Modelling undrained behaviour of sand with fines and fabric anisotropy

Abstract

Fabric anisotropy and fines content (f_c) in sands modify significantly their mechanical behaviour, particularly as related to static liquefaction under undrained conditions. The fabric anisotropy aspect, expressed by means of an evolving fabric tensor F, has been addressed in the recently developed Anisotropic Critical State Theory (ACST) that enhances the two critical state conditions on stress ratio (η) and void ratio (e) of the classical Critical State Theory by an additional condition on the critical state value of F in relation to loading direction; based on this concept it introduces the dependence of dilatancy on fabric anisotropy. Various models have been successfully developed within this framework for clean sands. The f_c aspect has been addressed within the Equivalent Granular State Theory (EGST) that substitutes a properly defined equivalent granular void ratio (e*) for e in any model for clean sand in order to obtain the response of sand with fines without any other change of the model structure and constants. Along these lines, a constitutive model is constructed in this work in order to address the effect of both F and f_c simultaneously, by a combination of these two powerful propositions. The idea is very simple: one takes a constitutive model developed within ACST for clean sands, hence it accounts for fabric anisotropy, and substitutes the e* for e, as well as the derivative quantities of such substitution, hence it accounts for f_c. The result yields a model that can simulate data on the undrained response for a range of f_c, with emphasis on static liquefaction. It is shown that the inclusion of fabric anisotropy improves previous similar simulations made within the EGST but without the framework of ACST.

Appendices

Appendix

Equivalent granular void ratio, e*

Rahman, Lo [50], by re-analysing the experimental data of McGeary [40] on binary packing studies and nine different sand-fines mixture from around the world, concluded that $b$ entering Eq. (3) is a function of both f_c and where D_s is the size of sand and d_f is the size of fines. Furthermore, the functional relationship, $b = F\left( {f_{c} ,\chi } \right)$, has to possess a number of mathematical attributes [51]. To simulate the required attributes, Rahman and Lo [50] proposed a semi-empirical equation expressed as below.

$$b = \left[ {1 - \exp \left( { - 0.3\frac{{{f_{c} \mathord{\left/ {\vphantom {f {f_{{{\text{thre}}}} }}} \right. \kern-\nulldelimiterspace} {f_{{{\text{thre}}}} }}}}{k}} \right)} \right] \times \left( {r\frac{{f_{c} }}{{f_{{{\text{thre}}}} }}} \right)^{r}$$

(28)

where r = χ⁻¹ = particle size ratio, d_f/D_S and k = 1 − r^0.25. Since sand and fines are generally not single-size materials, D_S/d_f was generalized to D_S,10/d_f,50 based on the argument in Ni et al. [44], where the subscripts denote fractile passing. f_thre can be obtained from the experimental data, where available, as outlined in Rahman et al. [52]. However, as an initial approximation, f_thre can be taken as 0.30, but it may be determined more reliably using the following equation developed by Rahman et al. [52].

$$f_{{{\text{thre}}}} = 0.40\left( {\frac{1}{{1 + \exp \left( {\alpha - \beta \chi } \right)}} + \frac{1}{\chi }} \right)$$

(29)

The parameters α and β are determined by curve fitting to eight databases for χ in the range of 2 to 42, and this gave α = 0.50 and β = 0.13 [53]. The value of α and β in Eq. (29) has been evaluated with new emerging datasets by independent researchers, and general acceptability was observed [31, 42, 11] and used in many other studies [55, 59, 3]. Therefore, α = 0.5 and β = 0.13 are not material-dependent parameters; rather, they are fitting constant for many soils. Note D_S,10 is 0.225 mm for Sydney sand and d_f,50 is 0.006 mm for the fines used in this study.