Figure 3.6 Separation of state parameter from overconsolidation ratio (from Jefferies and Shuttle, 2002, with permission Institution of Civil

In document Soil Liquefaction - Jefferies and Bean (Page 155-159)


developed around the conceptual framework of Drucker et al. (1957). The overconsolidation ratio R represents the proximity of a state point to its yield surface when measured along the mean effective stress axis. The variable p′max is the equivalent maximum mean effective stress experienced by the soil, which might be thought of as a preconsolidation pressure (although that term is commonly applied in the sense of vertical effective stress). The term equivalent indicates that p′max may not have been actually experienced by the soil if, for example, a K0 consolidation is involved (in which case p′max is the projection of the yield surface on the mean stress axis). Figure 3.6 illustrates the difference between ψ and R. Their respective definitions are:

[2.6 bis]


NorSand (Jefferies, 1993) was the first state parameter based model and generalized critical state theory. The familiar Cam Clay model is a special case of NorSand.

Subsequently other authors introduced the state parameter into bounding surface models (Manzari and Dafalias, 1997; Li et al., 1999), a simple hyperbolic plastic stiffness model (Gajo and Wood, 1999), a critical state-like model using Rowe’s stress dilatancy (Wan and Guo, 1998) and a unified clay and sand model (Yu, 1998).


The critical state needs to be formalized before being used as a basis of models and then its representation developed. Representing the critical state is usually broken down into two parts: the relationship between the critical void ratio and the mean effective stress;

and the relationship between the stresses in the critical state. Of course, the soil must meet both sets of criteria when shearing at the critical state.

3.3.1 Existence and Definition of the CSL

Critical state models are based on the existence of a unique critical state locus, formally expressed as an axiom (the First Axiom):


where C() is the function defining the CSL. There are two conditions in [3.3]: first, the volumetric strain rate must be zero; second, the rate of change of this strain rate must also be zero. This can equivalently be stated as the requirement on dilatancy expressed in Section 2.5.2: both dilatancy and rate of change of dilatancy must be zero during shearing at the critical state. There are no strain rate terms in C(), making the CSL identical to the steady state of Poulos (1981). Constant mean stress is invoked in [3.3] to avoid a less

Constitutive modelling for liquefaction 111

easily understood definition for the situation in which mean stress is increased while the soil is continuously sheared at the critical state.

Uniqueness of the CSL simply refers to C() being a single valued function of void ratio and mean effective stress. For any given mean effective stress, there is only one value of ec. This critical void ratio ec is independent of the strain conditions and direction from which the critical state is approached.

Why start with an existence axiom? The short answer is to avoid getting bogged down in arguments over the interpretation of experimental data on the CSL. In developing a mathematical framework, an axiom is invoked as the starting point for the theory and the relevance of the theory is justified when it is seen how well it matches the stress-strain data of soils. Only the relevance of the theory may be disputed, as starting from an axiom means that the framework is always correct provided that the mathematics and physics are consistent. Starting from axioms also allows behaviours to be predicted, as aspects of the theory become necessary for self-consistency with the axioms. This can provide enormous insight into soil behaviour, one such example being plastic yield in unloading (which is pervasive with soils, but not an obvious consequence of plasticity theory).

Critical state frameworks can be developed with strain rate dependency. Having a rate effect does not negate the theory put forward here. Of course, the way in which the CSL varies with shear strain rate will need to be defined and this will involve additional parameters. To date, there seems to be little data on what a strain rate effect might look like (other than studies for confinement of underground nuclear explosions, but these are not usual engineering material velocities), and the experience in usual soil mechanics tests on sands is that there is no measurable rate effect. Also note that the flow structure at the steady state postulated by Poulos (1981) is inadmissible mechanics—this flow structure is undefined and, apparently, not measurable.

3.3.2 Critical state in void ratio space

Conventionally, the critical state is represented in e—p′ space using the semi-log form:

[2.1 bis]

There is abundant data to show that the CSL is more complex than [2.1] when viewed over a wide range of mean stress. Figure 2.22, for example, shows that the critical state line of Erksak sand has a distinct “knee” or yield point (for want of a better phrase) at a mean stress of about 1000 kPa, while Figure 2.24 shows that a smooth curve provides a better fit for Guindon tailings. Li and Wang (1998) suggested plotting e against (p′)α, and that α=0.7 will generally linearize the relationship for sands as is done for Toyoura sand

on Figure 2.25. Verdugo (1992) points out that the location of the curvature of the critical state line can depend on whether one views the data on logarithmic or arithmetic scales and that a bi-linear relationship usually suffices. The semi-log representation is convenient for engineering of soils as it arises naturally when the stiffness is proportional to the mean stress. The knee in the Erksak sand CSL occurs when this is no longer the case and the stiffness becomes approximately constant.

Despite the additional complexity that can be captured in representing the CSL, for most practical engineering equation [2.1] is a sufficient approximation for commonly encountered stress levels in engineering of about 10 kPa <p′ < 500 kPa. More elaborate representations of the critical state, such as that shown on Figure 2.25, are relatively easily adopted and are merely additional details to the basic framework—they do not affect the reasonableness of the approach.

3.3.3 Critical Stress Ratio M(θ)

The critical state friction angle or is not especially helpful for constitutive modelling, as models are cast in terms of stress invariants, and the relationship between these invariants needs to be expressed for the critical state. The convention is to introduce a critical stress ratio, M, so that, at the critical state;


The parameter M (and it is deliberately not called a property) was initially viewed as a constant. However, it became obvious quite early (Bishop, 1966) that constant M

Constitutive modelling for liquefaction 113

Figure 3.7 Example of variation of critical friction ratio M with Lode

In document Soil Liquefaction - Jefferies and Bean (Page 155-159)

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