Mechanical Behaviors of Hybrid Composites with Orthogonal Spiral Wire Mesh and Polyurethane Elastomer

This work aims to significantly improve the mechanical properties of conventional rigid lattice structures under repeatable large deformations. A novel hybrid material is proposed based on the concept of interpenetrating composite materials. The material utilizes a woven TC4 orthogonal spiral wire mesh as the skeleton and PU elastomer (OSWM‐PU) as the matrix. The uniaxial tensile tests demonstrate that OSWM‐PU possesses the excellent load‐bearing capacity, allowing for large deformations (≥60%) while maintaining partial integrity even after matrix fracture. Optical measurements and simulation analysis reveal that Poisson's ratio can be adjusted within a certain range by manipulating the microscopic parameters (p, d) of the longitudinal helical filaments. Cyclic tensile experiments further demonstrate that OSWM‐PU exhibits exceptional energy absorption performance, multiple energy dissipation modes, and a more pronounced Mullins effect. The stress relaxation experiment reveals the significant influence of the volume fraction of the skeleton on long‐term loading conditions. The orthogonal spiral wire skeleton exhibits a superior hooking effect without dividing the matrix, enabling OSWM‐PU to possess enhanced collaborative deformation capability and inherent designability in the orthogonal direction. These characteristics make it highly promising for applications in various robot joints and as flexible aircraft skin, offering excellent prospects for utilization.

Although the flexible arm connection method can result in unexpected mechanical performance, the sparse mesh structure associated with it leads to inadequate air tightness and compromised aerodynamics. [21,22]Currently, there are several implementation methods for flexible skin in engineering applications, including the rotation and folding of rigid mechanisms, deformation mechanisms driving wing deformation, and the inherent elasticity of materials such as elastic rubber, shape memory polymers, or alloys. [23,24]Interpenetrating composite materials (IPCs) [25] can effectively combine the reinforcement provided by the skeleton with the sealing performance of the matrix, offering promising applications in the field of flexible skin.Materials with partition characteristics, such as honeycomb structures, cannot form a continuous matrix phase, and a closed skeleton separates the matrix material, making it challenging to achieve an interpenetrating effect. [26]Therefore, composite materials featuring porous interconnected metal skeletons or periodic point skeletons are generally the preferred configurations.
Foam aluminum, as a representative porous metal, finds numerous applications.Fan et al. [27] developed an aluminum/ polyurethane foam interpenetrating composite by filling it with hard foam, leading to improved compression performance and strain rate sensitivity.Zheng et al. [28,29] successfully created a new type of composite damping structure by combining wire mesh-based porous metal rubber with silica gel.However, the tightly woven metal wires pose a challenge as they hinder complete encapsulation of the entire sample during silicone filling.Yao et al. [30][31][32] prepared metal-resin interpenetrating composites by spontaneously infiltrating unsaturated polyester resin into porous stainless steel fiber preforms under vacuum conditions, but insufficient filling remains an issue in these composites.Periodic lattice structures show promising potential in composite-reinforced frameworks due to their favorable skeleton distribution and porosity, which contribute to improved filling efficiency.Feng et al. [33] reported a composite material prepared by mixing orthogonal arranged boron nitride thin sheets into polymer elastomers through rolling milling and other methods.Li et al. [34] manufactured a 3D lattice structure and epoxy resin ICM through selective laser melting.Wei et al. [35] employed natural infiltration to introduce epoxy resin into the 316L stainless steel lattice structure fabricated via selective laser melting.Xu et al. [36,37] produced a novel rubber-reinforced composite material using a woven lattice structure as the skeleton and rubber as the interpenetrating matrix, displaying a strong Mullins effect.Hajarian et al. [38] constructed a composite skin composed of two layers of silicone elastomer and one layer of nylon weft weave.Nevertheless, 2D lattice structures have limited potential to achieve a strong interlocking effect between the framework and the matrix.Therefore, greater attention should be directed toward 3D distributed periodic lattice structures as more suitable frameworks to enhance the interpenetrating composite effects.
This study focuses on the development of a novel biomimetic composite material that exhibits softness and deformability, inspired by the loading behavior of spiral microstructure networks present in extracellular matrix tissues immersed in matrix fluids.The composite material, named orthogonal spiral wire mesh composite material (OSWM-PU), was fabricated utilizing an orthogonal overlapping spiral wire mesh lattice structure as the underlying framework, which was subsequently infiltrated with interpenetrating polyurethane elastomer using a vacuum infiltration method.To comprehensively understand the mechanical properties of the composite materials, extensive investigations were conducted on the uniaxial tensile properties and trends in Poisson's ratio variation under various microparameter models through a combination of experimental and simulation methods.Additionally, the study analyzed the changes in load-bearing capacity, energy absorption characteristics, and effective modulus under cyclic loading at constant strain.The Mullins effect of the composite material was examined, and the Mullins damage model was employed to describe the damage behavior of OSWM-PU composites subjected to different cyclic loads.Stress relaxation experiments were also conducted, varying the wire diameters and pitches, and the relaxation behavior was characterized using the Prony series model.

Design and Fabrication
Inspired by the arrangement of spiral microstructures in living organism extracellular tissues, the replacement of traditional lattice structure rigid arms with spiral spring arms is expected to result in remarkable deformation capabilities.The OSWM represents an enhanced lattice structure that utilizes woven spiral wire components to construct effective beams, departing from the conventional monolithic or hollow-beam design.When a spring wire undergoes unidirectional tensile load, it generates both axial force and circumferential torque.Consequently, when the local plastic deformation threshold is reached, the spring wire lattice structure is expected to exhibit torsional deformation as a whole.In Figure 1, computerized spring pressing machines with specific forming tools control the wire feed trajectory, enabling the winding of spiral wires with different pitches and wire diameters.Utilizing these spring woven beams as building configurations in periodic unit cells, woven spring-arm lattice grilles are obtained, sharing the same link function as their monolithic and hollow counterparts.The effective beam junctions of the OSWM, where the overlap joint occurs, involve the orthogonal laying of spring wires through a specific connection process.This work considers various lapping types of spiral wires with different parameters to explore the mechanical response systematically.
A novel 2D elastic metamaterial was proposed, comprising an OSWM skeleton filled with polyurethane (PU) elastomer.This composite hybrid design offers a large deformation capacity and presents new possibilities for ICMs.Unlike conventional 2D lattice structures, the unique spatial arrangement of the OSWM creates a superior quasi-3D interpenetrating network skeleton, facilitating enhanced connectivity between the matrix and skeleton.The PU elastomer is selected as the matrix material due to its low moduli and excellent sealing performance, serving as an excellent complement to the OSWM skeleton.The OSWM, as a metal wire-woven skeleton, provides superior load-bearing capacity compared to the PU elastomer, acting as a strength supplement both before and after matrix failure.This novel composite material, designed to mimic human skin, exhibits improved load-bearing capacity, meets the requirements for large deformation capacity, and offers designability in the orthogonal direction.
To ensure the composite effect is not compromised by potential contamination of the metal wire during processing and transportation, it is crucial to conduct ultrasonic cleaning as an initial step.In the preparation of the OSWM, Ti6Al4V(TC4) spiral wires with a fixed pitch were overlapped in an orthogonal manner and securely bonded using adhesive at the junctions of effective beams.The skeleton structures consisted of 60 Â 130 mm tessellations of 10 mm unit cells, with intended wire diameters (d) ranging from 0.3 to 0.5 mm and pitches (p) ranging from 2 to 4 mm.To maintain consistent beam lengths, a constant cell size was maintained to accommodate variations in pitch and lap joint modality.The next step involved introducing the liquid-state PU elastomer into the OSWM skeletons using the vacuum infiltration method, followed by vulcanization (Figure 2).The vulcanization process involved solidifying the elastomer at room temperature for 24 h before removing it from the mound.The mechanical properties of TC4 at room temperature (25 °C) are presented in Table 1.The quasistatic unidirectional tensile experiments and cyclic tensile tests were conducted using a microcomputer-controlled tensile test machine (DK-5000).To prevent sample sliding or breakage due to clamping during mechanical stretching, small veins were machined on the clamping surface of the jigs.Displacement control mode was employed to regulate the displacement of different gauge lengths on the samples, with loading speeds set at 3/8/13 mm min À1 .A microcomputer program was utilized to control the chuck and achieve cyclic loading under constant strain and constant strain increase.For each composite with different microscopic model parameters, three samples were measured, and their average values were obtained as the final result.In order to measure the Poisson's ratio of OSWM-PU, deformation images were captured on the sample surface using a camera with pattern speckle during the stretching process.The GOM Correlate software was then employed to conduct a 2D plane strain analysis and extract the strain field from the images taken at different time points.The disparities between experimental measurements and simulation analyses in terms of Poisson's ratio were examined, utilizing the optical measurement system.The stress-strain curve of the PU elastomer under room temperature conditions is depicted in Figure 3.

Finite-Element Analyses
The deformations and stress-strain curves were calculated through element analyses using the commercial software ABAQUS.For PU elastomer solidified at room temperature, a hyperelastic constitutive relation based on the incompressible Ogden law was used in the finite element analysis (FEA).The material parameters of the constitutive model were determined by fitting the uniaxial tensile test data of the PU elastomer (Table 2).Mesh refinement was applied at the contact boundary to mitigate the impact of model geometry on the simulation results.Assuming a strong bond and negligible relative displacement between the OSWM and PU elastomer interface under loading, a tie constraint was utilized to effectively connect and integrate the interface.To optimize computational costs based on the periodic unit segment, the number of calculated lattices was controlled at 2 Â 3. The equivalent model calculation formula did not consider the influence of the built-in OSWM on the model and performed solid model calculations using material parameters fit with different experimental data to validate the applicability of the Mullins damage and viscoelastic Prony series model.

Uniaxial Tension Experiments on OSWM-PU
The mechanical response of the OSWM, PU elastomer, and OSWM-PU composite was analyzed through uniaxial tensile   tests.Figure 4 illustrates that the OSWM exhibits approximately linear growth at strains below 0.3, while the polymer already shows signs of softening.The measured data can be found in Supplementary Material-Part A. The combination of OSWM and PU elastomer significantly enhances the load-bearing capacity of the OSWM-PU composite, with the tensile strength approaching the sum of the individual components.However, while providing satisfactory strength performance, it also leads to a reduction in the material's ability to stretch.Mechanical response of the composite materials under different loading rates reveals a notable influence of the loading rate on their deformation ability.Prior to fracture, a region of force attenuation is observed at the peak load due to the gradual formation and expansion of cracks originating from the edges during the stretching process.Subsequently, the material experiences sudden fracture, leading to a significant decrease in stress and creating a noticeable disparity between the peak stress and the minimum stress after fracture.Despite the occurrence of fracture, the built-in OSWM maintains its connection, resulting in postfracture strength higher than that of the OSWM and PU elastomer alone.The deformation mode transitions from the joint deformation of the OSWM and PU elastomer to the straightening of the spiral wires and the friction between the spiral wire and PU elastomer.During this stage, the gradual detachment of the spiral wire from the polymer induces intense friction, leading to the formation of a stress plateau.To ensure accurate simulation of the quasistatic loading state while considering factors such as tensile stroke and experimental efficiency, a subsequent loading rate of 3 mm min À1 has been determined as a suitable compromise.
Further uniaxial tensile experiments and numerical simulations were performed on the OSWM-PU material, with variations in the microscopic model parameters (i.e., d and p), and images of the tensile process were recorded.As shown in Figure 5, the experimental results reveal that during the elastic deformation stage, an increase in the parameter p enhances the tensile strength and deformation ability of the material.Additionally, as p increases, the helix angle of the spring wire in the OSWM also increases.By analyzing the deformation mechanism of the spring, it can be inferred that, under identical conditions, the uniaxial force of the spring increases with the augmentation of the helix angle.The equivalent density ρ is originally defined by the following equation.
where V OSWM is the volume of OSWM and V is the volume of OSWM-PU.When p decreases, the ρ increases, more wires are staggered at the overlap.This causes the connection area of the PU matrix to become compact and prone to fracture, resulting in a decrease in deformation ability.Similarly, as d increases, the equivalent density ρ increases, resulting in a decrease in deformation capacity, but the load-bearing capacity continues to increase, with a peak tensile stress σ of ≈690 kPa.The numerical calculations of the elastic deformation mode and stress-strain curve demonstrate good agreement with the microscopic parameters model.Throughout the stretching process, the PU elastomer undergoes lateral shrinkage deformation and compression due to the transverse spiral wire compressing the matrix.Furthermore, the mechanical performance of OSWM-PU in the off-axis direction is also assessed through numerical simulation, that is, loaded at 45°with respect to the OSWM.Deformation characteristics of off-axis (45°) composite solution based on numerical simulation can be seen in the Supplementary Material-Part B. Unlike on-axis tensile loading, where the deformation mode is primarily uniaxial loading along the helical wires, loading in the 45°direction results in contraction toward the center along the fiber direction.Similar to fiberreinforced composites with orthogonal arrangements, loading in this manner leads to an overall decrease in stiffness and modulus.

Effect of the Longitudinal Variable Stiffness on Poisson's Ratio
The OSWM-PU composite material exhibits inherent design flexibility in the orthogonal direction, due to the presence of the embedded orthogonal overlapping metal wire mesh.Unlike traditional composite materials with rigid beam connections, which lack effective bonding with the matrix, resulting in poor deformation collaboration between the skeleton and the matrix, the inclusion of spiral wires increases the contact area with the matrix, effectively addressing this limitation.This collaborative deformation mechanism alters the lateral shrinkage behavior of the polymer during uniaxial tension due to the obstruction of the hook spring.Under the constraint of the skeleton, the soft matrix material undergoes unconventional flow during the stretching process.The spiral wire pattern impedes the movement of the matrix, restricting its longitudinal deformation.
Expanding on this concept, the relationship between Poisson's ratio and the lateral stiffness of the OSWM-PU composite is further elucidated by adjusting the longitudinal stiffness to limit the lateral shrinkage of the PU matrix.Figure 7 presents a contour map of the strain field in the OSWM-PU composite material obtained through optical imaging techniques.The curved frame region, which contains the skeletal structure, shows more pronounced interference with the movement of the matrix.Regions with the skeleton exhibit protrusions, while areas without the skeleton show sinking phenomena.Furthermore, the transverse and longitudinal strain data of representative points within the analyzed region were extracted, and a curve graph illustrating the relationship between Poisson's ratio and strain was plotted (Figure 8).It demonstrates good agreement and similar trends to the simulated curve.Therefore, for subsequent analyses, simulation methods provide a more accurate description of the influence of longitudinal stiffness on Poisson's ratio of the composite material.
Figure 9 shows that the Poisson's ratio decreases with increasing strain.Specifically, when the pitch is fixed at p = 3 mm, and the filament diameters are controlled at d = 0.3/0.4/0.5 mm, a larger filament diameter results in an increasing Poisson's ratio.Additionally, when the maximum filament diameter is set at d = 0.5 mm, and the pitch varies as p = 2/3/4 mm, a decrease in pitch corresponds to a decrease in Poisson's ratio.When the matrix contracts during the stretching process, it adheres to the mesh.A material with strong contraction combined with one that hardly contracts inevitably leads to a decrease in Poisson's ratio.Furthermore, as the longitudinal resistance to deformation of the mesh increases, the deformation of the matrix decreases correspondingly, leading to an increase in the diameter of the helical wires in the perpendicular load direction and a decrease in Poisson's ratio.Conversely, when the pitch decreases, the loading behavior during compression of the composite material is different from that during stretching.A decrease in pitch causes the helical wires to become more densely packed per unit length.As the pitch decreases, the equivalent density of the metal wires increases, thereby increasing the resistance to deformation during compression.During stretching, the perpendicular load direction manifests as compression behavior.Therefore, as the pitch decreases, the resistance to deformation increases, resulting in a decrease in Poisson's ratio.Due to the transformation in deformation mode inherent to OSWM, the longitudinal stiffness under off-axis loading is lower than that under on-axis loading.In the initial stages of loading, the Poisson's ratio may even be smaller than that of the PU elastomer.As the strain increases, the tensile angle decreases and the load-bearing capacity of the helical wires increases, further impeding the contraction of the longitudinal matrix.Consequently, the Poisson's ratio becomes smaller than that of the PU elastomer.However, the longitudinal stiffness under off-axis loading thoroughly remains lower than that under onaxial loading, leading to an increase in Poisson's ratio.

Mechanical Response of OSWM-PU upon Cyclic Tension
In this experiment, the deformation range limit set at a failure strain of 60%.To evaluate the response of OSWM-PU under cyclic deformation, repeated tension tests were conducted up   to a strain of 0.4, and the results were compared with those of OSWM and PU elastomer.The responses of PU elastomer and OSWM-PU exhibited significant attenuation softening in the first cycle, leading to distinct differences in subsequent cycles due to accumulated damage (Figure 10a).As the tensile cycles progressed, the loading response gradually stabilized, forming a hysteresis region of a certain magnitude, without any failure observed in PU elastomer or OSWM after ten cycles.The evolution of the effective Young's modulus per cycle in Figure 10b demonstrates that the effective moduli for the OSWM-PU composite stabilized after approximately four cycles, while the PU elastomer stabilized after two cycles.The effective modulus after cyclic stability is ≈10.5% higher than that of PU elastomer and 483.3% higher than that of OSWM.To quantify mechanical resilience of their stretching process, the dissipated energy density ΔU i under the i-th stretching cycle was calculated, which is the area surrounded by stress (σ) and strain (ε) curve under each cycle.
and Q i is defined as the ratio of the energy density of the i-th cycle to the first.
Figure 10c demonstrates that all three materials exhibit significant energy absorption in the initial cycle, followed by stable behavior in subsequent cycles.By the end of the tenth cycle, the OSWM-PU composite demonstrates an energy density dissipation value ≈100% higher than that of the PU elastomer with the same configuration.The lattice geometry of OSWM plays a crucial role as the spiral wires effectively engage with the matrix, resulting in a relatively high absolute energy absorption value.The energy density consumed by the composite materials in each cycle exceeds 5 kJ m À3 , indicating that the presence of OSWM increases energy consumption compared to a single material.
The composite mode leads to increased frictional energy consumption between the OSWM and PU elastomer, resulting in an overall increase in energy consumption.The normalized energy absorption represents the ratio of the dissipated energy in each cycle to the energy consumed during the loading process, serving as the energy consumption factor for each cycle (Figure 10d).
Although composite materials exhibit higher energy absorption per cycle compared to monolithic materials, their normalized energy absorption is lower than that of the PU elastomer, ≈0.29.This phenomenon of high energy absorption but low normalized energy absorption suggests that the presence of OSWM partially limits the cyclic softening effect of PU.OSWM, which demonstrates linear elastic deformation behavior, helps to restrain the nonlinear deformation of PU to some extent.This characteristic makes the new composite material promising for applications in the field of active and passive control of large deformations.Furthermore, the lower normalized energy absorption indicates that the energy dissipation in OSWM-PU primarily occurs in the first cycle, and as subsequent cycles stabilize, the energy dissipation decreases significantly.Therefore, it is important to consider preloading in practical applications to account for initial damage and ensure the proper performance and durability of the OSWM-PU composite material.
Figure 11 depicts the diverse energy dissipation mechanisms observed during the stretching process of the OSWM-PU composite material.In the elastic deformation region, the PU elastomer matrix demonstrates characteristic energy dissipation behavior due to the fracture of internal covalent bonds, resulting in irreversible stress softening.This internal molecular rearrangement and bond breaking contribute to energy absorption within the elastic deformation range.The connection between the metal wire and PU elastomer relies on the cohesive force of the polymer.As multiple loading cycles occur, adhesion failure takes place, leading to relative friction at the interface between the metal wire and the polymer, which contributes to energy dissipation.It is important to note that the simulation models the wire-polymer contact using tie constraints, assuming perfect adhesion.Consequently, the energy dissipation resulting from friction between the wire and the matrix is not accounted for.Failure of the bonding nodes in the OSWM causes misalignment of the orthogonal spring wires, resulting in energy consumption.Even after partial failure and damage to the matrix, the OSWM-PU continues to provide load-bearing capacity through the steel wire mesh.Upon fracture of the PU matrix, the wire mesh detaches from the polymer, leading to significant friction and the formation of a stress plateau.This indicates the absorption of a substantial amount of energy during subsequent stretching.These various energy dissipation modes enable the OSWM-PU composite material to exhibit enhanced load-bearing capacity, superior energy absorption capability, and improved resistance to damage and failure compared to a single material.

Mullins Damage Behavior Under Cyclic Tension
The Mullins effect is observed in rubber materials, characterized by an instant and irreversible softening of the stress-strain curve due to the material's previous maximum load history.However, the Ogden hyperelastic constitutive model employed in Section 4.1 is insufficient to accurately describe the damage behavior of composite materials subjected to cyclic loading.The pseudoelasticity theory introduces a continuous damage parameter η (0 < η ≤ 1) based on the constitutive relationship of rubber, so that the stress after unloading and reloading under the same strain is less than the stress in the main loading path.In the present study, the investigation focuses on the Mullins effect under uniaxial tension experiments, and its corresponding stress-strain state can be expressed as follows. [39] Elongation ratio λ i under uniaxial tension is After excluding volumetric forces from the equilibrium equation The Cauchy stress equation on the main loading path can be described as During the deformation process, η can be either active or inactive; switching between these two states, η remains continuous.When η is in an inactive state and remains constant η = 1, the material exhibits mechanical behavior along the main loading path.
Define η as then the strain energy function on the unloading path can be expressed as Due to the activation of η (0 < η ≤ 1) during unloading, the corresponding unloading Cauchy stress is Therefore, the nominal stress t is expressed as The maximum elongation ratio λ during loading determines the maximum strain energy function W m and damage variable η.The maximum strain energy function is When the elongation ratio λ = 1, that is, completely unloaded, the damage deformation η reaches the minimum value The choice of the damage function is flexible, but certain conditions must be adhered to ϕð1Þ ¼ 0 ðϕ 00 ðηÞ < 0Þ The following equation is a conditional damage function proposed by Ogden et al.
By combining Equation ( 9) and ( 15), the expression for η can be obtained as The damage degree of the initial state is represented by the parameter r, while the material damage parameters are denoted by m and β.The error function, Erf, can be mathematically expressed as follows.f ðxÞ ¼ The strain energy function of Ogden (N = 3) constitutive model under uniaxial tension is [40] where μ i is expressed in pressure and α i is a dimensionless constant.Combining Equation ( 18) and ( 9) yields the nominal stress under uniaxial tension t 0 Cyclic loading tests were conducted on both PU elastomer and OSWM-PU at strain levels of 0.2, 0.4, and 0.6.As shown in Figure 12, both PU elastomer and OSWM-PU exhibit a notable Mullins effect, which becomes more prominent with increasing cyclic deformation.This deviation in the main loading curve is attributed to the occurrence of plastic deformation between each loading-unloading cycle in practical experiments.As the cyclic strain increases, the normalized energy consumed correspondingly also increases, indicating that elongation ratio λ determines the damage changes during the cycling process.
To validate the suitability of the elastic damage model based on the Ogden (N = 3) constitutive law, parameter fitting was conducted using the experimental data of PU elastomer and OSWM-PU composite, resulting in the determination of the damage parameters as presented in Table 3.The detailed data can be found in Supplementary Material-Part C. The fitted hyperelastic damage parameters were then employed in the ABAQUS finiteelement analysis for equivalent calculations.The corresponding results, represented by the light purple dashed line in Figure 12, exhibit a good agreement with the experimental results.The R 2 fitness value of 0.991 indicates the excellent applicability of the model in describing the cyclic damage behavior of these materials.It is noteworthy that these two materials display noticeable differences in energy dissipation under identical cyclic loading conditions, suggesting that the presence of the wire mesh enhances the damage characteristics of the composite material.
From Equation ( 16), it can be seen that the damage variable η is related to the tensile ratio of each cycle.Here, the larger the tensile ratio, the smaller the damage variable during the unloading process.According to Equation (11), the damage variable η can be expressed as follows.
where t is the nominal loading stress, and t 0 is the nominal unloading stress.Figure 13 depicts the transformation process of damage variables η during unloading of OSWM-PU composite when different tensile strain conditions are applied.Compared to the η trend of single-polymer materials under the high strain level, OSWM-PU composite exhibits much greater attenuation under high-strain conditions.At higher levels of strain, the embedded OSWM experiences irreversible plastic deformation, leading to a reduction in stress during unloading.However, after three cycles of complete unloading, the difference in damage values is small and decreases in an approximate linear manner.

Stress Relaxation Analysis of OSWM-PU
In order to investigate the stress relaxation behavior of OSWM-PU, stress relaxation experiments were conducted at various strains and microscopic model parameters.The stress relaxation  deformation mode of the OSWM-PU composite was then analyzed using the generalized Maxwell viscoelastic model (Figure 14) in conjunction with ABAQUS simulation analysis. [41]e generalized Maxwell model provides a description of the stress relaxation behavior of polymer materials, as depicted by Equation ( 21) where τ i is the i-th Prony retardation time constant τ i ¼ η i =E i , η i is the viscosity coefficient.
To incorporate the generalized Maxwell model into ABAQUS, the stress relaxation behavior of linear viscoelastic polymer materials is commonly simulated using a Prony series representation of the stress relaxation function.When the strain of the material remains constant, the relationship between stress and strain in the presence of stress relaxation can be described as follows.
σðtÞ ¼ EðtÞε 0 (22)   where E(t) is the relaxation function of the rubber material, which can be expressed as the form of a Prony series.where E(0) = E 0 at t = 0, E 0 is the instantaneous modulus of rubber, g i is the ratio between shear relaxation modulus G i and shear traction relaxation modulus G 0 , which can be expressed as where υ is Poisson's ratio.Similarly, the shear modulus G(t) and bulk modulus K(t) can be obtained with the same method.τ i , k i , and g i are the coefficients of the Prony series in ABAQUS, and the rubber material is approximately incompressible, k i = 0.In this study, the Yeoh hyperelastic constitutive model and Prony series are employed to fit the experimental data.Figure 14 illustrates the stress relaxation curves of PU elastomer and OSWM-PU at a strain of 0.4.The viscoelastic model parameters obtained from fitting the experimental data are compared with the results of equivalent simulation experiments in ABAQUS (dashed line).The experimental results of PU elastomer show good agreement with the simulation results, while the simulation results of OSWM-PU tend to stabilize earlier, indicating some differences.During the stress relaxation process, the equivalent solid model used in the simulation effectively captures the relaxation behavior of PU elastomer, a single-component material.However, as a multicomponent material, the presence of OSWM-PU in the early stage of relaxation leads to a slower stress decay.With the progression of time, the connection between the OSWM and the matrix weakens, resulting in a continuous decrease in stress and even fracture of the matrix.The experimental results show that OSWM-PU exhibits significantly longer decay times and larger stress reductions compared to PU elastomer.Similar trends are observed in the stress relaxation curves at different strains.The stress relaxation curves of OSWM-PU composites with different wire diameters (d) at a strain of 0.3.For d = 0.4/0.5 mm, the stress retention time is ≈450 s, after which the matrix gradually fractures, leading to a sharp drop in stress.Following matrix fracture, the stress level of OSWM-PU with d = 0.5 mm remains higher than that of the composite with d = 0.3 mm.However, the stress of the composite with d = 0.4 mm sharply decreases and becomes lower than that of the d = 0.3 mm composite.
The stress reduction during relaxation in OSWM-PU composite increases with the wire diameter, indicating a positive correlation between the wire diameter and the material's relaxation softening behavior.This highlights the influence of wire component size and characteristics on the relaxation behavior of OSWM-PU and underscores the importance of considering time-dependent effects in its analysis.It is noteworthy that when the strain exceeds a certain threshold, an increase in wire diameter can further promote matrix fracture in OSWM-PU composites.This observation suggests that OSWM-PU composites may have limitations when subjected to long-term loading conditions.Among the different pitch values, OSWM-PU composites with a pitch of 4 mm exhibit better stress relaxation performance without experiencing a fracture.Conversely, the composite material with a pitch of 2 mm shows a sharp decrease in stress after matrix fracture, indicating inferior performance compared to the material with a pitch of 3 mm.This denser crosslinking reduces the occurrence of fracture, as observed in the case of a pitch of 4 mm.The stress relaxation behavior of OSWM-PU under various conditions (i.e., p, d, strain) before failure (t ≤ 520 s) is quantitatively described in Figure 15.The results show a positive correlation between the wire diameter (d) and strain with the amount of stress relaxation, while there is a negative correlation with the pitch (p).This can be attributed to the influence of the volume fraction of the wire mesh in OSWM-PU.As the volume fraction increases, the stress relaxation holding time decreases accordingly.The volume fraction of the wire mesh in the OSWM-PU composite plays a critical role in its stress relaxation behavior.As the proportion of the wire mesh increases, the ability of the matrix to sustain stress over time decreases, leading to a decrease in the stress relaxation holding time.Essentially, the increased presence of the wire mesh disrupts the connectivity and continuity of the polymer matrix, thereby affecting its ability to retain stress.

Conclusion
This study focuses on the development of a novel hybrid composite material consisting of a metal wire-woven orthogonal spiral mesh as the skeletal structure and a PU elastomer as the matrix.Experimental and simulation methods were used to investigate the effects of microscopic model parameters on the mechanical properties, energy dissipation, Mullins damage effects under cyclic tension, and stress relaxation behavior.
The key findings of this research are summarized as follows: 1) The OSWM-PU composite exhibits excellent tensile mechanical properties and remarkable elastic deformation ability (≤60%) compared to single-component materials.The tensile strength increases with the wire diameter and pitch, but higher volume fractions of the wire mesh result in reduced deformation ability.The experimental and simulation results show good consistency within the range of uniaxial tensile elastic deformation; 2) Optical measurement and simulation methods were used to study the variation of Poisson's ratio under longitudinal stiffness conditions.The results indicate that the spiral wire skeleton effectively connects with the matrix and induces re-entrant deformation, thereby limiting the longitudinal shrinkage of the PU elastomer.Increasing longitudinal stiffness leads to a decrease in Poisson's ratio, suggesting that it can be controlled within a specific range by adjusting the microscopic model parameters (p and d); 3) During cyclic tension, the composite material exhibits superior mechanical properties (effective modulus, energy dissipation) that gradually stabilize with an increasing number of cycles.The initial cycle experiences significant energy loss, highlighting the importance of considering preloading strategies in engineering applications.Multiple energy dissipation modes of the composite material under tensile conditions have been identified; 4) The hyperelastic Mullins damage model was fit using parameter fitting methods, and the simulation results of the equivalent model show excellent agreement with experimental findings.Under the same strains, the OSWM-PU composite exhibits more pronounced Mullins damage hysteresis compared to a single-component PU elastomer.Calculations of the damage variables different strains indicate a greater extent of damage as the strain increases; and 5) Composite materials show lower stress retention capacity compared to PU elastomers at high strain levels, leading to matrix tearing within a relatively short period.Increasing wire diameter and decreasing pitch result in premature relaxation and fracture.In other words, a higher volume fraction of the wire mesh increases the susceptibility to damage.The wire diameter (d) and strain demonstrate a positive correlation with stress relaxation, while the pitch (p) shows a negative correlation.

Figure 1 .
Figure 1.The OSWM design and fabrication process: a) schematic diagram illustrating the use of a computerized spring pressing machine to wind a spiral wire with a fixed pitch and b) the adhesive method employed to firmly secure the orthogonal overlapping spiral wires in place.

Figure 3 .
Figure 3. a) Uniaxial tensile test using digital image correlation measurement system and b) stress-strain curves of PU elastomer (25 °C).

Figure 6
displays the longitudinal displacement cloud map during uniaxial tension, with the black dashed line representing the boundary of longitudinal deformation in the dense PU elastomer.It is evident that the ordered structure of the woven skeleton partially restricts the longitudinal deformation of the matrix.By adjusting the pitch and wire diameter, the stiffness of the longitudinal spring can be controlled, allowing for a detailed analysis of the impact of different microscopic model parameters (i.e., p, d) on Poisson's ratio.

Figure 5 .
Figure 5. a) Uniaxial stress-strain curves of OSWM-PU with different pitches, b) numerical simulation and experimental mechanical results of OSWM-PU with different diameters and loading directions, c) optical images (left) of the OSWM-PU at different levels of stretching, and corresponding FEA results (right) based on periodic element numerical calculation methods.

Figure 7 .
Figure 7. Cloud maps of the uniaxial tensile strain field in OSWM-PU: a) strain cloud diagram in the direction of load.b) Strain cloud diagram in the vertical load direction.

Figure 8 .
Figure 8.Comparison of simulation and experimental Poisson's ratio for the same parameter.

Figure 6 .
Figure 6.Schematic diagram illustrating the calculation of Poisson's ratio and the longitudinal variable stiffness parameters during the uniaxial tensile process of OSWM-PU.

Figure 9 .
Figure 9. Analysis of Poisson's ratio under simulation conditions: a) variation diagram of Poisson's ratio with longitudinal wire diameter under the condition of p = 3 mm (off-axis loading d = 0.3 mm) and b) variation diagram of Poisson's ratio with longitudinal pitch change under the condition of d = 0.5 mm.

Figure 10 .
Figure 10.Cyclic tensile results at a strain of 0.4: a) stress-strain curves of OSWM, PU elastomer, and OSWM-PU during 10 tensile cycles, b) effective modulus of different materials across various cycles, c) dissipated energy density of different materials, d) normalized energy absorption of different materials.Error bars represent the standard deviation of the datasets.

Figure 11 .
Figure 11.Energy dissipation mechanisms in the stretching process of OSWM-PU composite: a) stress softening due to fracture of covalent bonds in PU elastomer, b) friction resulting from debonding between OSWM and the matrix, c) deformation of OSWM caused by stretching and compression from the PU matrix, with debonding failure at the lap joint, and d) significant friction between OSWM and polymer after fracture of the PU matrix.

Table 3 .
Damage model fitting parameters.