*Symbol Description * *How measured * *Applications *
*Γ, λ, *
*H, χ * Plastic modulus, stress

dilatancy

Triaxial tests; possibly self bored pressuremeter

Displacement calculations, finite element models

*G * Shear modulus Seismic shear wave velocity,
resonant column test, self
bored pressuremeter

Dynamic analyses, displacement calculations

*D** _{r}* Damping ratio Resonant column test Dynamic analysis

*CRR*Cyclic strength Cyclic triaxial, cyclic simple

shear tests low loads for large numbers of cycles. (cyclic mobility)

**2.4 DETERMINING THE CRITICAL STATE **

**2.4.1 Triaxial Testing Procedure **

The critical state comprises two aspects, a locus or line in void ratio-mean stress space, and the ratio between the stresses at the critical state. The void ratio aspect is the most difficult to measure, and so that is dealt with first.

Experience indicates that the preferred method of determining the critical state line is a series of triaxial compression tests on loose samples, generally markedly looser than the critical state. Loose samples do not form shear planes and do not have the tendency to localization that is normal in dense (dilatant) sands. Originally, the standard protocol followed Castro and concentrated on undrained tests. Undrained tests are more convenient and should always be the starting point for the practical reason that the strains required to reach the critical state are well within the limits of triaxial equipment for loose samples. Small strains result in large pore pressure changes, and therefore undrained samples can change state (i.e. move to the critical state) relatively quickly.

However, it turns out that it is quite difficult to obtained data on the CSL above about
*p′=200 kPa with undrained tests because samples have to be consolidated to p′>2 MPa *
prior to shear. Such high pressures are both inconvenient for most commercial triaxial
equipment as well as often involving grain crushing effects. Drained tests are therefore
used as well as undrained. In drained tests on loose samples, the sample moves to the
critical state at a much slower rate and displacements to the limits of the triaxial
equipment are required.

The goal in the undrained testing is to determine both the void ratio and the mean
stress at the critical state accurately over a range of critical state conditions. The test data
*are presented in q–ε*1* and ∆u–ε*1* plots and a q–p′ stress path for review and picking the *
critical conditions. Drained testing is useful to flesh out the higher stress end of the CSL.

*The test data are presented as both q–ε*1* and ε**v**–ε*1 plots. Often the sample may have to be
taken to 20% axial strain, and lubricated end platens are essential.

Successful critical state line testing is dependent on getting certain details of the triaxial testing correct:

• uniform samples must be prepared to a loose and predetermined void ratio (the operator must be able to achieve a desired void ratio)

• samples must be fully saturated

• the void ratio must be known accurately (to within about ±0.003)

• the measurement system must be capable of measuring low stresses, as well as pore pressures at a high rate with very little system compliance (a “liquefied” sample may be at a mean effective stress of ≈1kPa, derived as the difference between a measured total stress of 300 kPa and pore pressure of 299 kPa.)

The required procedures to deal with each of these aspects are covered in Appendix B.

By far the greatest aid to critical state testing of sands is a good computer-controlled testing system. Computer data acquisition is now a generally accepted tool, but the inclusion of feedback and control of the test by computer is not widely practised.

Computer control, such as that provided in the GDS testing systems (Menzies, 1988, Horsfield and Been, 1987) provides the flexibility to test along any desired stress path,

Dilatancy and the state parameter 73

under stress or strain rate control (as distinct from load or displacement rate control).

Measured data also needs correcting for membrane penetration and cross-sectional area changes during the test, and these corrections are easily done with computer controlled work. True constant volume tests compensating for membrane errors can be achieved.

Regardless of whether computer control is used, it is essential to use computerized data acquisition. At some point models are going to be fitted to the data to evaluate design parameters, and possibly calibrate parameters for numerical analysis. Doing this on a few data points or on data taken off paper traces is both tedious and loses information. Without computer-based data acquisition, testing is simply not to modern standards.

**2.4.2 Picking the critical state from test results **

Regardless of whether testing is drained or undrained, there is judgment involved in identifying critical conditions from the test data. Interpretation of the critical state from triaxial tests is conceptually straightforward, but it is also surprising how much confusion and disagreement there has been on the existence, or otherwise, of a critical state. Much of this disagreement is based simply on different interpretations of the test. It is appropriate firstly to repeat the definition of the critical state as the state at which the soil

*“continues to deform at constant stress and constant void ratio” (Roscoe, Schofield and *
Wroth, 1958). Implicit in the definition is the expectation that the sample will continue to
deform in the same way with further strain, so that a temporary condition where void
ratio and effective stresses are constant does not represent the critical state. The critical
state is defined very simply in terms of the dilatancy: both dilatancy and rate of change of
*dilatancy must be zero, i.e. D=0 and * It is this second condition that assures that
the true critical state has been reached and not a transient condition. Figures 2.19 to 2.22
show the results of a series of undrained tests on Erksak sand (Been et al, 1991) to
illustrate some of the details of test interpretation.

The stress-strain and pore pressure behaviour of a loose sample are shown on Figure 2.19. A clearly defined steady state is reached after about 8% axial strain, and the soil deforms at this constant state to 20% strain, at which time the test was terminated. In this test the critical state stresses can be determined unambiguously, but corrections to the void ratio for the effects of membrane penetration are important for the accuracy of the final critical state point (Appendix B).