b) crack
c) bend
c. Bend.
a. Indentation.
b. Crack.
Figure 15. Some practical cases in which nonuniform strain generates electric polarization of ice. The
arrows indicate the direction of motion of charge carriers.
(
)
r t r r r r
Ω = ∫ j1 - j2 - j3 + j4 dt.
amount of work against this pressure
(23)
0
Eai(P) = Eai(0) + Pγai
The quantities ηi are given by
(20)
where γai is the so-called activation volume.* For the
ηi = 1,1,1,1
for i = 1,2,3,4
(24)
defects that increase volume, γai is positive, and their
formation energy is raised proportionally to the applied
and
pressure. This results in an exponentially decreasing
Φ
= 3.85kBroo
concentration of such defects with increasing pressure.
(25)
T
Since the formation energy of defects Eai depends on
the applied pressure P according to eq 20, in tr e pres-
h
where roo = 2.76 is the oxygenoxygen distance in
ence of a pressure gradient, there is a force Fi acting
ice (Jaccard 1964). To find the electrical field strength
caused by the pressure gradient ∇P, we should solve
upon the defects
the system of eq 2223 under particular initial and
Fi = -grad(Eai ) = -γ ai grad(P) .
r
(21)
boundary conditions. Thus, in the case of one dominat-
ing carrier type and static ∇P
r
To find a resulting flux of the defects ji , we have to
add this force to the transport equation
γ ai∇P
E=
r
(26)
εε
(
)
ei + ∞ 0 ⋅ Φ
r
ji = ei E - ηiΦΩ ni i - Di grad ni
r
r
(22)
ei
while for two types of charge carriers, for example
where ei =
electric charge of the defects
H3O+ ions and L-defects, and static pressure gradient
E=
electric field strength
the electrical field is
ni =
defect concentration
i =
r (γ - γ a1)∇P
defect mobility
E = a4
Di =
defect diffusion coefficient
(27)
e
Ω=
r
configuration vector
(see Evtushenko et al. [1987] and Evtushenko and Pe-
trenko [1991]). Electrical polarization of ice caused by
Normally, in solids pressure is not hydrostatic, i.e., σ11 ≠ σ22 ≠ σ33.
*
stable and growing cracks was calculated in another
Then, instead of eq 20 we should use
publication (Petrenko 1993a). Electromagnetic emis-
Eai(σij) = Eai(0) + γai(σ11 + σ 2 + σ 3) = Eai(0) + γaiσjj
(20a)
2
3
sion generated by such cracks will be described in the
or
section on electro-fracture effects.
Eai(εij) = Eai(0) + αaiεjj
Equations 2627 show how the defect's activation
(20b)
volumes can be determined from measurements of the
where αai is a deformation-potential constant of ith type charge carrier.
12
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