Advantages and limitations of an α-plasticity model for sand

Abstract

The stress–strain response of sand was observed to depend on its material state, i.e., pressure and density. Successful modelling of such state-dependent response of sand relied on the correct representation of its state-dependent stress–dilatancy behaviour. In this study, an improved fractional-order $\left( \alpha \right)$ plasticity model for sands with a wide range of initial void ratios and pressures is proposed, based on a state-dependent fractional-order plastic flow rule and a modified yielding surface. Potential positive performances and negative limitations of the proposed approach in terms of the critical state of sand are discussed, based on the simulations of a series of drained and undrained triaxial tests of different sands. It can be found that unlike previous fractional models, the developed model can reasonably simulate the key features, e.g., strain softening/hardening, volumetric dilation/contraction, liquefaction, quasi-steady-state flow as well as steady-state flow, of sand for a wide range of initial states. However, due to typical forms of the critical-state lines being used, negative performances of the fractional approach could occur when simulating the undrained behaviour of very loose sand and the drained behaviour of very dense sand.

Appendix

In order to derive Eq. (17), the following analytical solutions of the power-law functions, $(\sigma^{\prime} - \sigma^{\prime}_{{\mathrm{c}}} )^{\mu }$ and $(\sigma^{\prime}_{{\mathrm{c}}} - \sigma^{\prime})^{\mu }$, are needed:

$${}_{{\sigma^{\prime}_{{\mathrm{c}}} }}D_{{\sigma^{\prime}}}^{\alpha } (\sigma^{\prime} - \sigma^{\prime}_{{\mathrm{c}}} )^{\mu } { = }\frac{\varGamma (1 + \mu )}{\varGamma (1 + \mu - \alpha )}(\sigma^{\prime} - \sigma^{\prime}_{{\mathrm{c}}} )^{\mu - \alpha }$$

(25)

$${}_{{\sigma^{\prime}}}D_{{\sigma^{\prime}_{{\mathrm{c}}} }}^{\alpha } (\sigma^{\prime}_{{\mathrm{c}}} - \sigma^{\prime})^{\mu } { = }\frac{\varGamma (1 + \mu )}{\varGamma (1 + \mu - \alpha )}(\sigma^{\prime}_{{\mathrm{c}}} - \sigma^{\prime})^{\mu - \alpha }$$

(26)

where $\mu$ is the power index. Details for deriving Eqs. (25) and (26) can be found in [44] and thus not repeated here for simplicity. Accordingly, Eq. (14) with $\beta = 0$ should be rearranged as:

$$\begin{aligned} f & = (q - q_{{\mathrm{c}}} )^{2} + 2q_{{\mathrm{c}}} (q - q_{{\mathrm{c}}} ) \\ & \quad + M_{{\mathrm{c}}}^{2} (p^{\prime} - p^{\prime}_{{\mathrm{c}}} )^{2} + 2M_{{\mathrm{c}}}^{2} p^{\prime}_{{\mathrm{c}}} (p^{\prime} - p^{\prime}_{{\mathrm{c}}} ) \\ & \quad - M_{{\mathrm{c}}}^{2} p^{\prime}_{0} (p^{\prime} - p^{\prime}_{{\mathrm{c}}} ) + q_{{\mathrm{c}}}^{2} + M_{{\mathrm{c}}}^{2} p_{{\mathrm{c}}}^{\prime2} - M_{{\mathrm{c}}}^{2} p^{\prime}_{0} p^{\prime}_{{\mathrm{c}}} \\ \end{aligned}$$

(27)

and

$$\begin{aligned} f & = (q_{{\mathrm{c}}} - q)^{2} - 2q_{{\mathrm{c}}} (q_{{\mathrm{c}}} - q) \\ {\kern 1pt} & \quad + M_{{\mathrm{c}}}^{2} (p^{\prime}_{{\mathrm{c}}} - p^{\prime})^{2} - 2M_{{\mathrm{c}}}^{2} p^{\prime}_{{\mathrm{c}}} (p^{\prime}_{{\mathrm{c}}} - p^{\prime}) \\ & \quad + M_{{\mathrm{c}}}^{2} p^{\prime}_{0} (p^{\prime}_{{\mathrm{c}}} - p^{\prime}) + q_{{\mathrm{c}}}^{2} + M_{{\mathrm{c}}}^{2} p_{{\mathrm{c}}}^{\prime2} - M_{{\mathrm{c}}}^{2} p^{\prime}_{0} p^{\prime}_{{\mathrm{c}}} \\ \end{aligned}$$

(28)

Then, substituting Eqs. (27) and (28) into Eqs. (25) and (26), respectively, one has:

$$d_{{\mathrm{g}}} = - \frac{{{}_{{p^{\prime}}}D_{{p^{\prime}_{{\mathrm{c}}} }}^{\alpha } f(p^{\prime})}}{{{}_{{q_{{\mathrm{c}}} }}D_{q}^{\alpha } f(q)}} = M_{{\mathrm{c}}}^{2} \frac{{(p^{\prime} - p^{\prime}_{{\mathrm{c}}} ) + (2 - \alpha )(p^{\prime}_{{\mathrm{c}}} - p^{\prime}_{0} /2)}}{{(q - q_{{\mathrm{c}}} ) + (2 - \alpha )q_{{\mathrm{c}}} }}\left[ {\frac{{p^{\prime}_{{\mathrm{c}}} - p^{\prime}}}{{q - q_{{\mathrm{c}}} }}} \right]^{1 - \alpha }$$

(29)

$$d_{{\mathrm{g}}} = - \frac{{{}_{{p^{\prime}_{{\mathrm{c}}} }}D_{{p^{\prime}}}^{\alpha } f(p^{\prime})}}{{{}_{q}D_{{q_{{\mathrm{c}}} }}^{\alpha } f(q)}} = M_{{\mathrm{c}}}^{2} \frac{{(p^{\prime} - p^{\prime}_{{\mathrm{c}}} ) + (2 - \alpha )(p^{\prime}_{{\mathrm{c}}} - p^{\prime}_{0} /2)}}{{(q - q_{{\mathrm{c}}} ) + (2 - \alpha )q_{{\mathrm{c}}} }}\left[ {\frac{{p^{\prime} - p^{\prime}_{{\mathrm{c}}} }}{{q_{{\mathrm{c}}} - q}}} \right]^{1 - \alpha }$$

(30)

Further substituting Eq. (2) into Eqs. (29) and (30), a unique stress–dilatancy equation is obtained:

$$d_{{\mathrm{g}}} = M_{{\mathrm{c}}}^{1 + \alpha } \frac{{(p^{\prime} - p^{\prime}_{{\mathrm{c}}} ) + (2 - \alpha )(p^{\prime}_{{\mathrm{c}}} - p^{\prime}_{0} /2)}}{{(q - q_{{\mathrm{c}}} ) + (2 - \alpha )q_{{\mathrm{c}}} }}.$$

(31)