3.1.1 Why model?
This chapter is about modelling soil stress-strain behaviour, which will lead into modelling liquefaction. Before doing so, an obvious question needs to be addressed: why model? There are several reasons and it would be an error to think of modelling only as a precursor to stress analysis using finite elements.
The understanding of liquefaction has been plagued by dubious “concepts”, many of which run counter to a sound appreciation of soil mechanics. This has led to mutually exclusive propositions, and the notion that grasping the subject of liquefaction requires great wisdom and many years of experience. Constitutive modelling, using an appropriate model (and what is appropriate follows later in this chapter), shows that liquefaction is simply another soil behaviour that is relatively easily understood. Given this understanding, which is accessible to anybody given a little diligence, how to engineer liquefaction resistant works becomes far less contentious. Explanations such as
“metastable particle arrangement” are tossed out in favour of conventional soil properties and proper geomechanics for engineering designs.
Modelling is also important because geotechnical engineers depend to a large extent on in situ tests to determine sand or silt properties but in situ tests do not really measure soil properties; rather they measure a response to a loading. Obtaining soil properties from in situ tests involves solving an inverse boundary value problem, and a model is required for this.
Modelling is also an excellent way to capture full-scale experience. Because civil engineering must rely largely on case histories from failures rather than testing of prototypes, a great weight is placed on such experience and properly so. But, full-scale experience needs to be understood using a sound framework and this framework necessarily comes from mechanics. Mechanics, in turn, is based upon understanding soil constitutive behaviour. Wroth (1984) made this point years ago, but it is still not always fully appreciated.
3.1.2 Why critical state theory?
A basic premise is that a proper constitutive model for soil must explain the changes in soil behaviour caused by changes in density. Despite the obvious nature of this premise, void ratio (or any related variable such as relative density) is rarely included as a variable in constitutive models for soil, as can be ascertained from the proceedings of a workshop (Saada, 1987) where some 30 different models for sand were represented. The exceptions
are models based upon critical state theory and this naturally sets up critical state theory as the preferred starting point.
There is more to critical state theory than independence of the properties relating to density. Constitutive models for soil cover a philosophical range from descriptive to idealized. Descriptive models are intrinsically curve-fitting and anchored to test data—
they can be very suitable for computing if the stress paths in the problem of interest are similar to the test conditions. However, the accuracy of descriptive models in representing a particular situation is too often offset by an absence of insight into the underlying physical processes. Idealized models, on the other hand, start from postulated mechanisms from which behaviours are then derived. Idealized models trade possibly reduced accuracy in a particular situation for a consistent (and known) physics.
3.1.3 Key simplifications and idealization
There are two key simplifications in what follows: isotropy and small strain theory.
Isotropy is familiar to everyone with a grounding in engineering science but there is also a litany that soils are intrinsically anisotropic. Isotropy will nevertheless be assumed because there is little point in getting involved in the complexity of anisotropic behaviour if the isotropic version is not functional—anisotropy is further detail, not a fundamental premise that will make or break the model. Further, anisotropy can be approximated in an isotropic model, as will be shown, if there is the supporting evidence. This then leads to the point that most practical engineering has enough difficulty obtaining characteristic parameters for a simple isotropic model and that anisotropy is therefore (at least presently) a distraction from more important issues.
Small strain theory is a far more important point. Almost all degree courses in engineering science teach small strain theory in which higher than first order terms are dropped in moving from displacement gradients to strain. This small strain approximation (small meaning major principal strain of the order of 0.1% or so) is very reasonable within the context of elasticity. However, soil behaviour may involve strains to failure of as much as 50%. The standard elasticity based small strain theory taught in engineering courses is not genuinely adequate for geomechanics and even routine work should invoke large strain theory of one sort or another.
To save layering additional information and complexity, what follows will be presented within the familiar small strain context with two exceptions. First, incremental volumetric strains will be integrated to obtain void ratio change. Tracking void ratio this way is a large strain approach and assures that the correct end state will be reached.
Second, large strain analysis will be used formally in evaluating in situ tests. In doing this, there is an implied assumption that the properties determined in calibrating with small strain definitions are sufficiently representative for reasonable subsequent use in large strain analysis. There is as well, of course, the ever-present problem with laboratory test equipment and observations where there is a limit to how accurately the strain can be determined and the accuracy of measurements at larger strains.
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3.1.4 Overview of this Chapter
Although this book is about liquefaction and it has just been suggested that liquefaction is to be modeled, that modelling will not be found in this chapter. Instead, the principle of effective stress will be invoked and the desired framework will be developed in a drained approach. This is done deliberately as undrained behaviour will never be properly simulated unless the drained response is properly understood and captured. Undrained behaviour arises because of boundary conditions, and is not in itself fundamental soil behaviour. Readers unable to restrain their enthusiasm, or wishing to verify that the NorSand model does describe liquefaction well, can flip to Chapter 6 (static liquefaction) and Chapter 7 (cyclic mobility) to look at the plots.
In presenting the material, a fair bit of mathematics is inevitable. However, the underlying ideas are rather easy to understand (most can be visualized geometrically). So, to avoid unnecessary confusion, most of the formal derivations have been bundled in Appendix D. This way the ideas can be presented more simply and the derivations left until the whole picture has been obtained.
Critical state ideas are indeed straightforward but they do have the feature that there is no practically useful closed form solution (i.e. an equation directly relating stress and strain) for the models that follow. This is the case even for something as straightforward as a drained triaxial compression test in which the whole sample is at the same stress state and a known stress path is applied. However, the incremental form of critical state models relating changes in stress to strain increments is delightfully simple and these relations can be integrated easily in a spreadsheet to provide the desired results. An example spreadsheet (NorSand.xls) has been provided on the website that implements the model. The routines required are written in VBA and can be found under the Visual Basic editor (use the “Alt”+“F11" keys) of Excel. Each routine is discussed in this book, comments are provided in the VBA code, and the code follows the variable use and equations derived in Appendix D. VBA is an open source code, so nothing is hidden.
A feature of critical sate theory is that the ideas were developed in the context of the triaxial test. This test has axial symmetry of strains and fixed principal stress directions, both substantial simplifications for a model to be used in real engineering. However, the fist step is to understand and appreciate the generalization of the state parameter framework with NorSand and how this computes real soil behaviour. Triaxial conditions are sufficient for this. On the other hand, much of real engineering involves plane strain.
The Chapter therefore concludes with a validation of the 3D version of NorSand under plane strain. The full derivation of this “industrial strength” NorSand is given in Appendix D. This chapter, then, provides enough to look at in situ tests and how the state parameter is determined in situ (Chapter 4) and gives the context for the ideas. The application of NorSand to the full-scale experience of static liquefaction follows in Chapter 6, and with cyclic-induced liquefaction being presented in Chapter 7.
3.2 HISTORICAL BACKGROUND
As always, the history of a subject matter helps to understand the formalism that has developed around it, and the reasons for such formalism. The history of critical state
theory goes back a surprisingly long way, and centers as much upon Cambridge, Massachusetts, as upon Cambridge, England.
Soil has two behaviours that are apparent to the casual observer: plasticity and density dependence. Plasticity is apparent because deformations imposed on soil are largely irrecoverable. Density dependence is obvious because soil can exist over a range of densities at constant stress, and dense soil behaves quite differently from loose soil. A most interesting, and fundamental, aspect of geotechnics is how these two behaviours may be represented within a single, complete constitutive model.
Most soil models are based on plasticity, which is in itself a macro-scale abstraction of the underlying micromechanical reality of grain realignments and movements. Plasticity theory is the dominant methodology for constitutive modelling of geomaterials, as it reasonably captures their behaviour in a computable way. Although purists might argue the necessity for a micromechanical approach, the reality is that micromechanical models are complicated and generally unusable. Much as engineering with metals uses plasticity, even though it is dislocation movements that matter, so too can soil mechanics use plasticity without worrying about the internal mechanisms of the soil skeleton. In fact, plasticity theory can be given a fundamental slant through thermodynamics (Drucker, 1951). Plasticity theory will be therefore be used for soil without further discussion of its relevance.
Plasticity is the idea that some (and usually most) strains are not recovered when a body is unloaded, an idea which dates back 150 years. Tresca (1864) first proposed a yield condition which distinguished between those stress combinations that cause yield (or irrecoverable strains) from those that do not. During yielding, strains are viewed as comprising two mechanisms, one elastic (denoted by the superscript “e”), and one plastic (denoted by the superscript “p”), with the relation
One of the differences between plasticity and elasticity is the treatment of strains. In elasticity, principal strain increments are in the same direction as principal stress increments. This relationship is intrinsic to the way the theory is developed. In plasticity, however, theoretical development first concentrates on identifying the stress conditions under which plastic strain occurs to define the yield surface. The magnitude and direction of the plastic strains require further thought.
A simple thought experiment to show the direction of plastic strains is shown in Figure 3.1. An ice hockey puck is sitting on the ice and is about to slide under the action of two forces, both of which are applied by strings acting at an angle. Sliding starts when the force on one string is increased slightly. One can immediately appreciate that the puck starts moving in the direction of the force resultant, not the force increment that
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initiated sliding. This is the simplest explanation of what is termed the normality principal (alternatively called associated flow). Plastic strain increments are directed normal to the stresses defining the yield surface, not to the stress increment that initiates the yielding. In a slightly more sophisticated explanation, Calladine (1969) shows that normality is a way for a material to maximize the energy absorbed during yielding.
With normality, the principal stresses and principal strain increment directions are aligned, and the net strain increment vector is normal to the yield surface at the stress state corresponding to the present yielding. This is illustrated in Figure 3.2, which plots