## 3 Experimental Results and Analysis
### Damage Evolution Analysis of Sandstone under Cyclic Loading and Unloading
As can be seen from Figure 13, the fitting equation has high accuracy, and the equation can well-describe the rock damage evolution process in the prepeak deformation stage of the rock. It can be seen that, with the increase of loading rate, the initial damage parameter \(a\) keeps decreasing; that is, the initial damage degree \(D_{0}\) keeps increasing. As shown in Figure 14, the endogenous growth rate of damage evolution r shows a good linear relationship with the initial damage parameter \(a\). Figure 12: Damage evolution curve. Figure 13: Damage evolution fitting curve. From the derivation of the damage evolution Equation (5), the differential form of the damage evolution equation can be obtained:  
\[\frac{\mathrm{d}D}{\mathrm{d}\sigma}=rD(1-\frac{D}{k}) \tag{6}\]  
If the value of \(D\) is measured once every stress interval by increasing \(\Delta\sigma\), and the \(n\)th value is denoted by \(D_{n}\), the original continuous variables \(D(\sigma)\) and \(\sigma\) become discrete variables (\(D_{0},\;D_{1},D_{2},\ldots\)) and \((n=0,1,2,\ldots\)), and the continuous differential Equation (6) becomes a discrete difference equation, as follows:  
\[D_{n+1}=(r\Delta\sigma+1)D_{n}(1-\frac{r\Delta\sigma}{(r\Delta\sigma+1)k}D_{n}) \tag{7}\]  
Define \(\mu=r\Delta\sigma+1\), \(D^{\prime}_{n}=\frac{r\Delta\sigma}{(r\Delta\sigma+1)k}D_{n}\); \(D^{\prime}_{n}\) is the generalized damage variable, and the above equation can be simplified as:  
\[D^{\prime}_{n+1}=\mu D^{\prime}_{n}(1-D^{\prime}_{n})=f(D^{\prime}_{n}) \tag{8}\]  
where \(n=0,1,2,3\ldots\), and the mapping f is a logistic mapping reflecting the generalized damage variable \(D^{\prime}_{n}\). Equation (8) is the classical one-dimensional logistic mapping, which has very complicated properties. If the control parameter \(\mu\in[0,4]\), the mapping f acts to still map any value \(D^{\prime}_{n}\in[0,1]\) to that interval, i.e., the interval \([0,1]\) is an invariant interval for the mapping \(f\). In the invariant interval, the sequence of \(D^{\prime}_{n}\) shows very different orbits with \(\mu\). As the control parameter \(\mu\) increases, the mapping \(f\) will go through the stable region (\(\mu<3\)), multiplicative period bifurcation region (\(3\leq\mu<3.5699\)), and finally enter the chaotic state (\(3.5699\leq\mu<4\)). The values of each bifurcation point are shown in the following Table 2:  
Figure 14: Relationship between parameter a and parameter r.  
\(0<D_{n}\leq k\leq 1\), so \(D^{\prime}_{n}=\frac{r\Delta\sigma}{(r\Delta\sigma+1)k}D_{n}\in(0,1)\), and \(\mu=r\Delta\sigma+1>1\). The rock damage evolution process will have the nature of multiplicative period bifurcation or chaotic dynamics. The rock damage evolution process is analyzed at a loading rate of 0.5 KN/s, and the damage evolution equation is  
\[D=\frac{k}{1+\mathrm{e}^{a-rc}},0\ <k<1 \tag{5}\]0  
The discrete equation for the generalized damage is:  
\[D=\frac{k}{1+\mathrm{e}^{a-rc}},0\ <k<1 \tag{5}\]1  
so \(\mu=0.134\Delta\sigma+1\), \(D^{\prime}_{n}=\frac{0.134\Delta\sigma}{0.134\Delta\sigma+1}D_{n}\). When \(0<\Delta\sigma<14.93\) MPa, there is \(1<\mu<3\); that is, when stress is increasing in this interval, the internal energy of the rock keeps accumulating, deformation keeps accumulating, accompanied by continuous energy dissipation, and the rock keeps being damaged; the generalized damage process will eventually enter a stable state, that is, the fixed point \(D^{\prime}_{n}=1-1/\mu\) formed after the contraction of phase space. When \(14.93\) MPa \(<\Delta\sigma<19.18\) MPa, there is \(3<\mu<3.5699\). The generalized damage process enters the multiplicative period bifurcation zone with the increase of stress, and a series of bifurcation points and corresponding limit rings appear; at first \(D^{\prime}_{n}\) oscillates between two values, and, with the increase of stress, the bifurcation performance oscillates between four values.