Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 300

z

z ′

y ′

ʳ

B 0

y

I

x ′

ʱ

ʲ

x

Fig. 7.10 A schematic plot of general coordinate systems .x;y; z / and local coordinate systems
.x 0 ;y 0 ; z 0 / in which the shear crack is in the plane .x 0 ;y 0 /

7.3.2 GMPs Due to Shear Cracks

The mechanics of shear cracks is a crucial factor in the theory of faulting and EQ
mechanics, since the growth and interaction of shear cracks are assumed to play a
key role in both the rock strength and development of fault zones (Scholz 1990 ). In
a sense, the fault zone as a whole can be considered as a gigantic shear crack which
is described by its seismic moment tensor (Aki and Richards 2002 ). In what follows
we show that not only tension but also the shear cracks can generate GMPs. Since
the displacement field around a shear crack is more complicated than that originated
from a tension crack, this results in a more complicated theory of electromagnetic
phenomena associated with shear crack growth (Surkov 2001 ).
As before, we consider the ground as a uniform elastic half-space with constant
conductivity .Letx-axis be horizontally directed along the magnetic meridian
while z -axis is vertically upward. The vector B 0 of geomagnetic field makes an angle
I with x-axis (I<0in the northern hemisphere) in the vertical plane containing
the magnetic meridian. Additionally, we use the local coordinate system .x 0 ;y 0 ; z 0 /
in which the shear crack is in the plane .x 0 ;y 0 / asshowninFig. 7.10 .Thex 0 -axis is
perpendicular to the shear crack plane and makes an angle with z -axis. In general,
the orientation of the axes x 0 , y 0 , and z 0 is defined by Euler angles Ǜ, LJ and .
Assuming for the moment that the rock shift points parallel to x 0 -axis, then the
seismic moment tensor of the shear crack has only two components m 13 D m 31 D
S Πu x , where is the shear modulus of the matter, and S is the crack area.
The displacement discontinuity at the crack surface, Πu x .t/ D u x .t; z 0 D 0 C /
u x .t; z 0 D 0 /, is considered as a given function of time. In the wave and interme-
diate zones, that is far away from the shear crack, the components of mass velocity
are given by (Aki and Richards 2002 )