# Research on the Accumulated Plastic Strain of Expansive Soil under Subway Loading
## 1 Introduction  
Expansive soil is widely distributed throughout the world and is known to exist in more than forty countries on six continents, of which China is one of the countries with the widest distribution and largest area of expansive soil [1, 2]. Although these countries are geographically different and have different climatic environments, they all face the same problem of expansive soil hazards. Expansive soil poses formidable challenges in engineering construction and operational safety. Highways and railways constructed in expansive soil regions become \"roadblocks\" and \"hard nuts\" to crack. They epitomize the saying \"Where there is a ditch, there is a slip; where there is a dike, there is a collapse\". Similarly, working with river channels in these areas is akin to \"Digging a foot during the day, only to find it stretches eight feet longer at night\". Expansive soil not only causes great obstruction to above-ground projects such as buildings, highways, railroads, and canal slopes, but also poses a serious threat to underground engineering projects such as underground railways, pipelines, tunnels, and foundation structures [3, 4, 5]. The repetitive nature, complexity, and long-term potential hazards are hallmark features of expansive soil disasters, posing persistent challenges to the engineering community. The magnitude and widespread impact of the damages resulting from the unfavorable properties of expansive soil often necessitate significant human, material, and financial resources to address this unique geological hazard. According to statistics, the United States allocates billions of dollars each year towards addressing the issue of expansive soil hazards, surpassing the cumulative losses from disasters such as floods, hurricanes, earthquakes, and tornadoes. Likewise, in China, the economic losses caused by expansive soil amount to over billions of dollars annually [6]. Hence, expansive soil has been recognized as a \"cancer\" that impedes engineering construction globally. So far, researchers have made significant strides in studying the composition of expansive soil minerals, their macro-micro structure, swelling-shrinkage mechanisms, and physical-mechanical properties. However, there remains a relative scarcity of research concerning the mechanical properties of expansive soil under dynamic loading conditions. [PERSON] et al., investigated the attenuation patterns of a dynamic elastic modulus of saturated expansive soil under various influencing factors. Based on this, they established a model for the attenuation of a dynamic elastic modulus related to compaction degree, reinforcement layers, confining pressure, and frequency [7]. [PERSON] et al., examined the dynamic shear modulus and damping ratio of expansive soil under freeze-thaw conditions. The results indicated that freeze-thaw cycles led to an increase in the dynamic elastic modulus of expansive soil, but a decrease in the damping ratio [8]. These researchers explored different experimental conditions to study dynamic modulus indices of expansive soil under dynamic loading. However, it is worth noting that under lower dynamic loads, the soil exhibits elastic deformation with closed hysteresis loops. In this scenario, dynamic shear modulus or dynamic elastic modulus can effectively describe the soil's ability to resist dynamic elastic deformation. Yet, under higher dynamic loads, the soil experiences irreversible strains, and hysteresis loops no longer close. Consequently, dynamic shear modulus or dynamic elastic modulus lose their significance. Considering this, [PERSON] and [PERSON] investigated the changing patterns of accumulated plastic strain under significant dynamic loading on expansive soil, proposing a corresponding strain model to predict strain growth during prolonged cyclic loading [9; 10]. However, their simulation of dynamic loading in indoor experiments lacked meticulous detailing or a comprehensive rationale for employing sine wave simulation of dynamic loading.