## 2 Materials and Methods
### Test Process
#### 2.2.4 Nanoindentation Test  
(1) Test Principle  
Nanoindentation testing is capable of evaluating mechanical properties such as harness, creep behavior, elastic modulus, and load-depth responses at the nanoscale. A standard test procedure typically involves three stages: loading, holding at peak load, and unloading. The Berkovich indenter, featuring a regular three-sided pyramidal geometry, is the most widely used tip in nanoindentation testing. When the indenter penetrates the surface of the specimen, the surrounding material first experiences elastic deformation. As the load continues to increase, plastic deformation progressively develops, eventually forming an indentation profile that mirrors the geometry of the indenter [23]. The indentation profile is shown in Figure 3a, where \(h_{\text{P}}\) is the residual indentation depth after unloading, \(h_{\text{m}}\) is the peak indentation depth, \(h_{\text{s}}\) is the sinking depth of the specimen surface, \(h_{\text{c}}\) is the contact depth under the peak load, and \(h_{\text{k}}\) is the creep depth generated during the load retention process. After unloading the load, the typical load-depth curve is shown in Figure 3b, where \(F_{\text{n}}\) is the peak load, \(h_{\text{r}}\) is the tangential depth of the unloading curve, and the slope S of the initial stage of the unloading curve is the tangential stiffness at the indentation position [18; 24; 25]. Nanoindentation tests were performed using the German Bruker Hysitron TI Premier system, which offers a maximum indentation depth of 5 \(\upmu\)m, an effective load resolution of 50 nN, and a loading range of 0-150 mN. The instrument employs an electrostatic drive mechanism and features excellent temperature drift control, rapid data acquisition, and high measurement accuracy, fully satisfying the experimental requirements. The classic Oliver-Pharr method was employed to determine the hardness and elastic modulus of the specimens. As this method does not account for the load-holding stage, a holding period of 10 s was applied to ensure stable unloading data [18; 24]. The specific calculation process is as follows:  
\[F_{\text{n}}=m(h-h_{\text{p}})^{n} \tag{1}\]  
where \(F_{n}\) is the load, mN; constants \(m\) and \(n\) are determined by fitting the unloading curve. Figure 3: Indentation profile and load–depth curve. According to the definition of contact stiffness, it can be obtained by deriving the power function fitting formula of the unloading curve at the peak load or peak depth position [18]. \[S=\left(\frac{\mathrm{d}F_{\mathrm{n}}}{\mathrm{d}h}\right)_{h=h_{\mathrm{max}}}=mn \left(h_{\mathrm{m}}-h_{\mathrm{P}}\right)^{n-1} \tag{2}\]  
As shown in Figure 3b, the contact depth can be calculated according to Formula (3). \[h_{\mathrm{c}}=h_{\mathrm{m}}-\eta\left(h_{\mathrm{m}}-h_{\mathrm{r}}\right) \tag{3}\]  
where \(\eta\) is a constant related to the geometry of the indenter. For the Berkovich indenter, \(\eta=0.7268\). For the Berkovich indenter used in this paper, the relationship between the contact projection area (\(A_{\mathrm{c}}\)) and the contact depth can be expressed by Formula (4). \[A_{\mathrm{c}}=24.5h_{\mathrm{c}}{}^{2} \tag{4}\]  
Nanoindentation hardness (\(H_{\mathrm{IT}}\)) and reduced modulus (\(E_{\mathrm{IT}}\)) are calculated by Formulas (5) and (6):  
\[H_{\mathrm{IT}}=F_{\mathrm{m}}/A_{c} \tag{5}\]  
\[E_{\mathrm{r}}=S\sqrt{\pi}/(2\alpha\sqrt{A_{c}}) \tag{6}\]  
where \(\alpha\) is a constant related to the shape of the indenter. For the Berkovich indenter, \(\alpha=1.034\). From the above Formulas (1)-(6), it can be seen that if the load is not maintained after the peak value to eliminate the creep effect, the deformation of the sample will continue to increase during the unloading stage, causing \(h_{\mathrm{p}}\) to be close to or equal to \(h_{\mathrm{m}}\), and the calculated parameters will seriously deviate from the actual values. When the non-rigidity of the indenter is neglected, the elastic modulus of the coal sample can be represented by \(E_{\mathrm{r}}\).