INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2007; 31:1–22Published online 24 May 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.521

A numerical model for flexible pavements rut depthevolution with time

Fatima Allou1, Cyrille Chazallon1,*,y and Pierre Hornych2,z

1Laboratory of Mechanics and Modelling of Materials and Structures in Civil Engineering, University of Limoges,Boulevard Derche, 19300 Egletons, France

2Materials and Pavements Structures Division, Laboratoire Central des Ponts et Chaussees, Route de Bouaye BP4129,44341 Bouguenais cedex, France

SUMMARY

A simplified method has been developed for the finite elements modelling of flexible pavements rut depthevolution with time. This method is based on the shakedown theory established by Zarka for metallicstructures. The yield surface of Drucker-Prager and the plastic potential of Von Mises have been used. Thesimplified method determines straightforwardly the purely elastic state or the elastic shakedown state or theplastic shakedown state. The calibration of the simplified method with two unbound granular materials forroads under repeated loads triaxial tests, is explained. Then, a finite elements modelling of a flexiblepavement has been carried out. Calculations of 2D and 3D have been performed and rut depth evolutionswith time are shown, which underline the capabilities of the model to take into account the accumulation ofplastic strains along the loading cycles. Copyright # 2006 John Wiley & Sons, Ltd.

Received 13 September 2005; Accepted 14 February 2006

KEY WORDS: elastoplasticity; elastic shakedown; plastic shakedown; repeated load triaxial tests

1. INTRODUCTION

Pavement design is a process intended to find the most economical combination of layerthicknesses and material types for the pavement, taking into account the properties of the soilfoundation and the traffic to be carried during the service life of the road. In many countriestraditional design methods are more or less empirical but there is, worldwide, an increasingdesire to develop analytical approaches and finite element modelling including the non-linearbehaviour of unbound or bonded materials.

*Correspondence to: Cyrille Chazallon, Laboratory of Mechanics and Modelling of Materials and Structures in CivilEngineering, University of Limoges, Boulevard Derche, 19300 Egletons, France.yE-mail: chazallon@unilim.frzE-mail: pierre.hornych@lcpc.fr

Copyright # 2006 John Wiley & Sons, Ltd.

In many other branches of engineering much progress has been made in understanding theresponse of structures to repeated loading and the nature of long-term failure and wear.Particularly relevant to the analysis of pavements, is the progress that has been made in theunderstanding of wear processes of metal surfaces subjected to rolling loads, sliding loads andwear (see References [1,2], for example). This last analysis uses the concept of the shakedowntheory to determine the long-term behaviour of the surface. The basic assumption is that thevarious possible responses of an elastic–plastic structure to a cyclical load history depends onthe load level: if the load level is sufficiently small, the response is purely elastic, no permanentstrains are induced, and the structure returns to its original configuration after each loadapplication. However, if the load level exceeds the elastic limit load, permanent plastic strainsoccur. When the load exceeds the elastic limit, the structure can exhibit three long-termresponses depending on the load level:

* elastic shakedown, where a fully elastic closed loop is obtained, although the response maybe plastic for a finite number of initial load applications;

* plastic shakedown when the ultimate response is a closed plastic loop; and* ratchetting.

The shakedown concept applied to pavements has been introduced first by Sharp et al. [3].Contributions of Yu et al. [4] and Maier et al. [5] are based on the fact that if ratchetting occurs,the structure will fail. The critical load level below which the structure shakes down and abovewhich it fails is called the shakedown load and it is this parameter that is the key design load.The work of Habiballah and Chazallon [6] is based on the theory developed by Zarka andCasier [7] for metallic structures subjected to cyclic loadings. Zarka defines the plastic strains atelastic 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 simplifiedmethod. Habiballah has extended the previous results to unbound granular materials with anon-associated elastoplastic model. To improve the finite element modelling of a flexiblepavement, the model takes now into account the initial stress state of the material, and in orderto describe the time-dependent evolution of the rut depth, a parameter linking time and cyclenumber has been added. This last parameter is determined with a ‘multi-stage’ procedure,developed by Gidel et al. [8]. This procedure is described in Section 3.3.

After reviewing the main approaches used for modelling of flexible pavements, the paperdescribes the simplified method based on the shakedown theory. A comparison of calculationswith experimental results of repeated load triaxial tests on two unbound granular materials isdiscussed. Finally, the results of finite elements modelling of the long-term behaviour of aflexible pavement with the simplified model will be presented, where the rut depth evolution withthe number of cycle is taken into account.

2. FLEXIBLE PAVEMENT MODELS

Most pavement design methods are based on mechanistic empirical pavement analysis using amulti-layer linear elastic model. The design criterion is generally a fatigue criterion for theasphalt layer based on the maximum tensile strain at the bottom of the asphalt layer. The onlyrutting criterion concerns the sub-grade soil, and consists in limiting the vertical elastic strains atthe top of the sub-grade. No criterion is applied for the unbound granular layers.

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DOI: 10.1002/nag

Models for sub-grade and unbound layers in pavement engineering have been split in twocategories dealing with the main mechanical behaviour which has to be taken into account:

* resilient behaviour. In research applications, the resilient behaviour of unbound granularmaterials for roads is studied by laboratory repeated load triaxial tests and modelled withnon-linear elasticity [9]. This resilient behaviour is obtained in pavements when thegranular base is adapted (elastic shakedown) or accommodated (plastic shakedown),

* long term elastoplastic behaviour. These models are developed from results of repeated loadtriaxial tests and the monotonic triaxial tests. These models are split in two categories:

* analytical models. Such material models have been proposed for the development of plasticstrains in unbound granular materials in pavement structures: Barksdale [10] found thepermanent axial deformation, ep1 at different stress states being proportional to the numberof load cycles N. Since 1972, many analytical models have been developed and most ofthem have been listed in an extensive review performed by Lekarp and Dawson [11].However, these models have never been used in pavement structure calculations, except theHornych model [12], which has been used with a Boussinesq solution by de Abdelkrimet al. [13], for a railway track platform.

* plasticity theory-based models* classical time-independent plasticity requires the definition of isotropic hardening laws

and simplified accumulation rules [14,15], or kinematic hardening laws [16]. The last twomodels have been used for finite elements modelling of pavements.

* viscoplastic equivalent models based on the equivalence between time and number ofcycles, have been developed by Suiker and de Borst [17] for a railway track platform andMayoraz [18] for the laboratory study of a sand. They are based on the Perzynaviscoplastic concept [19].

In the next section, we will recall the simplified shakedown approach for UGM (unboundgranular materials), and we will present the modifications developed to take into account theinitial stress state and the long-term pavement performance.

2.1. The constitutive model

Let us consider an elastoplastic structure. Its boundary G is subjected to imposed surface forcesFdi ðx; tÞ in the GFi

partition and to prescribed surface displacements Udj ðx; tÞ in the GUj

partition.The body forces Xd

j ðx; tÞ and the initial strains eIij ðx; t ¼ 0Þ are defined in the volume V. Thisstructure is supposed to satisfy the theory of small displacements and deformations. The generalproblem can be solved with the finite elements method as follows:

eijðx; tÞ ¼Mijklsklðx; tÞ þ epijðx; tÞ þ eIijðx; 0Þ ð1Þ

where the actual strain tensor eijðx; tÞ is kinematically admissible with Udj ðx; tÞ on GUj

and theactual stress tensor sijðx; tÞ is statically admissible with Fd

i ðx; tÞ on GFiand with Xd

j ðx; tÞ in V;epijðx; tÞ is the plastic strain tensor, eIijðx; 0Þ is the initial strain tensor, Mijkl is the complianceelasticity matrix. The general problem can be decomposed into an elastic and inelastic part.

2.1.1. Elastic problem. The response associated with the elastic part is expressed as follows:

eelij ðx; tÞ ¼Mijklselklðx; tÞ þ eIijðx; 0Þ ð2Þ

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where the elastic strain tensor eelij ðx; tÞ is kinematically admissible with Udj ðx; tÞ on GUj

and theelastic stress tensor selij ðx; tÞ is statically admissible with Fd

i ðx; tÞ on GFiand with Xd

j ðx; tÞ in V.

2.1.2. Inelastic problem. The inelastic problem is obtained by difference between the total andthe elastic problem. It can be expressed by the following equation:

eineij ðx; tÞ ¼ eijðx; tÞ � eelij ðx; tÞ ¼MijklRijðx; tÞ þ epijðx; tÞ ð3Þ

where eineij ðx; tÞ is kinematically admissible with 0 on GUj:

The residual stress field Rijðx; tÞ is obtained by the difference between the actual and the elasticstress fields as follows:

Rijðx; tÞ ¼ sijðx; tÞ � selij ðx; tÞ ð4Þ

It is statically admissible with 0 on GFiand with 0 in V.

With the knowledge of the plastic strain tensor epijðx; tÞ and the compliance elasticity matrixMijkl ; the inelastic problem is solved with a null stress boundary condition and the inelasticstrain field eineij ðx; tÞ is obtained. Then, the residual stress tensor is calculated with the followingexpression:

Rijðx; tÞ ¼M�1ijklðe

ineij ðx; tÞ � epijðx; tÞÞ ð5Þ

The main idea of the Zarka method [7] is to introduce an internal structural parameter to givean estimate of the stabilized state and of the inelastic component for metallic structuressubjected to cyclic loading. This approach has been applied by Habiballah and Chazallon [6] topredict the inelastic behaviour of unbound granular materials under large cycle numbers. Thesame framework has been used here and the simplified method is summarized in the nextsection. The Drucker-Prager yield surface with linear kinematic hardening is used. Its expressiontakes the following form:

f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ðSij � yijÞðSij � yijÞ

q� aI1ðsijÞ � k ð6Þ

where yij ¼ ð2H=3Þepij is the kinematic hardening tensor, H is the hardening modulus, Sij is the

deviatoric part of the actual stress tensor sij ; I1ðsijÞ is the first stress invariant, a and k arematerial parameters. The flow rule is non-associated and the plastic potential of Von-Mises isused:

g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ðSij � yijÞðSij � yijÞ

qþ cste ð7Þ

The actual deviatoric stress can be written as

Sijðx; tÞ ¼ Selij ðx; tÞ þ devRijðx; tÞ ð8Þ

Substituting it in (6), we obtain the field of transformed structural parameters:

Yijðx; tÞ ¼ yijðx; tÞ � devRijðx; tÞ ð9Þ

Then, the yield surface in the deviatoric plane can be expressed as follows:

f ðSelij � YijÞ40 ð10Þ

In the plane of the transformed structural parameters, the yield surface boundary becomes acircle centred in Sel

ij :With the transformed structural parameters field, the inelastic problem can

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be written as

eineij ðx; tÞ ¼M0ijklRklðx; tÞ þ

3

2HYijðx; tÞ ð11Þ

where M0ijkl is the modified elasticity matrix, defined by the following equality:

M0ijkl ¼Mijkl þ

3

2Hdev ð12Þ

where (dev) is the deviatoric operator.Thus, we obtain the residual stress tensor of an elastoplastic structure with

Rijðx; tÞ ¼M0�1ijkl eineij ðx; tÞ �

3

2HYijðx; tÞ

� �ð13Þ

Finally, the plastic strain epij is given by

epijðx; tÞ ¼3

2HðYijðx; tÞ þ devRijðx; tÞÞ ð14Þ

2.1.3. Response of a structure subjected to a cyclic loading. During such a loading, the elasticresponse of the structure can be written:

selij ðx; tÞ ¼ ð1� LðtÞÞselijminðxÞ þ LðtÞselijmax

ðxÞ ð15Þ

where selijminðxÞ and selijmax

ðxÞ are the minimum and maximum values, respectively, of the cyclicloading, LðtÞ is a monotonic periodic function of time.

The local stresses at the level of the plastic mechanisms are expressed as

esijðx; tÞ ¼ Sijðx; tÞ � yijðx; tÞ ð16Þ

In the local stress plane, the plasticity convex domains, (C0)min at the minimum stress stateand (C0)max at the maximum stress state, are fixed cones which are reduced, in the deviatoricplane, to circles centred on the isotropic stress axis (Figure 1). The normality law is written withthe Moreau’s notation [20]:

’epij 2 @cðC0ÞminðesijÞ with esij 2 ðC0Þmin ð17Þ

@cðC0ÞminðesijÞ is the subdifferential to the convex (C0)min at esij ; where the plastic strain rate is an

external normal to the convex (C0)min.At the maximum stress state and using (9), the transformed structural parameter at the level

of the inelastic mechanism is

Yij ¼ �esij þ Selijmax

ð18Þ

with

Yij 2 CðSelijmaxÞ and CðSel

ijmaxÞ ¼ ðC0Þmax þ Sel

ijmax

Equation (18) implies that Yij belongs the convex set CðSelijmaxÞ obtained from ðC0Þmax with the

translation Selijmax

(Figure 1). The normality law is

’epij 2 �@cCðSelijmaxÞðYijÞ with Yij 2 CðSel

ijmaxÞ ð19Þ

�@cCðSelijmaxÞðYijÞ is the subdifferential to the convex CðSel

ijmaxÞ at Yijðx; tÞ; where the plastic strain

rate is an internal normal to the convex CðSelijmaxÞ: This convex is locally built for each plastic

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mechanism. Thus, in the transformed structural parameters plane, the yield surface is a circlecentred in Sel

ijmaxðx; tÞ:

The nature of the limit state of the structure will depend on the elastic response. According tothe loading amplitude DSel

ij ; the convex set CðSelij Þ ¼ ðC0Þ þ Sel

ij moves linearly between CðSelijminÞ

and CðSelijmaxÞ: Two situations exist:

* elastic shakedown will occur when those two sets have a common part C1;* otherwise, plastic shakedown occurs.

2.1.4. Elastic shakedown. At each point of a structure in an elastic shakedown situation, theinitial transformed structural parameters ðY0Þij can be transported with the movement of theplastic convex. Three cases can be obtained:

* (Y0)ij is inside C1 and remains immobile (Figure 2(a));* (Y0)ij is such that after the first cycle, it reaches the boundary of C1 and remains immobile

(Figure 2(b)); and* (Y0)ij is transported with the movements of the convex to finish on the boundary C1 or C

01

(Figure 2(c)). In this case, the stabilized state is reached after several cycles.

Thus, the final position ðY1Þij determines the final cycle, which solves the inelastic problemand the general problem.

2.1.5. Plastic shakedown. The solution is obtained from geometrical considerations [7,6]. In thetransformed structural parameters plane, Yijmax

and Yijminbelong to extreme positions of the two

convex centred in Selijmax

and Selijmin; respectively. The final cycle is defined by the mean value

Figure 1. Evolution of plasticity criteria convex in the ðp; qÞ plane, deviatoric plane and in the transformedstructural parameters plane.

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ðepijÞmean and the range Depij : Thus, the values of the DYij and ðYijÞmean fields are, respectively,

DYijðxÞ ¼ DSelij ðxÞ 1�

%rminðxÞ þ %rmaxðxÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ðDSel

ij ðxÞDSelij ðxÞÞ

q0B@

1CA ð20Þ

and

ðYijÞmeanðxÞ ¼DSel

ij ðxÞ

21þ

%rminðxÞ � %rmaxðxÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ðDSel

ij ðxÞDSelij ðxÞÞ

q0B@

1CAþ Sel

ijminðxÞ ð21Þ

where %rmin and %rmax are the radii of the two convex centred in Selijmin

and Selijmax

(see Figure 3). Theirvalues are

%rminðxÞ ¼ affiffiffi3

pI1ðselijmin

ðxÞÞ þ kffiffiffi3

pð22Þ

%rmaxðxÞ ¼ affiffiffi3

pI1ðselijmax

ðxÞÞ þ kffiffiffi3

pð23Þ

Modifications have been added in order to describe the rut depth evolution with time. For that,a new function FðNÞ is defined, which is applied to the stabilized plastic deformation as follows:

epijðx;NÞ ¼ FðNÞ:epijðxÞ ð24Þ

The next sections present the determination of the model parameters.

(a) (b) (c)

Figure 2. Cases to asses ðY1Þij :

Figure 3. Value of Yijmeanand DYij during plastic shakedown.

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3. EVALUATION OF MODEL PARAMETERS

The accuracy of any numerical simulation depends not only on the capabilities of theconstitutive model, but also on the quality of the associated material parameters. The simplifiedmethod requires the elasticity parameters, Drucker-Prager parameters (the elasticity coneaperture c and the apex of the Drucker-Prager cone on the isotropic stress axis pn), thehardening modulus and the function FðNÞ: In order to determine the model parameters,monotonic and cyclic triaxial tests have been performed on Missillac sand (0/4mm) and on amicro-granite (0/10mm) from the quarry of Poulmarch by Hornych [21].

The main characteristics of these two materials are shown in Table I:

3.1. Elasticity parameters

The k–y model [22] is used to describe the resilient behaviour of UGM in repeated load triaxialtests. Its expression is

E ¼ k13p

pa

� �k2

ð25Þ

where E is the resilient modulus, p is the mean normal stress, pa is reference pressure equal to100 kPa; k1 and k2 are model parameters. In this model, the Poisson’s ratio n is constant.

The model parameters are obtained using repeated load triaxial test results. The testprocedure consists of a cyclic conditioning following a stress path with a high level of stress, tostabilize the permanent strains of the material and to attain the resilient behaviour. Then, shortloadings are performed to study the resilient behaviour.

With the k–y model and a given stress path, we are able to solve the elastic problem (Section2.1.1) with a step-by-step calculation and to obtain the initial and the final plasticity convexpositions. Then the inelastic problem (Section 2.1.2) is solved using a constant Poisson ratio andthe Young modulus given by

E ¼9GDKD

3KD þ GDwhere KD ¼

pmax � pmin

evmax � evminand GD ¼

qmax � qmin

3ðedmax � edminÞð26Þ

pmin, pmax and qmin, qmax are the minimum and maximum mean stresses (respectively deviatoricstresses) of the stress path respectively. evmin; evmax and edmin; edmax are the minimum andmaximum volumetric strains (respectively deviatoric strains) of the corresponding stress states.

Table I. Mechanical characteristics.

Material LA MDE Fines (%) rdOP (kg/m3) wOP (%)

Poulmarch 20 13 7.8 2200 6.3Missillac – – 7.5 2040 9.2

LA: Los Angeles value; MDE: Micro Deval test; rdOP and wOP: optimum dry density and watercontent (standard Proctor test for the sand, modified Proctor test for the granular materials).

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DOI: 10.1002/nag

3.2. Drucker-prager parameters

The parameter pn ¼ k=3a is the elasticity cone apex position on the isotropic axis. It is identifiedby the failure line of three monotonic triaxial tests. These tests give also the slope of thecharacteristic line: jdil separating contractant and dilatant behaviour. The parameter c ¼artgð3a

ffiffiffi3

pÞ represents the elasticity cone aperture in the (p,q) stress space. It is determined to

obtain a reduced initial elastic domain just before plastic flow, where the elastic strain must beequal to 10�5 for low stress ratios q/p. The Drucker-Prager parameters adopted for the Missillacsand and the Poulmarch micro-granite are listed in Table II.

3.3. Plasticity parameters

The plastic calculation requires two parameters: the hardening modulus H and the functionFðNÞ: These two parameters require an adjustment on repeated load triaxial tests results, withdifferent stress ratios. Two series of tests were conducted on the Poulmarch and Missillacmaterials. The multi-stage test procedure, developed by Gidel et al. [8], consists, in eachpermanent deformation test, to perform successively several cyclic load sequences, following thesame stress path, with the same q/p ratio, but with increasing stress amplitudes (Figure 4):

* stress ratios of 1, 1.5, 2, 2.5 and 3 for the Poulmarch material.* stress ratios of 1, 1.5, 2 and 3 for the Missillac sand.

Each loading stage was applied for 10 000 cycles.In Figure 5, for both materials, the permanent deformations tend to stabilize at the end of

each loading stage, except the last two stages of the stress paths q=p ¼ 2:5 and 3 for thePoulmarch material. For these two cases, the number of loading cycles is insufficient to reach the

Table II. Parameters for the Drucker-Prager model.

Material c (8) pn (kPa) jdil (8)

Poulmarch 15 15 54Missillac 15 12.8 41

0

20

40

60

80

100

120

140

0 50 100 150

p (kPa)

q (k

Pa)

q/p = 1

q/p = 1.5

q/p = 2

q/p = 3

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600

p (kPa)

q/p = 1q/p = 1.5q/p = 2q/p = 2.5q/p = 3

q (k

Pa)

(a) (b)

Figure 4. Cyclic loads applied during the staged loading tests: (a) Missillac sand;and (b) Poulmarch material.

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stabilized state. Hornych et al. [12] suggests the following equation to relate permanent axialdeformation with the number of cycles:

ep1 ¼ FðNÞ:A ð27Þ

where

FðNÞ ¼ ½1� ðN=100Þ�B� ð28Þ

ep1 is the vertical plastic deformation, N is the number of cycle, A is the limit value of ep1 when Ntends towards the infinite, B controls the shape of the plastic strains curve.

This model has been calibrated with the previous multi-stage triaxial test results. Thesimplified elastoplastic model is based on the shakedown theory and gives the stabilized plasticstrains. Thus, the hardening modulus calibration has to be performed using limit state plasticstrains (for N infinite). The calibration is shown Figures 6(a) and (b), with the experimentalplastic strain at the end of each stress stage and the model (27) at N ¼ 10 000 for thecorresponding sets of parameters ðA;BÞ: The vertical plastic strain ep1 increases as the mean stressDp and the stress ratio q=p increase.

The stabilized plastic strain, elasticity and Drucker-Prager parameters, are used to determinethe hardening modulus and the parameter B, with the simplified method. A linear evolution ofthe hardening modulus with the stress path length is assumed for each stress ratio q=p; in the

Logpmin

Dp

� �;Log

H

pa�Lmin

L

� �� �

plane (Figure 7), where Lmin ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2min þ q2min

qand L ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDp2 þ Dq2

p:

Thus, the hardening modulus is expressed hereafter:

Log10H

pa

� �¼ ða� 1Þ � Log

pmin

Dp

� ��

1

2Log

q

p

� �2

min

þ1

DqDp

� �2

þ1

0BBB@

1CCCAþ b ð29Þ

where a and b are material parameters, pa is the atmospheric pressure.

0

2

4

6

8

10

12

14

16

0 10000 20000 30000 40000 50000 60000

Number of cycles

Perm

anen

t axi

al d

efor

mat

ion

(10-3

)

q/p = 1

q/p = 1.5

q/p = 2

q/p = 3

0

5

10

15

20

40

45

5 000 10 000 15000 20 000 25 000 30 000 35 000 40 000 45 000

Number of cycles

q/p=1

q/p=1.5

q/p=2

q/p=2.5

q/p=

(a) (b)

325

30

35

Plas

tic a

xial

str

ain

(10-3

)

Figure 5. Variation of permanent axial deformation with the number of cycles: (a) Missillac sand;and (b) Poulmarch material.

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Parameters a and b are determined with linear regressions which are functions of the q=pratio.

In (29), the hardening modulus is function of the applied stresses and of the initial stress stateof the material. The relative position of the stress path with respect to the dilatancy line and itsinfluence on the amount of vertical plastic strain is taken into account in the evolution law ofparameters a and b by means of a break point, for a value of q/p corresponding to the slope ofthe characteristic line (Figures 8 and 9).

To estimate the rut depth evolution with time (number of cycles), the same approach is usedto determine the evolution law of B. Experimental results indicate a linear evolution of theparameter B with the stress path length, the applied stress and the initial stress state of thematerial for each stress ratio q=p; in the

pmin

Dp

� �; B �

Lmin

L

� �� �

plane (Figures 10(a) and (b)).

∆p(kPa) ∆p(kPa)

Perm

anen

t axi

al d

efor

mat

ion

(10-3

)

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120 140

q/p = 1

q/p = 1.5

q/p = 2

q/p = 3model

0

5

10

15

20

25

30

35

40

45

0 200 400 600

q/p = 1

q/p = 1.5

q/p = 2

q/p = 2.5

q/p = 3

model

Perm

anen

t axi

al d

efor

mat

ion

(10-3

)

(a) (b)

Figure 6. Calibration of the sets of parameters ðA;BÞ of the Hornych model at the end of each stress stage:(a) Missillac sand; and (b) Poulmarch UGM.

y = 0.9935x + 4.8574

R2 = 0.9904 (q /p =1)

y = 1.5076x + 4.6932

R2 = 0.9628 (q /p =2)

y = 1.8038x + 4.1752

R2 = 0.9981 (q /p =3)

y = 1.2814x + 4.8603

R2 = 0.9667 (q /p =1.5)

3.2

3.4

3.6

3.8

4

4.2

4.4

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Log

[(H

/pa)

(Lm

in/L

)] y = 1.3007x + 5.7097

R2 = 0.9996 (q/p=1)

y = 1.38x + 5.5304 R2 = 0.9926 (q/p=1.5)

y = 1.5553x + 5.5083 R2 = 0.9957 (q/p=2)

y = 2.1494x + 5.5764 R2 = 0.978 (q/p=2.5)

y = 2.5885x + 5.1286 R2 = 0.9753 (q/p=3)

2.5

2.7

2.9

3.1

3.3

3.5

3.7

3.9

4.1

4.3

4.5

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Log

[(H

/pa)

(Lm

in/L

)]

Log (p min / ∆p)Log (p min / ∆p)

Figure 7. Evolution law of the hardening modulus for each q/p ratio.

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a=0.2962(q/p)+0.9152R2= 1 (q/p > 1.7)

a=0.6014(q/p)+0.3893R2= 0.9983 (q/p < 1.7)

0.8

1

1.2

1.4

1.6

1.8

2

0 1 4

q/p

q/p=1.7 b = 0.0058 (q /p ) + 4.8516

R2 = 1 (q /p <1.7)

b = -0 .5254( q/ p) + 5.75

R2 = 0.9998 (q /p > 1.7)

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4

q/p

q/p=1.7

a b

(a) (b)

2 3 0 1 2 3

Figure 8. Variation of the parameters a and b with q/p for the Missillac sand.

a= 0.2546(q/p)+1.0301 R2=0.9746 (q/p<2.24)

a= 1.3188(q/p)- 1.3833 R

2=0.994 (q/p>2.24)

0

0.5

1

1.5

2

2.5

3

0 1 4

q/p

q/p=2.24

b= -0.454(q/p)+6.4906 R2= 1 (q/p< 2.24)

b=-0.1365(q/p)+5.78R2= 0.8081 (q/p >2.24)

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

q/p = 2.24

a b

2 30 1 4

q/p2 3

(a) (b)

Figure 9. Variation of the parameters a and b with q/p for the Poulmarch material.

B.(

Lm

in/ L

)

B.(

Lm

in/ L

)

pmin / ∆p pmin / ∆p

y R2 = 1 (q/p=1)

y = 0.3868x+ 0.0248 R2 = 0.9984 (q/p=1.5)

y = 0.2641x + 0.0419 R2 = 0.9993 (q/p=3)

y = 0.3599x + 0.0147 R2 = 0.9987 (q/p=2)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8

R2 = 0.9989 (q/p=1)

R2 = 0.9843 (q/p =1.5)

y = 0.2528x - 0.0053 R2 = 0.9979(q/p=2)

y = 0.193x- 0.0086 R2 = 0.9979 (q/p=2.5)

y = 0.2019x - 0.0215 R2 = 0.9999(q/p =3)

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

(b)(a)

y = 0.2338x + 0.0006

y = 0.3258x - 0.0064 = 0.5569x + 0.0025

Figure 10. Evolution law of the parameter B: (a) Missillac sand; and (b) Poulmarch UGM.

F. ALLOU, C. CHAZALLON AND P. HORNYCH12

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DOI: 10.1002/nag

Thus, B can be expressed by

B ¼ c:

DqDp

� �2

þ1

q

p

� �2

min

þ1

0BBB@

1CCCA

1=2

þd:L

Lminð30Þ

where c and d are material parameters.These two parameters are defined with linear regressions, which are functions of the q=p ratio.

Experiments do not show a clear dependence of the parameters c and d with q/p because theparameter B has not an understandable rheological meaning (Figures 11 and 12).

4. VALIDATION

In this section, a numerical simulation is presented to demonstrate the capabilities of theproposed model. The simplified method has been implemented in the non-linear finite elementscode CAST3M [23].

d = 0.0446 (q /p ) - 0. 0421R2 = 1 1<q/p<1.5

d = -0 .0202 (q / p)+ 0.0551R2 = 1 1.5<q/p<2

d = 0. 0272(q /p ) - 0.0397R2 = 1 2<q/ p<3

0

0.01

0 .0 2

0 .0 3

0 .0 4

0.0 5

0 1 2 3 4q/ p

c = -0.3402(q/p) + 0.8971 R2 = 1 1<q/p<1.5

c = -0 .0 958(q / p)+ 0. 5515

R2 = 1 2<q/p<3

c = -0 .0 538( q/ p) + 0. 46 75R2 = 1 1.5<q/ p<2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4

q/ p

c d

(b)(a)

Figure 11. Variation of the parameters c and d with q/p for the Missillac sand.

c = 0. 184(q /p) + 0. 04 98R2 = 1 1<q/ p<1.5

c = 0 .010 6(q/p) + 0.1701R2 = 1 2.5<q/p<3

c= -0.146(q/p) + 0.5448

R2 = 1 1.5<q/p<2

c = -0 .1124(q /p) + 0. 4776

R2 = 1 2<q/ p<2. 5

0

0 .0 5

0 .1

0. 15

0 .2

0 .2 5

0 .3

0 .3 5

0 1 2 3 4q/p

d = -0.014(q/p)+ 0 .0 14 6

R2 = 1 1<q/ p<1.5

d = -0 .0 24 6(q/p) + 0.0523

R2= 1 2.5<q/p<3

d = 0 .0 02 2 (q /p ) - 0 .00 97R2 = 1 1.5<q/ p<2

d = -0.00 78 (q /p ) + 0 .0 10 3R2 = 1 2<q/p<2.5

-0 .02 4

-0 .0 19

-0 .0 14

-0.00 9

-0.00 4

0 .0 010 1 2 3 4

q/p

c d

(a) (b)

Figure 12. Variation of the parameters c and d with q/p for the Poulmarch material.

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DOI: 10.1002/nag

The previous repeated load triaxial tests have been modelled, and Figures 13 and 14 show acomparison between the computed plastic strains with the proposed evolution law of thehardening modulus and the stabilized plastic strains. The best predictions are obtained for thestress paths q=p ¼ 1 and 3, for the Missillac material and for the stress paths q=p ¼ 1; 1.5, 2, forthe Poulmarch material. Even for the other stress paths a reasonable agreement withexperimental results is obtained.

With the evolution law of FðNÞ; the plastic strain can be determined for different numbers ofcycles, for several stages, with various q=p ratios. Figures 15–17 show a comparison between themeasured and calculated deformations. It can be seen that the proposed approach gives fairlygood results. In the tests q=p ¼ 1; 1.5, 3, the difference between the measured and calculated

(kPa)p∆

Perm

anen

t axi

al d

efor

mat

ion

(10-3

)

0

2

4

6

8

10

12

14

16

18

0 20 40 60 80 100 120 140

q/p = 1

q/p = 1.5

q/p = 2

q/p = 3

model

Figure 13. Comparison between the model (limit state) and the experimental results for ep1 (Missillac sand).

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600

q /p = 1

q /p = 1. 5

q /p = 2

q /p = 2. 5

q /p = 3

model

)(kPap∆

Perm

anen

t axi

al d

efor

mat

ion

(10-3

)

Figure 14. Comparison between the model (limit state) and the experimental resultsfor ep1 (Poulmarch material).

F. ALLOU, C. CHAZALLON AND P. HORNYCH14

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2007; 31:1–22

DOI: 10.1002/nag

0

2

4

6

8

10

12

14

16

0 20000 40000 60000Number of cycles

Pla

stic

axi

al s

trai

n (1

0-3

)q/p = 1

q/p = 1.5

q/p = 2

q/p = 3

model

500003000010000

Figure 15. Comparison between the model and the experimental results (Missillac sand).

0

2

4

6

8

10

12

14

16

18

10 000 20 000 30 000 40 000 50000 Number of cycles

Plas

tic a

xial

str

ain

(10-3

)

q/p = 1

q/p = 1 .5

q/p = 2

model

Figure 16. Comparison between the model and the experimental results (Poulmarch material),stress path ratios q=p ¼ 1; 1.5 and 2.

0

5

10

15

20

25

30

35

40

45

10000 20000 30000 40000 50000 Number of cycle

Axi

al p

last

ic s

trai

n (1

0-3)

q/p = 2 .5

q/p = 3

m o del

Figure 17. Comparison between the model and the experimental results (Poulmarch material),stress path ratios q=p ¼ 2:5 and 3.

FLEXIBLE PAVEMENTS RUT DEPTH EVOLUTION WITH TIME 15

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DOI: 10.1002/nag

deformations, for the Missillac sand, is less than 2� 10�4, except in the test with q=p ¼ 2; wherethe difference is less than 8� 10�4. For Poulmarch, the difference is 4� 10�4 with the stressratios q=p ¼ 1; 1.5, 2. With the stress ratios q=p ¼ 2:5 and 3, the difference is greater especiallyfor the highest stress levels, where no stabilization of the vertical plastic strain is observed, witha maximum value of 2� 10�3.

Thus, the proposed simplified method, which requires the evolution law of the hardeningmodulus and the function FðNÞ for each stress path, can be used to model the behaviour ofUGM material in road layers, as a function of the number of load applications.

5. NUMERICAL MODELLING OF A FLEXIBLE PAVEMENT STRUCTURE

The objective of this part is to present the results of the modelling of a pavement structure withthe simplified method and to predict the rut depth evolution with time. The finite elementsmodelling is performed with the finite elements code CAST3M involving three steps:

* the first step is the pre-processing where the finite elements mesh is generated, load andboundary conditions are assigned and material properties are defined;

* the second step is the elastic analysis where the minimum and maximum stress fields arecomputed; and

* the third step is the calculation of the plastic displacement and strain fields.

5.1. The finite elements analysis

The pavement structure consists of a 40mm of asphalt surface layer placed on 450mm ofPoulmarch unbound granular base over 3.51m of Missillac sub-grade soil. The 2D and 3Dpavement configurations are shown in Figures 18 and 19. In the 2D case, the pavement is

Figure 18. The 2D finite element mesh for pavement simulation (2D).

F. ALLOU, C. CHAZALLON AND P. HORNYCH16

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2007; 31:1–22

DOI: 10.1002/nag

discretized into 2600 eight-noded axi-symmetric quadrilateral finite elements. Due to thesymmetry, the 3D calculation is carried out on a quarter of the structure; 1000 cubical elementswith 20 nodes are used.

First, initial stresses in the pavement are determined with the materials unit weights and thelateral stress coefficient K0, which is assumed equal to 0.5. Then the pavement is subjected tocyclic applications of the French Standard axle load (dual wheel half-axle loaded at 65 kN). In3D analysis, the contact area between the tyre and the pavement can be represented by twosemicircles and a rectangle (Figure 19). This shape is converted to an equivalent rectangle,having an area of 0.5227L2 and a width of 0.6L, where L is a function of the load ðQ ¼ 65 kNÞand the tire pressure (P): L ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ=ðP � 0:5227Þ

p[24]. In 2D analysis, a circular shape is chosen so

that: R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ=ðP � pÞ

p:

The geometry and boundary conditions of the finite element simulations of the pavement areshown in Figure 18.

5.2. Material properties

The asphalt layer is considered homogeneous and linear elastic with a Young modulusE ¼ 5400MPa and a Poisson ratio n ¼ 0:35: The granular layer and the sub-grade areconsidered non-linear and described using the k–y model with k–y values obtained by Hornych[21] (Table III).

Figure 19. The 3D finite element mesh for pavement simulation and contact area betweentyre and pavement surface.

Table III. Parameters of the k–y model.

k1 (Mpa) k2 n g (kN/m3)

Poulmarch 58 0.623 0.25 22Missillac 102.8 0.0 0.2 20.4

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DOI: 10.1002/nag

The field of transformed structural parameters is determined using the maximum andminimum values of the stress fields, Sel

minðxÞ and SelmaxðxÞ; respectively, and the Drucker-Prager

parameters (Table II). Finally, the plastic displacements and strain fields are determined withthe evolution laws of the hardening modulus and of the function FðNÞ; presented in Figures7–12. The Young modulus field is given by (26) with a constant Poisson ratio.

5.3. Evolution of permanent deformations with the number of cycles

Using the simplified method and the proposed evolution laws, the inelastic displacement andplastic strain fields can be calculated. Figures 20 and 21 show the plastic strain fields obtained inthe 2D and 3D analysis, where the maximum values A1 (in the base course) and A2 (in the soil)are presented in Tables IV and V. The maximum value of the plastic strain is found in themiddle of the base course. In the sub-grade, the maximum value is at the top. The shakedowndomain (elastic and plastic) extends till 2m depth.

The plastic strains increase with the number of cycles. From N ¼ 10 000 to thestabilized state, the strains increase by 30% in the base layer, and by only 1% in thesub-grade.

Figure 20. Vertical plastic strain versus load cycle number (2D calculation): (a) N=10 000;(b) N=100 000; (c) N=1000 000; and (d) N=limit state.

F. ALLOU, C. CHAZALLON AND P. HORNYCH18

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2007; 31:1–22

DOI: 10.1002/nag

Figure 21. Vertical plastic strain field versus load cycle number (3D calculation): (a) N=10 000;(b) N=100 000; (c) N=1000 000; and (d) limit state.

FLEXIBLE PAVEMENTS RUT DEPTH EVOLUTION WITH TIME 19

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DOI: 10.1002/nag

5.4. Rut depth calculation in a typical flexible pavement

Figure 22 shows the maximum rut depth obtained in the 2D and 3D analysis, under the wheelcentreline, for three load values, 676, 696 and 716 kPa after 104, 105 and 106 cycles and at thelimit state. The rut depth increases with the number of load applications. For the three loads,the difference between the value of the rut depth at N ¼ 104 and the stabilized state is about19%, and 5% between N ¼ 106 and the stabilized state.

We can see that:

* there is a slight difference between 2D and 3D analysis, which does not exceed 6%. Thedifference between the two calculations is the shape of the contact area of the tire, which iscircular in 2D calculation and rectangular in 3D calculation.

* the 2D and 3D calculations show the tire pressure effect on the rut depth. For the same axleload (65 kN), an increase of the tire pressure creates a decrease of the contact area and anincrease of the rut depth. The same effect has been obtained by Hornych [21] on theFatigue Test Track of the L.C.P.C. in Nantes.

It should be noted that these analyses have been carried out under simple conditions:

* all the unbound materials remain at the optimum density* the Young modulus of the asphalt layer remains constant all along the cyclic loadings* the effect of water movements and their influence on moisture in unbound layers during the

pavement’s life is not taken into account.

6. CONCLUSION

Many analytical permanent deformation models have been developed and calibrated onrepeated load triaxial tests, but they remain unusable, with their present equations, to calculate

Table IV. Values of the maximum of the plastic strain (2D) (Pressure: 676 kPa).

Number of cycle 10 000 100 000 1 000 000 Limit state

A1 (10�3)

Z ¼ 0:14m 12.24 14.60 15.87 17.30

A2 (10�3)

Z ¼ 0:49m 6.50 7.14 7.37 7.55

Table V. Values of the maximum of the plastic strain (3D) (Pressure: 676 kPa).

Number of cycle 10 000 100 000 1 000 000 Limit state

A1 (10�3)

Z ¼ 0:14m 10.72 13.06 14.32 15.76

A2 (10�3)

Z ¼ 0:52m 7.46 7.62 7.66 7.68

F. ALLOU, C. CHAZALLON AND P. HORNYCH20

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2007; 31:1–22

DOI: 10.1002/nag

rutting of unbound granular layers of flexible pavements under large numbers of traffic loads.On the other hand, traditional incremental theory of elastoplasticity leads to a prohibitedly largenumber of time steps within each individual load cycle in the context of solving realisticboundary value problems with finite element modelling. In order to reduce the calculation timeof pavement finite element modelling within the framework of elastoplasticity, a simplifiedmethod based on the shakedown theory is developed.

This model requires three elasticity parameters and four plasticity parameters. Elasticityparameters are determined with the experimental procedure for the study of the resilientbehaviour of the materials, whereas plasticity parameters require the determination of themacroscopic cohesion and of the hardening modulus. Then, the evolution law of the hardeningmodulus and the temporal function F are determined with the calculation of the stabilizedplastic strain which requires a multi-stage procedure to test UGM under various stress pathsand at different stress levels. Finally, the finite elements modelling of a flexible pavement ispresented. This model captures the general trend of the mechanical behaviour of the structuresubjected to cyclic loading. The time influence on the rut depth is taken into account and now,the study has to be widened to other granular materials, tested at different water content. Thislast aspect should lead to take into consideration hydro-mechanical coupling.

3.5

4

4.5

5

5.5

6

6.5

0.0E+00 2.0E+05 4.0E+05 6.0E+05 8.0E+05 1.0E+06 1.2E+06 1.4E+06 1.6E+06

Number of cycles

Rut

dep

th (

mm

)

2D - 676kPa 3D - 676kPa 2D - LS - 676kPa 3D - LS - 676kPa

2D - 696kPa 3D - 696kPa 2D - LS - 696kPa 3D - LS - 696kPa

2D - 716kPa 3D - 716kPa 2D - LS - 716kPa 3D - LS - 716kPa

Figure 22. Rut depth evolution and limit state (LS).

FLEXIBLE PAVEMENTS RUT DEPTH EVOLUTION WITH TIME 21

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DOI: 10.1002/nag

Although the structural analysis is very encouraging, comparisons of the model predictionswith the results of a full-scale instrumented experiment need to be performed. Finally, thecalculation times obtained with this method are very satisfactory. For the 2D modelling, in spiteof the large finite elements number taken into account for structural analysis (2600), thecalculation time is about 15min, on a PC.

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Journal for Numerical Methods in Engineering 1980; 15:225–245.3. Sharp R, Booker J. Shakedown of pavements under moving surface load. Journal of Transportation Engineering

1984; 110(1):1–14.4. Yu HS, Hossain MZ. Lower bound shakedown analysis of layered pavements discontinuous stress fields. Computer

Methods in Applied Mechanics and Engineering 1998; 167:209–222.5. Maier G, Pastor J, Ponter ARS, Weichert D. Direct methods of limit and shakedown analysis. In Numerical and

Computational Methods, Chapter 12, vol. 3, de Borst R, Mang HA (eds). Elsevier, Pergamon: Amsterdam, 2003.6. Habiballah TM, Chazallon C. Cyclic plasticity based model for the unbound granular materials permanent strains

modelling of flexible pavements. International Journal for Numerical and Analytical Methods in Geomechanics 2005;29:577–596.

7. Zarka J, Casier J. Elastic plastic response of structure to cyclic loading: practical rules. Mechanics Today, vol. 6,Nemat-Nasser (ed.). Pergamon Press: Oxford, 1979; 93–198.

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traitees. Bulletin de Liaison des LPC 1993; 184:45–55.13. Abdelkrim M, De Buhan P, Bonnet G. A computational procedure for predicting the long term residual settlement

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structures. Proceedings of the 4th International Conference on Asphalt Pavements, Seattle, Washington, U.S.A., 1997;783–802.

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16. Chazallon C, Hornych P, Mouhoubi S. An elastoplastic model for the long term behaviour modelling of unboundgranular materials in flexible pavements. International Journal of Geomechanics (ASCE), accepted.

17. Suiker ASJ, de Borst R. A numerical model for cyclic deterioration of railways tracks. International Journal forNumerical Methods in Engineering 2003; 57:441–470.

18. Mayoraz F. Comportement mecanique des milieux granulaires sous sollicitations cycliques: application auxfondations des chaussees souples. These de Doctorat, Ecole Polytechnique Federale de Lausanne, 2002.

19. Perzyna P. Fundamental problems in viscoplasticity. Record in Advanced Applied Mechanic 1966; 9:243–377.20. Moreau JJ. Rafle par un convexe variable. Seminaire unilaterale, Universite de Montpellier, June 1971.21. Hornych P. Rapport Interne Confidentiel, L.C.P.C. de Nantes, 2003.22. Hicks RG, Monismith CL. Prediction of the resilient response of pavement containing granular layers using

nonlinear elastic theory. International Conference on Asphalt Pavement 1972; 1:410–429.23. CAST3M, see web site http://www-cast3m.cea.fr/cast3m. CAST3M is a research FEM environment; its development

is sponsored by the French Atomic Energy Commission, 2004.24. Huang YH. Pavement Analysis and Design. Prentice-Hall: Englewood Cliffs, NJ, 1993.

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DOI: 10.1002/nag