Geology Reference
In-Depth Information
4.6 Failure Criteria and Flow Laws for General Stress States
So far, we have discussed deformation behavior in situations involving a simple
homogeneous state of stress, usually specified by a single component of stress
(uniaxial stress state), with possibly the superposition of a hydrostatic component
to be taken into account. However, in practical applications, it is frequently nec-
essary to deal with general states of stress (complex, triaxial, or multiaxial stresses)
in which all three principal components and their orientations have to be taken into
account, as well as their variation from point to point. The problem then arises of
how to generalize the stress-strain-strain rate relationships so far discussed, that
(
1975
), Rice (
1970
),
1975
), and Kestin and Bataille (
1980
) for general consider-
ations on constitutive relations and their extension to time-dependent plasticity,
including the question of memory effects and the use of internal variables. There
are two distinct approaches to the problem, one being a generalization of the
treatment of viscous fluids and the other an extension of the mathematical theory
of plasticity, on which we shall now comment briefly.
The simplest approach through the theory of viscous flow is to retain the normal
treatment based on isotropic Newtonian or linear viscosity but to assign to the
dynamic viscosity an effective or equivalent value, given by g
¼
r
=
3e where r
;
e are
the normal stress and strain rate, respectively, measured in a uniaxial experiment
carried out to steady state within the ranges of stresses that is of interest (Griggs
1939
;
the factor three arises naturally from the constitutive equations for isotropic linear
viscous flow with zero bulk viscosity). However, this procedure is only suitable for an
approximate treatment of the broad aspects of the flow and it automatically
suppresses any characteristic features arising from the actual nonlinear nature of the
viscosity. The alternative is to solve the equations of viscous flow with a viscosity
that is no longer a constant but is itself a function of stress. Difficulties associated both
with establishing the form of this function and of solving the nonlinear equations
have largely inhibited development along these lines.
The theory of plasticity, as developed in its simplest form for isotropic metals at
low temperatures, generalizes the uniaxial stress-strain relations in terms of
''effective'' stresses and strain increments, these being scalar functions of the
actual stress and strain increment components that serve to describe the
mechanical behavior in the same form as in the uniaxial case (not to be confused
with the ''effective stress'' in situations involving pore pressure, Paterson and
Wong
2005
, p. 148, although the general concept is the same in that we seek to
specify that aspect of the stress state that is effective in the particular situation).
Following von Mises (Hill
1950
, p. 26), the effective stress r
is taken to be
p
r
1
r
2
h
i
1
=
2
r
¼
1
=
2
Þ
2
þ
r
2
r
3
Þ
2
þ
r
3
r
1
Þ
2
ð
ð
ð
where r
1
;
r
2
;
r
3
are the principal stresses. The corresponding effective plastic
strain increment de
is taken to be
