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in a sample. Therefore a plastic/craze zone is usually formed in the vicinity of the
crack tip. The high value of strain rate in the plastic zone gives rise to intensive
production/multiplication of dislocations and point defects. Taking the notice of
different mobility of the lattice defects one may expect that the charge separation
can occur in the plastic zone. A close analogy exists with SW, in which, as we have
noted, the intensive deformation results in both the production of the lattice defects
and the charge pile up at the SW front.
Suppose that there are two kinds of the defects with opposite charges
˙
q. Let us
multiply Eq. (
9.9
)forthefirst.i
D
1/ and the second .i
D
2/ kinds of defects by
q
1
D
q and by q
2
D
q, respectively. Then taking the sum of these two equations,
we come to the continuity equation for the electric current density
j
:
@
t
Cr
j
D
0;
(9.30)
where
j
D
V
qa
2
r
.
1
n
1
2
n
2
/
C
.
1
C
2
/
E
:
(9.31)
Suppose that the mobility and conductivity of the defects of the first kind are
greater than those of the second kind, i.e.
jr
.
1
n
1
/
j jr
.
2
n
2
/
j
and
1
2
.
Besides we assume that
1
is a constant value and neglect the term
V
which
describes the current due to the medium displacement with the velocity
V
.For
convenience, we shall omit the subscript 1 everywhere. Then combining Eqs. (
9.30
),
(
9.31
), and (
9.10
), we get
@
t
C
=
Cr
j
d
D
0;
j
d
D
qD
r
n;
(9.32)
where
D
""
0
= is the charge relaxation time, D
D
a
2
is the coefficient of
diffusion, and
j
d
is the diffusion current density. In the case of the dislocation one
should replace the above value of D by the following one: D
D
c
d
a.
Consider a thin plane crack growing at constant velocity
V
c
along the x-axis in a
sample, which is infinite in the direction of
z
-axis as shown in Fig.
9.5
.Lety-axis be
perpendicular to the crack surface. The normal component of total current density
must be continuous at the interfaces .y
D
0/ between crack sides and air/vacuum.
It follows from the symmetry of problem that E
y
D
0 in the space between crack
sides. Then the boundary conditions at y
D
0 read
@
t
E
y
C
E
y
qD@
y
n
D
0:
(9.33)
At the initial moment t
D
0 when the crack begins to grow the charge density
D
0. Given the function
j
d
and n,Eq.(
9.32
) can be solved for then
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