Laboratory Study of Rock Deformation and Fracture - Ultra and Extremely Low Frequency Electromagnetic Fields - page 355

9.2.7 Electric Field in Collapsing Pores

Deformation of heterogeneous materials is accompanied by an enhancement of
strain near pores and inclusions followed by the generation of local plastic and
craze zones around the inhomogeneities. For example, the fast pore compression
arises while the SW propagates in porous matter. Large shear strain near the pore
surface results in the intensive heating of matter in plastic zones. Sometimes the
shock compression is even accompanied by the formation of local melting zone
around the pores that results in both material strength degradation and collapse
of pores and micro-voids (Dunin and Surkov 1982 ). The fast deformation of the
dielectric material around the inhomogeneities can impact the electric effects caused
by the presence of space charges near the inhomogeneities. Below we show that the
accumulation of electric charges in the vicinities of micro-inhomogeneities, grains
and crazings leads to the generation of a strong electric field which may result in
local disruption of dielectrics.
Suppose that both the width of the SW front and typical distances between pores
are much greater than a typical size of pores. First of all we study the compression
of individual pores in the SW (Surkov 1991 ). The high compression of the matter
results in intensive production of the dislocations and point defects in plastic zones
which arise around the pores. As before we also suppose that there are only two
kinds of defects with charges of opposite signs. Considering the pore collapse one
should take into account primarily the fast displacement of the material around
the collapsing pore. In this notation the flux density of the defects is basically
determined by the velocity V of the lattice displacement. As a first approximation,
one can ignore other terms in Eq. ( 9.8 ) and substitute the flux density f i D n i V
(i D 1;2) into Eq. ( 9.9 ). If the matter is incompressible, i.e., r f i D 0, then the
solution of Eq. ( 9.9 ) is given by: n i D n i0 C M. Here we have neglected the
recombination of the defects. As the next approximation, we will search a small
correction ın i to the defect number density n i , i.e., ın i n i . Substituting the
density in the form of n i D n i0 C M C ın i into Eqs. ( 9.8 ) and ( 9.9 ) we get

1
q i r . i E / D 0;

@ t ın i Cr .ın i V / a 2

2 .n i0 C M/ i C

r (9.38)

where the conductivities i depend on the defect densities. Combining Eq. ( 9.38 )
for i D 1 and i D 2 we come to the continuity equation given by Eqs. ( 9.30 ) and
( 9.31 ) in which one should substitute n 1 D n 2 D n 0 C M, where n 0 D n 01 D n 02 .
For simplicity, all the pores are assumed to be ball-shaped with the same
radius b. As the porosity is small so that the pore interaction can be neglected,
then the distributions of the strain, electric charges, and fields around the pore are
approximately spherically symmetric. Combining Eqs. ( 9.10 ), ( 9.30 ), and ( 9.31 ), we
obtain

r 2 @ r r 2 E r C Ǜ 0 @ r Œn. 2 1 / C

V r E r

0 D 0;

@ t E r C (9.39)