# Modeling and Characterization of Complex Dynamical Properties of Railway Ballast
## 1 Introduction
They characterized the effect of the ballast stiffness at the rail and found it impacts the rail response at low frequencies, whereas it does not affect the structure at high-frequency ranges. [PERSON] et al. [41] characterized space-varying stiffness and distributions on the dynamic behavior of railway transition zones. [PERSON] [42] reviewed recent research and future directions of railway track substructures. He indicated that the key performance indicators for railway tracks are the track support stiffness, the vertical alignment or level, and the rate at which the level deteriorates due to plastic settlement associated with plastic strain. In addition, [PERSON] et al. [43] estimated complex ballast stiffness based on measurements during the dynamic track stabilization. [PERSON] et al. [11] comprehensively reviewed vertical stiffness measurement methods and values as well as effective ballasted track parameters and their contribution to railway track condition monitoring, indicating that vertical stiffness is one of the most influential factors for recognizing track vertical behavior. [PERSON] et al. [44] studied the effects of longitudinal track unevenness and space-varying track stiffness on railway-induced vibration. [PERSON] et al. [45] studied a benchmark of calibrated track models for the simulation of differential settlement in a transition zone using field measurement data. Despite research covering most aspects of the complex dynamical properties of ballast, the complex nonlinearity of ballast has not been comprehensively modeled. _Ballast_ is a homogeneously graded material that comes from hard rock. Describing the dynamic stiffness and damping of the time-varying nonlinear granular material is crucial to modeling the whole vehicle-track system. Analytical modeling of ballasted tracks has been developed for static and dynamic analysis. In static analysis, the ballast layer is mainly modeled as a simple linear elastic spring with a constant stiffness. In dynamic analysis, other structural elements, such as viscous dampers, are used to model the ballast layer analytically. Analytical models of ballasted tracks have been widely used to study ballasted track dynamic behavior due to their low requirement of computational time. The previous analytical modeling approaches have several drawbacks in quantifying the actual mechanical behavior of the ballast layer, such as the amplitude and frequency dependence of stiffness and hysteresis. It is noted that ballast exhibits complex nonlinear, time-dependent load-deformation behavior under cyclic loading as elastoplastic material with a rheology mechanism. The stiffness and damping properties of the ballast layer exhibit space-varying and time-varying features due to ballast densification and deterioration under complex operation conditions, such as cyclic loading, temperature change, wetness change, ballast particle size/shape changes, and fouling due to long term service/operations. These have been widely reported in previous experimental research, such as [21; 22; 23; 24; 25; 26; 27]. From a material perspective, the conventional analytical approach of modeling the ballast layer does not represent the real complex behavior of railroad ballast under cyclic loading, in which ballast behaves as nonlinear elastic-plastic-viscous material with nonlinear load-deformation behavior. However, in previous modeling, the track system, particularly the nonlinearity of ballast, has not been comprehensively included. Multi-body simulations of vehicle-track systems have been routinely carried out using commercial software such as Vampire 5.0, NUCARS 2024.2.0, and VI-Rail 16.0 [46; 47; 48; 49; 50]. These programs simulate the dynamic response of a vehicle on a track by simply assuming a simple uniform ballast and track support stiffness. As part of the track structure, ballast is made of a homogeneously graded material from hard rock, with a diameter ranging from 3 to 6 cm.