## 2 SDoF Model for Low-Velocity Heavy-Mass Impact
### Unloading Stage  
After attaining the maximum deflection, the beam rebounds due to the material elasticity. Equation (3) is rewritten to obtain the unloading path as  
\[ \begin{array}{ll}F_{u}=&\left[\frac{(M_{1m}-\Delta M_{1})+(M_{0m}-\Delta M_ {0})}{l_{1}}+\frac{(M_{2m}-\Delta M_{2})+(M_{0m}-\Delta M_{0})}{l_{2}}\right] +(N-\Delta N)\left(\frac{1}{l_{1}}+\frac{1}{l_{2}}\right)W_{u}\\ &=F_{m}-\left(\frac{\Delta M_{1}+\Delta M_{0}}{l_{1}}+\frac{\Delta M_{2}+ \Delta M_{0}}{l_{2}}\right)-\Delta N\left(\frac{1}{l_{1}}+\frac{1}{l_{2}} \right)W_{u}-N_{m}\left(\frac{1}{l_{1}}+\frac{1}{l_{2}}\right)(W_{m}-W_{u}) \end{array}  \tag{27}\]  
where \(M_{1m}\), \(M_{0m}\) and \(M_{2m}\) are the bending moments at \(x=-l_{1}\), \(0\) and \(l_{2}\) at the peak point \(W=W_{m}\), respectively; \(N_{m}\) is the axial force at the peak point \(W=W_{m}\); \(\Delta M_{1}\), \(\Delta M_{0}\) and \(\Delta M_{2}\) are the change of the bending moments at \(x=-l_{1}\), \(0\) and \(l_{2}\) during unloading; \(\Delta N\) is the change of the axial force during unloading; and \(W_{u}\) is the displacement of the beam at \(x=0\) during unloading. It is evident that the quantities \(F_{m}\) and \(N_{m}\) at the unloading point in Equation (27) can be determined from the theoretical analysis of the loading path, while the changes of the bending moment and the axial force are unknown, which is relevant to the change of the shape function of the beam during the unloading stage. Unloading is widely regarded as an elastic process [24], and thus the change of the displacement profile during unloading is  
\[\Delta w(x)=\left\{ \begin{array}{ll}\frac{\Delta W}{2}\Big{(}1+\cos\frac{ \pi x}{l_{1}}\Big{)}&-l_{1}\leq x<0\\ \frac{\Delta W}{2}\Big{(}1+\cos\frac{\pi x}{l_{2}}\Big{)}&0\leq x\leq l_{2} \end{array} \right. \tag{28}\]  
where \(\Delta W=W_{m}-W_{u}\). Accordingly, the changes of the bending moment and the axial force related to \(\Delta W\) are  
\[\Delta M_{1}=\frac{3 EII}{l_{1}^{2}l_{2}}(W_{m}-W_{u}) \tag{29}\]  
\[\Delta M_{2}=\frac{3 EII}{l_{1}l_{2}^{2}}(W_{m}-W_{u}) \tag{30}\]  
\[\Delta M_{0}=\frac{6 EI}{l_{1}l_{2}}(W_{m}-W_{u}) \tag{31}\]  
\[\Delta N=A\varepsilon\Delta\varepsilon \tag{32}\]  
where \(\Delta\varepsilon\) is the change of the axial strain, given as  
\[\Delta\varepsilon=\frac{1}{l}\int_{l}\frac{dw_{m}}{dx}\frac{d\Delta w_{u}}{ dx}\ dx+\frac{1}{2l}\int_{l}-\left(\frac{d\Delta w_{u}}{dx}\right)^{2}dx \tag{33}\]  
Figure 3: A summary of the loading regimes. where \(w_{m}(x)\) is the shape function at the maximum deflection \(W_{m}\), and \(w_{u}(x)=w_{m}(x)-\Delta w(x)\) is the shape function during unloading. For unloading from a maximum displacement \(W_{m}\leq W_{3y}\), the first term at the right-hand side of Equation (33), \(\Delta\varepsilon_{1}\), can be determined by using the shape function \(w_{1}(x)\) defined in Equation (5):  
\[\Delta\varepsilon_{1}=\frac{W_{m}(W_{m}-W_{u})}{8l}\bigg{(}\frac{1}{l_{1}}+ \frac{1}{l_{2}}\bigg{)}\pi^{2} \tag{34}\]  
For unloading from a maximum displacement \(W_{m}\geq W_{3y}\), \(\Delta\varepsilon_{1}\) can be determined by using the shape function \(w_{2}(x)\) defined in Equation (16), expressed as  
\[\Delta\varepsilon_{1}=\frac{8W_{m}-(8-\pi^{2})W_{3y}}{8l}(W_{m}-W_{u})\bigg{(} \frac{1}{l_{1}}+\frac{1}{l_{2}}\bigg{)} \tag{35}\]  
On the other hand, the second term at the right-hand side of Equation (33), \(\Delta\varepsilon_{2}\), only depends on \(\Delta w(x)\). Thus, \(\Delta\varepsilon_{2}\) can be determined when using Equation (28), expressed as  
\[\Delta\varepsilon_{2}=-\frac{\pi^{2}}{16l}(W_{m}-W_{u})^{2}\bigg{(}\frac{1}{l _{1}}+\frac{1}{l_{2}}\bigg{)} \tag{36}\]  
Now, \(\Delta N\) can be obtained from Equation (32) for unloading from any peak displacement \(W_{m}\). Thus, the unloading path from any loading regimes can be determined from Equation (27) when using the values of \(F_{m}\), \(T_{m}\), \(\Delta N\) and \(\Delta M\) in the corresponding regime.