## 3 Results
### Indentation Strain Rate (\(\dot{\epsilon}_{i}\))  
The ISE under different conditions must be clarified before the effects of strain rate and aging time on the indentation mechanical properties of PMMA are analyzed, and the characteristic depth marked as \(d_{sp}\) is defined as the critical depth that the mechanical properties of PMMA no longer depend on the depth. Moreover, it can be used to represent the mechanical properties of the bulk material. Moreover, no explicit evidence shows that aging has no effect the PMMA ISE. This finding also indicates that the statement \(\text{dsp}=1000\) nm in our previous paper [1] cannot be used directly in this paper. Therefore, the effects of hygothermal aging on the ISE of PMMA must be understood. Based on the investigation of the ISE of unaged PMMA in our previous paper [1], shear transformation-mediated plasticity is utilized to explain the plastic deformation of amorphous materials [34]. Additionally, the Arrhenius equation [35; 36] is applied to correlate the indentation \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\dot{\text{p}}/P\) (s\({}^{-1}\))} & \multicolumn{4}{c}{**Aging Time (h)**} \\ \cline{2-5}  & **240** & **312** & **600** & **720** \\ \hline
0.05 & 0.016 & 0.016 & 0.015 & 0.016 \\
0.1 & 0.00627 & 0.00698 & 0.00698 & 0.00648 \\
0.2 & 0.00204 & 0.00195 & 0.00258 & 0.00279 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitted parameter A of the aged PMMA under different loading rates. Figure 2: Force-time curves at the loading stage of the aged PMMA specimens under nanoindentation at aging times of 240 (**a**), 312 (**b**), 600 (**c**), and 720 (**d**) h.  
effective shear strain rate (\(\dot{\gamma}\)), shear yield stress (\(\tau_{s}\)), activation energy (\(\Delta F_{0}\)), shear strain of Shear transformation zone (STz) (\(\gamma^{T}=0.04\)), and activation volume (\(\Omega\)):  
\[\dot{\gamma}=\varnothing_{ss}\gamma^{T}v_{G}\exp\biggl{(}-\frac{\Delta F_{0}}{k _{B}T}\biggr{)}\sinh\biggl{(}\frac{\gamma^{T}\Omega\tau_{s}}{2k_{B}T}\biggr{)} \tag{2}\]  
where \(k_{B}\), T, and \(\varnothing_{ss}\) are the Boltzmann constant, absolute temperature, and a constant set to 0.5 for plastic state, respectively. \(v_{G}=10^{10}s^{-1}\) is the attempt frequency. The shear transformation-mediated plasticity is introduced in detail in many existing articles [1; 17; 25]. Thus, reintroduction is not needed. The \(\dot{\gamma}\) in the Arrhenius equation is the shear strain rate, rather than \(\dot{P}/P\) or \(\dot{\varepsilon}_{i}=\frac{dh}{hdt}\). \(\dot{\varepsilon}_{i}\) can be controlled by setting different \(\dot{P}/P\) values. In the indentation tests, the indentation effective \(\dot{\gamma}\) can be calculated [17; 25]:  
\[\dot{\gamma}=\sqrt{3}C\dot{\varepsilon}_{i} \tag{3}\]  
where C is 0.09 [37; 38; 39]. The \(\dot{\gamma}\) of the aged PMMA can be obtained during indentation (Figure 3) and decreases with increasing indentation depth until a convergence value is reached. In addition, the \(\dot{\gamma}\) values of 0.004, 0.008, and 0.016 s\({}^{-1}\) correspond to the \(\dot{P}/P\) values of 0.05, 0.1, and 0.2 s\({}^{-1}\), respectively, which is consistent with the result of the unaged PMMA. Aging time can also be considered to have almost no effect on it by the comparison of indentation results between unaged and aged PMMA because the indentation conditions are the same. For convenience, the effective indentation \(\dot{\gamma}\) is simplified into strain rate (SR) in the following text. The continuous contact stiffness can be obtained by using CSM (Figure 4). The contact stiffness S almost increases linearly with the increasing depth h. [PERSON] et al. gave the theoretical explanation i.e., \(S\propto h\)[40] based on the dimension analysis. In addition, the slope of S-h can be regarded as the  
Figure 3: Variation in the shear strain rate vs. the depth of the aged PMMA with different aging times: (**a**) 240 h, (**b**) 312 h, (**c**) 600 h and (**d**) 720 h.  
elastic property of the materials to a certain degree.