## 4 Stretchable MSP Antenna-Based Strain Sensor  
In general, antennas that perform in a single frequency band become a barrier of careful surveillance because of the harmful radiation of wireless components and the interruption of data signal transmission. Hence, the efficacy of the single-band antenna cannot to be ensured. For this reason, a dual-band antenna should be utilized to improve the reliability of the stretchable strain-based MSP antenna. In this example, a dual-band antenna functioning at 2.5 GHz and 5 GHz, usually exploited for wireless fidelity, was regarded. On the basis of composition of the MSP antenna, as shown in Figure 2, vital DVs and the material properties are listed in Table 1. Here, \(\mu\) and \(COV\) stand for mean and coefficient of variation, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Substrate** & **Patch** & **Feed Line** & **Ground** & **Source** \\  & (\(\mu\), \(COV\)) & & & & (\(\mu\), \(COV\)) \\ \hline Thickness (mm) & 0.97, 0.05 & 0.03 & 0.03 & 0.05 & \\ Width (mm) & 70 & 47 & 2.5 & 70 & 0.09, 0.1 \\ Length (mm) & 70 & 40 & 32 & 70 & 2.5 \\ Permittivity & 2.2 & 1 & 1 & 1 & 1 \\ Young’s Moduli (MPa) & 1.32 & 0.0124 & 0.0124 & 1.32 & \\ Conductivity (S/cm) & 0 & \(1.51\times 10^{4}\) & \(1.51\times 10^{4}\) & \(1.51\times 10^{4}\) & 0 \\ Dielectric Loss Tangent & 0.0009 & 0.01 & 0.01 & 0.01 & 0 \\ Magnetic Loss Tangent & 0 & 0.001 & 0.001 & 0.001 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric and material properties of the stretchable micro-strip patch (MSP) antenna. To investigate the behavior of stretchable antennas, a multi-physics analysis that includes both the strain estimation under mechanical deformation of the antenna and the R\({}_{\ell}\) estimation by electrical analysis is required. In the previous research [8], such a multidisciplinary analysis was performed for a stretchable antenna geometry and it was shown that the fluctuating thickness of the substrate resulted in a high return loss. This result was because the uncertainties of the substrate thickness led to the escalation of forged feed radiation and surface waves, which limit the range of the bandwidth. Therefore, in this proposed framework, a design optimization process was also utilized to find the optimal contour of the antenna patch to maximize antenna efficiency by reducing the return loss. The existing geometry in the work of [8] was used as the starting geometry in the optimization process; therefore, first, the dimension reduction and prediction using Steps 1 and 2 of the proposed framework were used for the starting geometry. In the analysis of the antenna, first, a displacement that results in a tensile strain along the width direction, which is denoted as the \(y\) direction on Cartesian coordinates, was applied on the antenna substrate, as demonstrated in Figure 4a. In the structural analysis to evaluate the deformation of the antenna under this tensile strain, a linear static finite element analysis (FEA) was employed in this study using the commercial software ANSYS(r). As the geometry and the applied displacements are also symmetric, while the material properties are assumed to be isotropic, symmetry boundary conditions were utilized, as shown in Figure 4b. In this study, it was assumed that the maximum applied tensile displacement on the stretchable antenna is 12 mm. The deformation of the stretchable antennas might be larger, but for demonstration purposes, 12 mm was considered in this study as the bound of the displacement of the stretchable antenna. The application can easily be extended for larger displacements in a future study. The substrate of the antenna was discretized into 202 rectangular regions that represent the fluctuating substrate thickness, as shown in Figure 4c by white bounds. The outer geometry of the patch shown by gray color was discretized into 57 points, as depicted by white nodes in Figure 4c.