Mechanical Behavior of Chain-Link Wire Nets under Pressure from a Warhead: Quasistatic Experiments and Simulations

. Wire nets woven from high-strength steel wires have been used as applique armor against attack by short-range weapons. In this study, the mechanical behavior of chain-link wire nets under pressure from a warhead was investigated by quasistatic experiments and simulations. First, the new rig, the warhead device, and the wire nets were designed; pressure tests were conducted; and the deformation, fracture of the wire nets, and pressure force vs. displacement curves were obtained and analyzed. Then, the numerical approach and Finite element (FE) model were developed, considering the contacts between the steel wires in the inner connections, the contacts between the warhead and the mesh of the wire nets, and the fracture of the steel wire material. By comparison with the experimental data, the numerical approach and FE model are shown to be reliable in predicting the behavior of the wire nets under pressure from a warhead. Finally, the parameters of the wire net size and the mesh angles were further investigated by using the validated numerical approach and FE model, and suggestions for the initial design of the wire nets are discussed.


Introduction
Wire nets woven by steel wire are three-dimensional metallic fabric structures, and various wire nets with different fabric techniques and geometry features have been developed (Figure 1), such as chain-link wire nets with diamondshaped or ring features, double-twisted wire nets with hexagonal features, and welded wire nets with rectangle features. Because of their lightness and unique mechanical properties compared to steel sheets or other protective devices [1,2], wire nets have been used in many engineering applications. In natural hazard protection systems, wire nets are used as key components for rockfall protection barriers [3,4], slope reinforcement [5], and ground surface support systems [6]. In thin-walled security screens, wire nets are used as security screen doors and window grills against unauthorized intrusions and windborne debris impacts [7,8]. In protective engineering applications, wire nets are employed to reinforce the concrete slab to protect it against explosion and projectile impacts [9,10]. With the increase in the strength and ductility of steel wires, wire nets woven from high-strength steel wires have been used as applique armor ( Figure 2) against attacks by short-range weapons such as rifle propelled grenades (RPGs), mortar shells, or heavy machine guns [11,12].
Many previous studies on the static mechanical properties of wire nets woven by high-strength steel wire were conducted, and these studies mainly focused on the natural hazard protection field and simulated the load deduced by shallow landslides and rock block impact conditions, including quasistatic in-plane tensile experiments [13,14], quasistatic out-of-plane pressure experiments [15,16], and supplementary numerical simulation [17][18][19]. Figure 3 shows the fracture of the wire nets under short-range weapon attacks and rigid warhead impacts with low velocity [12,20]. The contact area of the wire nets impacted by the short-range weapons or rigid warheads is limited to a single mesh, and this phenomenon is much different from those presented in studies simulating shallow landslides and rockfalls. To improve the efficiency of wire nets in intercepting shortrange weapons, the static mechanical behavior of wire nets under quasistatic pressure from a warhead is a basic research work, which is important for the initial design of wire nets to resist attack by short-range weapons. To date, little attention has been given to the mechanical behavior of chain-link wire nets under quasistatic pressure from a warhead. Due to the abovementioned research gaps, the aims of this paper are threefold, as follows: (i) To design and conduct quasistatic experiments of wire nets pressurized by warheads based on the results from shortweapon attack tests and rigid warhead impact tests and then investigate the mechanical behavior of the wire nets.
(ii) To develop the numerical approach and finite element (FE) model to simulate the wire nets pressurized by a warhead and validate the numerical simulation results by comparison with the experimental data.
(iii) To discuss the wire nets sizes and mesh angles by using FE simulation and then to discuss suggestions of an initial design for the wire nets.

Experimental Methodology
Several testing rigs for wire nets under out-of-plane pressure were developed [6,15,16]. Different from previous rigs used in out-of-plane pressure tests for wire nets, a new testing rig (Figure 4(a)) has been developed. The new rig can apply pretension on the wire nets to avoid any possible sag in the initial testing stage. The height of the rig is 1.5 m, and the length and width of the rig are 2 × 2 m. The low parts of the rig are composed of 4 plane sheets and 4 I-beams, which are connected to the counterforce grooves, the middle parts of the rig are composed of 12 I-beams, and the upper parts ( Figure 4(b)) of the rig are composed of sliding connections and U-type connectors. Rectangular wire nets of approximately 1 × 1 m could be installed on the rig using U-type  Modelling and Simulation in Engineering connectors. The x, y, and z displacements of the wire nets can be constrained by the connectors, and then the sliding connections can apply slight pretension to the wire nets before the tests begin. According to the shape and dimensions of the short-range weapon ( Figure 5), a warhead device of caliber 107 mm is designed and is shown in Figure 6, including warhead, support, and flange plate. The charging and internal structures of the warhead are ignored, and the homogeneous warhead device is manufactured from 304 stainless steel material. The support is designed to lengthen the warhead and apply vertical force, and the flange plate is designed to be connected to the hydraulic presser.
Because the inner cross connections of the wire nets woven by the high-strength steel wire cannot be fixed by electric welding or brazing with a filler metal [21], a fabric technique involving a loose connection type and repeating rhomboidal patterns is used for the chain-link wire nets, and the two wires bent in opposite directions are in loose contact through the inner connections and tied together through knotted connections at the boundaries [14]. The chain-link wire nets are designed, and the drawings of the chain-link wire nets are shown in Figure 7. The detailed dimensions of the chain-link wire nets are shown in Table 1.
To provide the large deformation of the wire nets under pressure from a warhead, a hydraulic presser with a 500 mm large stroke and a measuring range of 20 kN is employed. The warhead was connected to the hydraulic presser through the flange plant by nuts and bolts, and then the warhead needed to be aligned with the center position of the wire net mesh. The longitudinal axis of the warhead and the wire nets needs to be orthogonal during the pressure tests. The setup of the tests is shown in Figure 8. The hydraulic presser pressed the warhead device down at a constant velocity of 10 mm/min, and when the warhead contacted the wire nets, the force and the downing displacement were recorded by the built-in displacement sensor and load sensor of the hydraulic presser. The value of the force and the downing displacement is determined beginning from the warhead contacting the single mesh of the wire nets until the wire nets were fractured. When the wire nets were fractured, the tests were stopped.

Experimental Results and Analysis
The pressure testing processes and the deformations of the wire nets are shown in Figure 9. When the warhead device initially penetrated the center mesh, relative displacement occurred between the center mesh and the warhead until the center mesh constrained the warhead, and then the wire nets and the warhead device together moved downward. The deformation of the wire nets included the overall deformation and the local deformation. The overall deformation behavior of the wire nets fits the membrane stretching model, producing a funnel shape, and the local deformation referred to the expanded deformation of the center mesh when impacted by the warhead device.
The detailed local deformation and the fracture feature of the wire nets are shown in Figure 10. While the warhead device penetrated the single mesh, circumferential extrusion force was applied to the center mesh by the warhead, and the center mesh gradually deformed in this section in the form of the warhead. The pretensioning of the wire nets before the pressure tests began, together with the pressure on the wire nets applied by the warhead on the center mesh, resulted in symmetrical stress on the wire nets. Therefore, the relative sliding between the steel wires at connections A~D was not obvious. There was no significant relative sliding between the two steel wires at inner connections B and D, and only slight sliding at inner connections A and C was observed (Figure 10(a)). For the three-dimensional fabric technology and eccentricity geometry, the strength of the wires in the connections would be slightly decreased for the pre-bending, and the steel wires in the connections resist the bending moment, shear force, and torsional force. The combined stress on the wires at the connections would first exceed the limited stress; therefore, the fracture of the wire nets took place at the connections in the center mesh, and the fracture behavior of the steel wire presented inclined section failure under the combined stress ( Figure 10  occurred at the locations of the connections where the single mesh contacted the warhead. The measured nonlinear pressure force vs. displacement response curves are shown in Figure 11. The tangent of the curves exhibits a progressive increase from 0 to 100~125 mm, and after the 125 mm displacement, the tangent of the curves remains almost constant until failure.

FE Modeling of the Wire Nets under
Pressure from the Warhead Device Although there are several studies on FE modeling of chainlink wire nets for in-plane tensile mechanisms [13,17,18], the intricate three-dimensional geometry of the chain-link wire nets and the contacts between the single wires in the inner connections, as well as the contacts between the mesh and the warhead device, makes FE modeling of the wire nets under pressure from a warhead a challenging and distinctive task. Therefore, the chain-link wire nets under pressure from the warhead device were implemented in ANSYS 2022R1 to reproduce the deformation and fracture behavior of the wire nets.

Numerical Approach and FE Model.
There was no fracture or sliding of the knotted connections of the wire nets in the experiments, so the knotted connections in the FE model could be simplified as fixed connections. The warhead device was simplified in the FE model, the flange plant was ignored, and the warhead and support were considered in the FE model. The geometry of the wire nets and the warhead device had much influence on its mechanical behavior, and thus, the imports of the FE model for the chain-link wire nets and the warhead device are in accordance with the measurements used in the experiments. Because the wires in the inner connections must resist tension, bending, shear, and torsion, beam elements with quadratic order and flexible stiffness behavior were selected for discretizing the wires. Compared to the wires, the stiffness behavior of the warhead devices was rigid, and solid elements with rigid behavior were selected for discretizing the warhead device. The capture curvature method was used, and the curvature normal angle was refined for meshing and capturing the contact behavior of the wires in the connections. A mapped mesh with hard behavior was used for the mesh of the warhead device geometry.
The wires in the inner connections came into contact with each other when the wire nets were pressed by the warhead device. Two main methods have been developed to handle such problems. The multibody coupling algorithm can establish translational constraints and keep free rotation between the contact nodes of the beam elements in the inner connections, but the sliding and contacts between the beam  No. of transversal meshes n L = 11 pcs/m 5 Modelling and Simulation in Engineering elements cannot be considered. The implicit beam-to-beam contact algorithm could consider the contacts and sliding of the beam elements in the inner connections, but the normal stiffness in the contacted beam elements would be dramatically changed, which would cause considerable difficulties in convergence or require a more refined mesh in the area of contact and more calculation time. The sliding of the steel wires at the connections was not significant; therefore, to capture the main behavior of the wire nets under pressure by the warhead, a hybrid algorithm was applied to simulate the contact behavior of the wire nets to balance the calculation accuracy, calculation time, and convergence. On the one hand, frictionless beam-to-beam contact was used to simulate the contacts between the beam elements in the inner connections; on the other hand, the multibody coupling algorithm was used to constrain the sliding between the contact nodes of the beam elements. Furthermore, the penetration tolerance factor and normal stiffness factor for the beam-to-beam contact algorithm were refined and optimized.
The multiple to solid contact algorithm was used to simulate contacts and sliding between the warhead device and the wire nets. The frictional coefficient was also considered, and the edgy contact type of line segments was used for the multiple-to-solid contact algorithm. The boundary conditions to simulate the pressure tests were applied in the FE model. The translational freedoms of the simplified knotted connections were constrained, and free rotations were used for modeling the connections between the wire nets and the Utype connectors. Vertical displacement was imposed on the warhead device to perform the quasistatic tests. The FE model of the chain-link wire nets and the warhead device was established and is shown in Figure 12.

Materials and Fracture.
The chain-link wire nets are woven from the high-strength steel wire, and the engineering strain-stress curve of the steel wire under uniaxial tension is shown in Figure 13. As shown in Figure 13, highstrength steel wire is a ductile material with plastic hardening characteristics, the Young's modulus of the steel wire is approximately 1941270 MPa; additionally, the average yield strength (YS, 0.2% offset line) of the wire used is approximately 1550 MPa. The breaking force of the steel wire is approximately 13.58 kN, and the ultimate tensile stress is approximately 1984.66 MPa. The ultimate strain of the steel wire is approximately 0.032. The multilinear isotropic hardening constitutive relationship was employed to describe the steel wire material. For the high nonlinear characteristics of the steel wire material, the plasticity with large strains and large-deflection effects should be considered, and the true    7 Modelling and Simulation in Engineering strain-stress curve of the steel wire material was used in the numerical analysis. The true strain-stress curve of the steel wire material can be obtained by the following formulas [17]: where σ eng and ε eng can be derived from the engineering stress-strain curve and are shown in Figure 13. Although the wires in the connections resisted axial force, bending moments, shear force, and torsional force, the fracturing of the steel wire in the tests was mainly caused where σ 1 is the axial stress, σ 2 is the total bending stress, σ 1 and σ 2 could be obtained by the numerical simulation; σ lim   is the ultimate tensile stress, and σ lim is equal to 1984.66 MPa; and λ is the bending moment influence coefficient considering the eccentricity geometry and the stress coefficient of the cross-section, and λ is equal to 0.20.

FE Modeling
Results and Validation. The FE modeling results are compared with the experimental data to assess the reliability and accuracy of the model in predicting the behavior of the wire nets under pressure from the warhead. In particular, the deformation and fracture behavior of the wire nets, the pressure force vs. displacement curves, and the maximum displacements and peak pressure forces obtained from the FE modeling and experimental tests are discussed.
The equivalent stress contour, the deformation, and the death element location of the wire nets are shown in Figure 14. The deformation of the wire nets obtained by the FE modeling included the overall deformation, the local deformation, and the death element located in the inner connections of the center mesh. The deformation and fracture behavior of the wire nets coincided well with the experimental results (Figures 9 and 10). Figure 15 shows the stress-displacement curves of the death element, including the axial stress, total bending stress, and combined stress.
At the beginning, the center mesh was mainly deformed due to eccentric stretching by the surrounding mesh, and the bending stress increased much more rapidly than the axial stress. As the center mesh expanded due to pressure from the warhead, the circumferential extrusion force increased, the axial stress increased rapidly, and the bending stress gradually decreased. Comparing the axial stress and bending stress, the axial stress is the main factor causing fracturing of the steel wire, and the axial stress accounts for 80.0% of the combined stress. Figure 16 displays the pressure force vs. displacement curves obtained by the FE modeling simulation and experiments. The general trend of the numerical results matched well with those of the experiments. The discrepancy can be partially attributed to the initial pretensioned conditions of the wire nets and the sliding at the inner connections. In the tests, the wire nets were pretensioned before the warhead device was applied, which would increase the stiffness of the wire nets at the beginning, resulting in the pressure force being larger than that of the numerical simulation at the same displacement from 0 to 140 mm. With the increase in the displacement from 140 to 180 mm, the pressure forces obtained by the tests were lower than those obtained by the numerical simulation due to the sliding effects at connections A and C. Table 2 shows the maximum displacement and peak pressure force obtained by the tests and numerical simulation, and the maximum displacement and the peak pressure force obtained by the numerical simulation coincide closely with the experimental data. Therefore, the developed numerical approach and FE model are reliably able to predict the behavior of wire nets under pressure from a warhead device.   Figure 17. The results indicate that as the wire net size decreased, the wire nets were much stiffer, which means that their force-displacement curves had steeper rising slopes, but the peak pressure force did not significantly change. Therefore, we can reduce the wire net sizes to increase the stiffness of the wire nets without considering the reduction in the peak pressure force. The mesh angle is a key parameter when designing wire nets, and five mesh angles with the same incircle diameter of wire nets were designed. Because some of the parameters of the designed wire nets with different angles are not changed, only the changed parameters are presented in Table 3. Then, the relationships between the angle of mesh, areal density of the wire nets, and peak pressure force are analyzed by FE modeling, and the numerical simulation results are shown in Figure 18. As shown in Figure 18, the areal density and the peak pressure force increase with increasing mesh angle. To obtain a larger peak pressure force, the angle of the mesh would be close to 90°, but this would increase the areal density of the wire nets and the weaving difficulty. However, because the initial bending of the steel wire was relatively large, the initial bending stress increased, resulting in the peak pressure force rapidly decreasing when the mesh angle was less than 53.9°.

Conclusion and Outlook
This paper reported the mechanical behavior of chain-link wire nets under quasistatic pressure from a warhead. Quasistatic pressure tests were performed on the designed wire nets, and the deformation and fracture behavior and pressure force vs. displacement of the wire nets were investigated. Furthermore, FE modeling of the wire nets under pressure from a warhead device was developed using ANSYS 2022R1. A comparison between the numerical results and experimental data demonstrates the reliability of the numerical method and FE model in predicting the mechanical behavior of wire nets pressurized by a warhead. The effects of the wire net size and mesh angles were also investigated by using the FE model. The findings of this study can be summarized as follows:  x × y = 58:7 × 106:1 α = 48:9 n Q = 17 n L = 9 2 x × y = 59:8 × 97:5 α = 53:9 n Q = 17 n L = 11 3 x × y = 59:9 × 93:0 α = 58:9 n Q = 17 n L = 11 4 x × y = 62:3 × 84:5 α = 63:9 n Q = 16 n L = 12 5 x × y = 64:0 × 79:5 α = 68:9 n Q = 15 n L = 13 11 Modelling and Simulation in Engineering (i) The deformation of the wire nets included the overall deformation and the local deformation. The overall deformation of the wire nets fits the membrane stretching model with a funnel shape, and the local deformation referred to the expanded deformation of the center mesh impacted by the warhead device.
(ii) The fractures of the wire nets were located at the connections of the single mesh in contact with the warhead, and these phenomena were the same as those observed in the low-velocity impact tests or short-range weapon attack tests. The pressure force vs. displacement curves obtained by the tests could be identified for the initial design of the mechanical behavior of the wire nets.
(iii) The stiffness of the wire nets is related to the wire net sizes, but the peak pressure force of the wire nets under pressure from the warhead is independent of the wire net sizes. The peak pressure force is related to the mesh angle, and to obtain a larger peak pressure force, the areal density of the wire nets and the weaving difficulty would be increased. The optimal mesh angle for the incircle diameter 45 mm of the wire nets is 59.9°considering the areal density of the wire nets and the weaving difficulty.
However, the interaction between wire nets and shortrange weapons is a dynamic process, and the findings presented in this paper are only the initial wire net design suggestions. The performance of high-strength steel wires and short-range weapon materials under the high strain rates, impact energy (velocity, mass), and attack angles of the weapons requires further experimental and numerical analysis.

Data Availability
The authors confirm that the data supporting the findings of this study are available within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest. igure 18: Areal densities of the wire nets and mesh angles vs. peak pressure force curve.