Laboratory Study of Rock Deformation and Fracture - Ultra and Extremely Low Frequency Electromagnetic Fields - page 335

Here V x is the x-component of the material velocity in the SW, so that the term n i V x
describes defects transfer due to the material motion. The flux caused by the electric
field is determined by the last term in Eq. ( 9.7 ). Here q i is the particle charge for the
given type, i is ion conductivity, and E x is the x-component of the electric field.
Notice that all functions in the right-hand side of Eq. ( 9.7 ) are taken at the point
x and at the time t, besides the functions i and i depend on temperature and
mechanical stresses in the SW. The analysis shows that the difference between the
functions i and i does not qualitatively change the resulting effect. Therefore in
what follows we assume that i D i D i .
Generalizing Eq. ( 9.7 ) to 3D case we obtain the vector of the flux density

f i D a 2 r . i n i / C n i V C i E =q i : (9.8)

The first term of Eq. ( 9.8 ) describes, as before, the motion of the defects with
respect to atomic lattice. This term consists of two summands: a 2 i r n i and
a 2 n i r i . The first summand is usual diffusion flux due to the gradient of the
number density of the defects and the value a 2 i plays the role of diffusion
coefficient. The next term can occur at the constant value of n i . This flux arises
from the deformation and heating of the lattice, which affect the jump frequency, i ,
of the defects. Below such effects are discussed in more detail.
The multiplication/reproduction rate of the defects under plastic deformation is
proportional to the time-derivative of the strain deviator, d t . As it follows from the
observations, the number density of the point defects amounts to 10 16 -10 17 cm 3
per each percent of plastic strain in a SW (e.g., Klein 1965 ). The multiplication and
recombination of the defects can be accounted for in the continuity equation and is
given by

@ t n i Cr f i D M i d t ki n k n i : (9.9)

Here M i is the coefficient of defect multiplication, ki is a recombination coefficient
of vacancies and interionic/interstitial ions. The first term on the right-hand side
of Eq. ( 9.9 ) describes the multiplication rate of defects whereas the second one
describes their recombination. It follows from the charge conservation law that
M k D M i and ki D ik . Maxwell's equation for this case is

e D X
i

" 0 r ." E / D e ; q i .n i n i0 /; (9.10)

where " is dielectric permittivity, n i0 is initial number density of the defects, and e
is electric charge density. The coefficients ki are proportional to both the particle
scattering cross-section (of the order of 4a 2 ) and the particle velocity (of the order
of a), whence it follows that ki a 3 .