## Materials and Methods  
### Hyperelastic Sealing Material  
In this study, the production of a UMS using FKM, one type of rubber class designated in American Society for Testing and Materials (ASTM) D1418, as a hyperelastic material was considered. FKM is a sealing material that is widely used in the oil and gas industries [17]. It consists of 80% or more of fluoroelastomer, which is a synthetic rubber with very high thermal and chemical stability. Unlike [PERSON]'s law, which is used for general linear elastic behavior, a hyperelastic material shows nonlinear behavior in its stress-strain relationship. Therefore, the stress-strain relationship of a hyperelastic material is defined using the strain energy density function, \(W\). By differentiating the strain energy density function with respect to strain, a corresponding stress component can be obtained. The strain energy density function can be expressed in various ways: statistics-based, invariant-based, or stretch-based mechanics. Various hyperelastic material models are introduced in the literature [18]. In this study, the invariant-based hyperelastic material model based on continuum mechanics was applied. The general form of an isotropic hyperelastic material model can be expressed as follows [18; 19]:  
\[W=\sum_{i,\ j=0}^{\infty}C_{ij}(I_{1}-3)^{i}(I_{2}-3)^{j}, \tag{1}\]  
where \(I_{1}\) and \(I_{2}\) are the first and second invariants of the stretch tensor, respectively, and \(C_{ij}\) is the material constant. Although hyperelastic materials are theoretically incompressible, they actually show a slightly compressible behavior in reality. To implement this behavior into the analysis, the strain energy density was assumed to be composed of two different strain energy densities [18; 20; 21; 22]:  
\[W=W_{distortion}+W_{dilatation}, \tag{2}\]  
where \(W_{distortion}\) is the distortional (or shear) strain energy density, representing the constant volume in shape, and \(W_{dilatation}\) is the dilatational (or hydrostatic) strain energy density, indicating the change in volume. It is noted that the distortional energy density, \(W_{distortion}\), is related to the tangential component of the stress vectors, showing a linear correlation with the first invariant \(I_{1}\). The volumetric dilatation of isotropic materials can be affected only by the normal component of the stress vectors. Thus, the volumetric strain energy density, \(W_{dilatation}\), is related to the compressibility. Nonlinear elastomeric materials are generally known to have very high bulk moduli compared to shear moduli. This implies that their volume changes very little, even if the material is elongated or shrunk in the event of shear deformation. In other words, the material shows almost-incompressible behavior under volumetric compression. If an elastomeric material is assumed theoretically to be completely incompressible (bulk modulus \(k=\infty\)), with complex shape or load conditions, it can be difficult to obtain calculated results because of element locking when predicting the material behavior through numerical analysis [17; 23]. However, this ideal condition is not found in practice, and even if the material is theoretically incompressible, it shows slightly compressible behavior. Therefore, the compressibility of the UMS was defined as nearly incompressible, and the Poisson's ratio was set at 0.499. The nonlinear elastomeric behavior of FKM was described using the neo-Hookean model, which is one of the simplest hyperelastic models. For numerical analysis, in the case of the neo-Hookean constitutive equations, the [PERSON]'s ratio of 0.499 can be applied to describe the nearly incompressible behavior of an elastomeric material [1; 17]. In addition, this model shows very good conformity with experimental values up to approximately 40% strain under uniaxial tension [1; 17].