Category WATER IN ROAD STRUCTURES

Cane Cekerevac[24], Susanne Baltzer, Robert Charlier, Cyrille Chazallon, Sigur0ur Erlingsson, Beata Gajewska, Pierre Hornych, Cezary Kraszewski and Primoz Pavsic

Abstract This chapter presents laboratory and in-situ experimental techniques used to describe the mechanical behaviour of pavement material at different saturation stages. The use of repeated triaxial load testing to obtain stiffness characteristics as well as the ability of the material to withstand accumulation of permanent de­formation during cyclic loading is considered. For unsaturated soils, in addition to mechanical variables, it is shown that a moisture/suction control should be added. Several techniques are described to assist in this. A brief presentation of model pa­rameters and tests needed for model calibration are introduced. Evaluation of pave­ment structural capacity based on deflection measurements with non-destructive testing equipment are presented. Finally, some examples of laboratory and in-situ measurement are shown.

Keywords Laboratory testing ■ suction control ■ repeated triaxial test ■ CBR test ■ parameter calibration ■ field testing ■ laboratory and in-situ experimental results

10.1 Introduction

Besides proper construction design, the materials used in the construction of pave­ments should be such that the effects of excess moisture are minimized. To help achieve this purpose, several laboratory and in-situ methods can be used. For ag­gregates used within the unbound layer, performance, when subjected to traffic and moisture is mainly characterized by:

• the compaction properties;

• the amount of degradation;

• the composition (as determined by sieving analysis); and

• the evaluation of the quality of fines (in some cases).

All of these are index tests, which do not give us proper insight into the behaviour of the material under traffic loads and in different moisture conditions. A better under­standing of actual in-situ behaviour can be obtained with the cyclic load triaxial test and with the Falling Weight Deflectometer (FWD) and other in-situ tests. From June 2004 the manufacturers of aggregates in the EU are required to undertake the initial type test and Factory Production Control to ensure that their products conform with European Standard EN 13242 (CEN, 2002).

Routine pavement design is based on an elastic calculation, with a resilient modulus. The design criterion is, typically, a limit placed on the maximum vertical strain. More elaborated models take into account the irreversible behaviour, e. g.:

• The Chazallon-Hornych model is based on the Hujeux multi-mechanism yield surface improved by a kinematical hardening; and

• The Suiker and Mayoraz elasto-visco-plastic models evaluate the irreversible strains on the basis of an overstress (Perzyna theory) which is the distance be­tween the stress level and a visco-plastic potential.

Each of these elaborated models is then based on a yield surface, a potential surface, a limit surface, in all cases a surface typical of the granular soil mechanics, with a frictional mechanism, possibly a cap contractive mechanism, a dependency not only on the shear/von Mise’s stress but also on the mean stress (p — q plane), and on the Lode angle.

How can we adapt these models to take into account the suction variation effects? For routine pavement design, only the elastic moduli need to be adapted. For the higher-level models, the yield surface and hardening mechanism also need to be adapted.

During the two last decades a number of models for partly saturated soils have been proposed (forareview, see e. g. Laloui etal., 2001). Most of them are based on the suction as an additional variable, with the same status as the stress tensor.

The so-called Barcelona Basic Model – BBM, proposed by Alonso et al (1990) is probably one of the best known. It is now the reference for most new develop­ments in mechanics of geomaterials under partial saturation.

The BBM is based on the well-known CamClay model. It is written within the framework of the independent stresses state variables p — q — s defined in Section 9.5.

The BBM yield surface depends not only on p — q stresses but also on the in­dependent stresses state variables p — q — s. Two lines are added with respect with the modified Cam-Clay model. On a wetting path (a loading path along which the suction decreases), the Loading-Collapse, LC, line allows a normally consolidated material to support irreversible plastic strains and hardening, and the plastic slope to depend on the suction level (as there will be an increase of stiffness with suction). For low stress level, the cohesion only depends on the suction level. For the case illustrated in Fig. 9.10, a capillary cohesion is postulated, which depends linearly on the suction. Eventually, under very high suction (a consequence of the drying process) irreversible strains may also occur. This is at the plane SI in the figure – the suction increase surface.

However, neither do the BBM, nor the other published models, introduce any suction dependency into the elastic moduli formulations.

From this illustrative model, it appears that building a coupled model for re­peated loading and suction variation on granular soil material needs the following developments: [23]

• Using the generalised effective stress or the net stress approach allows develop­ing coupled moisture – mechanics models to be developed.

• For any development an experimental basis will be needed to calibrate and vali­date the models.

9.5 Conclusions

This chapter deals with the constitutive modelling of the effects of water on the me­chanical behaviour of pavements. It has been shown that routine pavement design is based on an elastic calculation, with a resilient modulus. The design criterion is, typically, a limitation of the maximum resilient vertical strain. Such design ap­proaches do not model in a realistic manner the observed irreversible behaviour seen as rutting and as other forms of distress. To achieve a design approach that can replicate more closely the observed behaviour is likely to require use of the concepts of elasto plasticity and visco-plasticity. More elaborate models of soil and granular material will be needed to take into account these concepts and several approaches have been reviewed that attempt to do this.

At present few of these newer approaches explicitly include the effect of wa­ter pressures and suctions within the soil or aggregate pores, so the chapter has discussed how the available constitutive models could be improved to take into ac­count suction and suction variation effects. Some research topics have, also, been suggested to enable further development.

The constitutive models introduced in the previous sections express the constitutive stress-strain relation of the material. As soon as the water is involved, the material has to be considered as a multi-phase porous media with two phases: the solid matrix (for which we introduced the stress-strain constitutive relations) and the water phase. The two phases are coupled since the pressure acting in the water may affect the mechanical behaviour of the material. Also, the material deformation may modify the pressure in the water. Such hydro-mechanical coupling is well represented by
the Terzaghi effective stress concept for saturated conditions (when water is filling all the pores) (Terzaghi, 1943). It shows the importance of the consideration of the water phase in the analysis of the mechanical behaviour of the material.

In the case of non saturation (water is no longer filling all the pores) the effective stress, expressed as a function of the externally applied stresses and the internal fluid pressures, converts a multi-phase porous media to a mechanically equivalent, single-phase, single-stress state continuum (Khalili et al., 2005). It enters the elastic as well as elasto-plastic constitutive equations of the solid phase, linking a change in stress to strain or any other relevant quantity of the soil skeleton; e. g. see Laloui et al. (2003). As a first approximation, let us consider a force Fn applied on a porous medium (constituted by a solid matrix and pores) through an area A. In this case, we can define a total stress:

If we consider only the part of the load acting on the solid matrix (and deforming it), we may define an effective stress as the part of the load acting on the solid area (XSi) (Fig. 9.9):

The effective stress may be simply defined as that emanating from the elastic (mechanical) straining of the solid skeleton:

ee = Cea’ (9.18)

in which ee is the elastic strain of the solid skeleton, Ce is the drained compliance matrix, and a’ is the effective stress tensor.

In a saturated medium, the effective stress is expressed as the difference between total stress, a, and pore water pressure, u (Terzaghi, 1943):

Fig. 9.9 An illustration of inter-granular stresses

a’ = a — u (9.19)

In an unsaturated granular material with several pressures of different fluid con­stituents, the effective stress is expressed as:

n

a’= a — ^2 amumI (9.20)

m = 1

in which am is the effective stress parameter, um is the phase pressure, and m = 1, 2,… n represents the number of fluid phases within the system. I is the second order identity tensor. This equation is close to the one of Bishop (1959) for a three – phase material (solid, water and air):

a’ = (a — Ua) + X (Ua — u) (9.21)

where u is the pore water pressure, ua is the pore air pressure, x is an empirical parameter, which has a value of 1 for saturated soils and 0 for dry soils. It represents the proportion of soil suction that contributes to the effective stress. Several at­tempts have been made to correlate this parameter to the degree of saturation and the suction (Bishop, 1959; Khalili & Khabbaz 1998). As the parameter x seems path-dependent, several authors, starting from Bishop and Blight (1963), proposed the use of two sets of independent “effective” stress fields combining the total stress a, and the pore-air and pore-water pressures, ua and u (Fredlund & Morgenstern, 1977). In the literature the net stress a = a — ua and the suction 5 = ua — u are com­monly chosen (Alonso et al, 1990). In general, this net stress concept will be defined in invariant terms using the independent stress variables p(= (ai + a2 + a3)/3), q and 5.

Another way to describe the behaviour is to use the “saturated effective stress” a’ = a — uw and the suction, 5 (Laloui et al., 2001). This combination has the advantage – among others – of permitting a smooth transition from fully saturated to unsaturated condition.

Continuing with the approach having two sets of independent stresses, the strain rate obtained for the elasto-plastic behaviour may be decomposed into elastic and plastic parts:

є ij = єу + єp

each of which results from mechanical and suction variations, as follows. The elastic increment єe. is composed of a mechanical and a hydraulic strain increment:

(9.23)

where ej is the elastic mechanical strain increment induced by the variation of the effective stress a’, 1 eevh is the reversible hydraulic strain increment, Ee is the classical elastic tensor and к is a proportionality coefficient which describes the hydraulic behaviour.

Similarly the plastic strain increment is also deduced from mechanical and hy­draulic loads by considering two plastic mechanisms derived from two yield limits:

(9.24)

pm

Where eij is the mechanical plastic strain increment, associated with the mechani-

1 ph

cal yield surface and з eV is the hydraulic plastic strain increment, associated with the hydraulic yield surface.

Using an effective stress approach together with the non-linear models presented in the preceding section allows users to partly take into account the moisture varia­tion effects on mechanical behaviour. However, more fundamental modifications are probably needed. The next section indicates some tools to advance in that direction.

New concepts have been developed to determine the long term mechanical be­haviour of unbound materials under repeated loadings. All these concepts are pre­sented in a special issue (Yu, 2005) of the International Journal of Road Materials and Pavements Design.

The shakedown concept applied to pavements was introduced first by Sharp and Booker (1984). The various possible responses of an elastic-plastic structure to a cyclical load history are indicated schematically in Fig. 9.8. If the load level is sufficiently small, the response is purely elastic, no permanent strains are induced

and the structure returns to its original configuration after each load application. However, if the load level exceeds the elastic limit load, permanent plastic strains occur and the response of the structure to a second and subsequent loading cycle is different from the first. When the load exceeds the elastic limit the structure can exhibit three long term responses depending on the load level (Fig. 9.8). After a finite number of load applications, the build up of residual stresses and changing of material properties can be such that the structure’s response is purely elastic, so that no further permanent strain occurs. When this happens, the structure is said to have shaken down: it is in the elastic shakedown region. In a pavement this could mean that some rutting, subsurface deterioration, or cracking occurs but that, after a certain time, this deterioration ceases and no further structural damage occurs.

At still higher load levels however, shakedown does not occur, and either the permanent strains settle into a closed cycle (plastic shakedown behaviour) or they go on, increasing indefinitely (ratcheting behaviour). Contributions of Yu & Hossain (1998), Collins and Boulbibane (2000), Arnold et al. (2003) and Maier et al. (2003) are based on the fact that if either of these latter situations occurs, the structure will fail. The critical load level below which the structure shakes down and above which it fails is called the shakedown load and it is this parameter that is the key design load. The essence of shakedown analysis is to determine the critical shakedown load for a given pavement. Pavements operating above this load are predicted to exhibit increased accumulation of plastic strains under long-term repeated loading conditions that eventually lead to incremental collapse. Those pavements operating at loads below this critical level may exhibit some initial distress, but will eventually settle down to a steady state in which no further mechanical deterioration occurs.

The direct calculation of the shakedown load is difficult. Indeed lower and upper bounds are usually calculated using Melan’s static or Koiter’s kinematic theorems, respectively. These procedures are similar to the familiar limit analysis techniques for failure under monotonic loading, except that now the elastic stress field needs to be known and included in the calculation. Finally, the material is assumed to be perfectly plastic with an associated flow rule.

Shakedown models require an elasticity framework and parameters, for example as provided by the Universal model, and the knowledge of the rupture parameters of the Drucker-Prager or Mohr-Coulomb surfaces. 2D finite element plane strain calculations of pavements have been carried out in this way.

Parameters needed are:

• Elastic behaviour: k1, k2, k3, v; and

• Plastic behaviour: c, y.

Contributions of Habiballah and Chazallon (2005) and Allou et al. (2007) are based on the theory developed by Zarka and Casier (1979) for metallic structures submitted to cyclic loadings. Zarka defines the plastic strains at elastic shakedown with the Melan’s static theorem extended to kinematic hardening materials. The evaluation of the plastic strains when plastic shakedown occurs is based on his simplified method. Habiballah has extended the previous results to unbound granu­lar materials with a non-associated elasto-plastic model. The Drucker-Prager yield surface is used with a von Mise plastic potential.

This approach requires the elasticity parameters of the k — 0 or other model, the rupture line and the law describing the development of the kinematic hardening modulus. This model requires a “multi-stage” procedure, developed by Gidel et al., (2001) which consists, in each permanent deformation test, of performing, succes­sively, several cyclic load sequences, following the same stress path, with the same q/p ratio, but with increasing stress amplitudes. Finite element calculations have been carried out under axi-symmetric condition and 3D. The initial state of stress is determined with the k— 0 or other model, then the plastic strains are calculated.

Parameters required are:

• Elastic behaviour: k1, k2, v (assuming that the k—0 model was selected); and

• Plastic behaviour: c, y, H(p, q).

Visco-plastic equivalent models based on an equivalent time: number-of-cycles re­lationship, have been developed by Suiker and de Borst (2003) for the finite element modelling of a railway track structure and by Mayoraz (2002) for a laboratory study of sand.

Suiker has developed a cyclic densification model. It is based on repeated load triaxial tests carried out on two ballast materials. The idea was to develop a model that captures only the envelope of the maximum plastic strain generated during the cyclic loading process. The unloading is considered as elastic. The plastic deforma­tion behaviour is composed of two different mechanisms namely: frictional sliding and volumetric compaction. In general, both mechanisms densify the material. This model is based on the Drucker-Prager yield surface and Cap surface. The stress space is divided into four regimes (see Fig. 9.7):

(i) the shakedown regime, in which the cyclic response of the granular media is fully elastic.

(ii) the cyclic densification regime, in which the cyclic loading submits the granular material to progressive plastic deformations.

(iii) the frictional regime, in which frictional collapse occurs, since the cyclic load level exceeds the static peak strength of the granular material.

(iv) the tensile failure regime, in which the non cohesive granular material instan­taneously disintegrates, as it can not sustain tensile stresses.

The model requires the following parameters:

• Elastic behaviour: k1, k2, v; and

• Plastic behaviour:

о Monotonic parameters: pt, hm, h0, p0, d0, dm, Zf, Zc.

о Cyclic plasticity parameters: pt, h0, hm, Zf, af, ac, yc, p0, nf, nc, d0, dm.

The monotonic parameters initialise the state of stress and strains in the railway track structure which are required for the cyclic model.

Mayoraz has developed a visco-plastic equivalent model based on the associated modified Cam-Clay model with no elastic part. Permanent deformations compar­isons with the results of repeated load triaxial tests performed on a sand have been carried out. This model requires only 5 parameters and is based on the Perzyna concept (Perzyna, 1966) developed for visco-plastic creep of clay.

Parameters needed for the model are:

• Plastic behaviour:

о Rupture parameters: M; and о Plasticity parameters: n2, Г, J°nit and в*.

The plasticity theory based models require the definition of yield surface, plastic potential, isotropic hardening laws, and simplified accumulation rules (Bonaquist & Witczak, 1997 and Desai, 2002), or kinematic hardening laws (Chazallon et al., 2006). Some of these models have been used for finite element modelling of pave­ment. Now, the main concepts of these models are presented.

The model developed by Bonaquist is based on the plasticity model developed by Desai et al. (1986). These two models differ from each other by the simplified calculation method for large number of cycles. Consequently, the basis of the Desai model (Desai, 2002) is first presented and the different accelerated analysis proce­dures of each model are introduced.

The Desai formulation is based on the disturbed state concept which provides a unified model that includes various responses such as elastic, plastic, creep and micro-cracking. The idea is that the behaviour of a deforming material can be ex­pressed in terms of the behaviour of the relatively intact or continuum part and of the micro-cracked part. During the deformation, the initial material transforms continuously into the micro-cracked state and, at the limiting load, the entire ma­terial element approaches the fully micro-cracked state. The transformation of the material from one state to another occurs due to the micro-structural changes caused by relative motion such as translation, rotation and interpenetration of the particles and softening or healing at the microscopic scale. The disturbance expresses such micro-structural motions. Under repetitive loading, an accelerated procedure exists. From experimental cyclic tests, the relation between the deviatoric plastic strain trajectory and the number of loading cycles can be expressed as a power function of the number of cycle. Pavement finite element modelling feasibility has been carried out with this model (Desai, 2002). The к-0 model was used for the elastic part and the model requires the following parameters:

• Elastic behaviour: кь к2, v.

• Plastic behaviour:

о Rupture and characteristic parameters: 3R, n, y.

о Plasticity parameters: (3, ab nb Nr, b.

Bonaquist’s approximate accelerated analysis is based on the total plastic strain at the end of each cycle and defined by a power function of the cycle number which depends on the ratio: maximum deviatoric stress and the corresponding deviatoric stress at rupture (for the same q/p ratio). Instead of the Nr and b parameters a parameter %b is required.

The model developed by Chazallon and Hornych (Chazallon et al., 2006) is based on the model of Hujeux (1985) in its simplest formulation. This formulation is a non-associated elasto-plastic model and reproduces the saturated monotonic behaviour of sand and clay. A kinematic hardening has been added to reproduce the accumulation of plastic strains under repeated loadings. Each cycle is calculated, nevertheless, by a simplified approach based on the decoupling of the calculation of the elastic strains and the plastic strains. A pavement finite element modelling feasibility has been carried out with this model, see Hornych et al. (2007).

The elastic part is computed with the anisotropic Boyce model and the model requires the following parameters:

• Elastic behaviour: Ka, Ga, у and n.

• Plastic behaviour:

о Rupture and dilatancy parameters: C0, M, and Mc. о Monotonic plasticity parameters: в, PC0, a, b.

о Cyclic plasticity parameters: rf, Puc, Plc.(where the subscript l and u indi­cate loading and unloading, respectively – see Figs. 9.5 and 9.6).

Fig. 9.6 Representation of the influence of the Puc/ic parameters on plastic strains when an un­loading and a reloading occur (Chazallon et al., 2006) With permission from ASCE

A few material models have been proposed for the development of plastic strains in unbound granular materials in a pavement structure. Lashine et al. (1971), and Barksdale (1972) tested unbound granular material in a repeated load triaxial test for 100 000 cycles. They found that the permanent axial deformation, єpb at different

stress states is proportional to number of load cycles, N (Fig. 9.4). Since 1971, many analytical models have been developed and most of them are listed by Lekarp and Dawson (1998).

However, these models have never been used with finite element (FE) calcula­tions except the Hornych model (Hornych et al., 1993) which has been used with a simplified finite element calculation by de Buhan (Abdelkrim et al., 2003) for a railway track construction and by Hornych et al. (2007) for a full scale flexible pavement.

The mechanical processes that form the basis of a flexible pavement’s perfor­mance and of a flexible pavement’s deterioration can be separated into two cate­gories, namely:

(i) short-term mechanical processes; and

(ii) long-term mechanical processes.

The first category concerns the instantaneous behaviour of a flexible pavement, as activated during the passage of a vehicle, thus the flexible pavement behaviour can be studied by means of (visco)elastic models. de Buhan (Abdelkrim et al., 2003) and Hornych et al. (2007) use, respectively, linear moduli with a Boussinesq stress analysis and the modified Boyce models with a simple FE analysis for the short term behaviour. The second category concerns the mechanical processes, typically characterized by a quasi-static time-dependency, such as long-term settlements un­der a large number of vehicle axle passages. Together, the approach requires the following parameters:

• Elastic behaviour: E, v or Ka, Ga, у and n.

• Plastic behaviour:

о Rupture (i. e. shear failure) parameters: m, s.

p0

о Plasticity parameters: e B and n.

The Hornych model is another formulation of the Paute model (Paute et al., 1988) which was adopted also in an European norm – EN 13268-7 (CEN, 2000). Some other alternatives were suggested in Lekarp and Dawson (1998), but no overall framework has been established yet to explain completely the behaviour of unbound granular materials under complex repeated loading.

Routine pavement methods are mechanistic-empirical design methods, based on linear elastic calculations. Usually, the only rutting criterion to be used concerns the subgrade soil, and consists in limiting the vertical elastic strains at the top of the subgrade. Rarely is a criterion applied for the unbound granular layers although Dawson and Kolisoja (2004) have shown that in roads with thin bound layers, the road’s rutting will largely be the consequence of plastic deformation in the granu­lar layers. Similarly, it is rare for a plastic criterion to be used in design probably because of the much greater difficulty in computing plastic strain fields.

Advanced pavement models are based on the main tests for unbound granular materials: the monotonic triaxial test and the repeated load triaxial test (this test and the models’ calibrations are presented in Chapter 10).

These models are split in four categories:

• Analytical models;

• Plasticity theory based models;

• Visco-plastic equivalent models; and

• Shakedown models.

The above methods define stiffness as a function of stress alone. Full incorporation of the effects of moisture (as a pressure or suction) should necessitate use of an effective stress framework (see Section 9.5). However a more simple approach, at least in principle, is to adjust the stiffness value calculated by one of the above rela­tionships using a factor that is dependent on the moisture (and, perhaps, other) con­dition. The AASHTO ‘Mechanistic-Empirical Pavement Design Guide’ (MEPDG) takes this approach, though it’s attention to many details makes the implementation rather complex (ARA, 2004). In this approach, the reference stiffness value, Mropt (the value of Mr at optimum conditions), is adjusted by a factor, Fenv, to allow for different environmental effects, with the value of each factor being computed for each of a range of depths, lateral positions and time increments. For moisture the adjustment factor is based on the equation

Mr

Mr oP^ min

Where km is a material parameter and Sr and Sropt = the actual saturation and the saturation ratio at optimum conditions, respectively. The actual saturation value is obtained from the use of the Soil Water Characteristic Curve (SWCC) see Chapter 2, Section 2.7.1. Other adjustments are included in the Fenv factor to allow for freez­ing, thawing and temperature. The full approach is too detailed to include here. Interested readers are directed to the relevant report (ARA, 2004).

Long et al. (2006) take another approach, relating modulus to suction and water content rather than to saturation ratio, but still including some stress influence:

(p – 0 ■ 5) + ^t34^ (1 + V)(1 – 2v)

Yh(0.435) (1 – v)

where p is the mean normal stress on the element of soil, 0 and w are the volumetric and gravimetric water contents, respectively, 5 is the matric suction pressure, S0 is the slope of the soil desorptive curve (the rate of change of the logarithm of 5 with the logarithm of 0), Yh is the suction volumetric change index (an indicator of the sensitivity of volume change to change in matric suction) and v is Poisson’s ratio.

The behaviour of unbound granular materials in a pavement structure is stress – dependent. For that reason the linear elastic model is not very suitable. A non-linear elastic model, with an elastic modulus varying with the stress and strain level is, therefore, needed.

For isotropic materials, moduli depend only on two stress invariants1: the mean stress level, p, and the deviatoric stress, q, which are given in the general, as well as the axi-symmetric case (cylindrical state of stress with o1 = oaxial and o2 = o3 =

oradial as:

General

P = T

q = 2 0ij0ij with 0ij = 0ij pSij

In a similar way strain invariants can be introduced. The volumetric strain ev and the deviatoric strain eq, are defined as:

An invariant has the same value regardless of the orientation at which it is measured.

(9.3)

The stresses and strains are interconnected through the material properties as stated in Eq. 9.1 and the elastic (resilient) response of the material can be expressed according to Hooke’s law as a diagonal matrix:

where E and v are the material stiffness modulus (or the resilient modulus, usually denoted Mr) and Poisson’s ratio respectively, defined as:

Дq Дє3

Mr or E = and v = – (9.5)

Дє1 Дє1

where Д stands for the incremental change during the loading. An alternative for­mulation is:

where Kr and Gr are the bulk and shear moduli of the material. The bulk and the shear moduli are connected to the resilient modulus and the Poisson’s ratio through:

for isotropic materials.

The resilient modulus for most unbound pavement materials and soils is stress – dependent but the Poisson’s ratio is not, or at least to a much smaller extent. Biarez (1961) described the stress-dependent stress-strain behaviour of granular materials subjected to repeated loading. Independently, similar work was performed in the United States (Hicks and Monismith, 1971). Both results presented the к-0 model, which is written with dimensionless coefficients like:

3 p k2

Mr = ki pa and v = constant (9.8)

PaJ

where Mr is the resilient modulus, p is the mean stress, pa is the reference pres­sure (pa = 100 kPa) and k1, k2 are coefficients from a regression analyses usually based on repeated load triaxial test results. This model has been very popular for
describing non-linear resilient response of unbound granular materials. It assumes a constant Poisson’s ratio and that the resilient modulus is independent of the devia – toric stress. To address this latter limitation the Uzan-Witczak model – often called the “Universal” model – has become widely promoted, especially, in recent years, by authors in North America, e. g. Pan et al. (2006). It takes the form:

Mr = kpa 1 + and v = constant (9.9)

Pa PaJ

Subgrade soils are also stress-dependent and can also be modelled by one of the k-0 approaches. The principle difference between granular materials and many soils is that the former exhibit a strain-hardening stiffness whereas the latter, typically, exhibit strain-softening behaviour under transient stress loadings. In practice, the incorporation of non-linearity into the stiffness computations for subgrade soils is often less important than for granular materials as the stress pulses due to traffic loading will be a far smaller part of the full stress experienced by the subgrade than is the case for the unbound granular layer. Thus the error introduced by ignoring subgrade non-linearity will be correspondingly smaller.

In 1980, Boyce presented some basis for subsequent work on the stress-dependent modelling of the resilient response of cyclically loaded unbound granular material. The Boyce model takes into account both the mean stress and the deviatoric stress, with the bulk and shear moduli, K and G, of the material calculated as:

where Ka, Ga, and n are material parameters determined from curve fitting of re­peated load triaxial tests results.

Anisotropy of pavement materials is increasingly being recognised as a property that must be modelled if the pavement is to be adequately described (e. g. Seyhan et al., 2005). The Boyce model was modified to include anisotropy in the early 1990’s (Elhannani, 1991; Hornychetal., 1998). Hornych and co-workers ntroduced anisotropy by multiplying the principal vertical stress, cti in the expression of the elastic potential by a coefficient of anisotropy у so that p and q are redefined as follows (c. f. axi-symmetric part of Eq. 9.2):

and the stress-strain relationships are defined as:

Дєq* = 2 (ДЄ1* – Де/) = — and

q 3 1 1 ! 3Gr

Д n*

Дє„ * = Дє1*+ 2Де3* = – .

Kr

Kr and Gr, the bulk and shear moduli, respectively as:

The k— model, Universal model, Boyce model, and the modified Boyce model must be considered in pavement modelling to ensure a valid stress, strain, and deflection evaluation in pavements. When subjected to repeated loading, two types of defor­mations are exhibited, linear or non-linear elastic (or resilient) and plastic deforma­tions. Models based on non-linear elasticity deal with resilient deformations only. Their biggest disadvantage is that permanent deformations cannot be modelled.