## 1 Introduction
proposed a multiphase enhanced pentamode microstructure by introducing a soft connecting layer between the skeleton and the counterweight, which effectively reduces their coupling. This decoupling effect not only decreases the manufacturing time cost and time consumption of the pentamode structure but also enhances its convenience for production [4; 32]. The concept of \"mp\" pentamode metamaterials decouples equivalent density and bulk modulus, allowing for independent design of these parameters. While there have been numerous studies on pure solid-phase pentamode metafluid, research on water-solid metafluid remains limited [28; 33; 34]. [PERSON] et al. developed a two-dimensional carpet cloak composed of multilayer copper plates and water with certain impedance matching through scale factor introduction; subsequently, they designed a three-dimensional stealth bucket using a steel block [34; 35]. In 2023, [PERSON] et al. [33] proposed a metafluid structural cell consisting of water surrounding a single-phase hollow metal skeleton. It is noted that the pentamode metafluid always possesses some shear modulus, which hinders complete elimination of shear waves in low band curves and affects the wideband performance of the structure. In contrast, water-solid metafluid eliminates shear waves within the structure while exhibiting excellent broadband performance. Furthermore, it has been observed that the equivalent density and equivalent bulk modulus of the fluid element can be independently adjusted by modifying parameters like the number of edges, the material composition, and the filling rate. This capability is distinct from the coupled behavior of these two equivalent parameters in pentamode metafluid. However, similarly, the thickness of the skeleton plays a critical role in determining the equivalent modulus of the cell when using the same skeleton material. To achieve lower moduli, thinner skeletons are required, which increases processing complexity, time, and cost; moreover, certain thicknesses may be constrained by current manufacturing precision, leading to fabrication challenges [4]. Therefore, for the practical preparation of acoustic metafluid in the future, it would be prudent to explore relatively thicker skeletons to achieve lower bulk moduli. One notable form of skeletal structure in mechanical metamaterials is the \"re\" skeleton [17]. This structure represents a specific type of auxetic mechanical metamaterial. Auxetic mechanical metamaterials exhibit various forms, including curling and perforated structures [36; 37; 38; 39; 40; 41]. These materials are characterized by their resistance to shear deformation while being more susceptible to volumetric deformation, earning them the moniker \"anti-rubber materials\". Therefore, the \"re\" skeleton has the characteristic of low equivalent volume modulus when the skeleton's thickness is the same. Specifically, the \"re\" skeleton achieves this property by bending inward to create cells with reduced bulk modulus. Therefore, when the skeletons' thickness is the same, the \"re\" skeleton should have a low equivalent bulk modulus compared to the \"ss\" skeleton. However, its application in underwater metafluid remains relatively scarce. In this paper, an \"ss\" skeleton metafluid cell (ss1) is first constructed, and its basic composition and equivalent parameter calculation method are introduced. On the basis of the ss1 cell, the concept of \"re\" is introduced, the \"re\" cell is constructed, and the degree of concavity of the cell is described with variables. It is determined that the band curve is a compression wave curve from the vibration of a single cell. Then, the variation of the cell-equivalent parameter with the \"re\" variable \"e\" is analyzed, and the conclusion is drawn that the equivalent volume modulus decreases with the increase of e. Then, the concept of \"mp\" is introduced into the metafluid cell containing the concept of \"re\", and the single-cell re1 is constructed.