Model setup

The FEM model used in the study consists of a block representing a section of rail with cracks at the surface of, and propagating into, the rail block and an air block above the rail section. The model geometry with a single semi-elliptical crack and the standard Cartesian coordinates are shown in Figure 3.1a. The current flows in the y-direction, which is perpendicular to the crack surface length. A perpendicular current orientation is selected to give the maximum Bx and Bz signals and represents the standard experimental procedure where the ACFM sensor is oriented parallel to the crack. A uniform magnetic field is induced above the rail surface mimicking the operation of the physical ACFM sensor. The uniform field model (different with the probe sensor model discussed in section 2.5) is selected because it provides a good simulate to the experiments and greatly decreases the computing time and data processing procedure (see discussion in section 2.5). The uniform magnetic field is generated by applying surface current boundary condition along the front, up and rear surfaces of the air block shown in Figure 3.1a. This allows a uniform current distribution at the interface between the air and the rail block.

57

Figure 3.1b shows the details of the meshing of the whole model; areas of refined mesh are applied to the crack, the rail surface and the domain from which the signals are extracted. The maximum element size applied to the crack surface and the crack refined area, i.e. the refined mesh block with size of 25×25×5 mm3 under the rail surface, are 0.1 mm and 0.5 mm, respectively and both have an element growth ratio of 1.5. The total number of elements for the model is in the order of 1.5×106. Crack width of 0.1, 0.3, 0.5, 0.7 and 1 mm were modelled; results shows that the difference of Bx value is 0.15 when the width increases from 0.1 to 0.5 mm whilst the difference is 0.65 when the width changes from 0.5 to 0.7 mm. It is impossible to introduce an extremely narrow crack (RCF cracks are of the order of several microns wide) into the model due to meshing and solving problems; therefore the crack width selected in the modelling is 0.5 mm, based on a compromise between solvability and the signal sensitivity to changes in crack width. This has previously been shown to give good results when comparing model and actual measurements for semi- elliptical cracks [20-22].

The crack shapes and dimensions used in the model are based on semi-ellipse shapes with certain ellipse ratios (e.g. 1, 1.25, 1.5 and 1.75) that have been shown to approximate real light to moderate RCF cracks (based on the Network Rail classification diagram, Figure 2.7) in rails removed from service [3, 20, 22]. The rail material considered in the model was 260 grade rail steel. It was assumed that the steel has an electrical conductivity of 5 × 106 S/m and relative permeability of 50 [22, 155]. The conductivity of air is assumed to be 50 S/m, as it aids the convergence of the model. The effect of varying the modelled permeability and conductivity values by 20% of their assumed values was relatively small: varying the permeability of 50 by ± 10 resulted in a change of Bx value of ± 0.18, and changing the electrical

conductivity by ± 1×106 _{changed Bx value by ± 0.19 [22].} .

58

Figure 3.1 (a) Model geometry (for a single semi-elliptical crack) with two refined mesh blocks for the crack area and area of data extraction; (b) meshing of the model with refined mesh on the crack surface, crack area, rail surface and the area of data extraction.

In the present study, the full mesh boundary (FMB) condition is used rather than the impedance boundary (IB) condition, which was used in previous work [20, 22]. The impedance boundary condition can greatly reduce the meshing requirements and solving time by introducing current flowing only along the boundary, that is, the crack surface and the interface between air and rail surface. The IB method model was verified against experimental data for pocket length measurement, using the Bx signal, for cracks with vertical angles larger than or around 30° [20]. The FMB method meshes the whole domain with refined meshes for the area around the crack and the surface/near surface region. The FMB and IB models give comparable results for cracks with vertical angles larger than 30° but discrepancies are observed for shallow angle cracks (i.e. crack vertical angle less than 30°). The difference in using the FMB and IB conditions for shallow angle cracks will be discussed in section 4.1.

As the ACFM sensor gives results in analogue to digital conversion (ADC) units while modelling results are in SI units, the normalised Bx (equation 24) and Bz (equation 25) are used to compare the experimentally and numerically determined signals. The normalised maximum ΔBx (equation 26) is used to determine the crack pocket length and the Bz trough-peak ratio (equation 27) is proposed in the present study to determine the crack vertical angle. Full details are given in section 4.3

59 _100% 0   Bx Bx Bx Normalised (24) 0 max 0 100 % Bz Bz Normalised Bz Bz Bz     _{(25) 100%} 0 min 0 max     Bx Bx Bx Bx Normalised (26) peak trough Bz Bz ratio peak trough Bz   (27)

where Bx0 and Bz0 are the background signals of the x and z-components of the magnetic fields, respectively; the Bxmin is the minimum value of the Bx signal; the

Bz0 takes a value of 0 % and the maximum value recorded by the measurement line,

Bzmax, denotes the signal strength of 100 %; Bztrough is the value at the trough of the Bz signal and Bzpeak is the value at the peak of the Bz signal.