Laboratory Study of Rock Deformation and Fracture - Ultra and Extremely Low Frequency Electromagnetic Fields - page 336

At the SW the source function M i d t plays a significant role due to the high
value of strain rate d t Š 10 7 s 1 . Taking the following typical parameters of SW:
M i Š 10 25 m 3 , n i Š 10 24 m 3 , a D 0:281 nm (for a NaCl lattice), we obtain that
the recombination term in Eq. ( 9.9 ) is small in comparison with the source function
up to a frequency i D 10 12 Hz. Therefore, we can neglect this term in further
calculations.
Now we estimate the different terms in Eq. ( 9.8 ) for the flux density of
the defects. If the condition ea 2 jr i n i j > i E is valid at the SW, then

the conductivity of the crystal can also be neglected. Using the expression

i D e 2 a 2 n i i C i =.2k B T/, where T is the temperature and k B denotes
Boltzmann constant, then the above condition reduces to: E<k B T=.e/, where
is typical size/width of the SW in the crystal. Thus, we obtain the estimation
E<10 5 -10 6 V/m for the following parameters: T Š 10 3 K and Š 10 7 -10 6 m.
In this approach we shall ignore both the relaxation processes associated with the
particle recombination and the influence of the conductivity at the front of the SW,
that is at the length where the gradients of all parameters are large. Although both
these processes can be very important just behind the SW. Substituting Eq. ( 9.8 )for
f i into Eq. ( 9.9 ), omitting subscript i, and considering the plane SW propagating
along the x-axis, we come to

@ t n C @ x .n/ a 2 @ x .n/ D Md t :

(9.11)

Here d t denotes the absolute value of the rate of shear plastic strain. Equa-
tion ( 9.11 ) belongs to Fokker-Planck type equation with the source term on the
right. For the case of constant the last term on the left-hand side of Eq. ( 9.11 )
reduces to the form which is typical for diffusion type equation with the diffusion
coefficient a 2 .
Notice that the same equation can be applied to the number density of defect
electrons/holes whose jump frequency can be very sensitive to the temperature at
the SW (Freund 2000 , 2002 ). For the case of charge transfer by edge dislocations,
the parameters n and M stand for the number of dislocations per unit area and
multiplication coefficient, respectively. The charge q now has the meaning of
electric charge per unit of dislocation length and D c d =a where c d is the velocity
of dislocation slip. The charge q is considered to be independent of dislocation
velocity. Using these re-designations, Eq. ( 9.11 ) can be applied to the description of
dislocation kinetics as well.
In what follows we deal with a stationary SW. The parameters of such a wave
depend solely on the variable D .x V s t/=, where V s is the speed of the SW
propagation and stands for the characteristic scale of the SW front. The solution
of Eq. ( 9.11 ) can be found as a power series of small parameter Ǜ D 0 a 2 =.V s /.In
the first approximation we obtain (Sirotkin and Surkov 1986 )

n n 0 D M Ǜ d
d 0 .n 0 C M/:

(9.12)