## 2 Calculation Model of Particle Impact Strength Corrosion
### The Proposal of the Calculation Model  
The storage space of the geothermal mining project in the abandoned mine consists of collapsed roof voids resulting from long-wall mining. The volume expansion of the collapsed rock layers is influenced by the expansion characteristics of the broken rock mass. Based on previous research [21], it can be inferred that the volume expansion of broken rock mass in goaf is directly associated with factors such as lithology, size grading of fragmented blocks after crushing, and accumulation state. The strength erosion rate is defined as the mass loss of the target material after being impacted by moving particles (unit: kg/(m\({}^{2}\)-s)) [29]. Multiple factors influence the strength erosion rate caused by particle impact on broken rock blocks, and [PERSON] [30,31] proposed a deformation mechanism-based calculation model for impact erosion rate in brittle materials, which can be expressed as:  
\[\varepsilon_{m}=\frac{1}{2}\frac{M_{p}\big{(}v_{p}\sin\alpha-K\big{)}^{2}}{ \delta_{T}} \tag{1}\]  
where \(M_{p}\) denotes the mass of a single impacting particle (unit: kg); \(v_{p}\) represents the impacting velocity of the particle (unit: m/s); \(\alpha\) is the impact angle (unit: \({}^{\circ}\)); \(K\) denotes the velocity component perpendicular to the surface of the target material (unit: m/s), and when the velocity is below \(K\), no significant erosion occurs; and \(\delta_{T}\) represents the energy required for unit volume of target material erosion (unit: kg:m/s\({}^{2}\)). The \(M_{p}\) can be calculated by \(\rho_{p}V_{p}\), in which \(\rho_{p}\) is the density of the impact particle and \(V_{p}\) is the volume of the impact particle. Because the impact particle is exfoliated from the target material, the density of the impact particle is the same as that of the target material \(\rho_{T}\). Then, Equation (1) can be modified as:  
\[\varepsilon_{m}=\frac{1}{2}\frac{\rho_{T}V_{p}\big{(}v_{p}\sin\alpha_{n}-K \big{)}^{2}}{\delta_{T}} \tag{2}\]  
According to the particle discrete element numerical simulation method, each basic element 'ball' that makes up the rock block is bonded by a 'contact' element. Similar to the study method of reducing the bonding strength of rock model contact due to chemical corrosion rate when a particle discrete element is used to simulate the creep phenomenon [20,25], the corrosion rate of bonding strength between contacts inside the rock block after particle impact is assumed to be the same as the erosion rate of Equation (2). Then, the remaining rate of the bonding \(v_{c}\) after corrosion can be calculated as follows:  
\[v_{c}=1-\frac{1}{2}\frac{\rho_{T}V_{p}\big{(}v_{p}\sin\alpha_{n}-K\big{)}^{2} }{\delta_{T}} \tag{3}\]  
Equation (3) is applicable only in the presence of impact corrosion. The corrosion rate is 1 when there are no impact phenomena and becomes 0 after the bond is broken. Combining with Equation (3), the bond radius multiplier of the particle mechanics model during cycling can be calculated by:  
\[\lambda(n)=\frac{D_{n}}{D_{n-1}}= \begin{cases}1&n=0\\ 1-\frac{1}{2}\frac{\rho_{T}V_{p}\left(v_{p}\sin a_{n}-K\right)^{2}}{\sigma_{T}} &0\leq n<n_{c}\\ 0&n>n_{c}\end{cases}  \tag{4}\]  
where \(n\) denotes the impacting times; \(\lambda\) is the bond radius multiplier; \(\alpha_{n}\) is the impact angle of the \(n\)th impacting; \(D_{n-1}\) and \(D_{n}\) represent the bond radius at the \((n-1)\)th and \(n\)th impacting, respectively (unit: m); and \(n_{c}\) refers to the effective impacting times when the contact bond is broken. The fracture energy of a unit volume of rock mass can be calculated using the following equation [32]:  
\[\delta_{T}=\frac{5\sigma_{T}\varepsilon_{T}}{12} \tag{5}\]  
where \(\sigma_{T}\) and \(\varepsilon_{T}\) denote the stress and strain of the target material, respectively. Then the final impacting strength erosion model can be obtained, and the bond radius multiplier \(\lambda\) can be computed as:  
\[\lambda(n)=