## 3 Numerical Results
### Comparisons of \(w(a/2,b/2)\), \(\sigma_{x}\) versus \(T\) and \(R_{n}\)  
The comparisons of \(w(a/2,b/2)\) versus \(T\) for \(c_{1}=0.925925/\mathrm{mm^{2}}\) and \(c_{1}=0.0/\mathrm{mm^{2}}\), respectively, under \(\theta_{L}~{}=~{}60^{\circ}\) are shown in Figure 4 at \(t=0.1\) s, \(M_{\infty}~{}=~{}2\), \(k_{c}c(t)=0\), \(L/h^{*}=5\), \(L/R~{}=~{}1\), and \(\overline{T}_{1}\) = 100 K. Figure 4a shows the curves of \(w(a/2,b/2)\) vs. \(T\) and \(R_{n}\) = 1 case; the maximum value of \(w(a/2,b/2)\) is 32.773151 mm for \(c_{1}=0.925925/\mathrm{mm^{2}}\), which occurs at \(T\) = 1000 K. The \(w(a/2,b/2)\) values are firstly decreasing versus \(T\) from \(T\) = 100 K to \(T\) = 600 K, then increasing versus \(T\) from \(T\) = 600 K to \(T\) = 1000 K for \(c_{1}=0.925925/\mathrm{mm^{2}}\). Thus, amplitude \(w(a/2,b/2)\) of the \(R_{n}\) = 1 under \(\theta_{L}~{}=~{}60^{\circ}\) cannot withstand the higher temperature \(T\) = 1000 K of environment for \(c_{1}=0.925925/\mathrm{mm^{2}}\). The values of \(w(a/2,b/2)\) versus \(T\) are located in the constant line for \(c_{1}=0.0/\mathrm{mm^{2}}\). Figure 4b shows the curves of \(w(a/2,b/2)\) vs. \(T\) and \(R_{n}\) = 2 case, the maximum value of \(w(w(a/2,b/2)\) is 1377.64099 mm occurs at \(T\) = 1000 K for \(R_{n}\) = 2. The \(w(a/2,b/2)\) values are slightly decreasing versus \(T\) from \(T\) = 100 K to \(T\) = 600 K then increasing versus \(T\) from \(T\) = 600 K to \(T\) = 1000 K for \(R_{n}\) = 2. Thus, amplitude \(w(a/2,b/2)\) of the \(R_{n}\) = 2 under \(\theta_{L}~{}=~{}60^{\circ}\) cannot withstand the higher temperature \(T\) = 1000 K of environment for \(c_{1}=0.925925/\mathrm{mm^{2}}\). The values of \(w(a/2,b/2)\) versus \(T\) are located in the constant line for \(c_{1}=0.0/\mathrm{mm^{2}}\). Figure 4c shows the curves of \(w(a/2,b/2)\) vs. \(T\) and \(R_{n}\) = 10 case; the maximum value of \(w(a/2,b/2)\) is 14.614171 mm, which occurs at \(T\) = 600 K for \(R_{n}\) = 10. The \(w(a/2,b/2)\) values are firstly increasing versus \(T\) from \(T\) = 600 K to \(T\) = 1000 K for \(c_{1}=0.925925/\mathrm{mm^{2}}\). Thus, the amplitude \(w(a/2,b/2)\) of the \(R_{n}\) = 10 under \(\theta_{L}~{}=~{}60^{\circ}\) can withstand a higher temperature \(T\) = 1000 K of environment for \(c_{1}=0.925925/\mathrm{mm^{2}}\). The values of \(w(a/2,b/2)\) versus \(T\) are located in the constant line for \(c_{1}=0.0/\mathrm{mm^{2}}\). The comparisons of \(\sigma_{x}\) versus \(T\) for \(c_{1}=0.925925/\mathrm{mm^{2}}\) and \(c_{1}=0.0/\mathrm{mm^{2}}\), respectively, are shown in Figure 5 at \(t=0.1\) s, \(k_{c}c(t)=0\), \(L/R~{}=~{}1\), and \(\overline{T}_{1}\) = 100 K. Figure 5 shows the normal stresses \(\sigma_{x}\) (GPa) vs. \(T\) at the center position of outer surface z = 0.5 \(h^{*}\) for \(M_{\infty}~{}=~{}2\) with \(L/h^{*}\) = 5 and \(\theta_{L}~{}=~{}60^{\circ}\). Figure 5a shows the curves of \(\sigma_{x}\) vs. \(T\) and \(R_{n}\) = 1 case; the maximum absolute value of \(\sigma_{x}\) is \(-2.8882\times 10^{-3}\) GPa, which occurs at \(T\) = 600 K for r \(R_{n}\) = 1. The \(\sigma_{x}\) absolute values are firstly increasing versus \(T\) from \(T\) = 100 K to \(T\) = 600 K, then decreasing versus \(T\) from \(T\) = 600 K to \(T\) = 1000 K for \(c_{1}=0.925925/\mathrm{mm^{2}}\). Thus, stress \(\sigma_{x}\) of the \(R_{n}\)= 1 under \(\theta_{L}~{}=~{}60^{\circ}\) can withstand the higher temperature \(T\) = 1000 K of environment for \(c_{1}=0.925925/\mathrm{mm^{2}}\). The absolute values of \(\sigma_{x}\) versus \(T\) are located in the straight line for \(c_{1}=0.0/\mathrm{mm^{2}}\). Figure 5b shows the curves of \(\sigma_{x}\) vs. \(T\) and \(R_{n}\) = 2 case; the maximum valueof \(\sigma_{x}\) is 1.4213 \(\times\) 10\({}^{-1}\) GPa, which occurs at \(T\) = 1000 K for \(R_{n}\) = 2. The absolute values of \(\sigma_{x}\) are increasing from \(T\) = 100 K to \(T\) = 1000 K for \(R_{n}\) = 2.