## Part 1 Introduction  
Unlike metal materials, the shear strength of natural silica sand is controlled by internal frictions. Subjected to a relatively low loading, the destruction of sand specimens is caused by relative sand particle movements, instead of particle crushing. Especially during an earthquake, much of the severe damage of geotechnical structures can be attributed to sand liquefaction, which results in large land deformations due to positive excess pore water pressure. Therefore, a better constitutive modellingof sand on various states should consider both dilatant and liquefaction behaviours. In the triaxial stress-strain space, the dilatancy \(\mathrm{d}\) is commonly described as the incremental ratio of the plastic volumetric strain to that of the deviatoric strain: \(\mathrm{d}=\dot{\epsilon}_{\mathrm{v}}^{\mathrm{P}}/\dot{\epsilon}_{\mathrm{q}}^ {\mathrm{P}}\), where \(\epsilon_{\mathrm{v}}=\epsilon_{1}+2\epsilon_{3}\), \(\epsilon_{\mathrm{q}}=\frac{2}{3}(\epsilon_{1}-\epsilon_{3})\), superscript 'p' stands for plastic and overdot stands for increment. On the basis of the experimental results of clayey soils, with the assumption of incremental plastic energy in the Cam-Clay model, [PERSON] and [PERSON] [1] proposed \(\mathrm{d}=\mathrm{M}-\mathrm{q}/\mathrm{p}^{\prime}=\mathrm{M}-\eta\mathrm{d }=\mathrm{M}-\eta\), where \(\eta\) represents stress ratio. Later, [PERSON] and [PERSON] [2] modified the dilatancy law in order to ensure that the direction of strain flux is horizontal while \(\eta=0\). However, [PERSON] and [PERSON] [3], and [PERSON] and [PERSON] [4] pointed out that the above equations for cohesive soils does not adapt to cohesionless soils because \(\mathrm{d}\) is expressed as a unique function of stress ratio with a constant material parameter \(\mathrm{d}=\mathrm{d}(\eta,\mathrm{M})\) in both the above versions. A state parameter containing both the effect of void ratio and confining pressure should be carefully taken into account in the dilatancy law \(\mathrm{d}=\mathrm{d}(\eta,\psi,\mathrm{M})\) for sand. The simplification of the former leads to multiple sets of model parameters because the same sands in loose and dense states are treated as totally different materials. Numerous experimental results show that the stress-strain response of sand is largely dependent on the relative density and effective confining pressure. For this purpose, a dimensionless scalar \(\psi=\mathrm{e}_{\mathrm{act}}-\mathrm{e}_{\mathrm{c}}\) was introduced by [PERSON] and [PERSON] [5], which estimates the distance between the current and critical void ratios under the same effective confining pressure, \(\mathrm{p}^{\prime}\). Thus, the sand contracts in loose state while \(\psi>0\), and the sand dilates in dense state while \(\psi<0\). To avoid the counting problem of positive and negative signs, some researchers propose using the ratio of the current and critical void ratios \(\psi=\mathrm{e}_{\mathrm{act}}/\mathrm{e}_{\mathrm{c}}\) instead of the difference measured in \(\mathrm{e}-\log\mathrm{p}^{\prime}\) plan. One of the fundamental issues in sand constitutive modelling is the precise representation of critical state line in terms of the effective confining pressure \(\mathrm{e}_{\mathrm{c}}=\mathrm{e}_{\mathrm{c}}(\mathrm{p}^{\prime})\). In general, plotting these lines of cohesionless sand in \(\mathrm{e}-\log\mathrm{p}^{\prime}\) plan makes the expression algebraically more difficult. Firstly, the real experimental data [6; 7] revealed that the critical state line from very loose to loose states falls within a spindle instead of a simple line. The simplification to a straight line leads to inaccurate calculations of sand instability behaviour. Secondly, the unification of the critical state of the same sand specimen under different confining pressures is still under discussion. Attempts have been made--[PERSON] et al.