## 2 Methods and Materials
### Theoretical Background for Data Processing
#### 2.1.2 Heat Equation and Infrared Thermography  
The origin and theory of IRT has been described in detail elsewhere [60; 61]. For the sake of completeness, the fundamentals of IRT-based methods are briefly described below. In the thermographic systems, equilibrium infrared radiation emitted by a body is detected in a non-contact way by an infrared detector. Assuming a black-body model and using Stefan-Boltzmann's law, the temperature of the solid is obtained. When the test object does not satisfy the black body assumption, its emissivity must be considered and accounted for. The infrared emissivity is a function of wavelength, but a IRT camera works in the spectral range where it does not distinguish between wavenumbers. To correct for the non-unit emissivity, a grey body model can be adopted under the assumption that the emissivity is uniformly distributed within the camera's spectral bandpass. Since metals are opaque, the in-band radiance (IBR) received by the camera's sensor can be divided into three part: (i) the radiance emitted by the object surface, (ii) the radiance reflected from the object, and (iii) the radiance emitted by the atmosphere. The last term is often negligible when the distance between the lens and the object is small, while the remaining terms can be approximated as  
\[\mathbb{L}_{m}\cong\overline{\varepsilon_{body}}\times\mathbb{L}_{Pl}(T_{ body})+[1-\overline{\varepsilon_{body}}]\times\mathbb{L}_{Pl}(T_{s}) \tag{6}\]  
where \(\mathbb{L}_{m}\) is the IBR measured by the system, \(\mathbb{L}_{Pl}(T)\) is the IBR of the Planck's function at temperature \(T\), \(\overline{\varepsilon_{body}}\) is the average emissivity in the operating spectrum, \(T_{body}\), and are the temperatures of the object of interest and surroundings, respectively. This equation assumes that the radiance emitted from the object and the surroundings, is represented by [PERSON]'s law. The IBR is an integral of radiation over the working spectrum of the camera-lens system. Rearranging these terms gives the following equation  
\[\mathbb{L}_{Pl}(T_{body})\cong\frac{\mathbb{L}_{m}-[1-\overline{\varepsilon_{ body}}]\times\mathbb{L}_{Pl}(T_{s})}{\overline{\varepsilon}_{body}} \tag{7}\]  
which allows calculating the black body equivalent IBR of the test object. From the last equation, the temperature of the object can be calculated since \(\mathbb{L}_{Pl}(T)\) is a bijective function connecting the radiometric temperature to IBR (for further information, interested readers are referred to [62]). According to [PERSON]'s law, the differential heat conduction equation for anisotropic materials takes a form  
\[\Phi_{q}=-k\
abla T \tag{8}\]  
where \(\Phi_{q}\) denotes the heat flux vector (the flow of heat through a unit area per unit of time), \(k\) is the coefficient of heat conductivity, which, in general, can be expressed in a symmetric tensorial form, and \(\
abla T\) stands for the temperature gradient. Applying this equation to the experimentally measured temperature field permits determining the heat flux. Equation (8) is sufficient to describe heat conduction under stationary conditions, i.e., when the temperature distribution across the solid and the heat flux remain constant in time. To account for the non-steady behaviour, a second independent equation is required. The energy conservation law is commonly used for this purpose in the form of the continuity equation [61]:  
\[\frac{\delta Q}{\delta t}=c_{p}\rho\frac{\partial T}{\partial t} \tag{9}\]  
where the change in the internal heat energy with time, i.e., the heat power, is denoted as \(\frac{\delta Q}{\partial t}\) (the symbol \(\delta\) here, and in what follows, highlights that the infinitesimally small increment of heat is not a perfect differential), and it is related to the rate of the temperature change, \(c_{p}\) is the specific heat capacity, and \(\rho\) is the material's mass density.