"Everything should be made as simple as possible, but not simpler."(Albert Einstein)

KEYWORDS : Constitutive theory, Elastic-plastic, Strain localization, Parameter calibration, In-situ soil tests, Soil liquefaction, Stochastic finite elements

I NT RODUCT I ON

Recent advances in digital computer technology and in numerical techniques such as the finite difference or the finite element methods, have rendered possible, at least in principle, the solution of any properly posed boundary value problems in soil mechanics. Further progress in expanding analytical capabilities in geomechanics depends upon consistent mathematical formulations of generally valid and realistic material constitutive relations. An increasing effort has thus been devoted since the 1960's to a more comprehensive description of soil behavior. Numerous formulations have been proposed in the soil mechanics literature. All rely on a better knowledge and understanding of mechanics in general, and continuum mechanics in particular, than has been common in traditional soil mechanics training. The results and progress in the field of constitutive relations have thus until recently been mostly ignored by the mainstream of soil engineering. However, recent progress and honest validation exercises have instilled confidence and finally attracted the attention of the practice. We should therefore see in the future more impact on the practice of this area of soil mechanics.

It is the purpose of this paper to provide an overview of soil constitutive models and related issues. The review is brief and makes no attempt to be exhaustive. In recent years, the

Constitutive Relations for Soil Materials

Jean H. Prevost

and

Radu PopescuDepartment of Civil Engineering and Operations Research

Princeton University, Princeton, New Jersey, USA

growing interest in constitutive relations has led to a number of conferences devoted exclusively to theoretical, experimental, numerical implementations and application problems associated with this field. It would have been impossible to record all the papers and discuss all the models that have been proposed. Also, the reader is referred to the extensive review paper by Scott (1985) for a historical review and discussion of constitutive theories.

PREAMBLE

Soils consist of an assemblage of particles with different sizes and shapes which form a skeleton whose voids are filled with water and air or gas. The word ``soil'' therefore implies a mixture of assorted mineral grains with various fluids. Hence, soil in general must be looked at as a one (dry soil) or two (saturated soil) or multiphase (partially saturated soil) material whose state is to be described by the stresses and displacements (velocities) within each phase. There are still great uncertainties on how to deal analytically with partly saturated soils. The stresses carried by the soil skeleton are conventionally called ``effective stresses'' in the soil mechanics literature (see e.g., Terzaghi (1943)), and those in the fluid phase are called the ``pore fluid pressures.''

In a saturated soil, when free drainage conditions prevail, the steady state pore-fluid pressures depend only on the hydraulic conditions and are independent of the soil skeleton response to external loads. Therefore, in that case, a single phase continuum description of soil behavior is certainly adequate. Similarly, a single phase description is also adequate when no drainage (e.g., no flow) conditions prevail. However, in intermediate cases in which some flow can take place, there is an interaction between the skeleton strains and the pore-fluid flow. The solution of these problems requires that soil behavior be analyzed by incorporating the effects of the transient flow of the pore-fluid through the voids, and therefore requires that a two phase continuum formulation be available for porous media. Such a theory was first developed by Biot (1955, 1956, 1957, 1972, 1977, 1978) for an elastic porous medium. An extension of Biot's theory into the non-linear inelastic range (see e.g., Prevost (1980)) is necessary in order to analyze the transient response of soil deposits. This extension has acquired considerable importance in recent years due to the increased concern with the dynamic behavior of saturated soil deposits and associated liquefaction of saturated sand deposits under seismic loading conditions. For that purpose, soil is viewed as a multi-phase medium and the modern theories of mixtures developed by Green and Naghdi (1965), and Eringen and Ingram (1967), are used. General mixture results can be shown through formal linearization of the field and constitutive equations, to reduce to Biot's linear porelastic model (see e.g., Bowen (1982)).

During deformations, the solid particles which form the soil skeleton undergo irreversible motions such as slips at grain boundaries, creations of voids by particles coming out of a packed configuration, and combinations of such irreversible motions. When the particulate nature and the microscopic origin on the phenomena involved are not sought, phenomenological equations are used to provide a description of the behavior of the various phases which form the soil medium. In multiphase theories, the conceptual model is thus one in which each phase (or constituent) enters through its averaged properties obtained as if the particles were smeared out in space. In other words, the particulate nature of the constituent is described in terms of phenomenological laws as the particles behave collectively as a continuum. Soil is thus viewed as consisting of a solid skeleton interacting with the pore fluids.

Microstructural Aspects

The particulate nature of soil materials is directly responsible for their complex overall behavior. Sands consists of an aggregate of particles with different sizes and shapes which interact with each other through contact forces (both normal and tangential) at the points of contact. Considering the particles essentially incompressible, deformation of the granular assembly occurs as the particles translate, slip and/or roll, and either form or break contacts with neighboring particles to define a new microstructure. The result is an uneven distribution of contact forces and particle densities that manifests in the form of complex overall material behavior such as permanent deformation, anisotropy and localized instabilities (Deresiewicz (1958), Oda (1972), Oda and Konishi (1974), Vardoulakis (1988)). Similarly, clays are composed of plate-shaped particles of clay mineral. Each plate is subject to gravity forces and electrostatic forces at the points of contact, which hold the particles together. Therefore, if one knew all about the particles' geometry (their shapes and contacts) and understood all the physics and mechanics of interparticle contacts, then in principle one should be able to predict the overall macroscopic response of the assembly. Such studies could lead to a rational explanation of the observed macroscopic behavior, and allow direct correlation between average macroscopic constitutive parameter values and microscopic entities (e.g., relate the overall friction angle to the individual particle-to-particle friction coefficient). A number of studies have been initiated along these lines. Several experimental (see e.g., Drescher and De Josselin de Jong (1972), Oda et al. (1982), Subhash et al (1991)), analytical (see e.g., Walton (1988), Goddard (1990)) and numerical models (see e.g., Cundall and Strack (1979), Scott and Craig (1980), Bashir and Goddard (1991), Chang et al. (1992), Ting et al. (1993), Thornton and Sun (1993, 1994), Hogue and Newland (1994), Zhuang et al. (1995), Wren and Borja (1995), Borja and Wren (1995)), have been proposed to study the effect of particle-to-particle interaction on the overall material response. Because of their oversimplifications, these studies have had, as yet, little or no impact on the relations used in current constitutive models. Eventually, this may change as the models are further defined.

The most popular numerical models, commonly called Distinct Element or Discrete Element Methods, derive from the pioneering work of P.A. Cundall (Cundall (1971), Cundall and Strack (1979)) or are adopted from the procedures used in numerical Molecular Dynamics for which the book by Allen and Tildesley (1987) is the standard reference. They consist in approximating the mechanical constraint of non inter-penetrability of particles (see e.g. Keller (1986)) by some close-range steep repulsion law. Every time interval for which two particles are close enough to be viewed as contacting (numerically, they usually are allowed to overlap a little), they are assumed to exert on each other some dissipative forces which mimic friction. Thus, the evolution problem is reduced to the integration of a system of second order differential equations, to which classical methods are applied. The steeper the approximate laws of interaction, the more realistic are the results, at the price of reducing the time-step length to ensure numerical stability. Some researchers (see e.g., Bashir and Goddard (1991), Borja and Wren (1995)) have recently advocated the use of semi-implicit procedures to avoid full integration of Newtonian dynamical equations when computing quasi-static responses of granular systems, at the price of having to approximate and/or neglect the complicated intermediate path-dependent phenomena involved when contact and friction occur between particles.

Macroscopic Aspects

Although very complex when examined on the microscale, soils as many other materials may be idealized at the macroscale as behaving like continua. At the macroscale, the various phenomena associated with the discrete soil entities such as sand grains, clay platelets, etc. ..., are integrated and averaged to the level of a homogeneous continuum model. In the macroscopic field, the averaging volume represents and characterizes a physical point.

Because the averaging volume is macroscopically infinitesimal, it is usually denoted by dV. The characteristic length D of the averaging volume is selected such that

(Whitaker, (1969)) where l is the microscopic scale of the porous medium and L is the scale of gross inhomogeneities. Typically: in sands, and in clays, whereas

. Paradoxically, despite the large differences in the nature and structure of materials such as metals and alloys, polymers and composites, concrete, soils, there is a great unity displayed in their macroscopic behavior. With different orders of magnitude, terms like elasticity, viscosity, plasticity, hardening, softening, brittleness and ductility can be applied to all these materials. Therefore, while still recognizing the underlying particulate nature of these materials, instead of studying them as discrete systems, it is more convenient to consider them as continua, and use concepts from continuum mechanics, thermodynamics and rheology to analyze and model their behavior. The phenomenological behavior of the material is therefore the standard of reference in this approach, and it is the one adopted in modern soil constitutive theory.

The constitutive equations are formulated for a material point subject to a homogeneous state of stress and strain. The prototype is frequently taken to be a soil sample in a carefully performed triaxial or cube test with perfectly lubricated shear stress-free top and bottom (and lateral if any) boundaries. The first attempt to rationalize the behavior observed in laboratory soil tests was done by the Cambridge group (Roscoe, Schofield and Wroth (1958)) early in the 1960's. The work was performed mostly on clays but several of the concepts they developed found applications to sands. The developments of the Cambridge model included the ``critical state'' theory and the ``Camclay'' model (Schofield and Wroth (1968), Roscoe and Burland (1968)). The Camclay model was the first (and simplest) modern elasto-plastic constitutive soil model. The model had been formulated in the p-q plane, using triaxial soil test data, where p = hydrostatic axis and q = shearing or deviatoric axis. Its generalization to stress states other than that of the triaxial apparatus, assumed that the yield and failure conditions in p-q plane could be rotated around the p-axis, so that in effect, the model corresponds to the extended von Mises failure surface originally proposed by Drucker and Prager (1952), with a cap closing it on the hydrostatic axis as originally suggested earlier by Drucker, Gibson and Henkel (1955), to account for permanent volumetric deformations observed under hydrostatic stress loading conditions. The Cambridge model could account for such experimental observations as: (1) permanent volumetric deformations occur under hydrostatic loading conditions; (2) there is a coupling between volumetric changes and changes in shear stress; and (3) dense soils expand in volume during pure shear whereas loose soils contract. The most obvious limitations of the model are: (1) it does not adequately model structural and stress-induced anisotropy; (2) it is not applicable to cyclic shear loading conditions where it is observed that the stress-strain response is highly nonlinear with hysteresis loops of different proportions depending on the extent of the unloading-reloading; (3) it does not reflect the strong dependency of the shear dilatancy on the effective stress ratio as observed mostly in cohesionless soils (see e.g., Rowe (1962), Luong (1980), Luong and Touati (1983)), but also in cohesive soils (see e.g., Hicker (1985)); and (4) it does not account for the viscous time-dependent stress-strain response of cohesive soils. The Camclay model has been applied to some simple boundary-value problems(see e.g., Carter et al. (1979)). Since the late 1960's more elaborate constitutive models have been constructed to remove some of the limitations of the original model.

CLAS S ES OF CONS T I T UT I VE MODELS

Considerable attention has been given since the late 1960's to the development of constitutive

equations for soil media, but although many different models have been proposed, there is not yet firm agreement among researchers. Further, many of the constitutive models presented seem unnecessarily arbitrary, and at this stage it is important to emphasize that, to be satisfactory, a material model idealization should possess the following necessary properties:

1. The model should be complete, i.e. able to make statements about the material behavior for all stress and strain paths, and not merely restricted to a single class of paths (e.g., axial symmetry or pure shear);

2. It should be possible to identify the model parameters by means of a small number of standard, or simple material tests;\

3. The model should be founded on some physical interpretation of the ways in which the material is responding to changes in applied stress or strain (e.g., the material should not be modeled as elastic if permanent deformations are observed upon unloading).

The first property is clearly essential if the model is to be of practical application, and the second property is very desirable. The third property linked with the other two helps to ensure that time is occupied on developing a useful model, and not merely on an elaborate curve-fitting exercise of limited application - of application in fact, no wider than the data that are being fitted. Finally, it must be added that a material model may only be deemed to be satisfactory when with its aid, it is possible first to determine the stress-strain-strength behavior of the material at hand in one piece of testing equipment (e.g., in triaxial tests), and then to predict the observed behavior of the same material in some other type of testing equipment (e.g., in simple shear tests).

Elastic (see e.g., Duncan and Chang (1970); Coon and Evans (1971)), endochronic (see e.g., Valanis and Read (1982); Bazant and Krizek (1976)), micromechanical and many elastic-plastic models with various degrees of sophistication and/or complexity have been proposed. Elastic(-visco)-plastic models appear to be the most promising. Although, experience has shown that simple nonlinear elastic stress-strain models like the hyperbolic model are not capable of modeling fundamental aspects of real soil behavior, they still remain popular in practice. Also, many ``empirical'' models (e.g., Finn et al. (1977)) which rely on analytical relations based on experience and/or experimental observations between quantities of direct interest (e.g., rate of pore-pressure build-up in a cyclic test) without going through a rigorous framework of constitutive formulation (e.g., elastoplasticity) are still used widely. These models are not discussed hereafter.

Elastic(-Visco)-Plastic Models

The most popular and widely used soil models are cap models (Roscoe and Burland (1968), Schofield and Wroth (1968), DiMaggio and Sandler (1971), Sandler et al. (1976), Baladi and Rohani (1979)) based on classical isotropic plasticity theory, and are variations and refinements of the basic Cap model pioneered by Drucker, Gibson and Henkel (1955). The most obvious limitations of these Cap models are:

1. they do not adequately model stress-induced anisotropy;

2. they are not applicable to cyclic loading conditions.

Similar limitations apply to other elastoplastic models based on isotropic plasticity (e.g., Lade (1977), Pender (1978), Nemat-Nasser (1984)). In most of these models, the material behavior is assumed to remain unchanged as the stress state rotates around the hydrostatic stress axis. This is known to be incorrect (especially for sands), and a yield surface which more closely

resembles the Mohr-Coulomb condition is used in Lade and Duncan (1975) and Lade (1977). A yield surface which very closely approximates the Mohr-Coulomb criterion was presented by Matsuoka (1974).

It may be argued that plastic models based on isotropic plastic hardening rules are adequate for situations in which only loading (and moderate unloading) occurs, however, it is unlikely that such restrictions can be met at every point in general boundary value problems. In order to account for hysteretic effects, more elaborate plastic models based on a combination of isotropic (Hill (1950)) and kinematic (Prager (1959)) plastic hardening rules have been proposed.

An important theoretical development in plasticity was made simultaneously by Mroz (1967) and Iwan (1967). They showed how continuous yielding could be represented by a set of nested yield surfaces in stress space. The notion, in combination with kinematic and isotropic hardening/softening plastic rules, can give rise to a material representation of considerable power and flexibility. The concept was adopted and enlarged upon by Prevost (1977, 1978, 1985) and Mroz (1980), while the concept of a material behavior dependent on the distance from a yield or ``bounding'' surface was constructed initially by Dafalias (1975, 1977, 1980, and 1986), and later modified and/or enlarged by many others (e.g., Aboim and Roth (1982), Ghaboussi and Momen (1982), Mroz and Pietruszczak (1983a, 1983b), Bardet (1986, 1990), Crouch and Wolf (1994)). Both theories suffer inherent limitations namely: storage requirements for the multi-surface theory, ``a priori'' selection of an evolution law and arbitrariness in the mapping rule for the bounding surface theory. This is further discussed in Prevost (1982).

In Zienkiewicz and Mroz (1984), a generalization of classical plasticity theory is proposed. The generalization consists of specifying the occurrence of plastic deformation for a stress rate which points in either direction with respect to a loading direction in stress-space. The concept of a loading direction implies the existence of a loading surface. Further, continuity requirements make the loading surface the locus of neutral loading. The concept has been found convenient for the description of reverse and cyclic loading (see e.g., Aboim and Roth (1982), Zienkiewicz et al. (1985) Pastor et al. (1985 and 1990), Hirai (1987)), but has not been confirmed experimentally.

Rate-dependency (viscous) effects in cohesive materials have been incorporated in elastic-plastic models using mostly hypoelasticity theory (Green (1956)) as in Davis and Mullinger (1978). Other formulations based on Perzyna's (1963, 1966 and 1971) viscoplasticity have also been used (see e.g.,. Adachi and Oda (1982), and Katona (1984)).

Micromechanically Based Models

These models are constructed by superposing or integrating the response of smaller units, either micromechanical or simply mechanisms of yielding in particular stress sub-spaces. Often, concepts of plasticity are stated at the level of the postulated micromechanism in order to characterize its kinetics. The most popular model in this category was first proposed by Aubry et al. (1982) and later refined by several others (see e.g., Pande and Sharma (1983), Matsuoka and Sakakibara (1987), Kabilamany and Ishihara (1990 and 1991)). As in the original Mohr-Coulomb soil model (see e.g. Coulomb (1972)), this model uses the concept of mobilized friction angle to define yielding. However, in contrast with the original Mohr-Coulomb model, it recognizes that there are usually three distinct Mohr circles in a three-dimensional stress-state, which are used to define their mobilized friction angles. Each mobilized friction angle is then used to define a yielding mechanism. In stress space, these yielding mechanisms define an elastic domain whose boundary is in general non smooth and

possesses corners (although each yield function is assumed to be smooth). An extension of classical plasticity due to Koiter (1960) and Mandel (1965) to accommodate non smooth yield surfaces is then used to compute the resulting plastic flow. The numerical implementation of such models is rather delicate as discussed and presented in Simo et al. (1987), and Prevost and Keane (1990).

More truly micromechanically based models have also been proposed. In these models soil is viewed as an assemblage of particles, and the unit micro-mechanism response is defined at the truly micromechanical level of contact forces with rolling and sliding kinematics among the particles, and given macroscopic counter parts by proper definitions and averaging procedures. Such models have been presented in Nemat-Nasser and Mehrabadi (1984), Oda et al. (1980), Christoffersen et al. (1981), Mehrabadi et al. (1993), Iai (1993).

I NT EGRAT I ON ALGORI T HMS

Plasticity theory has recently gained widespread acceptance in large-scale numerical simulations of practical geotechnical engineering problems, due to its extreme versatility and accuracy in modeling real engineering materials behavior. Building upon pioneering works of Drucker and Prager (1952) on soil plasticity, the modern trend has been toward the development of more and more elaborate and complicated elastoplastic constitutive models which resemble the behavior of real engineering materials more closely.

The numerical solution of elastic-plastic boundary value problems is based on an iterative solution of the discretized momentum balance equations. Typically, for every load/time step, solution involves the following steps: Give a converged configuration at step n:

1. The discretized momentum equations are used to compute a new configuration for step (n+1) via an incremental motion which is used to compute at every stress point incremental strains ;

2. At every stress point, for the given incremental strains , new values of the state variables ( ) and are obtained by integration of the local

constitutive equations;

3. From the new computed stresses, balance of momentum is checked and if violated iterations are performed by returning to step 1.

In this section, attention is focused on step 2 which may be regarded as the central problem of computational plasticity since it is the main role played by the constitutive equations in the computations. In finite difference / finite element computer codes the elastoplastic constitutive equations are usually incorporated through a separate set of constitutive subroutines. The purpose of these subroutines is the integration of the elastic-plastic constitutive equations. That is, at every stress point, given a deformation history, the role of the constitutive-equation subroutine is to return the corresponding stress history. Exact analytical solutions for the elastic-plastic evolution problem are available only for the simplest elastic-plastic models. The first exact solution was obtained by Krieg and Krieg (1977) for the case of the isotropic elastic-perfectly plastic von Mises model. Later, Yoder and Whirley (1984) extended the solution to apply to the von Mises model with arbitrary combination of kinematic and isotropic hardening. Later, Loret and Prevost (1986) developed an exact solution for the isotropic Drucker-Prager model with linear hardening and arbitrary degree of non-associativity. Although error-free, these solutions are computationally too slow to be used routinely in actual calculations. Further, exact analytical solutions are not available

for more complex models. Therefore, all elastic-plastic models are implemented in analysis programs with some error, via an integration algorithm called the stress-point algorithm. Evidently, the accuracy and stability of the global solutions is to be strongly affected by the accuracy and stability of the stress-point algorithm. Also, the cost of the analysis is most strongly affected by the efficiency of the stress-point algorithm. The best algorithm, the one to be favored, is therefore the one which combines computational efficiency with accuracy.

The first stress-point algorithm to be developed was the radial return algorithm proposed by Wilkins (1964) for the elastic-perfectly plastic von Mises model. The algorithm was subsequently extended by Krieg and Key (1976) to accommodate isotropic and kinematic hardening laws. The algorithms are analyzed in Krieg and Krieg (1977); Schreyer et al. (1979); Yoder and Wirley (1984); and Ortiz and Popov (1985). Algorithms for the Drucker-Prager model have also been proposed. Approximate elaborate subincrementation strategies with successive radial stress corrections have been proposed (see e.g., Nayak and Zienkiewicz (1972)). Other somewhat arbitrary stress corrections have also been attempted (see e.g., Chen (1975); Vermeer (1980)) to correct for the inherent stress drift away from the yield surface. However, all these procedures tend to be quite expensive and are not error-free. They are analyzed in Loret and Prevost (1986). Integration algorithms for more complex models have also been developed, typically on a case-by-case basis (see e.g., Sandler and Rubin (1979) for the cap model). However, no general framework for developing consistent, accurate and stable algorithms was available until recently, when the concept of an elastic predictor with a plastic return mapping was developed (see e.g. Ortiz and Popov (1985)). It was therefore difficult to assess in general the relative merits and/or shortcomings of the various proposed procedures.

Since the elastic-plastic evolution problem is of a strain driven nature, the integration process is split into an elastic predictor and a return map to restore plastic consistency. The returning mapping is achieved by integrating the nonlinear plastic evolution equations, and there are several ways this can be implemented (see e.g., Nguyen (1977); Simo and Ortiz (1985); Simo and Taylor (1986); Ortiz and Simo (1986); Simo and Hughes (1987)). Due to its semi-explicit feature, attention has been mainly focused on the so-called ``cutting-plane'' algorithm (see e.g., Simo and Hughes (1987)), in geomechanics.

PARAMET ER CALI BRAT I ON

In order to be of practical use, a constitutive model has to be calibrated, i.e. the various coefficients which appear in the constitutive relations have to be identified based on suitable soil tests. More than for any other engineering material, available data for soil materials in natural and artificial deposits are affected by a number of uncertainties, such as: (1) limited information, (2) sampling and testing errors, and (3) random spatial variability. Calibration procedures thus have to deal with these uncertainties, and often have to rely on engineering judgement and collateral information. Therefore, it is important to select or relate constitutive model parameters to traditional soil parameters, as opposed to using numerical coefficients with no physical meaning. Thus, whenever available experimental data are inadequate for a ``proper'' calibration, empirical correlations or even engineering experience can be used. In this respect, a comprehensive collection of empirical correlations formulae was presented by Kulhawy and Mayne (1990).

Using Laboratory Soil Test Results

One vs. Multiple Laboratory Soil Tests

A common practice is curve fitting the results of one laboratory soil test. Close matching may demonstrate the ability of the constitutive model to simulate simple stress states and is considered a condition sine qua non for model validation. However, it does not guarantee a correct model calibration. It is now well known that laboratory soil tests although conducted under relatively uniform conditions, exhibit a large scatter in their results, which affects their reliability. The scatter is due to errors related to sample disturbance (during sample collection, transport and preparation), spatial variability of soil properties in natural deposits, testing methods, human factors, etc. For example, the scatter in friction angles at failure derived from a series of drained triaxial soil tests conducted at The Earth Technology Corporation (1992) with Nevada sand at 60% relative density was found to be about 50 to 60% of the average value (Popescu and Prevost (1995b)).

The alternative to curve fitting the results of a single soil test is to use information from all available tests performed on the material. The reliability of the results of a series of tests performed under identical conditions (random sampling) increases with the number of tests. For example, a measure of the random error associated with the mean of a series of test results is the expected standard deviation of the mean, or root-mean-square error , which decreases with the number of experiments, N (e.g. Bendat and Piersol, (1986)): .

For a detailed discussion on the expected error magnitude of soil test results and its dependence on the number of tests the reader is referred to Popescu and Prevost (1995b).

When using the results of more than one soil test a series of values is estimated for each soil constitutive parameter. Next, for each required parameter, one would select either (1) the most probable value (e.g. the average) - if dealing with back-analyses or ``class A'' predictions (e.g. Popescu and Prevost (1993b), Iai et al (1993)), or (2) a ``conservatively assessed mean'' or characteristic percentile - for design. An example is presented in Figure 1, showing the estimation of friction angles at failure of Nevada sand based on the results of a relatively large number of laboratory soil tests (Popescu and Prevost (1993a)).

Figure 1. Estimation of the friction angle at failure of Nevada sand, using results of

undrained triaxial compression soil tests.

The data presented in Figure 1 is used to illustrate the importance of repeated tests. Most of the laboratory soil tests conducted for the VELACS project (Arulanandan and Scott (1993)) were performed by The Earth Technology Corporation (1992). Supplementary soil tests were performed by other soil laboratories, as shown in Figure 1. If, for example, one were to use the results of isotropically-consolidated undrained compression tests performed by The Earth Technology Corporation to estimate the friction angle at failure of Nevada sand at 60% relative density, one obtains an expected error magnitude of 0.85% with a scatter in friction angle of . (The expected error magnitude is defined by Popescu and Prevost (1995b) as the root-mean-square error normalized by the sample mean). If the results of tests performed by other laboratories are also used, although the scatter is much larger ( ), the expected error magnitude is reduced further by about 20% as a result of the increased number of tests. Similarly, the expected error magnitude of the estimated friction angle at failure is reduced by over 30% in the case of Nevada sand at 40% relative density.

Stress-Strain Curve Generation

The calibration of elaborate soil constitutive models, especially those using multiple yield levels (e.g. Mróz (1967), Prevost (1977, 1978 and 1985)), require that stress-strain curves (typically obtained from triaxial or simple shear soil tests) be available. However, budget constraints often prevent detailed laboratory tests to be conducted on every soil type present at a given site. Further, usual paucity of field information, randomness in spatial variability of natural deposits require in every design situation that parametric studies and/or Monte Carlo type simulations be conducted. Therefore, the generation of stress-strain curves required for analysis, using limited field information, is a common and significant problem.

The best known and most widely used function is a hyperbola (Kondner (1963)), which has been applied to some quite complex soil deformation problems, most notably by Duncan and Chang (1970). However, the failure condition is approached asymptotically by the hyperbolic stress-strain curves, whereas in reality failure (or peak stress state) occurs at finite strain values. Therefore, alternative functions for describing typical shear stress-strain behavior of soil materials have been proposed.

A modified hyperbolic function shown in Figure 2 was proposed by Prevost and co-workers, which is applicable to both simple shear (Prevost and Keane (1989)) and triaxial stress states (Griffiths and Prevost (1990)), for a wide range of shear strain levels.

Figure 2. Modified hyperbolic function (after Griffiths and Prevost (1990)).

As illustrated in Figure 2, a large palette of shear stress-strain curves can be generated, with shapes controlled by the factor , where - the shear stress at failure - is a

function of friction angle at failure, initial stress state and slope of the stress path in the soil test. A drawback of this model is that spanning various shapes of the stress-strain curve requires relatively large modifications of the initial shear modulus, , which is not always in

accord with experimental evidence.

Hayashi et al (1992) solved this problem with a hyperbola whose shape depends on another parameter, , which is function of the grain size distribution characteristics. They developed their model based on shear stress-strain curves obtained in a simple shear soil testing device, using soil specimens under condition. The proposed functional stress-strain relation is a

linear combination of hyperbolas, and therefore the condition of zero slope at finite strains (peak stress or failure) is not satisfied. The required parameters are: initial shear modulus,

, shear stress at failure, , and the stress-strain curve parameter, , depending on maximum grain size and uniformity coefficient. The dependence of on grain size distribution characteristics is derived on a purely empirical basis from results of soil tests conducted by the authors on ten sands and gravels with a wide variety of grain size distributions.

Tatsuoka and Shibuya (1992) proposed a similar relation called ``general hyperbolic equation'', appropriate for modeling the pre-peak stress-strain behavior. Tatsuoka et al (1993) connected the general hyperbolic equation to a post-peak strain softening relation to obtain a more general equation for isotropically consolidated sands. This formulation considers the effects of void ratio, stress level, and strength and deformation anisotropy. Homogeneous strain state during the pre-peak deformation and sudden strain localization at the peak stress state are assumed. The relation for isotropically consolidated sands was then transformed to accommodate the behavior of anisotropically consolidated sands. Despite its performances in simulating the true stress-strain behavior for a wide range of strain levels, the model proposed by Tatsuoka et al (1993) has a series of limitations: (1) is only applicable to the plane-strain compression state, (2) was developed based on soil tests conducted using only one type of sand, and (3) has a complicate formulation, requiring a relatively large number of parameters.

Liquefaction Strength

Any constitutive model for sand materials should have at least one or more constitutive parameters related to the shear strain induced dilatancy. Common calibration methods for this type of soil parameters employ curve fitting of one available laboratory soil test, with all the entailed limitations, as described in §5.1.1. These limitations can be avoided by use of the liquefaction strength curve, which comprises results from more than one soil tests, as shown hereafter.

The liquefaction strength curve (Figure 3) is a plot of the cyclic stress ratio as a function of the number of cycles necessary to induce initial liquefaction ( ) in an undrained cyclic

soil test. The cyclic stress ratio is expressed either as (for triaxial tests), where is

the single amplitude cyclic axial stress and is the initial effective confining stress, or as

(for simple shear tests), where is the double amplitude cyclic shear stress and

is the initial effective vertical stress. Initial liquefaction is usually defined as the state in which either (1) the pore water pressure builds up to a value equal to the initially applied confining pressure, or (2) axial strains of about 5% in amplitude occur (e.g. Ishihara (1993)). To obtain a unique liquefaction strength curve from both triaxial and simple shear laboratory test results, the relation:

may be used to relate the stress ratio in different experiments, where is evaluated on the basis of the criterion proposed by Castro (1975), which relates the effect of the cyclic loading to the ratio of the octahedral dynamic shear stress to the static effective octahedral normal stress.

Figure 3. Estimation of the dilation parameter of Nevada sand at 60% relative density, using results of laboratory soil tests performed at The Earth Technology Corporation (1992).

The liquefaction strength curve can be obtained from the results of undrained cyclic laboratory soil tests (markers in Figure 3), or from empirical correlations with in-situ soil test results, as discussed in section §5.2.3.

The parameters related to dilatancy can be estimated by back-fitting the liquefaction strength curve, using element tests. An example is presented in Figure 3 (from Popescu and Popescu and Prevost (1993)), where the dilation parameter ( ) of a multi-yield plasticity

constitutive model (Prevost (1985)) is derived based on the results of undrained cyclic triaxial and simple shear laboratory soil tests The Earth Technology Corporation (1992)). A similar method was employed by Iai et al (1993) to estimate five dilatancy parameters of a multi-mechanism constitutive model.

Using In-Situ Soil Test Results

In-Situ vs. Laboratory Soil Tests

Both the shear strength and liquefaction resistance of soils in natural, undisturbed state are found to be significantly larger than measured in laboratory (e.g. Seed and De Alba (1986), Yoshimi et al (1989)). Moreover, based on results of cyclic triaxial tests performed by Mulilis et al (1977), Ishihara (1993) shows that the resistance to liquefaction of samples from the same type of sand prepared at the same relative density can vary over a wide range depending on the nature of the fabric structure created by different methods of sample preparation.

In this context, some of the advantages of using field measurements rather than laboratory data for soil property estimation are summarized by Campanella (1994): the disturbances are minimized, the effects of fabric and aging on measured properties are preserved, and the soil is tested in its natural in-situ stress state. Sophisticated sampling techniques are now available to minimize sample disturbance and to attempt to preserve the original fabric structure of the soil in natural deposits, like freezing (e.g. Yoshimi et al (1989)). However, these procedures are still too expensive to be used in routine practice. Further, so called ``homogeneous'' soil layers are in fact only ``stochastically homogeneous'', i.e. there are usually significant variations of soil properties from one spatial location to another. This variability, which, as shown in §5.3.3, may have significant influence on structural behavior, can only be captured by in-situ soil tests.

Most common in-situ soil testing techniques (e.g. standard penetration test) can provide information on parameters related to soil strength and deformability. Supplementary data, coming from laboratory soil tests, are usually required to identify stratigraphy, grain size distribution, damping properties, etc. Recently, procedures have been proposed to identify a number of soil properties directly from the results of cone penetration type tests, without relying on laboratory soil tests. For example a procedure for soil type identification based on the interplay between friction ratio and cone tip resistance was proposed by Harder and von Bloh (1988). Jefferies and Davies (1993) made use of a "soil classification index", estimated from the results of piezocone (CPTU) tests to obtain information on the soil type and grain size characteristics. The seismic piezocone (SCPTU) test is known to allow reliable evaluation of small strain moduli and damping characteristics (e.g. Stewart and Stewart and Campanella (1993), Campanella et al (1994)), and the resistivity piezocone (RCPTU or RSCPTU) can be used to infer dilatancy characteristics (Campanella and Kokan (1993)). The seismic piezocone and the self-boring pressuremeter (SBPM) are considered by Campanella (1994) "the best current in-situ tools for determining earthquake engineering parameters". However, it should be pointed out that the current data base pertaining to these two types of field tests is still very limited and the available empirical correlations with various soil properties do not cover a wide range of soil types.

Correlations with Penetration Test Results

Various methods are available for in-situ evaluation of geotechnical properties. The most common are as follows:

1. Dynamic penetration tests - standard penetration test (SPT), Baecher penetration test (BPT), and dynamic cone penetration test (DCPT).

2. Cone penetration tests - mechanical CPT, electric CPT, piezocone (CPTU), seismic piezocone (SCPTU), resistivity piezocone (RCPTU), and horizontal stress cone (HSC).

Table 1. Comparison between SPT and CPT (after Campanella (1994))

The advantages and disadvantages of each type of test, as well as their applicability in estimating various soil properties are discused by Campanella (1994). Table 1 provides a comparison between the standard penetration test (SPT) and cone penetration test (electric CPT and piezocone), for which a large database for empirical correlations with geotechnical properties is available. Both types of tests have good applicability in estimating various soil properties, like moduli, strength, and liquefaction resistance. However, given its advantages - precision, repeatability, continuous logging, multiple channel measurements, as well as ease of use in offshore testing (Jefferies and Davies (1993)) - the cone penetration test is becoming increasingly popular for in-situ investigations (especially in Europe and Canada).

The results of penetration type soil tests represent the response of soil to an imposed deformation. Geotechnical properties can be then estimated using either (1) an appropriate constitutive model and an analytical solution, or (2) empirical correlations. Various theories have been developed to express analytical dependence between cone tip resistance and soil strength. It seems however that there is currently no satisfactory general solution for interpreting cone test results, therefore empirical correlations are still mainly used to derive soil properties. Numerous such correlations between the results of standard penetration and cone penetration tests and various soil properties have been developed. References to some of these correlations are listed in Table 2.

Table 2. Empirical correlations between soil properties and penetration test results(notations in Appendix 1)

In general, more than one empirical correlations are available to derive each particular soil property, and the accuracy of estimations is highly dependent on selection of the appropriate correlations for the respective material. Use of available laboratory soil test results, as well as any other available collateral information to select the right correlation formula for each soil material can increase significantly the confidence in the estimated parameters. The accuracy of parameter estimation can also be increased by taking advantage of both (1) the higher reliability of cone penetration test results and (2) the vast data base and larger experience with using the standard penetration test, by means of transformations.

References to some of these relationships are listed in Table 3. For more detailed reviews on transformations the reader is referred to Jefferies and Davies (1993)) and Stark

and Olson (1995).

Table 3. Transformations (notations in Appendix 1)

Liquefaction Strength

Seed and co-workers (1983 and 1985) developed a procedure for estimating the liquefaction potential of soil deposits under level ground conditions based on correlations between the

equivalent cyclic stress ratio, , necessary to induce liquefaction in a number

cycles (this will be denoted by in the following), and the normalized SPT blow

count, . A boundary line was proposed, separating field conditions (in terms of SPT

resistance) susceptible to produce liquefaction from conditions not causing liquefaction, in sandy soil deposits, during earthquakes with magnitude M=7.5. Similar relations have also been proposed by other researchers (some of which are referred in Table 4), based on case histories of liquefied and non-liquefied sites documented with SPT measurements. The proposed correlations correspond to an earthquake of a certain magnitude (usually M=7.5), relatively small initial overburden stress ( MPa), and level ground conditions.

Correction procedures are available for transforming these correlations to correspond to other earthquake magnitudes (e.g. Seed et al (1983)), larger initial overburden stress and sloping ground conditions (Seed and Harder (1990)).

Correlations between liquefaction potential and normalized cone penetration resistance, , were first developed using the SPT data base and converting SPT blow counts to CPT resistance (e.g. Robertson and Campanella (1985); Seed and De Alba (1986)). Later on, when the number of earthquake related case histories documented with CPT data increased significantly, direct such correlations were developed (e.g. Shibata and Teparaksa (1988); Stark and Olson (1995)). For comparative studies on correlations between liquefaction potential and normalized cone penetration resistance, the reader is referred to Mitchell and Tseng (1990), and Stark and Olson (1995).

Table 4. Correlations between observed field liquefaction behavior and normalized penetration resistance (notations in Appendix 1)

Apart from penetration resistance ( or ), the liquefaction potential of sandy soil

deposits is significantly influenced by the grain size characteristics. This influence is accounted for in all the charts referred in Table 4, in terms of the percent of fines (F= percent passing through the #200 sieve) for SPT related correlations, and average grain size, ,

for CPT related correlations. A detailed discussion is presented by Ishihara (1993). He also investigates the influence of the plasticity index of fines on the liquefaction potential of silty sand deposits.

All these correlations have been originally developed as tools for the empirical assessment of liquefaction potential of natural soil deposits (e.g. Das (1983)). However, they can also be used to estimate soil parameters related to dilatancy by means of liquefaction strength analysis, as illustrated in Figure 4, since they provide a point on the liquefaction strength

curve ( ). Such a procedure is presented in Popescu (1995).

Figure 4. Evaluation of the dilation parameter from the results of liquefaction strength

analysis: a. relationship between normalized penetration resistance and cyclic stress ratio which causes liquefaction in cycles; b. illustrative example of the parametric

studies using element tests.

As for the post-liquefaction behavior, two main directions of research for estimating the residual strength are referred to in the literature. The first approach is based on the principle of "critical state" soil mechanics (e.g. Schofield and Wroth (1968)), and uses laboratory soil tests to estimate the in-situ undrained residual (or "steady state") strength (e.g. Poulos et al (1985)). For a detailed discussion on the ``steady state'' approach, and useful charts for estimating the undrained residual strength ratio with respect to the initial confining stress, the reader is referred to Ishihara (1993). Due to the very high sensitivity of the residual strength to small variations in void ratio, the laboratory-based techniques do not appear to be sufficiently reliable for use in full scale engineering analyses (e.g. Seed et al (1988)). The second approach is more empirically oriented and is based on back-analyses of liquefaction failure case histories documented with SPT measurements (e.g. the chart by Seed (1987), updated by Seed and Harder (1990)). Based on such field case histories, Stark and Mesri (1992) proposed a linear correlation between the undrained residual strength ratio, ,

and the equivalent clean sand normalized SPT blow count, :

where (correction for fines content, according to Seed

(1987)).

Appendix 1: Notations used in Tables 2, 3 and 4

Accounting for Inherent Spatial Variability of Soil Properties

The natural variability of soil properties within geologically distinct and uniform layers is known to affect the soil system behavior itself (e.g. Ohtomo and Shinozuka (1990), Popescu (1995) and Popescu et al (1997a) for soil liquefaction; Griffiths and Fenton (1993) and Fenton and Griffiths (1996) for seepage; Paice et al (1996) for settlements), and therefore should be accounted for during model calibration. A first step in acknowledging the effects of uncertainties induced by random variability of soil properties was made by introducing the concept of characteristic values as ``cautious estimates'' of those parameters affecting the occurrence of limit state (e.g. Eurocode (1994)). At this stage, assessment of characteristic values as percentiles of recorded soil properties is based mainly on the degree of confidence in the mean values, as estimated from soil test results.

The consequences of spatial variability are not well understood yet, and their exploration requires use of stochastic field based techniques of data analysis. Due to (1) the large degree of variability exhibited by soil properties (coefficient of variation ranging from 20% to 60%, as reported by Phoon and Kulhawy (1996)), and (2) the strong nonlinear behavior of soil materials, Monte Carlo simulations which combine digital generation of stochastic fields with finite element analyses seem to represent the only appropriate analysis method at this time.

Stochastic Analysis of Field Data

Experimental evidence shows that the natural variability of soil properties within distinct and uniform layers is significant, even in the case of supposedly homogeneous man-made fills. This is illustrated in Figure 5a which shows the results of a series of piezocone tests performed in a hydraulically placed sand deposit (Gulf (1984)): the recorded cone resistance exhibits random fluctuations about some average values shown with thick lines in Figure 5. As expected, some degree of coherence between the fluctuations can be observed, which becomes stronger as the measuring points are closer together. This is emphasized in Figure 5a

by representing some loose pockets (where recorded values are consistently lower than the expected average) and denser zones.

The spatial coherence is mathematically captured by the concept of a correlation function, with its parameters expressed in terms of ``correlation distances''. The correlation distance can be thought of as representing a length over which significant coherence is still manifested. The ensemble of auto-correlation functions (characterizing the spatial variability of each soil property) and cross-correlation functions (spatial correlation between pairs of different soil properties) is represented by the so called ``cross-correlation matrix''. The cross-correlation matrix, together with the probability distribution functions exhibited by each soil property characterize the stochastic variability of soil properties over the domain of interest. A rich literature on estimating the characteristics of spatial variability of soil properties is currently available. Various methods for evaluating the correlation structure starting from a limited amount of field test results, as well as procedures for raw data filtering and transformations are described. For more details on available methods for stochastic analysis of field data the reader is referred to DeGroot and Baecher (1993).

Figure 5. a. In-situ recorded cone tip resistances (from Gulf (1984)); b. simulated values

obtained at the same locations from one sample function of a stochastic field with probabilistic characteristics estimated from field data analysis.

The probability distribution function of soil properties is another important characteristic of spatial variability. From the results of numerous studies on the probability distribution of soil properties reported in the literature it can be concluded that: (1) most soil properties exhibit skewed, non-Gaussian distributions, and (2) each soil property can follow different probability distributions for various materials and sites, and therefore the probability characteristics and the shape of the distribution function have to be estimated for each case. For a detailed presentation of field data analysis procedures leading to estimation of probabilistic characteristics of spatial variability of soil properties, the reader is referred to Popescu et al (1997b).

Digital Simulation of the Random Spatial Variability

A deterministic description of the spatial variability of soil properties is not feasible due to the prohibitive cost of sampling and to uncertainties induced by measurement errors. In a probabilistic approach the soil properties over the analysis domain are considered components of a vector stochastic field, with probabilistic characteristics estimated from analysis of field data. Sample functions of this stochastic field are digitally generated, according to a prescribed cross-correlation matrix and prescribed probability distribution functions. An example of such a sample function is presented in Figure 5b, and a methodology to digitally simulate sample functions of a multi-dimensional, multi-variate, non-Gaussian stochastic field is described by Popescu et al (1996b).

At this junction it is mentioned that complete probabilistic description of a non-Gaussian stochastic field requires knowledge of the joint probability distribution functions of all orders. Since these cannot be estimated in practice from real soil data, the stochastic spatial variability of soil properties can be characterized by way of the correlation structure (cross-correlation matrix) and marginal probability distribution functions of each soil property (e.g. Popescu et al (1996b and 1997a)).

Effects of Spatial Variability on Dynamic Behavior

Soil liquefaction seems to be particularly affected by random spatial variability of soil properties. Previous studies (e.g. Popescu (1995), Popescu et al (1996a and 1997a)) have shown that both the extent and pattern of pore water pressure build-up in saturated soil deposits subjected to seismic excitation are different when predicted by Monte Carlo simulations, as compared to deterministic finite element analyses using the average values of soil properties.

A numerical example is presented in Figure 6 (from Popescu et al (1996a)). A loose to medium dense soil deposit (Figure 6a), with geomechanical properties as well as spatial variability characteristics estimated from in-situ (piezocone) soil test results, is subjected to a deterministic seismic excitation. Monte Carlo simulations are performed, with soil parameters corresponding to four sample functions of a stochastic field with probabilistic characteristics

derived from field data analysis. Each sample function represents a possible realization of the spatial distribution of in-situ recorded soil properties over the analysis domain. The multi-yield plasticity model (Prevost (1985)) implemented in the computer code DYNAFLOW (Prevost (1995)) was used for finite element analyses. A deterministic analysis is also performed. The soil constitutive parameters for the deterministic analysis are variable with depth, corresponding to the assumed linear variation exhibited by the average values of the field measurement results (Figure 5), and have the same values as the average values used for Monte Carlo simulations.

Figure 6. Excess pore pressure ratio in a soil deposit subjected to seismic excitation. Results of: a. Monte Carlo simulations; b. deterministic analysis using the average values of soil parameters; c. deterministic analyses using various percentiles of soil strength.

A comparison between the results of the deterministic and stochastic parameter input computations is presented in Figure 6 in terms of predicted excess pore pressure ratio with respect to the initial effective vertical stress. Four different responses in terms of excess pore pressure ratio are predicted by the Monte Carlo simulations (Figure 6a). There are however a series of common features of the stochastic input computational results, as compared to the results of the deterministic input analysis shown in Figure 6b: (1) for the same average values of the soil parameters, more pore pressure build-up is predicted by the Monte Carlo model than by the deterministic model; (2) in the case of the Monte Carlo simulation results, there are patches with large excess pore pressure, corresponding to the presence of loose pockets in the material (this predicted pattern of excess pore pressure build-up explains better the phenomenon of sand boils observed in areas affected by soil liquefaction).

Characteristic Percentile of Soil Strength

An explanation of the fact that more pore water pressure build-up is consistently predicted by Monte Carlo simulations than by the deterministic analyses is that liquefaction is triggered by the presence of loose pockets in the soil mass. Since deterministic analyses cannot account for such small scale variability in soil properties, to avoid non-conservative results geotechnical practice relies on factors of safety in the form of conservatively assessed mean values or, more recently, percentile values of soil strength (e.g. Eurocode (1994)). These values are recommended based on engineering judgement and experience, and there is currently no deterministic analytical procedure to predict how well they will work in various situations. Therefore, the recommended percentile values are most often too conservative. On the other hand, Monte Carlo simulations are much too expensive to be routinely employed in design, and therefore correctly defined characteristic percentiles, based on exploring the interplay of soil property distributions and overal behavior using an advanced soil constitutive model (as suggested by Jefferies et al, (1988)) are much needed in geotechnical design.

An attempt to define such a characteristic value of soil strength to be used in deterministic analyses of soil liquefaction potential is illustrated in Figure 6c (after Popescu et al (1996a)). Under the reserve of the (statistically) low number of Monte Carlo simulations, the 80-percentile value was recommended to be used in deterministic dynamic analyses, for the type of soil investigated in that study. For a more detailed presentation of this subject, addressing the interplay between soil property distribution, frequency content of the seismic excitation and dynamic characteristics of the soil deposit, the reader is referred to Popescu et al (1996a).

VERI FI CAT I ON AND VALI DAT I ON

In order to be useful to the engineering practice, a numerical model requires verification and validation by comparison of its predictions with observed full-scale in situ performance. A common problem in geotechnical engineering is that much data are unlikely to be obtained soon, because of the scale of the structures involved, the cost of testing and the low probability of having a particular instrumented geotechnical system subjected to design load. Consequently, some form of model study is desirable to enable alternative analysis procedures to be checked and validated.

The VELACS Project

The VELACS (VErification of Liquefaction Analysis by Centrifuge Studies) project (Arulanandan and Scott (1993)) offered a good opportunity to verify the accuracy of various analytical procedures. This NSF sponsored study on the effects of earthquake-like loading on a variety of soil models was aimed at better understanding the mechanisms of soil liquefaction and at acquiring data for the verification of various analysis procedures. The numerical predictions were intended to be ``class A'' predictions, and thus were made before the relevant experiments were performed. The verification and validation of the various analysis procedures were carried out by comparing their predictions with the measurements recorded in the centrifuge experiments, in terms of excess pore water pressure, acceleration and displacement time histories. A total of nine geotechnical models, shown in Figure 7, were tested. The tests were performed at five centrifuge laboratories.

Figure 7. Centrifuge geotechnical models used for the VELACS project (the dimensions are at the prototype scale).

Validation Study

For seven of the nine centrifuge models of the VELACS Project, some of the experimental errors were minimized by duplicating the centrifuge experiments. The duplicating experiments were intended to be performed under conditions identical to the primary experiment, using the same type of equipment and following the same specifications for sample preparation. The primary and duplicating tests for each model were carried out at different laboratory facilities, to minimize the bias in results induced by human factors. From a study on the reliability of the centrifuge soil tests performed for the VELACS project (Popescu and Prevost (1995a and 1995b)) it was concluded that: (1) four out of the seven centrifuge experiments which had been duplicated provide reliable excess pore water pressure results, and (2) the experimental records in terms of displacements and accelerations have considerably lower reliability than the recorded excess pore pressures. Therefore, it was decided that meaningful comparisons between VELACS "class A" predictions and centrifuge experimental results can be performed in terms of excess pore pressures. The results of the comparison study are presented in Popescu and Prevost (1995a).

A summary of the results are presented in Figure 8. A total of 56 numerical predictions, representing 75% of the "class A" predictions submitted by various analysts have been available for the comparative study presented in Figure 8a. The markers indicate the magnitude of normalized root-mean-square errors (see Figure 8b) of predicted excess pore pressures with respect to the average of centrifuge experimental recordings, for each of the nine centrifuge models involved in the VELACS project. For each centrifuge model, the root-mean-square errors estimated for each predictor are averaged over all pore pressure transducers.

Figure 8. Performance of various soil constitutive models in predicting excess pore pressure build-up: a. averages of the root-mean-square error index; b. evaluation of the root-mean-

square error between predicted and recorded excess pore pressure time histories (after Popescu and Prevost (1995a)).

CLOS URE

In this brief description of such a rich and complex subject matter, it has not been possible to cover all aspects and issues related to constitutive equations. In particular, one issue the writers have been compelled to omit is the strain localization phenomenon in soil materials, and other issues related to unstable material behavior. Strain localization phenomena have been examined experimentally in the laboratory by several investigators (e.g., Vardoulakis (1980); Mulhaus and Vardoulakis (1987)). The phenomenon is known to occur in real soils in nature very frequently and to often lead to catastrophic failures. However, its modeling presents several formidable difficulties and is the subject of much current research.

Strain localization is the phenomenon by which the deformations in solids localize into narrow bands of intense straining. The phenomenon is observed to occur in many materials (polycrystals, polymers, geomaterials, etc.). The width of the bands are known to be related to the microstructure of the materials.

Numerical simulation of strain localization requires that a characteristic length scale be incorporated into the governing equations. Further, it requires that the multi-scale nature of the phenomenon be addressed. Shear band widths in soils are of the order of millimeters while the engineering macrostructure of interest may be several meters in dimension. Previous works have been able to account for multi-scale effects, but have used simplified phenomenological constitutive equations in the band. However, it is clear that the details of the constitutive behavior inside the band strongly affect the response of the macrostructure.

Strain localization involves many aspects of material behavior. Its emergence, in some simple cases, can be predicted by a local analysis, viz. a loss of hyperbolicity of the dynamical equations (Hadamard (1903); Thomas (1961); Hill (1962); Mandel (1963); Rice (1976)). This analysis also allows to obtain the band orientation with respect to the axes of loading. However, the analysis cannot predict the exact location nor the thickness of the band. In numerical simulations by finite elements, several difficulties occur: the computed shear band has a width determined by the size of the elements; and the results exhibit a strong mesh dependency.

The main reason for these difficulties was identified about 10 years ago, and is due to the fact that the constitutive equations used do not incorporate a characteristic length scale parameter in the governing equations. This difficulty has been alleviated by having recourse to various regularization procedures (see e.g., Bazant (1984); Triantafyllidis and Aifantis (1986); Muhlhaus and Vardoulakis (1987); Needleman (1988); Loret and Prevost (1990a, 1990b and 1991)); Sluys and de Borst (1992)). Although involving radically different mechanisms, i.e., by adding gradient or viscous effects in the constitutive equation, each of the aforementioned works aim at restoring hyperbolicity by incorporating a characteristic length scale in the governing equations.

Once the band thickness can be incorporated into the theory, several other difficulties still remain. They are related to the multi-scale nature of the phenomenon. The shear band width is related to the microstructure of the material (in a sand for instance, it spans about 15 to 20 grains), and is several order of magnitude smaller than the usual macrostructure of interest in engineering. Hence any meaningful numerical discretization scheme must be able to accurately resolve the deformation field inside the band, and at the same time resolve the field within the macrostructure. This requires a very heterogeneous multi-scale computation for which neither homogenization theories nor usual finite elements are properly suited. Two complementary procedures have been developed to perform the multi-scale computations using finite elements. In one approach the shear band is captured by successive adaptive refinements of the finite element mesh (e.g., Deb et al. (1996a and 1996b)), in another the mesh is enriched by adding additional equations to the localized elements (e.g., Fish and Belytschko (1990), Loret et al. (1995)). These two approaches are necessary steps to allow for the resolution of the appropriate length scales necessary to study the critical phenomena which occur in the bands. Up to this stage, simplified phenomenological constitutive equations have been used to describe the material behavior within the shear bands. An attempt to more accurately describe the material behavior within the shear bands by using more realistic micro-mechanical constitutive equations will bring to evidence the effects of the microstructure on the next highest length scale. For instance, for a sand, the ``micro-mechanical filter'' should allow to retain the effects of the statistics of the grain size distribution, the directions of contacts and the sliding motions. This is the subject of much

current research.

ACKNOWLEDGEMENT S

The writers have been supported in work related to this paper by the National Science Foundation (Dr. Clifford J. Astill, Program Director) and Kajima Corporation (Japan). These supports are most gratefully acknowledged.

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