# Use of Surface Acoustic Waves for Crack Detection on Railway Track Components--Laboratory Tests
## 2 Test Specim and Methods
### Simulation of Elastic Wave Propagation via EFIT  
In contrast to the situation of an ideal surface crack in an ideal infinite material half-space, as discussed in the previous chapter, both the bulk waves generated by the transducer and those due to mode conversion at the crack can impede the detection of the reflected/transmitted surface acoustic wave intensities. To optimize the measurement setup and to clarify the wave propagation behavior, 2D and 3D elastodynamic finite integration technique (EFIT) software was developed and customized for the rail geometry. The elasto-dynamic equations in velocity-stress (\(\overset{\rightarrow}{\mathbf{v}}\), \(\sigma\)) representation (consisting of the Cauchy momentum equation \(\rho\frac{\overset{\rightarrow}{\partial\varphi}}{\partial t}=\overset{ \rightarrow}{\
abla}\cdot\sigma\), together with the isotropic linear elastic material law \(\dot{\boldsymbol{\sigma}}\approx E/[2(1+\mu)]\ tr(\dot{\boldsymbol{\epsilon}})+2 \mu\ \dot{\boldsymbol{\epsilon}}\), where the linear strain rate is given by \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mass Density \(\rho\)** & **Young’s Modulus \(E\)** & **Poisson’s Ratio \(\mu\)** \\ \hline
7800 kg/m\({}^{3}\) & 200 GPa & 0.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters of the railway steel as input for the simulation. Figure 4: Theoretical results for the scattering of Rayleigh waves at a surface breaking crack (data taken from [PERSON] and [PERSON] 1984 [11], for a Poisson’s ratio \(\gamma\) = 0.3. Reflection (**a**), transmission (**b**), energy conversion to bulk waves (**c**) depending on the angle of incidence, as well as reflection (**d**) and transmission (**e**) as a function of crack depth d divided by the wavelength \(\lambda_{R}\). \(\dot{e}\approx[(\overrightarrow{\
abla}\otimes\overrightarrow{v})+( \overrightarrow{\
abla}\otimes\overrightarrow{v})^{T}]/2\) and the material parameters Young's modulus \(E\) and Poisson's ratio \(\mu\) are given in Table 1)were discretized on a staggered grid and the time evolution executed by an explicit update of the stress tensor and velocity vector in an alternating manner (leap-frog time integration, see Figure 5). For details on the corresponding theory and equations for the elastodynamic finite integration technique (EFIT) the interested reader is referred to the publication by [PERSON] et al. [12]. A demonstration for a damped elastic wave simulation was recently published by [PERSON] et al. [13]. The software implementation was undertaken in Python using TensorFlow [14] as a tool for a convenient parallelization, enabling computation on multiple CPUs combined with GPU. The computations shown in this work, i.e., a 2D model of the rail cross-section, were performed using an Intel Xeon CPU E5-1650 v4 @ 3.60 GHz workstation with 6 cores and 30 GB RAM with a typical calculation and postprocessing time of \(\sim\)22 h.  
The rail geometries were drawn with the CAD package DraftSight and imported as a regular 2D cartesian grid with square edge length of 0.15 mm (i.e., a resolution of 20 grid points per wavelength) into the EFIT software. The values for the material parameters are listed in Table 1.  
The initial conditions were a zero stress and velocity field in the whole domain. The excitation was implemented by setting a time-dependent velocity boundary condition at the location of the sending transducer--left bottom of the rail head (position HBL)--corresponding to the contact of the transducer comb (sender) with the rail body. The time dependence of the input signal was determined to match the waveform found experimentally. Three different crack constellations with 1-3 cracks, as shown in Figure 6, were used in the simulations.