(1) For a finite height $h0>C,$$C>0,$ we define real functions
$hX≔h0,|X|a,$
(1)
and
$FX,hX≔E˜XX,hX−E˜XaX,hX.$
(2)
It is true that in general the Fourier transform of $FX,hX,$
$F*ζ≔∫0∞FX,hXcosζXdX≠f*ζ,h*ζ,$
(3)
where
$f*ζ,Y≔∫0∞FX,YcosζXdX,$
(4)
$h*ζ≔∫0∞hXcosζXdX.$
(5)
However, when $h0→0,$
$∫0∞FX,0cosζXdX=f*ζ,0.$
(6)
In other words, we expect
$limh0→0 F*ζ→f*ζ,0.$
(7)

During this limiting process, the four corners of the slit will merge and become the two crack tips at $X=±a,0.$ One of the main technical difficulties of fracture mechanics of piezoelectric materials is how to correctly describe this limiting process.

Ref. 1 suggests that in the Fourier transform domain the limiting process may be approximated as
$limh0→0 F*ζ→limh0→0 f*ζ,h*ζ=f*ζ,h0 sinaζζ→f*ζ,0$
(8)
which, the author believed, is plausible in an asymptotic sense.

Moreover, the approximation (8) becomes exact when $FX,Y$ is a linear function with respect variable Y, which is the difference of $E˜XX,Y$ and $E˜XaX,Y.$ To require the same restriction on $E˜XX,Y$ and $E˜XaX,Y$ may be too strong.

(2) From the perspective of classical fracture mechanics, it is also true that for a finite rectangular slit, there is no singularity for the electrical/mechanical fields at $X=±a$ an $Y=0.$ The singularities will appear at the four corners of the rectangular slit, $±a,±h0,$ with a singularity power index different from −1/2.

Nonetheless, it has become a consensus now that the fracture process of a piezo-electric ceramic is in fact a coupled multiscale phenomenon. This can be argued based on both its physical nature and its mathematical structure.

Ref. 1 tried to explore the asymptotic multiscale structure of the problem. Intuitively, the crack-tip field was viewed as the outer problem, and it was assumed that it has the form of the classical solution with respect to the “slow” coordinate variables (therefore there is basically no slit there). On the other hand, the electrostatic problem inside the crack was viewed as an inner problem that is controlled by the slit height, $h0,$ which is the length scale of the problem and it is associated with the “fast” coordinate variable.

The essential idea of this approach is using Eq. (8) to match the outer (macro) solution with the inner (micro) solution. Of course, the asymptotic multiscale analysis could be done differently.

1.
Li
,
S.
,
2003
, “
On Global Energy Release Rate of a Permeable Crack in a Piezoelectric Ceramic
,”
ASME Journal of Applied Mechanics
,
70
, pp.
246
252
.