The Critical State Model

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.