## 5 Discussions and Numerical Results
### The Influence of visco-Pasternak Foundation Characteristics on the Results  
As far as the author is aware, there has been no research that has been conducted examining the surface effects of microbeam vibration analysis when it is under the influence of a visco-Pasternak elastic basis and considering the effect of temperature change based on the generalized view of thermoelasticity. Therefore, in this section, the interest is in investigating how microbeams, which are grown under the influence of a visco-Pasternak elastic foundation, behave when they are exposed to laser pulses. These changes will be explained in Table 1.  
Table 1 shows the variation of the different dimensionless field variables concerning the dimensionless Winkler modulus \(K_{1}\) for different values of the shear stiffness of the elastic substrate (Pasternak's modulus) \(K_{2}\), as well as the effect of mechanical relaxation time due to viscosity (viscosity) or the so-called damping coefficient \(K_{0}\) to show the influence of the Winkler parameters and Pasternak on the behavior of a thermoplastic substrate. The Moore-Gibson-Thompson (MGITE) generalized thermoelastic model was considered to investigate and obtain the numerical results of the physical distributions. For comparison purposes, we will also take the values of other parameters as constants, such as \(K^{*}=200\), \(\omega=5\), \(t_{p}=0.01\), \(\sigma_{0}=1\), \(\tau_{0}=0.05\). At single values on the phase of the beam axis at point \(x=0.2\), numerical calculations are performed with different values of the foundation parameters. As a special case of the current work, in setting the foundation parameters \(K_{0}=K_{1}=K_{2}=0\), the present results lead to the calculations of the problem of a thermoelastic beam that does not contain a flexible visco-Pasternak base. According to the numerical values in Table 1, the obvious influences of the Winkler and Pasternak parameters \(K_{1}\) and \(K_{2}\) on the values of deflection \(w\) are shown. Changes in the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(K_{0}\)} & \multirow{2}{*}{\(K_{1}\)} & \multirow{2}{*}{\(K_{2}\)} & \multicolumn{6}{c}{**Studied Field Variables**} \\ \cline{3-8}  & & & \(w\) & \(\theta\) & \(u\) & \(\sigma_{xx}\) & \(M\) \\ \hline \multirow{4}{*}{0.3} & 0.0 & 0.0 & \(-0.430283\) & \(0.0508297\) & \(-0.0528164\) & \(0.0106182\) & \(0.0796363\) \\  & 0.05 & 0.1 & \(-0.387255\) & \(0.0451819\) & \(-0.0475348\) & \(0.0112818\) & \(0.0836181\) \\  & 0.1 & 0.2 & \(-0.365741\) & \(0.0423581\) & \(-0.044894\) & \(0.0119454\) & \(0.0875999\) \\  & 0.2 & 0.3 & \(-0.344227\) & \(0.0395342\) & \(-0.0422531\) & \(0.0126091\) & \(0.0915817\) \\ \hline \multirow{4}{*}{0.5} & 0.0 & 0.0 & \(-0.473809\) & \(0.0548417\) & \(-0.0535052\) & \(0.0096383\) & \(0.0578298\) \\  & 0.05 & 0.1 & \(-0.419509\) & \(0.0482245\) & \(-0.0509786\) & \(0.0124999\) & \(0.0749994\) \\  & 0.1 & 0.2 & \(-0.393998\) & \(0.0451462\) & \(-0.0491843\) & \(0.0135331\) & \(0.0811986\) \\  & 0.2 & 0.3 & \(-0.374298\) & \(0.0426381\) & \(-0.0467251\) & \(0.0142097\) & \(0.0852585\) \\ \hline \multirow{4}{*}{0.7} & 0.0 & 0.0 & \(-0.521105\) & \(0.0598587\) & \(-0.0546791\) & \(0.00650655\) & \(0.0371803\) \\  & 0.05 & 0.1 & \(-0.468990\) & \(0.0532077\) & \(-0.0514308\) & \(0.00681639\) & \(0.0408983\) \\  & 0.1 & 0.2 & \(-0.416884\) & \(0.0465567\) & \(-0.0487240\) & \(0.00743606\) & \(0.0446164\) \\  & 0.2 & 0.3 & \(-0.390828\) & \(0.0432313\) & \(-0.0460171\) & \(0.00805573\) & \(0.0464754\) \\ \hline \multirow{4}{*}{1.0} & 0.0 & 0.0 & \(-0.630129\) & \(0.0751540\) & \(-0.0555853\) & \(0.000403841\) & \(0.0025442\) \\  & 0.05 & 0.1 & \(-0.573774\) & \(0.0681880\) & \(-0.0539835\) & \(0.00357501\) & \(0.0214501\) \\ \cline{1-1}  & 0.1 & 0.2 & \(-0.516397\) & \(0.0606115\) & \(-0.0485851\) & \(0.00357501\) & \(0.0224715\) \\ \cline{1-1}  & 0.2 & 0.3 & \(-0.487708\) & \(0.0530351\) & \(-0.0458859\) & \(0.00374525\) & \(0.0234929\) \\ \hline \end{tabular}
\end{table}