Acta metall, mater. Vol. 41, No. 2, pp. 361-368, 1993
Printed in Great Britain
0956-7151/93 $6.00+ 0.00
PergamonPress Ltd
EXPERIMENTAL R-CURVE BEHAVIOR IN PARTIALLY
STABILIZED ZIRCONIA USING MOIRI~
INTERFEROMETRY
K. E. PERRY l, G. B. M A Y 2, J. S. EPSTEIN 2, H. O K A D A 3 and S. N. ATLURI 3
1Department of Process and Mechanical Engineering, University of Strathclyde, Strathclyde G1 1XJ,
Scotland, 2Fracture Behavior Section, Idaho National Engineering Laboratory, P.O. Box 1625,
Idaho Falls, ID 83415-2218 and 3Computational Modelling and Infrastructure Rehabilitation
Center, Georgia Institute of Technology, Atlanta, GA 30332-0355, U.S.A.
(Received 3 February 1992; in revised form 1 July 1992)
Abstract--Moir6 interferometry is employed to study toughening in partially stabilized zirconia (PSZ).
Energy to fracture as a function of crack growth curves (R-curves) is derived from mode I compliance
calculations and from near tip fitting of the moir6 fringes. The effect of the tetragonal to monoclinic phase
transformation in the zirconia is found by comparing the bulk compliance R-curves to the locally derived
moir6 R-curve. Localized strain field plots are produced from the moir6 data for the PSZ zirconia. The
observed transformation zone height compares favorably with that predicted by Okada et aL [Acta metall.
mater. 40, 1421 (1992)] in a companion paper, as does the qualitative nature of the R-curve with
predictions by Stump and Budiansky [Acta metall. 37, 3297 (1989)].
INTRODUCTION
Transformation toughening in PSZ due to the presence of stationary and propagating cracks is a rich
subject in terms of theoretical investigations. Experimental descriptions of the local crack tip transformation phenomena also exist, although considerably
less than theory. This paper reports an experimental
investigation on the crack tip induced transformation
toughening in PSZ that is used by Okada et al. [1] as
calibration for a computational investigation of the
crack tip transformation stress state.
This paper begins with a brief review of crack tip
induced transformation toughening in PSZ. The
reader is referred to the works cited for a more
detailed explanation. We will then describe the
experimental techniques employed: moir6 interferometry, acoustic emission and a fixed displacement
differential micrometer load frame coupled with a
capacitance transducer load cell. Results will then
be presented in terms of the local crack tip moir6
strain and displacement fields for a propagating
crack in the PSZ. The strain fields results will be
compared to the linear elastic fracture mechanics
(LEFM) predictions. Finally, we will present energy
to fracture as a function of crack increment (commonly referred to as "R-curves"). The R-curves
to be presented are developed with bulk load
compliance techniques and derived from the local
crack tip moir6 data, which is novel. Conclusions will
be drawn relative to the theoretical computational
predictions of the transformation toughening
phenomenon.
Zirconia transformation mechanics in stationary crack
stress fields
Using controlled processing techniques, a specific
distribution of second phase particles having a tetragonal crystal structure (t-ZrO2) can be precipitated
within a matrix of monoclinic zirconia (m-Zr02).
When a macrocrack forms in such a PSZ, it is well
known that the singular stress field triggers transformation of the metastable t-ZrO 2 to the monoclinic
phase. The phase transformation is accompanied by
both a shear deformation and a volume dilatation
(about 4%). When constrained within an elastic
matrix, a compressive residual stress developes at the
crack tip and across the fracture surface. The effective
stress intensity factor is reduced and R-curve behavior is observed.
It is generally assumed that the shear component
of the transformation is nullified by multiple
twinning or internal slipping within each particle.
In the transformed ZrO 2 particle, alternating bands
of shear may be formed that relieve the transformational shear stress. There is microscopic evidence
that this is approximately true around cracks, but
not when the transformation is induced by remote
external loading in the absence of macrocracks [3].
The assumption of negligible deviatoric transformation strains implies that the particle's affect
on the surrounding matrix is merely a compressive
hydrostatic stress, which simplifies both the determination of the critical transformation conditions and the resulting change in stress intensity
factor.
361
362
PERRY et al.: R-CURVE BEHAVIOR IN PSZ
CrackResistance
CrackExtension
Fig. 1. The theoretical R-curve for the PSZ by Stump and
Budiansky [2].
Several experimental investigations of transformation toughening have been presented. Mori et al. [4]
used Raman microprobe techniques to determine the
distribution of transformed tetragonal grains. They
found that there was no discrete transformation zone,
rather, the proportion of transformed grains gradually decreased away from the crack tip. Cox [5]
measured out-of-plane displacements using interference microscopy to verify that the transformation
zone is gradual. Dadkhah [6] performed a coupled
experimental and numerical analysis of transformation toughening in the spirit of the results to be
presented in this work. Residual strains were
measured within transformation zones using both
in-plane moir6 interferometry displacement fields and
an image correlation method for displacements below
the resolution limit of moir6 interferometry. These
results were used with a finite element analysis to
assess surface relaxation effects.
P S Z propagating crack fields: R-curve behavior
Stump and Budiansky [2] presented R-curve results
for a model of transformation toughening based on
the supercritical transformation criterion, i.e. the
existence of a totally transformed zone of material
surrounding the crack tip. The region surrounding an
advancing crack tip was modeled as a zone that has
undergone a uniform and complete dilatational phase
transformation. The zone was allowed to evolve
around the crack tip under conditions of increasing
far field load while the tip was maintained at a critical
stress intensity necessary for fracture. This procedure
revealed that the maximum zone height occurred
after a finite amount of crack growth. Since the
increased toughening is directly dependent on the
height of the transformation zone, the toughness also
exhibits a maximum for a finite amount of crack
growth. Thus, the R-curves generated by Stump and
Budiansky indicate that the toughness is not a
monotonic function of the crack advance but increases to a peak and lowers to a steady state value.
Figure 1 illustrates a typical R-curve generated by
their procedure.
EXPERIMENTAL DESIGN
The material employed was Nielsen's MS grade
PSZ. Approximate material properties are a dilatational strain for phase transformation of 0.5%, a
Young's modulus of 208 MPa, a Poisson's ratio of
0.23, and a transformation stress of 280 MPa. In
addition, one control sample was heat treated to
almost a fully monoclinic state before testing. This
was accomplished by a heat treatment at 1350°C for
4 h; X-ray diffraction analysis confirmed the monoclinic phase transformation.
The experimental design emphasized two critical
aspects: (i) a specimen configuration that would be
primarily two dimensional in the area of the phase
transformation zone and yield non-curved crack
growth; and (ii) controlled loading and diagnostics.
Each aspect will be discussed separately.
Specimen design
The specimen design chosen was a double cantilever beam (DCB); the dimensions are given in Fig. 2.
The overall specimen dimensions were selected by
optimizing the far field asymptotic crack terms to
produce straight crack growth [7]. The double
cantilever beam measures the effect of crack growth
in the presence of a wake zone of a preexisting crack,
unlike the typical bend test which measures only
initiation toughness [8].
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 loading area
on the specimen. A second cut by a vee-sharpened
I0
Therme
~
1200I/ramCrolMmd
DiffractionGrating
~
SpecimenThlckne~= 2.0
AllDimensionsIn mm
Fig. 2. The double cantilever beam geometry for the PSZ.
PERRY et al.: R-CURVE BEHAVIOR IN PSZ
L..
363
4Ocm
~l
Coarse Ad|l
Load S c r n
L.vau
~l.lwlll
Fig. 3. The single piece high compliance load frame.
0.7 mm thick diamond saw extended the opening
further into the sample. F o r the samples, a starter
crack was carefully introduced by a surgically sharp
blade combined with diamond paste, which produced
a fine crack whose notch root radius was on the order
of 10/tm. The samples were polished in successive
stages to a 1 # m finish.
Specimen loading and diagnostics
Rather than the traditional wedge opening technique for high compliance crack loading, direct pin
loading to open the crack mouth was chosen so that
the loads could be measured directly. Wedge loading
produces friction at the wedge/crack mouth contact
region, invalidating the load measurement. A stiff
load frame is essential to the validity of pin loading
data. The single piece load frame designed is shown
in Fig. 3. The crack mouth opening was controlled by
a fixed displacement differential micrometer with a
resolution of 0.2/~ m. The actual load was monitored
by a stiff calibration bar of aluminium in the load
train with a mounted capacitance transducer 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 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 in excess of 5 mm in the I-2 mm thick samples
using this combination of load frame and acoustic
emission.
AM 41/2--D
Moirb interferometry
Moir6 interferometry or diffraction grating interferometry uses a diffraction grating intimately
bonded to the surface under study. The grating
frequencies are variable, typically ranging from 300
to 12001/mm, and crossed gratings are generally used
in order to sample the X and Y displacement fields
during loading. The high frequency and thickness of
the replicated grating (approximately 20/lm) permits
resolution of displacement in small spatial areas,
typically on the order of 50ym. The grating is
illuminated by two temporally coherent laser beams
from the same source. The beams are oriented such
that the respective first order diffraction beams leave
the grating coaxially and normal to the plane of the
grating (Fig. 4). If the grating is undistorted and the
Dif froction orders
.3 2 ~
Y//.I
-1
tX
J" I"
~.Y
Fig. 4. Two beam interferometry with a phase diffraction
grating.
364
PERRY et al.: R-CURVE BEHAVIOR IN PSZ
incident beams are well collimated, the two first order
beams interfere to give a single broad fringe. Any
deviation from perfect alignment or a distortion in
the grating will cause the exit beams to diverge,
producing interference fringes. A review article by
Post [9] has described this technique both theoretically and practically. Livnat and Post [10] have also
shown the correspondence between the mathematical
description of classical (coarse) moir6 and diffraction
moir& That is
U~ = N~/F
-y
(1)
where: F i s the grating frequency, typically in lines per
mm; Niis the fringe order for the X(i = 1) or Y(i = 2)
direction; and Ui is the displacement in the i direction.
For this investigation, 12001/mm aluminized
crossed diffraction gratings were replicated. A threemirror system [9] was employed with a multiplication
of 2 for an in-plane displacement sensitivity of
0.4/~m. Both X and Y fields were recorded on
100 x 125ram Kodak T M A X 400 ASA film at
514nm, 0.25 W illumination with a 1/32s shutter
speed. The nominal magnification while observing
the crack growth was 40 x .
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 [11]. As noted, the DCB specimen
employed used a three-stage slot with a gradual
reduction in width and not a single slot of one
thickness as found in Tada et al. [11]. The boundary
element analysis performed for the actual specimen
yielded a K differing from Tada by 6%, or within the
data scatter. The second method employed was local
to the crack tip. The local method fitted the moir6
interferometry displacements with a postulated
L E F M mixed mode displacement field. The method
was first used by Barker et al. [12]; an initial extension
to mixed mode fields was given by Chona and
Matsumoto [13]. Because the local field is inherently
mixed mode, the method was developed in full
for this work. The L E F M mixed mode expansion is
given as
2400 I/mm
Fig. 5. The Y displacement field for the tetragonal to
monoclinic PSZ specimen.
usually over deterministic, having more data points
than unknown variables.
RESULTS
Figure 5 shows the Y displacement field for the
propagating crack in the PSZ. The curvature or
gradient of the Y displacement field fringes as they
approach the crack face are indicative of the residual
crack face strains in the transformation wake zone.
Figure 6 shows the Y displacement field for the
propagating crack in a heat-treated monoclinic control sample. The control sample displacement pattern
does not show the presence of the rear tip wake zone.
This absence does not imply the complete depletion
of tetragonal particles. The spatial resolution of the
moir6 interferometry is 50 # m and the particle size is
1-2/~m. Thus particles may be present that were not
transformed during the heat treatment and whose
subsequent transformation effect, due to the crack, is
below the spatial resolution of the moir6 data. For
Fig. 5, the transition from the crack face to the
IX
Ui (r, O) = [Kl F l (0) + K n H i (O)]r 1/2
+ ~ [A.F.(O) + B.H.(O)]r "/2 (2)
n=2
where i = 1, 2 for the X ( U ) and Y ( V ) displacements,
respectively.
To extract the KI and K n from (2), the crack tip
moir~ data is input as Ui(r, 0), A least squares fit to
the right hand expansion is performed by adding
terms until the rms error decreases and the values of
Kj and Kn stabilize. In general, this expansion may be
formulated for plane strain or stress and is dependent
on Young's modulus and Poisson's ratio. The fit is
4V"
I/ram
Fig. 6. The Y displacement field for the heat-treated monoclinic phase PSZ sample.
P E R R Y et al.:
R-CURVE
BEHAVIOR IN PSZ
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Fig. 7. (a-c) The radial, 0 and radial-0 strains for Fig. 5. from -180 to 180°. The open squares are PSZ
and solid dots are LEFM predictions.
uniform Y field fringes yields a zone half height of
0.12 mm (for steady state growth). Computations by
Okada et al. [1] for the Y displacement field yield a
gradient zone half height of 0.16 mm. The agreement
between computation and experiment imply a gradient in the transformation from tetragonal to fully
monoclinic particles. More closely, the numerical
results indicate that the crack face gradient region in
the Y field moir6 fringes comprise only 50% of the
total transformation zone. This gradient observation
is consistent with the results of Budiansky et al. [14]
and Mori et al. [4] in that the transformation zone is
a gradual transition from fully transformed particles
to tetragonal phase particles. The magnitude of the
steady state zone size has been observed by others,
particularly Mori et al. [4] and is dependent on the
aging of the PSZ,
It is interesting to note that in the Y displacement
fields (Figs 5 and 6) an asymmetry (shear) of 1 fringe
exists on the crack face. Two hypotheses were considered as explanations of this asymmetry. First, the
loading pins may have been misaligned. The photographs were checked carefully; no misalignment was
discovered. A second possible reason for the misalignment is slight crack deflection; this is apparent in
Figs 5 and 6 and the local moir6 K extraction yielded
a g i l of approximately 3-5% of the K I.
To determine near tip strain fields the X and Y
moir6 displacement fields are splined as surfaces in
terms of fringe order (or displacement), and their X
and Y coordinates. These surfaces are then transformed to a radial coordinate system. Derivatives are
then taken of these surfaces, which give radial-0
strains. Figure 7(a-c) show the radial, 0, and radial-0
strain fields for a 250 # m radius superimposed with
the theoretical L E F M mode I strains for comparison.
It is seen that considerable lowering of the PSZ
strains occurs due to the transformation toughening.
Figure 7(a) shows an interesting bimodal shape for
radial strain. Figure 7(b) shows an opposite trend in
0 strain due to the transformation effect.
Figures 8 and 9 show the plane stress, K~, as a
function of crack length (R-curve) for two different
fracture specimen undergoing the crack tip tetragonal
to monoclinic phase transformation. Figure 10 shows
the R-curve behavior for the heat treated monoclinic
control sample. It should be noted that a K, was
extracted from (2) but is not shown as it falls within
the 3-5% data scatter range. In particular, Fig. 9 also
details the locally extracted Kt from the moir6 data
and equation (2). Of distinct interest is the rising and
falling R-curve behavior, similar to the results of
Stump [2]. The energy values from the near tip moir6
data are lower due to the shielding effect of the
366
PERRY
R-CURVE BEHAVIOR IN PSZ
et al.:
4
K I (MPa (m) 1/2)
3
•
Bulk Compliance
Specimen
,
I
I
i
,
,
1
o
I
,
I
I
[
2
Aa ( m m )
I
I
ZR-2
I
3
4
Fig. 8. The R-curve for the PSZ.
transformation particles. Figure 9 shows a pronounced initial rise for the moir6 data K~. The more
pronounced rise for the local moir6 data, as compared to the bulk compliance derived K~, is to be
expected as the moir6 data was extracted in close
proximity to the tip and hence the transformation
region. For Fig. 8, the initiation fracture toughness is
below the reported fracture toughness of the MS
grade material (10-12 MPa(m) ~/2 [3]), while for Fig. 9
it is in the reported range. As expected the monoclinic
sample in Fig. 10, shows considerably lower fracture
toughness values as the tip propagates in the transformed zirconia matrix material. The magnitude of
the K values in Fig. 10 are higher than reported
transformed matrix values and may be attributed to
tetragonal particles that were not fully transformed to
the monoclinic phase during heat treatment. Finally,
Fig. 11 shows, for comparison, the moir6 derived Rcurves for two transformation samples and the heattreated sample. The consistent rising in K values for
later stages of propagation, past the initial effect, is
unexplained. The later rise may be influenced by the
15
[]
[]
[]
[]
ra
[]
[]
10
K I (MPa (m) 1/2)
I:]
Bulk Compliance
Crack Tip Moire
•
Specimen ,Z,R-3
,
0
,
, I t
0.5
,
,
,
I
1
,
,
,
,
I
1.5
,
,
,
,
I
,
.
2
.
,
I
2.5
.
.
,
.
I
,
3
Aa ( r a m )
Fig. 9. The bulk compliance K vs the locally obtained moir~ K.
,
,
i
I
3.5
i
4
PERRY et al.:
R-CURVE BEHAVIOR IN PSZ
367
7-
•
6-
•
•
•
•
5K I (MPa (m) 1/2)
4321
•
Moire Specimen Zr.1
::::::::::::::::::::::::::::::::::
0
0.5
1
1.5
2
2.5
3
3.5
Aa(mm)
Fig. 10. The local moir6 R-curve for the heat-treated monoclinic specimen.
rear boundary as the crack grows towards the free
end, in turn affecting the zone shape. It is also
conjectured that the rising R-curve is due to a small
cyclic loading effect in the transformation zone
during growth. The loading was one where the crack
stability was controlled by coupling the acoustic
emission transducer with the differential micrometer.
When high acoustic events occurred, indicating the
beginning of a growth instability, a slight unloading
then occurred by the micrometer operator, usually
decreasing the crack mouth opening by 50/~m. This
slight loading and reloading to control crack growth
may induce this nonlinearity.
CONCLUSIONS
Moir6 interferometry has been used to study Rcurve effects in partially stablized zirconia. Detailed
strain plots of the steady-state transformationtoughened PSZ have been produced. The steady-state
11
O
8.25-
O
O
0
O
O•
K 1 (MPa (m) 1/2)
O
[]
5.5 ~
[]
[]
[]
[]
[]
[]
2.75-
[]
O
•
K 1 (MPa (m) 1/2) Monocllnlc
K 1 (MPa (m) 1/2) Zr-2
K 1 (MPa (m) 1/2) Zr-3
Local Moire Data
o
::::::::::::::::::::::::::::::::::::::::
0
o.s
1
1 .s
2
2.s
3
3.5
4
Aa(mm)
Fig. 11. A comparison of the local moir6 R-curves for the heat-treated monoclinic (Zr-1) and tetragonal
to monoclinic transformation specimens (Zr-2, 3).
368
PERRY et al.: R-CURVE BEHAVIOR IN PSZ
zone height is similar to computational predictions by
Okada et al. [1]. The observed R-curve exhibits an
initial rise to a peak consistent with the results of
Stump and Budiansky [2]. The cause of the later
rising portion in the R-curve is open to debate; it may
be due to free edge effects interacting with the tip or
small amounts of loading and unloading. Indications
of the transformation shielding effect have been
obtained from the crack tip moir6 data. The moir6
data was taken within a 200--2000 # m radius centered
at the crack tip. If the inner data zone were decreased,
higher shielding effects would have most likely been
observed.
Acknowledgements--The experimental 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 Field Office, Idaho, Contract No.
DE-AC07-76ID01570. Mr G. Lassahn of the INEL Applied
Analysis Unit is gratefully acknowledged for developing the
mixed mode moir6 extraction algorithm. We wish to thank
Dr D. Stump for the INEL for the helpful comments in
preparation of the manuscript.
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