## 1 Introduction  
Due to rapid operating trains, high-speed railway raises more rigorous requirements for smoothness and stability of rail track. To ensure the service performance of rail track, it is critical to make effective control of cumulative deformation that occurs at substructures (subgrade bed and subsoil) and ground. The subgrade bed, which refers to the uppermost structure that directly bears the cyclic load from train and rail track, generally have a thickness ranging from 2.0 m to 3.0 m [1; 2; 3]. Since subsoil and ground are mainly composed of fine-grained soils, greater post-construction deformation is likely to occur at these positions compared with subgrade bed. Therefore, the cumulative deformation of ballast layer and subgrade bed should be more strictly controlled to make space for the deformation at other positions. To investigate the deformation property of ballast layer and subgrade bed, various kinds of research means, such as laboratory test, filed instrumentation, discrete element method, and finite difference method, have been adopted by researchers to study the mechanical behavior of coarse-grained fillings. A three-dimensional discrete element model was established by [PERSON] et al. [4] to simulate the realistic ballast particle shapes. With such a model, it can be concluded that load frequency has a significant effect on ballast permanent deformation when it is beyond 15 Hz. Moreover, both laboratory test and numerical analysis are performed by [PERSON] et al. [5] to study the deformation and degradation of ballast when subjected to cyclic loading. A numerical model, which can be adopted to predict the deformation response of rail track, was developed. Since many theoretical models about track settlement and ballast deformation were developed in the previous literature, [PERSON] [6] makes a summary to demonstrate the advantages and disadvantages of the models when they are used to predict the long-term deformation of substructure. Although many tests and models have already been developed to evaluate the deformation produced at rail substructures, most of them mainly focus on the deformation at ballast layer or the whole system. Currently, two kinds of technical measurements, improving both the quality and compaction degree of fillings, are adopted to restrain the cumulative deformation that occurs at subgrade bed. In Japan, surface layer of subgrade bed is constructed layer-by-layer with asphalt concrete and graded gravel (or blast furnace slag) so that it can suffer the dynamic train load [1]. Meanwhile, it is stipulated in Germany that the fillings in surface layer of subgrade bed should be dominated by coarse-grained soils. Moreover, compaction degree is restricted to be not less than 100% for surface layer and 97% for bottom layer [3]. In China, well graded angular gravel is recommended to fill surface layer, while round gravel, pebble, gravelly sand, and soil should be adopted to build the bottom layer of subgrade bed. The thickness of subgrade bed is set to be 3.0 m for the ballasted track and 2.7 m for the unballasted one [2]. Although the design criteria have already been described in detail in the codes of some countries, evolvement characteristic for cumulative deformation under cyclic load is not sufficiently considered in these design codes. In particular, ballast bed, which underlies the rail track in the ballasted railway, has a better adaptive capacity for deformation compared to the concrete slab in the unballasted one. Furthermore, new railway ballast can be replenished to ballast bed to raise track to the original location for the ballasted railway once relatively large plastic deformation has occurred. It is remarkable that the ballasted track is much easier to maintain and preserve than the unballasted one. Thus, the requirements of cumulative deformation should be different for the ballasted and unballasted railway, which is worthy of consideration in the design process.