2.3.1. Analytical Description

The natural frequencies and mode shapes of membranes are a function of the structural properties and boundary conditions. Let us consider a circular membrane with a uniform thickness *t* and firmly clamped at its periphery. If the displacement of the membrane along the *z*-axis is small compared to the diameter, the stress state is isotropic and the e ffect of the fluid (air) is negligible [19]. Therefore, in small deflection case, an ideal membrane is described by assuming that the residual stress is higher than the bending rigidity. Thus, the membrane can support only tensile loads [33]. The mode shapes are described by two spatial coordinates and identified by the nodal line (m, n). Here, m and n are the index relative to the modal diameter lines and the number of nodal circle lines, respectively [33]. The natural frequencies of vibration for a circular membrane can be described as:

$$f\_{(m,n)} = \frac{\alpha\_{mn}}{2\pi a} \frac{\sqrt{\sigma\_0}}{\rho},\tag{2}$$

where *a* is the membrane radius, σ0 is the residual stress, ρ is the density of the 4H-SiC thin-film and α*mn* values are derived from the roots of the first order Bessel functions. Zeros of the Bessel functions can be computed and tabulated [17].

#### 2.3.2. Finite Element Method Approach

A behavioral model of the membrane, based on the finite element method (FEM), was implemented to theoretically determine the mechanical properties of the fabricated film. Therefore, FEM calculations were used as virtual experiments to determine the natural frequencies and mode shapes of the membrane. The numerical model was built with the commercial COMSOL Multiphysics software package using eigenfrequency and solids mechanics modules. In the simulation procedure, a three-dimensional pre-stressed circular membrane was created. Perfectly clamped boundary conditions were used corresponding to a null mechanical displacement at the outer edge, in *x, y,* and *z* directions. The mesh settings were defined using the physics-controlled mesh option proposed by COMSOL Multiphysics, giving hence the following parameters: Tetrahedral shape elements and 8767 meshing nodes. These parameters were validated by comparing theoretical resonance frequency values of circular membranes to values obtained from FEM, a difference lower than 1% was obtained.

The residual stress and Young's modulus values extracted from the bulge test measurements were used as input parameters.
