
6
A 12
N 20
020
Samples
cut in
half
crystal
14 o
45
(hrs)
x 6
A 12
40
A
&
C)
•
ADL
single
35
crystal
mQA
N
AO
30
-4-
a)
12
C)
C
I
I
25
10
0.1
1.0
10
100
3p(K/min)
(a)
E-1
20
*3
I
I
0
I
0.1 1.0 10
I00
P,(K/min)
(b)
Indent impression and crack size as a function of cooling rate
a) and b) load = 3N.
I
0 2O
0 20
Samples
cut in
half
0
0
0
-
25
20 N
A
E
24
crystal
SA
_N
4-
c0
03
A ADL
single
"A
0
0
Anneal
x
A 12
S20
0 20
0[ Samples
x
cut in
half
S70
A
*
O
60
I
I
I
S
-0
U
x
(d)
(c)
cont. c) and d)
A
0
0.1 1.0 10 I00
P( K/min)
0.1 1.0 10 100
P( K/min)
E-I
S20
0 20
Samples
cut in
half
E-
o
65
2:
I
o 4
x 6
A12
crystal
E 75 F
N
S
Anneal
(hrs)
ADL
single
6
23
22
120 N
80
(hrs)
0 4
Load = 20N
0
0
*
M
16
13N
3(K/min)
* 0.1
x 1.0
A 10
m 100
I3NI
45
U
14
-
0
a)
a)
Samples x
U
c-4
35
x
-0
x
N
cut in IN
half
8(K /min)
* 0.1
x 1.O0
A 10
m 100
40
E
0
*
A
-z
30
A
x
A
12
Samples
cut in
E-
C
xm
25
IC)
• V
•
•Mph,
I
15
Anneal (hours)
10
(a)
E-2
m
20
20
I
I
I
I0
15
Anneal (hours)
I
20
(b)
Indent impression and crack size as a function of anneal time
a) and b) Load = 3N.
oo
120NI
I
-
25
8(K/min)
a 0.1
x 1.0
A 10
A
L
N
x
Samples
cut in
half
aC
U)
"V
70
I
o
L.
0
U)
23 .T
x
I
C)
I
II
•
15
10
Anneal (hours)
•
20
65
cont. c) and d)
Load = 20N.
X
A
Samplest
60
SI
l
cut
halfin
I
15
I0
Anneal (hours)
(d)
(c)
E-2
A
x
I
22
./(K/min)
0 0.1
x 1.0
A 10
] 100
75
A
.=.,.
--
80
inn
Ivy
U
0
24
120N
.
I
120
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Biographical Note
The author was born on May 9, 1952 in Albany, New York and raised
in Delmar, New York.
graduated in 1970.
He attended Bethlehem Central High School and
He received a Bachelor or Science in Ceramic Science
and a Bachelor of Arts in Mathematics from Alfred University in 1974.
He received a Master of Science in Ceramics from Massachusetts Institute
of Technology in 1976.
He is a member of Keramos and Sigma Xi.