## 2 Simple Model for Structural Elasticity of Polymer Layer  
A simple model of a cylinder-shaped polymer layer is described here to achieve a reliable estimation of the mechanical properties for GFPB structures. The mathematical approach for the cylinder-shaped polymer layer follows a thick-walled cylinder formulation based on the theory of elasticity and considers a differential element on a polymer layer. The thick-walled cylinder formulation is generally used to analyze the stress and strain relation of pressure vessels [22; 23]. Figure 2 presents the coordinate system of a thick-wall cylinder model and normal stresses on a differential element. Cylindrical coordinate system (\(r\), \(\theta\), \(Z\)) is introduced for the cylinder, where \(r\), \(\theta\), and \(Z\) are the radial, circumferential, and axial coordinates, respectively. In this study, the polymer layer of a GFPB is modeled as a thick-wall cylinder having an inner radius (\(r_{in}\)), outer radius (\(r_{o}\)), and length (\(L\)). The cylinder is loaded by an internal fluid-film pressure (\(p_{in}\)) on the inner surface of the cylinder. The outer surface is in contact with the bearing housing. Therefore, a uniform internal film pressure is assumed along the \(\theta\) and \(Z\) coordinates. A differential element is defined by an angular increment (\(d\theta\)), a radial increment (\(dr\)), and an axial increment (\(dZ\)) at the radial position (\(r\)). The original shape is element \(abcd\) and deformed shape by the internal film pressure is element \(a^{\prime}b^{\prime}c^{\prime}d^{\prime}\). The radial deformation of an element (\(u\)) is given by \(aa^{\prime}\) and \(bb^{\prime}\). By circular symmetry, considering the equilibrium of the differential element in the radial direction leads to an equilibrium equation, as follows:  
\[(\sigma_{r}+d\sigma_{r})(r+dr)d\theta=\sigma_{r}rd\theta+\sigma_{\theta}d \theta dr \tag{1}\]  
If the axial length of the polymer layer is considerably longer than the thickness, the axial strain of the element becomes relatively small so that the element could be regarded under a plane strain condition. Moreover, the axial strain of the polymer layer is significantly constrained because of the frictional force between the polymer layer and other sub-structure, thus leading to \(\varepsilon_{Z}=0\). From [PERSON]'s law, the stress and strain equations for the condition of plane strain in the cylindrical coordinates are given by [22]  
\[\left\{ \begin{array}{c}\sigma_{r}\\ \sigma_{\theta}\\ \sigma_{Z}\end{array} \right\}=\frac{E_{p}}{\big{(}1+\
u_{p}\big{)}\big{(}1-2 \
u_{p}\big{)}}\left[ \begin{array}{ccc}1-\
u_{p}&\
u_{p}&\
u_{p}\\ \
u_{p}&1-\
u_{p}&\
u_{p}\\ \
u_{p}&\
u_{p}&1-\
u_{p}\end{array} \right]\left\{ \begin{array}{c}\varepsilon _{r}\\ \varepsilon_{\theta}\\ 0\end{array} \right\} \tag{2}\]  
where \(E_{p}\) is the [PERSON]'s modulus of the polymer material, \(\
u_{p}\) is the [PERSON]'s ratio of the polymer material, and the radial and circumferential strains are expressed as \(\varepsilon_{r}=du/dr\) and \(\varepsilon_{\theta}=u/r\), respectively. Boundary conditions for the element are applied by considering the actual operating conditions of the polymer layer in the GFPB with the radial stress \(\sigma_{r}\) and radial deformation \(u\); they are  
\[\sigma_{r}(r_{in})=-p_{in},\ u(r_{o})=0 \tag{3}\]  
Figure 2: Cylindrical coordinate system of a cylinder-shaped polymer model loaded by a uniform internal film pressure and normal stresses on a differential element. which indicates that the inner surface of the cylinder is loaded by the internal film pressure, and the radial deformation at the outer surface of the cylinder is constrained by the bearing housing.