VALIDATION AND IMPLEMENTATION OF HASP CONSTITUTIVE MODEL FOR OVERCONSOLIDATED CLAYS

Značajan deo u oblasti konstitutivnog modeliranja tla predstavlja opisivanje naponsko-deformacijskih relacija prekonsolidovanih glina. Prekonsolidovane gline su u prošlosti bile opterećene vertikalnim efektivnim naponom koji je veći od tekuće veličine vertikalnog efektivnog napona. Prekonsolidacija može biti i posledica izvođenja različitih građevinskih radova na tlu i u tlu. U poređenju s normalno konsolidovanim glinama, imaju manji koeficijent poroznosti i veću smičuću čvrstoću. U prirodi su najčešće ispucale, što dovodi do nehomogenog polja deformacija. Iz tog razloga, ispoljavaju složen oblik ponašanja pri lomu. Veliki broj konstitutivnih modela za prekonsolidovane gline razvijen je koristeći koncept kritičnog stanja [35, 38] i Modifikovani Cam Clay (MCC) model [36]. MCC model se može, pri monotonom opterećenju, koristiti s velikom pouzdanošću za normalno konsolidovane i lako prekonsolidovane gline. Za jako prekonsolidovane gline, MCC model precenjuje smičući napon pri lomu i predviđa nagli prelaz iz elastične oblasti u elastoplastičnu oblast, što nije u skladu sa eksperimentalnim podacima koji pokazuju postepeno smanjenje krutosti prilikom opterećivanja. Za prevazilaženje nedostataka MCC modela, korišćeni su različiti koncepti. Zienkiewicz i Naylor [52] u relacije konstitutivnog modela uveli su matematički opis površi Hvorsleva, što su u svojim modifikacijama sledili i


INTRODUCTION
A significant part in the area of constitutive soil modelling is the description of the stress-strain relationships of overconsolidated clays.In the past, overconsolidated clays were exposed to the vertical effective stress that is greater than the current magnitude of vertical effective stress.Overconsolidation can also be a consequence of carrying out various construction works on the soil and in the soil.Compared to the normally consolidated clays, they have a lower void ratio and higher shear strength.In nature, they are mostly cracked, leading to a nonhomogeneous field of strains.For this reason, they exhibit a complex form of shear failure.
A large number of constitutive models for overconsolidated clays has been developed using the critical state concept [35,38] and Modified Cam Clay (MCC) model [36].The MCC model can be used for normally consolidated and lightly overconsolidated clays under monotonic load, with great certainty.For heavily overconsolidated clays, the MCC model overestimates the failure shear stress and predicts a sudden transition from elastic to elastic-plastic region, which is not in accordance with experimental data that indicate a gradual stiffness reduction during loading.
In addition, the concept of Multi Surface Plasticity -MSP [16,26] has been developed, which describes more specifically the hardening rule, a gradual transition from elastic to elastic-plastic region, mechanical behaviour of overconsolidated soil, as well as soil behaviour at cyclic loads.It was a general framework in which many constitutive models were developed.The boundary surface concept -Bounding Surface Plasticity -BSP [8,9,22] is based on the MSP concept and has been an improvement in describing the gradual transition from elastic to elastic-plastic region.The basic idea is to define, instead of the classic Cam Clay yield surface that limits the elastic region, the boundary surface within which development of plastic strain is allowed.The advantage of this concept is taking into account previous history of loads.Also, simulation of the soil behaviour under a cyclic load is made possible, since the yield surface that limits the elastic region can be moved within the boundary surface.Numerous constitutive models for overconsolidated soil are based on MSP or BSP concepts: Bubble model [2], MIT-E3 [48], 3 SKH model [45], Two Kinematic Hardening Constitutive model [12], Modified 3 SKH model [24], SANICLAY model [10], UHmodel [50].These models are mathematically more complex than the MCC model and have a greater number of material parameters.The mathematical complexity requires advanced numerical methods and appropriate software, which is not a problem because such commercial software is available to engineers in practice.Much greater problem for practical application of these models is that additional material parameters mostly cannot be obtained from standard laboratory tests.Due to the simplicity and easy identification of model parameters, the MCC model is still most often used in analysis of geotechnical problems, although the prediction of stress-strain relations do not correspond to the real behaviour of overconsolidated clays.One way to improve the constitutive model, without increasing the number of material parameters, is to use internal variables that adequately define soil state as an essential determinant of its mechanical behaviour.One such variable is state parameter, which is used insufficiently in constitutive modelling.

STATE PARAMETER CONCEPT
The state parameter concept was first introduced by Been and Jefferies [4] to describe the behaviour of sand.Instead of the void ratio that was used as an essential characteristic of the sand behaviour, they suggested to use the state parameter as the fundamental variable.The size of the mean normal effective stress p' significantly influences the behaviour of the soil, so that the coarse-grained soil for the given void ratio, at a large value of the mean effective stress behaves as loose, while for lower values of the mean effective stress behaves compacted.This means that besides the void ratio, the magnitude of the mean effective stress is also necessary for the characterization of the coarse-grained napona.Parametar stanja predstavlja razliku izmeču trenutnog koeficijenta poroznosti e i koeficijenta poroznosti ec na liniji referentnog (kritičnog) stanja, pri istom srednjem normalnom efektivnom naponu (Slika 1a): soil.The state parameter is the difference between the current void ratio e and void ratio ec on the reference state (critical) line at the same mean effective stress (Figure 1a): Ovakav koncept podrazumeva da postoji referentno stanje (steady state condition) koje treba da ima jedinstvenu strukturu.Za konstitutivne modele, definisane u okviru teorije kritičnog stanja, referentno stanje jeste upravo kritično stanje, kada se smičuće deformacije razvijaju bez promene zapremine i efektivnog napona.Takoče, mora biti ispunjen uslov da je linija kritičnog stanja CSL u v-p' ravni jedinstvena, gde je v specifična zapremina tla.
For the initial value of the state parameter greater than zero, characteristic for loose and normally consolidated soil, point A in Figure 1a, the soil volume is decreasing (contraction) until the critical state is reached (Figure 1b).This leads to plastic shear failure (Figure 1d).If the initial value of the state parameter is less than zero, as it is the case with compacted and overconsolidated soil, point B in Figure 1a, after the initial compression the soil will tend to increase the volume (Figure 1b).The soil exhibits a brittle failure, which implies an increase in the shear stress up to the maximum value (peak shear strength), and then decrease in shear stress (softening) during further deformation to the constant value (Figure 1d).In undrained conditions, characteristic effective stress paths are shown in Figure 1c.Konstitutivni modeli za pesak -nastali iz koncepta parametra stanja -jesu: Nor-Sand model [17], Severn-Trent sand model [11], model koji su razvili Li & Dafalias [23].
An analogy can be established between the behaviour of compacted granular materials and behaviour of overconsolidated clay, i.e. between compactness and overconsolidation ratio, so that the state parameter can also be used successfully to describe the behaviour of overconsolidated clays.One such model is the CASM model (Clay and Send Model) [51].

FORMULATION OF THE HASP MODEL
The HASP (HArdening State Parameter) model [18] was developed within the state parameter concept.The starting point for formulating a new constitutive model is the Modified Cam Clay model.Within the bounding surface concept [9] a modification of the hardening rule was made by using the state parameter.The bounding surface is the MCC surface, the size of which is defined by the value of maximum mean effective stress  0 p (Figure 2).The bounding surface can be called the surface of normal consolidation: gde je η -trenutni naponski odnos, a M -nagib linije kritičnog stanja (CSL) u naponskoj ravni.
where η is the current stress ratio and M is the slope of the critical state line (CSL) in the stress plane.

Zakon ojačanja HASP modela
Zakon ojačanja MCC modela zavisi samo od zapreminske plastične deformacije.Generalni zahtev za prekonsolidovana tla je prelaz iz kompresije u ekspanziju pre dostizanja vršne čvrstoće.Zakon ojačanja -koji je u funkciji samo zapreminske plastične deformacije -ne omogućava adekvatno opisivanje dilatancije i ojačanja kod prekonsolidovanih glina.Da bi površ tečenja nastavila da se širi i za vrednosti naponskog odnosa M<η<Mf, potrebno je koristiti kombinovani zakon ojačanja i formulisati ga u funkciji i plastične smičuće deformacije [28,50]: Associated flow rule applies, i.e. plastic strain increment vector is always normal to the yield surface.Bounding surface possesses all the characteristics of the MCC surface.For stress ratio below the critical state line, the volume decreases and the surface expands, while for stress ratio above the critical state line, the volume increases and the surface shrinks.On the other hand, yield surface expands (hardening) until peak strength is reached at stress ratio η=Mf, after which it shrinks (softening) until critical state is reached η=M.
U izrazu za koeficijent ojačanja (9) odnos ξ/d je definisan preko parametra stanja.Parametar stanja za trenutnu naponsku tačku (Slika 3) može se izraziti kao: The hardening coefficient ω directly affects the value of the plastic strains, and thus, with the adequate formulation of the hardening coefficient, it is possible to significantly reduce the plastic strains of overconsolidated clay in the initial load phase, when the MCC model predicts only elastic strains.It is then possible to assume that soil deforms plastically from the very beginning of loading.As the overconsolidation ratio of soil decreases in the deformation process, the hardening coefficient ω also decreases (ω→1) and plastic strains become dominant.When reaching the peak strength (transition from hardening to softening) the maximum volume change gradient is observed -maximum dilatancy and from expression (8) it can be concluded that ω=0.Then the relation applies, which means that parameter ξ is the maximum dilatancy value at peak strength in drained conditions [29].
The part of the expression (15) in parenthesis controls the sign of the hardening coefficient and with the overconsolidated ratio determines the magnitude of the hardening coefficient and hence affects the magnitude of plastic strains according to expression (10).For normally consolidated clays, the HASP model automatically transforms into the MCC model since Ψ =Ψ and the hardening coefficient is ω = 1 .For the description of stress-strain relations, five material parameters (M, λ, κ, Γ, μ -Poisson's coefficient) are needed, just like with the MCC model, and all parameters can be determined from the conventional triaxial test, direct shear test and oedometer test.By introducing the state parameter as an internal variable, the HASP model overcomes many deficiencies of the MCC model, while keeping the same set of input parameters, which is an advantage in engineering implementation compared to other constitutive models for overconsolidated clays.

VALIDATION OF THE HASP MODEL
The HASP model validation is performed by comparing the results of simulation of laboratory tests with published experimental results with different total stress paths.In order to confirm the HASP model efficiency, comparison was also made with the prediction of the MCC model.Clays with different overconsolidation ratios were selected, for which in literature there are well-documented triaxial test results and for which parameters of the MCC model have already been determined (Table 1).These parameters are at the same time the parameters of the HASP model.Naponsko-deformacijske relacije (Slika 4) i promene pornog pritiska (Slika 5), dobijene HASP modelom pokazuju veoma dobro slaganje sa eksperimentalnim rezultatima, za sve stepene prekonsolidacije pri triaksijalnoj kompresiji i ekstenziji.Može se uočiti da MCC model ne opisuje adekvatno ponašanje prekonsolidovane gline u nedreniranim uslovima.Vrednosti devijatora napona i pornog pritiska znatno su precenjene i odstupanja su veća što je veći stepen prekonsolidacije.
Stress-strain relations (Figure 4) and changes in pore water pressure (Figure 5) obtained using the HASP model correspond well to the experimental results, for all overconsolidation ratios at triaxial compression and extension.It can be seen that the MCC model fails to adequately describe the behaviour of overconsolidated clays in undrained conditions.Values of deviatoric stresses and pore water pressure are significantly overestimated and deviations are bigger with greater overconsolidation ratio.
Figure 6 shows the results of drained triaxial compression tests (CD tests) on kaolin clay [5] with overconsolidation ratios 8, 4 and 2.
The behaviour of overconsolidated clays during hardening is very well described with the HASP model.For samples with overconsolidation ratios 8 and 4, the HASP model predicts a drop in strength -softening, at strains greater than about 10% (Figure 6a).For highly overconsolidated samples (OCR=8, 4), after the initial compression of the samples, there is an increase in volume (Figure 6b) which is in accordance with experimental results, and excellent prediction of the change in volumetric strains is observed.Deficiencies of the MCC model in describing mechanical behaviour of overconsolidated clays can also be seen in drained conditions.The peak strength is overestimated up to twice the real value.Detailed overview of the validation of the HASP model on several overconsolidated clays with different overconsolidation ratios is shown in the paper [18].

MCC model HASP model
Test results

MCC model HASP model
Test results

Konsolidacija sloja gline
Kao primer implementacije HASP modela, urađena je analiza konsolidacionog sleganja tla usled fazne izgradnje nasipa na površini terena (primer u knjizi Applied Soil Mechanics with Abaqus Applications [13]).Model se sastoji od sloja gline, debljine 4.6 m, koji leži na nepropusnoj i nestišljivoj podlozi.Nivo podzemne vode se nalazi na površini terena, kao što je prikazano na Slici 7. Nasip se gradi u tri jednaka sloja debljine 0.6 m.Ukupna visina nasipa iznosi 1.8 m.Konstrukcija nasipa se izvodi po fazama/slojevima, a izgradnja jednog sloja traje dva dana, dok izgradnja čitavog nasipa traje šest dana.U modelu, konsolidacija gline nakon izgradnje nasipa traje još 200 dana.by using known variables at the end of the increment, in configuration t+Δt.The procedure generally consists of two steps: estimate of the elastic solution for the given increment (elastic prediction) and return to the yield surface (plastic corrector).This approach was later used and further developed by numerous authors and so the class of integration procedures was created, called return mapping [30,41,32,6,7,42,14].The implicit integration scheme that is called the Governing Parameter Method (GPM) was developed by Kojić and Bathe [19][20][21].It is a generalization of the radial return method which was introduced by Wilkins [49].The basic principle is that all unknown variables are expressed in the function of one parameter (the governing parameter) and the problem is reduced to the solving of one nonlinear equation with respect to the governing parameter.For the HASP model, the mean effective stress p' [47] was selected as the governing parameter as a value with clear physical meaning and with defined interval of possible values.The HASP model is implemented in Abaqus/Standard [1] using the user subroutine UMAT and GPM as numerical procedure for stress integration.

Consolidation of clay layer
As an example of implementation of the HASP model, analysis of the soil consolidation as the result of phased construction of the embankment on the clay surface was performed (example in book Applied Soil Mechanics with Abaqus Applications [13]).The FEM model consists of a layer of clay, 4.6 m thick, which lies on impermeable and incompressible base.The ground water table is on the clay surface, as shown in Figure 7.The embankment is built in three equal layers, 0.6 m thick.Total height of the embankment is 1.8 m.The structure of the embankment is made by phases/layers and construction of one layer takes two days, while the construction of the entire embankment takes six days.The consolidation of clay after construction of the embankment takes another 200 days.
The possibility of the HASP model to predict the change of pore water pressure, as well as the value of the consolidation settlement of the embankment and clay layer was performed.

Tabela 2. Parametri nasipa
The analysis was performed with different initial conditions, i.e. different initial overconsolidation ratios of clay layer (Table 4) in five calculation steps.In the first calculation step, the embankment is removed from the finite element mesh.The next three steps consist of simulation of the construction of the embankment in three layers, where each subsequent layer was added to the already deformed previous one.The fifth step is consolidation of clay and the embankment for a period of 200 days.

Results
For analysis of individual results, the results for the overconsolidation ratio OCR=5 were selected as an illustration.Figure 8 shows the timeline of the settlement under the centre of the embankment (surface of the clay layer) in semi-logarithmic plot.Strains develop most quickly (the highest gradient) during the first 6 days, which is how long the construction of the embankment lasts, and about 50% of the total settlement occurred by that time.Figure 9 shows the history of development of pore water pressure in the middle of the clay layer under the centre of the embankment.The pore water pressure increases during the construction of the embankment (six days) and during the consolidation process its full dissipation occurs.Slika 9. Razvoj pornog natpritiska u sredini sloja gline ispod centra nasipa, OCR=5 Figure 9. Development of pore water pressure in the middle of the clay layer under the centre of the embankment, OCR=5 Raspodela pornog natpritiska i disipacija tokom vremena data je na Slici 10.Usled brzog opterećivanja sloja zasićene gline male vodopropusnosti, ispod nasipa se odmah nakon nanošenja opeterećenja razvija porni natpritisak.S obzirom da je omogućeno dreniranje vode samo preko gornje površine, do najbrže disipacije dolazi upravo na gornjoj površini sloja gline.
Distribution of the pore water pressure and dissipation over time is shown in Figure 10.As the result of rapid loading of the layer of saturated clay of low water permeability, the pore water pressure develops under the embankment immediately after placing the load.Since water draining is enabled only over the upper surface, the fastest dissipation occurs exactly on the upper surface of the clay layer.Sleganja sloja gline tokom 206 dana ispod centra nasipa, za sve stepene prekonsolidacije, data su na Slici 12. Najveća sleganja, kao što se i očekuje, dobijena su za blago prekonsolidovane gline.
Settlements of clay layer over 206 days under the centre of the embankment for all overconsolidation ratios are shown in Figure 12.The largest settlements were, as expected, obtained for lightly overconsolidated clays.

Comparison with the MCC model
The same boundary value problem was analyzed by using the MCC model, which already exists as a standard material model in Abaqus.It predicts similar change in pore water pressure, settlements and shear strains over time, while the main difference is in the magnitude of the volumetric and shear strains.By using the HASP model, generally higher values of deformations are obtained compared to those from the MCC model, especially for lower overconsolidation ratios.Such results are expected, since the HASP model predicts elastic-plastic behaviour from the very beginning of the deformation process, while the MCC model predicts only elastic behaviour within the initial yield surface.
For higher overconsolidation ratios (in the given analysis OCR>12), similar values of settlements are predicted for both models (Figure 13).The HASP model, due to the high value of hardening coefficient ω for high overconsolidation ratio, predicts small values of plastic strains and total values of strains are not much different from the values of elastic strains.For lower values of overconsolidation ratio, differences in strain magnitude are more pronounced.While the material described with the MCC model remains in the elastic zone for the given loads and for the lower values of the overconsolidation ratio also, the HASP model predicts higher values of plastic strains as the result of lower values of hardening coefficient ω.In the presented analysis, the differences in the consolidation settlements are up to 20-25%.

CONCLUSIONS
The HASP model successfully overcomes many deficiencies of the MCC model when describing the mechanical behaviour of overconsolidated clays, while keeping the simplicity of the MCC model and the same number of parameters.By using the combined hardening rule in the function of plastic volumetric and shear strain and state parameter, the hardening coefficient has been formulated which controls all elements of the mechanical behaviour of overconsolidated clays.The hardening coefficient is at the same time the reduction coefficient for plastic strains, which allows elastic-plastic behaviour from the very beginning of deformation process.
In drained conditions, the model predicts gradual transition from contractive to dilatant behaviour before the peak strength is reached, as well as gradual transition from hardening to softening without additional mathematical description.In undrained conditions, the model predicts effective stress path of "S" shape, as well as negative failure pore pressure for highly overconsolidated clays.The higher the values of state parameter and overconsolidation ratio, higher the value of the hardening coefficient and the model predicts stiffer response.For normally consolidated clays, the HASP model automatically transforms into the MCC model, because the hardening coefficient equals one.
In the model validation process, the presented results of test simulations at different total stress paths are very well aligned with experimental results for all overconsolidation ratios.In comparison with the prediction of the MCC model, a significant progress was achieved in the following elements: a) the HASP model predicts gradual development of plastic strains from the very beginning of the deformation process; b) there is a pornog pritiska za prekonsolidovana tla.
gradual transition from elastic into elastic-plastic region; c) there is good prediction of failure shear stress, as well as pore water pressure for overconsolidated soil.
The HASP model is implemented in Abaqus/Standard through the available user subroutine UMAT.For numerical integration of constitutive relations, the Governing Parameter Method was used very successfully.
Through the discussed example of consolidation of saturated overconsolidated clay layer, as the result of phased construction of the embankment, the ability of the HASP model to predict the changes of pore water pressure, volumetric and shear strains was presented.The results were compared with the MCC model.As the result of rapid increase of load on the saturated clay layer with low permeability, the HASP model predicts the increase of pore water pressure during the construction of the embankment and full dissipation of the pore water pressure in the process of consolidation.Strains develop most rapidly during the construction of the embankment and greatest amount of settlement were obtained for slightly overconsolidated clays.In comparison with the MCC model, the main difference is in the magnitude of the volumetric and shear strains.By using the HASP model, higher values of strains are generally obtained against the MCC model, since the HASP model predicts elastic-plastic behaviour from the very beginning of the deformation process, while the MCC model predicts only elastic behaviour within the initial yield surface.
Based on the presented results, it can be concluded that the HASP model has good balance of sophistication and simplicity, which allows its wide practical use in solving various geotechnical problems.

Sanja JOCKOVIC MirjanaVUKICEVIC
There are two important conditions for the wide application of constitutive models for soil in contemporary engineering practice: a) the model should predict sufficiently well the soil behaviour at different stress paths; b) the material constants of the model can be determined from standard laboratory tests.Taking into account both conditions, a HASP model has been formulated to describe the mechanical behaviour of the overconsolidated clays, using the critical state theory and the boundary surface concept.The HASP model in a simple way overcomes many deficiencies of the Modified Cam Clay model, without introducing any additional material parameters.The formulation of the hardening rule in the function of the state parameter and overconsolidation ratio, allows the description of numerous elements of the mechanical behaviour of the overconsolidated clays.The HASP model has been implemented in software Abaqus using the Governing Parameter Method for the numerical integration of constitutive relations.The paper presents validation of the HASP model in comparison with the published results of triaxial tests as well as the possibilities of the model to adequately predict the behaviour of the overconsolidated clays through the analysis of the boundary value problem using the finite element method.The problem of the clay settlements due to phased construction of the embankment on the saturated clay surface was analyzed, assuming different overconsolidation ratios.
Key words: constitutive model, overconsolidated clays, state parameter Figure 1.a) State parameters b) Change of the void ratio c) Effective stress paths in undrained conditions d) Stressstrain relations

Slika 12 .
Sleganje sloja gline za različite stepene prekonsolidacije posle 206 dana Figure12.Settlements of clay layer for different overconsolidation ratios after 206 days [3] results shown are from two undrained triaxial compression tests on remolded samples of Cardiff clay[3]with overconsolidation ratios 5 and 12, as well as results of two undrained triaxial extension tests with overconsolidation ratios 6 and 10 (CU tests).

Table 2 .
Parameters of the embankment

Table 3 .
Parameters of the HASP model