The Influence of Stiffness Anisotropy on the Behaviour of a Stiff Natural Clay

Abstract

The stress–strain response of stiff plastic clays is known to be highly non-linear and anisotropic. However, whilst there have been many investigations on the non-linearity of stiff plastic clays, this is not the case for stiffness anisotropy. It is generally believed that, at least at small strains, soil stiffness is anisotropic and can be interpreted within the framework of cross-anisotropic elasticity. However, very few measurements of all cross-anisotropic stiffness parameters have been reported in the literature. Recent research on stiff plastic London Clay, from London’s Heathrow Airport Terminal 5 (T5), is one of the few examples where these independent stiffness parameters have been quantified. On the basis of these research findings, this paper uses a cross-anisotropic non-linear elasto-plastic model to simulate the behaviour of London Clay. The model is used to simulate a number of laboratory tests and the predictions are compared with available experimental data from the T5 investigation. The predictions of the cross-anisotropic non-linear elasto-plastic model for the laboratory tests are also compared with those of an isotropic model, which has been used successfully in the past to simulate the behaviour of stiff plastic clays. Predictions obtained with the two models are then compared in the numerical analyses of a tunnel constructed within the London Clay. Both short-term undrained and long-term drained conditions are examined in fully coupled analyses.

Appendices

Appendix 1

1.1 Isotropic Non-Linear Elasto-Plastic Model

The equations describing the variation of the tangent shear modulus, G, and tangent bulk modulus, K, in the non-linear range are as follows:

$$ \frac{3G}{p '} = A + Bcos\left( {\beta X^{\gamma } } \right) - \frac{{B\beta \gamma X^{\gamma - 1} }}{2.303}sin\left( {\beta X^{\gamma } } \right) $$

(2)

$$ \frac{K}{p '} = R + Scos\left( {\delta Y^{\mu } } \right) - \frac{{S\delta \mu Y^{\mu - 1} }}{2.303}sin\left( {\delta Y^{\mu } } \right) $$

(3)

where

$$ X = log_{10} \left( {\frac{{E_{d} }}{{\sqrt {3{\text{C}}} }}} \right)\,\, and\,\, Y = log_{10} \left( {\frac{{\varepsilon_{v} }}{T}} \right) $$

(4)

In the above equations A, B, C, R, S, T, β, γ, δ and μ are material constants, p′ is the mean effective stress, ε _v is the volumetric strain and E _d is the deviatoric strain, defined as follows:

$$ p' = \frac{{\sigma '_{1} + \sigma '_{2} + \sigma '_{3} }}{3} $$

(5)

$$ \varepsilon_{v} = \varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3 } $$

(6)

$$ E_{d} = \frac{2}{\sqrt 6 }\left[ {\left( {\varepsilon_{1} - \varepsilon_{3} } \right)^{2} + \left( {\varepsilon_{2} - \varepsilon_{3} } \right)^{2} + \left( {\varepsilon_{1} - \varepsilon_{2} } \right)^{2} } \right]^{1/2} $$

(7)

The above equations are derived from differentiation of the secant equations given in the original publication (Jardine et al. 1991). Owing to the trigonometric nature of Eqs. 2 and 3, minimum (E _dmin , ε _vmin ) and maximum (E _dmax , ε _vmax ) strain limits are set, below and above which the tangent shear and bulk moduli vary only with mean effective stress. Furthermore, the magnitude of the stiffness is prevented from falling below specified minimum values, G _min and K _min .

The non-linear elastic model is combined with a Mohr–Coulomb yield surface and a non-associated flow rule.

1.2 Anisotropic Non-Linear Elasto-Plastic Model

This model (Franzius 2004; Franzius et al. 2005) combines the transversely anisotropic formulation of Graham and Houslby (1983) with a non-linear response based on the isotropic non-linear model described previously. This appendix gives the basic equations of the model. The tangent vertical Young’s modulus $E_{v}^{\prime }$ is defined as:

$$ \frac{{E_{v}^{\prime } }}{{p^{\prime } }} = A + Bcos\left( {\beta X^{\gamma } } \right) - \frac{{B\beta \gamma X^{\gamma - 1} }}{2.303}sin\left( {\beta X^{\gamma } } \right) $$

(8)

where

$$ X = \log_{10} \left( {\frac{{E_{d} }}{{\sqrt {3C} }}} \right) $$

(9)

In the above equations A, B, C, β and γ are constants, p′ is the mean effective stress and E _d is the deviatoric strain which has been defined in Eq. 7. Owing to the trigonometric nature of Eq. 8 minimum (E _dmin ) and maximum (E _dmax ) strain limits are set, below and above which the tangent vertical Young’s modulus varies only with mean effective stress. Moreover, the magnitude of the stiffness is prevented from falling below a specified minimum value $E_{v}^{\prime }$ _min . The model has two additional parameters: the drained Poisson’s ratio v′ _hh and the anisotropic scale factor α defined in Eq. 1.

Similar to the isotropic non-linear model, this anisotropic non-linear formulation is combined with a Mohr–Coulomb yield surface and a non-associated flow rule.

Appendix 2

Gasparre (2005) and Gasparre et al. (2007) describe in detail how the five independent stiffness parameters of a cross anisotropic soil can be calculated by combining BE measurements and static TX. This Appendix gives the equations referred to in Fig. 2. These are as follows:

$$ \frac{{\nu_{hv}^{\prime } }}{{E_{h}^{\prime } }} = \frac{{\nu_{vh}^{\prime } }}{{E_{v}^{\prime } }} $$

(10)

$$ \nu '_{hv} = - \frac{{\delta \varepsilon_{v} }}{{\delta \varepsilon_{h} }} \frac{{\left( {1 - \nu '_{hh} } \right)}}{2} $$

(11)

$$ \nu '_{hv} = - \frac{{E'_{h} }}{2} \frac{{\delta \varepsilon_{v} }}{{\delta \sigma '_{h} }} $$

(12)

For an explanation on how these equations are derived the reader is referred to the original publications, i.e., Gasparre (2005) and Gasparre et al. (2007).