Techniques of thermomechanics, based upon the use of internal variables, are used to develop a systematic procedure for deriving elastic/plastic models of soils and granular materials. Fundamental thermodynamic state variables are defined and used to formulate free energy and dissipation potentials. These are used to determine yield conditions and flow rules. It is demonstrated that it is necessary to distinguish between plastic work and plastic energy dissipation. It is suggested that the stored energy associated with the plastic deformations is due to the fact that only a proportion of the intergrain contacts are actually plastic in a plastically deforming continuum element. The stored plastic energy of the continuum model arises from lockedin elastic energy on the micro scale. Some wellknown existing criticalstate models are reexamined and some of their shortcomings are highlighted. New models are proposed that overcome some of these objections. These models are able to predict nonassociated flow rules, contractive behaviour and prepeak failure for ‘loose’ soils, and aspects of static liquefaction, and can predict the position of the failure, phase change, instability and ultimate state lines. In some extreme cases the yield surfaces are found to contain concave segments, and dilatant behaviour can occur below the critical (characteristic) state line.
There are a variety of ways of constructing mathematical models of the mechanical behaviour of soils. One is to curvefit experimental data and formulate the basic equations needed to describe the model in terms of these empirically derived functions. However, even in this empirical approach, it is necessary to have some underlying theoretical framework that enables the model to have the necessary predictive qualities. The welldeveloped theory of rateindependent elastic/plastic materials is by far the most popular such background framework currently in use, and the theory developed by Lade and coworkers is one of the best known of such empirically based models (e.g. Lade, 1975).
A second approach to modelling is to start with some specific theoretical assumptions, such as a dilatancy law, an expression for ‘plastic work’ or ‘energy dissipation’, as in the original Cam clay studies (Schofield & Wroth, 1968; Gens & Potts, 1988; Wood, 1990), or to make some other micromechanical assumptions, as in the recently developed models involving particle crushing by McDowell & Bolton (1998) and McDowell (2000). These Cam clay, criticalstate models and the vast hierarchy of extensions are excellent examples of this approach to modelling. Because these theories are based on conceptual models of macro or micro behaviour, such models are capable of giving deep physical insights into the engineering behaviour of soils, although quite frequently at the expense of detailed numerical accuracy. The pedagogical merits of this approach are elegantly argued in Wood (2000). The choice of complexity of the model depends on the practical purpose underlying the model construction. There are now some very complex models available, involving large numbers of material parameters, but, as pointed out by Kolymbas (2000), such models are seldom ‘transportable’ from one research group to another.
A basic requirement of all such models is that they satisfy the basic laws of physics. For example, any equations describing the properties of isotropic materials must be expressed in terms of the invariants of stress, strain, strain increment, etc. The second law of thermodynamics is one of these basic laws that govern the dissipative behaviour of materials, but it is seldom invoked in geomechanical theories. Until recently thermodynamics and ‘heat flow’ were seldom seen as relevant to geotechnical problems. A noticeable exception is the overview paper by Mitchell (1991). However, this situation is now changing, particularly as a result of the development of geoenvironmental engineering: see Smith (2000) for a recent review of some of the application areas. In the early days of the development of the theory of elastic/plastic materials, quasithermodynamic postulates were introduced in an effort to ensure that the dissipation of energy was always positive in a closed cycle of stress (Drucker's postulate) or strain (Il'iushin's postulate).
However, it was soon realised that these postulates were actually classifications of types of material behaviour and not, in any sense, equivalent to the second law (Drucker, 1988; Lubliner, 1990).
In the last 20 years, however, there have been major developments in the theory of thermomechanics of continua, which have thrown new light on many longstanding issues in a number of application areas. In the case of isothermal deformations of rateindependent elastic/plastic solids, the new theories have shown that, under certain minimal assumptions, knowledge of the material's free energy and dissipation rate potentials is sufficient to uniquely determine the elasticity law, the yield function, the flow rule and the hardening rules (both isotropic and kinematic) of the material. The simultaneous determination of both the yield function and the plastic potential is particularly significant, since, in geomechanics, these two functions are normally chosen independently. (A notable exception to this is the modelling process used by Chandler (1985, 1988) and Chandler & Song (1990), whose approach using the theory of envelopes has several similarities to the more general thermomechanical procedure.)
The early developments of these general ideas were due to Ziegler (1983) and Ziegler & Wehrli (1987), who noted some applications of their general theory to the classical Coulomb model. A rigorous general theory was then developed by a number of researchers in France, accounts of which can be found in the books by Maugin (1992, 1999), Maugin et al. (2000), Besseling & van der Giessen (1994) and Lemaitre & Chaboche (1990). Applications of these ideas to soil mechanics were pioneered by Houlsby (1981, 1982, 1993). A comprehensive analysis of the isothermal thermomechanics of geomaterials was given by Collins & Houlsby (1997), who demonstrated that a nonassociated flow rule is a necessary property of a ‘frictional material’, in which the plastic deformations are governed by stress ratios rather than by the magnitudes of certain yield stresses. Houlsby & Puzrin (2000) generalised some aspects of this work to nonisothermal conditions, and in a series of papers have developed a family of sophisticated models, based on the use of internal functions (e.g. Puzrin & Houlsby, 2001b, 2001c). These authors have proposed the term ‘hyperplasticity’ to describe theories in which the irreversible plastic behaviour is determined by a dissipation potential function, by analogy with the wellestablished term ‘hyperelasticity’ used in the analogous situation in elasticity theory.
The object of this paper is to present a systematic procedure for establishing a hierarchy of models, starting from the familiar Cam clay models, which include internal friction, nonassociated flow rules and a variety of biased volumetric/shear hardening laws. These models have the advantage of automatically satisfying the second law of thermodynamics. This is not as trivial an exercise as may at first be thought, since, as will be seen, the wellknown original Cam clay model actually violates the second law, even though it satisfies Drucker's postulate. It will also be shown that some models with significant amounts of internal friction actually have yield surfaces parts of which are concave, a property that violates Drucker's postulate but not the second law.
In the next section the basic thermodynamic concepts of state and state variable are discussed in a geomechanics context. The following section is a brief outline of the procedure for constructing the yield function and flow rule of the material from a known dissipation function. A key step in this procedure is the use of the intermediate dissipative stress space. The wellknown elementary models for isotropic compression, Coulomb friction and original and modified Cam clay are then reviewed in the light of this new procedure. This review suggests ways in which the original dissipation functions for these simple models can be modified to include internal friction and shear hardening. Some illustrative examples are given in the final section.
There are several differences between the fundamental properties of metals and soils, which require a number of modifications to be made to the standard elastic/plastic theories for metals, if they are to be extended to soils. Here we concentrate on two of these factors:
A soil does not have a ‘natural state’ to which it returns when all the stresses are removed. Here we shall adopt the convention of introducing an effective pressure, p′_{R}, which is the pressure in an arbitrarily chosen reference state.
Since a soil is a twophase material, any description of the ‘state’ of a soil must include some volume fraction parameter, such as voids ratio, e, or specific volume, υ. However, knowledge of υ alone is not sufficient to determine the state uniquely. Another state variable is needed. This is frequently taken to be the effective pressure, so that a state is defined by knowledge of both υ and p′ – the coordinates of a point in the standard isotropic compression diagram.
When constructing thermomechanical descriptions of states, it is usual to start with ‘observable’ quantities such as volumes or strains, which are directly measurable geometric quantities, as the fundamental, independent state variables. Other state variables such as pressure and stress, which describe ‘what is being done to the specimen’, are viewed, at least initially, as the dependent state variables. These variables are frequently interchanged in the subsequent theory development, the general theory of which is most elegantly expressed in terms of Legendre transformations (Callen, 1960; Collins, 1996; Wilmanski, 1998). There are, however, rules governing these interchanges. Most importantly, it is impossible to have a geometric variable and its work conjugate force variable as a pair of independent state variables. By work conjugate variables we mean those variables whose product gives the work done. For example, the product of pressure and volume strain increment give the work done, so that pressure and volume strain are conjugate variables, as indeed are p′ and υ, since p′δυ is also the work increment, under the standard assumption that the solid phase of the material is incompressible. These work conjugate variables are related through a constitutive equation, or equation of state, and cannot both be chosen as the primary independent state variables.
In a thermomechanical formulation of geomaterials we cannot therefore choose υ and p′ as our initial independent state variables. In the standard development of isotropic compression of soils p′ is frequently replaced by p_{c}, the normal consolidation pressure, and υ and p_{c} are taken as the fundamental pair of state variables. The variables in this formulation are still mixed, however. A completely geometric description requires us to find a geometric parameter that defines the position of the elastic swelling line in place of p_{c}. Iwan & Chelvakumar (1988) use the voids ratio corresponding to p_{c} in their strainspace formulation of clays. Here, however, we prefer to follow Hashiguchi (1995) and introduce the reference, plastic specific volume, υ^{p}, defined to be the specific volume of the sample, attained when it is unloaded from its current state to the reference pressure, p′_{R}. This is an example of an internal variable, as it is ‘observable’, but not ‘controllable’. We hence take υ and υ^{p} as the defining independent volume fraction variables. This is in line with modern thermomechanical developments of plasticity theory, where the total and plastic strains are taken as the defining state variables.
The relationship between the true stresses p′ and q and the dissipative stresses π′ and τ depends on the form of the freeenergy function. It follows from equations (9) and (10) that if the free energy function depends only on the elastic strains—that is, just on the difference between the total and plastic strains—then the dissipative and true stresses are identical. The yield surface and flow rule can hence be readily transferred to true stress space. However, Collins & Houlsby (1997) showed that in the special case of a frictional material, where the dissipation function and hence the yield function in (π′, τ) space also depend explicitly on p′, the normality property of the flow rule is lost when it is transferred to true stress space.
Once the yield function and flow rule have been determined, the incremental form of the constitutive equations needed for modelling drained and undrained stress and strain paths can be found by the wellestablished procedures. The yield condition is differentiated to give the consistency equation. The flow rule is then used to determine the hardening modulus and the magnitude of the strain increments and their relation to the stress increments (Wood, 1990; Lubliner, 1990). An overview of the structure of this theory, indicating the steps needed to deduce the form of the yield function, flow rule, incremental form etc. from the free energy and dissipation functions, is given in Fig. 2.
Here we examine a number of the standard one and two dimensional models in the light of the above thermomechanical considerations. We shall consider only cohesionless materials, so that the effective pressure is always positive.
The presence of this shift stress in the modified Cam clay model was noted by Houlsby (1981) and Collins & Houlsby (1997). Here it has been shown to be a natural part of any model with different yield stresses in isotropic tension and compression. (It is not, however, a feature of the original Cam clay model, as was also noted in these two references. This apparent contradiction will be resolved below.) If the model is desired to have a nonzero tensile strength, t_{s} say, the shift stress must be chosen to be ρ′ = π_{c} − t_{s}, in which case p_{c} = 2π_{c} − t_{s}.
As will now be seen, the modified Cam clay model is not open to these objections. However, in this modification the frictional mechanisms of energy dissipation, as embodied in the original model, are abandoned. This was not of course the intention of its originators, Roscoe & Burland (1968), and the correct relationship between a dissipation function and the yield condition and flow rule had not been developed at that time. It will be argued that a more natural generalisation of the original Cam clay model, which preserves the frictional mechanism for energy dissipation, is the new ‘alpha’ model described below.
A feature of the above models is that, in the dissipative stress plane, the criticalstate line, defined to be the line on which the volumetric plastic strain increment is zero, is always a segment of the τaxis (Figs 3 and 4). This is because of the symmetry of the yield loci and the fact that the flow rule is always associated in this plane. When we transform to the true stress plane, the criticalstate line is now q = Mp′. In the case of modified Cam clay, this transformation is accomplished by the addition of the shift stress ½p_{c}, which varies linearly with distance along the τaxis. However, for the linear frictional model this same transformation is accomplished, as a result of the fact that the yield condition in (π′, τ) space involves p′ as a parameter. As a result the yield surface is ‘sheared’ when π′ and are identified with p′ and q. In addition the flow rule is no longer associated. This immediately suggests a procedure for generalising the modified Cam clay model to one involving nonassociated flow rules, by superposing these two types of model.
This model has much in common with those proposed by Nova & Wood (1979) and Chandler (1985). Two primary mechanisms of producing plastic deformations are envisaged
compaction, dominating at low stress ratios, involving the deformation of contact bonds between the particles and eventually bond fracture and particle crushing
particle rearrangement, which dominates at high stress ratios, resulting from the sliding and rolling of the soil particles.
In both these papers yield surfaces and flow rules were proposed for both regimes. However, these surfaces had to be joined together in an artificial manner at a certain transitional stress ratio. One advantage of the present approach is that the presence of the shift stress, which has emerged naturally from the thermomechanical formulation, gives rise to a model in which this transition is continuous. The relative importance of the two mechanisms is determined by the value of the parameter α.
From the flow rule (equation (35)) we observe that, in the ‘dense region’ where p′ <, p_{c}/2, the plastic volume strains are dilatant, even though the yield locus lies below the criticalstate line. In this model therefore it is possible to have plastic dilation below the criticalstate line. As the value of α is increased, at least part of the yield curve in this ‘dense region’, p′ ≤ p_{c}/2, lies above the criticalstate line. For values of α in the interval 0 to 0·172 there are two such arcs, necessarily convex, separated by a concave arc, which still lies below the criticalstate line. For higher values of α the whole of the yield locus in the ‘dense’ region, lies above q = Mp′, as in the classical models.
Before discussing the predicted form of the drained and undrained stress paths, we discuss a modification to the classical volumetric hardening model, which includes the effect of shear strains.
The main object of this paper has been to demonstrate a procedure for model construction, based upon the laws and techniques of thermomechanics. The alpha and beta models have been introduced to illustrate this general procedure. It is appreciated that, to be useful, these models must be fully validated against experimental data, and the results of such comparisons will be discussed elsewhere. There is a particular difficulty in trying to use these types of model for sands, owing to the problematic nature of the normal consolidation line, as discussed in Pestana & Whittle (1995) and Jefferies & Been (2000) for example. For sands it may well be preferable to refer the thermomechanical state parameters to the criticalstate line, as in the analyses of Been & Jefferies (1985).
Although modern developments in thermomechanics have had a large influence on many branches of mechanics, this is not yet true of geomechanics. Perhaps, in part at least, this is due to the commonly held view that nonassociated flow rules are not covered by modern thermomechanics theories. However, Collins & Houlsby (1997) showed that this was not true, and that such flow rules arise naturally for frictional materials, where the plastic dissipation depends on the effective pressure. The main achievements and conclusions of the present analysis are as follows:
A set of thermodynamically consistent state variables, which describe the state of the soil, has been developed. In particular it has proved helpful to define a reference, plastic specific volume, by analogy with the plastic strain, which is used as a key internal state variable in modern plasticity theories.
The construction of the plasticity models from the free energy and dissipation functions is a twostage process. The yield condition and plastic potential are first constructed in dissipative stress space, where the flow rule is always associated. The yield condition and flow rule are then constructed in true stress space, either by using a shift stress and/or by using the implicit dependence of these functions on the true stress variables, such as the effective pressure. It is important to distinguish between plastic work and plastic energy dissipation. Not all the plastic work is dissipated. The stored plastic energy gives rise to these shift stresses. These arise naturally when modelling isotropic compression with singlesurface models. This stored energy can be interpreted on the micro scale as lockedin elastic energy, which can accompany the macrolevel plastic deformations.
The twostep process has been used to analyse a number of the basic extant models, including Drucker–Prager and the original and modified Cam clay models. As a result of the understanding obtained from these analyses, a family of new models has been proposed. These incorporate some of the observed features of real soils and granular materials, not predicted by the classical models, such as nonassociated flow behaviour at high stress ratios, contractive behaviour at low stress levels, static liquefaction and instability. Yet these models maintain much of the familiar, simple structure of the classical criticalstate theories. They also demonstrate that concave yield surfaces are possible at low stress levels.
There are a large number of models, which have extended the classical criticalstate theories in a number of directions, available in the literature. These include CANA Sand (Poorooshasb, 1994), NOR Sand (Jefferies, 1993), Superior Sand (Boukpeti & Drescher, 1999), Severn–Trent Sand (Gajo & Muir Wood, 1999), CASM (Yu, 1998), and other ‘anonymous’ models due to Pender (1978), Krenk (2000) and Manzari & Dafalias (1997) etc. The special feature of the present approach is that the models are developed using ideas of modern internal variable thermomechanics, and are based on the fundamental physical concepts of work, energy and dissipation. In many ways this procedure is closest to the original ideas of criticalstate soil mechanics.
For simplicity and relevance, this paper has dealt only with the formulation of these models for the analysis of triaxial tests. It is not difficult to generalise the general theory to fully general threedimensional situations. The main limiting factor, however, is that the algebra of the relationship between the yield function and the dissipation function for specific models can become prohibitive, once one departs from the simple quadratic form of the yield locus. We have also been concerned only with singlesurface models. More realistic models using bounding surfaces, multiple yield surfaces, or subloading surfaces can be analysed in a similar fashion. They require the introduction of multiple internal variables. The internal function models of Puzrin & Houlsby (2001b, 2001c) represent one such generalisation.
ACKNOWLEDGEMENTS
The authors are grateful to Koichi Hashiguchi, Guy Houlsby, Poul Lade, Mick Pender, Andrew Schofield and David Smith for many helpful discussions on the fundamental problems of modelling soil behaviour related to this study.

e  voids ratio 
e_{1}, e_{2}, e_{3}  principal components of logarithmic, finite, strain tensor 
total, elastic and plastic, volumetric, finite, logarithmic strains  
total, elastic and plastic, finite shear strains  
L  length of test specimen 
M  slope of criticalstate line 
N  constant determining position of normal consolidation line 
p′  effective pressure 
p_{c}  normal consolidation pressure 
p′_{o}  initial effective pressure 
p′_{R}  reference effective pressure 
q  shear stress invariant 
R  radius of test specimen 
u  pore pressure 
U  internal energy 
V  volume of test specimen 
V_{s}  volume of solid phase 
V^{p}  plastic volume of test specimen 
υ  specific volume 
υ^{p}  reference, plastic specific volume 
δW  work increment 
α  parameter in nonassociated model 
β  parameter in weighted workhardening model 
γ  = λ − κ 
δ^{p}  = 
η  = q/ p′, stress ratio 
η  = q/Mp′ 
κ  slope of elastic loading/unloading line 
λ  slope of normal consolidation line 
Λ  = (L/R)^{2/3} aspect ratio 
δμ  scale parameter in flow rule 
π′  effective, dissipative pressure 
π_{c}  normal consolidation pressure in dissipative stress plane 
σ′_{1}, σ′_{2}, σ′_{3},  principal effective stresses 
τ  shear stress invariant of dissipative stress 
Φ  rate of dissipation function 
Ψ  free energy function 
Ω  = 1 − (1 − 2β)η^{2} 
ω  = 1 − η^{2} ≡ 1 − (q/ Mp′)^{2} 
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