Numerical simulation of rebar corrosion in track bed of shield tunnel under stray current

Track bed is the important support structure at the bottom of shield tunnel, during metro operation, the leakage of stray current will lead to corrosion of rebar in the track bed, which will affect the safe operation of metro. In this paper, the corrosion calculation model of track bed rebar under stray current is established, and the effects of stray current input mode and concrete resistivity on the corrosion rate of track bed rebar are studied. The result shows the corrosion rate of track bed rebar decreases nonlinearly with the increase of concrete resistivity, and the reduction gradient of corrosion rate in bilateral input mode is greater than that in unilateral input mode. With the decrease of concrete resistance, the corrosion area will cover the left half in unilateral input mode and cover both ends of track bed rebar in bilateral input mode. When the input point voltage increases by 40%, the corrosion current density in the middle of track bed will increase by about 61%.


Introduction
Shield tunnel is the main structural form of metro tunnel [1], the track bed is poured at the bottom to support the sleeper.During metro operation, due to the decline of the insulation performance of rail fasteners, stray current will flow from track bed to tunnel structure, result in rebar corrosion inside track bed.As the corrosion product of rebar is 4-8 times that of the original rebar [2], when the corrosioninduced expansion pressure exceed the concrete tensile strength, the cracks will penetrate the concrete of track bed, which will reduce the stiffness of track bed and affect safe operation of the metro tunnel.
Different calculation methods and model tests have been used to study the corrosion mechanism of rebars in underground structure [3][4][5][6][7].Through several groups of model tests, Bertolini et al. [3] studied the corrosion rate of rebar in underground concrete under the action of a direct current (DC) and alternating current (AC).Dolara et al. [4] developed a numerical model for a simplified shield tunnel and determined the corrosion area of rebars exposed to stray current.Lei et al. [5] studied the critical time of rebar corrosion in a shield tunnel segment using a model test and provided a durability evaluation model for shield tunnel segments.Based on real engineering examples and genetic programming, Gao et al. [6] proposed a calculation method for predicting the service life of tunnel structures subject to chloride-induced corrosion.Li et al. [7] established a corrosion calculation model of tunnel segment contained only a single row of main bars, research shows the 2D rust layer is distributed in an eccentric circle along the circumferential direction.
In the aforementioned studies, most research object focus on the main structure of the tunnel.However, relevant research shows that stray current has a great influence on the corrosion of track bed rebar, and the distribution form of rebar in the model test were unlike those of real track bed [8].Some studies [9] have shown that the corrosion characteristics are not only affected by the stray current magnitude, but also by the distribution form of rebar.Therefore, the existing model cannot accurately describe the corrosion characteristics of track bed rebar.
In our research, the details of shield tunnel and track bed are introduced, and the corrosion calculation model of track bed rebar under stray current is established.At last the effects of stray current input mode and concrete resistivity on the corrosion rate of track bed rebar are studied.

Details of shield tunnel and track bed
Each ring of shield tunnel consists of six pieces.It includes one key block (K), two adjacent blocks (B1, B2), and blocks (A1, A2, A3).The key block has a centre angle of 21.5°, the adjacent blocks have centre angles of 67.5°, and the standard blocks three standard have centre angles of 68°, the two adjacent blocks is connected by bending bolts, as shown in

Corrosion calculation model of track bed rebar
According to Ohm's law [10], the current density vector T i and S i in concrete electrolyte can be expressed as: Where the T i and S i are current density vector in track bed concrete and segment concrete respectively, the T  and S  are conductivity of track bed concrete and segment concrete respectively, the T E and S E are electric field intensity vector of track bed concrete and segment concrete respectively, the i is 0. When the DC current flows through the tunnel structure and track bed, the generated potential difference will affect the electric potential e E of the rebar surface.When the outside oxygen is sufficient, stray current will cause electrochemical corrosion to track bed rebar.The corrosion rate is closely related to the corrosion current density corr i , which can be expressed according to the Butler-Volmer equation [11], as shown in Eq (3).
Where corr i is the corrosion current density; , Fe exc i is the anodic exchange current density, e E is the electric potential; , Fe eq E is the anodic equilibrium potentials, Fe  is the anodic Tafel slope.In the process of corrosion reaction, the corrosion rate is determined by the reaction area, as follows [12]: Where AFe is the mole mass of iron,  Fe is the density of iron, the electrochemical calculation parameters as shown in Table 1.The software used for numerical calculation is COMSOL.Table 1.Electrochemical calculation parameters [13].During metro operation, due to the reduction of the rail fastener insulation performance, part of the stray current will flows from the rail to the track bed and segment structure, the current may flow from any point under the rail fastener.In the calculation model, it is assumed that the input location of stray current is the four points parallel on both sides of rail, selected point of X as the analysis object, as shown in Fig 5

Conclusion
In this paper, the effects of stray current input mode and concrete resistivity on the corrosion rate of track bed rebar are studied.The result mainly includes: (1) corrosion rate of track bed rebar decreases nonlinearly with the increase of concrete resistivity, and the reduction gradient of corrosion rate in bilateral input mode is greater than that in unilateral input mode.
(2) Under unilateral input mode, the corrosion area of track bed rebar concentrate near the input point when the concrete resistance is 1000 Ω/m, with the decrease of concrete resistance, the corrosion area gradually cover the left half of track bed rebar.
(3) Under bilateral input mode, the corrosion area gradually expands from top to bottom, and finally cover both ends of track bed rebar, when the input point voltage increases by 40%, the corrosion current density in the middle of track bed will increase by about 61%.

Fig 1 .
The mesh type in the model is tetrahedral mesh element, and each segment contains rebar inside, as shown in Fig 2. The diameter of segment rebar and track bed are 18mm and 16mm respectively, and the diameter of track bed reinforcement is 16mm, as shown in Fig 3.The distribution of rebar in track bed structure as shown in Fig 4.

Fig 7 Fig. 7 . 2 )Fig. 8 .
Fig 7 shows the corrosion current density distribution of track bed rebar under different concrete resistivity when unilateral input, it can be seen that when the concrete resistance is 1000 Ω/m, the corrosion area of track bed rebar concentrate near the input point.With the decrease of concrete resistance, the corrosion area extend from the upper to the lower rebar, and finally cover the left half of track bed rebar.Unit:mA/m 2

Fig. 9 .
Fig. 9. Relation curve between corrosion current density of point X and input pointvoltage under unilateral input.

4. 2 .Fig. 10 .
Fig. 10.Voltage distribution of track bed structure under different concrete resistivity when bilateral input.Fig 11 shows the corrosion current density distribution of track bed rebar under different concrete resistivity when bilateral input, it can be seen that when the concrete resistance is 1000 Ω/m, the corrosion area of track bed rebar concentrate near the upper rebar, with the decrease of concrete resistance, the corrosion area gradually expands downward, and finally cover both ends of track bed rebar.Fig 12 shows the relation curve between point X voltage and input point voltage under different concrete resistivity at bilateral input, it can be seen that the voltage at point X increases linearly with the input point voltage, when the input point voltage increases by 40%, the voltage at point X will increase by about 35%.Fig 13 shows the relation curve between corrosion current density of point X and input point voltage under different concrete resistivity at bilateral input, it can be seen that the corrosion current density at point X increases nonlinearly with the input point voltage, when the input point voltage increases by 40%, the corrosion current density at point X will increase by about 61%.Fig 14 and 15 shows the relation between input voltage, corrosion current density and concreteresistivity, the result shows under the bilateral input mode, the influence of concrete resistivity on rebar voltage and corrosion rate is more sensitive than that under the unilateral input mode.

Fig. 12 .Fig. 13 .Fig. 14 .
Fig. 12. Relation curve between point X voltage and input point voltage under bilateral input is the current density caused by external current, without considering the influence of other external currents, it is assumed that B B i