Modelling stress-induced anisotropy in multi-phase granular soils

Abstract

In this paper, we outline a constitutive model capable of describing anisotropy and many other features of the behaviour of multiphase granular soils, together with the computational framework that enables its numerical implementation. The constitutive model is formulated within the framework of bounding surface plasticity. It can simulate monotonic and cyclic loading for a wide range of stress and saturation states, it includes enhanced descriptions of wetting and drying processes, of anisotropy and of changing compressibility. These features are captured using a single set of parameters by using a combination of isotropic and kinematic hardening. The model is formulated based on the concept of effective stress for unsaturated states that guarantees smooth transitions between unsaturated and fully saturated states. Furthermore, we present unified formulations for saturated and unsaturated states in which the isotropic hardening law and the critical state line are described in a bi-logarithmic space defined by the logarithms of the mean effective stress and void ratio. Moreover, the constitutive model is coupled with a soil water characteristic model that allows consideration of the hysteretic nature of the saturation degree changes upon wetting/drying reversals. The paper describes the numerical implementation, which includes several smoothing techniques to enhance the constitutive model’s performance in numerical modelling during transitions between kinematic hardening and isotropic hardening and drying/wetting reversals. The numerical implementation also includes automatic error control and sub-stepping techniques, suitable for explicit integration algorithms, that give users additional control over the accuracy and speed of the analysis. Lastly, several examples are provided to demonstrate the range of application of the computational framework.

Appendices

Appendix 1: Additional details on the equations of SWCC

The equation of the SWCC is obtained from [81, 83]

$$ S_{w}^{\alpha } = S_{rw} + \left( {S_{ra} - S_{rw} } \right)*\left( {\ln \left[ {\exp \left( 1 \right) + \left( {\frac{{p_{c}^{*} }}{{P^{\alpha } }}} \right)^{{n_{x}^{\alpha } }} } \right]} \right)^{{ - m_{x}^{\alpha } }} \left( {\alpha = w,d} \right) $$

(69)

Tables 3 and 4 show the constitutive model and the SWCC data, respectively. Note that a star symbol (*) in front of a number denotes that the value of this parameter is assumed as there was insufficient data to directly obtain the parameter. Also, the SWCC parameters given in Table 4 are obtained by taking Eq. (69), as $S_{w}^{\alpha }$ in Eq. (39).

Table 3 The model parameters for Toyoura and Kurnell sands

Table 4 The SWCC parameters for Kurnell sand

All the numerical examples in Sect. 11 have adopted the mechanical parameters of Toyoura sand, the SWCC parameters given in Table 5 and $b_{1}$ and $b_{2}$ have been set to 2.0 and 1.5, respectively.

Table 5 The SWCC parameters used in Sect. 11

In Table 6, the first label, L(oose) or D(ense), indicates the initial sand states and the second label, D(rained) or U(ndrained), indicates the type of test.

Table 6 The initial conditions of the tests reported in [88]

Appendix 2: Isotropic hardening law

$f\left( \xi \right)$ is a function of both suction and volume changes (due to the dependency of saturation degree on void ratio). After the expansion of $f\left( \xi \right)$ in Eq. (9), we obtain

$$ d\varepsilon_{v}^{{}} = \frac{e}{{\left( {1 + e} \right)}}\left( {\lambda \frac{{dp^{{\prime }} }}{{p^{{\prime }} }} + \frac{\partial f\left( \xi \right)}{\partial e}\frac{1}{f\left( \xi \right)}\left( {1 + e} \right)d\varepsilon_{v}^{{}} - \frac{\partial f\left( \xi \right)}{{\partial p_{c} }}\frac{1}{f\left( \xi \right)}dp_{c} } \right)\left( {1 - \delta_{p}^{\theta } } \right) + \frac{{dp^{{\prime }} }}{K}\delta_{p}^{\theta } $$

(70)

By assuming ${\mathcal{C}} = - e\frac{\partial f\left( \xi \right)}{\partial e}\frac{1}{f\left( \xi \right)}\left( {1 - \delta_{p}^{\theta } } \right)$ and $\kappa = \frac{{p^{{\prime }} }}{K}\frac{1 + e}{e}$, and after some manipulations, the following is obtained

$$ d\varepsilon_{v}^{{}} = \frac{1}{{1 + {\mathcal{C}}}}\frac{e}{{\left( {1 + e} \right)}}\lambda \frac{{dp^{{\prime }} }}{{p^{{\prime }} }}\left( {1 - \delta_{p}^{\theta } } \right) - \frac{1}{{1 + {\mathcal{C}}}}\frac{e}{{\left( {1 + e} \right)}}\frac{\partial f\left( \xi \right)}{{\partial p_{c} }}\frac{1}{f\left( \xi \right)}\left( {1 - \delta_{p}^{\theta } } \right)dp_{c} + \frac{1}{{1 + {\mathcal{C}}}}\kappa \frac{{dp^{{\prime }} }}{{p^{{\prime }} }}\frac{e}{1 + e}\delta_{p}^{\theta } $$

(71)

By subtracting the elastic strain rate from the total strain rate, the plastic volumetric strain rate is obtained as follows

$$ d\varepsilon_{v}^{p} = \frac{1}{{1 + {\mathcal{C}}}}\frac{e}{{\left( {1 + e} \right)}}\lambda \frac{{dp^{{\prime }} }}{{p^{{\prime }} }}\left( {1 - \delta_{p}^{\theta } } \right) - \frac{1}{{1 + {\mathcal{C}}}}\frac{e}{{\left( {1 + e} \right)}}\frac{\partial f\left( \xi \right)}{{\partial p_{c} }}\frac{1}{f\left( \xi \right)}\left( {1 - \delta_{p}^{\theta } } \right)dp_{c} + \frac{1}{{1 + {\mathcal{C}}}}\kappa \frac{{dp^{{\prime }} }}{{p^{{\prime }} }}\frac{e}{1 + e}\delta_{p}^{\theta } - \kappa \frac{{dp^{{\prime }} }}{{p^{{\prime }} }}\frac{e}{1 + e} $$

(72)

After some manipulations and the replacement of $p^{{\prime }}$ by the size of the loading surface, we obtain

$$ d\alpha_{iso}^{l} = \frac{{\left( {1 + e} \right)}}{e}\frac{{\alpha_{iso}^{l} \left( {1 + {\mathcal{C}}} \right)}}{{\left( {\left( {\lambda - \kappa } \right)\left( {1 - \delta_{p}^{\theta } } \right) - \kappa {\mathcal{C}}} \right)}}d\varepsilon_{v}^{p} + \frac{{\alpha_{iso}^{l} \left( {1 - \delta_{p}^{\theta } } \right)}}{{f\left( \xi \right)\left( {\left( {\lambda - \kappa } \right)\left( {1 - \delta_{p}^{\theta } } \right) - \kappa {\mathcal{C}}} \right)}}\frac{\partial f\left( \xi \right)}{{\partial p_{c} }}dp_{c} $$

(73)

Appendix 3: Transition mechanism based on the gradient of the loading surface

It is also possible to use a general form of the vector of the gradient of the yield surface, $A^{\sigma }$ as follows

$$ \left[ {A^{\sigma } } \right] = \left[ {\frac{{\partial f_{1} }}{{\partial p^{{\prime }} }}\zeta + \frac{{\partial f_{2} }}{{\partial p^{{\prime }} }}\left( {1 - \zeta } \right) \quad \frac{{\partial f_{1} }}{\partial q}\zeta + \frac{{\partial f_{2} }}{\partial q}\left( {1 - \zeta } \right)} \right] $$

(74)

where the first and second components of the vector are the volumetric and deviatoric terms, respectively. It may be noted that $f_{1}$ and $f_{2}$ are the loading surfaces constructed based on the radial mapping and deviatoric rule, respectively. Note that when $\zeta = 1$ the radial mapping rule will be activated whereas for $\zeta = 0$, the deviatoric mapping rule will be the active mechanism.