## 2 Theoretical Background
### Curvature Ratio and Conformity Effects  
The seat curvature radius \(R_{s}\) does not directly participate in initial contact but significantly influences the deformation pattern and stress distribution of the plunger as pressure increases. The geometric constraint effect and conformity relationships (Equations (9)-(12)) discussed in this section build upon classical contact mechanics principles, with similar formulations found in the established literature [20]. \(R_{s}\) affects stress distribution through the \"geometric constraint effect\" in structural mechanics. This effect can be quantified by the following equation:  
\[\eta_{c}=\frac{R_{s}-R_{p}}{R_{s}+R_{p}} \tag{9}\]  
where \(\eta_{c}\) is the contact constraint coefficient, which determines the contact pressure distribution and stress concentration phenomena. Figure 4: Characteristic pressure distribution in Hertzian contact theory. In the conformity theory of contact mechanics, the ratio of \(R_{p}\) to \(R_{s}\) acts as a key factor determining the conformity of contact. The conformity coefficient (\(\gamma\)) is defined as follows:  
\[\gamma=\frac{R_{p}}{R_{s}} \tag{10}\]  
This conformity coefficient serves as a dimensionless parameter representing the geometric characteristics of the contact geometry, influencing the distribution of contact pressure and maximum equivalent stress. Contact area A varies with load F as follows:  
\[A=\pi a^{2}=\pi\bigg{(}\frac{3 FR_{eq}}{4E^{*}}\bigg{)}^{\frac{2}{3}} \tag{11}\]  
Equation (11) presents the explicit formula for determining the circular contact area, A, which is derived from the contact radius \(a\) (as defined in Equation (4)) and incorporates the material properties \(E^{*}\) and the equivalent radius of curvature \(R_{eq}\). This formulation quantitatively expresses the general Hertzian principle, proportionally indicated by Equation (6), wherein the contact area exhibits a nonlinear expansion with increasing load. Such nonlinear expansion underpins the \"self-adjustment\" mechanism characteristic of the concave contact geometry. Consequently, the contact characteristics, which are influenced by this area expansion, vary depending on whether \(\gamma<1\) (\(R_{p}<R_{s}\)) and when \(\gamma>1\) (\(R_{p}>R_{s}\)). A small R_s value can restrict lateral deformation of the plunger material, potentially causing stress concentration, while a large \(R_{s}\) value allows the plunger material to deform over a wider area, contributing to stress distribution. PAI has a significantly lower elastic modulus (4.5 GPa) than aluminum bronze alloy (117 GPa), resulting in most deformation occurring on the plunger side during contact. This asymmetric deformation characteristic affects the strain compatibility condition expressed by the following equation:  
\[\frac{\delta_{p}}{\delta_{s}}=\frac{E_{s(1-\frac{1}{p})}}{E_{p(1-\frac{1}{p})}} \tag{12}\]  
where \(\delta_{p}\) and \(\delta_{s}\) are the deformation amounts of the plunger and seat, respectively. Substituting the material properties of PAI and aluminum bronze alloy yields \(\delta_{p}/\delta_{s}\approx 26\), indicating that deformation on the plunger side is approximately 26 times greater than that of the seat. The eccentricity e is another critical design variable determining contact pressure distribution and maximum equivalent stress location. Eccentricity determines the initial contact position, and as eccentricity increases, the contact point moves further outward toward the seat edge. It also influences the distribution pattern of contact pressure, and appropriate eccentricity induces the contact pressure concentration required at low-pressure conditions. The proposed \(R_{p}>R_{s}\) configuration fundamentally challenges conventional contact mechanics principles. Traditional sealing approaches have assumed that conforming geometries (\(R_{p}<R_{s}\)) provide optimal stress distribution through geometric compliance, a paradigm validated primarily through empirical testing.