CONSTITUTIVE MODEL FOR CEMENTED QUICK CLAY IN DEEP-MIX STABILIZATION

Based on experimental results of a laboratory study on lime and cement stabilized quick clay, carried out at NTNU, Norway, a two yield surface constitutive model has been formulated. The model reproduces the mechanical behavior of the stabilized soil in the p’-q stress space. In the stress space the two yield surfaces appear as a “cap” and a “wedge” respectively. The formulated model “QUICKSTAB” has satisfactorily been able to simulate the overall geomechanical behaviour of stabilized quick clay and it agrees very well with the results established by the laboratory study. Typical model parameter values are also presented herein for various soil-binder mixtures. An extensive parametric study has been recommended to thoroughly explain the behavior of the model, including tests following different stress paths and extension of the study to include anisotropic cap-surface.


INTRODUCTION
Stabilization process involves complete remoulding of the soil and mixing the binder materials in.Remoulding of the extremely sensitive quick clay changes it into a viscous fluid before the binder material is added.The eventual engineering properties of the stabilized material depend on the fabric and particle cementation resulting from the chemical reaction of the binders.Stabilized clays have been shown by many researches (e.g.Balasubramaniam & Buensuceso (1989); Broms, 1999;Kivelö & Broms, 1999;Baker, 2000;Kwan et al., 2005) to closely behave similar to other structured and over-consolidated soils.Such soils display a substantial amount of intrinsic stiffness and strength when subjected to an external loading.When the external load exceeds the bond strength, the bonds break as the material yields leading further to a complete collapse of the microstructure.Prediction of the mechanical behaviour of engineering geo-materials requires a well formulated constitutive model that can best capture the non-uniformity, heterogeneity and anisotropy of soils, and some uncertainties in geotechnical assumptions.
This paper presents a constitutive model (QUICKSTAB) for stabilized quick clay, which has been proposed based on the results of an investigation on stabilization of quick clay and the potential for utilization of wastepaper sludge ash for deep-mix stabilization of quick clay.The study was carried out at NTNU, Norway and is explained more in Bujulu (2008).The model takes into account the actual mechanical properties of the stabilized material and simulates fairly well the overall behavior of this material obtained from the laboratory experiments.Typical parameter values are also presented for various binder mixtures.

MODEL CONCEPT
A multiple yield surface model is proposed to model the mechanical behaviour of the stabilized soil material.In the p'-q stress space the two yield mechanisms will appear as a "cap" and a "wedge" respectively.The cap is mainly meant to model the volumetric (oedometric) behaviour and the wedge is meant to model the deviatoric behaviour.The cap is identical to the isotropic cap used in the Hardening Soil Model (Brinkgreve et al., 2006).However, with a relevant rotational rule, rotation of the cap surface may be introduced in a similar manner to the S-Clay1 model (Wheeler et. al., 2003).The wedge yield criterion, or cone surface in full stress space, is formulated as proposed by Søreide (2003) and Dafalias & Manzari (2004).In Figure 1 the model principle is shown in the p'-q space.

MATHEMATICAL FORMULATION
The model is formulated in the triaxial stress space.The model principle is shown in the p'-q space in Figure 1.Formulation is discussed under three sub-sections, namely, the cap yield surface, hardening rules for the cap and cone yield and potential surface p' q "wedge" "cap" a w Failure line (Bounding line)

Cap Yield Surface
In equation (1) the cap surface is given as: where Μ (Greek capital μ) is an internal parameter related to the earth pressure coefficient under virgin loading, K 0 NC ; p m ' is the size of the cap and a c is the attraction for the cap.It may be shown, due to the strain requirement in oedometric condition and through an associated flow rule, that M will be given by Equation (2), assuming infinite elastic stiffness.K 0 NC is in this case obtained by including the a c term for the stresses.Μ is easiest obtained by a stress condition in which a c << p'.

Hardening Rules for the Cap
Two hardening rules are used for the cap surface.Firstly, an isotropic hardening rule which, as opposed to the models like the Modified Cam Clay Model (Roscoe and Burland, 1968), includes the possibility for a constant term.The expression will then take the form: 3) where ζ and Ζa are parameters controlling the hardening of the cap.The second cap hardening rule is the attraction softening.Under oedometeric condition (isolating the cap behavior) two types of attraction softening may generally be experienced.Type I may be given by a destructuration rule, see e.g.Karstunen et al. (2006).In this case the observed stiffness in the NC domain is experienced to be lower than the one for unbounded material (we may scale our stress down by a structural parameter).The structure is reduced during straining (destructuration rule), and for large strain the unbounded (remolded) stiffness is experienced.Type II is associated with the loss of the a c term in the yield criterion for the cap.The parameter a c will in this case scale our stress upwards, giving higher stiffness than for the unbounded material.A softening rule for the attraction, a c , will then give us the proper variation of stiffness with strain.In Figure 2  Hence we may propose the following softening rule for the attraction due to the cap type plasticity.
where μ ac is a hardening parameter.The C term is added to the vertical axis label (Figure 2) to take care of the "attraction" in the oedometer test results for stabilized quick clay (ref.

Cone yield and potential surface
The cone yield surface will in the p'-q stress space appear as a wedge.Equation ( 5) gives an expression for the wedge yield surface. 5) where α is the rotation of the wedge in the p'-q space; m is the size of the wedge; a w is the attraction for the wedge (see Figure 1).
The size parameter m will be a small number and will be given a default value of 0.01•M c,f.Where M c,f is the maximum value of q/(p'+a w ) in a triaxial compression test.The potential surface is not directly needed; only its derivatives with respect to the stress components (Equation ( 6)) are used.
For f α we will have the following requirements The function given by equation ( 9) may be used in triaxial stress-strain space.9) where q b is the bounding deviatoric stress In the triaxial shearing tests the material shows "attraction softening" (cohesion softening), ref.

Elastic Properties
For simplicity an isotropic elastic behaviour is chosen.As shown by Equation (11) below, the shear stiffness is assumed to be given by the Poisson ratio.This is done based on the oedometer tests.11) where κ*, Κ a and ν are the elastic input parameters LABORATORY DATA Four types of mixtures were investigated as shown in Table 1.For sample preparation and testing methods the reader is referred to Bujulu (2008).Results from undrained triaxial compression and incremental loading oedometer tests for the four mixture types at 90 days curing period were used to calibrate the proposed model.The results are presented in the same plots with the respective simulation curves for comparison and model evaluation purposes; ref.

MODEL SIMULATION
The material model is implemented in a numerical scheme, which allows for violation of several yield criteria.The rule for adding response from several yield criteria and plastic potential functions is called the Koiter rule (Schanz et al., 1999).The scheme allows only the plastic multiplier for the violated yield criteria to be activated.Simulations for the different mixtures can be obtained by varying the parameter values.Typical material properties used for the reported simulations are given by Table 2.  4 and Figure 6 show results from simulation of undrained triaxial shearing for different isotropic consolidation cell pressure.The practically minor discrepancy observed between experimental data and simulations, mainly in the triaxial test plots (Figure 4 and Figure 6), can be explained by the localization problem in the laboratory experiments due to wedging and locking in the failure plane, typical of the brittle cemented quick clay samples (ref. Figure 8).All triaxial tests on stabilized quick clay revealed a substantial reduction in cohesion in the post-peak loading range, i.e. cohesion softening, but the angle of friction remained essentially unchanged (ref.Figure 9).Due to this phenomenon, the experimental stress-strain behavior for larger strains can therefore not be trusted.Broms (1999) suggests neglecting cohesion when the residual strain is used in the design of lime-cement columns.

Figure 1 :
Figure 1: Model principle, visualized in p'-q space these two types of stiffness degradation are shown.Natural quick clay shows type I of softening while stabilized quick clay shows type II (ref.Figure 3).

Figure 2 :
Figure 2: Two types of stiffness degradation Figure 3: Typical oedometer modulus curves for natural and stabilized quick clay Figure 9. Equation (10) gives a mathematical expression for how this softening rule may look like. is a hardening parameter.

Figure 5 andFigure 5 :
Figure4and Figure6show results from simulation of undrained triaxial shearing for different isotropic consolidation cell pressure.Figure5and Figure7show results form oedometer simulations.The simulation curves

CONCLUSIONA
constitutive model for L-(C)-WSA stabilized quick clay has been formulated.The model (QUICKSTAB) has simulated well the overall geomechanical behaviour of stabilized quick clay and has shown good agreement with experimental results.The model parameters depend on the level of cementation (i.e.compressive strength) of the stabilized material and the confining pressure.With carefully selected parameters, the model functioned equally well for the tested four mixes.This suggests that the model may generally be applied to all similar mixes of cemented quick clay.However, an extensive parametric study should be done in order to thoroughly explain the behavior of the model.Different laboratory results should also be used for calibration of the proposed model.The isotropic cap surface should be validated by test following different stress paths and, if necessary, an anisotropic cap-surface may be included.Further, a nonassociated flow rule may be considered for the cap in order to control the levels of stress mobilization.The proposed additional tests should also be used to verify the calibration of the potential surface and the hardening rule for the cone (wedge).