An efficient optimization method for identifying parameters of soft structured clay by an enhanced genetic algorithm and elastic–viscoplastic model

Abstract

Soft structured clays usually exhibit complex behaviors, which can lead to difficulties in the determination of parameters and high testing costs. This paper aims to propose an efficient optimization method for identifying the parameters of advanced constitutive model for soft structured clays from only limited conventional triaxial tests. First, a new real-coded genetic algorithm (RCGA) is proposed by combining two new crossover and mutation operators for improving the performance of optimization. A newly developed elastic–viscoplastic model accounting for anisotropy, destructuration and creep features is enhanced with the cross-anisotropy of elasticity and is adopted for test simulations during optimization. Laboratory tests on soft Wenzhou marine clay are selected, with three of them being used as objectives for optimization and others for validation. The optimization process, using the new RCGA with a uniform sampling initialization method, is carried out to obtain the soil parameters. A classic genetic algorithm (NSGA-II)-based optimization is also conducted and compared to the RCGA for estimating the performance of the new RCGA. Finally, the optimal parameters are validated by comparing with other measurements and test simulations on the same clay. All comparisons demonstrate that a reliable solution can be obtained by the new RCGA optimization combined with the appropriate soil model, which is practically useful with a reduction in testing costs.

Appendix: Anisotropic elastic–viscoplastic model “ANICREEP”

Based on Yin et al. [49], the main constitutive equations are listed as follows:

$$\dot{\varepsilon }_{ij} = \dot{\varepsilon }_{ij}^{\text{e}} + \dot{\varepsilon }_{ij}^{\text{vp}}$$

(12)

$$\dot{\varepsilon }_{ij}^{\text{vp}} = \mu \,\left( {\frac{{p_{m}^{\text{d}} }}{{p_{m}^{\text{r}} }}} \right)^{\beta } \frac{{\partial f_{\text{d}} }}{{\partial \sigma_{ij}^{\prime } }}$$

(13)

$$f_{\text{r}} = \frac{{\frac{3}{2}\left( {\sigma_{\text{d}}^{{\prime {\text{r}}}} - p^{{\prime {\text{r}}}} \alpha_{\text{d}} } \right):\left( {\sigma_{\text{d}}^{{\prime {\text{r}}}} - p^{{\prime {\text{r}}}} \alpha_{\text{d}} } \right)}}{{\left( {M^{2} - \frac{3}{2}\alpha_{\text{d}} :\alpha_{\text{d}} } \right)p^{{\prime {\text{r}}}} }} + p^{{\prime {\text{r}}}} - p_{\text{m}}^{\text{r}} = 0$$

(14)

$${\text{d}}\alpha_{\text{d}} = \omega \left[ {\left( {\frac{{3\sigma_{\text{d}} }}{{4p^{\prime } }} - \alpha_{\text{d}} } \right)\left\langle {{\text{d}}\varepsilon_{\text{v}}^{\text{vp}} } \right\rangle + \omega_{\text{d}} \left( {\frac{{\sigma_{\text{d}} }}{{3p^{\prime } }} - \alpha_{\text{d}} } \right){\text{d}}\varepsilon_{d}^{\text{vp}} } \right]$$

(15)

$$p_{\text{m}}^{\text{r}} = (1 + \chi )p_{\text{mi}}$$

(16)

$${\text{d}}p_{\text{mi}} = p_{\text{mi}} \left( {\frac{{1 + e_{0} }}{{\lambda_{i} - \kappa }}} \right){\text{d}}\varepsilon_{\text{v}}^{\text{vp}}$$

(17)

$${\text{d}}\chi = - \chi \xi \left( {\left| {{\text{d}}\varepsilon_{\text{v}}^{\text{vp}} } \right| + \xi_{\text{d}} {\text{d}}\varepsilon_{\text{d}}^{\text{vp}} } \right)$$

(18)

where $\dot{\varepsilon }_{ij}$ denotes the (i, j) component of the total strain rate tensor and the superscripts e and vp represent, respectively, the elastic and the viscoplastic components. The elastic behavior in the proposed model is assumed to be isotropic, as in the modified Cam Clay model. The $p_{\text{m}}^{\text{d}}$ is the size of the dynamic loading surface. The $p_{\text{m}}^{\text{r}}$ and $p_{\text{mi}}$ are the size of the reference and the intrinsic yield surfaces, respectively. The initial reference preconsolidation pressure $\sigma_{p0}^{{{\prime }{\text{r}}}}$ obtained from an oedometer test can be used as an input to calculate the initial size p _m0 using Eq. (14).

The slope of the critical state line M is expressed as follows:

$$M = M_{\text{c}} \left[ {\frac{{2c^{4} }}{{1 + c^{4} + \left( {1 - c^{4} } \right)\sin 3\theta }}} \right]^{{\frac{1}{4}}}$$

(19)

where $c = (3 - \sin \,\phi_{\text{c}} )/(3 + \sin \,\phi_{\text{c}} )$ according to the Mohr–Coulomb yield criterion (ϕ _c is the friction angle); $- \pi /6 \le \theta = (1/3)\sin^{ - 1} ( - 3\sqrt 3 \bar{J}_{3} /2\bar{J}_{2}^{3/2} ) \le \pi /6$ using $\bar{J}_{2} = (1/2)\bar{s}_{ij} :\bar{s}_{ij}$, $\bar{J}_{3} = (1/3)\bar{s}_{ij} \bar{s}_{jk} \bar{s}_{ki}$ with $\bar{s}_{ij} = \sigma_{\text{d}} - p^{\prime } \alpha_{\text{d}}$.

The model was implemented as a user-defined model in the 2D version 9 of PLAXIS for a coupled consolidation analysis based on Biot’s theory (see details in [49]).

During consolidation coupled analyses, the permeability k varies with void ratio e:

$$k = k_{0} 10^{{(e - e_{0} )/c_{\text{k}} }}$$

(20)

Soil constants and state variables are summarized in Table 8 with their recommended methods of determination (see details in [49]) (Fig. 20).

Table 8 State parameters and soil constants of elastic–viscoplastic model

Fig. 20

Definitions for the model in a p′ − q space; and b one-dimensional compression condition