## 1 Introduction
* They are susceptible to damage such as explosive decompression, swell-ups, and tear-off because they cannot effectively withstand when subjected to pressure cycles and fast pressure changes because of the geometrical limitation of the O-rings. * They cannot maintain proper mechanical sealing if the insulating joints undergo deformations (e.g., bending, elongation, compression, or torsion) from external loads because of the short contact length of the O-rings. To resolve the disadvantages of O-rings, some different geometries have been proposed by researchers. [PERSON] et al. [5] applied nonlinear theory to investigate the performance of Y-ring seals under the influence of hydraulic pressure. Based on the failure criterion (i.e., von Mises stress), the location showing the maximum contact stress and the largest deformation was suggested, which could cause aggravated wear. [PERSON] et al. [6] investigated the contact behavior of an X-ring seal under a uniform squeeze rate and internal pressure. Both contact and internal stresses were analyzed using the hybrid photoelastic experimental method, in which the sealing performance was investigated based on the maximum shear failure criterion. [PERSON] et al. [7] performed FEA to obtain distribution curves of sealing contact pressure of bud-shaped composite sealing rings in a diverse pressure environment. The analytic 3D pressure distributions were compared experimentally using a pressure-sensitive film. In spite of their improved sealing performance compared to O-rings, these methods still have limitations over a broad range of operating conditions and surface finishes because of the geometric constraints. Another solution has recently been introduced to improve the disadvantages of O-ring sealing systems, namely the U-shape monolithic seal (UMS) or double seal. The UMS shows a high contact stress, with a substantially longer contact length and a larger cross-sectional area compared to conventional O-ring systems. Accordingly, it has the potential for widespread use, irrespective of the size and pressure ranges of pipelines. The structural advantages of the UMS are that it can maintain sealing even if subjected to structural deformations such as bending, elongation, compression, or torsion. Although the application scope of UMSs has been increasing, their exact behavior in the manufacturing process and different pressure environments is not yet fully understood for the following reasons. Firstly, compared to conventional O-ring sealing systems, because of a more complex geometry and highly restrained deformation conditions, it is difficult to predict analytically the nonlinear hyperelastic behavior of a UMS. In the case of hyperelastic materials, the dramatic change between softness and stiffness under various loading conditions because of the Mullins effect makes analysis difficult. Secondly, the bulk moduli of UMSs are very large compared to their shear moduli. As a result, they show incompressible behavior, thus causing a sharp increase in stress compared to the strain in a compression environment. Thirdly, they exhibit different behaviors depending on whether they are in a uniaxial, biaxial, or planar axial deformation state [8; 9; 10; 11]. Many researchers have proposed various mathematical constitutive models to predict the nonlinear response of hyperelastic materials under various deformation conditions [12; 13; 14; 15; 16]. Thus, it is necessary to predict the behavior of nonlinear hyperelastic UMS by a finite-element method (FEM) in the design stage, which can economically and effectively realize the integrity of the UMS. The aim of the present study was to analyze the hyperelastic behavior of a UMS numerically for various compression and pressure conditions, from assembly to use, in insulating joints. To overcome the difficulty of analyzing the nonlinearity of the UMS, a nonlinear constitutive equation was applied and a two-dimensional (2D) axisymmetric model was introduced to the analysis.