## 3 Modeling Test Cases  
### Inductive Heating  
We are dealing with a 3D heat conduction problem for a non-adiabatic solid body with internal heat sources. The cylindrical geometry of a pole (Figure 6a) was replaced with a 3D Cartesian model shown in Figure 6b and numerically solved by using the ThermoSource software from Tomsk Polytechnic University (Figure 6c). The model simulates a 55 mm thick pole shell made of reinforced concrete (pole type SK8 by the Russian nomenclature). It is worth noting that in real poles, plenty of rebar bunches each including four 5 mm diameter steel rebars are regularly placed within a pole, as shown in Figure 6a. Each rebar bunch was simulated as a parallelepiped-like defect (cross section: 20 \(\times\) 5 mm) and placed at depths of 10 and 20 mm from the external pole surface, and high-frequency powerful induction heating was simulated as energy discharge in a rebar (released power 10 MW/m\({}^{3}\), heating duration 20 s). The mathematical formulation of such TNDT problems was thoroughly discussed elsewhere [12; 13]. Three characteristic areas are specified on the pole surface: a defect-free area (area 1 in Figure 6c), diminution of the rebar cross section by 25% (area 2), and the same rebar thinning but additionally accompanied with a 1 mm thick air-filled delamination (area 3). The latter defect is to simulate the practical situations, where the process of corrosion wear is accompanied by corrosion products that appear between concrete and reinforcement and provide pressure on the adjacent concrete, thus causing voids and cavities.  
Two IR thermograms taken at 60 and 120 s are shown in Figure 7 to demonstrate that, over defect-free areas, the surface excess temperature \(\Delta T\) reaches 8 \({}^{\circ}\)C (at 120 s if the defect depth is 10 mm); the rebar thinning results in a weak decrease of surface temperature because of the lower released energy while a significant temperature decrease occurs over the air-filled delamination because of its high thermal resistance. Temperature evolutions at three characteristic points (see Figure 7) in time (\(\tau\)) are presented in Figure 8. Similarly to diffusivity measurement by using the known Parker technique, the fronts of temperature signals in Figure 8 depend on material thickness and thermal properties. This is also illustrated by Figure 9, where steel rebars are located at a depth of 20 mm. The thicker concrete layer diminishes both the excess surface temperature (down to 3 \({}^{\circ}\)C) and the rate of the temperature change.  
The same dependencies are shown in Figure 9 for the case where steel rebars are located at a depth of 20 mm. The thicker concrete layer diminishes the surface temperature up to 3 \({}^{\circ}\)C and makes the rate of change in temperature slower.  
Figure 6: Railway pole 3D numerical model (wall thickness 55 mm, 5 mm diameter steel rebars at 10 (or 20) mm depth, 1—defect-free case, 2—defective case, rebar cross section diminished by 25%, 3—same as (2), additional 1 mm–thick air-filled delamination): (**a**) pole scheme; (**b**) Cartesian model; (**c**) ThermoSource software model, two rebars shown).  
Figure 7: Synthetic IR thermograms of railway pole at 60 s (**a**) and 120 s (**b**) (model in Figure 6, reinforcement depth 10 mm, released power 10 MW/m\({}^{3}\), inductive heating duration 20 s).