A mathematical model describing the frozen deformation of unsaturated clayey soils in a closed system

The deformation pattern of frozen soil is closely related to its saturation, exhibiting shrinkage at low saturation and expansion at high saturation. While there have been experimental reports on this phenomenon, research on theoretical models remains relatively scarce. Based on the analysis of the laws governing soil freezing deformation and its physical mechanisms, a mathematical model has been established to uniformly describe the shrinkage-swelling deformation of unsaturated soil in a closed system. This work includes the following key aspects: (1) The proposal of effective coefficient η and effective saturation Sre to quantify the influence of pore size and saturation on freezing deformation, along with deriving the mathematical relationship between freezing volumetric strain and void ratio as well as degree of saturation. (2) Based on the freeze shrinkage mechanism induced by cryogenic suction and ice phase expansion, the freezing deformation process can be described into three stages: cold shrinkage, frost shrinkage, and ice expansion. Additionally, the critical frost shrinkage temperature Ti is defined to estimate the deformation generated by gas phase during soil freezing, thereby achieving a unified explanation for the swelling-shrinkage deformation under freezing. This model is characterized by its simplicity, effectiveness, and minimal parameters; it can effectively reflect patterns in freezing deformation and accurately predict its critical degree of saturations. Application of this model to predict results from some reported freezing tests conducted on fine-grained clay has validated its rationality and effectiveness.