International Journal of Fracture 59: 361-375, 1993.
~ 1993 K luwer Academic Publishers. Printed in the Netherlands.
361
Nonsingular toughening and R-curve behavior
in nominally pure alumina
G.B. MAY, 1 K.E. PERRY, 2 J.L. SHULL, 3 J.S. EPSTEIN, 4 H. OKADA, 5 C. SCOTT 6 and
S.N. ATLURI 7
Department of Mechanical Engineering. University of Washington, Seattle, WA 98195 USA;
ZDepartment of Process and Mechanical Engineering, University of Strathclyde, Strathclyde G1 1X J, Scotland, UK;
3Department of Materials Science, University of Illinois, Urbana-Champaign, IL 61801, USA;
4Fracture Behavior Section, Idaho National Engineering Laboratory, Idaho Falls, P.O. Box 1625.
Idaho 83415-2218 USA;
SVehicle Research Lab., Nissan Motor Co., I, Natsushima-cho, Yokuska, 237, Japan;
6Lighting Products Div., General Electric Corp., Cleveland, Ohio 44112, USA;
7Computational Modelling and Infrastructure Rehabilitation Center, Georgia Institute of Technology, Atlanta,
Georgia 30332-0356 USA.
Received 10 December 1991; accepted in revised form 25 September 1992
Abstract. Moire interferometry is employed to study toughening in medium to large grain size nominally pure alumina.
The fracture scale length, which is characterized by the grain size of the alumina, is systematically varied from 35 to
102 p,m. R curves are derived from bulk mode I compliance calculations for the differing grain sizes and from the near
tip moire fringes. The level of material toughening that arises from the nonsingular processes of crack bridging and
grain boundary friction are found by comparing the bulk and near tip moire R curves.
1. Introduction
R curve testing is well established in the metallic fracture community [1] for plane stress
fracture. For metals, the principal mechanism of energy dissipation is dislocation damage. For
monolithic non-transforming ceramics, the mechanisms of subcritical crack growth or R curve
behavior are known to be sensitive to size scale(s) local to the crack tip [2, 3, 4, 5, 6, 7, 8, 9]. In
particular, grain bridging, microcracking, and crack deflection and accompanying friction
processes operate on a local scale to affect the overall R curve behavior of the ceramic. However,
the above reports infer the local toughening mechanisms from bulk sample measurements with
some qualitative microscopic studies.
We wish to report in detail our recent experimental findings regarding the explicit toughening
mechanisms that account, for in some instances, the bulk rising R curve behavior in alumina
[5, 6, 7, 8, 9]. Pursuant to this goal, we have employed a unique high compliance, low load
capacity, miniature load frame to obtain bulk compliance R curves via direct pin loading of the
alumina samples. During the same experiment we have also employed high sensitivity moire
interferometry local to the growing crack tip in order to obtain directly the energy to fracture
that the crack tip by itself experiences. In order to study the effect of grain size as a controlling
size scale on the overall R curve behavior, we have carefully fabricated heats of nominally pure
alumina with 35, 47, 60, 74, 75 and 102 micron grain sizes. For the differing alumina grain sizes,
we will compare the two measurements, moire displacement derived and bulk compliance
derived energies of fracture, with micrographs of the growing crack, to infer the explicit
mechanisms of toughening in nominally pure medium to large grained alumina.
362
G.B. M a y et al.
2. Materials
The nominally pure A1203 under study is produced by General Electric under the brand
name Lucalox. This material is doped with ~ 500ppm of MgO as a grain stabilizer. Other
impurities from the raw alum-derived powder, such as potassium and calcium, are boiled out
of the material during sintering in a dry hydrogen environment between 1870 and 1890°C. The
chemical analysis of the pre-sintered product is given in Table I using Inductively Coupled
Plasma (ICP) Spectroscopy Analysis. The raw material is extruded and sintered to 12mm
square bar stock of variable length. Single-pass sintering for 3 to 4 h produces a uniform grain
structure with grains in the 30 ~tm size range. Figure 1 is a micrograph of Lucalox from a
Table I. Impurity analysis and concentration for pre-sintered Lucalox via
inductively coupled plasma spectroscopy analysis
Element
ppm*
B
2.0--3.0
ND-4.0
1.0-4.0
70.0
0.6
ND-2.0
30.0
52.0
ND-2.0
1.0
500t
Ca
Fe
K
Li
Mn
Na
Si
V
Zr
MgO
* p p m = parts per million. N D = not
detectable.
t Un-sintered state but is boiled off
after 15 successive firings 200 ppm.
Fig. 1. Thermally etched Lucalox, 200X.
Nonsingular toughening and R-curve behavior
363
single sintering pass that has been thermally etched at 1600°C for 4 h in argon. To obtain grain
sizes ranging from 35 to 102 ~m, samples were given 1, 3, 5, 7, 9, or 15 sintering passes; measured
grain sizes are given in Table II. ICP was not performed on the final sintered product.
Figure 2 is a white light transmission micrograph of the 15-pass Lucalox. The fine traces in
Table II. The spatial distribution of microcrack density
surrounding the grown crack
Specimen
Grain size
(microns)
Microcrack
densities
1 Pass
35.6
0.088
0.092
0.088
Tip
Wake
Far
3 Pass
47.6
0.122
0.119
0.095
Tip
Wake
Far
5 Pass
60.9
0.136
0.148
0.122
Tip
Wake
Far
7 Pass
74.0
0.197
0.199
0.166
Tip
Wake
Far
9 Pass
75.7
0.202
0.190
0.144
Tip
Wake
Far
15 Pass
102.4
0.291
0.322
0.308
Tip
Wake
Far
Fig. 2. White light transmission micrograph of 100 micron grain size Lucalox detailing microcracks, 500X.
364
G.B. May et al.
Fig. 2 show the grain boundary triple point microcracks. The range of processing-induced
microcrack density versus grain size may be obtained using the microcrack density approximation given by [10]
37~
- 16E(k)K(k)M(12),
(1)
where M is the number of crack traces in a 2-D micrograph, (I) is the average microcrack trace
size (in this case the assumed grain facet length), E(k) is the complete elliptic integral of the
second kind, and K(k) is the complete elliptic integral of the first kind. The as processed
microcrack densities are the far field values in Table II. These microcrack densities were
obtained by imposing a microscope focal plane grid divided into 5 x 5 squares at 500X over the
near tip, wake, and far field regions, then counting the number of traces in the grid, and
averaging over 20 readings in each area E11]. The 5 x 5 grid corresponds to an overall region of
244 ~tm x 244 tam directly on the specimen. In theory, according to (1), the microcrack density
will level out at a saturation density of ~ or 0.56.
Using a 5 MHz transducer in pulse echo mode [12], on the 12mm square sintered bar
stock, Young's modulus was determined to be 4 0 G P a and Poisson's ratio 0.23. In theory,
Young's modulus should vary with microcrack density; however, the 5 M H z transducer's wavelength within the Lucalox was greater than the microcrack sizes (1(~100 p.m) and
hence not affected by the microcrack density variation. Hence only bulk properties were
measured.
3. Experimental design
The experimental design emphasized two critical aspects,
(i) a specimen configuration that would be two dimensional in the damage distribution
(minimal thickness effects on crack growth) and
(ii) controlled loading and diagnostics.
Each aspect will be discussed separately.
3.1. Specimen design
The specimen design chosen was a double cantilever beam (DCB) geometry with a width to
thickness ratio approximating generalized plane stress for the specimen overall but not
necessarily for the local crack tip fields. The specimen dimensions are given in Fig. 3.
The overall specimen dimensions were selected by optimizing the far field asymptotic crack
terms to produce straight crack growth [13]. The double cantilever beam geometry measures
the effect of crack growth in the presence of a rear tip microcrack or frictional wake zone
due to the preexisting crack, unlike the typical bend test which measures only initiation
toughness [14].
A crack was introduced into the specimen in three stages. First, a large, 5 mm diameter,
rounded diamond saw cut was made to produce the pin contact region. A second cut by a
Nonsingular toughening and R-curve behavior
-I
"-
C
A
BI
Th°rme,
C eck
365
Specimen Thickness : E
All Dimensions In mm
Crossed
Diffraction Grating
1200 Ymm
Lucalox
A
19.0
R
C
1 0 . 1 7.6
D
12.7
E
2.0
Fig. 3. The double cantilever beam geometry for the Lucalox.
vee-sharpened 0.7 mm thick diamond saw extended the notch further into the sample. Finally, a
thermal crack was introduced using a mini torch and an acoustic emission transducer to
monitor initiation. The thermal crack length was measured using transmissive white light. The
crack typically extended 50 lam from the vee notch. The samples were polished in successive
stages to a 1 lam finish.
3.2. Specimen loading and diagnostics
Rather than the traditional wedge opening technique for high compliance crack loading, the
crack mouth was opened by two contacting inner load pins (0.8 mm diameter) connected to the
load train so that the loads could be measured directly. This method is in contrast to the wedge
loading which produces friction at the wedge/crack mouth contact region, invalidating any load
measurement. The load frame is shown in Fig. 4. The outer frame was machined from a single
piece of aluminum stock. The crack mouth opening was fixeddisplacement and controlled by a
differential micrometer with a resolution of 0.2 p.m. The actual load was monitored by a
calibration bar of aluminum in the load train with a capacitance transducer (Capacitec model
HPB-75 transducer and 4100S amplifier/clock) to monitor gage length separation. The load
transducer sensitivity was 100 g. This form of load transducer does not sacrifice critical overall
frame stiffness as does a typical low load transducer, which consists of a flexible bar with a strain
gage. A 450 kHz resonant transducer was fixed behind the specimen opposite the crack. The
acoustic emission (A/E) activity was monitored on a 500 MHz bandwidth scope, permitting
feedback to the load operator for control of stable crack growth. Straight cracks were grown
stably in excess of 5 mm in the sample (1-2 mm thick) using this combination of frame and
acoustic emission. Finally, 12001/mm crossed diffraction gratings were applied directly to the
specimen surface in the region where crack growth was expected. A three-mirror system [15]
was employed with a multiplication of 2 for an in-plane displacement sensitivity of 0.4 ~tm. Both
X and Y displacement fields were recorded on 100 × 125 mm Kodak TMAX 400 ASA film at
514 nm, 0.25 W illumination with a ~ second shutter speed. Displacement moire interferograms
were taken after the crack growth had been arrested by reducing the load until all A/E activity
ceased.
366
G.B. May et al.
J~.
40 crn
I
Coarse Adjt
Load Screw
Micrometer
Load Screw
Fi,q. 4. The single piece load frame.
4. Data reduction
The energy to fracture as a function of crack growth was obtained by two methods. The first
method used bulk mode I compliance calculations for the DCB specimen [16]. The DCB
specimen employed used a three-stage sl,3: and not a single slot of one thickness as found in
Tada [16]. A boundary element analysis performed for the actual specimen yielded a K differing
from [16] by 6 percent overall.
The second method employed fitted the moire fringes with an assumed L E F M mixed mode
displacement field. The method was first used by Barker [17] for mode I loading. An extension
to mixed mode fields was given by Chona [18]. Because the displacement field is inherently
mixed mode (due to grain boundary deflection), the method was developed in full for this work.
The local L E F M mixed mode eigenfunction expansion is given as
U,(r, O) :- [KIFI(O) + KnH,(O)]r½ + ~ [A,F,(O) + BnHn(O)]r2,
(2)
n=2
where i = 1, 2 for the X(U) and Y(V) displacements respectively.
The crack tip moire field is given as Ui(r, 0). One then simply performs a least squares fit
to the right hand expansion, adding terms until the rms error decreases and the values of K~
and K , stabilize. The fit is over deterministic as more data points are collected than
unknowns. It is noted that our digitizing procedure utilized the entire frontal moire region
which is basically a square starting at the crack tip and extending out to the free specimen
boundaries and hence is influenced by the overall plane stress crack tip solution which was
utilized in (2).
Nonsingular toughening and R-curve behavior
367
5. Results
The Lucalox K~c, measured as a function of the fracture energy versus grain size, is plotted in
Fig. 5. This K~c variation as a function of grain size has also been shown by Evans et al. [14]
and is attributed to the hexagonal close pack structure of the alumina, which yields high
anisotropy and residual stresses at the grain boundaries. Figure 5 shows that the fracture size
scale is controlled by the grain size.
A critical aspect of microcracking theories is that upon unloading residual strains exist [4, 14]
near the traction free boundary of the crack due to volume expansion effects and the inherent
residual stresses at the grain boundaries (Fig. 6). If this residual strain effect exists, there should
be a gradation of microcracking density around the crack as shown in Fig. 7. Table II gives the
spatial variation of microcrack density as a function of the grain size for the Lucalox specimens
tested. Interestingly, the microcrack density does not fully saturate to the value of 9 .
Figure 8 shows the Y displacement field for an elastic traction-free crack with no residual stresses
at the crack faces. The traction free face characteristics manifest themselves in that the Y fringes
++
4
KIC (MPa (rn)112)
0
+
+
3
[
"
20
I
+
•
,,i
40
,
60
,
,
I
I
S0
100
J
i
120
Grain Size, Microns
Fi9. 5. The variation of K with grain size for the Lucalox.
Microcrack Saturated
Stress
/
/ Unloading
/
P
"---] Residual
'Strain
Fig. 6. Hysteresis generated by microcracking,
Fig. 7. Predicted microcracking densification at a crack
tip, The theoretical saturation limit is ~ .
368
G.B. M a y et al.
I I J GratingDirecUon,0.4 micronsper Y fringe
I
Steel
'
I
Y
X
I
50 mm
Fig. 8. Linear elastic Y field displacement fringes.
approach the crack face at 90°. The perpendicular nature of the Y fringes at the crack face
indicate that the derivative of the Y fringes, in the Y direction, yield no strain and hence no Y
stress consistent with the traction free crack face. Figures 9a and b show the displacement fields
for the 35 and 47 ~tm grain size samples. The Y displacement field in this case approaches the
crack face with a strong gradient unlike Fig. 8. This strong shear gradient of the Y displacement
ResidualDisplacement~ '
(a)
ffl
GratingDirection, 0.4 micronsper Y fringe
Y
f
Figs. 9. Y displacement fields for the (a) 35. (b) 47, (c) 74 and (d) 102 lam grain size Lucalox.
Nonsingular toughening and R-curve behavior
369
fringes reveals compressive residual stresses approaching the crack face for the 35 I.tm grains
(Fig. 9a). Figure 9a contains an added frequency mismatch to improve the spatial resolution
[153.
Figures 9c and d show the moire Y field for the 74 and 102 gm grain size Lucalox. In both
cases, no residual wake effect is evident. However, highly local mixed mode effects at the crack
tip are revealed by the asymmetry of the crack tip Y displacement field in that the fringes on
either side of the crack face do not meet. This local mixed mode effect in a bulk mode I test is
due to crack deflection at the grain boundaries, which are inherently weak [14] from the
thermal mis-match and grain anisotropy at this large grain size.
Micrographs of the 35, 47 and 75 gm grain size Lucalox after thermal etching (Figs. 10a, b and
c) show the deflection and bridging. At the 102 gm grain size, the boundaries were so weak that
the grains literally pulled out.
Figure 11 gives the bulk R curve data of the Lucalox samples as a function of grain
size. Figures 12a-e show the moire data R curves versus the bulk values. It should be noted
that total energy (plane stress) is plotted (K~ + K~). Figures 13a and b detail the moire
derived K1 and K2 R curves for the 60 and 102 gm grain size Lucalox. The Kn in Figs. 13a
and b is on the order of 10 percent due to crack deflection at grain boundaries. As expected, the
moire extracted energy value is considerably lower than that of the bulk and more consistent
with the intrinsic fracture toughness without rear tip bridging or friction. The bulk R curve
reflects the cumulative effect of the crack bridging and friction, which increases with crack
length.
Figs. lOa, b and c. Transmission micrographs of the 35, 47 and 74 gm grain Lucalox, 400X.
370
G.B. M a y et al.
Figs. lOa, b and c.--contd.
Nonsingular toughenin9 and R-curve behavior
I0
r,O M
h:&roll
47 Mic#pn
74 Micron
•
*"
371
O
A
M
~
8
O
~, A~, ~
£
O
•
6
~4
0
0
....
*I
•
..:, .¢
A3s Micron
•
eee
102 Micron
i ....
i ....
i ....
i ....
i ....
0
I
2
3
4
An ( r a m )
Fig. 11. R
curve
behavior
of the
Lucalox
as a function
of grain
compliance
size from
measurement.
120
30
0
•
G (Jim 2) Compliance
O (Jim2) Moire
O
•
25
G (Jim 2) ComplhlnC!
O (Jim 2)
Moire
IO0 I
0
0
80
20
O
O
rl
•
E
•A
0
D
D
r~
m
c3
[]
60
•
t~
(3
[] O
L"I
rl
47 mlcton Alumlnl
40
20
3$MlcronAlumlnl
, j , , i
....
i ....
0,5
i ....
I
i ....
1,5
t ....
2
,
,
i
•
2.5
l
. . . .
I
. . . .
!
0.5
I
. . . .
J
1.S
. . . .
I
2
A a (ram)
. . . .
2.5
I
. . . .
3
|
l
i
i
i
3.5
Aa ( r a m )
Fig. 12a. The local and bulk R curves for the 35 ~tm grain
size alumina.
Fig. 12b. The local versus bulk R curves for the 47 lain
grain size alumina.
100
13
•
120
D
•
O (J/m:) ColnplJance
G (J/m2) Moire
G (J/m:1) Moire
G (J/m 3) Compliance
IOO
++
6O
II0
(3
O
o
13 11
4o
o
0 0
60 Mlcro+1 Alllllrllrll
0
4o
O
0
n
20
2o I
0
.
.
.
.
l
o.s
.
.
.
.
l
I
.
.
.
.
I
.
.
.
.
I
l~Sa ( m m )2
.
.
.
.
I
2.5
.
.
.
.
I
3
•
i
i
. . . .
I
:),s
J . . . .
i
t
2
.
.
.
,
I
3
?4 MJC;'OrlAIundnll
. . . .
1 . . . .
4
~a I m m l
Fig. 12c. The local and bulk R curves for the 60 !am grain
size alumina.
Fig. 12d. The local and bulk R curves for the 74 pm grain
size alumina.
G.B. M a y et al.
372
25
rl
O (J/hi l ) Compliance
O (J/m ;/) Moire
20
15
O
•o
D
Q
lO
13
D
102 Micron Alumina
. , , i
. . . .
03
I
I
. . . .
i
. . . .
1 .S
I
. . . .
2
~a ( r a m )
i
. . . .
i
2,5
. . . .
3
n . . , ,
3.5
Fig. 12e. The local and bulk R curves for the 102 lam grain size alumina.
•
•
•
•
KI
K2
4"
4
60 M i c r o n Moire K 1 lind K 2
=f
3.5.
3.5-
3"
3-
2.5-
2.S-
I.S"
I.S,
KI
Ki
:102 MIcron kto~l K I and Klt
o
o
1-
o
o
ILS0
0,S~
o
::::::::::::::::::::::::::::::::::
O.S
1
los
2
2.5
0,:::
3
0.5
3,S
Aa(mm)
Fig. 13a. The moire derived K~ and K, R curve for the
60 lam grain Lucalox.
::::I::::I:::::
1
1.5
.....
2
::::,::::'
2.5
3
I::::
3.5
Aa(mm)
Fig. 13b. The moire derived K~ and KII R curve for the
102 ~tm grain Lucalox.
6. Discussion
6.1. Load frame effects
The micrographs in Fig. 10 showed extensive crack deflection and friction between the deflected
crack path(s). We did not find any evidence of intact crack bridges behind the tip, even for the 47
to 74 lam range where the rising R curve usually is attributed to this effect [8]. We attribute this
lack of intact bridging to two features of our experiment,
(i) very thin samples, 10 to 30 grains thick depending on the grain size, which ensured through
- thickness crack growth, and
(ii) a loading system that was extremely sensitive i.e. using a differential micrometer for
opening the crack combined with an acoustic emission feedback to the load operator to
control the rate of loading or unloading.
Nonsingular toughening and R-curve behavior
373
One could increase the load and hear A/E without seeing any crack growth in the moire
interferograms (under a magnification of 40). Thus, when the crack was observed to grow in the
interferograms, we were assured that the crack front was through the entire specimen thickness.
As the crack advanced, as observed in the moire interferograms, or when the A/E activity
occurred at a higher rate, the load was decreased until the A/E ceased entirely. This was the load
at which K was obtained for the R curve.
6.2. Bulk R curves
Three classes of bulk R curves are shown in Fig. 11. The R curve for the 35 jam grains, the R
curves for the 47-74 jam grains and finally the R curve for the 102 jam grains. Each will be
discussed separately.
For the 35 jam grain size, Fig. 11 shows that the R curve stabilizes and correspondingly a
wake or residual compressive zone is evident in the moire displacement fields (Fig. 9a). The wake
zone or residual stress zone height of Fig. 9a does not change during steady state growth, nor is
there any indication of microcracking change from Table II. This constant zone height in Fig. 9a
corresponds to the stabilized portion of the R curve in Fig. 11. The stable R curve and constant
wake zone height are analogous to a plane stress test for metals where the R curve levels with
the steady state plastic zone height during propagation.
Between 47 and 74 jam grain sizes as shown in Table II indicate that microcracking occurs but
not up to the theoretical density of ~ . It should be noted that in counting microcrack density
some error may be present in distinguishing between a grain boundary triple point microcrack
and spurious grain cracking from rear tip bridging. A continuum damage wake zone is not
dominant in the corresponding moire displacements (Figs. 9b and c). An important added effect
is that crack deflection does become dominant as evidenced in the moire interferograms (Figs.
9b and c) and the micrographs (Figs. 10b and c). This deflection causes non-steady state friction
and local mixed mode shear behavior which is evidenced in the asymmetry of the interferograms
(Figs. 9b and c). The result is that the toughening mechanism is discrete in nature and not steady
state. The R curve for this class of grain sizes in Fig. 11 does not stabilize.
At the 102 jam grain size, microcracking occurs extensively due to the residual stress weakened
grain boundaries, Table II. The 102 jam R curve of Fig. 11 does stabilize but at a lower value
than the R curve for the 35 jam grain size Lucalox. The stabilization occurs because the
extremely weak grain boundaries 1-14] open fully, minimizing contact friction between the
deflected crack faces.
6.3. Moire R curves
Interestingly in Fig. 11 where the bulk R curves stabilize at 35 and 102 jam, the moire obtained
energy to fracture falls within the bulk data (Figs. 12a and e). (It should be noted that if all the
curves were plotted together the overall scatter would be small; however, for clarity these curves
have each been plotted separately.) For 35 p.m (Fig. 12a) this implies that bulk toughening from
ligament bridging and grain friction are not dominant - the near tip intrinsic toughness values
are close to the bulk. For the 102 jam grain size (Fig. 12e), the near tip values again fall within
scatter of the bulk. At 102 jam, residual stresses have weakened the boundaries 1-14] so that they
separate easily.
374
G.B. May et al.
For the intermediate values, 47 to 74 jam (Figs. 12b-d), the moire energy values are essentially
level and on the order of the initiation toughness of the single grain. As the moire data was
digitized ahead of the crack tip, the rear tip mechanisms of grain boundary friction and ligament
bridging which accounts for a large percentage of the energy dissipated during growth are not
accounted for in the local K solution, (2).
7. Conclusions
Moire interferometry has been used to delineate, quantitatively through the comparison of the
moire and bulk R curve data, the amount of toughening in medium to large grain size
alumina due to crack bridging and grain boundary frictional sliding. Microcrack toughening
does not necessarily operate for these grain sizes. This is to be expected; the size of the
singular field that can trigger microcracking is small (100 jam) relative to even the smallest
grain size (35 jam) for a continuum average type approximation to occur. At 35 jam there are
only 3 grains within the singular field instead of the large number required for continuum type
microcracking models.
We have shown, by comparing bulk and local moire energy measurements in nominally
2-D samples, that the toughening effects are not associated with the crack tip singularity field
and can account for a large percentage of bulk toughening in Lucalox. Our micrographs
indicate that frictional sliding takes place along the deflected path. We did not observe
intact ligaments or grains behind the tip. We attribute this to our thin samples, which produce a single through-thickness crack, and the high resolution fixed displacement loading
coupled with an acoustic emission feedback to fully propagate the crack through the sample
thickness.
These nonsingular toughening effects are discrete in nature and hence do not yield a steady
state R curve effect typical of a crack in a metal during its propagation. The result of this
discrete nonsingular toughening is a rising R curve behavior, even after 5mm of growth. Finally,
because toughening is due to the nonsingular rear face we postulate that the R curve should
differ depending on the overall specimen geometry tested.
Future work will continue on the spatial variation of friction or microslip that we have
measured along the crack face using moire interferometry to understand its relationship in the
overall toughening of a quasi brittle material.
Acknowledgments
The experimentation phase of this work was supported by the Solid Mechanics Division, Office
of Naval Research, Dr. R. Barsoum program manager. The data analysis phase of this research
was supported by the U.S. Department of Energy, Office of Energy Research, Office of Basic
Energy Sciences, Dr. O. Manley program manager, under DOE contract number DEAC07-76ID01570. Mr. G. Lassahn of the INEL Applied Analysis Unit and Mr. J. Smith of
Texas A&M University are gratefully acknowledged for developing the mixed mode moire
extraction algorithm. Dr. L. Lott of the INEL is gratefully acknowledged for performing the
ultrasonic property measurements.
Nonsingular toughening and R-curve behavior
375
References
1. 'Recommended Practice for R Curve Determination', ASTM Annual Book of Standards, Part 10, E561-80, American
Society for Testing Materials, Philadelphia, PA (1980).
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