## 3 Temperature-Dependent Time-Hardening Creep Model  
According to the time-hardening theory, the decrease of the creep rate during creep shows that the main factor of material hardening is time, which has nothing to do with creep deformation. When the temperature is constant, there is a certain relationship between stress, \(\sigma\), the creep rate, \(\varepsilon_{\sigma}\), and time, \(t\)[25].  
\[\Phi\big{(}\dot{\varepsilon}_{cr},\sigma,t\big{)}=0 \tag{1}\]  
\[\left\{\begin{array}{l}\varepsilon=f_{1}(\sigma)\cdot f_{2}(t)\\ f_{1}(\sigma)=\sigma^{n}\\ f_{2}(t)=\frac{A}{m+1}t^{m+1}\end{array}\right. \tag{2}\]  
\[\varepsilon_{cr}=\frac{A}{m+1}\cdot\sigma^{n}\cdot t^{m+1} \tag{3}\]  
where \(f_{1}(\sigma)\) is the stress function; n is the exponential parameter of the stress power function; \(f_{2}(t)\) is the time response function; and A and m are the coefficients and exponential parameters of the time power function, respectively.  
To consider the environmental temperature effect, a temperature-dependent time-hardening model has been proposed by the author as follows,  
\[\dot{\varepsilon}_{\rm{cr}}=A\sigma^{n}t^{m} \tag{4}\]  
\[\varepsilon_{cr}=A\sigma^{n}t^{m+1}+C \tag{5}\]  
\[\dot{\varepsilon}_{\rm{cr}}=\Big{(}a_{1}T^{3}+b_{1}T^{2}+c_{1}T+k_{1}\sigma+d _{1}\Big{)}\sigma^{n}t^{(a_{2}T^{2}+b_{2}T+k_{2}\sigma+c_{2})} \tag{6}\]  
where \(\dot{\varepsilon}_{\rm{cr}}\) is creep strain rate; \(\sigma\) is the uniaxial stress or equivalence stress in MPa; \(t\) is the total time in second; \(T\) is the temperature in \({}^{\circ}\)C; and \(a_{1}\), \(b_{1}\), \(c_{1}\), \(k_{1}\), \(d_{1}\), \(n\), \(a_{2}\), \(b_{2}\), \(c_{2}\), and \(k_{2}\) are the material parameters.  
Similar to Formula (5), Formula (7) is obtained by integrating the creep strain of Formula (6), where the integral constant \(C\) depends on the elastic strain before loading. The definitions of \(A_{1}\) and \(m_{1}\) are shown in Equation (7).  
\[\left\{\begin{array}{l}\varepsilon_{cr}=A_{1}\sigma^{n}t^{m_{1}+1}+C\\ A_{1}=a_{1}T^{3}+b_{1}T^{2}+c_{1}T+k_{1}\sigma+d_{1}\\ m_{1}=a_{2}T^{2}+b_{2}T+k_{2}\sigma+c_{2}\end{array}\right. \tag{7}\]  
Figure 2: Basic structural parameters of frustum observation window.  
As depicted in Figure 3, the strain exhibits a time-dependent variation. The experimental data distinctly illustrate that strain increases over time, and this effect is more pronounced at higher temperatures. During the loading period, the strain shows a linear upward trend. Under sustained loading, the strain experiences an initial rapid increase followed by a gradual reduction in the strain rate. This behavior corresponds to the first and second stages of creep. With the increase of temperature, the time of the first creep stage becomes longer, and the creep rate also increases.  
Utilizing the compression creep test data for the PMMA material at different temperatures, the nonlinear least square method is employed to fit the multiple parameters required for the improved time-hardening creep model. First, the individual curve parameters at each temperature are fitted, and then the parameters are fitted. Assuming that \(n\) is a fixed value, \(A_{1}\) and \(m_{1}\) will change, and the parameters are fitted by MATLAB. The parameters of the improved model are obtained by fitting the parameters of \(A_{1}\) and \(m_{1}\), and the results are verified. The fitting results are summarized in Table 2. As shown in Figure 3, the goodness of fit for each curve consistently reaches or exceeds 0.95, with an average value of 0.98292. This shows that the improved time-hardening model is suitable for various temperature conditions at a single stress level.