## 5 Discussions and Numerical Results
### The Influence of visco-Pasternak Foundation Characteristics on the Results

Table 1: Dimensionless physical field variables with Winkler basis (\(K_{1}\)), Pasternak foundation (\(K_{2}\)), and viscous damping coefficient (\(K_{0}\)). visco-Pasternak foundations \(K_{1}\) and \(K_{2}\) have a significant effect on both bending moment \(M\) and deflection \(w\). Increasing the values of the fixed base factors reduces the absolute values of the deviation \(w\) for each analyzed case. This is due to the fact that the system becomes more robust as the microbeams become more rigid. This occurrence and behavior were previously discussed in the paper by [PERSON] and [PERSON] [45], and by comparison, it was found that there was great agreement in the results. Looking at Table 1, it can be noticed that raising the values of the dimensionless Winkler foundation (\(K_{1}\)) and Pasternak foundation (\(K_{2}\)) moduli leads to an increase in the amount of both dimensionless heat stress \(\sigma_{xx}\) and the bending moment M in the microbeam. The reason for this occurring is because a rise in the values of \(K_{1}\) and \(K_{2}\) leads to a rise in structural rigidity and, as a consequence, the stability of the microbeam. On the other hand, the displacement \(u\) and the temperature change \(\theta\) are less influenced by the variations in the coefficients of the viscous basis of Pasternak. This is because the absolute value of the displacement \(u\) and the amount of temperature change are very sensitive to changes in the foundation as well as the viscous basis. The absolute value of the displacement \(u\) and the amount of temperature change decrease by raising the values of the basis factors \(K_{1}\) and \(K_{2}\). We also see from the table that the quantities of the various distributions of the physical areas of research decrease when the Winkler and Pasternak factors \(K_{1}\) and \(K_{2}\) remain unchanged, and the viscous damping coefficient \(K_{0}\) undergoes an increase. Additionally, when there is a change in the viscous damping coefficient \(K_{0}\), there is a corresponding minor change in temperature as well as displacement. The absolute value of the deflection \(w\) increases in proportion to the decrease in the viscosity coefficient \(K_{0}\). The non-dimensional stress \(\sigma_{xx}\) and the bending moment \(M\) both grow as the viscous damping coefficient \(K_{0}\) and other foundations are increased. On the other hand, and by checking with the results that are in Table 1, it becomes clear that the quantities of the different distributions of the studied physical fields change dramatically when the Winkler and Pasternak coefficients \(K_{1}\) and \(K_{2}\) remain unchanged, and the viscous damping coefficient \(K_{0}\) is subject to different values. In addition, when there is a rise in the value of the viscous damping coefficient \(K_{0}\), there is an increase in the amount of both temperature change \(\theta\), displacement \(u\), and deflection \(w\). When the viscosity coefficient \(K_{0}\) is lowered, the absolute values of the heat stress \(\sigma_{xx}\) and bending moment \(M\) rise to a higher level. The amount of temperature change grows when there are higher values of the viscous damping coefficient \(K_{0}\) and with the stability of other foundations. For this reason, this observation must be considered when manufacturing and designing small-sized devices. It has been investigated that the reduction of the dimensionless thermophysical domains can be achieved by increasing the modulus of the viscoelastic structural damping of the beams. This is due to the fact that an increase in the structural damping factor causes the structure to be less rigid. The effect of the viscoelastic structural damping coefficient is amplified when the Winkler and Pasternak coefficients \(K_{1}\) and \(K_{2}\) are fixed.