# Modeling and Characterization of Complex Dynamical Properties of Railway Ballast
## 1 Introduction
It is crucial to describe the complex dynamic properties of this granular material to model the whole track and vehicle-track systems. However, the research on the theoretical modeling of ballast has not comprehensively characterized the nonlinear stiffness properties. In principle, ballast stiffness relies on interactions of particles/granules and frictional interlocking, which renders complex nonlinear properties in ballast under dynamic loading. The simple idealization of ballast stiffness has been used for multi-body simulations and to develop a non-destructive field-testing technique for monitoring railway components. The existing modeling of ballast stiffness is only considered as part of the nonlinear properties of ballasts [22; 24; 25; 26; 29; 43]. Numerical modeling of railway ballast has been investigated to simulate the complex dynamical behavior of ballasts observed in experiments. Most finite element models (FEM) of ballasted tracks in the literature model the ballast layer as a continuum of homogenous material by discretizing the ballast layer into very small elements [6; 51]. Although the continuum assumption works in most cases, the insight visualization of stresses and displacement cannot be correctly evaluated using FEM because the discontinuous ballast layer is modeled as a continuum material. The challenge of FEM is to choose an appropriate material model that simulates the complex discontinuous ballast material behavior. The discrete element method (DEM) is a numerical method used to solve mathematical problems associated with discrete characteristic materials like granular material [52; 53; 54; 55; 56]. DEM has been used to model ballast where discontinuous property is considered. DEM can visualize the microscopic mechanical behavior of ballast under loading. The initial settlement, volumetric change, particle breakage, force chain distribution, particle displacement, particle shape, particle size distribution, and ballast fouling can be investigated using DEM. The research based on DEM correlates well with the ballast rheology mechanisms and the effect factors from the experiments. DEM is a powerful tool that can analyze the mechanical behavior of ballast microscopically and macroscopically. The main limitation of DEM is the massive requirement of computational time. In the literature, DEM is used mainly to validate experimental tests of railway ballast on a small scale. However, previous analytical modeling of railway ballast for multi-body simulations and condition monitoring/evaluation needs to comprehensively cover the complex mechanical behaviors of ballast from experiments and numerical analysis. To fill this gap, this paper proposes a new model to comprehensively quantify the nonlinear, rheological, elastic-viscous-plastic properties of ballast and cover the experimentally observed phenomena of time/space-varying, displacement-dependent, and frequency-dependent stiffness and hysteresis. On the other hand, the conventional ballast models used in multi-body simulations have been mainly idealized based on conventional experiments. The standstill track stiffness measurement can be performed with the traditional jack-loading method, impact hammer method, falling weight deflectometer method, and track-loading vehicle method. The continuous methods to measure track stiffness include the unbalanced loading laser displacement method, deflection basin deformation rate method, eccentricity excitation method, and the GPR method, which has been widely applied recently. The dynamic stiffness of railway ballast is considered a time-varying quantity, considering the effect of contaminations, fouling, degradations, and wet content. The ballast stiffness also has a feature of high spatial variability due to section-by-section ballast deterioration and maintenance along the railway network. Railway ballast experiences high deterioration under loading and environmental conditions. By nature, some of the most notable track deteriorations are located on the ballast, which has high nonlinearities.