Tímarit Verkfræðingafélags Íslands - 01.12.1983, Page 9

The same set of parameters apply to
the cavity volume, that is, the volume
elastance
Ev = dV/dp (6)
and the volume elasticity
ev = (1/V)dV/dp (7)
Moreover, the volume dilatance p = 0
Dv = dV/da (8)
and the dilaticity
dv = (l/V)dV/do (9)
In the case of Hookean solids the
above parameters E, e, D and d are con-
stant. Under botli internal and external
stress, the cavity volume increment is
iV = pEv + aDv (10)
and the cavity volume strain
AV/V = pev + odv (11)
(3.ii) Daia for symmetric cavities. The
theory of the elastic parameters of
spherical, cylindrical and penny-shaped
cavities is briefly reviewed in section
(12) below. The above parameters are
easily worked out and are given in Table
I.
It is of interest to point out that at the
conditions given, the dilatance of the
spherical and cylindrical cavity forms is
notably larger than the elastance. In
fact, the dilatance/elastance ratio is
dependent on the degree of stress focus-
ing and is therefore greatest in the case
of the spherical cavity. In the case of the
penny-shaped cavity under unidirec-
tional axial stress, the ratio is probably
close to unity and the difference be-
tween dilatance and elastance of little
practical significance for the present
purpose.
At this juncture, it should be pointed
out that under more general force field
conditions, the elastic parameters of
cavities of an arbitrary shape will de-
pend on the orientation of the cavity
with regard to the imposed stress field.
We are then confronting a very complex
situation that has not received much at-
tention.
(3.iii) Flat cavities of an arbitrary con-
tour. Exact results for flat cavity
elastance are available only in the case
of the penny-shaped cavity. Other con-
tours have to be discussed on the basis
of approximations. A fairly efficient
method for this purpose is given in sec-
tion (12) below. As already stated, there
are good reasons for assuming that the
dilatance of flat cavities under uniform
stressing perpendicular to the cavity
plane is about the same as the elastance
under internal fluid pressure.
(3.iv) Inclusions. In continuation of the
above development, it ts of interest to
consider the pressure response of
cavities that are filled with a fluid or a
solid of different elastic properties than
the surrounding rock. Assuming impos-
ed external stress of the type defined
above, the cavity or inclusion pressure
(or stress field) is then a reaction to the
stress that has to be derived on the basis
of the given conditions. Considering
first the case of a fluid inclusion of a
bulk modulus kf, we find that the inclu-
sion pressure p is given by
p = -kf(aV/V) (12)
and hence with the help of equation (11)
p = -kf(pev + odv) (13)
and that
P = -kfOd^/O+k^) (14)
or
P = -odv/(cf + ev) (]5)
where Cf=lkf is the fluid compressi-
bility.
Considering the case of the spherical
cavity and using the data in Table I, we
find
p = -1.35(kf/u)o/tl + 0.75(kf/u)t, (16)
It is of interest to note that equation
(16) holds also for a homogeneous and
isotropic spherical inclusion of the bulk
tnodulus kf. Let the inclusion be stiff
compared to the surrounding rock such
that k//i»l. We find then that
p = -1.8o (17)
The stress in the inclusion is thus con-
siderably larger than in the surrounding
rock. This is a case of stress enhance-
ment by stress focusing.
In the following, it is convenient to
define the quantity
Ec = cfV+Ev (18)
as the cavity capacitance and the ratio
(see equation (15))
Y = dv/(cf + ev) = Dv/Ec (19)
as the cavity pressure reactance.
Consider now a cavily of volume V
containing a lluid of density />. The
cavity pressure increment p due to the
addition of a lluid mass m and external
(hydrostatic) stress o is then
Table I
Elastance/dilatance data assuming Poisson’s relation \ =/u
Cavity Spherical Cylindrical Penny-shape
Radius/width R R* R, W**
External stress radial radial axial
Internal fluid pressure
Radial elastance Er 0.25R//V 0.5R///
Central width elastance Ew R///
Radial elasticity eR 0.25/p 0.5///
Central width elasticity eW R/W//
Volume elastance Ev 7tRV// 7tR2/// 2R3///
Volume elasticity ev 0.75 //u \/p 2R/?tW//
External stress
Radial dilatance Dr 0.45R/// 0.75R ///
Central width dilatance Dw ~R/p
Radial dilaticity dR 0.45/// 0.75///
Central width dilaticity dw ~R/W //
Volume dilaticity DV l.8R3/// 1.5ttRV// ~2RVp
Volume dilaticity dv 1.35/// 1.5/// ~2R/nWp
* Data for the cylindrical cavity obtained by assuming zero axial strain.
** Elastance/dilatance estimated equal in the case of the penny-shaped cavity (see Section (12)).
TÍMARIT VFÍ 1983 — 89