SAMARIS. Draft report

Transcription

1 SAMARIS Work Package 5 - Performance based secifications Selection and evaluation of models for rediction of ermanent deformations of unbound granular materials in road avements Draft reort Pierre Hornych Absamad El Abd LCPC Pavement Materials Divison February 24

2 . INTRODUCTION MECHANICAL BEHAVIOUR OF UNBOUND GRANULAR MATERIALS PERFORMANCE OF UNBOUND GRANULAR MATERIALS IN PAVEMENTS CYCLIC BEHAVIOUR OF UNBOUND GRANULAR MATERIALS RESILIENT BEHAVIOUR The non-linear elastic Boyce model Simlified descrition using the K-Theta model PERMANENT DEFORMATION BEHAVIOUR Exerimental behaviour Influence of stress level Influence of material characteristics MODELLING OF PERMANENT DEFORMATIONS OF UNBOUND GRANULAR MATERIALS EMPIRICAL RELATIONSHIPS Relationshis describing the influence of the number of load cycles Relationshis describing the influence of the level of stress ELASTO-PLASTIC MODELS The elasto-lastic model of Chazallon EVALUATION OF SELECTED PERMANENT DEFORMATION MODELS EMPIRICAL APPROACHES Variation with stress level Variation with the number of load cycles ELASTO-PLASTIC MODEL Prediction of monotonic triaxial tests Prediction of cyclic triaxial tests METHODS OF PREDICTION OF PERMANENT DEFORMATIONS IN PAVEMENT STRUCTURES GENERAL PRINCIPLES AND ASSUMPTIONS MODELLING OF RESILIENT BEHAVIOUR AND DETERMINATION OF STRESS PATHS IN GRANULAR PAVEMENT LAYERS Finite element modelling of the resilient behaviour of avement structures Tyical stress aths under moving wheel loads PREDICTION OF PERMANENT DEFORMATIONS Evaluation of the risk of rutting using a stress criterion Simlified layer strain rocedures Rational rut deth rediction methods EVALUATION OF A SIMPLIFIED RUT DEPTH PREDICTION METHOD BIBLIOGRAPHIC REFERENCES

3 Selection and evaluation of models for rediction ermanent deformations of unbound granular materials in road avements. INTRODUCTION The objective of this reort is to make a review of existing models for the rediction of ermanent deformations of unbound granular materials, in order to select models for the SAMARIS roject, which will be used for the rediction of ermanent deformations in different exerimental avement structures. The reort resents first some general asects of the behaviour of unbound granular materials, and makes a review of available models to describe ermanent-deformation behaviour. Both simle, emirical relationshis and more elaborate incremental, elastolastic models are reviewed. Then, several models are selected and evaluated, by comaring their redictions with results of existing reeated load triaxial tests. Based on this evaluation, two models are selected for the modelling work in SAMARIS. Finally, methods for alying these models for calculation of rut deths in avement structures are discussed. 2. MECHANICAL BEHAVIOUR OF UNBOUND GRANULAR MATERIALS 2. erformance of unbound granular materials in avements Unbound granular materials (UGMs) are continuously graded granular materials, consisting in general of crushed rock articles. They usually contain a certain amount of fines (tyically 4% to %) and water (they are generally artially saturated). Such materials are used for mainly for base and subbase layers of low traffic avements, and also for caing layers. In Pavements, the erformance of unbound granular materials is characterised by the accumulation of ermanent deformations, leading to rutting of the avement. Desite this imortance of rutting of unbound granular materials, esecially in low traffic avements, there is resently no well established method to study the ermanent deformations of unbound granular materials in the laboratory, and to redict there rutting in the avement. In the absence of satisfactory method to redict the rut deth, the design of avements with unbound granular layers remains, in most design methods, very emirical. They are considered as linear elastic materials, the values of their elastic moduli are often determined on the basis of emirical rules, and the design criterion used for these materials is generally a criterion limiting the maximum vertical elastic strains at the to of the unbound layers and/or at the to of the subgrade, of the form : e z b A N () This criterion assumes that rutting is governed by the vertical elastic strains at the to of the unbound material (granular base or subgrade), and is indeendent of the nature of the material (arameters A and b have a unique value, indeendent of the material). 3

4 Thus, today, due to these over-simlified design methods, it is not ossible to take fully advantage of the real erformance of unbound granular materials in avements. There is a strong need to imrove this situation, and to develo and introduce in current ractice : aroriate mechanical erformance tests to determine the resistance to ermanent deformations of unbound granular materials ; more aroriate models to redict their ermanent deformations in avements. These new aroaches are articularly needed with the increasing use, in avements, of other sources of materials than crushed rock unbound granular materials, like recycled materials (demolition waste) and various by-roducts. For these novel materials, the emirical knowledge available for traditional materials, based on simle emirical identification tests (aggregate resistance tests, cleanliness of fines, etc..), and conventional design rocedures cannot be emloyed. 2.2 Cyclic behaviour of unbound granular materials UGMs are materials consisting of individual grains, with little or no cohesion. Therefore, these materials have very little or no tensile strength and their behaviour deends strongly on the stress state, and also on the grading of the material and the shae of the grains. The most widely used test to study the mechanical behaviour of unbound granular materials is the cyclic triaxial test. The advantage of this test is the ossibility to study the behaviour of the material under cyclic loadings, simulating accurately the in-situ conditions. Cyclic triaxial aaratuses for UGMs are now widely available in Euroe, but they differ largely by their characteristics (secimen size, loading caabilities). The rincile of the cyclic triaxial test is recalled on figure. In the tests erformed at LCPC, and which will be used for this roject, the material is submitted to a cyclic vertical stress σ and a cyclic horizontal stress σ 3 which vary in hase (this tye of loading is called variable confining ressure loading (VCP)). But other, simler test equiment, allow to aly only a cyclic vertical stress, the horizontal stress being held constant (constant confining ressure loadings (CCP). VCP tests resent the advantage of simulating more closely the in situ loading conditions. Figure 2 resents a tyical examle of resonse of a granular material in a cyclic triaxial test. The resonse of the material is essentially elasto-lastic, and it can be observed that : The ermanent strains accumulate raidly during the first load cycles, but then they tend to stabilise (at least for stress levels under the critical state), and the resonse of the material becomes essentially elastic. The elastic art of the resonse of the material is strongly non-linear (stress-deendent). 4

5 q σ3 stress σ3 q cycle cycle 2 time σ3 ermanent strain resilient strain strain 25 Figure. Princile of the reeated load triaxial test gq cycles : Axial stress σ (kpa) Axial strain ( - 4 ) Figure 2. Examle of stress-strain cycles obtained in a reeated load triaxial test on a granular material The aroach generally used to study this comlex behaviour consists in studying and modelling searately : The stable resilient behaviour obtained after a large number of load cycles, which can be described by non-linear elastic models. Such models begin to be imlemented in avement analysis rograms and used for avement modelling and design The accumulation of ermanent strains, which is more comlex to describe. Develoment of realistic ermanent deformation models (elasto-lastic models) remains a subject for research. 2.3 Resilient behaviour This review concerns mainly the rediction of ermanent deformations. However, as the objective of this work is to develo methods for analysis and design of avements, it is also necessary to take into account, in our modelling, the resilient (or elastic) art of the behaviour of UGMs. Modelling of the resilient behaviour of unbound granular materials was largely studied in the Euroean IV th framework roject COURAGE [999]. In this roject, a review of available resilient behaviour models was erformed, and several, widely used models were evaluated, by comarison with triaxial test results on different unbound granular materials. 5

6 It was concluded that the non-linear elastic model roosed by Boyce [98], and modified by Hornych and al. [998], to take into account anisotroy, describes well the resilient behaviour of unbound granular materials. Therefore, it is roosed to use this model to describe the resilient behaviour of unbound granular materials in the work of SAMARIS. At LCPC, this model has been imlemented in the finite element rogram CESAR-LCPC, and thus can be used for avement structure calculations. Alications of these finite element calculations are resented in Hornych and al. [22] The non-linear elastic Boyce model This Boyce model is exressed in terms of Bulk Modulus, K, and Shear Modulus, G, with : K = and v q G = (2) and (3) 3 q with : : mean normal stress, q : deviator stress; v : volumetric strain, v = + 2 3, q : shear strain, ( ) 2 = q 3 3 The values of K and G are stress deendent according to the following relationshis : n 2 n q β K = Ka and G = G a. a (4) and (5) a K β a (6) 6Ga Ka, Ga, n : arameters of model; a : constant equal to kpa; with : = ( n) To introduce anisotroy in the model, Hornych and al. [998] roosed to multily the rincial stress, σ by a coefficient of anisotroy γ. This leads to the following stress-strain relationshis : v ( n ) n 2 * K = + a q * * and K n a a 6G a * with : * ( γσ + 2 ) 3 σ3 = and q* = γσ σ3 2 * = γ + 3 q* = γ 3 and ( ) v 2 * q * q* = (7) and (8) 3G n a * In this model, the vertical elastic modulus E v and the horizontal elastic modulus E h are linked by the relationshi : Eh 2 = γ (9) Ev In triaxial tests, values of γ are generally lower than, which means that the material is stiffer in the vertical direction (E v > E h ). An examle of adjustment of the anisotroic Boyce model on results of a reeated load triaxial test on an unbound granular material is shown on figure 3. This figure shows the adjustment of volumetric resilient strains and shear resilient strains (strains during unloading) obtained for different stress aths q/. 3 a n 6

7 v ( -4 ) Volumetric strains P (kpa) q/ = q/ =2 q/ = 3 q/ = anisotroic model 2 Shear strains 5 q ( -4 ) 5-5 q/ = q/ = 2 q/ = 3 q/ = Anisotroic model P (kpa) Figure 3 : examle of adjustment of exerimental results with the anisotroic Boyce model Simlified descrition using the K-Theta model A roer determination of the arameters of the Boyce model requires reeated load triaxial tests with variable confining ressure, and with measurement of radial strains. However, some triaxial aaratuses do not have these caabilities, and allow to do only constant confining ressure tests, and measure only axial strains. In this case, the most suitable model to describe the test results is the K-Theta model, although this model is less accurate than the Boyce model. The exression of the K-Theta model is : n 3 E = K () a with E : elastic modulus (also called resilient modulus) K, n model arameters, a = kpa 7

8 It should be noted that the K-theta model reresents in fact a articular case of the Boyce model, when the arameter β is equal to zero. Therefore, modelling aroaches based on the Boyce model, which will be used in SAMARIS, can also use the simler k-theta model. 2.4 Permanent deformation behaviour Exerimental behaviour The most common rocedure to study ermanent deformations of UGMs using the reeated load triaxial aaratus consists in alying a large number of load cycles ( 5 and more), with a single level of stress. Figure 4 shows examles of curves of variation of ermanent axial strains with the number of load cycles obtained in such ermanent deformation tests. Usually, for stress levels found in granular avement layers, the ermanent deformations increase raidly during the first few thousand load cycles, and then tend to stabilise (shakedown). For higher stress levels, this stabilisation is not observed, and ermanent deformations continue to increase, eventually leading to failure. DEFORMATION PERMANENTE AXIALE ( -4 ) 75 q = 2 kpa 5 25 q = 8 kpa q = 4 kpa N CYCLES ( 3 ) Figure 4 : Evolution of ermanent axial strains with the number of load cycles, for different levels of deviatoric stress (after Martinez, 98) Influence of stress level Generally, in reeated load triaxial testing, the alied stresses state is described in terms of mean stress and deviatoric stress q. Early research on ermanent deformations has shown that ermanent axial strains increase with increasing deviator stress q, and decrease with increasing mean stress. This has led to roose relationshis, relating the ermanent axial strains or shear strains to and q, such as : where : 2,8 q s = fn ( N ).L. (Pain, 979) () s is the ermanent shear strain, 8

9 f n (N) is a shae function, deending on the number of cycles N, L = 2 + q 2 Other researchers have tried to relate the accumulation of ermanent strains with the shear strength of the material, assuming that the level of ermanent strains is a function of the roximity of the alied stresses to the failure line of the material. Barksdale [972], for examle, exressed the ermanent axial strain as a function of the stress ratio R f = q/q f, where q is the alied shear stress and q f the shear stress at failure for the same confining stress σ 3. This tye of aroach has been questioned by Lekar and Isacsson [998], who argued that failure in granular materials under reeated loading is a gradual rocess, and not a sudden collase as in monotonic failure tests, and that there is no direct relationshi between the monotonic shear strength, and the accumulated ermanent strains. More recently, Arnold an al. [22] and Werkmeister and al. [22], have erformed ermanent deformation tests with different stress level, and have identified three different tyes of evolution of ermanent strains with the number of load cycles, deending on the level of stress, as shown on figure 5. Range A, called the lastic shakedown range, where a comlete stabilisation of ermanent strains is observed after a finite number of load cycles, and the behaviour becomes entirely resilient (lastic shakedown). Range B, called the intermediate range. In this range, the ermanent strain rate er cycle either continuously decreases, or becomes constant, but the ermanent strains continue to increase, at a very slow rate, without comlete stabilisation. Range C, called the incremental collase range. In this range, after a first decrease, the ermanent strain rate er cycle starts to increase, leading rogressively to failure (very high strains). Figure 5 : different tyes of ermanent deformation behaviour, deending on stress level. (after Arnold and al., 22). Having defined these 3 ranges of behaviour, Arnold and al. have tried to determine the boundaries between these different ranges, in the (,q) stress sace. An examle of results, obtained for a granodiorite crushed rock granular material is shown on figure 6. The boundaries are defined as straight lines in the (,q) sace. 9

10 Figure 6 : Range A and range B behaviour boundaries in the (,q) stress sace for a granodiorite unbound granular material (after Arnold and al., 22). Another imortant factor, for the accumulation of ermanent deformations, is the rotation of rincial stresses. In the avement, the material is subject to moving wheel loads, which roduce a change in the direction of the rincial stresses. This rotation is not simulated in cyclic triaxial tests. However, research using a hollow cylinder triaxial aaratus (Chan 99), has shown that ermanent strains increase significantly when a rotation of the rincial stresses is alied (for the same rincial stress amlitude). Similar observations have been made by Hornych and al. [2], in an accelerated loading exeriment on a low traffic avement. They comared the effect of a moving wheel load, and of a cyclic load alied on a fixed circular late (with the same vertical stress level and load duration), and found that the moving wheel load roduced about two times higher ermanent deformations than the cyclic late load Influence of material characteristics Research erformed in France, on a large variety of unbound granular materials, has shown that the following arameters have a significant influence on the resistance to ermanent strains : The water content : Permanent deformations increase significantly when the water content increases, articularly at high water contents (around or above otimum water content). This moisture sensitivity is strongly related to the fines content of the material. Figure 7 shows an examle of the influence of water content on the characteristic ermanent axial strain A c (obtained after a standard loading rocedure), for three crushed rock granular material of different origin (hard limestone, soft limestone and micro-granite). For all three materials, there seems to be a critical water content, above which the ermanent strains start to increase raidly.

11 Permanent strain A c ( -4 ) 5 Hard limestone Soft limestone Microgranite w - w OPM (%) Measured water content in granular base layer Figure 7 : Sensitivity of ermanent axial strains to water content, for 3 different unbound granular materials (after Gidel and al., 2) The density : Clearly, an increase of the degree of comaction of the material reduces the void ratio, and increases the number of contacts between the grains, thus increasing the resistance to ermanent deformations. More generally, the resistance to ermanent deformations strongly deends on the void content. For the same degree of comaction, a modification of the grading, reducing the void content, increases the resistance to ermanent deformations. The mineralogical nature of the material : The mineralogy influences the shae of the articles, their surface roughness, and also the quality of the fine fraction and its sensitivity to water, and this can lead to significant differences in behaviour, for the same grading. 3. MODELLING OF PERMANENT DEFORMATIONS OF UNBOUND GRANULAR MATERIALS The resilient resonse of unbound granular materials has been widely studied, and is well described by the non-linear elastic models resented reviously. Modelling of ermanent deformations is less advanced, for several reasons : The exerimental study of ermanent deformations Is much more time-consuming than the study of the resilient behaviour. Whereas the resilient behaviour of a material can be determined using a single triaxial test (with different stress aths), the study of ermanent deformations requires an imortant number of tests (in each test, only one stress ath is generally alied, to avoid influence of revious stress-history), with large numbers of load cycles ( 5 or more). In addition, results of ermanent deformation tests resent a much larger exerimental scatter than results of resilient behaviour tests.

12 Permanent deformation characteristics are often considered to be less imortant than resilient behaviour : resilient strains are directly related to fatigue cracking of the bituminous layers, which is almost imossible to remedy once it has occurred, whereas ermanent deformations only create rogressive deformations of the surface, and can be remedied by overlaying the avement Finally, ermanent deformation models are more difficult to aly to avement structure calculations than resilient behaviour. Modelling of rutting requires the simulation of large numbers of load cycles, with varying environmental conditions (temerature, moisture content, different loads). Existing ermanent deformation models are mostly based on reeated load triaxial testing, and are of two main tyes : Emirical laws, describing the variation of ermanent strains with the number of load cycles and the maximum alied stresses. Incremental models, generally based on the theory of elasto-lasticity. 3. Emirical relationshis 3... Relationshis describing the influence of the number of load cycles One of the first interretation alied to results of ermanent deformation tests in the triaxial consisted in describing the variation of ermanent deformations (generally only axial deformations) with the number of load cycles N. A summary of various relationshis of this tye, roosed by different authors, is resented in table. Table : Relationshis describing the variation of ermanent axial deformations with the number of load cycles Author relationshi Parameters Barksdale [972] = a b log( N) Khedr [985] + -b N = A.N Paute and al * A N [988] = N + D Sweere [99] b = a N B Hornych and al. [993] * N = A - Vuong [994] r a c = N b a, b A,b * ermanent deformation after the first cycles A arameter function of stress level, D a, b * ermanent axial strain after the first cycles A, B r a, b, c resilient axial strain 2

13 Wolff and Visser [994] -b N ( c N + a)( - ) = e N N D N A.( Huurman [997] P ( ) = ) B + C.(e ) a, b, c A, B, C, D arameters function of the level of stress The first well known relationshi describing the variation of ermanent axial deformations with the number of load cycles is that roosed by Barksdale [972], who found a linear increase of with the logarithm of the number of load cycles N : = a + ( ) b log N Later, other researchers (Khedr [985], Sweere [99], Vuong [994]) found similar results, indicating a linear relationshi between log( ) and log(n), leading to relationshis like that roosed by Sweere [99] : () b = a N (2) Hornych and al. [993], who tested three French unbound granular materials, found that for tyical stress levels found in avement foundations, ermanent axial strains stabilise after a large number of load cycles (about 5 ). They obtained good redictions with the following relationshi, which assumes that has a finite limit for an infinite number of cycles : B * N = A - (3) Relationshis describing the influence of the level of stress The relationshis describing the variation of ermanent deformations with the number of load cycles resented above cannot be alied to the rediction of ermanent deformations in avement structures, because they do not take into account the alied stresses. Other researchers have followed another aroach, trying to relate the ermanent deformation after a given number of cycles to the alied stresses (generally the maximum stresses ). Relationshis of this tye are listed in table 2. Some of these relationshis also try to coule the effects of both stresses and number of load cycles, but they are only very few of them (see table 2). Table 2 : Relationshis describing the variation of ermanent axial deformations with alied stresses Author relationshi Parameters Lashine et al. q [97] = a σ 3 a, σ 3 confining stress, q deviator stress 3

14 = q aσ Barksdale [972] R.q. ( sinφ) ( ) f 2 C.cosφ + σ3sinφ Shenton [974] q = K σ Pain [979] = f ( ) s n max 3 a q N.L. 2,8 b 3 a, b, R f = ratio of alied deviator stress q to deviator stress at failure q f, φ friction angle, C cohesion K,a, q max maximum alied deviatoric stress s ermanent shear strain, f n (N) shae function, deending on the number of cycles N, mean normal stress, L = 2 + q 2 Lentz et Baladi [98] * Paute et al. [994] = f( N) Nishi [994] Lekar et al. [998] ( ) ( ) ln -,5 n q S =,95S q S + - m( q S),ult q = k ( Nref ) ( L ) b m a b q = a q ( + *) q ( ) + * b ( ) ln N,95 S axial strain at 95% of the deviatoric stress at failure, m sloe of the failure line, S deviatoric stress at failure * ermanent axial strain after the first cycles b,*, m sloe of the failure line in,q sace f(n) function of the number of cycles N a, b,ult strain ultimate ermanent axial a,b ( N ref ) : ermanent strain after a reference number of cycles N ref L = 2 + q 2 reference mean stress Several relationshis are very simle, and are exressed as a ower relationshi between and the stress ratio q/ or q/σ 3 (for examle q = K b or a q = K ) b Recently, Gidel and al. [2], erformed tests on two granular materials, and tried to model variation of ermanent axial strains with stress level and number of load cycles. By describing searately the variation of with the number of cycles N and the maximum cyclic stresses max and q max, he arrived to the following exression : 4

15 B n N L (N) max = N a s q (m + max ) max max with : L 2 q 2 max = max + max, a = kpa,, B, n, m, s arameters of the model. 3.2 Elasto-lastic models In the field of soil mechanics, various elasto-lastic models simulating accurately the monotonic and cyclic behaviour of soils and granular materials have been develoed. The e lasticity models searate the strains into elastic and lastic arts : = + where is the total strain, e is the elastic strain and is the lastic strain. The models link the stress tensor to the strain tensor with the incremental equation: d σ = H d, where H is a fourth order tensor. This equations system is solved by incremental calculations. The major advantage of this incremental aroach is that it is ossible to describe the resonse to any tye of loading history (whereas the emirical models describe only the resonse to a cyclic loading of constant stress amlitude). The modelling of cyclic behaviour requires elaborate elasto-lastic models, which can generate lastic strains during loading and also during unloading, like models with kinematic hardening. Various models of this tye exist in soil mechanics, but generally, the available models are used to describe relatively low numbers of load cycles (less than cycles), and would need to be adated to avement loading conditions, where the number of loads is much larger (tyically 5 to 7 ). Another characteristic of these models is that they generally have much more arameters than the emirical relationshis resented in 3., which need to be determined using several different tyes of tests (monotonic and cyclic tests, with different stress aths). Few authors have develoed secific elasto-lastic models for avement alications. Recent develoments in this field have been roosed by Bonaquist and Witczak (997), Hicher and al. (999) and Chazallon (2) The elasto-lastic model of Chazallon The model develoed by Chazallon (2, 22) is based on the work of Hujeux (985). It is an elasto-lastic cyclic model with kinematic hardening, alicable to large numbers of load cycles. The model uses the yield function and lastic otential of the non-associated model of Hujeux (Hujeux 985) in its simlest formulation. The formulation used by the author and roosed by Hujeux is a one-mechanism model used for monotonic loading of sands based on the critical state concet (Schoffield et al 968). The model has been first resented in (Chazallon 2), and some modifications have been added. To take into account the unsaturated state of the material, leading to a macroscoic cohesion, a constitutive arameter C has been added. It aears in the exression of the yield surface, lastic otential, the kinetic laws and the elasticity law, by adding C to the mean stress. The elastic art of the behaviour is considered non-linear and is described by the anisotroic Boyce model [Hornych and al., 998] described reviously. In the exression of this model, the mean stress is relaced by : 5

16 ( γσ + 2σ = 3 + C 3 where : C is the cohesion due to the unsaturated state of the material The yield function f is written: / 2 3 rcbmi ( ~ X ~ ) = + I ( ~ X ~ ) II( ~ X ~ σ σ f S σ ) ln (4) 2 3 * 3c where I is the Trace oerator, S II is the deviatoric stress oerator, b is a arameter which controls the shae of the yield surface in the (,q) lane. M is the sloe of the critical state line in the (,q) lane. c * is the critical ressure corresonding to the actual void ratio. The hardening is given by: c = c ex( β v ) (5) where v is the volumetric lastic strain. c and β are constant which determine the osition of the critical state line in the (e,ln()) lane. c is the initial critical ressure corresonding to the initial critical void ratio e c. r c is a hardening variable associated to the deviatoric lastic strain. el r d c = + r a + d The kinetic of the hardening variable r c is: dr d 3( r ) 2 / ami ( ~ X ~ c = λ c σ ) (7) (6) el el Initially, under monotonic loading r c = r m, where r m reresents the initial elastic domain, (r c ) and a = a m governs the evolution of the hardening variable r c. When unloading occurs, el r c = r c reresents the initial elastic domain and a = a c governs the evolution of the hardening variable r c. dλ is the lastic multilicator, and it can be determined by the consistency condition df = and dλ. The former equations are comleted by the definition of a nonassociated lastic otential g and the following kinetic: g( σ X ~ ) d~ dλ ~ = ~ (8) σ / 2 27 I where : = + ( ~ X ~ ) II( ~ X ~ ( ~ X ~ σ g S σ )/MI σ ) ln (9) 2 * 3c X ~ is the tensorial kinematic hardening variable. Its kinetic is given by d X ~ = µ d ~ σ and µ (2) If one follows a stress ath AB from A to B, and one unloads from B to C, X ~ is the stress of the origin of the new stress lane (,q) during unloading. 6

17 ~ X ~ = ~ σ P * B + uc / lc c I and Puc / lc (2) Where I ~ is the second order identity tensor, P uc and P lc are two arameters which take into account the osition of the yield surface and the lastic otential during unloading (subscrit uc) or reloading (subscrit lc). Figure 8 shows the yield surface of the model, and its evolution during a cyclic loading between oints A and B in the (,q) stress sace. During the first loading from A to B, the origin of the surface is at oint O, and the size of the surface grows during loading (isotroic hardening). When unloading starts, from oint B, the origin of the surface moves to oint O 2, and then the osition of the surface continues to change, during unloading and subsequent reloading (isotroic and kinematic hardening). q (loading) M New origin of yield surface (unloading) B O2 O A Pc (loading) Yield surface at oint A Yield surface at oint B, when unloading starts q (unloading) Figure 8 : yield surface of the model of Chazallon, and hardening mechanism. The arameters of the model are summarised in table 3. They can be divided into 4 nonlinear elastic arameters, 7 lastic arameters describing monotonic loading and 5 lastic arameters describing cyclic loading; these three grous of arameters are uncouled, and can be determined searately. Table 3 : arameters of the elasto-lastic model of Chazallon Non-linear elastic arameters n Ka MPa Ga MPa γ C kpa Monotonic loading arameters M c MPa β a m b el r P m uc Cyclic loading arameters P lc r el c a c µ The elastic arameters (K a,g a, n) and the anisotroic arameter γ are determined using the results of cyclic triaxial tests used to determine the resilient behaviour. These tests include a 7

18 cyclic conditioning, which aim is to stabilise the ermanent deformations, and then cyclic loadings with various (q/) ratios. The monotonic loading arameters are determined using monotonic triaxial shear tests till failure at different initial confining ressures (for avement alications, low confining ressures of kpa, kpa and 2 kpa are used). C is the cohesion. M is the sloe of the critical state line in,q sace, searating contractant ad dilatant behaviour. In the Hujeux model, the initial critical ressure c and the lastic comressibility modulus β require, for their determination, isotroic comression tests erformed till at least 2 MPa, which can not be erformed using triaxial cells for unbound granular materials for roads. Alternatively, their values can be estimated using correlations (Rahma 992). The hardening arameters a m and b are determined by fitting the σ - stress strain curves of the monotonic triaxial tests. el The cyclic arameters (P uc, P lc, r c, a c, µ) are determined using cyclic triaxial tests with large numbers of load cycles. The model assumes that the lastic strains tend to stabilise el (elastic shakedown or lastic shakedown). The cyclic lastic arameter r c is determined from the level of axial lastic strain obtained at stabilisation in the (N, ) lane. The arameters P uc and P lc determine how raidly the shakedown state will be attained. 4. EVALUATION OF SELECTED PERMANENT DEFORMATION MODELS In SAMARIS, the objective is to roose two aroaches for rediction of ermanent deformations. One routine level aroach, based on emirical relationshis, relating ermanent deformations to the alied stresses and number of load cycles, and one more advanced aroach, based on elasto-lastic modelling. As a first ste, several different models have been selected from the literature study and evaluated, by comarison with results of existing reeated load triaxial tests. For the evaluation, it was decided to use two series of cyclic triaxial tests, erformed on a /2 mm granular material (crushed microgranite), and a /4 mm sand, containing aroximately 8 % of fines. The microgranite secimens were comacted at a density equal to 97 % of the Modified Proctor Otimum density, and a water content 2 % below otimum. The test rogram is summarised in table 4. Five tests were erformed, each following a different stress ath q/, and each test included several loading stages, with increasing stress levels. The stress levels are reresentative of those found in the granular base layer of a low traffic avement. Table 4 : rogram of ermanent deformation tests on the microgranite Test Loading sequence Number of cycles N max (kpa) q max (kpa) q/ ( -4 ) , , , , , ,6 8

19 , , , , , ,5 53, ,5 7, ,5 47, ,5 56, , , , ,5 The tests on the sand were erformed on secimens comacted at the normal Proctor otimum density, and at the otimum water content (w = %). Four tests were erformed, using a test rocedure similar to that used for the microgranite. The alied stress levels and the ermanent strains obtained for each load sequence are resented in table 5. Here, the stress levels are reresentative of values found in a caing layer, or a subgrade. Table 5 : rogram of ermanent deformation tests on the sand Test Loading sequence Number of cycles N max (kpa) q max (kpa) GSM GSM q/ GSM GSM ( -4 ) 9

20 Emirical aroaches The idea is to roose, on the basis of the literature review, a relationshi describing the variation of ermanent axial strains with both stress level and number of load cycles, using an exression of the form : ( N) = f( N) g(,q ) max max where : f(n) is a function of the number of load cycles, N; g( max,q max ) is a function of the maximum mean stress max and maximum deviatoric stress q max. The two functions f and g will be selected from those resented above Variation with stress level To describe the variation of ermanent deformations with stress level, the three relationshis roosed by Nishi [994], Lekar and al. [998], and Gidel and al. [2] have been evaluated. The relationshis, and the arameter values obtained for each tested material, are shown in table 6. Table 6 : emirical ermanent deformation relationshis selected for evaluation, and arameters obtained for the two materials. = L. max ) n (. a Gidel (2) =. Nishi (994) a qmax b max = L. max q.( max ) b a max Lekar (998) R : correlation index Relationshi UGM Sand s qmax ( m + - ) max max (22) (23) (24) = 29,32 n =,62 m = 2,4 s = 46,82 R =,83 =,2 a = 5,55 b = 4,4 R =,69 =,83 b = 3,76 R =,64 = 3,45 n =,36 m =,6 s = 26,2 R =,8 =,8 a = 3,3 b = 2,7 R =,76 = 23,78 b = 2,42 R =,76 2

21 The relationshis were adjusted on the final strain values obtained at the end of each loading stage, using a least squares method. The coefficient R is a correlation index, indicating the quality of the adjustment. It is defined by : (y 2 i f(xi)) R = i R 2 ( yi y) i where : y i are the exerimental values obtained for each stress level, f(x i ) are the values redicted by the model, and y is the average of the exerimental values y i. As can be seen in table 6, the relationshi roosed by Gidel gives the best results, for both materials. The redictions obtained with this relationshi are shown on figures 9 and. These figures comare the exerimental strains obtained at the end of each loading stage (for each of the tests, with different values of q/), with the redicted values. This relationshi will be retained for the rest of the work q/=3 q/=2,5 3 e (-4) 25 2 q/=2 mesures modèle 5 q/=,5 5 q/= (kpa) Figure 9 : Predictions obtained with the model roosed by Gidel and al., for the /2 mm UGM 2

22 3 q/= ( -4 ) 5 q/=3 mesures modèle q/=,5 5 q/= (kpa) Figure : Predictions obtained with the model roosed by Gidel and al., for the sand. Similarly, figures and 2 show the results obtained with the model roosed by Nishi (the second, in terms of values of correlation index). The redictions remain accetable for the sand, but not for the UGM e (-4) 25 2 q/=3 q/=2,5 q/=2 mesures modèle 5 q/=,5 5 q/= (kpa) Figure : Predictions obtained with the model roosed by Nishi for the /2 mm UGM. 22

23 3 25 q/=2 2 ( -4 ) 5 q/=3 q/=,5 mesures modèle (kpa) Figure 2 : Predictions obtained with the model roosed by Nishi for the sand. q/= Variation with the number of load cycles For describing the variation of axial ermanent strains with the number of load cycles, two relationshis have been tested: the one roosed by Sweere [99], which is very widely used, and the one roosed by Hornych and al. [993] (see 3..). The main difference is that the relationshi of Hornych redicts a stabilisation of ermanent strains (finite limit for N infinite), contrary to that of Sweere. The two relationshis, and the arameter values and correlation indexes obtained with them, for the two materials, are resented in table 7. As reviously, the adjustments were erformed using a least squares method. The two functions give relatively similar correlations indexes R. Table 7 : Parameters obtained for the two selected functions of the number of load cycles, and correlation indexes Relationshi UGM Sand f(n) = A.[-(N/N ) -B ] (N >N =) Hornych (993) (25) f(n) = A.N B Sweere (99) (26) R : correlation index A = 2,67 B =,4 R =,66 A =,98 B =,75 R =,62 A =,4 B =,6 R =,63 A =,773 B =,28 R =,66 23

24 Finally, figures 3 and 4 show the results obtained for rediction of the exerimental results on the microgranite, using the comlete relationshi, ( N) = f( N) g( max,q ), with max the function g roosed by Gidel (selected in 4..), and the two functions f(n) selected here (Hornych, and Sweere). For the microgranite, the evolution with the number of load cycles seems better redicted by the function f(n) of Sweere q/= q/=2,5 e(-4) 25 2 mesure modèle 5 q/=2 q/=,5 5 q/= Nombre de cycles (-) Figure 3 : Microgranite - Predictions obtained with the function of variation with the number of load cycles f(n) roosed by Hornych 6 5 q/=3 4 q/=2,5 e (-4) 3 mesure modèle 2 q/=2 q/=, Nombre de cycles (-) Figure 4 : Microgranite - Predictions obtained with the function of variation with the number of load cycles f(n) roosed by Sweere. q/= 24

25 Similarly, figures 5 and 6 show the results obtained for rediction of the exerimental results on the sand, with the two different f(n) functions. The two functions give similar results, and redict the exerimental results relatively well, excet for the highest q/ ratio (q/ = 3)/. 3 q/= e(-4) 5 q/=3 mesure modèle q/=,5 5 q/= Nombre de cycles (-) Figure 5 : Sand - Predictions obtained with the function of variation with the number of load cycles f(n) roosed by Hornych. 3 q/= e(-4) 5 q/=3 mesure modèle q/=,5 5 q/= Nombre de cycles (-) Figure 6 : Sand - Predictions obtained with the function of variation with the number of load cycles f(n) roosed by Sweere 25

26 4.2 Elasto-lastic model As for the emirical relationshis, it was also decided to evaluate the elasto-lastic model, by comarison with exerimental results; However, for this model, the revious tests could not be used, because the determination of the model arameters requires not only cyclic load triaxial tests, but also monotonic triaxial shear tests. So, in connection with another research roject, a secific test rogram was set u, in order to evaluate the elasto-lastic model. These tests were erformed on a / mm crushed gneiss material, at a density of 97 % of the Modified Proctor Otimum density, and at a water content of 6 % (close to otimum) Prediction of monotonic triaxial tests 4 monotonic triaxial tests were erformed on the material, with confining ressures of,, 2 and 5 kpa, in drained conditions (the material is unsaturated). These tests allow to determine the monotonic loading arameters of the model. The values of the monotonic arameters are given in table 8. The values of the arameters P c and β, imossible to determine with the test equiment used, have been assumed identical to those used by Chazallon and al.[22]. The redictions obtained with the model for the axial stress axial strain (σ = f( ) ) monotonic loading curves are shown on figure7. The model redicts quite well the exerimental results, for the different confining ressures. Table 8 : values of the elasto-lastic model arameters obtained for the gneiss Non-linear elastic arameters n K a MPa G a MPa γ C kpa Monotonic loading arameters M c MPa β P a b el m r m uc Cyclic loading arameters P lc el r c a c µ , Pcf = 5 kpa q (Pa) 5 Pcf = 2 kpa mes modèle 4 Pcf = kpa es 26

27 Figure 7 : rediction of monotonic triaxial shear tests with different confining ressures using the elasto-lastic model Prediction of cyclic triaxial tests Four cyclic triaxial tests were also erformed on the crushed gneiss material, with a rocedure similar to those used for the other materials. Three tests were erformed to determine the ermanent deformation behaviour; each test included several load levels, with the same stress ratio q/, but with increasing stress amlitudes. The alied stress aths are summarised in table 9. These tests allowed to determine the cyclic loading arameters, which values are given in table 8. A last cyclic triaxial test was used to determine the nonlinear elastic arameters (see table 8). This test included a cyclic conditioning, followed by short load sequences ( cycles) following different stress aths. Table 9 : rogram of ermanent deformation tests on the crushed gneiss Test 2 3 Loading sequence Number of cycles N max (kpa) q max (kpa) q/ An examle of rediction of axial strains, for the ermanent deformation test with a stress ratio q/ = 2, is resented on figure 8. With the selected arameter values, the model redicts better the behaviour for the lower stress levels (where the ermanent strains stabilise more raidly). However, due to the large number of arameters of the model, the adjustment rocedure is relatively comlex. The study will have to be continued, to see if a better adjustment of the results can be obtained for the highest stress levels. 27

28 9 8 7 ( -4 ) mesure modèle Nb cycles Figure 8 : Prediction of a cyclic ermanent deformation test with the elasto-lastic model. 5. METHODS OF PREDICTION OF PERMANENT DEFORMATIONS IN PAVEMENT STRUCTURES 5. General assumtions The work erformed in this task WP 5.2 of SAMARIS deals only with the ermanent deformations of unbound avement layers. Therefore, in this reort, only the rediction of the ermanent deformations of unbound granular avement layers is considered, and no ermanent deformations are suosed to occur in the other avement layers (bituminous layers and subgrade. Parallel work on modelling of bituminous materials is carried out in task WP 5.3 of the roject. At a later stage, the ossibility to associate the modelling of rutting of both bituminous and unbound materials will be examined. For the rediction of ermanent deformations of unbound avement layers, several general assumtions will be made concerning the cyclic behaviour of UGMs, based on the exerimental observations.. As in all the reort, it will be assumed that the behaviour of unbound granular materials is elasto-lastic and strains will be divided into an elastic art and a lastic art : e = + 2. The second assumtion is that for one load cycle, the increment of ermanent strains is very small, comared with the elastic strains : δ e << Exerimentally, this is verified excet for the very first load cycles (less than cycles). 28

29 3. The third assumtion is that the elastic roerties of the granular materials are indeendent of the number of load cycles. In triaxial tests, this is also well verified, after the first few hundred cycles (where an increase of elastic modulus is generally observed). With these assumtions, the rediction of ermanent deformations in the avement layer can be erformed in two indeendent stes : Calculation of the stress field in the avement, considering only the resilient behaviour. Use of the resilient stresses to calculate the ermanent strains in the granular layers, and then the resulting dislacements. Possible aroaches for the calculation of the stress field in the avement, and of the ermanent strains and dislacements are resented and discussed below. 5.2 Modelling of resilient behaviour and determination of stress aths in granular avement layers Finite element modelling of the resilient behaviour of avement structures The determination of the stress field in the avement, using suitable models for the behaviour of the avement materials, and realistic material roerties, is an essential ste for the accurate rediction of ermanent strains. Although most current avement design methods are based on linear elastic calculations, it is well known that these calculations are not satisfactory for low traffic avements, where the granular layers reresent the main structural element; In such avements, linear elasticity often leads to unrealistic stress states in the granular layers, in articular tensile stresses at the bottom of the unbound granular layers, which cannot be sustained by these materials. At LCPC, an imortant research rogram has been carried out in the ast few years, to study the resilient behaviour of unbound granular materials, in connection with the Euroean research rojects COST 337 and COURAGE. This work led to the choice of the non linear elastic model of Boyce [98], modified to take into account anisotroy, to describe the resilient behaviour of unbound granular materials (see art 2.3 of the reort). It has also led to the develoment of a new module of the finite element rogram CESAR-LCPC, dedicated to the modelling of the resilient behaviour of avements under moving wheel loads (Heck and al, [998], Hornych and al. [998], Heck [2]). This module, called CVCR includes several secific models for avement materials : Two non linear elastic models for unbound granular materials (and also granular soils): the modified Boyce model, and the k-theta model (also described in art 2.3). The viscoelastic model of Huet-Sayegh for bituminous materials (Huet [963], Sayegh [965]), which gives very good redictions of comlex modulus measurements. The module CVCR has been validated by comarisons with results of several exeriments on low traffic avements, with granular bases (Courage [999], Hornych and al. [2, 22]). These validations indicated that CVCR leads to a much better rediction of the non linear, load-deendent resonse of these thinly surfaced avements, and much more realistic stress distributions in the granular layers than classical linear elastic calculations. Therefore, the module CVCR will be used in Samaris for the calculation of stress fields in avement structures. In addition, because the objective is to determine only the stress fields in the granular layers, a linear elastic behaviour will be assumed in the calculations for 29

30 bituminous layers and for the subgrade soil, and only the UGM layers will be considered non linear elastic Tyical stress aths under moving wheel loads In our study, it is assumed that the roerties of the avement are constant along the Ox axis (direction of dislacement of the vehicles), and that the seed of the vehicles is constant. With these assumtions, the ermanent deformations need to be calculated only in the lane (,y,z) erendicular to the Ox direction, due to symmetry. At a oint M, of coordinates (, y, z) in this lane, the assing of one vehicle roduces a loading history, or stress ath σ Μ (t). For the sake of simlicity, and because our aroach does not take into account rincial stress rotation, this stress ath will be described only by the variations of the mean stress M (t) and of the deviatoric stress q M (t). In our case, where all the materials are elastic, the determination of the stress aths from the results of the CVCR finite element calculations is articularly simle. In the referential of the moving load, the stress ath at M (, y, z) corresonds to the variation of the stresses along the x axis, M (x) and q M (x), for constant values of y and z. In order to illustrate the stress aths obtained in granular avement layers, the aroach described above was alied to the modelling of two tyical low traffic avements, consisting of : a bituminous wearing course, with two different thicknesses : 4 cm and 8 cm ; a granular base and subbase, with a total thickness of 4 cm ; a subgrade soil, with a total thickness of 23 cm. The CVCR calculations were erformed in 3D, with a mesh comrising 296 elements (describing /4 of the structure, due to symmetry), and a load consisting of 2 twinned wheels, with a total load of 65 kn (see figure 9). In the calculations, the soil and the bituminous concrete were considered linear elastic, and the granular material was described using the Boyce model with arameter values corresonding to a /2 mm crushed rock material (crushed gneiss), of medium erformance. 3