## 2 Problem Statement  
A domain is divided into several small cells, with each cell being a set of carbon fibres generated by an algorithm, allowing us to obtain a virtual material with properties close to the real ones (Figure 3) [12]. The fibres stretch in the \(z\) direction and are randomly distributed in \(x\) and \(y\) directions in the real material. Due to the random positions of the fibres in the composite material, contact between fibres is also possible and random. The presence of contacts has a major impact on the induced current because the contact allows the current to pass from one fibre to another. To explain the electrical conductive behaviour of composite materials made up of conductive charges and insulating matrices, percolation theory can be applied. As the conductive charges increases, the composite undergoes a transition from insulator to conductor. The percolation threshold is reached when the measured electrical conductivity of the composite material jumps. Below the percolation threshold, electrical paths do not exist and electrical properties are dominated by the material matrix. The concentration  
Figure 3: An example of virtual material with the same microscopic behaviour as real material. of conductive charge must be higher than the percolation threshold in order to obtain conductive networks [24]. In the virtual material, when one generates fibres, fibres are divided into layers, and nodes are defined as the centre of fibres at each layer. Nodes are randomly distributed into the layer but with the constraint of limited overlapping between fibres. The layers are built sequentially, with the position of a node in a layer being obtained by the perturbation of the node of the same fibre in the previous layer. A resistance network will be deduced from fibres' geometry and contact between them, as shown in Figure 4.  
To simulate the electrical behaviour of CFRP, a global model is given by the following equation:  
\[\mathbf{J}=-\sigma\
abla V-\sigma j\omega\mathbf{A}, \tag{1}\]  
where \(\mathbf{J}\) is the density of current, \(\sigma\) is the conductivity of the carbon, \(V\) is the electric potential to be solved, \(j\) is the imaginary unit, \(\omega\) is angular frequency, and \(\mathbf{A}\) is the magnetic potential vector. This model is obtained from  
\[\mathbf{E}=-\frac{\partial\mathbf{A}}{\partial t}-\
abla V, \tag{2}\] \[\mathbf{J}=\sigma\mathbf{E} \tag{3}\]  
with  
\[\mathbf{B}=\text{rot}\mathbf{A}; \tag{4}\]  
where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic induction as a source term, and \(t\) is the time. From the Equation (1), one has  
\[\frac{\mathbf{J}}{\sigma}=-\
abla V-j\omega\mathbf{A}. \tag{5}\]  
When there exists contact between two nodes, the segment connecting them is defined as edge \(\ell\). From the volume integral on a volume \(\ell\times S\) of Equation (5), one has  
\[\int_{\ell}\iint_{S}\frac{\mathbf{J}}{\sigma}\text{d}S\text{d}\ell=-j\omega \int_{\ell}\iint_{S}\text{Ad}S\text{d}\ell-\int_{\ell}\iint_{S}\
abla V\text{ d}S\text{d}\ell, \tag{6}\]  
where \(S\) is the cross-sectional area of a fibre. Then,  
\[\frac{I\ell}{\sigma}=-j\omega S\int_{\ell}\mathbf{A}\cdot\text{d }\ell+S\Delta V,\] \[\frac{I\ell}{S\sigma}=-j\omega\int_{\ell}\mathbf{A}\cdot\text{d} \ell+\Delta V, \tag{7}\] Figure 4.0  
Figure 4.1 Equivalent resistance network deduced from fibres’ geometry. where \(R\) is the resistance of segment.