## 1 Introduction  
An unexpected failure of the sealing system in an insulating joint can create serious problems in oil and gas pipelines. Insulating joints are critical components installed in underground natural-gas and oil pipelines. Installed at appropriate intervals between the pipes, insulation joints can reduce the amount of current flowing in the pipelines and maintain the electrical potential difference between pipe sections, thereby preventing pipe corrosion due to electrical or chemical reactions and increasing the service life of pipelines. Furthermore, they can provide electrical protection from the risk of explosion by static electricity or lightning. Elastomeric O-ring seals are widely used in various environments and surface conditions because they can provide sealing in many static or dynamic applications at industrial sites. In particular, O-ring sealing systems have traditionally been used in insulating joints. Figure 1 shows a three-dimensional (3D) schematic, with a 1/4 cross-sectional cut, of an insulating joint based on an O-ring sealing system. To structurally separate the external pipes that are connected at either end, insulating spacers (laminate thermoset or high-density epoxy resin and glass-fiber shape-composite material) and insulating fillers (epoxy resin or cold curling resin), which consist of nonmetal materials, are installed inside the pipelines. Elastomeric O-ring seals (Viton rubber, fluorine rubber, or nitrile butadiene rubber) are mounted to prevent leaks of gas or oil between the metal and insulation components. To prevent undesired failures of a sealing system, a number of studies have been conducted on elastomeric O-ring sealing systems using finite-element analysis (FEA). These have been focused on predicting the behavioral characteristics of O-ring seals in diverse and complex compression and pressure environments [1]. [PERSON] et al. [2] suggested an analytical stress-distribution model to describe the stress distribution of O-ring seals by adapting two-stress complex functions. The sealing performance of O-ring seals was described by the analytic stress-distribution functions under various squeeze rates and internal pressures. [PERSON] et al. [3] introduced end-face deformation of the combined contact model of the flexible ring and O-ring based on numerical simulations. In the combined seals, fractional compression of the O-ring and the elasticity modulus of the flexible ring had the largest effect on the end-face deformation. [PERSON] et al. [4] exposed a rubber O-ring to cyclic high-pressure hydrogen gas to investigate the fracture behavior. The crack damage due to degradation of the mechanical properties of the O-ring was characterized in terms of the hydrogen pressure, using a developed durability tester. To summarize the previous studies, the sealing capability of O-rings depends on:  
Figure 1: Example of insulating joint with a double-O-ring sealing system. * the magnitude of stress acting on the O-ring;
* the contact stress between the O-ring and the wall;
* the contact area (or length) for sealing between the O-ring and the wall. O-ring seals used in oil and gas pipelines are exposed to extreme environments such as large changes in temperature, pressure, and conveyed fluid [5]. Furthermore, depending on the geometry of O-ring seals, they are exposed to highly complex operating conditions as they are compressed and constrained. In such situations, it has been reported that leakage can start when the pressure acting on an O-ring is greater than the contact stress or when there is a gap in the contact length [6]. In this regard, even though O-ring sealing systems have been used extensively in industry, they have the following shortcomings:  
* Their resistance to high-pressure environments is limited because they have low contact stress because of the small cross-sectional area of the O-rings.