## 2 Theoretical Background
### Contact Mechanics  
Hertzian contact theory provides a fundamental model for quantitatively describing stress distribution and deformation characteristics occurring during contact between curved surfaces. The mathematical formulations presented in this section (Equations (1)-(8)) are based on classical Hertzian contact theory, as presented in comprehensive references [17; 18]. Figure 3: Geometric parameters of concave contact geometry. Figure 2: Geometric characteristics of contact geometries: (**a**) rectangular contact geometry; (**b**) concave contact geometry. These established relationships form the theoretical foundation upon which our contact geometry analysis is built. In the concave contact geometry, initial contact occurs between the curved surface of the plunger (\(R_{p}\)) and the edge rounding of the seat (\(R_{r}\)). The equivalent radius of curvature (\(R_{eq}\)) at this initial contact stage is defined as follows:  
\[\frac{1}{R_{eq}}=\frac{1}{R_{p}}+\frac{1}{R_{r}} \tag{1}\]  
The equivalent elastic modulus (\(E^{\star}\)) considering the elastic properties of the two contacting materials is given by the following:  
\[\frac{1}{E^{\star}}=\frac{1-\
u_{p}^{2}}{E_{p}}+\frac{1-\
u_{s}^{2}}{E_{s}} \tag{2}\]  
where \(E_{p}\) and \(E_{s}\) are the elastic moduli of the plunger and seat, respectively, and \(\
u_{p}\) and \(\
u_{s}\) are their Poisson's ratios. In Hertzian contact, the pressure distribution follows a semi-elliptical distribution:  
\[p(r)=p_{0}\sqrt{1-\left(\frac{r}{a}\right)^{2}} \tag{3}\]  
where _p(r)_ is the contact pressure at distance r from the center, \(p_{0}\) is the maximum contact pressure at the center, and \(a\) is the contact radius. The contact radius \(a\) is determined as a function of the equivalent radius of curvature, equivalent elastic modulus, and contact force:  
\[a=\left(\frac{3 FR_{eq}}{4E^{\star}}\right)^{\frac{1}{3}} \tag{4}\]  
where \(F\) is the contact force. The maximum contact pressure \(p_{0}\) is calculated as follows:  
\[p_{0}=\left(\frac{6{FE^{\star}}^{2}}{\pi^{3}R_{eq}^{2}}\right)^{\frac{1}{3}} \tag{5}\]  
Figure 4 illustrates this characteristic pressure distribution in Hertzian contact theory, showing how pressure maximizes at the center and gradually diminishes towards the contact boundaries. This is a fundamental aspect of the contact pressure equalization mechanism [13]. A key characteristic of Hertzian contact theory is the nonlinear expansion of the contact area with increasing load. In curved surface contact, the relationship between contact area and load follows:  
\[A\propto F^{\frac{2}{3}} \tag{6}\]  
From this, the average stress per unit area \(\sigma_{avg}\) is derived as follows:  
\[\sigma_{avg}=\frac{F}{A}\propto F^{\frac{1}{3}} \tag{7}\]  
Equation (7) demonstrates a self-limiting characteristic where the rate of stress increase diminishes as load increases. For example, when the load increases by a factor of 8, the average stress increases by only a factor of 2. This characteristic plays a crucial role in preventing maximum equivalent stress from exceeding the material's compressive strength in high-pressure environments. The maximum shear stress (\(\tau_{max}\)) occurring under high-pressure conditions has the following relationship with the maximum contact pressure [19]:  
\[\tau_{max}\approx 0.31p_{max} \tag{8}\]  
This equation predicts the magnitude of internal shear stress caused by contact pressure. The maximum shear stress typically occurs at a depth of approximately 0.48 below the surface and directly affects the yielding behavior of the material. These principles of Hertzian contact theory are essential for understanding the relationship between contact pressure at low-pressure conditions (2.0 MPa), which determines valve sealing performance, and maximum equivalent stress at high-pressure conditions (87.5 MPa), which determines durability.