# Stress-Strain Model for Freezing Silty Clay under Frost Heave Based on Modified [PERSON]'s Equation
## 1 Introduction  
Frost heave is a primary cause of the failure of critical infrastructure in cold regions, such as underground gas pipelines and roadways, including bridges--especially those with their footings in frost-susceptible soils [1, 2]. Therefore, frost heave research on freezing soils is relevant in today's scenario [3, 4]. Additionally, in cold regions where this is a major problem, engineers are attempting to find practical solutions. The efforts to explain the mechanism of frost heave began in the 1920s [5]. Since then, numerous experimental and theoretical efforts have been undertaken to systematically investigate the frost heave characteristics of a variety of soils, including the mechanism, amount, and pressure [4, 6, 7, 8]. Various methods have been proposed to describe the frost heave process, such as the capillary model [9], rigid ice model [10, 11, 12, 13, 14], segregation potential model [15], thermomechanical models [16], and semi-empirical approaches [17, 18]. In addition, recent research interests in frost heave and frost action in frost-susceptible soils have been renewed in China [19, 20]. Indeed, the frost heaving of frost-susceptible soils is a coupled hydro-thermomechanical (THM) and thermomechanical (TM) interaction process. The development of mathematical modeling and computing technology has promoted the advancement of the TM and THM methods. Recently, based on the discrete ice lens theory, [PERSON] and [PERSON] (1991) [16] established a TM model to explain the water heat transfer and mechanical characteristics of freezing soils. [PERSON] and [PERSON] (1996) [21] used the newly developed two dimensional (2D)-segregation potential (SP) frost action model and provided distributions of ice contents, stresses, and displacements. [PERSON] and [PERSON] (2006) [22] proposed a TM model using a porosity rate function within the theoretical framework of heaving soil depending on the stress state and temperature gradient (\(\
abla T\)). [PERSON] and [PERSON] (2011) [23] adopted the THM theory to simulate an unsaturated porous medium during frost heave. [PERSON] et al. (2012) [24] proposed a practical method for three-dimensional frost heave simulation based on [PERSON]'s equation. [PERSON] et al. (2014) [25] took the displacement, temperature, and porosity of the soil as variables and developed a coupled THM model to characterize the one-dimensional (1D) freezing process of saturated soil. [PERSON] and [PERSON] (2014) [26] developed a two-dimensional (2D) numerical THM model by expanding [PERSON]'s equation from 1D to 2D to evaluate the frost heave process. For simplification, several analysis methods of the TM and THM models of freezing soil under frost heave are based on the assumption that the compressive modulus (\(E_{\mathrm{s}}\)) of the soil is a constant, or an empirical formula is formulated in which it is a function of temperature only [27]. As such, from a macroscopic mechanical perspective, the expansion associated with phase change during freezing causes compressive stresses to develop in the freezing zone of the soil. Accordingly, freezing-induced stress and strain are important features because they cause the upward swelling of soil and damage the underlying infrastructure [4]. Meanwhile, numerous factors, including the temperature gradient, overburden pressure, and cooling temperature, significantly influence the \(E_{\mathrm{s}}\) of freezing soil. In engineering, a more effective analysis method is one that uses an appropriate stress-strain model with an accurate \(E_{\mathrm{s}}\) for freezing soil to better predict the maximum potential deformations and stresses resulting from frost heave at a satisfactory level of confidence. In this regard, the key research questions of a stress-strain analytical model should be investigated and an accurate \(E_{\mathrm{s}}\) value for freezing soil should be determined. This study describes a series of one-dimensional multistage freezing experiments with silty clay columns.