A Coupled Discrete-Finite Element Method for Shear Strength Analysis of Geogrid-Reinforced Railway Ballast

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Introduction
In order to ensure the stability of geotechnical materials, appropriate reinforcement measures are very necessary [1]. As an economic and effective technique, geogrid reinforcement of railway ballast is an alternative way to minimize ballast deformation [2]. Geogrids can provide lateral and vertical confinement to ballast, thereby reducing its settlement and minimizing particle breakage [3]. Available literature has provided an insight into the performance of reinforced ballast [4,5]. ese studies show that the rate of permanent deformation associated with lateral ballast spreading can be reduced effectively by the use of geogrid reinforcement. e reinforcement effect is generally attributed to the tensile strain generated by the interlocking action between geogrid and surrounding aggregate.
Laboratory test and numerical simulation are two important ways to study the mechanical behavior of geomaterials [6,7]. In recent years, numerical methods have developed rapidly with the development of computer. As an effective numerical approach, the discrete element method has been used to analyze the geogrid-reinforced ballast behaviors [8,9]. Miao et al. [10] presented the pull-out behavior of a triangular geogrid embedded in ballast under special consideration of the particle size using the discrete element method. e results unveiled the optimal reinforcement scheme. Chen et al. [11] simulated the behaviors of geogrid-reinforced ballast under confined and unconfined conditions under cyclic loading. e responses of geogridreinforced ballast with different geogrid positions, geogrid apertures, and geogrid numbers were investigated. e finite element method (FEM) has been widely used in various fields through decades of development [12,13]. You et al. [14] evaluated the load-carrying capacity of a prestressed concrete sleeper using LS-DYNA. e results of this study lead to better insight into the influences of rail seat abrasion more clearly and improve track maintenance and inspection criteria. e coupling method can give full play to the advantages of different numerical methods. In recent years, the coupling method has attracted more and more attention of researchers [15,16]. Nishiura et al. [7] developed a quadruple discrete element method for viscoelastic multibody dynamics. e sleeper motion modeled using the quadruple discrete element method was coupled with the rail motion modeled using a finite element method. is study suggests that the proposed QDEM-FEM can provide greater insight into the impact response of ballasted railway tracks. Bai et al. [17] established a coupled thermo-hydro-mechanical mechanism in view of the soil particle rearrangement for saturated/unsaturated soils under the framework of granular thermodynamics. e established model spans the complete process from unsaturated to saturated soils and is verified by the typical test results. e interaction between geogrid and ballast is an interaction of structure with granular media. e coupling of the finite element method (FEM) with the DEM is an effective approach for such problems [18][19][20]. e finite element method is used to model the continuum structure, and the discrete element method is used to model discrete particles. Researchers have conducted some useful studies on the coupled discrete-finite element method. Tran et al. [21] developed a coupled finite-discrete framework to investigate the behavior of a biaxial geogrid sheet embedded in granular soil and subjected to pull-out loading. e detailed responses of the geogrid and surrounding soil were investigated. e contact algorithm at the interface of structure and granular media is the key of the coupled discrete-finite method.
In this paper, a coupled discrete-finite element method is developed to analyze the shear strength of geogrid-reinforced ballast by direct shear tests and pull-out tests. e DEM and FEM are employed to simulate ballast and geogrid, respectively. A contact algorithm is proposed at the geogridballast contact surface. e reinforcement effect of geogrid is investigated by contrasting the results of direct shear tests and pull-out tests.

Generation of Ballast Particle Shapes.
e irregular particle shape has a significant impact on the macromechanical behavior of geotechnical materials [22][23][24]. In order to generate the irregular shape of ballast stone, an arbitrary convex polyhedron is firstly generated and its surfaces are extracted as the ballast particle surfaces. e size and shape of ballast are both considered during the polyhedron generation. e polyhedron is then filled with spherical particles with uniform particle size. In order to obtain a more accurate ballast particle shape, each particle will expand continuously. e system is finally stabilized by a cycling process until the sphere radius reaches its maximum and the location of each sphere does not change distinctly.
In consideration of the computational efficiency, one ballast stone contains several spheres. e real ballast stones and the generated clumps are shown in Figure 1. Each ballast particle consists of a number of overlapped spheres and behaves as a rigid body. e mass, center of mass, and moment of inertia of a clump in local coordinates are determined using the finite segment method [25]. e diameters of generated ballast particles range from 21 mm to 63 mm with the mean size of approximately 40 mm. e ballast sizes and gradation used in the numerical simulations are shown in Figure 2.

Contact Force Model.
e contacts between clumps are contacts of two regular spheres [26]. Based on Hertz's theory of the particle-particle contact of two elastic spheres, the normal contact force consists of elastic and viscous forces and can be written as [27] Without considering the viscous force and considering the Mohr-Coulomb friction law, the tangential contact force can be determined as [28] where x n and _ x n are the normal deformation and deformation rate, x s is the shear deformation, K n and K s are the normal contact stiffness and tangential contact stiffness, μ is the friction coefficient, and A is a material constant depending on Young's modulus, viscous coefficients, and Poisson's ratio of the material and can be determined by the restitution coefficient of particle collisions at a certain speed. K n and K s can be calculated as and G are Young's modulus, Poisson's ratio, and shear modulus of the material, respectively. R A and R B are the radii of the two contiguous particles. e maximum time step in the nonlinear DEM can be determined by e real time step in the calculation is less than the maximum and is determined by where α is an empirical coefficient. Normally, with higher coordination number (>4), α � 0.2, and with lower coordination number (<4), α � 0.4. In this study, we set α � 0.2.

Finite Element Model of Geogrid.
Geogrid is a kind of typical geosynthetics with unique structural properties and is widely used in the reinforced soil [29,30]. In order to study the behavior of geogrid in ballast, the finite element method was used to simulate the dynamic character of geogrid. In this study, a geogrid with an aperture of 40 mm × 40 mm is modeled by an elastic two-node beam element with circular section as shown in Figure 3. e radius of each element is 3 mm. e finite element model of geogrid contains totally 54 elements and 45 nodes. e central difference method is employed to obtain the numerical solutions of the dynamic responses of the geogrid. e computational parameters are shown in Table 1.

Coupling of Discrete and Finite Element Methods
e transmissions of the mechanical variables at the interface between ballast and geogrid are important for the coupled DEM-FEM analysis of shear strength of geogrid-reinforced ballast. e contact forces obtained from the discrete element method are treated as the force boundary condition in the finite element method. e deformations of the geogrid are treated as the displacement boundary condition in the discrete element method. However, the contact forces calculated from the DEM do not always act on the nodes of the finite element. To solve the problem, a contact algorithm based on the static equilibrium is proposed to obtain the equivalent nodal loads of the finite element.
A coordinate system x 1 x 2 x 3 is established as shown in Figure 4, and the equivalent nodal forces components at both ends can be calculated as where R A and R B are the equivalent nodal forces on the beam element, P is the external load obtained from the DEM, L is the length of beam element, a is the distance between contact point and end point A, and b is the distance between contact point and end point B. e time step of the finite element is based on the natural period T of the geogrid vibration and can be calculated as

Advances in Materials Science and Engineering
Δt ≤

Numerical Models of Direct Shear Test and Pull-Out
Test.  Table 2.

Shear Stress-Displacement
Analysis. e shear stresses against shear displacements of direct shear test without and with geogrid and pull-out test are shown in Figure 6. For all specimens, the strain-softening behaviors are observed, and the peak stress increases with the increase of normal stress. e reinforcement effect of geogrid is illustrated through the comparisons of simulation results of the three tests. e    Due to the large particle size, the fluctuation of shear stress could be observed in Figure 6. e particles on the shear surface skip over the particles beneath during the shear process. e bearing skeleton changes with the dislocation between particles. Once the bearing skeleton changes, the shear stress decreases rapidly.
Researchers have conducted some useful studies on geogrid-reinforced ballast. Qian et al. [31] carried out the triaxial test of ballast material reinforced by triangular and square aperture geogrids. Miao et al. [10] carried out the pull-out test of geogrid-stabilized ballast based on the discrete element method. Chen et al. [11] proposed a discrete element model for cyclic loads of geogrid-reinforced ballast under confined and unconfined conditions. ese studies studied the reinforcement effect of geogrid on ballast materials. In this paper, the reinforcement effect of geogrid is further studied by comparing the direct shear test and pullout test results of ballast materials.

Ballast Particles' Displacements.
In order to gain insight into the interlock effect between ballast particles and geogrid, ballast particles' displacement vectors are investigated as shown in Figure 7 for all the three tests under the normal stress of 500 kPa. In the direct shear test without geogrid, upper ballast particles move rightwards along with the upper shear box. Ballast particles along the shearing surface move rightwards obviously due to the friction between ballast particles. Rotation movements are presented during the shear process, especially for those particles below the shearing surface.
In direct shear test with geogrid, ballast particles in the upper shear box have a similar behavior to those in direct shear test without geogrid. However, ballast particles along the shearing surface no longer move along with the upper box due to the constraint of geogrid. e movements of ballast particles in the lower box show more stability under the protection of geogrid. e stability of ballast particles is enhanced obviously with the installation of geogrid.
In the pull-out test, ballast particles on the shearing surface produce large displacements following the movements of geogrid. e ballast particles above and beneath the shear surface move up and down due to the drive of particles on the shear surface. e relative motion between particles is not obvious.

Nominal Volumetric Strain.
In order to reveal the mechanism of ballast-geogrid interlock, the nominal volumetric strain is employed to investigate the relative position changes of the clumps. Figure 8 shows the relative position changes of two adjacent clumps. X i and x i are global and local coordinate system, respectively. X n A and X n B and X n+1 A and X n+1 B are the centroid coordinates of clumps A and B at time steps t n and t n+1 in the global coordinate system. x n A and x n B and x n+1 A and x n+1 B are the centroid coordinates of clumps A and B at time steps t n and t n+1 in the local system. Accordingly, the distance between clumps A and B in global and local systems can be determined as e relationship between the global and local coordinates is written as where where α 1 and α 2 are the angles of X i and x i at time steps t n and t n+1 . e deformation gradient of the relative position of clumps A and B is defined in local system as where where l AB and l A′B′ are the modules of Δx n BA and Δx n+1 BA .
Advances in Materials Science and Engineering 5 Substitute (9) into (11): e matrix of displacement derivatives can be written as where I is the unit matrix. We define the strain of clump A and clump B as Suppose clump A had N neighbors, and the strain of clump A is defined as e nominal volumetric strain is thus calculated as e nominal volumetric strain of ballast particles is shown in Figure 9. For direct shear tests, simulation results are quite different between reinforced and unreinforced cases. In unreinforced specimen, large strains concentrate on both ends of the shearing surface, and only several dislocations occur in the inner part along the shearing surface. In contrast, in the reinforced specimen, the strain distribution is mainly located above the geogrid, and an obvious shear band can be observed. e dislocation of ballast particles is not obvious during the shearing process under the unreinforced case, and the nominal volumetric strain of ballast particles on the shearing surface is not obvious. e installation of geogrid provides horizontal and lateral constraints to ballast particles along shearing surface, which results in an obvious slide of particles and shear band in the upper box. e geogrid is helpful for the stability of particles in the lower box. In the pull-out test specimen, large strain distributes along both sides of the shearing surface due to the movement of geogrid.
e nominal volumetric strain of particles given in this part is only used to show the deformation of granular materials. In the discrete particle model, the description of particle properties is reflected by the force-displacement model at the contact point. e physical properties of a single particle are meaningful only at the mesolevel. e physical properties of granular materials composed of a large number of particles need to be described by constitutive relationship, which is related to the physical properties and arrangement of single particles. However, this paper focuses on using the discrete particle model to directly simulate the deformation of granular materials and does not discuss how to establish the nominal constitutive relationship of granular materials in the sense of continuum model.

Conclusions
Shear strength of geogrid-reinforced ballast is modeled by a coupled discrete-finite element method through direct shear tests and pull-out tests. e discrete element and finite element are employed to model ballast and geogrid, respectively. A contact algorithm based on the static equilibrium is proposed at the geogrid-ballast contact surface. e behaviors of the geogrid-reinforced ballast are simulated and compared with those without geogrid. e responses of the reinforced ballast are investigated by ballast particle displacements and nominal volumetric strains under microscale. e installation of geogrid provides horizontal and lateral constraints to ballast particles along shearing surface, and ballast particles along the shearing surface no longer move along with the upper box in the direct shear test. e strain distribution is mainly located above the geogrid, and obvious shear is formed. Geogrid can improve the stability of ballast materials. e coupled discrete-finite element model proposed in this paper could reveal the mechanism of interaction between geogrid and ballast under microscale. It provides a new numerical method for dealing with similar problems. e mechanical behavior of geogrid is simulated by the elastic constitutive model, which is not very precise. e modeled shear test and pull-out test of reinforced ballast are still far away from modeling the practical functions of geogrid-reinforced ballast in engineering practice. In further work, the nonlinear constitutive model will be considered to analyze the behavior of geogrid.
Data Availability e underlying data supporting the results of this study are uploaded as supplemental files.

Conflicts of Interest
e authors declare that they have no conflicts of interest.