*2.1. Diffusion Mechanism*

The corrosion factors diffuse from the outside to the interior of the cable due to the impact of the concentration gradient difference of corrosion factors within and outside the cable. The diffusion forms of corrosion factors in the cable can be divided into two types: (1) The corrosion factors diffuse radially along the gap between steel wires until the concentration of corrosion factors reaches saturation in each layer of steel wire on the section. (2) Corrosion factors will diffuse upwards in the longitudinal and circumferential directions of the cable due to the gaps between the layers of steel wires, eventually causing three-dimensional damage to the cable. The diffusion mechanism is shown in Figure 1.

**Figure 1.** Transfer form of corrosion factors in the cable: (**a**) radial transmission; (**b**) circular and axial transmission.

The diffusion mechanism of corrosion factors is mainly described by Fick's law. Fick's law is made up of the first and second laws of Fick. Fick's first law is proposed based on the concentration gradient difference of diffusing substances, which believes that the greater the concentration gradient difference of the diffusing substance, the greater the flow of the substance per unit of time through the unit cross-sectional area perpendicular to the direction of diffusion, proportional to the concentration gradient at the cross-section. However, Fick's first law is only applicable to the analysis of steady-state diffusion processes, in which each volume element at any moment has an equal quantity of incoming and exiting material and a constant concentration throughout the process. This circumstance is not typical, in actuality. The diffusion of substances will be influenced by both environmental factors and the properties of the materials themselves. The diffusion rate is likely to alter as diffusion time and depth change. Based on Fick's first law, the second law to describe the non-stationary diffusion of substances was proposed to explain the diffusion law of substances in actual processes more accurately. The law of variation of concentration of

corrosion factors with diffusion time and depth was obtained. Fick's second law can be expressed as the following equation:

$$\frac{\partial \mathcal{C}}{\partial t} + \nabla \times (-D \times \nabla c) = 0 \tag{1}$$

where *C* is the concentration of corrosion factors, *D* is the diffusion coefficient of corrosion factors, *c* is the gradient difference of concentration of corrosion factors, and *t* is the corrosion time.

The more significant the gradient difference in concentration of corrosion factors, the faster the diffusion rate of corrosion factors, which is more likely to cause damage to parallel steel wires in the cable. For spatial diffusion issues, the concentration gradient difference of corrosion factors and the Hamiltonian operator in Equation (1) can be expressed as follows [44].

$$\begin{cases} \nabla \mathcal{L} = \frac{\partial \mathcal{L}}{\partial x} + \frac{\partial \mathcal{L}}{\partial y} + \frac{\partial \mathcal{L}}{\partial z} \\ \nabla = \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \end{cases} \tag{2}$$

By substituting Equation (2) into Equation (1), the analytical solution model of Fick's second law's error function can be produced, as shown below:

$$\mathcal{L}(\mathbf{x}, y, z) = \mathbb{C}\_0 + (\mathbb{C}\_S - \mathbb{C}\_0) \left( 1 - \text{erf} \frac{\mathbf{x}}{2\sqrt{D \times t}} \times \text{erf} \frac{y}{2\sqrt{D \times t}} \times \text{erf} \frac{z}{2\sqrt{D \times t}} \right) \tag{3}$$

where *C*(*x*,*y*,*z*) represents the concentration at any spatial point within the component, *Cs* is the concentration of corrosion factors on the element's surface, *C*<sup>0</sup> is the concentration of corrosion factors within the structure, and *erf* is the error function.

The calculation method is shown the following equation:

$$\operatorname{erf}(\mathbf{x}) = \frac{2}{\sqrt{\pi t}} \times \int\_0^z \exp\left(\beta^2\right) d\beta \tag{4}$$

The concentration of initial corrosion factors within the structure is relatively low in general. The effect of the concentration of corrosion factors within the structure on the predicted results is neglected to make the solution easier. Equation (3) is further simplified to obtain the following equation:

$$\mathbb{C}(x, y, z) = \mathbb{C}\_{\mathbb{S}} \times \left(1 - \text{erf}\frac{x}{2\sqrt{D \times t}} \times \text{erf}\frac{y}{2\sqrt{D \times t}} \times \text{erf}\frac{z}{2\sqrt{D \times t}}\right) \tag{5}$$

The key to constructing a spatial diffusion model of corrosion factors, according to Equation (5), is getting the concentration and the diffusion coefficient of corrosion factors on the surface of the cable. The concentration and the diffusion coefficient of corrosion factors on the surface of the cable can be calculated using the procedures below.

The initial defect size of a square cable is shown as an example for demonstration. Create a three-dimensional coordinate system using the cable's defect position as the center point, then characterize the cable's damage range. Any point within the damage range can be represented as follows. Define the distance from the coordinate origin at the center position of different sampling points as *Ri*, as shown in Figure 2. The concentration of corrosion factors at each site is measured using electrochemical measuring equipment. After completing the test of the concentration of corrosion factors at each location of each working condition, the concentration and the diffusion coefficient of corrosion factors on the surface of the cable under various operational circumstances can be obtained by substituting results into Equation (5).

**Figure 2.** Schematic diagram of sampling points for corrosion factors.

The following are the specific procedures for building the spatial diffusion model of corrosion factors using the ML method: (1) Using the testing procedures mentioned above, determine the concentration of corrosion factors at each location in the cable space along the path above. (2) Using ambient temperature, humidity, cable inclination angle, and cable defect area as input variables, the prediction model for diffusion coefficient and concentration of surface corrosion factors is created based on the ML approach. (3) According to Fick's second law, the spatial diffusion model of corrosion factors is established by comprehensively considering the influence of environmental and material factors on the diffusion rate of corrosion factors. (4) Combine the results of the salt spray corrosion test and the prototype cable corrosion test to verify the accuracy of the prediction model.

#### *2.2. Test Method*

Due to the influence of the gradient difference of the concentration of corrosion factors inside and outside the cable, the corrosion factors diffuse from the outside to the inside. The diffusion forms of corrosion factors in the cable can be divided into two types: (1) The corrosion factors diffuse along the steel wire and the gap between the steel wires in the radial direction until the concentration of corrosion factors in each layer of the steel wire on this section reaches saturation. (2) Due to the gap between the layers of steel wires, corrosion factors will spread upward along the longitudinal and ring of the cable, resulting in three-dimensional damage to the cable. In order to clarify the diffusion law of corrosion factors in the cable, the concentration of corrosion factors on the surface of each layer of steel wire in the cable was measured by the BION-1881 chemical analyzer under different periods, defect areas, ambient temperature, and humidity. The test procedure is as follows: Firstly, the coordinate system is established with the cable defect position as the origin. Secondly, after the cable was corroded for a corresponding time, it was dissected and cut into sections of 5 cm, and its spatial position in the cable was sorted out and recorded, respectively. Finally, an electrochemical analyzer was used to measure the concentration of corrosion factors on the surface of each cut steel wire. The test procedure of the concentration of corrosion factors is as follows: (1) Each layer of corroded steel wire is processed and cut into a 5 cm section. After the processing is completed, it is numbered; (2) Wash the cut steel wire with water and repeatedly brush it with a brush to completely dissolve the corrosion factors on the surface of the steel wire in water; (3) Completely immerse one electrode rod of the electrochemical analyzer into a beaker, test the concentration of corrosion factors in the water, and take readings after stabilization to obtain the concentration of corrosion factors on the surface of the corroded steel wire. The testing process of corrosion factor concentration is shown in Figure 3. According to the test methods above, the different distribution laws of corrosion factors along the cable in radial, circumferential, and axial directions and the distribution forms of corrosion factors in three directions can be obtained, respectively.

**Figure 3.** Testing process of corrosion factor concentration.
