# Research on the Influence of Loading Frequency on the Dynamic Response of Concrete Sleopers
## 1 Introduction  
During the operation of a train, the force between the wheels and rails is transmitted downward along the track structure, and from the steel rail to the sleeper. When the dynamic response between the wheel and rail increases or when the rail is damaged, the load transferred to the sleeper increases. Heavy-duty railways are mainly ballasted track structures. Under the action of periodic train loads, the ballasted tracked will settle. Since the ballast bed belongs to the granular body, the ballast bed has uneven settlement, which will cause the sleeper to exhibit an empty hanging phenomenon, thereby increasing the interaction between the wheel and rail, and increasing the force applied on the sleeper [1; 2; 3]. When the loading force load exceeds its bearing capacity, high stress appears inside the sleeper, and then cracks appear. Under long-term dynamic action, the cracks gradually expand until they penetrates the entire sleeper, eventually leading to loss of its bearing capacity, as shown in Figure 1.  
As a component of the track structure, sleepers have been studied extensively. [PERSON] [4] conducted on-site tests and simulations of the slab tracked of a heavy-duty railway wide sleeper and analyzed the influence of the tracked slab on the mechanical properties of the wide sleeper at the macro and micro levels, [PERSON] [5] analyzed the stress and crack propagation of prestressed concrete sleepers and composite sleepers by using digital image technology, [PERSON] et al. [6] established a sleeper model with the finite difference method and compared and discussed the mechanical properties of elastic sleepers and ordinary sleepers, [PERSON] et al. [7] took the Sichuan-Tibet Railway as the research object to explore the influence of friction sleepers on the lateral resistance of the tracked, [PERSON] et al. [8] used the finite element method to simulate concrete sleepers to explore the crack propagation of sleepers under three-point bending loads, [PERSON] et al. [9] studied the strain state and damage of prestressed concrete sleepers under dynamic and static loads, [PERSON] et al. [10] used the finite element method to simulate the sleeper and study its response on a heterogeneous basis, and [PERSON] [11] used the finite element method to establish the sleeper model and analyzed the influence of the bending and shear stiffness of the sleeper on the track stiffness. To study the influence of different wear depths on prestressed concrete sleepers, [PERSON] et al. [12] established a finite element model of the sleeper to facilitate subsequent research and analysis. [PERSON] et al. [13] used ABAQUS finite element software to carry out numerical simulation and analysis on the bearing capacity of prestressed concrete sleepers. [PERSON] et al. [14] simulated the ballast bed with the finite element method and explored the effects of traditional concrete sleepers and new X-shaped sleepers on the settlement, lateral displacement, and dynamic characteristics of the ballast bed. [PERSON] et al. [15] used a finite element to simulate the rail, under-rail cushion, sleeper, and ballast bed, established a three-dimensional solid finite element model, and studied the influence of rail structure parameters such as sleeper mass and loss factor, ballast bed stiffness, and damping on the vibration performance of the sleeper. In the literature, the simulation of concrete sleepers commonly relies on the finite element method or the finite difference method, both of which treat the sleeper as a continuum; however, concrete sleepers are made of a mixture of coarse and fine aggregates, cement, minerals, etc. From a microscopic point of view, concrete is composed of fine particles. In essence, the load transfer and crack damage in concrete depend on the interaction between the internal particles. Therefore, the discrete element method can analyze the micromechanical properties of sleepers. It has reference value to fundamentally solve the practical problems of concrete.