Heinrich J.G., Aldinger F. (Eds.) Ceramic Materials and Components for Engines
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A
combination of eqs.
(2)
and
(3)
for a gives:
In principle, with known values of
x
and
Y,
eq.
(6)
allows direct determination of the R-curve by monitor-
ing the crack length
c
as
0,
is increased. Several meth-
ods have been proposed to evaluate the parameter
x
form special graphic representations of eq.
(6) [6,9].
In analogy to eq. (5a),
eq.
(6)
can be solved for the
maximum stress at failure,
o,s(P),
by applying eq.
(4)
and a suitable hnction
for
K,(c).
The analytical analysis
of KRAUSE
[3]
who used a power law function
as
a
representation of the R-curve shows that the presence of
an R-curve leads to
oIs
IX
P”
with
n
<
113.
Comparable
results. i.e.
a
deviation from the power law behavior
with
n
=
1/3
are achieved by using more physically
based relationships for the R-curve
[
10-1 11.
The following simplifications are implied in the pre-
sented analysis concerning the geometry factor
Y
and
the parameter
x:
a) The geometry factor
Y
is assumed to be constant
during the stable growth of the crack. It has been recog-
nized that this is not necessarily true for half-penny
surface cracks, but it has also been argued that the
changes
in
Yare small enough to be neglected
[3].
Only
a few experimental data exist on the evolution of the
crack shape
[9, 12, 131.
They indicate that
Y
decreases
during stable crack extension up to
as
much as
25%.
b) The parameter
x
is also considered to be constant
during crack growth. There is some empirical
[13, 141
as well as some theoretical
[
151
evidence that increasing
the applied load causes the residual stress to decrease.
Furthermore, a dependence of
x
on the indentation load
cannot be excluded
[16].
It is the aim of this work to investigate the influence
of possible changes of
Y
on the indentation strength and
IS-plots with the help
of
simple model calculations.
To
back up the theoretical considerations, experiments to
determine the evolution of the crack shape of indenta-
tion cracks during stable crack growth are performed
using a multiple decoration technique.
EXPERIMENTAL
Experiments were performed on a commercial gas
pressure sintered silicon nitride doped with approx.
3
weight% MgO provided by
ESK,
Kempten, BRD.
Young’s modulus
E
=
306,s
f
0,3
GPa was determined
by a resonant beam technique
[17],
Fracture toughness
K,
=
6,4
f
0,6
MPadm was measured using the SEW-B
method
[
181,
hardness is
H
=
14,6
GPa.
Flexural specimens
(3~4x45
mm3)
were machined
from large plates and mirror polished to
1
pm diamond
finish on the tensile surface.
Vickers indentations were placed
in
the center of the
tensile surface with the indentation diagonals aligned
parallel and normal to the specimen’s long axis. Loads
between
49
N and
294
N were used. During the inden-
tation process the cracks to be used to study the evolu-
tion of the crack geometry were decorated with lead
acetate solution according to a procedure described
elsewhere
[19,20].
The growth of indentation cracks was monitored by
interrupted indentation strength (IS-) tests on an electro-
mechanical universal testing machine. The decorated
specimens were loaded to different load levels below
the indentation strength and then immersed into satu-
rated lead acetate solution to decorate the newly formed
portion of the crack which is in each case typical for the
applied load level. After drying, these specimens were
broken to make an investigation of the crack profile
possible. By placing two indentations within the region
of constant bending moment it was possible to produce
stably grown cracks at relative applied stresses up to
o,/o1s=
0,97.
Due to the flat trend of the o,(c)-curve
in this region, these cracks may still be considerably
shorter than
cnl.
Note that the indentation strength of the
decorated specimens was the same
as
for non-decorated
specimens. All experiments were performed in air at
room temperature.
MODEL
CALCULATIONS
Incorporating the possible influences mentioned
above into eq.
(2)
leads to a general expression:
(7)
Applying the conditions for equilibrium and stable
crack growth, eqs.
(3)
and
(4),
allows the calculation of
oIs
and
c,,,
and of corresponding pairs
of
o,
and
c
values
for known trends of
K,(c),
~(c,
P),
Y(c)
and chosen
values for
P.
The starting crack length
co
is calculated
from the condition
Kt,(O,
CO,
P)
=
K,(CO).
Calculations following this scheme were performed
for three conditions. Where it was necessary to define
material parameters or specimen geometries, values that
correspond to the material and specimens described in
the experimental part were chosen.
i)
crack-length independent
Y:
K,,
x
and
Y
are inde-
pendent of
c.
K,
=
7
MRadm,
x
=
0,07
(with
E
and
H
of
the investigated material after ANSTIS
[7])
and
Y
=
1,26.
The results for this case should be identical with ana-
lytical solutions presented earlier.
ii)
crack-length dependent
Y:
the geometry factor de-
creases with crack length during stable extension. The
trend for this change has to be calculated separately, a
possible procedure will be proposed below.
K,
=
7
MPadm is constant.
iii)
crack-length dependent
Y
&
R-curve:
both, the
fracture toughness and the geometry factor are consid-
ered to depend on crack length. Any possible interac-
tions between the decrease of
Y
and the increasing
fracture toughness are neglected.
A
pronounced R-curve
in the form of an exponential law was assumed (for
c
in
mm and
K,
in MPadm):
Since there is not much quantitative knowledge
about the influence of crack length and indentation load
122
on
x,
the factor
x
was kept constant during the model
calculations.
The change
of
crack shape and geometry factor
Y
during stable crack growth
NEWMAN
&
RAJU
[21] proposed an empirical equa-
tion to calculate the geometry factor
Y
for semi-elliptical
cracks in bars with rectangular cross section subjected
to flexural loading. They defined
Y
as
Y(alc,
alh,
clb,
cp).
where
c
is the crack-half-length at the surface,
a
the
crack depth,
h
the bar's height,
b
the bar's half-width and
cp
the angular coordinate, see fig. 1.
For
many usual
crack geometries (i.e.
eo
=
a0
l
co
close to
eo
=
1,
crack
length small compared to the specimen dimensions) the
value
Ys
=
Y(cp
=
0)
at the specimen's surface is larger
that
Yo
=
Y(cp
=
7d2) at the deepest point of the crack.
Therefore such cracks tend to grow more along the
surface of the specimen than depthwise. Their aspect
ratio
e
changes during stable extension.
An indirect evidence of such an decrease can for ex-
ample be found in a paper by
BRAm
ET.
AL.
[4]. Typi-
cal as-indented cracks with
eo
P
1
have
Y(co)
=
1,27
calculated with the NEWMAN
&
RAJU
formula. The
value
Y(c,)
=
0,77 they determined for critical cracks
via a calibration procedure can only be obtained for
cracks with a lower aspect ratio.
Direct experimental evidence of the decrease of the
aspect ratio of indentation cracks has recently been
given by SGLAVO [9].
2c
1
El.
J
D
II
Fig.
I:
Schematic of a semi-elliptical indentation
crack in a flexural specimen.
The change of the crack shape can be determined
experimentally but the experimental expense of such a
procedure calls for a method to estimate the crack shape
with the help of calculations Some possibilities to cal-
culate the evolution of the crack geometry with the help
of this formula have been proposed [12, 13, 21, 221 but
they do not take into account the starting geometry of
the cracks.
It is reasonable to assume that the stress intensity is
constant along the crack front during stable crack exten-
sion.
For
the investigated indentation cracks this can be
expressed as:
K,,(cp
=O)=K,,(cp
=x/2)=Kc
.
(9)
For a first approximation, and to avoid too many
as-
sumptions in the calculation the residual stress intensity
will be neglected here,
so
that eq.
(9)
reduces to:
aacx
For
a given specimen cross section (i.e. fixed
h
and
b)
eq. (10) can be solved for the crack depth
a
numeri-
cally, whereby the crack half-length
c
is varied within
the validity range of the NEWMAN
&
RAJU
equation:
cdb
I
clb
5
0,5,
as
already proposed [22]. Here the as-
indented shape of the crack is additionally taken into
consideration by allowing changes in the crack depth
a
only if they lead to a crack extension,
a
2
ao.
The only
experimental data that have to be known for this calcu-
lation are crack length
co
and crack depth
a0
of the as-
indented crack.
An example of the change of the crack aspect ratio
e
with crack length is plotted in fig. 2 for starting cracks
with different sizes but the same aspect ratio
eo
=
1.
co
=
50
pm
co
=
200
pm
h=2mm
h=3mm
k
075
l,o
1,5
2.0
2,5
38
c
I
C"
Fig.
2:
Typical change of the crack aspect ratio
for crack with
co
=
50
pm and
co
=
200 pm and
eo
=
1
in
a
flexural specimen with
b
=
2
mm
and
h=3mm.
RESULTS AND DISCUSSION
Evolution
of
crack shape and geometry factor
Y
grown for some amount is shown in fig. 3.
An example of a decorated crack that has stably
Fig.
3:
SEM
micrograph of a multiple-decorated
crack
(P
=
294
N)
with a length
co
<
c
<
cnl.
The
contour of the as-indented crack is indicated by a
dashed line
The hardness impression can be made out at the ten-
sile surface of the specimen. The dark zone below the
impression is not decorated because of the high com-
pressive stresses within this zone. The geometry of the
crack immediately after the indentation
is
indicated by a
dashed line (the geometry of as-indented cracks
is
de-
termined separately [23], the contour indicated here
123
12
"
'
'
'
"
' '
'
"
''__
'.
' '
'.
49~
1,1
-
0
98N
.u
0
A
196N
.
V
294N
.
0
1,o
-
.-
Po
'
0,9
A
0-
mO
1
0,8
0
v.r..o
Y
AAv
0
g
0,7
-
c
I
c,
Fig.
4:
Evolution of the crack aspect ratio during
stable crack extension for cracks introduced with
different loads.
190
-
-.
-_
.
--
--
-_
-_
--_
--
r
The lines correspond to the calculations, the symbols
represent values for
Ys
calculated with the experimental
data from fig.
4.
The correspondence is reasonable, the
deviations are less than
6%
in all cases. The same cal-
culation was repeated for alumina
data
from literature
[9]
with an similar good correlation.
The bi-linear trend of the curve in figs. 2 and
5
is the
main difference that discerns the present calculation of
the crack geometry change from previous ones [12,
21,
221. It can be explained by the influence
of
the
as-
indented geometry of the cracks. According to eq.
(10)
an equilibrium crack aspect ratio exists for every crack
length
c.
During the first segment of the bi-linear trend,
this configuration is not yet reached, the crack depth
u
is
too deep to fulfill the condition imposed by eq.
(10).
As
soon
as
the equilibrium aspect ratio is reached, the crack
grows strictly according to eq. (10). A calculation that
does not take into account the as-indented geometry
results in the right-hand segment of the bi-linear curve
only and will therefore only predict smaller changes in
e
(or
Y)
as
observed.
Because of the excellent results achieved with this
simplified approach (which neglects the influence of the
residual stresses on crack growth) there seems to be no
need to improve the calculation by proceeding accord-
ing to eq.
(9).
Since
K,,
is
rapidly decreasing with crack
length and also with increasing
Kappl
[15]
it will only
influence the begin of the stable crack growth. The
influence of
K,
tends to favor larger crack aspect ratios.
The characteristic bend in the curve tends to disappear
but the aspect ratio of critical cracks will not change
much.
Another change
in
the presented trends for the evo-
lution of the crack shape during stable crack growth
may arise if the possible R-curve effect is taken into
consideration. The rising fracture toughness will have
more influence on portions of the crack that have al-
ready grown to a further extent. R-curve effects tend to
operate against the influence
of
the gradient
in
the
bending stress and reduce the tendency towards a de-
creasing crack aspect ratio.
Calculation
of
indentation strength data
for
model
cases
The indentation strength
qs
and the critical crack
length
c,
are affected by the crack-length dependences
of
Y
and
K,.
Fig. 6 shows the relative critical crack
length as a function of starting crack length (expressed
as
indentation load
P)
for the three investigated cases.
-c-
Y
=
Y(c),
K,
=
const.
49 98 196
294
indentation
load
P
Fig.
6:
Influence of a decreasing geometry factor
and a R-curve on the relative length
of
critical
cracks.
It is interesting to note that for all but the simplified
case cracks reach more than 2,52c0 during the stable
extension. Pronounced stable crack growth is favored by
the R-curve but also by the decrease of
Y.
Since the
cracks for the case iii) are longer than for the other case,
the indentation strength in lower. Critical cracks with
124
c,
>
2,52
co
have been reported by several authors
[4,
13, 161.
Fig.
7
shows the indentation strength plot for all
three calculated cases. As expected the data of the sim-
plified case can be found on a straight line with the
slope
ns
=
1/3. Data for the case ii) show a slope
ny=
0,26.
The data for case iii) can be fitted with
nRy
=
0,l
1.
_r---
Y
=
const.,
K,
=
consf.
-
‘
I
I
q..
a;”
400
t
~=const.,~~=const.
0
A
Y
=
Y(c),
K,
=
const.
Y
=
Y(c)
&
Rcurve
.-
I’
J
20
40
60
80
100
200
400
indentationload
P
Fig.
7:
Indentation strength plot for the three
in-
vestigated cases.
According to the simplified theory a slope less than
1/3
in
the indentation strength plot and critical cracks
longer than
2,52
co
are only expected if the investigated
material shows R-curve behavior. The model calcula-
tions show that at constant fracture toughness a typical
decrease of
Y
alone (case ii)) may lead to IS-data that
are typical
for
an R-curve material according to the
simplified theory
-
i.e. a slope
<
1/3. If the evaluation of
such data is carried out using the principles of the sim-
plified theory an R-curve will be deduced even though
none exists. It is also confirmed, that if the tested mate-
rial shows R-curve behavior, neglecting the decrease of
Y
leads to an overestimation of a possible R-curve effect
deduced from IS-experiments.
Examples that confirm these conclusions can be
found in literature. For a fine
(4
pm) grained SSiC, the
control material used by PADTURE
&
LAWN
[24],
an
alumina [9],
or
on the sapphire presented by
COOK
ET
AL.
[lo]
the IS-plots exhibit slopes less than
n
=
113
even though the materials are believed to have no R-
curve. On the other hand there are also examples that
perfectly match the simplified approach with a
Y
inde-
pendent of crack length, as for instance the glass data in
[S].
This is not necessarily a contradiction to the results
of the model calculations. How much the crack aspect
ratio and the geometry factor change during the stable
extension depends strongly on the as-indented geometry
and the specimen cross section. Cracks that are close to
the equilibrium shape in the as-indented state do not
change their shape a lot. Together with the influence of
the residual stresses this may lead to a geometry factor
that is in fact constant during IS-experiments.
If the residual stress intensity K,, is smaller than the
indentation stress intensity at
full
load,
i.e.
if the left-
hand equity
in
eq.
(1)
is not fulfilled, then cracks will
not grow instantly once the external load is applied [16].
Consequently such cracks will not reach the expected
critical crack length and will not experience the pre-
dicted change in shape.
For the practical evaluation of indentation strength
tests it seems to be indispensable to account for the
change of
Y
with crack length. Since the actual decrease
of
Y
is material and geometry-dependent it should at
least be checked if
Y
does indeed change. The meas-
urement of the indentation strength as a hnction of
indentation load alone will not suffice to decide if this is
the case. A simple means to do
so
is offered by the
representation of crack growth data proposed by
DRANSMANN
ET
AL.
[6]. Rearrangement of eq.
(7)
under
the condition
K,,
=
Kc(c)
yields
P
/
c3’*
[
M
Padm]
Fig.
8:
Model crack extension plot after [6] for
case i), dashed line, and case ii), trough line. the
error bars correspond to a measurement uncer-
tainty off
yo5
for crack lengths.
Included
in
the diagram are error bars which refer to a
f
5%
error
in
the measurement of crack lengths. The
scatter between test runs with different specimens may
even be exceed this value. It becomes clear, that the
measurement error ‘and data scatter prevent one from
spot an increasing slope from such diagrams.
As
the most efficient way to get information about a
possible change of the crack shape and the geometry
factor
Y,
direct investigations of as-indented and
(nearly) critical cracks have to be recommended.
CONCLUSIONS
AND
SUMMARY
The stable growth of indentation cracks
in
silicon
nitride has been monitored using a multiple decoration
technique.
A
decrease of the crack aspect ratio occurs
for the
four
investigated indentation loads.
It was possible to calculated the decrease of the
crack aspect ratio assuming that the stress intensity
125
factor along the crack contour is constant. The
as-
indented crack shape was taken into consideration. The
correspondence of the calculated decrease with the
experimental one is good.
Model calculations
of
the stable crack extension
of
indentation cracks were performed to investigate the
influence of the observed change of the crack shape and
the consequent decrease of the geometry factor Yon the
indentation strength.
The results show that such a decrease has an addi-
tionally stabilizing effect on the cracks during the stable
extension. As a consequence the commonly accepted
relation between indentation strength and indentation
load,
01s
a
P"
with
n
=
113
is not longer fulfilled for
materials with a single-valued toughness and an expo-
nent
n
<
113
is observed. An evaluation of such data
according to the standard theory will lead to the errone-
ous conclusion that the material exhibits R-curve be-
havior. For materials that exhibit a true rising fracture
resistance, the decrease of
Y
will lead to an overestima-
tion
of
the R-curve effect.
The complex mutual influences of a possible vari-
able residual stress parameter, the tendency of indenta-
tion cracks to change their geometry during flexural
loading and a possible R-curve on the evolution of the
geometry factor
Y
during stable crack extension depend
on the tested material
as
well
as
on the test-geometry. It
is recommended to investigate the evolution of the crack
shape if fracture properties are to be deduced from sta-
ble crack growth experiments on indentation cracks.
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126
PREDICTION
OF
THERMAL SHOCK RESISTANCE
OF
COMPONENTS
USING THE INDENTATION-QUENCH TEST
M.
Collin* and
D.
Rowcliffe
Materials Science and Engineering, Royal Institute
of
Technology,
100
44
Stockholm, Sweden
ABSTRACT
Two applications of the indentation-quench test have
been studied. The thermal shock resistance is evaluated
for three ceramic materials (alumina, silicon carbide
whisker reinforced alumina and silicon nitride) and the
ranking is in agreement with results in the literature.
The results show that the sensitivity of the indentation-
quench test
is
higher compared to other methods. The
indentation-quench method is then applied to
a
simpli-
fied component (tapered plate). It is demonstrated how
the test can be used for making
a
rough estimate of the
pattern of thermal stresses generated on the surface at
quenching.
INTRODUCTION
Ceramic materials have generally good high tempera-
ture properties and are prime candidates for many
applications involving thermal excursions. Examples
are as varied as engine components, electronic devices
and cutting tools. In many of these applications the
ceramic material will meet rapid temperature changes,
which will cause transient thermal stresses and risks
for thermal shock damage. For simple geometries, the
transient stresses at the surface will be tensile on rapid
cooling and compressive on rapid heating. Normally
rapid cooling is the most dangerous temperature
change for ceramics, because the tensile strength is
lower than the compressive strength. The components
will fracture if the tensile stress exceeds the strength of
the material and insufficient thermal shock resistance
often limits the application area of ceramics. It is thus
necessary
to
consider the thermal shock resistance and
there are needs both for evaluation of the thermal
shock resistance of materials themselves and for
evaluation of the risks for thermal shock damage in
components.
A simple approach to estimating the thermal shock
resistance of ceramic materials is to use the traditional
thermal shock resistance parameters
[
1,2].
These pa-
rameters permit ranking of materials for general
working conditions and are useful for the first selection
among materials. However the parameters treat heat
transfer and material properties in
a
very simplified
way and the parameters are of limited value for aniso-
tropic materials and fiber composites. For more exten-
sive investigations, practical measurements are needed.
A common approach is the quenching-strength method
[3,4].
Heated samples are quenched into a coolant,
generally water, and the remaining strength after
quenching is evaluated by bending. The remaining
strength
is
plotted
as
function
of
the temperature diffe-
rence,
AT,
over which the sample
is
quenched, and the
critical temperature difference,
AT,,
is defined as the
point at which the material shows a drastic drop
in
remaining strength. The drop in strength is caused by a
limited growth of at least one of the most harmful
defects. The harmfulness of a defect depends on both
size and location and both of these are statistically
distributed. A consequence of this is that a great num-
ber of specimens are needed for good statistics in the
quenching-strength test. The specimens have to be
manufactured with standardized shape and size. Alter-
natively, it has been suggested to use NDE techniques
for evaluation of thermal shock damage after quench-
ing
[5].
Examples of such techniques are measuring
Young’s modulus and internal friction by sonic-
resonance
[6].
NDE evaluation is particularly advanta-
geous for studying cumulative effects of cyclic thermal
shock. However, most NDE techniques demand a
rather large number of specimens in order to obtain
good statistics. Statistical effects can be reduced by
introducing precracks with known size and location
and an indentation-quench test to evaluate the thermal
shock resistance of brittle materials
is
currently being
explored in detail
[7-lo].
This indentation-quench test
treats localized cracks with known crack geometry,
which makes modeling possible. As the artificial
cracks are made in the center of the sample surface,
edge effects are avoided. Another advantage is that the
method in most cases can be applied directly to
components.
127
Table 1. Material properties
Alumina" (130°C) Reinforced Aluminab (200°C) Silicon Nitride (400°C)
Elastic Modulus (GPa)
E
410 430 300d
Poisson's ratio
V
0.23
0.22 0.24d
Thermal Expansion
(K'
)
a
5.4
x
10"
5.2
x
10"
3.1
x
Thermal Conductivity (W/(m
K))
k
24" 24'
-
'Ref.
[ll].
bCalculated from
the
components. 'Ref.
[lo].
dRef.
[12].
'Ref.
[13].
For
the component engineer it is important to
evaluate if there is any risk for thermal shock damage
in the component and what part of the component that
is most at risk. The first step is usually
a
finite element
calculation of transient thermal stresses. Commonly the
next step is practical measurements using thermal
shock or thermal cycling of components under severe
conditions and evaluation by inspection for visible
cracks. In
a
similar way
as
for the quenching-strength
type of tests, many components are needed for good
statistics. By using the indentation-quench test it
should be possible to obtain good statistics from one
single component. Originally the indentation-quench
test was designed and evaluated for cutting tool inserts
but the results suggest that the test could be used for
more types of components.
In this paper the indentation-quench test is used to
compare the thermal shock resistance of three ceramic
materials. Further, the potential of usage of the method
as
a
diagnostic tool for components is illustrated by
measurements and calculations on
a
tapered plate.
test. The definitions of the crack length,
c,
and the
crack depth,
a,
have been given elsewhere
[lo].
The
quenches were made by heating the specimens in a
furnace with air atmosphere and then quenching them
by free fall into water at
30°C.
The furnace tempera-
ture was selected to give the desired temperature
difference,
AT.
The water bath did not show any meas-
urable change in temperature during quenching. In all
cases the specimens were kept in the furnace for
20 minutes to assure temperature uniformity before
quenching.
SIMPLIFIED COMPONENT
An airfoil-like tapered plate was manufactured. The
material was the same grade of silicon nitride as speci-
fied above. In all, 12 indents arranged in 4 rows were
made in the plate. The peak load and the mean crack
size after indentation were 200
N
and 200 pm. Figure
1
shows the arrangement of the indents and the dimen-
sions of the plate.
EXPERIMENTAL PROCEDURE
Row
COMPARISON OF MATERIALS
Three different materials were investigated:
(1) high-purity densely sintered fine-grained alumina
with an average grain size
<
5
pm (Procera Sandvik),
(2) alumina reinforced with 30 vol.% of silicon carbide
whisker (Sandvik Coromant), and
(3)
silicon nitride,
grade CC690, (Sandvik Coromant). The samples were
in the form of plates
(1)
13
mm diameter
x
4
mm,
(2) 13 mm square
x
4
mm and
(3)
25 mm square
x
8
mm. Material properties are listed in Table
1.
The measurements were performed using the pro-
cedure previously established for the indentation-
quench test
[7,
101. Precracks were made through in-
dentation with a Vickers diamond for 20
s.
The peak
load and the mean crack length after indentation for
each material were (1) 35
N
and 114 pm,
(2)
60
N
and
102 pm, and (3)
70
N
and 105 pm respectively. Appro-
ximately
16
indentation cracks were measured in each
mm
Fig.
1.
Diagram
of
the tapered plate showing the arrangement
of
the
indents.
128
RESULTS AND DISCUSSION
COMPARISON
OF
MATERIALS
Figure
2
shows the crack growth as
a
function of
the temperature difference,
AT,
for thermal quenches of
precracked samples of the three materials. The mean
percentage crack length increase, with respect to the
as-indented crack length, has been calculated from the
growth of the individual cracks along the surface. The
error bars show 95% confidence level.
m
ATu
=
160°C
+
I
al
300
60
80
100 120 140 160
Temperature
Difference,
"C
100
150 200 250
300
Temperature Difference,
"C
300
0 100
200
300
400
500
600
Temperature Difference,
"C
Fig.2. Crack growth in indented alumina (a), reinforced alu-
mina
(b)
and silicon nitride (c) quenched over a range of
temperature differences. The
bars
show
95%
confidence level
of the mean value.
Regime
A.
At very low
AT
no
significant crack growth
can be detected.
Regime
B.
In
a
medium
AT
interval the crack growth is
stable. The variation in percentage crack growth
between the individual cracks is rather small as shown
by the mean values at
95%
confidence level. One
reason for the scatter between the individual crack
sizes is the variation in the microstructure at the tip of
each crack. It has been observed that the cracks grow
in a stepwise manner at the microstructural scale [lo].
Regime
C.
At
a
certain
AT
some of the cracks grow
unstably out to the sample edge or stop at another in-
dent, while the other cracks still grow stably or do not
grow at all. Within the tested interval of temperature
difference, Regime
C
is reached for alumina and rein-
forced alumina but not for silicon nitride.
The regimes have been modeled and explained
elsewhere
[lo].
The major reason for the absence of
crack growth in Regime A is the very low surface heat
transfer coefficient when the temperature
on
the sur-
face
of
the sample is below the boiling temperature of
water [14]. The regime with stable crack growth occurs
due to the combination of residual stress from the in-
dent and thermal stress from the quench. The stable
crack growth is mainly advantageous, because it makes
it possible to define specific values of
AT
for the
evaluation of the test, such as the lowest temperature
difference to cause
a
stable crack length increase of
lo%,
ATlo.
Another suggested way to evaluate the test
is the temperature difference when Regime
C
starts,
ATu.
Values for
ATlo
and
ATu
are listed in Table
2.
A
high value of
ATlo
or
ATu
indicates a high thermal
shock resistance. The ranking of the materials accord-
ing to thermal shock resistance is: Silicon nitride
>
Reinforced alumina
>
Alumina. This result is in accor-
dance with literature data using other techniques
[5,6,15] and shows the potential of the indentation-
quench test for ranking materials. It is also important to
notice that thanks to the residual stress, the sensitivity
of the indentation-quench test is higher than that of
other methods. This can be illustrated by comparing
our results for alumina
(AT10
=
120°C
and
ATu
=
160°C) with results reported in the literature
from investigations using other techniques for evalua-
tion of thermal shock damage after quenching alumina
samples.
In
these investigations higher temperature
differences are needed
to
get significant response. The
critical temperature difference
(ATc)
was determined to
be
200°C
in
a
quenching-strength investigation (4mm
thick samples,
22°C
water)
[15].
In another investiga-
tion changes in retained strength, elastic modulus
and internal friction were reported to occur for
AT
=
250-300°C (2
mm thick samples, ice water)
U61.
The pattern
of
crack growth is similar for
all
mate-
rials and can be divided into three regimes:
129
Table
2.
Fracture toughness and thermal shock resistance
Alumina Reinforced Alumina Silicon Nitride
Toughnessc=lOOpm (MPadm)
KIc.100
2.9
a
Toughnessc=ZSOpm (MPadm)
KIC.ZS0
3.9
a
6.4
a
7.6a
6.9
-
Thermal Shock Resistance
(“C)
ATIO
120 180
260
160 280
-
Thermal Shock Resistance
(“C)
ATU
a
Ref.
[lo].
An expression for the prediction of
ATu
has been
derived elsewhere
[lo]:
(1-v)
1
1
AT,
=
A[K/c(~u)]---
Ea
f’p
&
Here
KI,
is the fracture toughness,
cu
and
au
are the
crack length and the crack depth at the onset of
unstable crack growth
(250
pm and
175
pm respec-
tively when the precracks are
100
pm
[17]), v
is Pois-
son’s ratio,
E
is the elastic modulus,
a
is the thermal
expansion coefficient,
p
is the Biot number,
h
is the
surface heat transfer coefficient,
r
is half the thickness
of the sample,
k
is
the thermal conductivity,
F
is
a
geometric function
(F
=
0.71
for semielliptical surface
cracks with
a/c
=
0.6-1.0 [18])
and finally
a,,
b,
,
c,
and
d,
are constants with the values
3.15, 1.33, 0.266
and
51.4
for an infinite plate
[lo].
The parameter
f
2
has
a
value between
0
and
1
and shows the part of the
total temperature difference that determines the maxi-
mum stress at the surface. In this investigation we have
used Equation
1
for estimations off
>
,
B
and
h
and the
results are summarized in Table
3.
Table
3.
Predicted values
off’B,
P
and
h
Alumina Reinforced Alumina
f‘s
0.40
0.44
P
2.7 3.4
Wk (m-’)
1350
1700
h
(W/(mZK))
32
000
41
000
The estimated values for the surface heat transfer
coefficients are
32
000
and
41
000
W/(m2K) for alu-
mina and reinforced alumina respectively. There is an
encouraging agreement between the estimated values
and values found in the literature. By using
a
combina-
tion of fast response thin film thermocouples and cal-
culations according to the lumped capacitance method
the surface heat transfer coefficient for alumina has
been determined to be
10
00040
000
W/(m2K) in the
temperature range
10O-25O0C
[
191.
This illustrates
how Equation
1
can be
a
valuable tool to roughly
estimate the surface heat transfer coefficient at
AT”.
If
we instead knew the Biot number and all material
properties we could use Equation
1
for prediction of
ATu.
Below,
f;
for silicon nitride will be estimated to
be
0.27.
By using this value together with Equation
1,
ATu
for silicon nitride can be predicted to be
990OC.
Calculated Value
off
>
B
200
400
600
800
Temperature Difference,’C
Fig.3. Values of
f
2
calculated from stable crack growth
in
non-tapered and tapered silicon nitride plates using
Eq.
2.
Figure
3
shows
f
>
calculated for silicon nitride
using the stable crack growth in Regime B. These cal-
culations
are
based on
an
assumption that the sum
of
residual and thermal stress intensity is equal to the
fracture toughness of the material after each quench:
Here
KIc,
F,
E,
a,
v,
f
have the same meaning as
in Equation
1.
The constant
x
is called the residual
stress factor and is a material dependent constant. In
this paper we have used
a
value of
0.106
for silicon
130
nitride in accordance with values used for silicon
nitride in the literature
[20,21].
P
is the indentation
load,
c
is the crack length at the surface, AT is the
temperature difference and
a
is the crack depth. For
measurement of the crack depth, fracturing of the
specimen is required. This has been done for alumina
and reinforced alumina and it has been suggested that
the crack depth can be estimated from the crack length
according to the following expression
[17]:
Here
co
is the initial crack length and
c
is the actual
crack length. It is assumed that the crack depth can be
estimated in the same way for silicon nitride.
Measurements with crack growth below
10%
have
been excluded because the figures of crack growth are
uncertain. In the AT-interval
300450°C
the mean
value off
>
is
0.27
for the non-tapered sample, which
corresponds to a pvalue of
1.4
and
a
value of
h/k
of
350
m-I. This value is much lower than the corre-
sponding values for alumina and reinforced alumina
(1350
and
1700
m-' respectively). Thus the value of
h
is lower and/or the value of
k
is higher for silicon
nitride. According to the heat transfer literature a lower
value of
h
could be reasonable. The heat transfer situa-
tion when
a
plate is quenched is very complicated and
there is no easy way to predict the surface heat transfer
coefficient,
h.
The heat transfer mechanism changes
while the excess temperature above the boiling point
(the temperature difference between the surface and the
boiling temperature of the liquid) increases. Initially
the heat transfer rate will increase, but after the critical
excess temperature is reached, the heat transfer rate
will decrease
[14].
The conditions when quenching
alumina and reinforced alumina are close to the critical
excess temperature and
a
high value of
h
can be
presumed, while the quenching of silicon nitride is
above the critical excess temperature and the value of
h
can be assumed to be lower.
SIMPLIFIED COMPONENT
Figure
4
shows the results from quenching the
tapered plate with
AT=
100,
200,
....,
600°C.
As
predicted, the crack growth increases with increasing
severity of the quenches and the crack growth is
greatest in the thickest part of the plate. These results
illustrate how the indentation-quench test can be used
for components.
When quenching with
AT=
500°C
an edge crack
started to grow close to Row
4.
As
the crack inter-
sected the indent cracks of Row
4,
further measure-
ments on this row would not have been significant. It is
interesting to note that the edge crack did not grow at
the relatively mild quenches with AT=
100-400
"C.
Thus, if there is
a
risk for edge cracks the investigation
should be planned with mild quenches.
%
Crack Length Increase
1
2
3
4
(Rownumber)
3
4.5
6
7.5
(Thickness,
mm)
Fig.
4.
Mean crack growth
in
indented tapered silicon nitride
plate quenched over a range of temperature differences.
Rows
1-4
are described in Fig.
1.
Values off
>
have been calculated for Row 1-3
(Equation
2)
and the results are included in Figure
3.
The mean values off
>
in the temperature interval
300400°C
are
0.15,
0.20
and
0.35
respectively. The
value at Row
3
is surprisingly high compared to the
value obtained for the non-tapered plate. We have not
found the explanation for this yet.
The crack growth pattern
is
effectively linked to the
thermal stress pattern and could therefore be used for a
rough estimation of thermal stresses. The maximum
transient thermal stress at the surface of a plate,
CT~~,,,~~,
can be calculated using the following
expression
[
101:
(4)
Here
E,
a,
v
and
f
>
have the same meaning as in
Equation
1.
Equation
4
has been derived for an infinite
plate, but it can also approximately be utilized for a
tapered plate. Using Equation
4
and values off
from
Figure
3,
the maximum thermal stresses at the surface
after quenching with AT
=
400°C
have been calculated
as
73,
98 and 169 MPa for Row
1,
Row
2
and Row
3
respectively. This illustrates how the indentation-
quench test can be used to obtain information about the
thermal stress pattern and it is also possible to compare
these results with FEM calculations. The test was
originally developed for cutting tools, which have a
very suitable shape for polishing and indentation. We
131