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\ctrline{\bf SURFACE ACOUSTIC WAVE MEASUREMENTS OF SURFACE CRACKS IN CERAMICS}
\vskip24pt
\ctrline{J.\ J.\ W.\ Tien, B.\ T.\ \hbox{Khuri-Yakub}, G.\ S.\ Kino}
\ctrline{Edward L.\ Ginzton Laboratory}
\ctrline{Stanford University}
\ctrline{Stanford, CA  94305}
\vskip12pt
\ctrline{and}
\vskip12pt
\ctrline{A.\ G.\ Evans, D.\ Marshall}
\ctrline{Department of Materials Science and Mineral Engineering}
\ctrline{University of California}
\ctrline{Berkeley, CA  94720}
\vskip24pt
\ctrline{\bf ABSTRACT}
\vskip6pt

                    
\ctrline{\vbox{\hbox par 14cm{We have extended our earlier investigation
of scattering from surface
cracks.  In particular, we have studied the change in the reflection
coefficient of a Rayleigh wave incident on a half-penny shaped surface
crack along with the corresponding change in the acoustic crack size estimates
as the cracked sample is stressed to fracture.  We have examined in this
manner both cracks in annealed samples and \hbox{as-indented} cracks.  We
have found that the fracture behavior for cracks in these two types of
samples differ quite significantly, with the cracks in the annealed samples
exhibiting a partial crack closure characteristic and the cracks in the
\hbox{as-indented} samples displaying both crack closure and crack growth
effects.}}}
                        
\vskip12pt
\ctrline{\bf INTRODUCTION}
\vskip6pt
     This work is an extension of our earlier study$\null↑1$ aimed at the
establishment of procedures for locating and characterizing surface cracks 
in structural ceramics.  The basic technique we have been using consists
of launching a Rayleigh wave on the surface of the ceramic and observing the
reflections of the acoustic surface wave from the crack.  The particular
type of crack we have been studying is a half-\hbox{penny} shaped surface
crack introduced at a given orientation into the ceramic surface by a
Knoop hardness indentor. 
In a previous paper,$\null↑1$ we described
a scattering theory valid in the low frequency regime based on the model
of an open half-\hbox{penny} shaped crack.  This theory related the reflection
coefficient measured in the experiment with the crack size and fracture stress
of the cracked sample.  The theory was shown there to give predictions for 
the fracture stress which were in good agreement with experimental results.
However, predictions for the crack size were observed to be considerable less 
accurate, with crack size estimates being smaller by factors of two to three
than the actual crack sizes measured after fracture.
  
     Since that time we have extended the theory and correlated our results with 
what would be expected from fracture mechanics studies of ceramics.  A series
of experiments on annealed and unannealed samples has demonstrated the power
of acoustic techniques for studying cracks in ceramics.  The work indicates 
that cracks in annealed samples tend to be in contact over most of their
cross-\hbox{section} and open up with applied stress.  At the point of fracture, 
a very slight growth in crack radius is observed.  On the other hand,
cracks in unannealed as-\hbox{indented} samples exhibit partial closure at
the sample surface.  Upon application of stress, crack growth occurs,
with crack radii tending to increase on the order of 50\%. Upon release
of stress, the cracks partially close, with their effective radii decreasing
by less than 10\%.  As stress is further applied, further crack growth is
observed, terminating in fracture at lower applied
stresses than for equivalent annealed samples.  This hysteresis effect has
been observed for the first time acoustically.  Both the initial crack radius,
$C↓0$, and final radius, $C↓m$, can be measured after
fracture.  The results obtained can be predicted accurately via acoustic
techniques and agree well with the predictions of earlier theories and
experiments by Marshall.$\null↑{2,3,4}$

     An important result of our work is the demonstration that it is highly
desirable to anneal ceramics in which surface cracks are likely to be
present.  Annealing inhibits further crack growth by relieving the
residual stresses that are present.  As evidence, we note that when
unannealed samples are stressed, the crack radius increases by a
factor of approximately two.  Annealing, however, causes an increase
in the fracture stress observed by a factor of approximately $1.5$.

\vskip12pt
\ctrline{\bf THEORETICAL REVIEW}
\vskip6pt
     A general theory of scattering from flaws developed by Kino and 
Auld$\null↑{5,6}$ forms the basis for our work.  The scattering configuration 
considered is shown in \hbox{Figure 1}.  The reflection coefficient,
$S↓{21}$, is defined as the amplitude ratio of the reflected
signal from the flaw, $A↓2$, received by transducer 2, to the
incident signal, $A↓1$, transmitted by transducer 1, at the
terminals of the transducers.   In terms of the input power to the
transmitting transducer,$P$,  the Rayleigh wave displacement field when
the receiving transducer is used as the transmitter, 
$u↑{(2)}↓j$, and the applied stress in the vicinity of the flaw before
the flaw is introduced, $\sigma↑{A(1)}↓{ij}$, $S↓{21}$ is given by
$$S↓{21}={A↓2\over A↓1}={j\omega \over 4P}
\int↓{S↓c}u↑{(2)}↓j\sigma ↑{A(1)}↓{ij}n↓i\, dS\eqno(1)$$
for the case where the flaw is a void.  Here, the integral is taken
over the entire surface of the void, $S↓c$.  
     
For the situation of a Rayleigh wave normally incident on a half-\hbox{penny}
shaped surface crack of radius $a$  located in the \hbox{$x$-$y$} plane
(\hbox{Figures 2 and 3}), the reflection coefficient may be written as
$$S↓{11}={j\omega \over 4P}\int↓S\,\Delta u↓z\sigma↓{zz}↑A\,dS\eqno(2)$$
where $\Delta u↓z$ is the discontinuity in the Rayleigh wave displacement field 
across the crack and $\sigma↓{zz}↑A$ is the applied stress.  The integral
is now taken over just the semi-circular area, $S$.  The theory
we have developed for this configuration is strictly valid only in
the low frequency regime (i.\ e. we require the maximum depth to which
the crack extends below the sample surface to be much less than
an acoustic wavelength).  To take into account the effect of imaging
at the surface in increasing the value of the stress intensity
factor near the surface, as well as the variation with depth of the Rayleigh
wave stress fields, we use the results of Budiansky and O'Connell$\null ↑7$
to write the surface integral in Eq.\ (2) in terms of a contour
integral of the square of the mode $I$ stress intensity factor,
$K↓I$, around the crack circumference $C$ (Figure 3)
$$\int↓S\Delta u↓z\sigma↓{zz}↑A\,dS={2(1-\nu↑2) \over 3E}
\int↓CaK↓I↑2(\theta)\,d\lscr\eqno(3)$$
Here, $\nu$ is Poisson's ratio and $E$ is Young's modulus.
The angular dependence of $K↓I$ caused by the surface
imaging forces may be approximated from the results of Smith,
Emery, and Kobayashi.$\null↑8$  Smith, {\it et al.} considered
the case of a half-\hbox{penny} shaped surface crack in a beam of 
thicknes $2c$ subject to a bending load.  The applied stress then takes the
linear form
$$\sigma↑A↓{zz}(y)=A(1-{y\over c})\eqno(4)$$
where $A$ is a constant and $y$ is the distance from the sample surface
\hbox{(Figure 2)}.  The corresponding stress intensity factor is given by
$$K↓I(\theta)=2\sqrt{a\over \pi}A[\psi(\theta)-({a\over c})\psi(\theta)]
\eqno(5)$$
where the functions $\psi(\theta)$ and $\psi(\theta)$ were numerically
evaluated by Smith, {\it et al.} (Figure 4).  To make
use of these results for our case where the stress is due
to the Rayleigh wave and not a bending load, we made a linear
approximation to the Rayleigh wave stress field and so evaluated
effective values for the constants $A$ and $c$ appearing in Eq.\ (4).
Our result for the normalized reflection coefficient $wk|S↓{11}|\,$,
for a semi-\hbox{circular} crack, as a function of the normalized crack
depth, $2\pi a/\lambda$, is given by the dashed curve
in \hbox{Figure 5}.  Here $k={2\pi / \lambda}$ is the propagation
constant of the Rayleigh wave, and $w$ is the width of the acoustic beam at 
the crack.

\vskip 12pt
\ctrline{\bf EXPERIMENTAL RESULTS}
\vskip 6pt
In our experimental studies, we used a commercial hot-\hbox{pressed}
silicon nitride (NC-132) ceramic.


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