The Critical State Model
The critical state
concept was introduced in §1.8 in general terms. We are now in a
position to set up a
model for critical state behaviour, postulating the existence of an ideal
material that flows as
a frictional fluid at constant specific volume v
when, and only when,
the effective
spherical pressure p and
axial-deviator stress q satisfy
the eqs. (5.22
bis) and
q
= Mp
v
= Γ − λ ln p (5.23
bis).
Fig. Associated Flow for Soil at Critical StaThis
concept was stated in 1958 by Roscoe, Schofield and Wroth4
in
a slightly
different form, but
the essential ideas are unaltered. Two hypotheses are distinguished: first
is the concept of yielding
of soil through progressively severe distortion, and second
is the
concept of critical
states approached after severe distortion. Two levels of difficulty
are
recognized in testing
these hypotheses: specimens yield after
a slight distortion when the
magnitudes of
parameters (p, v, q)
as determined from mean conditions in a specimen can
be expected to be
accurate, but specimens only approach the critical
state after severe
distortion and
(unless this distortion is a large controlled shear distortion) mean conditions
in the specimen can-
not be expected to define accurately a point on the critical state line.
It seems to us that
the simple critical state concept has validity in relation to two
separate bodies of
engineering experience. First, it
gives a simple working model that, as
we will see in the
remainder of this chapter, provides a rational basis for discussion of
plasticity index and
liquid limit and unconfined compression strength; this simple model is
valid with the same
accuracy as these widely used parameters. Second,
the critical state
concept forms an
integral part of more sophisticated models such as Cam-clay, and as such
it has validity in
relation to the most highly accurate data of the best axial tests currently
available. Certain
criticisms5,6 of the simple
critical state concept have drawn attention to
the way in which
specimens ‘fail’ before they reach the critical state: we will discuss
failure in chapter 8.
The error introduced
in the early application of the associated flow rule in soil
mechanics can now be
cleared up. It was wrongly supposed that the critical state line in
Fig. 6.9(a) was a
yield curve to which a normal vector could be drawn in the manner of
§2.10: such a vector
would predict very large volumetric dilation rates
v& p vε& = M.However,
we have seen that the set of points that lie along the critical state line
are not
on one yield curve: through each critical state point we can
draw a segment of a
yield curve parallel
to the p-axis in Fig. 6.9(b). Hence it is
correct to associate a flow
vector which has with
each of the critical states. At any critical state very large
distortion can occur
without change of state and it is certainly not possible to regard the
move from one
critical state to an adjacent critical state as only a neutral change: the
critical
state curve is not a yield curve.
Steady state
theory
An alternate to the critical
state concept is the steady state concept.
The steady state strength is
defined as the shear strength of the soil when it is at the steady state
condition. The steady state condition is defined as "that state in which
the mass is continuously deforming at constant volume, constant normal
effective stress, constant shear stress, and constant velocity." Steve
Poulos built off a hypothesis that was formulating towards the end of his
careerSteady state based soil mechanics is sometimes called "Harvard soil
mechanics". It is not the same as the "critical state"
condition.
The steady state occurs only
after all particle breakage if any is complete and all the particles are
oriented in a statistically steady state condition and so that the shear stress
needed to continue deformation at a constant velocity of deformation does not
change. It applies to both the drained and the undrained case.
The steady state has a slightly
different value depending on the strain rate at which it is measured. Thus the
steady state shear strength at the quasi-static strain rate (the strain rate at
which the critical state is defined to occur at) would seem to correspond to
the critical state shear strength. However there is an additional difference
between the two states. This is that at the steady state condition the grains
position themselves in the steady state structure, whereas no such structure
occurs for the critical state. In the case of shearing to large strains for
soils with elongated particles, this steady state structure is one where the
grains are oriented (perhaps even aligned) in the direction of shear. In the
case where the particles are strongly aligned in the direction of shear, the
steady state corresponds to the "residual condition."
Two common misconceptions
regarding the steady state are that a) it is the same as the critical state and
b) that it applies only to the undrained case. A primer on the Steady State
theory can be found in a report .Its use in earthquake engineering is described
in detail in another publication.