## 3 Results  
This problem has been simulated on a computing cluster at Riga Technical University using 4 \(\times\) 36 cores. Each simulation was run for 42,884 cycles, after which the solver failed to converge. The results are shown in Figure 10 (cross-section in the longitudinal direction) and Figure 11 (cross-section in the transversal direction).  
The simulation results are compared with the rail squat field measurements in [14]. Cracks have been measured at specified grid points on a tangent rail over a span of 3 years using a handheld ultrasonic testing device with the accuracy range of \(\pm\)0.1 mm. The field measurement results can be seen in Figures 12 and 13.  
Two loadings--pressure and shear traction--have been applied to the model. The magnitude of the shear traction is not symmetric around the coordinate origin, even although the load area is. Traction is then applied in the positive x direction, and the values change as shown in Figures 8 and 9. This can cause damage to develop slightly asymmetrically against the y-z plane, which is best seen in Figure 10a. Damage develops faster on the positive side of the x axis (the right side in Figure 10). Against the x-z plane, in Figure 11, damage developed symmetrically, because both the pressure and the shear traction are symmetric. The same asymmetric crack growth has been observed in the field measurements (see Figure 13). In reality, such asymmetry happens because the shear traction from a wheel rolling forward is applied in the rolling direction.  
Figure 11a clearly shows that damage first develops close to the location of maximum pressure (the middle of the rail in transversal direction). In addition, the maximum damage remains under the same area (see Figure 11b-d). This is consistent with the field measurements shown in Figure 12. Cracks are deeper closer to the center of the rail head and shallower closer to the sides. This shows that they first initiated and have been growing for longer under the rolling surface.  
The simulation ended unexpectedly quickly, because the fatigue resistance of a rail without any defects should definitely be above 42,884 cycles. Loads have thus been applied to a three node (one horizon) thick layer on top of the rail head. As bonds extending to nodes below this layer are broken, the applied loads are no longer transferred downwards and the loaded nodes simply moved through the layers below them; this can be seen in the c and d parts of Figures 10 and 11. This is the so-called \"matter-interpenetration\" problem, and usually it is solved through different contact models. The simplest model--short-range force--was introduced in [31], and a better description is available in [39; 49]. Other contact models and properties are presented in [50; 51; 52; 53].  
Figure 10: The cross-section (x-y plane) along the middle of the rail head in the longitudinal direction. Damage is shown in the top part of the model after: (a) 37,000; (b) 42,500; (c) 42,850; and (d) 42,884 cycles.  
Figure 11: The cross-section (x-z plane) along the middle of the rail head in the transversal direction. Damage is shown in the top half of the model after: (**a**) 37,000; (**b**) 42,500; (**c**) 42,850; and (**d**) 42,884 cycles.  
Figure 12: An ultrasonic rail squat measurement: (**a**) crack depths at each grid point; (**b**) top view of the rail surface.  
While a shot-range force contact mode is available in Peridigm, it has been implemented only for explicit and not quasi-static simulations and it does not consider contact between nodes that are bonded initially. As damage develops, it is important that the contact model reconsiders the contact between two nodes that were bonded initially but are not anymore. This limitation has been explored, and it is possible to resolve it. Future work will concentrate on how to efficiently pass data between Peridigm's damage and contact models.