The constants αaj are usually called the deformation
The rest of the surface charge relaxes very slowly
with a relaxation time τ2
potential constants and for a quite wide variety of mate-
rials these constants are about 10 eV.
εsε 0
τ2 ≈
The force acting on charge carriers in the presence
(37)
σs
of a strain gradient is
which is about 0.1 second in pure ice. That means that
F = -grad(U ) = α grad(ε ii ) = α∇ε ii .
r
(42)
a crack moving with 103 m/s velocity can have a 100-
m long electrically charged tail.
This force initiates a flux of the charge carriers, i.e.,
Since the intrinsic electrical field is an oscillating
electrical currents, and as a consequence an electrical
function of the crack length (see Fig. 17), so is the sur-
potential difference appears, as in the case of the bend-
face charge and the resulting electrical dipole moment
ing of ice samples (Evtushenko et al. 1984, 1987).
of the crack. When a crack grows, its total dipole mo-
To find the electrical field in the arr a around a crack,
e
ment oscillates and generates an EME. An average fre-
r
quency of EME f relates to the crack velocity vcr and an
sider ice with four types of charge carriers. Then
average grain size g as
4
r
v
∑ ε j jj
P r
˙
f = cr .
=P=
r
(38)
(43)
g
t
j =1
This relationship provides an investigator with a
r
where fluxes jj are determined by the transport equa-
simple method of crack velocity determination using
tion
measurements of f and g (Petrenko 1992a, Gluschenk-
ov and Petrenko 1993). The dielectric relaxation reduc-
σj
(
)
es λs and should be taken into account. Readers can
jj = ejE - ηjΦΩ + αaj∇εii
- Dj∇nj
r
r
r
(44)
2
find more details of the theory of such EME in some
ej
previous papers (Petrenko 1992a, Gluschenkov and
Petrenko 1993).
where σj = |ej| j nj are the partial conductivities and Dj
are the diffusion coefficients.
EME from the pseudo-piezoelectric effect
Inside the bulk material
The second important mechanism resultingin crack-
induced electrical polarization of ice is the pseudo-
div E = -div P / ε ∞ε 0 .
r
r
(45)
piezoelectric effect discussed above (see eq 2027 and
Taking into account that curl E = 0 (quasi-statir nary
Fig. 15b). A theoretical description of such polariza-
r
r o
approximation) and that at infinity (r = ∞) E = P = 0 ,
tion was developed earlier (Petrenko 1993a).
When a crack appears and expands, it generates
we can find
strains in the surrounding material that we define as
r
r
∇ϕ = - E = P / ε 0ε∞ .
We have to find the electrical field strength E(r , t) and
rr
(46)
rr
electrical potential ϕ(r , t) that are caused by these elas-
r
If for a particular crack, the deformation ε ii (r , t) is
r
tic strains. The charge carriers have to move because
known, then the system of eq 4346 is complete. In the
their energy U depends on strains
presence of dielectric/conductor or conductor/conduc-
tor interfaces, we have to join the proper boundary con-
U = U0 + U1 (εij) .
(39)
ditions to this system of equations. The system of eq
4346 has analytical solutions for many cases of practi-
In a linear approximation for an isotropic material
cal interest that will be shown in the following sections.
Figure 18 illustrates three types of elementary
3
U - U0 - αaj ∑ ε ii = U0 - αaj ⋅ ε ii , ε ii << 1 (40)
cracks. Notice that the mode III crack does not produce
~
an electrical field since, for an isotropic material, εii = 0
i =1
(see for example Hellan 1984).
where we use the Einstein summation convention
For a mode I crack
3
θ
∑ ε ii = ε ii .
(1 - ν)
cos
(41)
2
εii = KI
⋅
(47)
i =1
2πr
E
15
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