# Influence of Structural Parameters on Deformation Behavior of Metal Rhombic Chain Mesh
## 2 Materials and Methods  
### Numerical Model of Metal Netting  
Metal diamond chain netting is a three-dimensional porous structure with small-diameter wires, formed by interlacing and combining multiple spirally curved metal wires. It represents one of the most widely used netting types in aquaculture cages. Chain netting is typically arranged and installed along the x-direction according to metal wire length. This assembly configuration provides a certain bending resistance, which helps reduce netting deformation, as shown in Figure 1.  
To analyze the bending deformation capability of chain netting, we first examine the strain energy generated by the deformation of a single mesh in the metal netting. An imaginary force F is applied to one half of a mesh unit, with the other end being rigidly fixed. The total strain energy \(U\) comprises deformation energies from bending, shearing, torsion, and axial deformation. Since axial and shear forces are significantly smaller than bending and torsional moments, the strain energy U can be approximated as the sum of deformation energies from bending and torsional moments. \[U_{Total}=\int_{0}^{L}\frac{M^{2}(x)}{2E\Gamma_{zz}}dx+\int_{0}^{L}\frac{T^{2} (x)}{2C\Gamma_{0}}dx \tag{1}\]  
\[M_{Total}=\int_{0}^{L}\frac{M^{2}(x)}{2E\Gamma_{zz}}dx=\frac{F^{2}cos^{2}\theta _{0}^{local\theta}\,x^{2}dx}{2E\Gamma_{zz}}=\frac{F^{2}\beta\cos^{2}\theta}{6E \Gamma_{zz}} \tag{2}\]  
\[T_{Total}=\int_{0}^{L}\frac{T^{2}(x)}{2C\Gamma_{0}}dx=\frac{F^{2}\int_{0}^{L \theta}x^{2}dx}{2E}\times\frac{2(1+v)}{J_{0}}=\frac{F^{2}(lcos\theta)^{3}(1+v )}{3E\Gamma_{0}} \tag{3}\]  
where \(x\) represents the distance along the warp direction; \(L\) is the total length of metal wire along the warp direction; \(M\) denotes the bending moment; \(T\) represents the torsional moment; \(E\) is Young's modulus; \(G\) is the shear modulus; \(I_{zz}\) is the moment of inertia; and \(J_{0}\) is the polar moment of inertia. Assuming linear elastic behavior, the total strain energy can be described as follows:  
\[U_{Total}=M_{Total}+T_{Total}=\frac{F^{2}(lcos\theta)^{3}}{6E}\left[\frac{2(1+ v)}{J_{0}}+\frac{\cos^{2}\theta}{I_{zz}}\right] \tag{4}\]  
where \(v\) is Poisson's ratio. The definitions of \(\theta\) and l are shown in Figure 1.  
\[I_{eq}=\left[\frac{2(1+v)}{J_{0}}+\frac{\cos^{2}\theta}{I_{zz}}\right]^{-1} \tag{5}\]  
Figure 1: Geometry and force diagram of metal mesh (2a: Mesh width; \(l\): Mesh side length). The equivalent moment of inertia of the mesh wire cross-section represents the bending resistance capacity of the netting. It can be observed that the bending stiffness depends on the wire diameter, material Poisson's ratio, and netting weaving angle. The numerical simulation of the metal planar netting structure is conducted in ANSYS Workbench 2022 using the finite element method. To analyze the influence of different mechanical models at chain netting nodes on netting deformation, two modeling approaches--node coupling and contact connection--are established to simulate wire interactions at connection points. The element types and local mesh divisions for both models are shown in Figure 2. The physical model of the metal netting is created using Rhino 8.0 software and imported into ANSYS for numerical simulation. The node coupling model (M1) considers two adjacent nodes at the metal wire connection points as coupled into a single node, representing a connection model where nodes neither separate nor slip, and is simulated using beam elements in ANSYS. The contact connection model (M2) considers the mutual contact pressure between connecting nodes and is simulated using solid elements in ANSYS. The contact boundary problem between two adjacent nodes in metal netting connections with solid contact constraints can be described as finding the displacement field \(X\) within the region that minimizes the system's potential energy under contact boundary conditions, expressed as follows:  
\[\left\{ \begin{matrix}\Pi(X)=\frac{1}{2}X^{T}KX-X^{T}F\\ s.t.\Delta\geq 0\end{matrix} \right.