Geology Reference
In-Depth Information
J
2
¼
1
=ðÞ
r
ð
2
and e
ij
;
r
ij
are components of the general plastic strain rate and
stress tensors, respectively (see, for example, Johnson et al.
1962
; Smith and
Nicolson
1971
). However, little has been done to extend these approaches to
anisotropic materials.
The procedures just described, which assert in general that the flow behavior is
determined entirely by the shear or deviatoric stress components, are based on
many observations showing that the mean stress or hydrostatic component has
little influence in plastic deformation. These observations have been mainly on
metals and mainly at low temperatures. However, even with metals it is known
that there is a small increase in flow stress with increase in mean stress, which
becomes more obvious when the mean stress is large compared with the shear
stresses. Further, it has been observed that a significant influence of mean stress
can appear in high-temperature creep of metals when cavity growth occurs (Dyson
et al.
1981
; Lonsdale and Flewitt
1977
; Needham and Greenwood
1975
). Similar
pressure effects have been observed in polymers and glasses, where they are often
rationalized in terms of a concept of ''free volume'' in the structure. Finally, in
granular flow of soils and cataclastic flow of rocks, it is always necessary to take
account of the role of the pressure component, this is being covered in the simplest
case by the Coulomb failure criterion (Jaeger and Cook
1976
, p. 95; Paterson and
Wong
2005
, p. 24). The factor common to all these cases is that effects associated
with volume change during the flow can no longer be neglected. This situation
tends to occur in a number of situations with rocks, so that the hydrostatic com-
ponent of the stress will often have to be taken into account in treating flow under
general states of stress, as in: (1) low-temperature cases where some cataclastic
component of flow is involved, as already indicated; (2) cases involving changes in
porosity which may include generalized types of high-temperature cavitation, even
under predominantly compressive conditions when fluids are present; (3) situations
in the lower crust and mantle of the earth where the hydrostatic component of the
stress becomes very large relative to the deviatoric components and influences the
elastic constants significantly.
The role of pressure is commonly incorporated in the failure criteria and flow
laws for general states of stress at high temperature by assuming that it can be
expressed through an exponential multiplying factor exp
pV
=
R
ð Þ
as in uniaxial
tests (see
Sect. 4.4.2
). This multiplying factor can be applied respectively to the
equivalent viscosity and to the universal strain rate versus stress relationship in the
two simple approaches for general stress states, mentioned above. More generally,
in the plasticity theory approach, the pressure dependence can be allowed for by
postulating a form of yield surface which, instead of being a cylinder parallel to an
axis equally inclined to the three principal stress axes in stress space, is a cone that
opens in the direction of increasing compressive stresses in a way expressing the
appropriate linear (Coulomb), exponential, or other form of pressure dependence.
