# Study on the Mechanical Criterion of Ice Lens Formation Based on Pore Size Distribution
## 1 Introduction  
With the development of national defense and economic construction, an increasing number of foundation engineering projects have been built in cold regions in China, such as the Qinghai-Tibet Highway, Qinghai-Tibet Railway, Qinghai-Tibet Transmission and Harbin-Dalian high speed railway [1; 2; 3]. These projects have promoted the research on frost heave. With the gradual improvement of engineering grades and the strict requirement of deformation, a large number of important foundation projects have put forward for the purpose of preventing and controlling frost heave. According to the frost heave ratio, different measures are used to solve the problem of frost heave. Therefore, to obtain an accurate evaluation of frost heave is an important issue for engineers. It was found that frost heave was caused by water migration into the freezing zone and forming ice lenses [4; 5; 6]. The early frost heave model can predict frost heave, but it cannot reflect the formation of ice lens. To simulate the formation of ice lens, many frost heave models including the criterion of ice lens formation have been proposed [7; 8; 9]. These criterions can be divided into two types: the thermal method and the mechanical method. Generally, these criterions of the mechanical method can be divided into four groups. (1) Segregation pressureIn 1972, [PERSON] [10] proposed the frozen fringe theory (second model of frost heave). The frozen fringe theory assumed that water migration was blocked by the frozen fringe. Consequently, the rate of frost heave was lower than that in the capillary theory. Based on the frozen fringe theory, some scholars pointed out that when the ice pressure is greater than the separation pressure, the soil matrix will break and a new ice lens will be formed [11; 12]. The segregation pressure \(P_{\text{sep}}\) can be expressed as:  
\[P_{\text{sep}}=P_{\text{ob}}+\frac{2\sigma_{\text{iw}}}{R}f(P_{\text{R}}) \tag{1}\]  
where \(P_{\text{ob}}\) is the overburden pressure, \(\sigma_{\text{iw}}\) is the solid-water interfacial free energy, and \(R\) is the soil particle radius. (2) Pore ice pressure  
Different from [PERSON]'s result, [PERSON] believed that when the ice pressure bore the entire external load, the ice lens was formed. The ice pressure can be calculated by using the generalized Clapeyron equation [13]:  
\[P_{\text{i}}(T)=\rho_{\text{i}}\left(\frac{P_{\text{w}}(T)}{\rho_{\text{w}}}- L\frac{T}{T_{0}}\right) \tag{2}\]  
where \(\rho_{\text{i}}\) is the density of ice (\(\rho_{\text{i}}\) = 917 kg/m\({}^{3}\)); \(\rho_{\text{w}}\) is the density of water; \(T_{0}\) is the freezing point of water; \(T\) is the temperature and \(L\) is the latent heat of solidification per unit mass (\(L\) = 334.56 kJ/kg). \(P_{\text{i}}(T)\) and \(P_{\text{w}}(T)\) are the distribution functions of ice pressure and water pressure. (3) Neutral stress (pore pressure)  
[PERSON] pointed out that when the effective pressure in the soil drops to zero, the ice lens will form [10]. Due to the existence of unfrozen water, there is pore water pressure. By introducing [PERSON]'s formula, the neutral stress can be expressed as:  
\[P_{\text{n}}=\chi P_{\text{w}}+(1-\chi)P_{\text{i}} \tag{3}\]  
\[\chi=n^{-1}W_{\text{n}} \tag{4}\]  
where \(P_{\text{n}}\) is the neutral stress (or pore pressure); \(\chi\) is the correlation coefficient; \(n\) is the pore volume and \(W_{\text{n}}\) is the pore water content. This neutral stress has been adopted and further investigated by other researchers [5; 14]. Based on the principles of equilibrium, continuity and energy in the multi-phase porous medium, [PERSON] et al. [15] thought effective stress, water pressure and ice pressure combined to form the load:  
\[\sigma=\sigma_{\text{s}}+P_{\text{i}}\frac{A_{\text{i}}}{A}+P_{\text{w}}(1- \frac{A_{\text{i}}}{A}) \tag{5}\]  
where \(\sigma\) is the total stress; \(\sigma_{\text{s}}\) is the effective stress; \(A\) is the pore area and \(A_{\text{i}}\) is the contact area of ice particles.