# A Variable-Order Dynamic Constitutive Model for Clay Based on the Fractional Calculus
## 1 Introduction  
Clay, such as soft clay or silt. is an elastic-viscoplastic material with high water content and compressibility [1; 2; 3]. Due to its low bearing capacity, a large secondary settlement in infrastructure constructions (expressways, metro tunnels, airstrips, etc.) built on a clay foundation occurs readily under the action of cyclic loading, which directly affects operational usage and can generate considerable reconstruction expense [4; 5]. Consequently, it is essential to conduct research about accumulative strain of clay subjected to cyclic loading. Accumulative strain of clay subjected to cyclic loading will increase along with loading duration, and various constitutive models designed to predict the accumulative strain upon soil have been proposed by many scholars. To date, these models can be roughly separated into theoretical models and empirical models, according to their method of data acquisition; the former are generally more complex compared to the latter, and a great diversity of models have been obtained due to the various theories adopted by scholars. Based on elastoplastic theory, many dynamic constitutive models capable of predicting accumulative strain of soil have been proposed [6; 7; 8; 9]. With the introduction of numerical simulation approaches, several authors have developed constitutive models to capture the effects of cyclic loading on traffic settlement [10; 11; 12]. Furthermore, based on general thermodynamic principles and [PERSON]'s laws of motion, [PERSON] and other authors [13] presented a constitutive model to explore the dynamic response of soft clay under dynamic loads. Although these theoretical models have good ability in predicting the accumulative deformation of soil, the complexity of their equations has inhibited their popularity inengineering contexts, Empirical models are obtained by fitting mass data to curves of accumulative plastic strain-times of curve, and an empirical model was first exhibited by [PERSON] [14]. Subsequently, taking into consideration the differences in the mechanical properties of various soils, scholars have proposed different dynamic constitutive models for fine-grained soils [15], saturated soft clay [16], peaty soil [17], silt sand [18], frozen clay [19], compacted loess [20] etc. In addition, scholars have carried out extensive experiments to explore the effects of intermittent loading, loading characteristics, variable confining pressures, principal stress rotation and other factors affecting the mechanical properties and deformation characteristics of soil [21; 22; 23; 24]. Because of their simplicity and desirable ability to predict material behavior, empirical model formulas are now widely applied in practical engineering. Nevertheless, the parameters in empirical models have the weakness of being susceptible to experimental circumstances which greatly limits the ability of these models to predict accumulative deformation of soil. Element models, as one of the traditional rheological constitutive models, are utilized by engineers and researchers for describing the deformation characteristics of geotechnical materials because of their simple concepts and explicit physical meanings. For instance, [PERSON] et al. [25] derived a fractional-order derivative viscoelastic-viscoplastic constitutive model capable of estimating delayed deformations characterized by squeezing. Based on the Nishihara model, [PERSON] et al. [26] constructed a fatigue deformation model by introducing a nonlinear function to reflect the entire process of the irreversible accumulation deformation of limestone under osmotic pressure and cyclic disturbance coupling. [PERSON] et al. [27] proposed a new nonlinear creep constitutive model by introducing a damage variable into the viscoplastic element to accurately describe the whole creep process for frozen sand with different dry densities and grain size distributions under different shear stress levels and temperatures.