## 1 Introduction  
Existing theories of rock and soil mechanics majorly focus on the strength and deformation characteristics of soil under high compressive stresses. The descriptions of stresses and deformations during shearing are limited to strain hardening and shrinkage. However, the soil is often subjected to low stress and tensile stress in engineering problems [1; 2; 3], and the stresses and deformations during shearing often exhibit peaks and dilatancy. In such cases, the soil mechanics theories that are based on the effect of a high compressive stress will no longer be applicable. Thus, it is necessary to establish an appropriate stress-strain relationship model to describe the actual working conditions with an improved accuracy. The soils that are involved in geostechnical engineering practice are mostly over-consolidated or compacted unsaturated soils [4]. Presently, the main research ideas on the stress-strain relationship of unsaturated soils--starting from the Barcelona basic model (BBM) proposed by [PERSON] et al. [5]--which are mostly based on the critical state theoretical framework, can be unified as the Cambridge-type model. It is generally challenging to accurately describe the high peak stress ratio and dilatancy behaviour of the soil using this type of model, and the research on an appropriate stress-strain model is still lacking. An appropriate dilatancy prediction equation (stress-dilatancy relationship) is the key to establishing a stress-strain constitutive model of soil [6]. Initially, the researchers expressed the dilatancy rate as various functional relationships of the current effective stress ratio \(\eta^{\prime}\) and the critical state friction constant \(M\). An experimental comparison then revealed that the dilatancy equation should also consider the soil compaction state variablesthat reflect the influences of the void ratios or the confining pressures. In the past two decades, experimental studies on the mechanical properties of the unsaturated soils have shown that the dilatancy is also a function related to the change in the water content under the unsaturated state [7; 8]. However, little attention has been paid to the stress-dilatancy relationship of the unsaturated soil, and the function of the saturated soil is mostly directly adopted when deriving the constitutive model. To better describe the dilatancy characteristics of soil, researchers have introduced multi-mechanism models, such as the boundary surface model [9] and the sub-loading surface model [10], in which the prediction method of the minimum dilatancy rate during the shear deformation process was introduced. Initially, based on the summary of the experimental patterns, [PERSON] [11], as well as [PERSON] and [PERSON] [12], expressed the minimum dilatancy rate \(D_{\text{min}}\) through their newly defined state parameters (stress state and compacted state), \(I_{\text{R}}\) (relative dilatancy index), and \(\psi_{\text{ct}}\) (volume deformation state parameter), respectively, as follows. \[D_{\text{min}}=\alpha\cdot I_{\text{R}} \tag{1}\]  
and  
\[D_{\text{min}}=X\cdot\psi_{\text{ct}} \tag{2}\]  
where \(\alpha\) and \(X\) represent the dilatancy coefficients under the corresponding state parameters (they can also be referred to as the minimum dilatancy coefficient). Moreover, the minimum dilatancy coefficient has been obtained as a function of the soil fabric, as demonstrated in previous studies [11; 12; 13]. The changes in the water content in unsaturated soil will induce changes in the soil fabric, such as the transition from a single-pore distribution to a dual-pore fabric under aggregation [14; 15; 16]. However, there are relatively few studies on the relationship between the minimum dilatancy rate and change in the water content. At the same time, the multi-mechanism models of unsaturated soils that have been attempted to be established in recent years have mostly adopted either suction or degree of saturation as the state variable for the water content [14; 17].