a constitutive model for chemically sensitive clays

A Constitutive Model for Chemically Sensitive Clays

Nathalie Boukpeti1, Robert Charlier1, Tomasz Hueckel2, and Zejia Liu3

1 Department GeomaC, University of Liege, Belgium [email protected] Tel: +32-4 366 91 43 Fax: +32-4 366 95 20 2 Department of Civil and Environmental Engineering, Duke University, USA 3 The State key Laboratory for Structural Analysis of Industrial Equipment, Dalian University of Technology, China

Abstract. This paper deals with a chemo-hydro-mechanical (CHM) model for unsaturated clays. The chemo-mechanical effects are described within an elasto-plastic model using the concept of chemical softening proposed by Hueckel (1997). The constitutive behaviour of partially saturated clays is modelled following Alonso-Gens’ formulation. The equilibrium equations with the chemo-hydro-mechanical constitutive relations are combined with the governing equations for liquid transfer and contaminant transport, and are solved numeri-cally using finite elements. Numerical examples are presented to analyse the chemical effects upon the response during wetting of a clay specimen, and during an excavation.

Keywords: constitutive model, unsaturated clay, plastic mechanisms.

1 Introduction

It has been recognized that the presence of certain chemicals in the pore fluid of clayey soils affects their hydro-mechanical behaviour. Understanding of the chemical effects is essential for the design of clay barriers or the assessment of borehole or tunnel stability. Both expansive and contractive strain have been measured on clay specimens permeated with organic liquids, depending on the chemical concentration and the stress level (Fernandez and Quigley 1991). Com-paction of clay specimens upon exposure to salt solutions has also been observed and is explained by the osmotic effect (Di Maio 1996). It was shown that part of this compaction is irreversible. The chemo-mechanical behaviour of clay has been described by means of models based on the micro-structure (e.g., Guimaraes et al. 2001). The macroscopic behaviour can also be represented in a phenomenological manner using the framework of plasticity. Within this framework, the concept of chemical softening was proposed by Hueckel (1997) to describe strains occurring in clay under constant stresses due to the presence of a single chemical. It is as-sumed that the yield surface shrinks as the contaminant mass concentration in-creases. In order to predict adequately the chemical expansion observed during permeation of clay with organic liquids (ethanol and dioxane) at low external

Robert Hack, Rafig Azzam, and Robert Charlier (Eds.): LNES 104, pp. 255–264, 2004.c© Springer-Verlag Berlin Heidelberg 2004

256 Nathalie Boukpeti et al.

stress, Hueckel (1997) formulated a chemo-elastic strain of expansion as a func-tion of concentration. Similar concepts were used in the model presented by Boukpeti (2003). To represent the behavior of unsaturated soils, various ap-proaches have been considered ( e.g., Matyas and Radhakrishna 1968). The model presented in this paper follows the ‘Alonso-Gens’ formulation (Alonso et al. 1990), which considers the effects of suction (difference between air and water pressure) on the compression coefficient, on the pre-consolidation pressure, and on the cohesion. The model presented in this paper is based on the constitutive equations proposed by Collin (2003) and Collin et al. (2002) for unsaturated soils, applicable in particular to chalk. The yield surface is composed of an elliptical cap, a frictional line and a traction limit. The effects of suction and chemical con-centration are incorporated into the model. The chemo-hydro-mechanical model results in the coupling of the governing equations for equilibrium (including the constitutive equations), flow and mass transport. The fluid flow model takes into account unsaturated conditions and is described by Collin (2003). For the chemi-cal mass transport, only the advective and diffusive effects are considered in this paper. A complete description of the mass transport model is given by Radu et al. (1994) and Li et al. (1999). In the following section, the constitutive equations for the chemo-hydro-mechanical behaviour of unsaturated clays are presented. Next, two numerical examples are discussed; wetting of a clay specimen with a chemi-cal, and an excavation in chemically sensitive clay. The numerical simulations were performed using the finite elements program LAGAMINE.

2 Constitutive Model

2.1 General Formulation

The constitutive equations, expressed in rate form, relate the strain tensor εij to the

stress tensor σij, the suction s, and the chemical mass concentration c. A positive strain corresponds to compaction. The suction is defined by the difference be-tween the air and water pressure (s ≥ 0). The mass concentration is the ratio of mass of chemical to the total mass of fluid (c ≤ 1). In the formulation given be-low, the stress tensor designates alternatively the net stress (difference between total stress and air pressure), in the case of unsaturated conditions, or the effective stress in the case of full saturation. The strain rate is decomposed into an elastic (reversible) part and a plastic (irreversible) part. The elastic part can also be de-composed into mechanical, suction, and chemical contributions, namely:

, , ,e p e m e s e c pij ij ij ij ij ij ijε ε ε ε ε ε ε= + = + + +� � � � � � � (2.1-1)

The plastic deformation occurs when the yield condition on stresses is verified. However, this plastic strain rate can be generated by variations in suction or con-centration at constant stress state. The mechanical elastic stress-strain law is given by:

A Constitutive Model for Chemically Sensitive Clays 257

,e m ekl klij ijCσ ε= �� (2.1-2)

with the compliance elastic tensor eklijC defined as:

( )1 22

3 3eklij ik jl mm ij kl

eC G Gδ δ σ δ δ

κ+

= + −

(2.1-3)

where G is the elastic shear modulus, e is the void ratio and κ is the elastic volu-metric coefficient. The elastic deformation induced by suction is given by:

( ) ( ), 1

3 1s e esij ij ij

at

sh s

e s p

κε δ= =+ +

�� (2.1-4)

where e is the void ratio, pat is the atmospheric pressure and κs is the elastic stiff-ness parameter for changes in suction defined in the Alonso-Gens’model. This isotropic deformation is contractive for an increase in suction s (drying) and ex-pansive for a decrease of s (wetting). :The reversible chemical strain is also iso-tropic:

, 1

3c e eij ij ijc l cε β δ= − =� � � (2.1-5)

The coefficient β is the chemical expansion coefficient defined by Hueckel as:

( )[ ]

−+−−= 1

1ln1exp 000 c

ccF βββ (2.1-6)

where F0 and β0 are material constants dependent on the soil and the chemical. The plastic deformation is described within the framework of strain-hardening/softening plasticity. The yield criterion reads:

( ), 0ijf σ θ ≤ (2.1-7)

where θ is an internal variable depending on plastic strain, suction and chemical concentration. During plastic loading, the yield function f verifies the consistency condition:

0ijij

f ff σ θ

σ θ∂ ∂= + =

∂ ∂� �� (2.1-8)

The evolution of the internal variable q is described by the general hardening law:

pp

s cs c

θ θ θθ εε

∂ ∂ ∂= + +∂ ∂ ∂

� � � � (2.1-9)


In the particular plastic models described in the next section, either the volu-metric or the shear component of the plastic strain rate is considered in the harden-ing law. The plastic strain vector is defined by the non-associated flow rule:

p pij

ij

gεσ∂= Λ

∂�� (2.1-10)

where g is the plastic potential and pΛ� is the plastic multiplier.

Fig. 2.1. Yield surface in the stress plane (q, p).

2.2 Plastic Behaviour

Three plastic mechanisms are considered: pore collapse or plastic volumetric compaction, frictional failure with possible dilation, and tensile failure. These three mechanisms are represented by three plastic models with the following yield criteria:

( )( ) ( )2 21 0 0( ) 0 ; - /2 s sf q M p p p p p p pβ= + + − = > (2.2-1)

( ) ( )2 t 0( ) 0 ; - / 2s sf q M p p p p pβ σ= − + = < < (2.2-2)

3 0tf p σ= + = (2.2-3)

where p0 is the pre-consolidation pressure, ps is a measure of the cohesion (ps ≥

0), σt is the tensile strength, and M(β) is a parameter related to the friction angle and function of the Lode angle as defined by Van Eekelen (1980). The stress in-

variants p and q are defined as: / 3kkp σ= and 3 / 2 ij ijq s s= , where sij is the

deviatoric part of the stress tensor. In the stress plane (q, p) f1 is represented by an


ellipse with intercepts on the p-axis -ps and p0 (see Fig. 2.1). The yield function f2

is plotted as a friction line of slope M and intercept on the p-axis -ps. The tensile

failure criterion f3 = 0 is represented as a vertical line in Figure 1. In this case, an associated flow rule is used with no hardening. The description of the pore col-lapse model and the frictional failure model is completed below.

2.2.1 Pore Collapse Model The pore collapse model is based on the Cam clay model; the yield surface f1 is

elliptical, the flow rule is associated, and the internal variable p0 evolves with the volumetric component of the plastic strain. The effects of suction and chemical concentration on the internal variable p0 is described by:

( )( )( )0

00 , ( )

s

cc

pp s c p S c

p

λ κλ κ

−∗ −

=

(2.2-4)

where cp is a reference pressure, p0* is the pre-consolidation pressure for s = 0

and c = 0, and λ (0) is the compression coefficient at zero suction. The compres-sion coefficient λ(s) is a decreasing function of s:

( ) ( ) ( ) ( )0 1 exps r s rλ λ β ′= − − + (2.2-5)

where r is a constant representing the maximum stiffness of the clay (at large suction) and β’ is a constant controlling the stiffness increase with suction in-crease, as defined in the Alonso-Gens’ model. For a vanishing value of concentra-tion, equation (2.2-5) defines the Loading Collapse line (LC) represented in Fig. 2.2a. The chemical softening function S(c) is of the form proposed by Hueckel:

( ) exp( )S c ac= − (2.2-6)

where a is a constant representing the strength of chemical softening. The chemi-cal softening line (CHS) is depicted in Fig. 2.2b. The evolution of p0

* follows the volumetric hardening law:

*0 0

1 pv

ep p ε

λ κ∗ +=

−�� (2.2-7)

2.2.2 Frictional Failure Model Frictional failure takes place when the state of stress verifies the yield criterion f2. Plastic strain obey a non-associated flow rule with the plastic potential:

( )2 ’ 0 rg q M p p= − + = (2.2-8)


where M’(β) is related to the dilatancy angle and a function of the Lode angle, and pr is determined from the requirement that the stress point lies on g2. The internal

variable ps evolves as a linear function of suction:

*s s sp p k s= + (2.2-9)

where ks is a constant, and *sp is the value of the variable ps for saturated condi-

tions. This type of hardening law for the effect of suction was proposed in the Alonso-Gens’model and is illustrated in Fig. 2.2a.

3 Numerical Examples

3.1 Wetting of Clay with Organic Liquid

In this example, we analyse the chemical effect of an organic contaminant on the behaviour of a partially saturated clay specimen during wetting. The cylindrical specimen of 2 cm thickness and 5.38 cm diameter is in oedometric conditions (uniaxial deformation). Initially, an external vertical load of 133 kPa is applied on the specimen, and the coefficient K0 = 0.625. The air pressure is at atmospheric pressure, pa = 100 kPa, and is assumed constant during the test. The initial pore pressure is pw = -78.5 MPa, which corresponds to a suction s = 78.6 MPa, and a saturation degree Sr = 0.49. The initial chemical concentration is zero. The chemo-hydraulic boundary conditions consist of a fixed value of chemical concentration at the top boundary, c = 0.6, and fixed values of suction at the top and bottom boundary, s = -28.6 MPa at the top, and s is equal to its initial value at the bottom. The parameters of the CHM system are listed in Table 1. Values of the mechanical and transport coefficients (kw,int is the intrinsic permeability and dm is the molecu-lar diffusion coefficient) are typical for clay; the chemical parameters are chosen

Fig. 2.2. Yield surface; a) in the plane (s, p), b) in the plane (c, p).


within the range of values given by Hueckel (1997) for clay and organic liquids. Figure 3.1 depicts the evolution of suction and chemical concentration along the specimen with time. Equilibrium of flow is reached much faster than chemical equilibrium. The volumetric deformation of the specimen due to wetting with water and with an organic liquid is represented in Fig. 3.2, with a positive strain representing expansion. In the absence of chemical effects, swelling occurs as a result of a decrease of suction, which is governed by the parameter κs. Chemical effects are of two types. In the case of a highly over consolidated clay (large value of p0), the model predicts expansive strain controlled by the parameters F0 and β0. On the contrary, if the clay is slightly over consolidated or normally consolidated, additional chemical compaction is predicted, which is governed by the plastic yield criterion (f1 = 0) and the parameter α. It is apparent in Fig. 3.2 that the chemical plastic compaction is larger at the top of the specimen than at the bot-tom. This is correlated with smaller value of suction, and therefore smaller pre-consolidation pressure p0, at the top than at the bottom of the clay specimen.

Table 1. Model parameters typical for clay and water solution of organic liquid.

κ λ κs e G

MPa

Μ ps

MPa

p01

MPa

p02

MPa Mechanical

0.02 0.1 0.11 0.66 10 1.2 2.078 2000 0.2

F0 β0 a Chemical

-0.04 1.32 3.45

kw,int

m2

dm

m2/s

Transport

4.7·10-18 10-9

1 highly overconsolidated clay; 2 slightly over consolidated clay

Fig. 3.1. Wetting of clay with organic liquid: (a) suction, (b) chemical concentration.


Fig. 3.2. Volumetric response to wetting with organic liquid; p01 is for a highly over con-

solidated clay, p02 is for a slightly consolidated clay.

3.2 Excavation in Chemically Sensitive Clay

The perturbations caused by an excavation may lead to chemical reactions occur-ring around the opening, e.g. oxidation to air, or significant swelling due to water circulation. In this section, we consider the problem of an excavation in hard clay soil in unsaturated conditions, with the presence of a contaminant diffusing in the soil from the tunnel boundary. The tunnel is circular with 5 m diameter and is located 220 m deep. The geometry, initial and boundary conditions are depicted in Fig. 3.3. Assuming a soil unit weight of 21 kN/m3, the initial state of stress at the tunnel’s depth is 4.6 MPa, and is assumed to be isotropic. The initial pore pressure distribution is in equilibrium under gravity, with pw = -78.5 MPa at the top bound-ary of the model. The air pressure is pa = 100 kPa and is assumed to remain fixed. The excavation is modelled by reducing the stress applied to the tunnel’s face from its initial value to zero. Simultaneously, the chemical concentration is in-creased from c = 0 to c = 0.6. The model parameters are similar to those listed in Table 1 except for: G = 200 MPa, ps = 72.75 MPa, p0

1 = 2500 MPa, p02 = 10 MPa,

kw,int = 4.7·10-14 m2, dm = 3·10-8m2/s. Figure 3.4a shows the concentration distribu-tion along the horizontal line passing through the tunnel centre as a function of the distance to the tunnel face after 2300 days. It is apparent that the chemical has diffused into the soil over a distance of approximately 10 m. The volumetric strain distribution is depicted in Fig. 3.4b. The volumetric changes in the absence of chemical are close to zero. Chemical effects induce an expansion of about 2.5% close to the tunnel’s face, in the case of a highly over consolidated clay (p0 = 2500 MPa). In the case of a soil with a smaller pre-consolidation pressure, some chemi-cal compaction occurs in addition to the expansion. However, with the value of p0 = 10 MPa chosen here, the plastic chemical compaction is not significant.


Fig. 3.3. Geometry of the tunnel, initial and boundary conditions.

Fig. 3.4. (a) chemical concentration distribution; (b) volumetric strain distribution.

4 Concluding Remarks

A constitutive model for the chemo-hydro-mechanical(CHM) behavior of unsatu-rated soil has been presented. The chemical effects are described using the con-cepts of chemical softening and elastic expansive strain as proposed by Hueckel (1997). The formulation uses the framework set by Alonso et al. (1990) to de-scribe the behavior of unsaturated soil. The model has been implemented into the finite elements code LAGAMINE and applied to two numerical examples. It was shown that the predicted chemical strain depend on the value of the pre-consoli-


dation pressure, and therefore on the suction level (or saturation degree). This type of results suggests to investigate experimentally the chemical effects on the me-chanical behavior of clay in various saturation conditions.

Acknowledgements

This work is sponsored by Ministry of Education of Belgium under the inter-national joint research project “Qualité et durabilité de la protection des nappes aquifères sous sites d'enfouissement technique avec barrières argileuses d'étanché-ité” and The National Natural Science Foundation of China under the project No. 59878009. The support is greatly acknowledged.

References

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a constitutive model for chemically sensitive clays

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